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## Material Information- Title:
- Extended Logistic Model of Crop Response to Applied Nutrients
- Creator:
- BROCK, KELLY HANS (
*Author, Primary*) - Copyright Date:
- 2008
## Subjects- Subjects / Keywords:
- Analysis of variance ( jstor )
Crops ( jstor ) Datasets ( jstor ) Fertilizers ( jstor ) Irrigation ( jstor ) Logistics ( jstor ) Mathematical independent variables ( jstor ) Modeling ( jstor ) Parametric models ( jstor ) Standard error ( jstor )
## Record Information- Source Institution:
- University of Florida
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- University of Florida
- Rights Management:
- Copyright Kelly Hans Brock. Permission granted to University of Florida to digitize and display this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Embargo Date:
- 4/30/2004
- Resource Identifier:
- 55893187 ( OCLC )
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EXTENDED LOGISTIC MODEL OF CROP RESPONSE TO APPLIED NUTRIENTS By KELLY HANS BROCK A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2004 Copyright 2004 by Kelly Hans Brock This work is dedicated to my parents, Gene and Joann Brock. ACKNOWLEDGMENTS I would first like to thank Dr. Allen Overman, without whose support, encouragement, and guidance I quite likely would have not entered into an advanced degree program. His support and patience have been invaluable during my graduate career. His insight and wisdom into the scientific and engineering communities have greatly broadened my view of science and the world. I would also like to thank the professors who served on my supervisory committee in addition to Dr. Overman: Dr. Roger Nordstedt, Dr. Frank Martin, Dr. Raymond Gallaher, and Dr. Paul Chadik. Special thanks go to Dr. Richard V. Scholtz, III, whose constant help, insight, and guidance have allowed me to survive graduate school. His professionalism and selfless devotion to excellence in engineering and academia are unparalleled. Finally, I would like to thank all of my friends and family who have stuck with me and encouraged me throughout my academic career. I especially thank my parents, Gene and Joann Brock, whose devotion to hard work has been an inspiration, and who have always given me the freedom to explore my own paths. TABLE OF CONTENTS page A C K N O W L E D G M E N T S ................................................................................................. iv L IST O F T A B L E S .................................................................... .......................... .. vii LIST OF FIGURES ............................... ... ...... ... ................. .x A B S T R A C T .......................................... .................................................. x iii CHAPTER 1 IN TRODU CTION ................................................. ...... ................. 2 LITER A TU R E REV IEW ............................................................. ....................... 5 The Extended Logistic M odel .................................. ...................................... 5 Alternative M odels and Approaches ........................................ ....... ............... 12 Arkansas Bermudagrass and Tall Fescue Study.................................... ................... 14 3 M ATERIALS AND M ETHOD S ........................................ ......................... 21 Solutions for a Single Data Set: First Method.........................................................21 P rim ary P aram eters ......................... .. .................... .. ...... ........... 2 1 S econ dary P aram eters .............................................................. .....................22 L in e a r M o d e l ................................................................................................. 2 2 N onlinear M odel .................. .... .. ............................ ..... ......... 23 Corresponding M odel for Y ield ........................................ ....... ............... 23 Linear Regression for Linear Model Parameters..............................................23 Nonlinear Regression for Nonlinear Model Parameters ...................................24 Initial Estimates of Nonlinear Parameters (Linearization)..............................28 Statistical Analyses for a Single D ata Set ................................. ............... 30 Solutions for a Single Data Set: Second M ethod............................................ 33 Solutions for Multiple Data Sets and Commonality of Parameters............................35 N onlinear Case 1: Com m on b ................................ ......................... ........ 36 N onlinear Case 2: Com m on c ................................ .. ............................... 43 Nonlinear Case 3: Common bn and c ............. ...........................................47 Nonlinear Case 4: Common An, bn, and cn .................. ......................... 52 Linear Case A: Common Ab................................ ................................. 55 Linear Case B : Com m on Nc ...................................................... ..................61 v Linear Case C: Common Ab and N ....................................... ............... 66 Generalized Equations for All Analyses................................... ...................... 68 Variable D efinitions ....................................................... 69 General Equations for Nonlinear Portion.........................................................69 General Equations for Linear Portion............................................................. 73 Analysis of Variance (ANOVA) for Commonality of Parameters (F-test) ..............75 4 RESULTS AND DISCU SSION ........................................... .......................... 78 A n aly sis Strategy ................................................................................... 7 8 O overview of R results ....................................... ............... .............. 80 Round One .................................... ................................ .........80 R found Tw o ................................................................... ........... 84 R found Three ............................................. ........................... 86 Discussion of Commonality of Parameters .............. ............................................86 V alidity of the F-test.. ....................................................... .......... ................. 86 Commonalities Suggested by the F-test ...................................... ............... 89 Beyond the F-test: Commonality Based on Goodness of Fit .............................91 5 SUMMARY AND CONCLUSIONS ........... ................................. ...............132 Sum m ary of R research .......................................................... ...............................132 K ey Observations and Conclusions................................... .................................... 133 F u tu re W ork ...................................... .............................................. 13 8 Closing R em arks .................. ............................. ....... ................. 139 APPENDIX: SUPPLEMENTAL DISCUSSION...........................................................150 M atrix In v ersio n ............................................................................. ........ .............. 1 5 0 Sensitivity A analysis ........................ .................. ........................... 153 L IST O F R E FE R E N C E S ........................................................................ ................... 16 1 BIOGRAPHICAL SKETCH ............................................................. ............... 163 LIST OF TABLES Table p 2-1 Application schedule for commercial fertilizer treatments (listed as N-P205-K20, each in k g h a-1) ..................................................................... 19 2-2 Broiler litter content analysis for 1982-1985 (in terms of kg nutrient Mg-1 litter)...20 2-3 Equivalent N application rates for broiler litter (kg ha-1)..................................20 4-1 M ode descriptions. ........................................... ... .... ........ ......... 93 4-2 Optimized parameters for bermudagrass with commercial fertilizer....................93 4-3 Statistics for optimized parameters for bermudagrass with commercial fertilizer...94 4-4 Optimized parameters for bermudagrass with broiler litter. ...................................94 4-5 Statistics for optimized parameters for bermudagrass with broiler litter ...............94 4-6 Optimized parameters for tall fescue with commercial fertilizer.............................95 4-7 Statistics for optimized parameters for tall fescue with commercial fertilizer.........95 4-8 Optimized parameters for tall fescue with broiler litter. ......................................95 4-9 Statistics for optimized parameters for tall fescue with broiler litter.....................96 4-10 Arithmetic average of parameters over select data groups.................. ............96 4-11 Standard deviations of parameters over select data groups............... ................ 97 4-12 Analysis of variance for parameters for irrigated bermudagrass with commercial fertilize er. .......................................................... ................ 9 8 4-13 Analysis of variance for parameters for nonirrigated bermudagrass with com m ercial fertilizer. ....................................... .......................... 99 4-14 Analysis of variance for parameters for bermudagrass with commercial fertilizer (both irrigated and nonirrigated). ........................................ ......................... 99 4-15 Analysis of variance for parameters for average of bermudagrass data with commercial fertilizer (split into two groups: nonirrigated and irrigated).............100 4-16 Analysis of variance for parameters for irrigated bermudagrass with broiler litter. 100 4-17 Analysis of variance for parameters for nonirrigated bermudagrass with broiler litter. .............................................................................. 10 1 4-18 Analysis of variance for parameters for bermudagrass with broiler litter (both irrigated and nonirrigated)..................... ........ ............................. 102 4-19 Analysis of variance for parameters for irrigated tall fescue with commercial fertilizer. .......................................................................... 103 4-20 Analysis of variance for parameters for nonirrigated tall fescue with commercial fertilizer. .......................................................................... 104 4-21 Analysis of variance for parameters for tall fescue with commercial fertilizer (both irrigated and nonirrigated)..................... ........ ............................. 104 4-22 Analysis of variance for parameters for average of tall fescue data with commercial fertilizer (split into two groups: nonirrigated and irrigated). ...............................105 4-23 Analysis of variance for parameters for irrigated tall fescue with broiler litter.....106 4-24 Analysis of variance for parameters for nonirrigated tall fescue with broiler litter. 106 4-25 Analysis of variance for parameters for tall fescue with broiler litter (both irrigated and nonirrigated). ........................................... ... .... ................. 107 4-26 Analysis of variance for select parameters for all bermudagrass...........................107 4-27 Analysis of variance for select parameters for all irrigated bermudagrass. ...........107 4-28 Analysis of variance for select parameters for all nonirrigated bermudagrass. .....108 4-29 Analysis of variance for select parameters for all fescue..................................... 108 4-30 Analysis of variance for select parameters for irrigated fescue ...........................109 4-31 Analysis of variance for select parameters for nonirrigated fescue. ......................109 4-32 Analysis of variance for select parameters for all irrigated samples....................110 4-33 Analysis of variance for select parameters for all irrigated samples with commercial fertilizer. ................................................................ ..... .......... 110 4-34 Analysis of variance for select parameters for all nonirrigated samples with com m ercial fertilizer. .............................................. ..... .. .... .............. .. 111 4-35 Summary of results for single parameter commonality F-test ...........................111 5-1 Parameters for bermudagrass with commercial fertilizer assuming invariance of cn, Ab, and Nc1 across all data sets in this group. ................................. ............... 141 5-2 Statistics for parameters for bermudagrass with commercial fertilizer assuming invariance of cn, Ab, and Nc1 across all data sets in this group. .............................141 5-3 Parameters for tall fescue with commercial fertilizer assuming invariance of cn, Ab, and Nc~ across all data sets in this group.............. .............................................. 142 5-4 Statistics for parameters for tall fescue with commercial fertilizer assuming invariance of cn, Ab, and Nc~ across all data sets in this group.......................142 LIST OF FIGURES Figure page 4-1 Dry matter yield (Y), N uptake (Nu), and N concentration (Ne) versus N application rate (N) for 1982 nonirrigated bermudagrass with commercial fertilizer. This was the data set with the best overall model fit in this category of data. .....................1112 4-2 Phase plot for 1982 nonirrigated bermudagrass with commercial fertilizer ..........113 4-3 Dry matter yield, N uptake, and N concentration versus N application rate for 1984 irrigated bermudagrass with commercial fertilizer. This was the data set with the poorest overall model fit in this category of data ................................................. 114 4-4 Phase plot for 1984 irrigated bermudagrass with commercial fertilizer ..............115 4-5 Dry matter yield, N uptake, and N concentration versus N application rate for the average of all irrigated and nonirrigated bermudagrass with commercial fertilizer.. 116 4-6 Phase plot for average of all irrigated and nonirrigated bermudagrass with comm ercial fertilizer.. .............................................. ...... ...... .. .......... .. 117 4-7 Dry matter yield, N uptake, and N concentration versus N application rate for 1983 irrigated bermudagrass with broiler litter. This was the data set with the best overall m odel fit in this category of data ................................................ ........ ....... 118 4-8 Phase plot for 1983 irrigated bermudagrass with broiler litter.............................119 4-9 Dry matter yield, N uptake, and N concentration versus N application rate for 1982 irrigated bermudagrass with broiler litter. This was the data set with the poorest overall model fit in this category of data.............. .............................................. 120 4-10 Phase plot for 1982 irrigated bermudagrass with broiler litter..............................121 4-11 Dry matter yield, N uptake, and N concentration versus N application rate for 1981-2 irrigated tall fescue with commercial fertilizer. This was the data set with the best overall model fit in this category of data. ..............................................122 4-12 Phase plot for 1981-2 irrigated tall fescue with commercial fertilizer................. 123 4-13 Dry matter yield, N uptake, and N concentration versus N application rate for 1982-3 irrigated tall fescue with commercial fertilizer. This was the data set with the poorest overall model fit in this category of data. ............... .......... .........124 4-14 Phase plot for 1982-3 irrigated tall fescue with commercial fertilizer................. 125 4-15 Dry matter yield, N uptake, and N concentration versus N application rate for the average of all irrigated and nonirrigated tall fescue with commercial fertilizer. ...126 4-16 Phase plot for average of all irrigated and nonirrigated tall fescue with commercial fertilize e r. ............................ .. ................ ................ ................ ............. ........ 12 7 4-17 Dry matter yield, N uptake, and N concentration versus N application rate for 1983-4 irrigated tall fescue with broiler litter. This was the data set with the best overall model fit in this category of data.... ...................................128 4-18 Phase plot for 1983-4 irrigated tall fescue with broiler litter ..............................129 4-19 Dry matter yield, N uptake, and N concentration versus N application rate for 1982-3 irrigated tall fescue with broiler litter. This was the data set with the poorest overall model fit in this category of data.... ...................................130 4-20 Phase plot for 1982-3 irrigated tall fescue with broiler litter.............................131 5-1 Dry matter yield, N uptake, and N concentration versus N application rate for 1982 bermudagrass with commercial fertilizer using common cn, Ab, and Nc1 for all bermudagrass with commercial fertilizer (1982-1985). .............. .... ..............143 5-2 Dry matter yield, N uptake, and N concentration versus N application rate for 1983 bermudagrass with commercial fertilizer using common cn, Ab, and Nc1 for all bermudagrass with commercial fertilizer (1982-1985). .............. .... ..............144 5-3 Dry matter yield, N uptake, and N concentration versus N application rate for 1984 bermudagrass with commercial fertilizer using common c, Ab, and Nc~ for all bermudagrass with commercial fertilizer (1982-1985). .............. .... ..............145 5-4 Dry matter yield, N uptake, and N concentration versus N application rate for 1985 bermudagrass with commercial fertilizer using common cn, Ab, and Nc~ for all bermudagrass with commercial fertilizer (1982-1985). .............. .... ..............146 5-5 Dry matter yield, N uptake, and N concentration versus N application rate for 1981- 2 tall fescue with commercial fertilizer using common Cn, Ab, and Nc~ for all tall fescue with commercial fertilizer (1981-1984).....................................................147 5-6 Dry matter yield, N uptake, and N concentration versus N application rate for 1982- 3 tall fescue with commercial fertilizer using common cn, Ab, and Nc~ for all tall fescue with comm ercial fertilizer (1981-1984).....................................................148 5-7 Dry matter yield, N uptake, and N concentration versus N application rate for 1983- 4 tall fescue with commercial fertilizer using common cn, Ab, and Nc1 for all tall fescue with commercial fertilizer (1981-1984).....................................................149 A-1 Model dry matter yield, N uptake, and N concentration versus N application rate demonstrating the effect on the models of a change of 20% in the parameter An. 156 A-2 Model dry matter yield, N uptake, and N concentration versus N application rate demonstrating the effect on the models of a change of 20% in the parameter bn. 157 A-3 Model dry matter yield, N uptake, and N concentration versus N application rate demonstrating the effect on the models of a change of 20% in the parameter cn. 158 A-4 Model dry matter yield, N uptake, and N concentration versus N application rate demonstrating the effect on the models of a change of 20% in the parameter Ab. 159 A-5 Model dry matter yield, N uptake, and N concentration versus N application rate demonstrating the effect on the models of a change of 20% in the parameter Ncl. 160 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy EXTENDED LOGISTIC MODEL OF CROP RESPONSE TO APPLIED NUTRIENTS By Kelly Hans Brock May 2004 Chair: Allen R. Overman Major Department: Agricultural and Biological Engineering Crop response in terms of crop yield and plant uptake of nutrients under varying conditions is a key concern from the perspectives of both agricultural production and water reuse. To adequately design nutrient management systems such as crop-based wastewater reuse systems, engineers need the ability to estimate production of dry matter and levels of nutrient removal as a function of crop species, soil characteristics, climate, and additional inputs, including irrigation and applied nutrients. In order for a method of estimation to be most useful to an engineer, it should be relatively simplistic and broadly applicable to many scenarios, and yet maintain an adequate level of accuracy. The search for such a method has led to the development of the Extended Logistic Model (ELM), which was used in this study to describe total seasonal dry matter production and nitrogen uptake in response to applied nitrogen. The ELM was applied to data obtained from a 1988 study in Fayetteville, Arkansas, involving response of bermudagrass [Cynodon dactylon (L.) Pers.] and tall fescue [Festuca arundinacea Schreb.] to varying levels of applied commercial (inorganic) fertilizer and broiler litter. A nonlinear/linear regression scheme was developed into a program called DAEDALUS to aid in conducting multiple analyses on multiple parameters and data sets efficiently. Analysis of variance was used to test for commonality of parameters across data sets. The overall intent of this research was to lend further insight into the ELM and bring it one step closer to efficacy for practical use by engineers, especially by searching for patterns in each of its five parameters and comparing results to previous findings. The research reaffirmed that the ELM is a consistently good descriptor of crop response to applied N. Results also suggested that the parameter Nc1, the lower theoretical plant N concentration limit, is a strong function of crop type-as has been found in previous studies-and is affected little by water availability or fertilizer source. CHAPTER 1 INTRODUCTION One may ask why an agricultural engineer whose academic focus has been primarily on soil and water engineering would choose a mathematical crop modeling topic for his dissertation. The historical tie between the two areas may help to answer such a question. Research that ultimately led to the Extended Logistic Model, the focus of this dissertation, began when Dr. Allen Overman began serving as an engineering advisor to the city of Tallahassee, FL, regarding its wastewater treatment and reuse planning. The ultimate result of this was the more than 800 hectare Tallahassee Southeast Farm, a large wastewater reuse operation that is a joint project between the city and a contracted farmer. In this system, which handles approximately 64 million liters per day, treated wastewater is applied to crops (including forage crops and corn) as a method of effective wastewater reuse. Early in its development questions were asked regarding the best crops to use, and with each of these crops, how much yield could be expected at a given nutrient application rate, and what kind of nutrient uptake by the plant could be expected. These questions initiated the journey that has led to a series of crop models developed by Overman and his associates over the past 30 years, including the Extended Logistic Model. It should be noted here that this dissertation uses the term "nutrient uptake" to emphasize nutrient removal from the environment and incorporation into the harvested portion of the plant. A more commonly accepted term for this is plant nutrient content (Soil Science Society of America, 1997). 2 Today, especially in Florida, there is an ongoing balancing effort between meeting the needs for an ever growing population with an ever growing standard of living and effective preservation of the environment, especially Florida's precious aquifer-based water supply. As more concern arises about the impact of the various chemicals our society uses and wastes it produces, including those from agricultural operations, more environmental laws are put into place in an effort to preserve the integrity of natural resources and ecosystems. TMDLs (total maximum daily loads), CNMPs (comprehensive nutrient management plans), and BMPs (best management practices) are a few of the key acronyms that reflect an effort to be more responsible in agricultural and industrial operations with regard to the environment. The research behind the Extended Logistic Model has been an attempt to search for patterns that may lend insight into how a plant responds to its environment, especially nutrient uptake and dry matter accumulation in response to the plant's interface with soil and water (and the nutrients contained therein). More knowledge gained with respect to this plant-soil-water system can lead to improved management practices, optimizing the balance between desired agricultural goals (whether maximum dry matter yield, maximum plant quality, or some other factor) and environmental goals (whether minimal impact, maximum nutrient uptake, or some other factor). The primary objectives of this research were 1) to develop a modified nonlinear/linear regression scheme for use in applying the Extended Logistic Model to multiple data sets 2) incorporate this scheme into a computer program for efficient analysis 3) analyze data from a 1988 forage study from Fayetteville, Arkansas 4) determine whether previous basic patterns and internal consistency for the Extended Logistic Model hold for this study and 5) search for general patterns and commonality of parameters between the data sets in this study. In Chapter 2 of this dissertation, a qualitative discussion over the Extended Logistic Model, including its origins, strengths, and criticisms, is given. The forage study upon which the analyses in this dissertation are based is also discussed there. Chapter 3 provides a very extensive review of the equations of the Extended Logistic Model and the nonlinear/linear regression methods employed to solve for its five fundamental parameters. The reader may wish to focus on the most critical points in this chapter, such as the discussion regarding solutions for a single data set at the beginning of the chapter, and the general equations and F-test presented at the end of the chapter. The four special cases regarding simultaneous analysis of multiple data sets, discussed in the middle of the chapter, are presented in rather great detail but are really simple extensions of the case for a single data set, and are all represented through the general equations presented near the end of the chapter. Chapter 4 discusses the results of the analyses and includes extensive tables, as well as figures illustrating results of the analyses for individual data sets. Table 4-2 through Table 4-9, as well as Table 4-12 through Table 4-34, are presented to cover in great detail the results of each analysis. The most important and useful tables in Chapter 4, however, are Table 4-10, Table 4-11, and Table 4-35, which provide useful summaries of results and help provide insight into patterns. Chapter 5 draws on the results from Chapter 4 and compares them to previous studies. This chapter provides a good overall view of the results and key conclusions found in the research behind this dissertation. An illustration of the effect of the assumption of commonality of some parameters is given through a series of seven figures in this chapter. The Appendix 4 should not be overlooked, especially the section on "Sensitivity Analysis," which provides useful illustrations for how each of the five parameters affect the behavior of the equations in the Extended Logistic Model. CHAPTER 2 LITERATURE REVIEW This chapter briefly reviews the history of the Extended Logistic Model, the reasoning behind it, and advantages and disadvantages of its use. Alternative models and approaches are also discussed, and the forage grass experiment upon which the analyses in this study were based is reviewed. Discussion of the Extended Logistic Model in this chapter is largely qualitative, leaving more technical discussion to Chapter 3. The Extended Logistic Model Development of the Extended Logistic Model was first sparked by an interest in the Mitscherlich equation, developed in the early 20th century to describe dry matter yield in response to applied nitrogen (N). Overman proposed the logistic equation as an alternative model to the Mitscherlich equation for describing dry matter yield (Overman and Scholtz, 2002). The logistic equation is a function with three parameters, one linear (A) and two nonlinear (b and c). The application of the logistic equation as a model is not new. Its characteristic S-shape curve gives it broad applicability to a variety of processes that exhibit such sigmoidal behavior, including those observed in the fields of agriculture, biology, economics, and engineering (Ratkowsky, 1983). Overman has shown its applicability to human population dynamics over time, as first applied by Pearl et al. (1940) to the total population of the U.S. The logistic equation, in fact, is commonly referred to as the Verhulst-Pearl model. Overman chose the logistic model over Mitscherlich in part because of this sigmoidal behavior (Overman and Scholtz, 2002). However, the recognition of sigmoidal growth in different types of vegetation is not new either. Ratkowsky (1983) points out the broad range of vegetative growth that exhibits sigmoidal behavior over time, including pasture regrowth yield, onion (Allium cepa Cepa Group) yield, and area of cucumber (Cucumis sativus L.) cotyledons (Mills and Jones, 1996). He also notes that the logistic equation is not the only model that can describe sigmoidal behavior; the Gompertz, Richards, Morgan-Mercer-Flodin, and Weibull distribution-based models all have sigmoidal characteristics. The sigmoidal pattern in the response of dry matter yield to levels of applied N was noted early in the 20th century by Russell (1937) when examining barley (Hordeum vulgare, Mills and Jones, 1996) data from a previous study. Overman extended the logistic concept to include a description of N uptake by the plant, observing that it also followed sigmoidal behavior and so could also be described by a logistic equation. Earlier versions of the model assumed this equation had three different parameters (An, bn, and c,) compared to the logistic equation for dry matter yield (with parameters A, b, and c). Analyses of data for several different crops demonstrated that the c and c, parameters were not significantly different (Overman et al., 1994; Overman and Scholtz, 2002). With the new assumption that there was really only one "c" parameter (termed c,), mathematical analysis of this tie between yield and N uptake led to a new perspective for the model, now termed the Extended Logistic Model, based on just two postulates (Overman and Sholtz, 2002). The first postulate states that plant N uptake follows a logistic response to applied N, as described by N, = 1 [2-1] 1 + exp(b, c,N) where N, is plant N uptake (or content), Nis applied N, and the remaining three factors are parameters of the model discussed later. The second postulate states that dry matter yield is a hyperbolic function of N uptake, as described by A" N Y N, [exp(Ab)- 1] "[ Y= [2-2] A +N, exp(Ab)- 1l where Y is dry matter yield, N, is plant N uptake, and the other three factors are parameters discussed later. These two relationships mathematically explain why dry matter yield response to applied N can be expressed as a logistic equation as well. With these postulates, the Extended Logistic Model is a five parameter model that can describe dry matter yield, N uptake, and N concentration in terms of applied N, as well as dry matter yield and N concentration in terms of N uptake (the latter two being so-called phase relationships, since the independent variable applied N is only implicit in these cases). The five parameters are defined in Chapter 3; for the discussion here it is sufficient to simply list them: An, bn, cn, Ab, and Nc1. There are several mathematical, logical, and practical advantages to using the Extended Logistic Model. Overman and Scholtz (2002) point out it is relatively simple to use (once parameters are known) and is a well behaved function. "Well behaved" denotes it has limited (non-infinite) bounds and increases monotonically; it is continuously differentiable at all points. Such properties give it more physical meaning than simple linear, quadratic, or high order polynomial equations, or even the Mitscherlich equation (which, like the logistic equation, contains an exponential term). For example, with the Extended Logistic Model, dry matter yield and N uptake approach zero with decreasing applied N, never going negative, which would make no real world sense, and both approach an upper limit with increasing applied N, which is more sensible than a model that approaches infinity. Another key indicator of the strength of the model is that the ratio of N uptake to dry matter produces another, well behaved function that sensibly describes N concentration versus applied N. Many models fail such a ratio test, resulting in infinite singularities or other nonsensical anomalies. The resulting phase relationship between Y and N, and between Nc and N, also works well. The Extended Logistic Model has been shown to apply very well to a variety of different crops, illustrating its key strength as a broad-based model. There are also a few criticisms of the use of the Extended Logistic Model. At extremely high values of applied N, the upper limit of the N uptake and dry matter logistic equations likely do not make physical sense, since at such high levels, one would expect the N uptake or dry matter yield to begin to decrease with increasing applied N. However, such "toxic" levels of applied N are rarely, if ever, encountered in real world situations, and so within the bounds of practical limits, the upper limits of the logistic equations function adequately. The model has also been criticized as not being "mechanistic" (Overman and Scholtz, 2002) and simply "an exercise in nonlinear regression" (Boote et al., 1996, p 711). The term "mechanistic" implies an approach in which the model is developed beginning with very fundamental (e.g., molecular level or even quantum level) relationships, although Monteith (1996) gives it the more general definition that a mechanistic crop model is one "in which all quantified processes have a sound physical or physiological basis" (p 695). While there are compartmental models, as discussed in the next section, it can be argued that no truly mechanistic model of crop response has been created or is even possible. There is no doubt that the Extended Logistic Model is not mechanistic at such a fundamental level. The approach in developing it has been more inductive than deductive, i.e., a search for general patterns through analysis of real world data rather than an approach based on initial hypotheses for fundamental relationships. It can be argued, however, that this "top down" approach is equally as valid, if not more so, than one that attempts to be more "mechanistic." In fact, Monteith (1996) states that the philosophy that "descriptions of the natural world" (p 695) should be based on "facts derived from observation" (p 695), not "speculation and dogma" (p 695), has been the foundation upon which modern science, in all its success, has been built. Analyses using the Extended Logistic Model are also often criticized for a lack of "validation," which Jones and Luyten (1998) define as "the process of comparing simulated results to real system data not previously used in any calibration or parameter estimation process" (p 25). This lack of "validation" reflects a difference in philosophy behind the analyses involving the Extended Logistic Model versus what has become the "norm" in the crop modeling community. The key difference in this philosophy lies in differentiating validation from calibration, the latter of which Jones and Luyten (1998) define in the following statement: "Calibration consists of making adjustments to model parameters to give the best fit between simulated results and results obtained from measurements on the real system" (pp 24-25). In essence, calibration based on some data is advocated, followed by validation of the model using other data. This leads to an interesting exercise (albeit a somewhat questionable approach) in testing the "predictive" ability of a model, which is often the objective behind modeling. This, however, is not currently the primary objective or philosophy behind research involving the Extended Logistic Model. Rather, the focus has been on whether the Extended Logistic Model is a good, consistent descriptor of crop response to applied N. Numerous applications to a variety of data sets have shown this to be true (Overman and Scholtz, 2002). The criticism, however, is that the model is so universal or generic that its parameters are not invariant; thus they must be adjusted with each new set of data, varying from site to site and often year to year (Boote et al., 1996). Does the fact that parameters change mean a model is unsuitable? The Extended Logistic Model does not assume universal invariance of any of its five parameters. Rather than a devotion to "validation" and "calibration," research involving the Extended Logistic Model has been a more wholistic search for patterns that may ultimately lead to understanding how (and, more tentatively, why) its parameters vary from situation to situation, as well as cases in which they do not vary. Jones and Luyten (1998) describe two broad, fundamental objectives in biological simulations. The first is the desire to better understand the behavior of a system and the various interactions and relationships therein. The second is the desire to better predict system behavior so that a system may be better managed for a particular goal. The current research behind the Extended Logistic Model is still focused primarily on the first goal, although perhaps this has not always been clearly stated in papers regarding it. This focus reflects the philosophy that jumping into a predictive game without a better understanding of behavior would be, at the least, premature and, more likely, irresponsible. The search for patterns has led to some interesting developments and evolution of the Extended Logistic Model, however, since the early articles of over a decade ago that Boote et al. (1996) referred to when criticizing it. One of the key areas of research has been the search for commonality of parameters for a given set of conditions, leading to evidence that certain parameters may be functions of only one or two influences (e.g., genetics, soil type, water availability, etc.) Research by Scholtz (2002) on corn (Zea mays L.) and ryegrass (Lolumperenne L.) data sets concluded that the parameter Ab is a genetic based parameter, likely a function of plant species and possibly specific plant variety. Scholtz (2002) also tentatively suggested that the product Nc1 exp(Ab), known as the secondary parameter Ncm in this dissertation (see Chapter 3), may also be a function of crop type, although some fluctuations from year to year were observed for perennials such as bermudagrass (Cynodon dactylon (L.) Pers., Huneycutt et al., 1988), which he suggested may be due to annual fluctuations in region temperature or differences in the plant's initial quality at the start of the season. Wilson (1995) also concluded from her dissertation research that there was evidence to support the idea that Ab and Nm, were functions of crop type. If both Ab and N,, are functions of crop type alone, it can be logically concluded that Nc~ is also a function of crop type only. Wilson (1995) also noted that c, may be related to crop type, citing cases for ryegrass, bahiagrass (Paspalum notatum, Mills and Jones, 1996), and corn where each was grown at different locations, but for which Cn was invariant with respect to location and only varied with crop type. She noted, however, a study in which bermudagrass did not exhibit the same Cn when grown at different sites, and in other cases two different grasses grown at the same location had the same value for Cn. The evidence so far, therefore, seems to suggest Cn is likely not a function of crop type only. Alternative Models and Approaches There are a myriad of models relating to crop behavior. These range from incredibly simplistic linear equations to complex, computer driven, compartmental models with multiple subcomponents describing different aspects of the plant and environmental influences thereupon. Monteith (1996) quite humorously points out the different philosophies from different disciplines that have influenced the development of crop models. He states that physicists tend to take an extreme reductionist approach, employing the philosophy of Occam's razor, while biologists tend to take the extremely overly complex approach, attempting to model a biological system in virtually infinite detail. Agronomists tend to adhere to the relatively simplistic linear, quadratic, and sometimes higher order polynomials (3rd or 4th degree). A perfect example is the original analysis by Huneycutt et al. (1988) of the data from their forage experiment, which are the data subjected to analyses in this dissertation. Dry matter yield response to applied N was classified as linear or quadratic, with some tendency to vary between the two from year to year, although some patterns were observed; e.g., broiler litter applied to nonirrigated bermudagrass tended to evoke a linear response in dry matter yield, while that of irrigated produced a quadratic response (Huneycutt et al., 1988). Often, these models can be quite adequate within the range of data, and if one shares the philosophy that extrapolation of a model beyond data is never a good idea and that interpolation between data points should be done with extreme caution, then there is no need for concern over the often illogical behavior these models exhibit in such situations. In fact, Ratkowsky (1983) points out that if given the choice between several models that fit data equally well, the one that is closest to a linear model is generally preferred. Linear and polynomial models do have drawbacks, however, if attempts are being made to determine a physical or rational basis for crop response, and they often lack broad-range applicability. Jones and Luyten (1998) outline the popular philosophy and techniques behind computer simulation in present day crop modeling. With this approach, it is common for a system (however it is defined) to be broken into compartments whose behavior is described mathematically. The system is then usually modeled by a set of first-order differential equations that are functions of time (temporal elements). Distributed system modeling adds spatial elements to this type of analysis. Various numerical techniques (most often using computers) are employed to solve the resulting system of differential equations. Such analyses are subject to numerical errors. Jones and Luyten point out that the interactions of all subsystems in biological systems are incompletely understood and incredibly complex, and so models that attempt to describe or predict these behaviors usually contain some level of empiricism, which is one reason calibration is usually required. Monteith (1996) reaffirms this idea by noting that some of the submodels (or compartments) are based on "firm experimental evidence" while others are based on "arbitrary assumptions" (p 696). The Extended Logistic Model is in a way the median and perhaps even a potential intercessor between the two extremes of the simplistic models that describe macroscale observations adequately but without much physical basis and the rather complex compartmental models that attempt to simulate, to varying degrees of success, the underlying processes resulting in such observations. It should be noted that many crop models or crop modeling software packages attempt to account for several environmental effects, such as impact of insects, sunlight, temperature, carbon dioxide concentration, etc. (Jones and Luyten, 1998). This goes far beyond the applied N effects that are the primary focus of the Extended Logistic Model in this study, and yet it is conceivable that some of the parameters in the Extended Logistic Model (most notably the linear parameter A,) could potentially be tied to such models or their successors to account for such environmental effects. Monteith (1996) argues for a "balance between simplicity and complexity" (p 696) in the vast field of crop modeling. He states that modelers need to aim for the right balance between (i) restricting algorithms to the minimum for a comprehensible model that allows current problems to be explored in the simplest possible way and (ii) making the structure of the model adaptable enough for more complex interactions to be introduced as new ideas and needs develop. (p 696) It can be argued the Extended Logistic Model was developed with this balanced approach in mind. It is relatively simple and broadly applicable to a variety of situations, yet it is acknowledged that the physical significance of all of its parameters is not well understood, and that as more information is assessed, functions that describe or predict those parameters could be incorporated into the model with relative ease. Arkansas Bermudagrass and Tall Fescue Study The data analyzed in this study are derived from a multiyear experiment with forage grasses conducted by Huneycutt et al. (1988) at the Main Experiment Station in Fayetteville, Arkansas. There were three factors considered in this study: crop type, fertilizer source (and application level), and irrigation treatment (irrigated versus nonirrigated). Its objective was to find yield responses to fertilizer treatments and irrigation treatments. Measurements taken included cumulative seasonal dry matter yield and crude protein concentration. For the analyses in this dissertation, crude protein was converted to terms ofN concentration (or specific N), and together with yield data, this was used to calculate N uptake by the plant. All data, which were in English customary units in the original study, were converted to SI units for the analyses here. The two crops considered were Tifton 44 bermudagrass (Cynodon dactylon (L.) Pers.), a warm season grass with a harvest season from April through September; and Kenhy tall fescue (Festuca arundinacea Shreb.), a cool season grass with a harvest season of September through August. Both are used as forage grasses for cattle. A third crop combination of tall fescue with clover was also included in the experiment, but it was not considered in the analyses here to maintain the focus on a single cropping system. Two types of fertilizer were used in the study: commercial fertilizer and broiler litter. There were six treatment levels for commercial fertilizer, broken into sub- treatments spread out over the season, as outlined in Table 2-1. The initial treatment each season included application of a 13-13-13 fertilizer. Additional N, P, and K were supplied as necessary to meet the experiment's levels by using ammonium nitrate, concentrated super-phosphate, and muriate of potash, respectively. There were three treatment levels of broiler litter, applied in terms of total mass at 4.5, 9.0, and 13.5 Mg hal-. A new batch of broiler litter was used each year, and the nutrient concentration varied as shown in Table 2-2. This resulted in different nutrient application rates each year, as outlined in Table 2-3. Unlike commercial fertilizer, all broiler litter was applied at the beginning of each season for each crop. It should be noted here that since the season ran from September through August of the following year, its season is denoted by two years, e.g., 1981-1982. Thus in the tables, the year in parentheses denotes the beginning of the fescue season, while that outside of the parentheses denotes the end of the fescue season and the entire season for bermudagrass (which occurred within the same year). Irrigation treatments consisted of a nonirrigated treatment, in which the crops only received natural precipitation, and an irrigated treatment, in which crops received enough supplemental irrigation via perforated-pipe sprinklers to bring their weekly precipitation levels to a minimum of 3.8 cm for June through September of each season. The two grasses were established in separate, contiguous plots on Captina silt loam soil with an initial (1980) pH of 6.2. Captina silt loam is of the taxonomic class fine-silty, siliceous, active, mesic Typic Fragiudults and used mainly in production of pasture and hay (United States Department of Agriculture [USDA], 2003). At the end of the experiment in 1985, the pH had shifted slightly, to 6.1 for the bermudagrass trials and 6.4 for the tall fescue trials. Each crop type was configured in a randomized complete block design (RCBD) that included all fertilizer treatments (a zero plot, three levels of broiler litter, and six levels of commercial fertilizer) and three replications of each treatment. Two separate RCBD's were set up within each crop, one irrigated and one nonirrigated. Plots were fertilized before initial sprigging/planting for all trials. The analyses in this dissertation neglected the first year of data from the Arkansas study with the idea that plots would have not been well established within the first year. It should also be noted that while there were three replications for each treatment, only the average values of the three replications were reported by Huneycutt et al. (1988), and so it was these averaged values that were used in the data analysis in this dissertation. There are a few confounding points from this experiment that could have had a potential impact on the results of the analyses included in this dissertation. The Extended Logistic Model as employed in this study is only used to describe N response and does not account for any response to P or K. Although a modified form of the model exists, termed the triple Extended Logistic Model, which can account for all three factors, it could not be applied here since N, P, and K effects were not experimentally separated. In general, as N application increased, so did P and K application; thus increases in yield could have been partially due to increases in P and K application, not just N. While this was true for commercial fertilizer, the broiler litter cases were even more confounded since nutrient content varied from year to year, and since there were also micronutrients present that could have had positive or negative effects on plant response depending on their concentrations. The assumption in the analyses here was that the response due to change in N application was far greater than any response due to change in P or K application. This is generally true, except in the cases where there is either a substantial P or K deficiency. It was assumed that the supplemental P or K was sufficient to meet the needs of the plant at the given N application rate, essentially making N the limiting nutrient, although no plant tissue analyses were available to clearly substantiate this claim. The authors made several interesting observations and conclusions from the Arkansas study. In general, yields declined over the years, especially in nonirrigated crops where stress due to drought in later years of the study likely impacted yields. Response to broiler litter was less pronounced than that to commercial fertilizer in both crops. Possible reasons given include incomplete mineralization of the organic N in broiler litter (previous studies had suggested only 60% is mineralized after 300 days in the soil), upfront application of broiler litter versus spread out application of commercial 18 fertilizer over the season, and losses in available N in the litter due to volatilization, leaching, or immobilization. The authors also note that the nutrient ratios in broiler litter differed significantly from those in the commercial fertilizer. Table 2-1. Application schedule for commercial fertilizer treatments (listed K20, each in kg ha1) Treatment Beginning of season Time of Application After 1st After 2nd harvest harvest as N-P205- After 3rd harvest 112-112-135 225-112-202 337-112-270 449-146-337 562-180-404 674-213-472 112-112-135 225-112-202 337-112-270 449-146-337 562-180-404 674-213-472 56-112-67 84-112-67 112-112-101 140-146-112 168-180-135 225-213-135 56-112-67 84-112-67 112-112-101 140-146-112 168-180-135 225-213-135 Data adapted from Huneycutt et al. (1988) Trial Bermuda- grass Tall Fescue 56-0-67 84-0-67 84-0-84 112-0-90 140-0-101 168-0-135 56-0-67 84-0-67 84-0-101 140-0-90 168-0-135 225-0-135 56-0-67 84-0-84 112-0-67 140-0-101 168-0-135 56-0-67 84-0-67 84-0-67 112-0-67 112-0-101 56-0-0 84-0-67 112-0-67 112-0-67 56-0-0 84-0-67 112-0-67 112-0-101 Table 2-2. Broiler litter content analysis for 1982-1985 (in terms of kg nutrient Mg-1 litter) Year N P205 K20 (1981-)1982 45 37 31 (1982-)1983 37 29 42 (1983-)1984 33 23 30 1985 28 23 30 Data adapted from Huneycutt et al. (1988) Table 2-3. Equivalent N application rates for broiler litter (kg ha-1) Year (1981-)1982 (1982-)1983 (1983-)1984 1985 Litter Rate (Mg/ha) 4.5 202 166 148 126 9.0 404 332 297 252 13.5 607 499 445 377 CHAPTER 3 MATERIALS AND METHODS Analyses of data sets in this study include linear regression, nonlinear regression, and several statistical analyses. This chapter begins with two methods for analyzing a single data set. This concept is then expanded to analyzing special "cases" for multiple data sets in which some parameters are assumed common between data sets. Finally, a general set of equations is presented that can be used in all cases and for any number of data sets. Solutions for a Single Data Set: First Method Primary Parameters The five primary parameters of the logistic model can be considered as A,, b, c,, Ab, and Ncl. These parameters do have a physical basis. Ais the maximum theoretical plant N uptake for the system. The intercept parameter for plant N uptake, bn, relates to the initial condition of the soil; it can account for unmeasured N that is initially present. The parameter c, is the N response coefficient, which accounts for some of the plant's response to applied N. The parameter Ab is the natural logarithm of the ratio of the upper theoretical N concentration limit, Ncm, to the lower theoretical plant N concentration limit, Ncl; it also relates b, to a corresponding secondary parameter in the yield model, termed b (presented later). Secondary Parameters Useful secondary parameters include A, b, Kn, Y, and Nam At least three of these have significant physical meanings: A is the maximum theoretical dry matter yield for the system; b is the intercept parameter for dry matter yield, related to initial soil conditions; and Nam is the maximum theoretical plant N concentration limit. Written in terms of the primary parameters, these are defined as A= An [3-1] N,, exp(Ab) b = b, Ab [3-2] A KAn [3-3] Sexp(Ab)-1 An 1 [ m Nj exp(b)- 1 [3-4] Ncm No, exp(Ab) A [3-5] A The logistic model can be broken into two parts: a linear portion and a nonlinear portion, as demonstrated by Overman and Scholtz (2002). Linear Model N K = +( N, [3-6] No = N, + N [exp(Ab)-l [3-7] Here No is N concentration (or specific N, typically in units of g kg-1) and N, is N uptake (typically in units of kg ha-1). From this relationship and linear regression one can determine the parameter N,, as well as secondary parameters K, and Y,. Once nonlinear regression (presented later) is used to determine An, then Ab can also be determined from the above relationship. Nonlinear Model N, A= [3-8] 1 + exp(b c,,N) Corresponding Model for Yield Y = [3-9] 1 + exp(b cN) Linear Regression for Linear Model Parameters From least squares regression for intercept and slope, respectively: N = (,N)( )- (-N) (2 )) [3-10] Y n NZ N- N X )2 N l=1 n N Ne)-W NzXv S[exp(Ab)- 1] 1 n)-) [3-11] A Ym n N IN. )2 where n is the number of data points in the data set Solving for Ym and K,: m = (N N ( N N [3-12] n(_ (N.NcN))- X N)[) Nc t N.2) N, N 2(N, N, K = ( [3-13] n( (NNj)) (INJXZNC) Solving for Ab (including the parameter A,): Nc' [exp(Ab)-l]= 1 [3-14] A, Y, [exp(Ab)-1]=- [3-15] Y.Nl A exp(Ab)= A +l [3-16] Ab = In I" +1 [3-17] \Y N \ Nonlinear Regression for Nonlinear Model Parameters This method is based on that outlined by Overman and Scholtz (2002). We want to minimize the error sum of squares, given by E= Z[N -N^2 [3-18] where N, is the true (data) value for N uptake and N is the corresponding estimated (model) value for N uptake. Substituting the nonlinear model into this equation: 2 E=A N.- ," [3-19] E 1+ exp(b -cnN) In order to optimize the three parameters, we must set each partial derivative of the error sum of squares equal to zero: dE = 0 [3-20] OAn dE =0 [3-21] bb dE = 0 [3-22] Solving for the linear parameter fro the first partial derivative: Solving for the linear parameter A, from the first partial derivative: E = 2 N A 0 [3-2 OA LL 1+ exp(b cN) 1+ exp(bn cN) -1 I F[N 1 + exp(bn cN) [ L )]-N [3-2 A 1 A rr1 + exp(bn cN) 1+ exp(b cN), 0 AnZ[[1+exp(b -c N)]2 ] 1+exp(b$ -cN)] [3-2 An = [3-2 1 [+ exp(b- cN)]2 Using the same method for the nonlinear parameters bn and cn, however, only leads to implicit solutions: OE = 0 Sb. 0 2n [r1+ exp(b _cnN)] 2 F A 7" 2X N -1+ ep(bA, -[ exp (b c N)] Nlpbexp(b-c N)] 0I [1+ exp (bAb cN) ] [1 + exp(bb cN )] Nu A" exp( E 1+ exp(b,, c N)e I1+ exp(b, c,)N)] c -" A l+exp(b -c N) [I + exp(b -c N)]2 0 2A NNexp(b-cn- N)] 2' Nexp(b- cAN) [1 + exp(b, cN) ]2 An [ [+ exp(bn -cnN)]3 3] 4] 5] 6] [3-27] [3-28] [3-29] [3-30] Since these solutions are implicit, the 2nd order Newton Raphson method is a suitable iterative method for finding values of b, and c,. If reasonable initial guesses for b, and c are made, b'n and c',,, respectively, the partial derivatives with respect to b, and c, can be approximated for the "new" estimates of b, and c,, b', +-b, and c',+6, , respectively, as O2E aE + acE r E -b --, + -y Ab ) + \ ---- &b [ 3 3 1 dE SE 2E a2 E C n ^ ^ \c k,. + OW.O (, + & ae b'+Ab, c', +Ac, K e b',,c', b ,,)c' be',,c Since the value of both of these partial is zero at the minimum sum of squares, the following system of equations applies: A ) K & C ( 2E 2 E ( E b2 6,bn + :e cn =- - n n ) b',,c, c K b',C' , ] [3-32] [3-33] [3-34] Note the relationship cE acnabn )b',-c',n S2E bnn)b, -c wb -8c .b,-.', has been employed here. Removing iterative subscripts for simplicity and defining the following variables: aE Jbn b" b aE OCn [3-35] [3-36] [3-37] 02E H b [3-38] n 82E Hn 2 [3-39] a2E Hb. = H = [3-40] The system of equations can be simplified to the matrix form: Hbnbn Hbnn ]bn L Jbn [3-41] Hbncn H c &n cn Solving this system using Cramer's Rule (Kolman and Shapiro, 1986): ~bn Hbn -Jc HCc b CJb H +J H -bH I cn H bcn = H= H[3-42] H H H, H -H2 Hbnbn bncn bnbn Cncn bncn Hbncn HCnn - HL c JJ -J HJ, [34H3 These values of 8bA and &, can then be used to update the previous "guesses" for bn and c iteratively until Eqs. [3-31] and [3-32] are equal to zero (or within a specified tolerance.) Solving for the second partial and cross derivatives: 02E Hbb Hbnbn 2 2 abn [3-44] a8 2A N. exp(b -cN) -2A" exp(b, -cN) b_ L j[l i +exp(b -c N)]2 "L[1+ exp(bn -cN)]3 Hb = 2A N exp(b cnN) 2N, [exp(b cN)]2 [1 + exp(b cN)]2 [ + exp(b- c N)]3 -2A2 exp(b cN) 3[exp(b, -cnN)]2 S[1+ exp(bn -cN)]3 [1 + exp(b -c N)]4 82E H - cncC n 2 [3-46] S2A -NNexp(b -cN) +2A2 Nexp(b -c N) c 0c [1+ exp(bn cN)]2 [1 +exp(b -cN)]3 H =2A1 NFN' 2 exp(b c- N) 2NN2 [exp(b- c N)]2 SL [1 + exp(b cn )]2 [ + exp(b cn)]3 [3-47] +2A -N2 exp(bn -cN) 3N2 [exp(bn -cN)]2 Z [1 + exp(b cN)]3 [I + exp(bn Cn)]4 82E HBcB =H S2An N exp(b. cN)-2 N[+ exp(b,- c N)3 cc [1 + exp(b, cN)]2 [I + exp(bn cN )]31 H =2A N ,Nexp(bn c N) 2NN[exp(bn c N)]2 bncn ".1. + [exp(bn -cn)]2 [1 +exp(b -cAN)]3 [3-49] 2A1[ -Nexp(b -cN) 3N[exp(b -c )]2 [1 + exp(b, c,)]3 [ + exp(b c N)]4 Initial Estimates of Nonlinear Parameters (Linearization) Initial estimates of parameters in the nonlinear model can be found by linearizing the model and performing linear regression. The following steps show how to linearize the model: A N = [3-50] 1 + exp(b, cN) S=l+exp(b- cnN) [3-51] Nn 1 -1 exp(b, cN) [3-52] Nn In A-1= b -cN [3-53] N. In AN N =c.N-b [3-54] A -N To conduct this type of analysis, an initial guess of the An parameter must be given and then the parameters c and bn can be solved by simple linear regression. For the method employed in this analysis, 110% of the maximum value of N, for a given data set was used as a rough estimator for A This is generally sufficient to generate reasonable initial estimates of c and bn using the above linearization method. The solutions for b, and c from linear regression, respectively, are: N I N N A NN ) ln-N A)' N b 2)2 [3-55] l n N2 _~ln) n I n.A -N N In A -NN n 2 ( [3-56] nZN2 ( N)2 Iteration using the previously described nonlinear regression scheme can then be used to find optimized values for all three parameters. Statistical Analyses for a Single Data Set Statistical measures of key interest include correlation coefficient, variance, covariance, and standard error of the parameters. The correlation coefficient is defined as (Cornell and Berger, 1987): _-(N. N)2 [3-57] where N, is the actual N uptake data, N, is the estimator for N uptake based on the optimized logistic model, and N, is the average N uptake based on the data. Variance is defined as n2 s2 = N "- ) [3-58] n-p p= where n is the number of data points, and p is the number of parameters in the model (three, in this case, since only the nonlinear logistic model is being considered here). To find the covariance and standard error in the three parameters involved in nonlinear regression, three additional derivatives must first be found: a2E HAA- [3-59] 82E HA = Hbn =Ab [3-60] 32E H Ac = H CA [3-61] A c nAn OAnoc Previously it was found from Eq. [3-23] that: E = 2 N -1 [3-62] lAn r 1+exp(b- c N) +exp(b cN)_ Expanding and taking the second derivative: OE = _2 "1l exp( -c N ) _ +An 2~[ I[ 1 + exp(bn cN)[3-63] + 2 y A n ] 1 S_ 1+ exp(b cnN) 1 + exp(b cN) ] OE= 2 f [[N. +2y [[ -An [3-64] OA, I+ exp(b, c N) Z [I+ exp(b- c- Nc)]2 H2 = 2 [3-65] A 2En [1+ exp(bn -cN)]2 Previously it was found from Eq. 3-27 that: E 4 N. exp(b- cnN) 2 exp(b- cnN) b, + exp(b -cn)]2 [ + exp(bn -cN)] [3-66] Taking the derivative of this with respect to A : 82E H Anbn = HbnAn , = 2 [N +exp(b -cN) -4An exp(bn -cN) [3-67] N. [1+ exp(bn -c N)]2 [1 + exp(b -c N)]3 Previously it was found from Eq. [3-30] that: OE F2A NNexp(b -cnN) +2A Nexp(b -cnN) [3-68] c -2 A [ + exp(b -c N)] 2 n I+ exp(b -cN)]3 Taking the derivative of this with respect to A : H =H HAc2 E Ancn nAn B4Acn [3-69] = 2 -NNexp(b -cN) +4A Nexp(b -c cN) 3-69] S[1+ exp(b cN)]2 1 + exp(b cN)]3 The resulting Hessian matrix for this model is (Overman and Scholtz, 2002): HAnAn HAnbn HAncn H= HbA Hbnb Hbnc [3-70] H cnAn H CA H cnn To compute covariance and standard error, the inverse of [3-70] must be obtained (see Appendix). The covariance for two differing parameters can then by found by (Overman and Scholtz, 2002): COV(a,b)= s2[Hb 1] [3-71] where COV(a,b) is the covariance between parameter a and parameter b, and [H -1] is the corresponding element from the inverse of the Hessian matrix. For example, for the covariance between An and bn: COV(A,, b)= s2 [HAbn 1] [3-72] The standard error is given by (Overman and Scholtz, 2002): SE(a) =(s 2H )1/2 [3-73] where SE(a) is the standard error of parameter a and Haa1 is the corresponding element from the inverse of the Hessian matrix. Note for the linear portion of the model, the correlation coefficient and variance can readily be calculated from the N concentration (rather than uptake) analogs to Eqs. [3-57] and [3-58], respectively. Because parameters in the linear portion are solved for directly in the method outlined above, standard error and covariance for the linear portion cannot be assessed using this method (unless the corresponding derivatives are taken). However, the actual method of regression used in this study, even for a single dataset, was actually based on the general equations presented near the end of this chapter, which are based on the method for multiple data sets presented later. These equations include the appropriate derivatives and cross derivatives to obtain covariance and standard error for both the nonlinear and linear portions of the model, so a complete set of statistical measures can be obtained. Solutions for a Single Data Set: Second Method Suppose the error sum of squares is redefined to include all three data measures (N,, N, and Y), as first proposed by Scholtz (2002). The non-normalized version of this is: E= [N -^J. + -[N+-N +[Y-2 [3-74] As a result of the new error sum of squares, only nonlinear regression is used to optimize all five primary parameters simultaneously rather than the "split" of the model into linear and nonlinear sections as in the first method. Advantages of this approach include a more balanced influence of data and the ability to readily calculate sensible standard errors for all five primary parameters. Inserting the model into the new error sum of squares: E=- N A,, E I 1+ exp(b cN) [375] SA,( + exp(b cN)) A cAN c A(1+exp(b, -cN))_ 1 +exp(b-cN) Note here the model for N9 is presented as a function ifN only rather than Nu, as was used in the first method. The model for N9 in this form is simply the ratio of N, to Y . Normalization of this error sum of squares is sensible since it includes three different data types (N,, Nc, and Y), which generally are of different magnitudes. The normalized version of the error sum of squares requires dividing the three elements of Eq. [3-75] through by An, N,,, and A, as shown: -N -- A- A 2 E 1+ exp(b, cN) A, Simplifying further and rewriting in terms of primary parameters only: A,, 1 + exp( b c ,N) ) +' Nc A,,(+ exp(b -cN)) 1+ exp(bY ,Ncm NmA(1+exp(bn -CN))_ A +exp( A, 1+ exp(b, cN) + Nc (1+ exp(b Ab- cN)) 2 + N exp(Ab) (1+ exp(b, cN)) + YNc exp(Ab) _1 2 S A, 1 + exp(b, Ab cN) Just as presented in the first method, to minimize the error sum of squares, the first derivatives with respect to all five basic parameters should be zero: E =0 OA. [3-76] .N) [3-77] -c N) [3-78] [3-20] dE =0 [3-21] Obn dE = 0 [3-22] acc dE S= 0 [3-79] 8Ab dE = 0 [3-80] This is the point where the second method becomes rather complex. Compared to the first method, these first derivatives alone are quite lengthy. Second derivatives and cross derivatives becoming increasingly complex, and if commonality of parameters for multiple data sets is to be considered, this method becomes quite unwieldy. Thus, while the second method may be more ideal and unbiased than the first method, it is less practical in application. For this reason the first method was used in this study. Solutions for Multiple Data Sets and Commonality of Parameters Of key interest in crop behavior analysis is the search for commonality of model parameters between different data sets. Individual parameters of highest concern are b c, Ab, and Ncl. A is not considered in this study (except in the case of total commonality) because previous studies have shown it varies with water application and other environmental factors. Combinations of these parameters are also considered. For the cases presented below only two data sets (denoted by "1" and "2" suffixes) are considered for simplicity. However, the points presented can be easily expanded to analysis of several data sets simultaneously, as was done in the program for this study. Indeed, the cases presented below can be grouped into a set of general equations (for any number of data sets). These more formal equations better reflect the basis used for the algorithms in the regression program, but the cases are presented as they are in this section for better clarity on how each is derived. The corresponding general equations appear near the end of this chapter. Regression for the nonlinear portion is presented first, followed by regression for the linear portion, since the latter is partially dependent on the results of the former, as will be shown. Nonlinear Case 1: Common b, In general, the error sum of squares for the nonlinear portion for the two data sets would be: E= [N1, -N1,, + -[N2, -N2, [3-81] For the case of common b : Al A2 E =ANl ", + N2, "A2, [3-82] E Zr 1+ exp(b cl N1) 2 1+ exp(b, c2, N2) Strictly speaking, it may be sensible to normalize this error sum of squares so that in the case that the scale of one data set is much greater than the other, it will not unduly bias the results. However, early attempts at using a normalized method in programming revealed computational instability. Therefore, the non-normalized approach was used, as is presented here. Since only one data type is involved, and data sets are usually within the same magnitude, the biasing effect of not normalizing was assumed to be minimal. Ratkowsky (1983) does not mention normalizing when analyzing multiple sets of data in tests for commonality of parameters. Taking derivatives with respect to Al, and A2,, setting equal to zero, and solving: =E _2 N1 -1 = 0 [3-83] ,A1 [ 1+ exp(b -cl,% N1l)L + exp(b -cl-c Nl) jNlu +y ]p,[l1" ] =O 0 [3-84] 1 + exp(b cl Nl) [I + exp(b, cl N1)]2 Al 1 )]2 =j[ %" [3-85] [1 + exp(b -cl N1)]2 1 + exp(b clN) [3-85] A r+ exp(bn cl ) [3-86] Al,= [3-86] [1 + exp(b, -clN1l)]2 Similarly, 2, 1+ exp(b c2n N2) A2 = [3-87] y1 2 [1+ exp(b c2 N2)]2 Once again, only the A, parameters can be solved for directly. Other solutions are implicit only and so the nonlinear regression procedure illustrated previously must be applied to solve for the remaining parameters. Taking the appropriate first, second, and cross derivatives: OE Al Al exp(b cl NI) J E =2 N1F-77 Jbn = l+ exp(b, -c1,N1) (l+exp(b cl N1))2 [3-88] A22 A2N exp(b c2 N2) +2 [ 1 + exp(b -c2 N2) I (1 + exp(b -c2n N2))2 aiE FF;i A1 l Al N1exp(b cl N1) c L = l+ exp(b cln)]L (l+Iexp(b -cl NI))2 [3-89] N2 A2,, OE !_ 2 1+ exp(b c2. N1) Jc2, -2C [3-90] J" Oc2, A2 N2exp(b -c2 N2) L( + exp(b- c2 N2))2 HAI Al 2 = 21 [3-91] H a A1n ( + exp(b cl N1))2 HA2 A2 2'- 1 [3-92] 2 BA2 ~ (+ exp(b, -c2 N2))2 H 2E 2A Nl- l exp(b -cl N1) 2Nl [exp(b -cl N1)]2 6bnn b 2 [l +exp(b -cl N1)]2 [ + exp(b -cl N1)]3 2A2 exp(b -cl Nl) 3[exp(b -cl N1)]2 S[ + exp(b -cl N)]3 [ + exp(b -cl N1)]4 [3-93] +2A2 N2L exp(b -c2 N2) 2N2 [exp(b -c2 N2)]2 + [l 1+ exp(b c2 N2)]2 [I + exp(b -c2 N2)]3 2A22 exp(b -c2 N2) 3[exp(b -c2 N2)]2 S[ + exp(b -c2 N2)]3 [I + exp(b -c2 N2)]4 N1 N12 exp(bn -cl N1) S 2E [ + exp(b, -clN1)]2 H = 2A1l clcl" 1 2 2N1 Nl2 [exp(b cl.N1)]2 + [3-94] [1+ exp(b, -cl N1)]3 +2A -F N12 exp(b, cl, N) + 3N12 [exp(b- cl, N1)]2 S" 1 [1 + exp(bn -cl NSl)] [1 + exp(bn -cl. Nl)]4 -2E c2n N2 N22 exp(b -c2 N2) S2A[ +exp(b -c2g N2)]2 = 2A2, C [3-95] 2N2 N2 2[exp(bn -c2nN2)]2 S [+ exp(b -c2,N2)]3 -N22 exp(b c2 N2) [+ + exp(b -c2g N2)]3 +2A22 C 3N2 [exp(b, c2,N2)]2 + [ +exp(bn -c2nN2)]4 H = H =0 [3-96] H2E 22E H1,b" Ab 1 2 bbnAln b2 EbA1 A b [3-97] 2 Nl,, exp(b- c1'N1) 4A1_ exp(b c1lN1) S[1 + exp(bn -cl, Nl)]2 "j [1 + exp(b cl, N1)]3 82E =H 2E A2,b" 8A2 Hb b"A2" 8b A2 S N2, exp(b -c2,N2) 4Aexp(b -c2 N2) [3-9 Z [1 + exp(bn c2,,N2)]2 1[ + exp(b c2,N2)]3 d2E d2E H = H, - dAl1 c l "C' cc AlA1 [3-99] Nl,,Nlexp(bn -clN1) N+4AI I exp(b -cl,,N1) [3-99] L [1+exp(bn- clNl)]2 1 [1 +exp(b, -cl N1l)] d2E d2E H A2cl -- H 2- 0 [3-100] n2, c" 8A2 acl ,'"A2" cl, 2A2 82E 82E H Al H =- 2 0 [3-101] A," 8Al c2 c2 1 c2 Al 82E 82E H A2 c2, A2 c 2nA2n Hc2 ,A2 -N2,N2exp(b -c2 N2) [3-102] [l1+ exp(b -c2 N2)]2 + 4A2, N2 exp(b -c2 N2) S[1+ exp(b, -c2 N2)]3 H2E 82E H =H bcln Bb b cl- cl b" -cl b = 2A1 N1 Nl exp(b cl, N1) 2N1 Nl[exp(b c N)]2 [3-103] S[l+exp(b -cl N1)]2 [l+exp(b -cl N1)]3 2A2[ N1exp(b cl Nl) 3Nl[exp(b -cl N1)]2 1 [l+exp(b -cl N1)]3 [l+exp(b -cl N1)]4 82E 82E b"nc2" -3b c2, c2"b" 3c2 b =2A2 I N2 N2 exp(b -c2, N2) 2N22 N2[exp(b -c2 N2)]2 [3-104] [l+exp(b c2 N2)]2 [1+exp(b, -c2N2)]3 2A22 N2exp(b, -c2 N2) 3N2[exp(b c2 N2)]2 [1 + exp(b, -c2 N2)]3 [l +exp(b, -c2 N2)]4 82E 82E H a2- 2 H 2 =2 0 [3-105] cn cl, 1c2, Oc2, cl1 In order to solve for the nonlinear parameters, the Newton Raphson procedure outlined in the first method for solutions of a single data set can be employed here. Using similar notation as in that section, in this case the following system of equations applies: b 2 db, &1+ +.b2 E&2 b',cl',c2' b'c'c2' b'c'c2'[ b b',,cl',,c2', 82E 82E +2E S 2a ab )b',,cl',,c2' n 2 b',,cl',,c2', c% c2 )b',cl',c2', [3 [3-107] OE nE b'ncl'nc2'n K 2E 8 2E K 2E 0c 2 Eb 0c 2 Ec 1 c 2 E n2 , ,b',,cl',,c2', 2 cb',,cl',,c2' c2 b',cl',c2' [3-108] K c2, b',,cl',,c2'' Replacing these derivatives with the "J' and "H" notation for first and second partial derivatives (respectively) used previously, this system of equations in matrix form becomes: Hbnb, Hbnc1n Hbnc2n n bn Hclbn Hclcl Hclc2n -c [3-109] Hc2nbn Hc2ncn Hc2n2n C2 -Jc2 The extension of this matrix system in the case for more than two data sets should be fairly obvious. For each additional data set, one additional row and column would be added to the Hessian matrix, and one additional element would be added to each of the remaining matrices. For example, for three data sets, there would be one additional row, column, and element to reflect the inclusion of c3n and &3n. Since a matrix inversion routine is already necessary when determining standard error, the method of inversion is convenient to use to solve this system: H^ H- H -J1 L ( Hb Hbncln Hbnc2n -bn n = Hclb, Hc2,cl, Hcc2,n cn [3-110] c&2n, Hc2nbn Hc2n cln Hc2nc2n -c2n This solution gives the values which can then be used to update parameter values and iteratively solve for the parameters until all first partial derivatives are within a specified tolerance of zero, as presented in the first solution for a single data set The Hessian matrix for standard error analysis in this case is: H AlAl H HA HA2 HAlb HAl4cl, H 41c2, HA2 A1 HA2A2n H A2bn H A2ncln H A2nc2n H= HbA1, HbA2n Hbnbn Hbncl Hbnc2n [3-1 HlAl HA2 Hb H 1 H 2 c'nA'n cnAclA2n cnclb l clnc n cll c2n H H H H H Hc2,A1, Hc2,A2, Hcl2b, Hc2,cl, Hc2,c2, This matrix can be readily simplified by noting the cross derivatives between two differing data sets must be zero: HAl Al 0 HAl} H Al rc 0 HA 0 H ,b H1 0 = H Hb Hb c10 H [3-1 H O,, 1,, O H c,,b,, H O,,ex,, O 0 H 2,,A2,, c2,,b, H 2,,c2,, 11] 12] Once the inverse of this matrix is obtained, the equations presented earlier for covariance and standard error (Eqs. [3-71] and [3-73], respectively) can be used. It should be noted, however, that the variance used now includes both sets of data: Al1 2 1 rN1 1+ exp(b, clN1) s=- [3-113] n-p A21 +y N2 --- - + 2 1+ exp(b -c2,N2) where n is now the total number of data points in both data sets, andp is the total number of parameters (five in this specific case). Nonlinear Case 2: Common c, The error sum of squares for this case is: Al A2 E= N "Al +A + N2- --- _] [3-114] = "1 l+exp(bl, cN1) 2 l+exp(bl, -cN2) Using the same approach as in the previous case: OE Al -1 Al, L- l+exp(bl cN1) l+exp(bl cN1) [3-115] A1 = + 2 "- =0 [3-116] OA [l + exp(bl c1 c N)] -l+ exp(bl2 cN. ) A2 = [3-117] y1 A-1 + exp(b2n cnN2)] [3-117] S[l+ exp(b2 cN2)]2 Taking the remaining first, second, and cross derivatives: bE Al= 2 AlN exp(b1 cN1) E =2 1 N l+exp( c N) (l+exp(bl-cN1))2 [3-118] Jb" 81 O 1 + exp(bl, cN1) ](I + exp(bl cn N1))2 aE Jb2, 2 Ob2n 2 N2 A2n A2 exp(b2 cN2) [3-119] ZL 1+ exp(b2n cN2) ](1+ exp(b2- c.,N2))2 EN Aln AN1 + exp(bl. -cnNX ) c" c LL" 1 + exp(bl c lN1) (1 + exp(bl cN1)) r r ) 2 [3-120] S N2A2n A2nN2exp(b2n cNN2) 21[ 1+ exp(b2,, c,,N1) (1+ exp(b2, cN2)) a2E H 02 E HA2 A2 -- - 2z 1 2 !(1+ exp(bl c,,N))2 + b1 N)) S(1 +exp (b2, cN2))? a2 E H blnbln -- 2 [2A Nl, exp(bl- c.N1N) 2N1, [exp(bl c.N1)]2 [3-123] l" [1 + exp(bl -cN1)]2 [1 + exp(bl -cN1l)]3 Sexp(bl. cN1) 3[exp(bl, cN1)]2 -2 12-- [+ exp(bl cN1)]3 [1+ exp(bl cN1)]4 a2E Hb2nb2n -b2n2 wn ~b2n = 2A2 N2, exp(b2 cN2) 2N2, [exp(b2, cN2)]2 [3-124] L[1 + exp(b2,, cN2)2 [1 + exp(b2, cN2)] 22 exp(b2, -cN2) 3[exp(b2, -cN2)]2 Z .L [1 + exp(b2 cN2)T [1+ exp(b2, cN2)r 82E H 2 cncn Cn2 Nl,, N1 exp(bl cN1) 2N1, N12 [exp(bl,- cCNl)]2 2AI [1 + exp(bl, c-N1)2 + [1 + exp(bl c, N1)] 2A1 '- N12 exp(bl cN1) 3N12 [exp(bl, cN1)]2 21 + exp(bln cN1)]3 [1 + exp(bl, -cN1)] N2, N22 exp(b2, -c,,N2) 2N2 N22 [exp(b2, -cN2)]2 + A [1+ exp(b2, -cN2)]2 [+ exp(b2- c N2)] "+'2A2 N22 exp(b2 -cN2) + 3N22 [exp(b2 cN2)]2 + L [+ exp(b2, -cN2)]3 [+ exp(b2, -cN2)]4 [3-125] H2E H 2 E A "A'" dA 2,A = A2 ,1 [3-121] [3-122] S2E H AI, A2 A A2 [3-126] H 2E H 2E HAl,,abl n bA l b n bln Aln bl Al [3-127] 2! Nl,, exp(bl -cN1) 4A exp(bl -cN1) [3-127l) [1 + exp(bl, clN)]2 41 [I + exp(bl,, -cNl)] HbE H0E H b2 Hb-AI =b2 E1 0 [3-128] b2" Al Ob2 n b2 nAl 2E H a Al"c" 8Al Sc 02E H c"Al" Bc A41 [3-129] 2 Nl,, Nexp(bl, cN1) 4A1,'I N1exp(bl, -cN1) [3129] [1 + exp(bl,,- c,,N1)]2 J 1 [+ exp(bl -cN)] a2E H, 2 )1" A2 bln4 a2E Hb b, A2 6bln OA2 [3-130] H2E =2E HA2nb2n A2 b2 = Hb2nAn 8b2 EA2 N2,, exp(b2n -cN2) exp(b2 Z [1+ exp(b2, cN2)]2 14A2Z 1 + exp(b2, Q2E dA2 c,, S [3-131] cN2)] - c N2)] 2E H A2 . Sc,,8A2,, S- N2,,N2 exp(b2n -cN2)' 1 [l+ exp(b2n cN2)]2 + 4A2 N2exp(b2n-cnN2) [l1+ exp(b2n -cN2)]3J [3-132] a2E H b b2 b2nbl" 8b2,8bl, b 2E H 2 O2, oblnb2, [3-133] 82E -2E H =H bl"c" Obl Cn c"bl" nOcbl1 = 2A N1, Nlexp(bl cNl) 2N1l N[exp(bl cnN1)]2 [3-134] [l+exp(bl- c N1)]2 [ + exp(b1 cN1)]3 2A2- NI exp(bl c N) 3N[exp(bl -c N1)]2 Z1 [1+ exp(bln -cN1)]3 [1 + exp(bl -c N1)]4 82E 82E H = H Hb2ncn b2 cn c"b2" =cC, b2 -N2 N2 exp(b2 -c N2) [ + exp(b2n -cN2)]2 = 2A2 [3-135] S22N2 N2[exp(b2 c N2)]2 [l + exp(b2 cnN2)]3 2A2 [ N2exp(b2 c N2) 3N2[exp(b2n -cnN2)]2 S[l+ exp(b2 -cnN2)]3 1[ + exp(b2 cN2)]4 Using an approach very similar to Case 1, the solution to the corresponding system of equations for this case is: Hbi HH H H -J n H blbln Hblnb2n Hb1nc, bl b2, Hb2nbl Hb2,b2 Hb2c Jb2 [3-136] &L cn H Hbln H b2n H -JC The extension of this matrix system in the case for more than two data sets would be very similar to such an extension in Case 1 (except, of course, additional elements would be the result of additional "bx," parameters rather than "cxn" parameters). The Hessian matrix for standard error analysis in this case is: HA1 A1l HA1'A2n HAlnbln HAlnb2n HAl ,c HA2A1 HA2nA2n HA2nbln HA2,b2n HA2nc, H= HblA1n HblnA2n Hblnbln Hblnb2n Hbl1nc [3-137] Hb2nA1n Hb2nA2n Hb2nbln Hb2nb2n Hb2ncn HcAln HcA2n Hcbln Hcb2n HCncn This matrix can be readily simplified by noting that the cross derivatives between two differing data sets must be zero: H 0 HA1,,bl 0 HA ,,C, 0 HAA2A2n 0 HAZb2n HA2ncn H= HblAlo 0 Hblbl 0 Hb1,, [3-138] 0 Hb2,A2, 0 Hb2,b2, Hb2,c, H, H 2 HcAb H Hcb2 H c,, Once the inverse of this matrix is obtained, the equations presented earlier for covariance and standard error (Eqs. [3-71] and [3-73], respectively) can be used. Again, however, the variance now includes both data sets: Aln 2 1 1+ exp(bl, -cN1) [ s = [3-139] +n-p N 1 +exp(b2n -cN2) where n is the total number of data points in both data sets, andp is the total number of parameters (five in this specific case). Nonlinear Case 3: Common b, and c, The error sum of squares for this case is: Al A2 E= Nl-- Aln +A N2- [3-140] 1 1 + exp(bn cN1) I 1 + exp(b cnN2) Using the same approach as in the previous case: E =21-- Nl- C 10 [3-141] MAn 1+ exp(bn -c l)N1 1 + exp(b cN1) [ NI. 1+ exp(bn cN1)[3 Al = c [3-142] 1 [ + exp(b -cN1)]2 N 2 , A2 xpb 2)] [3-143] Z [l+exp(b,-cN2)]2 Taking the remaining first, second, and cross derivatives: OE Al Al exp(b c N1) J c, = 21 Nl- . Jbn b 1 + exp(b cN1) (1 + exp(bc -cN1))2 [3-144] + 2 N2A2 A2n exp(b -c N2) LLU 2" 1+exp(b, -cN2)I (i + exp(b -c N2))2i lE A 1Aln N1Nexp(b cN1) c 1 + exp(b- c l) (1+ exp(b cN1))2 2 [[N2 uA2 A2 N2exp(b -cN2) IU 1 + exp(b cN1)IL(1+ exp(b c N2))2 H 2E _2- 1 [3-146] A A A12 21 (I + exp(b, cN1))2 H 2E =2 + [3-147] A2n BA2n Al2+expp(b -cN2))2 SNl, exp(b,, cN1) 2N1,, [exp(b n [1 + exp(b,- cN1 )]2 [I + exp(b,, 12A2 exp(bn -c.N1) n [I + exp(b, -cN1)] + 2A2 N2, exp(b -cN2) +2A2 2. [L + exp(b cn-N2)]2 _2A22.[ exp(bn -cnN2) [1 + exp(bn -cN2)]3 3[exp(b - [l + exp(b, 2N2, [exp(bn [1 + exp(b- 3[exp(b [1 + exp(b, - cl)l]2 cN1)f - cN1l)]2 [3-148] - c,,N2)]2 cN2)]3 cnN2)]2 - cN2)]4 H 2E H a Cncn Cn 2 '[Nl N12 exp(b,, -cNl) 2N1,, Nl [exp(b,, -c,,)]2 2 [1+ exp(b, cN1I)]2 [1+ exp(b cN1)]3 S2A -Z N12 exp(b cN1) + 3N12 [exp(b, c, N1)2 [1+ exp(b, cN1)]3 [ + exp(b, -cN1)]4 +2A2 N2, N22 exp(bn cN2) 2N2, N22 [exp(bn c-NN2)]2 [1 + exp(b cN2)]2 [1 + exp(b, cN 2)] 2 N22 exp(b cN2) +3N22 [exp(b- cnN2)]2 + 2A2 [l+exp(b cN2)]3 [1+ exp(b, cN2)]4 - a2E H ,A2 "' A A2, H2E H', A2, 'Al 2E 2E .1"A lnAbn, Ob "Aln^ S21 Nl,, exp(b, cN1) 4A1 exp(b cN1) [lI [+ exp(b -CnN1)2 )] 1[1+ exp(b) -cNI)] H 82E H a- E H =H Alnc" 9A1, c^ CA"' 9c ,.11 S aAab N [3-152] 2 Nl,, N1 exp(b- clN) + 4A N1exp(b, cN1) 1 + exp(b -c,,Nl)]2 [+ exp(b, -cnNl)]3 8 2E H b bAb- Sb,- [3-149] [3-150] [3-151] 2 E H E 2 N2, E exp(b, c,2) -A2 exp(b, c,2) Z [1 + exp(bn cN2)]2 n 11 + exp(b c N2)]3 82E 82E H A2ncn" A2 H cA2n" c A2 -2 N2 N2 exp(bn c N2)1 [3- 1 [l+ exp(b [-c X2)]2 [+ 4A p 'N2exp(b -c N2) 1 [l+exp(bn -cN2)]3 2zE -82E H b 2 H CA 8 b ncn obc cnbn O;nCb = 2Al N1, Nlexp(bn cN) 2N1l Nl[exp(b cnN1)]2 1+ exp(b, -cN1)]2 [1+exp(b -cN1)]3 -2Al Nexp(bn CNl) 3Nl[exp(bn CnN1)2 [3-155] n [1 + exp(b -CnN) 1] + [exp(b -c Nl)]4 + 2A2 N2,N2exp(b -N2) 2N2, N2[exp(b, -cN2)]2 2 Z [1 + exp(b cN2)]23 [1 + exp(b, cN2)]3 2A22 N2 exp(b cN2) +3N2[exp(b -cnN2)]2 [1 + exp(bn c N2)] [I +exp(b -c N2)] Using an approach similar to Case 1, the solution to the corresponding system of equations for the iterative "3 values for this case is: 5n Hbnbn Hbncn Jbn [-156 L I [ cr,: Z J ] [3-156] Note that symbolically this is the same as the first method for the solution for a single data set. For conformity with Case 1 and Case 2, it is presented here in the "inverted" solution form rather than the Cramer's Rule solution presented in that section. The extension of this case to more than two data sets would not alter this final system symbolically, but additional elements would be present in the underlying derivatives since for each new data set, a new element would be added to the error sum of squares equation. The Hessian matrix for standard error analysis in this case is: H AlnAn H AlA2n HAlnbn HAlncn H HA2nAn HA2nA2 HA2nb HA2 [3-157] HbnA2n Hbnbn Hbln, H H ,. HCbn Hncn This matrix can be simplified by noting that the cross derivatives between two differing data sets must be zero: H AlnAn 0 HAlnbn HAlcnc O H H H S 0 HA2A2n A2b HA2nc [3-158] HbnAl HbnA2n Hbnbn Hbnc H Aln HcA2n Hcbn H nn Once the inverse of this matrix is obtained, the equations presented earlier for covariance and standard error (Eqs. [3-71] and [3-73], respectively) can be used. Again, the variance now accounts for both sets of data: 2 Al 2 1 N 1 + exp(b cN1) s [3-159] n p NbA2 2 I 1+ exp(b c N2) where n is the total number of data points in both data sets, andp is the total number of parameters (four in this specific case). Nonlinear Case 4: Common An, b., and c, The error sum of squares for this case is: A2 A2 E= N% A,,- +y rN2- A" [3-160] 1 + exp(bn cN1l) 1+ exp(b cN2) Using the same approach as in the previous case: OE A 1 ,, 2Z N + exp( clNI) + exp(b,- cnN1) LL 1[3-161] +21 N% "--c = 2ZI 1+ exp(b, cN2) 1+ exp(b, c2) 1 Np IN2 ) S1+ exp(b, cN1) 1+ exp(b, c N2) A [3-162] 1 1 S[1 + exp(b cN1)]2 + [1 + exp(b cN2)]2 Taking the remaining first, second, and cross derivatives: Jb = Z + exp(bn cN1) I (1+ exp(b -cN1))2 A,, A,exp(b, c,Nl) [3-163] +2A A2 exp(b, cN2) aZL =2 1+ exp(b, -cN2) L(1 + exp(b cN2))2 ] F FE A1 A N12exp(b cN1) [3-164] (A A + exp(bn -cN2)) LZL" 1 + exp(b, cNl) I (1 + exp(b, cN2))2 2 1 E +I H 2 -A = 21 ]_ 1 +2y [3-165] A EA 2 ~ 1+ exp(b, c,N1))2 1 + exp(b, c,N2))2 2 2E N1, exp(b -c N1) 2N1 ,[exp(b Hbb 2 2A abb 2b [2 + exp(b, cN1)]2 [1+ exp(b - 2A exp(b -cNl) 3[exp(b- n2 [1+ exp(b cN1l)]3 [1+ exp(bn S2AI N2,, exp(b -cN2) 2N2, [exp(bn + z [1+ exp(b cN2)]2 [+ exp(b- -2A exp(b cN2) 3[exp(bn 2A2I [ + exp(b, cN2)]3 [+ exp(bn -cN1)]2 ~ c.N1)]3 - cN1)]4 [3-166] -Cn2)]3 c.NN2)] -c.N2)]2 -c.N2)]4 82E H a2) cncn aCn2 2A Nl N12 exp(b- clN1) 2N, N12 [exp(bn c Nl)]2' [1 + exp(b -c N1)]2 [1+ exp(bn -c N1)]3 + 2A N12 exp(bn -clN1) 3N2 [ep(b cN1)]2 S[1+ exp(b -c l1)]3 [1 + exp(b -c N1)]4 + 2A N2, N22 exp(b cnN2) 2N2 N22 [exp(bn -cN2)]2 n [1 + exp(bn -CN2)]2 [+ exp(b -c N2)]3 2A --N22 exp(bN -cN2) 3N22 [exp(b cN2)]2 [1 + exp(b -c N2)]3 [1 + exp(b -c N2)]4 82E H2E H Hb nb" A Ob bn- BA b ~aA 2 Nl,, exp(bn -c Nl) exp(bn -c Nl) Z[ [1+ exp(b cN1)]2 1 b [i + exp(b, cN1)]3 +2 N2,, exp(b, -cn2) -4An exp(b, -cnN2) [1+ exp(b cnN2)]2 [1+ exp(b cN2)]3 82E H2E Ancn A, c, cnAn 8 H ACOAn ACc H 'Ac ,-i 21 F- N1, N1exp(b cN1)+ Nlexp(b- cN1) S[+ exp(b- cN1)]2 ] [l + exp(b l -cN1)]3 21 N2 N22exp(b -cN2)] [ N2exp(b -cnN2) S[1 + exp(b- cN2)]2 [1+ [exp(b cnN2)]3 [3-167] [3-168] [3-169] a2E 02E H =HC bnc" B c bn c cBcbn =2An Nl, N exp(bn- cnN) + 2N1 N1[exp(b clN)]2 [1 + exp(bn -cN1)]2 [l +exp(bn -cN1)]3 -2A2 Nlexp(bn cNl) 3N1[exp(bn cN1)]270] [1+ exp(b cN1)]3 [1 + exp(bn cnN1)]4 [ + 2A~ -N2 N2 exp(bn -cN2) + 2N2 N2[exp(bn -c N2)]2 S [ + exp(bn -cnN2)]2 [I+ exp(b, -cN2)]3 _-A -N2exp(b- c ,N2) 3N2[exp(b- cN2)]2 2A [1+ exp(bn -C N2)]3 [1 + exp(bn -cN2)]4 Since An can still be solved for directly as long as b, and c, are known, the solution to the corresponding system of equations for the iterative values is symbolically the same as Case 3: bn Hbnbn Hbncn b[1 L 1I L H .Jb [3-156] &n LHbncn Hcncn Jcn As in Case 3, the extension of this case to more than two data sets would not alter this final equation symbolically, but would add elements to the error sum of squares and the corresponding derivatives. The Hessian matrix for standard error analysis in this case is the same symbolically as that in the case of a single data set: H AnAn HAnbn HAncn H= HbnA Hbb Hbncn [3-70] HcnAn HcAbn Hcncn Once the inverse of this matrix is obtained, the equations presented earlier for covariance and standard error (Eqs. [3-71] and [3-73], respectively) can be used. Again, the variance includes both sets of data: 2 1 1 1+ exp(bn cNNl) [3-171] n-p N 1+ exp(bn cN2) where n is the total number of data points in both data sets, andp is the total number of parameters (three in this case). Linear Case A: Common Ab Recall the linear portion of the model is: N, = N,1 + NcN [exp(Ab)-1] [3-172] An The corresponding error sum of squares is: E =Z[N -N,' -Z N N-~'N N [exp(Ab) -] [3-173] A, For the case of two data sets ("1" and "2") the error sum of squares is: E= [Nl ~, + [N2c N2c [3-174] E=Z N1- N 1, Nl l E= NlN -Nli- N "iNiu [exp(Ab)-1] Al 12 [3-175] + N2N2c N2cN2, [exp(Ab)-l] A2n Note here it is assumed the An's of the two data sets differ. For the purpose of this analysis, the values of A,are assumed "known". The values of A,used here would be those found in the previous nonlinear regression. Therefore, depending on the case, they may be the same or different, but here the more general case of unique values is presented. Simplifying back to primary parameters: Again, the derivatives with respect to the unknown parameters should be set to zero in order to minimize the error sum of squares. Rearranging the error sum of squares and then taking the derivative with respect to Ab: N1,, N N1 N1, E _. NI N, N1 1 exp(Ab) Al Al SA2 [3-176] N2c N2^ N2c1N2, + N2 -N2,1 N2- exp(Ab)+ N2,1N2 A2. A2. OE 8Ab N N1 NN1l N l, NI, N1 N exp(Ab)+ NIN AI AA -2y 1 [3 -177] AAb exp(Ab) N N1 2 +[ N2c, N2 A2 A2, -2V x N exp(A) N [3-178] A2,, 1 A2, Setting equal to zero, rearranging, and solving for Ab: -0 SAb ---N N1 N[3-178] = 2exp(Ab)-2.,tt M- N1,z -- NN2, N2, N2, N2, -^h A2 A2 N1. N1 -,, N2 ,, N2,, exp(Ab) [ Nl,, + [ 2" =]2] N, N1lA, + I,, [3-179] + N2F N2, + N2, N2, I N2, N2, S Al,, L A2,, Z C N A1, I A1 +Z[[N2- N21 + N2,1 N2, ][N2,c N2A Al, A21 Taking the derivatives for the two remaining parameters: -2y-] [3-182] NE !, x NlI r Nl,, exp(Ab) [3- n Al n A1 N l +N, exp(Ab)-1 Al41 Al1 v Setting equal to zero, rearranging, and solving for Nl,: =E 0 = 2N1c [ N, exp(Ab) 11 N1-, Al Ail NL 1 [3-183] + 1[N,[Nl_ NI +2N N1C NI. exp(Ab)- Al Al Z[FNl1u Ni, lexp0xb N1lL = LI A2 [3-184] N1, N1, exp(Ab)-l Al A1, Similarly: SN2C [N2 N2, exp(Ab) O2E +IA2n A2n = 2Y [3-185] A2n A2n 2 N2 N2 exp(Ab)-1 N2,, -z[N2 exp 2 [3-186] A2 "exp(Ab)-l I IA2n A2 I Since all parameters can be solved for explicitly, each parameter can be updated by iterating through the sequence of equations. In other words, N1, and N2, can be updated each time using Eqs. [3-184] and [3-186], respectively, and these new values of Nl~, and N2,, can then be used in Eq. [3-181] to update Ab, which can then be used to update Nl, and N2, again. This process is repeated until all first partial derivatives are within a specified tolerance of zero. The extension to cases of more than two data sets should be fairly obvious. The corresponding error sum of squares and the derivatives thereof would simply have an additional element for each data set, and an additional equation for each Nx,, would be formed. Standard error analysis will require the second derivatives and cross derivatives. The first derivative with respect to Ab, written slightly differently here, is: OE OAb 2N1, Nl, + N2,, N2 [3-187] =2exp2LAb) 1 +N A-2 [3-187] Al A2, rNl["- Ni+ N1 + NNI N1N 1Nl" -2exp(Ab) A + [N2 -N2, +N2cN2, N2N2N2 A2, A2 Taking the second derivative: 02E NA I ,N ]2 + N2,N2, a =E 4exp2(Ab)t.r|IN +NH 2 OAb2 Al A2 SNNl Nl 1 ~ 1 [3-188] Z' ( 1 Aln l Al1 ] 2 exp(Ab A 1- N2I{-N2 + N2~cN2, N2 cN2] + N 2 c N 2 A2, A2 The first derivate with respect to N1,, rearranged here, is: 2 OE N% N% S2N N exp(Ab) - NIc I Ai [3-189] +2Z[N1c[N NI "exp(Ab)-1 S A1 A1, Taking the second derivative: S2E aN1 2 I A Similarly for N2,: S2E 2:-[N2, SN2,1 L A2, Taking the cross derivative of Ab and N1c,: a2E 2E 8N1, 8Ab 8AbaNl, 8Ab Al% NI exp(Ab)- Al Completing the derivative: 2E -_41,FN1. Nl exp(Ab)- Nl exp(Ab) NIcl ,Ab [-LA A, IL A 1e J Similarly: 2 E =-4 [N2, 8N2I 8Ab LI A2, 2 e( b N2" exp(Ab) A2, 1L N2 exp(Ab) A2,7 The Hessian matrix for standard error analysis in this case is: HAbA H= HNIAb HN2,, Ab H AbN1c HAbN2c, H NlcNi H NlcN2 H 1, N N2, N2c,N2 This matrix can be simplified a bit by noting the cross derivatives between two differing data sets must be zero: H AbAb HNIcAb HN2cAb N2,, A6 H AbNl1c H AbN 2 HNciNlcl 0 0 HN2cN 2c, [3-196] [3-190] N2 N exp(Ab)- I A2 I [3-191] NI exp(Ab) Al 12 [3-192] [3-193] [3-194] [3-195] Once the inverse of this matrix is obtained, the equations presented earlier for covariance and standard error (Eqs. [3-71] and [3-73], respectively) for the nonlinear portion can now be used for the linear portion as well. It should be noted, however, that a modified form of the variance must be used, based on the normalized error sum of squares for the linear portion: N1c N1- N11 N1 [exp(Ab)-i1 2 1 Al s = [3-197] S+ N2 -N N21 N2- [exp(Ab)- l] n-p Ac A2 where n is now the total number of data points in both data sets, andp is the total number of parameters (three in this specific case). Linear Case B: Common N1 The normalized error sum of squares for this case is: E=Z_ Nl-Nc- NNl- exp(Abl)+ NcINI 1 2 [3-198] + N2 N- NcN2 exp(Ab2)+ Nc N2 A2 A2 Taking the derivative with respect to Abl: Nlc N, NcINI exp(Abl)+ N-1N1l OE A' n Al1 A -2Y A [3-199] Setting equal to zero and solving: NIc N- N-NI" exp(Abl)+ NNI Al N AlA 0 = -2 A x [ NcNI. exp(Abl) NAI exp(Abl)j AN 1NI N1,c LA1. ''[NI S[N1, Abl = In N,1 + Al z N,,Nlu 2 Al Similarly for Ab2: ZLN2 -Nc,+ NN2] NN2 Ab2 = In ANN2, i A2, A2, I Taking the derivative with respect to Nc NlN + Nc, [N +Nl Nl~ A, Aie N2c + Nct N2 A2,, N1, exp(Abl) Al, xp( EL N2 A2,, Abl)-1 N2 exp(Ab2)- A2,, N2A exp(Ab2)-l A2,, Setting to zero, rearranging and solving for Nc,: OE 8Abl [3-200] Nc N1,A Nj1 N1, A,1L Al, [3-201] NANI, Al [3-202] [3-203] =2 +2Y [3-204] 1, exp(Abl)-1 A Al N2 N2 : NNd + A exp(Ab2)-12 + N2,N2 AS 2 exp(Ab2)-1 +AI IA, NA2, [3-205] Nl Nu N exp(Abl)-1 A A, A + IN2c N" N2 exp(Ab2)- i[h A2n A2 Nc I I- 2 n 2A2 [3-206] 1 N exp(Abl)-l 2+ exp(Ab2)-1 Since all parameters can be solved for explicitly, each parameter can be updated by iterating through the sequence of equations until all of the first derivatives are within a selected tolerance of zero. The extension to cases of more than two data sets should be fairly obvious. The corresponding error sum of squares would simply have an additional element for each data set, and an additional equation for each Abxx would be formed. Standard error and covariance analysis will require the second derivatives and cross derivatives. The first derivative with respect to Ncl, expanded out here, is: N1 exp(Abl)- OE Al, Aln = 2N, NC, I N2, exp(Ab2)-1 2 A, A2 [3-207] +l Al exp(Abl)- +2 +[ 72 N u N2 exp(Ab2)-l A2 A2 The second derivative is: S A4eb)(Abl)- 82E Al, Al, 1 -b2 [3-208] 2exp+ (Ab N N exp(Ab2)- 1 A2l A2 The second derivative with respect to Ab A is: =4exp2 (Ab2l) - 8Abl2 A\ Abl2 A12 [3-209] -2exp(Abl)- N1-Nc- t [I b [A2, Al,, Similarly, the second derivative with respect to Ab2 is: -=4exp2(Ab2) AZb22 'L^ A2 LA2[3-210] -2exp(Ab2) N2c- N~1 + U ] The cross derivative between NC and Ab1 is: 82E N N1 N =--4 N "exp(Abl)-1 Nlexp(Abl [3-211] 8NAb1 L A, A1 JIL A Similarly, the cross derivative between No, and Ab2 is: 2E = -4 N2U N2 exp(Ab2)-IN exp(Ab2)l [3-212] cN,8OAb2 IA2 A2, A2,, The Hessian matrix for standard error analysis in this case is: HAblAbl HAblAb2 HAb1lNe H= HAb2Abl HAb2Ab2 HAb2Nc [3-213] HN,,Ab1 HN,,Ab2 HNc,,N This matrix can be simplified a bit by noting the cross derivatives between two differing data sets must be zero: HAblAb1 0 HAbINi H= 0 HAb2Ab2 HAb2N [3-214] HNcAbl HNcAb2 HN,,N Once the inverse of this matrix is obtained, the equations presented earlier for covariance and standard error (Eqs. [3-71] and [3-73], respectively) can be used. The variance used in this case is: NI Noz NNI" [exp(Abl)-] 2 1 Al 11 s 1 L "1 A [3-215] n-p + N2- N- NCN2 [exp(Ab2)-1 Lc A A2 where n is the total number of data points in both data sets, andp is the total number of parameters (three in this specific case). Linear Case C: Common Ab and N1 The error sum of squares for this case is: E Z[LN1 Nl NIN"I [exp(Ab)-1] L A [3-216] + Zr N2 NI NN2" [exp(Ab)-1] A2 Taking the derivative with respect to Ab: OE 8Ab N1 No INI exp(Ab)-+ N NI. Al A1 -2 [i e ) [3-217] x I exp(Ab) Ai N2 N N2 exp(Ab)+ NIN2" rN2C N NA SA2 A2, -2z x NIN2. exp(Ab)- I A 2,, A2 Setting equal to zero and solving: rNl N 1+ A I 1] AL Al1 +1 Nc-c+ NN2 FNcN27 Ab = 1n A2 [3-218] c N,,N1 NcIN2] Taking the derivative with respect to Nt,: NI, + N,j[Nlu, Nl,, exp(Ab) - 1. NAl, AI N2 +N,, N2u exp(Ab) A2" A2 + x N 2 N 2" exp(Ab)- 1 - A2,, A2, [3-219] 1 Setting to zero and solving: [ INl NI- exp(Ab)-1 Z Aln, Al + 2e [N2CN2, N2 exp(Ab) 1- L N2i AA2 A2 N,= I2, 2 [3-2 L A A, exp(Ab)-1 +i] 1 N2, exp(Ab) -1 The extension to more than two data sets would simply involve additional elements in above equations corresponding to additional datasets (Nx, Nx,, and Ax). Standard error analysis will require the second derivatives and cross derivatives. The second derivative with respect to Ab is: 82 E = e 2 It N1N,, N,1N2 = 4exp (Ab} +Y Ab 2A, [3-2A, L [N1, N1,, + N N1,, NN1, [3-2 L L Al iL Al, } -2exp(Ab) + 2 N2,1N2 I NN2 + -N2c -N2,1+ ,,. ,,1 A2,, 11A2,, 20] the 21] The second derivative with respect to N,, is: NI- N N1I exp(Ab)-[l2 82E -Al A Al n, S2 = A% A [3-222] dNt N2 N2 aNC2 + I 2 N exp(Ab)-1 2 The cross derivative between NC, and Ab is: a E Nl N% Nl 2 -4 l exp(Ab)-1 exp(Ab LL Ab Al, Al IAl, [3-223] N2 N2 AN2 -4 L N2 N2 exp(Ab)- I exp(Ab) [I- A2, A2 1 A2, The Hessian matrix for standard error analysis in this case is: H= AbAb HAbN [3-224] HNcAb HNciNc Once the inverse of this matrix is obtained, the equations presented earlier for covariance and standard error (Eqs. [3-71] and [3-73], respectively) can be used. The variance for this case is: N1 Nc NNl [exp(Ab)-1] 2 1 Al s = [3-225] n-p + N2c N-, [exp(Ab)- ] Lc A A2 where n is the total number of data points in both data sets, andp is the total number of parameters (two in this case). Generalized Equations for All Analyses Mainly for programming purposes, a set of general equations can be derived to encompass all of the above cases. Then, with only small addendums, these equations can be used to fit any case. The resulting equations are presented in this section. Little description or derivation is shown since these are merely generalizations of the patterns apparent in previously presented cases. It should be noted that these general equations also apply to cases involving a single data set. Indeed, because of slight discrepancies observed between the non-normalized and normalized approaches when analyzing data, the normalized approach was used even when analyzing a single data set for consistency. This had the added benefit that standard error analysis could easily be extended to the linear portion for a single data set without having to derive any more equations. Variable Definitions Let i denote the current data set andj the current data point. Let nds equal the number of data sets in the analysis and ndp(i) the number of data points in data set i. Let N(i,j) denote the N uptake value for data point within data set i. Similarly, let N,(ij) denote the N uptake value for data point within data set i. Similarly, let Nc(i,j) denote the N concentration value for data point within data set i. Let An(i) denote the value of An for data set i. Similarly let b,(i), c,(i), Ab(i), and Ncl(i) denote the values of b,, c,, Ab, and Nc~, respectively, for data set i. Let EN, and ENc denote the total normalized error sum of squares for the nonlinear portion (based on N uptake) and linear portion (based on N concentration), respectively. The "J' and "H" notation used previously will also be used here to represent first and second (or cross) derivatives, respectively. The suffix (i) will denote that the derivative applies specifically to data set i. General Equations for Nonlinear Portion For all cases the following equations apply: nd ndp() A ) (i) E A = I N (i, j) [3-226] 1-1 J-1 I 1+ exp(b, (i) c, (i)N(i, j))_ "ndp(z) N. (i, j) E1+ exp(b (i) -c (i)N(i, j))2 + exp(b(i) ndp(L) 1 L l+exp(b ( i) c (i)N(i, j))]2 OE cCn(i) A, (i) N. (i, j) A (i) ndp() 1 + exp(b (i)- c(i)N(i, j)) -2 -= x A (i)N(i, j) exp(b Cn(i)- ()N(i, j)) (I + exp(b (i) c (i)N(i, j))) H (i,i) A (~ 2 A (i8 )2 ndp () 1 2 2 (J I + exp(b, (i) c, (i)N(i,j)) [3-230] [3-231] Hb,.b (i, i) = Ob, (i) SN. (i, j) exp(b (i) c,, (i)N(i, j)) ndp( ) [ + exp(b,(i) c,(i)N(i, j))]2 = 2A()))) =1 2N,(i)[exp(b,,(i)-c,,(i)N(i, j))] [1 + exp(bn,, (i)- c,, (i)N(i, j))]3 exp(b, (i)- c,, (i)N(i, j)) S ndp( [1+ exp(b,, (i)- c, (i)N(i, j))]3 2A, (i) C j = 3[exp(b,(i)-c,(i)N(i,j))] [1 + exp(b (i)- c, (i)N(i, j))]4 Jb,(i) [3-227] [3-228] [3-229] J,, (i)= S 2E H (1,1)- Oc (i) 2 N. (i, j)N(i, j)2 exp(b (i) c, (i)N1) ndp(i) [1 + exp(bn (i)- c (i)N(i,j))]2 = 2An (i) J 1 2N (i, j)N2 (i, j)[exp(b (i)- c (i)N(i, [1+ exp(bn (i) (i)N(i, j))]3 N2 (i, j) exp(bn (i)- c (i)N(i, j)) Sndp(i) [1 + exp(bn (i) c(i)N(i, j))]3 +2A 2 )]2 J 3N (i, j)[exp(b(i)-c(i)N(i,j))] [1+ exp(b (i)- (i)N(i, j))]4 S2E 2E H (1, = 02-- = H (i, ) = 02 E- S N,(i, j) exp(b (i)- c(i)N(i, j)) [1+ exp(bn (i) n (i)N(i, j))]2 S [1+ exp(b(i)- cn(i)N(i, j))]3 H (2 8E H (i2E Ancn ) A (i)ac (i) An n (i)A (i) ndp N, (i, j)N(i, j) exp(b (i)- c (i)N(i, j)) 2 . jr N [I + exp(b (i) (i)N(i, )]2 ) ndp( N(i, j) exp(b (i)- cn (i)N(i, j)) + 4An [) I exp ,1 [1+ exp(b,(i)-c,(i)N(i, j))]3 [3-232] j))]2 [3-233] [3-234] S2E 2E H (i, i)- H (i, i) = bn (i)enc(i) cn(i)8b,(i) N (i, j)N(i, j) exp(b (i) c,, (i)N(i, j)) ndp() [I +exp(bn (i-c (i)N(i, j))]2 = 2An(i) ] [3-235] S 1 2N (i, j)N(i, j)[exp(bn ((i)- N(i,))] [1 + exp(bn (i)-c(i)N(i, j))]3 N(i,j)exp(bn(i) -cn(i)N(i, j)) Sndp(,) [ + exp(b (i) c (i)N(i, j))]3 2An2 (i) 1 3N(i, j)[exp(bn(i) -c(i)N(i,j))] + [1 + exp(bn ((i) N(i, j))]4 For common bn, each bn(i) would be assigned the same value bn, and the following equations would also apply: nds Jbn = Jbn (i) [3-236] l=1 nds Hbb, = Hbb (ii) [3-237] i=1 For common cn, each c,(i) would be assigned the same value cn, and the following equations would also apply: nds Jn,, = n, (i) [3-238] i=1 nds H,,, = (i,i) [3-239] 1=1 Note Eqs. [3-236] through [3-239] also apply for both common bn and cs, as does the following equation. nds Hb,,c = Hcb = aHbcn (i,i) [3-240] -=1 For common An, bn, and cs, all previous equations would apply, as well as the following: nds ndp(z) N, (i, j) nd ndp(() 1 1- [ + exp(b (i) c (i)N(i, j))]2 nds HAnAn = ZHAnAn (i,1i) nds z1i nds 1=1 HAcGner Hc AE =r HAniLi r i) General Equations for Linear Portion nds ndp(') EMc = Nc (i, j) 1-1 J=1 NcO OE ndp) N = -2 + OAb(i) J 1 ab 2E H AbAbb OAb (i ) 2 (-N (i)N(i, j) [exp(b(i)) An (i) S(i) (i, j) A ( ') ndp(i ) / *A ) Iexp2 (Ab(i))N Nc (i)N (i, -1 A, (i) Nc (i, j) Nc, (i) + Nc1 (i)N (i,j) ndp(I) A,(i) 2 exp(Ab(i)) J,=1 [N (ANN. (i,j)] [3-241] [3-242] [3-243] [3-244] [3-245] [3-246] [3-247] .( ). N (i, j) N exp(A )) - I A, (i) A n (i) An(i A exp(Ab ) -( A, (i)\ A, (i) r ~ 2dp(')N (i, j) J An (i) Sexp(Ab(i)) An, (i) d2E HAbN (i,)0 b(i (i= HO Ab(i)(N,(i) dp (z) N. (i, j) 4 A AJ N i) HNcAb ( ,) (iexp(Ab(i)) A, (i) For cases with independent Ab's, the following would apply ndp(() J-LJL N N(i) N(i)N (i', j) A, (i) I ndp, N d (i)N (i, j) A,- (i) For cases with independent NAi's, the following would apply: p( Ni, j) N (i, j) N (i, j) exp(Ab(i)) - A, (i) A, (i) N, (i) = i) A [3-252] N Ni) = j N ndp(') [ j) (ij)A N(i'J)exp(Ab(i))-l1 -1 An (i) A, (i) For common Ab, each Ab(i) would be assigned the same value Ab, and the following equations would also apply: BE _d" aE =b OAb ,_, OAb(i) [3-253] OE 8N,, (i) [3-248] (2E) /N', (i> [3-249] 02E aNc, (i)8Ab(i) 1][N(i) exp(Ab(i)) An (i)0 [3-250] Ab(i) = In N (i)N (i, j) An (i) [3-251] nds HAbAb = Hbb(i,i) [3-254] Sn A A(i) Ab = In 2 [3-255] nds ndp. i Xc (i)Xo (i, j) 12 =1 =l An(i) For common Nc1, each NAl(i) would be assigned the same value Nc1, and the following equations would also apply: OE d" n E ^= [3-256] ON1 ,, 11 N,, (i) nds HN, = HN,, (i,i) [3-257] 1 1 nds ndp(i) (, j NA (i,) N. (i, j) exp(( 1 I N(ij)A (i A (i) N 1 = [3-258] Nz1 ) N N(i, j) N. (i, j) exp(Ab(i))-1 S1 An (i) A, (i) For common Ab and Nc1, all of the above equations would apply as well as the following: nds HAbN = HAbN (,) [3-259] 1=1 Analysis of Variance (ANOVA) for Commonality of Parameters (F-test) In this study, several data sets are analyzed in multiple ways, both independently and with various parameters held in common for a selection of data sets. To test whether or not parameters should be considered common, an F-test is employed similar to that outlined in Ratkowsky (1983). For the purposes of this study, the F-test was based on a normalized residual sum of squares that included all data (N,, N,, and Y). This way the fit of different models (with varying common parameters) to the complete data set could be compared. Normalization was employed since N,, Nc, and Y are often of different orders of magnitude. The general form for the normalized residual sum of squares, RSS, using the conventions presented in the previous section is: ndp(l) ndp(l) . N,, (i, j) N, (i, j) N, (i, j) N, (i, j) ]=1 ]=1 2 + J-1 dSS A,2 (i) N2 (i)exp (Ab(i)) RSS = [3-260] S NC (i) exp (Ab(i)) [Y(i,j)- Y(i,j)1 + 1 A2 (i) where N,, N9, and Y are model-calculated values (as functions of applied N only): Aj(i) N (i, j) =Ai) [3-261] 1 + exp(b (i)- c,, (i)N(i, j)) S(N, j) (i) exp(Ab(i))(1 + exp(b, (i) Ab(i) c, (i)N(i, j))) [3262] N(ij) [3-262] 1 + exp(b (i)- c,, (i)N(i, j)) Y(i,j) =i) [3-263] Nc1 (i) exp(Ab(i))(1 + exp(b, (i) Ab(i) c (i)N(i, j))) Note that the value for the Nc here is merely the ratio of N, to Y The general form of the F-test is presented below. Let A denote a test case in which no common parameters are assumed between data sets; i.e., all parameters are individual. Let B denote a test in which one or more of the parameters are common. LetpA and pB denote the number of parameters estimated in each test case A and B, respectively. Let dfA and dfB denote the degrees of freedom in each test case A and B, respectively. Let n denote the total number of data points in the analysis (from all data sets included in the analysis, counting N,, Nc, and Y points separately). Let RSSA and RSSB denote the residual sum of squares in each test case A and B, respectively. Let F denote the calculated F-test statistic (or variance ratio) and Fa the critical value for F at significance level a. With these definitions, the following equations apply: dfA n PA [3-264] df, = n p [3-265] F = (RSS RSSA)/(df dfA [3-266] RSSA /dfA Once F is calculated, it is compared to Fo(dfB-dfA, dfA, a) (which is obtained from the appropriate F distribution tables). If the following holds true: F > F, [3-267] then the null hypothesis is rejected for a test of level a (Wackerly et al., 2002). The null hypothesis, in all cases, is that the parameters assumed to be common are so; the converse of this hypothesis is that all parameters are unique. Therefore, if Eq. [3-267] holds true, it cannot be assumed that the parameters in question are common at the selected test level a. CHAPTER 4 RESULTS AND DISCUSSION In this chapter, the strategy and results of analysis are discussed in detail, including evidence of commonality of parameters. Numerical results are listed in the tables at the end of the chapter, followed by figures used to exemplify some results. Broad conclusions comparing the results of this study to other studies with respect to the Extended Logistic Model are reserved, for the most part, for the next chapter. Analysis Strategy Admittedly, this analysis was in some ways daunting because of the large amount of data and the complexity of the experiment. Indeed, altogether there were some 28 data sets with a total of 140 individual data points (or 420 if the three data types of N uptake, N concentration, and dry matter yield are counted separately) spread across multiple years. These data were characterized based on combinations of three different factors: crop type (bermudagrass or tall fescue), irrigation treatment (irrigated or nonirrigated), and fertilizer treatment (commercial fertilizer or broiler litter application). Clearly, the potential number of combinations of analyses was massive. In the first round of analyses, each data set was analyzed individually to obtain its own optimized set of parameters and standalone statistics. In the second round of analyses, data were isolated based on crop type and fertilizer treatment. Irrigation treatment was considered both separately and together. Analyses were not conducted within single years but rather across all years. This was done partially because data from a single year could be extremely limited, especially in the case of broiler litter where there were only four application rates (including zero). This was also done to simplify analyses somewhat, and because the interest lies primarily in commonality of parameters that extends beyond a single season. In this second round of analyses, a group of data was first analyzed with all parameters independent (the most general model) in order to establish a baseline residual sum of squares for use in the F-test. After this analysis, termed Mode (0), there were 15 additional modes that could potentially be considered, split into four levels. Mode descriptions can be found in Table 4-1. In the first level (1), one of the parameters was assumed invariant and the others allowed to vary between data sets. Parameters whose invariance could not be rejected in this first level were then grouped together into the varying combinations of levels (2)-(4) and analyzed further to determine whether two or more parameters could be considered simultaneously invariant between data sets. Note this round also included special cases for commercial fertilizer where data of a given crop type and a given N application rate were averaged over the years into two averaged data sets of nonirrigated and irrigated data. This was possible only for commercial fertilizer because while N application rates for commercial fertilizer were the same scheme each year, N application rates for broiler litter varied from year to year because broiler litter was applied based on schemes of total tonnage, not N rate. The results for the second round were then perused to identify possible parameters that could be common on even broader scales (for example, across both crop types or across both types of fertilizer). The groupings chosen for the third round were based on these results, but often additional parameters were tested for commonality even if commonality was not suggested by the second round, since such a test was very convenient anyway. Overview of Results Round One The results of first round analyses can be seen in Table 4-2 through Table 4-9. Note that covariance is not included in these tables. This is partially for brevity. In the vast majority of cases, covariance between parameters was very small, often very close to zero. However, in cases where standard errors were relatively high (note in a few cases the standard error was more than 50%!), a significant covariance between parameters was observed. The effect the standard errors have on the resulting model equations is discussed and illustrated in figures for all five parameters in the "Sensitivity Analysis" section of the Appendix. Examples of results from each data category can be seen in Figure 4-1 through Figure 4-20. As noted in these figures, they include the best fit example and the poorest fit example from each group of data. The averaged data values for irrigated and nonirrigated data with commercial fertilizer are also among these figures. The figures demonstrate that, even with the "worst" fits, the model describes the data consistently well. It is also difficult to deny that on an individual data set scale, the Extended Logistic Model describes the data extremely well if the nonlinear correlation coefficient, R, is used as the chief measure of goodness of fit. In only five out of 34 cases did RN, (the correlation coefficient for the N concentration portion of the model) fall below 0.90. Granted, two of these R values were exceptionally low (0.57 and 0.68, from the tall fescue with broiler litter data of 1982-3). The RN, value (the correlation coefficient for the N uptake portion of the model) never fell below 0.97 and in the vast majority of cases was 0.99 or greater. Only once was the Ry value (the correlation coefficient for the dry matter yield model) less than 0.97, and even then it was better than 0.95. What is not surprising here is that the N uptake model had the highest R values. This can be attributed, at least in part, to the fact that three of the five parameters, An, bn, and cn, are optimized solely against the N uptake data, and so in this sense the regression scheme gives extra weight to properly fitting N uptake at the potential expense of fit to the other two data types. What is perhaps more noteworthy is the extremely good fit to dry matter yield data, even though these data are not directly involved in the regression scheme. This evidence further solidifies the concept of the strong interconnectedness of N uptake, N concentration, and yield in response to applied N. For a preliminary search for patterns and potential commonality among parameters, each parameter was arithmetically averaged across various groups of data, as shown in Table 4-10. The simple standard deviation was also taken for the parameters in the groups, as shown in the corresponding Table 4-11. Note this should not be confused with standard errors calculated in regression analysis. Following is a discussion of each of the parameters. While the linear parameter An was not the focus of this study (because it is known to vary in response to several different environmental inputs), some patterns were observed in the averaging procedure described above. This was a rather simplistic and qualitative approach, however, since no statistical tests of significant difference were performed for An. More focus is given on An in this section than the other four parameters since it was not considered in the round two and round three analyses and discussions. First, it is rather clear there is a difference in response due to crop type: the average An value for all bermudagrass was 404 kg ha-1, while for all tall fescue it was 337 kg hal-, about 17% lower than for bermudagrass, although both had high standard deviations. This difference is more apparent if only the commercial fertilizer treatments are considered (note that broiler litter treatments had far fewer treatment levels, four versus seven, and much higher standard errors in An overall than commercial fertilizer treatments). Bermudagrass treated with commercial fertilizer had a mean An of 490 kg ha-1, while tall fescue with commercial fertilizer had a mean An of 342 kg ha-1, 30% lower than for bermudagrass; however, there was a very large standard deviation of 36% for tall fescue. There is also evidence of differences in An due to fertilizer source. The average An for all commercial fertilizer treatments was 398 kg ha-1, and that for all broiler litter treatments was 324 kg hal-, about 19% lower than that for commercial fertilizer, although both had high standard deviations as well. This apparent difference is even greater if only bermudagrass is considered (490 kg ha-1 for commercial fertilizer and 318 kg ha-1 for broiler litter), but it is less apparent if only tall fescue is considered (342 kg ha-1 for commercial fertilizer and 331 kg ha-1 for broiler litter, well within the standard deviations of each other). Evidence of differences in An due to water availability is also present. The average An for all irrigated treatments was 406 kg ha-1, and that for all nonirrigated treatments was 345 kg ha-l, about 15% lower than that for irrigated treatments, but these are within standard deviations of each other. If only commercial fertilizer treatments are considered, differences are more apparent: 528 kg ha-1 for irrigated bermudagrass versus 452 kg ha-1 for nonirrigated bermudagrass, and 433 kg ha-1 for irrigated tall fescue versus 251 kg ha-1 for nonirrigated tall fescue, although the latter did have a very high standard deviation of 45%. Overall, there appear to be possible differences in bn due to both irrigation and crop type. Effects on bn due to fertilizer source are less apparent. Although there is some overlap when considering standard deviations, from the averages alone for both bermudagrass and tall fescue, bn tended to be more of a function of fertilizer source in an irrigated situation than in a nonirrigated situation, where the values of bn were about the same within each crop. Taking standard deviations into account, bn was much more variable in tall fescue than in bermudagrass. The most marked difference in c, was due to fertilizer source, with a value of 0.00836 ha kg-1 for commercial fertilizer and 0.00625 ha kg-1 for broiler litter, although both exhibited standard deviations of 32%. In many cases, cn was rather highly variable, especially with tall fescue, in which very high standard deviations were often observed for this parameter. The parameter Ab was most highly variable in broiler litter groups, and especially so when tall fescue was the crop. With such high standard deviations, there was no obvious factor that influenced Ab most strongly. Irrigation treatment, fertilizer source, and crop type all could have played potential roles in its value. There is strong evidence that Nc1 is likely a function of crop type for the most part, with an average value of 13.2 g kg-1 for all bermudagrass and 18.2 g kg-1 for all tall fescue, each with very low standard deviations (6% and 9%, respectively) compared to other parameters. Most differences due to fertilizer source were easily within standard deviations of each other, although there was some evidence of small differences due to irrigation. Round Two The results of the second round of analyses can be found in Table 4-12 through Table 4-25. Each of these tables includes the value of parameters assumed invariant (according to mode) and the resulting F-values from comparison of the variances of each mode to the most general one, mode (0). Critical F-values (obtained from an F-value calculator similar to an F distribution textbook table) at the a = 0.10 level of significance (or 90% confidence) are listed in the tables as well, each being a function of the number of degrees of freedom in the most general model, and the difference in degrees of freedom of the general model from the mode being tested. Recall from Chapter 3 that if the calculated value ofF falls below the critical value ofF, then the null hypothesis cannot be rejected at the stated confidence level. Thus in this analysis, if the calculated F- value fell below the critical F, then the null hypothesis (that the selected invariant parameters are truly common) could not be rejected with 90% confidence, and so commonality of the stated parameters could be assumed. An F-value above the critical F- value meant one could reject the null hypothesis with 90% confidence and so commonality could not be assumed. The 90% level was chosen in this study because it is considered standard and is more rigorous than higher levels of 95% or 99%. This may seem counter-intuitive at first thought, but in essence it is easier to reject a null hypothesis with 90% confidence than with 95% or 99% confidence, thus, it is harder to statistically support commonality (the null hypothesis) at 90% versus higher levels. In other words, if the null hypothesis is supported (i.e., cannot be rejected) at 90%, it will also be supported at 95% and 99% confidence levels. More discussion on the F-test, including anomalous observations, is presented in the next major section of this chapter. The tables for round two all include the four possible versions of mode (1) (holding each parameter common one at a time). If a parameter failed this initial test for commonality, it was not considered in further combinations with other parameters for the data set grouping in question. An exception to this is Table 4-12, in which additional combinations were included to illustrate the point that even if a parameter fails the initial test (as cn did in this case), it may not fail when combined with other parameters (note cn paired with either Ab or Ncl passed the commonality test). This is likely because one of the parameters that passes the commonality test alone can in some circumstances "compensate" for the one that fails the test so that both appear common when paired together. This is the primary reason why if a parameter failed a commonality test on its own, it was not considered further. To clarify, the concurrent commonality pairings in Table 4-12 that include c, are not considered valid and are not considered when discussing commonality pairs from this point forward. In a few cases where a large number of pairings were tested, some pairs that failed the commonality test were not included in the table for brevity. Out of the 14 data groupings analyzed in round two, common ban's within a group occurred three times; common c,'s occurred six times; common Ab's occurred eight times; and common Nc's occurred 12 times. Concurrent commonality occurred far less often, however. There was only one instance in which Ab and Nc~ were concurrently common, and only once were cn, Ab, and Nci concurrently common. The parameters bn and Ab were concurrently common twice, as were the pairs bn, Nc1 and cn, Ab. The pair bn, Nc~ was concurrently common five times. Round Three Results for round three can be found in Table 4-26 throughTable 4-34. The groupings chosen for these analyses were based in large part on the potential broader commonalities suggested by examining results from round two. In the nine analyses of round three, common ban's were observed twice, as were common can's. Common Ab's were observed three times, and common NA's were observed five times. Concurrent common pairs bn, Ab; cn, Ab; and cn, Nc~ were observed once each. The concurrent common pair bn, Nc~ was observed twice. Discussion of Commonality of Parameters Validity of the F-test The rare occurrence of negative F-values, a theoretical impossibility, in regression results for some groupings of data reveals a statistical quandary. This anomaly can be explained, but it brings into question the validity of the F-test as it was employed in this study to test for commonality of parameters. The F-test, when used to test general models against more simplified ones, assumes that in each model all parameters are perfectly optimized, generating the lowest residual sum of squares possible using that model. This was the case in analyses performed by Ratkowsky (1983), which served as the basis for the approach used in this study. Even with perfect, least squares optimization of parameters, Ratkowsky (1983) points out that tests for invariance of parameters in nonlinear models are only approximate due to |

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PAGE 1 EXTENDED LOGISTIC MODEL OF CROP RESPONSE TO APPLIED NUTRIENTS By KELLY HANS BROCK A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2004 PAGE 2 Copyright 2004 by Kelly Hans Brock PAGE 3 This work is dedicated to my parents, Gene and Joann Brock. PAGE 4 iv ACKNOWLEDGMENTS I would first like to thank Dr. A llen Overman, without whose support, encouragement, and guidance I quite likely w ould have not entered into an advanced degree program. His support and patience have been invaluable during my graduate career. His insight and wisdom into the sc ientific and engineering communities have greatly broadened my view of science and the world. I would also like to thank the professors who served on my supervisory committee in addition to Dr. Overman: Dr. Roger Nordstedt, Dr. Frank Martin, Dr. Raymond Gallaher, and Dr. Paul Chadik. Special thanks go to Dr. Richard V. Scho ltz, III, whose constant help, insight, and guidance have allowed me to survive gradua te school. His professionalism and selfless devotion to excellence in engineer ing and academia are unparalleled. Finally, I would like to thank all of my fr iends and family who have stuck with me and encouraged me throughout my academic car eer. I especially thank my parents, Gene and Joann Brock, whose devotion to hard work has been an inspiration, and who have always given me the freedom to explore my own paths. PAGE 5 v TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iv LIST OF TABLES............................................................................................................vii LIST OF FIGURES.............................................................................................................x ABSTRACT.....................................................................................................................xi ii CHAPTER 1 INTRODUCTION........................................................................................................1 2 LITERATURE REVIEW.............................................................................................5 The Extended Logistic Model......................................................................................5 Alternative Models and Approaches..........................................................................12 Arkansas Bermudagrass and Tall Fescue Study.........................................................14 3 MATERIALS AND METHODS...............................................................................21 Solutions for a Single Data Set: First Method............................................................21 Primary Parameters.............................................................................................21 Secondary Parameters.........................................................................................22 Linear Model.......................................................................................................22 Nonlinear Model..................................................................................................23 Corresponding Model for Yield..........................................................................23 Linear Regression for Linear Model Parameters.................................................23 Nonlinear Regression for Nonlinear Model Parameters.....................................24 Initial Estimates of Nonlinear Parameters (Linearization)..................................28 Statistical Analyses for a Single Data Set...........................................................30 Solutions for a Single Data Set: Second Method........................................................33 Solutions for Multiple Data Sets and Commonality of Parameters............................35 Nonlinear Case 1: Common bn............................................................................36 Nonlinear Case 2: Common cn............................................................................43 Nonlinear Case 3: Common bn and cn.................................................................47 Nonlinear Case 4: Common An, bn, and cn..........................................................52 Linear Case A: Common b ................................................................................55 Linear Case B: Common Ncl................................................................................61 PAGE 6 vi Linear Case C: Common b and Ncl...................................................................66 Generalized Equations for All Analyses.....................................................................68 Variable Definitions............................................................................................69 General Equations for Nonlinear Portion............................................................69 General Equations for Linear Portion..................................................................73 Analysis of Variance (ANOVA) for Co mmonality of Parameters (F-test)................75 4 RESULTS AND DISCUSSION.................................................................................78 Analysis Strategy........................................................................................................78 Overview of Results...................................................................................................80 Round One...........................................................................................................80 Round Two..........................................................................................................84 Round Three........................................................................................................86 Discussion of Commonality of Parameters................................................................86 Validity of the F-test............................................................................................86 Commonalities Suggested by the F-test..............................................................89 Beyond the F-test: Commonality Based on Goodness of Fit..............................91 5 SUMMARY AND CONCLUSIONS.......................................................................132 Summary of Research...............................................................................................132 Key Observations and Conclusions..........................................................................133 Future Work..............................................................................................................138 Closing Remarks.......................................................................................................139 APPENDIX: SUPPLEMENTAL DISCUSSION............................................................150 Matrix Inversion.......................................................................................................150 Sensitivity Analysis..................................................................................................153 LIST OF REFERENCES.................................................................................................161 BIOGRAPHICAL SKETCH...........................................................................................163 PAGE 7 vii LIST OF TABLES Table page 2-1 Application schedule for commercial fe rtilizer treatments (listed as N-P2O5-K2O, each in kg ha-1).........................................................................................................19 2-2 Broiler litter content analysis fo r 1982-1985 (in terms of kg nutrient Mg-1 litter)...20 2-3 Equivalent N application ra tes for broiler litter (kg ha-1).........................................20 4-1 Mode descriptions....................................................................................................93 4-2 Optimized parameters for bermudagrass with commercial fertilizer.......................93 4-3 Statistics for optimized parameters fo r bermudagrass with commercial fertilizer...94 4-4 Optimized parameters for bermudagrass with broiler litter.....................................94 4-5 Statistics for optimized parameters for bermudagrass with broiler litter.................94 4-6 Optimized parameters for tall fesc ue with commercial fertilizer.............................95 4-7 Statistics for optimized parameters fo r tall fescue with co mmercial fertilizer.........95 4-8 Optimized parameters for ta ll fescue with broiler litter...........................................95 4-9 Statistics for optimized parameters for tall fescue w ith broiler litter.......................96 4-10 Arithmetic average of parameters over select data groups.......................................96 4-11 Standard deviations of para meters over select data groups......................................97 4-12 Analysis of variance for parameters fo r irrigated bermudagrass with commercial fertilizer....................................................................................................................9 8 4-13 Analysis of variance for paramete rs for nonirrigated bermudagrass with commercial fertilizer................................................................................................99 4-14 Analysis of variance for parameters fo r bermudagrass with commercial fertilizer (both irrigated and nonirrigated)..............................................................................99 PAGE 8 viii 4-15 Analysis of variance for parameters for average of bermudagrass data with commercial fertilizer (split into tw o groups: nonirrigated and irrigated)...............100 4-16 Analysis of variance for parameters for irrigated bermudagrass w ith broiler litter.100 4-17 Analysis of variance for parameters for nonirrigated bermudagrass with broiler litter........................................................................................................................1 01 4-18 Analysis of variance for parameters for bermudagrass with broiler litter (both irrigated and nonirrigated)......................................................................................102 4-19 Analysis of variance for parameters fo r irrigated tall fescue with commercial fertilizer..................................................................................................................103 4-20 Analysis of variance for parameters for nonirrigated tall fescue with commercial fertilizer..................................................................................................................104 4-21 Analysis of variance for parameters for ta ll fescue with commercial fertilizer (both irrigated and nonirrigated)......................................................................................104 4-22 Analysis of variance for parameters for average of tall fescue data with commercial fertilizer (split into two groups : nonirrigated and irrigated)..................................105 4-23 Analysis of variance for parameters for irrigated tall fescue wi th broiler litter.....106 4-24 Analysis of variance for parameters for noni rrigated tall fescue with broile r litter.106 4-25 Analysis of variance for parameters for ta ll fescue with broiler litter (both irrigated and nonirrigated)....................................................................................................107 4-26 Analysis of variance for select parameters for all bermudagrass...........................107 4-27 Analysis of variance for select para meters for all irrigated bermudagrass............107 4-28 Analysis of variance for select para meters for all nonirrigated bermudagrass......108 4-29 Analysis of variance for sele ct parameters for all fescue.......................................108 4-30 Analysis of variance for select parameters for irrigated fescue.............................109 4-31 Analysis of variance for select parameters for nonirrigated fescue.......................109 4-32 Analysis of variance for select pa rameters for all irrigated samples......................110 4-33 Analysis of variance for select paramete rs for all irrigated samples with commercial fertilizer..................................................................................................................110 4-34 Analysis of variance for select para meters for all nonirrig ated samples with commercial fertilizer..............................................................................................111 PAGE 9 ix 4-35 Summary of results for singl e parameter commonality F-test...............................111 5-1 Parameters for bermudagrass with comme rcial fertilizer as suming invariance of cn, b and Ncl across all data sets in this group..........................................................141 5-2 Statistics for parameters for bermuda grass with commercial fertilizer assuming invariance of cn, b and Ncl across all data sets in this group...............................141 5-3 Parameters for tall fescue with commer cial fertilizer assuming invariance of cn, b and Ncl across all data sets in this group.................................................................142 5-4 Statistics for parameters for tall fescue with commercial fertilizer assuming invariance of cn, b and Ncl across all data sets in this group...............................142 PAGE 10 x LIST OF FIGURES Figure page 4-1 Dry matter yield (Y), N uptake (Nu), and N concentration (Nc) versus N application rate (N) for 1982 nonirrigated bermudagrass with commercial fertilizer. This was the data set with the best overall mo del fit in this category of data.......................112 4-2 Phase plot for 1982 nonirrigated bermuda grass with commercial fertilizer..........113 4-3 Dry matter yield, N uptake, and N concen tration versus N application rate for 1984 irrigated bermudagrass with commercial fert ilizer. This was the data set with the poorest overall model fit in this category of data...................................................114 4-4 Phase plot for 1984 irrigated bermud agrass with commercial fertilizer................115 4-5 Dry matter yield, N uptake, and N concentr ation versus N application rate for the average of all irrigated and nonirrigated bermudagrass with comm ercial fertilizer..116 4-6 Phase plot for average of all irri gated and nonirrigated bermudagrass with commercial fertilizer..............................................................................................117 4-7 Dry matter yield, N uptake, and N concentr ation versus N application rate for 1983 irrigated bermudagrass with broiler litter. Th is was the data set with the best overall model fit in this category of data............................................................................118 4-8 Phase plot for 1983 irrigated be rmudagrass with br oiler litter...............................119 4-9 Dry matter yield, N uptake, and N concentr ation versus N application rate for 1982 irrigated bermudagrass with broiler litter. This was the data set with the poorest overall model fit in this category of data................................................................120 4-10 Phase plot for 1982 irrigated be rmudagrass with br oiler litter...............................121 4-11 Dry matter yield, N uptake, and N concen tration versus N app lication rate for 1981-2 irrigated tall fescue with commercial fertilizer. This was the data set with the best overall model fit in this category of data..................................................122 4-12 Phase plot for 1981-2 irrigated tall fescue with commercial fertilizer...................123 PAGE 11 xi 4-13 Dry matter yield, N uptake, and N concen tration versus N app lication rate for 1982-3 irrigated tall fescue with commercial fertilizer. This was the data set with the poorest overall model fit in this category of data.............................................124 4-14 Phase plot for 1982-3 irrigated tall fescue with commercial fertilizer...................125 4-15 Dry matter yield, N uptake, and N concentr ation versus N application rate for the average of all irrigated and nonirrigated tall fescue with commercial fertilizer....126 4-16 Phase plot for average of all irrigated and nonirrigated tall fescue with commercial fertilizer..................................................................................................................127 4-17 Dry matter yield, N uptake, and N concen tration versus N app lication rate for 1983-4 irrigated tall fescue with broiler litt er. This was the data set with the best overall model fit in this category of data................................................................128 4-18 Phase plot for 1983-4 irrigated ta ll fescue with broiler litter.................................129 4-19 Dry matter yield, N uptake, and N concen tration versus N app lication rate for 1982-3 irrigated tall fescue with broiler litt er. This was the data set with the poorest overall model fit in this category of data................................................................130 4-20 Phase plot for 1982-3 irrigated ta ll fescue with broiler litter.................................131 5-1 Dry matter yield, N uptake, and N concen tration versus N application rate for 1982 bermudagrass with commercial fertilizer using common cn, b and Ncl for all bermudagrass with commercial fertilizer (1982-1985)..........................................143 5-2 Dry matter yield, N uptake, and N concen tration versus N application rate for 1983 bermudagrass with commercial fertilizer using common cn, b and Ncl for all bermudagrass with commercial fertilizer (1982-1985)..........................................144 5-3 Dry matter yield, N uptake, and N concen tration versus N application rate for 1984 bermudagrass with commercial fertilizer using common cn, b and Ncl for all bermudagrass with commercial fertilizer (1982-1985)..........................................145 5-4 Dry matter yield, N uptake, and N concen tration versus N application rate for 1985 bermudagrass with commercial fertilizer using common cn, b and Ncl for all bermudagrass with commercial fertilizer (1982-1985)..........................................146 5-5 Dry matter yield, N uptake, and N concen tration versus N application rate for 19812 tall fescue with commerci al fertilizer using common cn, b and Ncl for all tall fescue with commercial fertilizer (1981-1984)......................................................147 5-6 Dry matter yield, N uptake, and N concen tration versus N application rate for 19823 tall fescue with commerci al fertilizer using common cn, b and Ncl for all tall fescue with commercial fertilizer (1981-1984)......................................................148 PAGE 12 xii 5-7 Dry matter yield, N uptake, and N concen tration versus N application rate for 19834 tall fescue with commerci al fertilizer using common cn, b and Ncl for all tall fescue with commercial fertilizer (1981-1984)......................................................149 A-1 Model dry matter yield, N uptake, and N concentration versus N application rate demonstrating the effect on the models of a change of 20% in the parameter An.156 A-2 Model dry matter yield, N uptake, and N concentration versus N application rate demonstrating the effect on the models of a change of 20% in the parameter bn.157 A-3 Model dry matter yield, N uptake, and N concentration versus N application rate demonstrating the effect on the models of a change of 20% in the parameter cn.158 A-4 Model dry matter yield, N uptake, and N concentration versus N application rate demonstrating the effect on the models of a change of 20% in the parameter b .159 A-5 Model dry matter yield, N uptake, and N concentration versus N application rate demonstrating the effect on the models of a change of 20% in the parameter Ncl.160 PAGE 13 xiii Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy EXTENDED LOGISTIC MODEL OF CROP RESPONSE TO APPLIED NUTRIENTS By Kelly Hans Brock May 2004 Chair: Allen R. Overman Major Department: Agricultura l and Biological Engineering Crop response in terms of crop yield and plant uptake of nutrients under varying conditions is a key concern from the perspe ctives of both agricu ltural production and water reuse. To adequately design nutrie nt management systems such as crop-based wastewater reuse systems, engineers need th e ability to estimate production of dry matter and levels of nutrient removal as a function of crop species, so il characteristics, climate, and additional inputs, including ir rigation and applied nutrients. In order for a method of estimation to be most useful to an engineer, it should be relatively simplistic and broadly applicable to many scenarios, and yet maintain an adequate level of accuracy. The search for such a method has led to the development of the Extended Logistic Model (ELM), which wa s used in this study to describe total seasonal dry matter production and nitrogen uptake in response to applied nitrogen. The ELM was applied to data obtained from a 1988 study in Faye tteville, Arkansas, involving response of bermudagrass [ Cynodon dactylon (L.) Pers.] and tall fescue [ Festuca arundinacea Schreb.] to varying levels of applied commercial (inorganic) PAGE 14 xiv fertilizer and broiler litter. A nonlinear/lin ear regression scheme was developed into a program called DAEDALUS to aid in c onducting multiple analyses on multiple parameters and data sets efficiently. Anal ysis of variance was used to test for commonality of paramete rs across data sets. The overall intent of this research was to lend further insight into the ELM and bring it one step closer to efficacy for practical use by engi neers, especially by searching for patterns in each of its five parameters a nd comparing results to previous findings. The research reaffirmed that the ELM is a cons istently good descriptor of crop response to applied N. Results also suggested that the parameter Ncl, the lower theoretical plant N concentration limit, is a strong function of crop typeÂ—as has been found in previous studiesÂ—and is affected little by water availability or fertilizer source. PAGE 15 1 CHAPTER 1 INTRODUCTION One may ask why an agricultural engineer whose academic focus has been primarily on soil and water engineering would choose a mathematical crop modeling topic for his dissertation. The historical tie between the tw o areas may help to answer such a question. Research that ultimately led to the Exte nded Logistic Model, the focus of this dissertation, began when Dr. Allen Overman be gan serving as an engineering advisor to the city of Tallahassee, FL, regarding its wastewater treatment and reuse planning. The ultimate result of this was the more than 800 hectare Tallahassee Southeast Farm, a large wastewater reuse operation that is a joint project between the city and a contracted farmer. In this system, which handles approximately 64 million liters per day, treated wastewater is applied to crops (including forage crops and co rn) as a method of effective wastewater reuse. Early in its development que stions were asked regarding the best crops to use, and with each of these crops, how much yield could be expected at a given nutrient application rate, and wh at kind of nutrient uptake by the plant could be expected. These questions initiated the journey that has led to a series of crop models developed by Overman and his associates over the past 30 years, including the Extended Logistic Model. It should be noted here that this di ssertation uses the term Â“nutrient uptakeÂ” to emphasize nutrient removal from the environm ent and incorporation into the harvested portion of the plant. A more commonly accepte d term for this is plant nutrient content (Soil Science Society of America, 1997). PAGE 16 2 Today, especially in Florida, there is an ongoing balancing effort between meeting the needs for an ever growing population with an ever growing standard of living and effective preservation of the environment, especially FloridaÂ’s precious aquifer-based water supply. As more concern arises about the impact of the various chemicals our society uses and wastes it produces, including those from agricultural operations, more environmental laws are put into place in an effort to preserve th e integrity of natural resources and ecosystems. TMDLs (total ma ximum daily loads), CNMPs (comprehensive nutrient management plans), and BMPs (best management practices) are a few of the key acronyms that reflect an effort to be more responsible in agri cultural and industrial operations with regard to the environment. The research behind the Extended Logistic Mo del has been an attempt to search for patterns that may lend in sight into how a plant responds to its environment, especially nutrient uptake and dry matter accumulation in response to the plantÂ’s interface with soil and water (and the nutrients contained therei n). More knowledge gained with respect to this plant-soil-water system can lead to improved management practices, optimizing the balance between desired ag ricultural goals (whether maximum dry matter yield, maximum plant quality, or some other factor ) and environmental goals (whether minimal impact, maximum nutrient uptake, or some other factor). The primary objectives of this resear ch were 1) to develop a modified nonlinear/linear regression scheme for use in applying the Extended Logistic Model to multiple data sets 2) incorporate this scheme into a computer program for efficient analysis 3) analyze data from a 1988 forage study from Fayetteville, Arkansas 4) determine whether previous basic patterns and internal consistency for the Extended PAGE 17 3 Logistic Model hold for this study and 5) s earch for general patterns and commonality of parameters between the data sets in this study. In Chapter 2 of this dissertation, a qualit ative discussion over the Extended Logistic Model, including its origins, strengths, and criticisms, is given. The forage study upon which the analyses in this dissertation are based is also discussed there. Chapter 3 provides a very extensive review of the equa tions of the Extended Logistic Model and the nonlinear/linear regression methods employe d to solve for its five fundamental parameters. The reader may wish to focus on th e most critical points in this chapter, such as the discussion regarding solu tions for a single data set at the beginning of the chapter, and the general equations and F-test presented at the end of the chap ter. The four special cases regarding simultaneous analysis of multiple data sets, discussed in the middle of the chapter, are presented in rather great detail but are really si mple extensions of the case for a single data set, and are all represented th rough the general equations presented near the end of the chapter. Chapter 4 discusses the re sults of the analyses and includes extensive tables, as well as figures illustrating results of the analyses for individual data sets. Table 4-2 through Table 4-9, as well as Table 4-12 through Table 4-34, are presented to cover in great detail the results of each analysis. The most important and useful tables in Chapter 4, however, are Table 4-10, Table 4-11, and Table 4-35, which provide useful summaries of results and help provide insi ght into patterns. Chapter 5 draws on the results from Chapter 4 and compares them to previous studies. This chapter provides a good overall view of the results and key conc lusions found in the research behind this dissertation. An illustration of the effect of the assump tion of commonality of some parameters is given through a series of seven figures in this chapter. The Appendix PAGE 18 4 should not be overlooked, especially the se ction on Â“Sensitivity Analysis,Â” which provides useful illustrations for how each of the five parameters affect the behavior of the equations in the Extended Logistic Model. PAGE 19 5 CHAPTER 2 LITERATURE REVIEW This chapter briefly reviews the histor y of the Extended Logistic Model, the reasoning behind it, and advantag es and disadvantages of its use. Alternative models and approaches are also discussed, and the fora ge grass experiment upon which the analyses in this study were based is reviewed. Discussion of the Exte nded Logistic Model in this chapter is largely qualitative, leaving mo re technical discussion to Chapter 3. The Extended Logistic Model Development of the Extended Logistic Model was first sparked by an interest in the Mitscherlich equation, deve loped in the early 20th century to describe dry matter yield in response to applied nitrogen (N). Overma n proposed the logistic equation as an alternative model to the Mitscherlich equati on for describing dry matter yield (Overman and Scholtz, 2002). The logistic equation is a function with three parameters, one linear ( A ) and two nonlinear ( b and c ). The application of the logistic equation as a model is not new. Its characteristic S-shape curve gives it broad a pplicability to a variet y of processes that exhibit such sigmoidal behavior, including t hose observed in the fi elds of agriculture, biology, economics, and engi neering (Ratkowsky, 1983). Overman has shown its applicability to human populat ion dynamics over time, as first applied by Pearl et al. (1940) to the total population of the U.S. The logistic e quation, in fact, is commonly referred to as the Verhulst-Pearl model. PAGE 20 6 Overman chose the logistic model over Mi tscherlich in part because of this sigmoidal behavior (Overman and Scholtz, 2002 ). However, the recognition of sigmoidal growth in different types of vegetation is not new either. Ratkows ky (1983) points out the broad range of vegetative grow th that exhibits sigmoidal behavior over time, including pasture regrowth yield, onion ( Allium cepa Cepa Group) yield, and area of cucumber ( Cucumis sativus L.) cotyledons (Mills and Jones, 1996). He also notes that the logistic equation is not the only model that can de scribe sigmoidal behavior; the Gompertz, Richards, Morgan-Mercer-Flodin, and Weibu ll distribution-based models all have sigmoidal characteristics. The sigmoidal pattern in the response of dry matter yield to levels of applied N was noted early in the 20th century by Russell (1937) when examining barley ( Hordeum vulgare Mills and Jones, 1996) data from a previous study. Overman extended the logistic concept to include a description of N uptake by the plant, observing that it also followed sigmoida l behavior and so coul d also be described by a logistic equation. Earlier versions of the model assu med this equation had three different parameters ( An, bn, and cn) compared to the logistic equation for dry matter yield (with parameters A b and c ). Analyses of data for several different crops demonstrated that the c and cn parameters were not significantl y different (Overman et al., 1994; Overman and Scholtz, 2002). With the new a ssumption that there wa s really only one Â“ c Â” parameter (termed cn), mathematical analysis of this tie between yield and N uptake led to a new perspective for the model, now te rmed the Extended Logistic Model, based on just two postulates (Overman and Sholtz, 2002). The first postulate states that plant N uptake follows a logistic response to applied N, as described by N c b A Nn n n u exp 1 [2-1] PAGE 21 7 where Nu is plant N uptake (or content), N is applied N, and the remaining three factors are parameters of the model discussed later. The second postulate states that dry matter yield is a hyperbolic function of N uptake, as described by u n u cl nN b A N b N A Y 1 exp 1 exp [2-2] where Y is dry matter yield, Nu is plant N uptake, and the other three factors are parameters discussed later. These two rela tionships mathematically explain why dry matter yield response to applied N can be expr essed as a logistic equation as well. With these postulates, the Extended Logistic Model is a five parameter model that can describe dry matter yield, N uptake, and N concentrati on in terms of applied N, as well as dry matter yield and N concentration in terms of N uptake (the latter two being so-called phase relationships, since the independent va riable applied N is only implicit in these cases). The five parameters are defined in Chapter 3; for the discussion here it is sufficient to simply list them: An, bn, cn, b and Ncl. There are several mathematical, logical, and practical advantages to using the Extended Logistic Model. Overman and Scholtz (2002) point out it is relatively simple to use (once parameters are known) and is a we ll behaved function. Â“Well behavedÂ” denotes it has limited (non-infinite) bounds and incr eases monotonically; it is continuously differentiable at all points. Such properties give it more physical meaning than simple linear, quadratic, or high order polynomial e quations, or even the Mitscherlich equation (which, like the logistic equa tion, contains an exponential te rm). For example, with the Extended Logistic Model, dry matter yield a nd N uptake approach zero with decreasing PAGE 22 8 applied N, never going negative, which w ould make no real world sense, and both approach an upper limit with increasing applie d N, which is more sensible than a model that approaches infinity. Anot her key indicator of the strengt h of the model is that the ratio of N uptake to dry matter produces a nother, well behaved function that sensibly describes N concentration versus applied N. Ma ny models fail such a ratio test, resulting in infinite singularities or other nonsensical anomalies. The resulting phase relationship between Y and Nu and between Nc and Nu also works well. The Extended Logistic Model has been shown to apply very well to a vari ety of different crops, illustrating its key strength as a broad-based model. There are also a few criticisms of the use of the Extended Logistic Model. At extremely high values of applied N, the upper limit of the N uptake and dry matter logistic equations likely do not make physical sense, since at such high levels, one would expect the N uptake or dry matter yield to be gin to decrease with increasing applied N. However, such Â“toxicÂ” levels of applied N are rarely, if ever, encountered in real world situations, and so with in the bounds of practical limits, th e upper limits of the logistic equations function adequatel y. The model has also been criticized as not being Â“mechanisticÂ” (Overman and Scholtz, 2002) and simply Â“an exercise in nonlinear regressionÂ” (Boote et al., 1996, p 711). The term Â“mechanisticÂ” implies an approach in which the model is developed beginning with very fundamental (e .g., molecular level or even quantum level) relationships, although M onteith (1996) gives it the more general definition that a mechanistic crop model is on e Â“in which all quantified processes have a sound physical or physiological basisÂ” (p 695) While there are compartmental models, as discussed in the next section, it can be argued that no truly mech anistic model of crop PAGE 23 9 response has been created or is even possi ble. There is no doubt that the Extended Logistic Model is not mechan istic at such a fundamental level. The approach in developing it has been more inductive than de ductive, i.e., a search for general patterns through analysis of real world data rather than an appro ach based on init ial hypotheses for fundamental relationships. It can be argued, however, th at this Â“top downÂ” approach is equally as valid, if not more so, than one that attempts to be more Â“mechanistic.Â” In fact, Monteith (1996) states that the philosophy that Â“descriptions of the natural worldÂ” (p 695) should be based on Â“facts derived fr om observationÂ” (p 695), not Â“speculation and dogmaÂ” (p 695), has been the foundation upon which modern science, in all its success, has been built. Analyses using the Extended Logistic Model are also often critic ized for a lack of Â“validation,Â” which Jones and Luyten (1998) define as Â“the process of comparing simulated results to real system data not prev iously used in any ca libration or parameter estimation processÂ” (p 25). This lack of Â“v alidationÂ” reflects a difference in philosophy behind the analyses involving the Extended L ogistic Model versus what has become the Â“normÂ” in the crop modeling community. The key difference in th is philosophy lies in differentiating validation from calibration, the latte r of which Jones and Luyten (1998) define in the following statement: Â“Calibrati on consists of making adjustments to model parameters to give the best fit between simulated results and results obtained from measurements on the real systemÂ” (pp 24-25). In essence, calibration based on some data is advocated, followed by validation of the model using other data This leads to an interesting exercise (albeit a somewhat ques tionable approach) in testing the Â“predictiveÂ” ability of a model, which is often the objective behind mode ling. This, however, is not PAGE 24 10 currently the primary objective or philos ophy behind research involving the Extended Logistic Model. Rather, the focus has been on whether the Extended Logistic Model is a good, consistent descriptor of crop response to applied N. Numerous applications to a variety of data sets have shown this to be true (Overman and Scholtz, 2002). The criticism, however, is that the model is so uni versal or generic that its parameters are not invariant; thus they must be adjusted with each new set of data, varying from site to site and often year to year (Boote et al., 1996). Does the fact th at parameters change mean a model is unsuitable? The Extended Logistic Model does not assume universal invariance of any of its five parameters. Rather than a devotion to Â“valida tionÂ” and Â“calibration,Â” research involving the Extended Logistic M odel has been a more wholistic search for patterns that may ultimately le ad to understanding how (and, more tentatively, why) its parameters vary from situation to situation, as well as cases in which they do not vary. Jones and Luyten (1998) describe two br oad, fundamental objectives in biological simulations. The first is the desire to bette r understand the behavior of a system and the various interactions and relati onships therein. The second is the desire to better predict system behavior so that a system may be bett er managed for a particular goal. The current research behind the Extended Logistic Model is still focused primarily on the first goal, although perhaps this has not always been clearl y stated in papers regarding it. This focus reflects the philosophy that jumping into a predictive game without a better understanding of behavior would be, at the least, prematur e and, more likely, irresponsible. The search for patterns ha s led to some interesting developments and evolution of the Extended Logistic Model, however, since th e early articles of over a decade ago that PAGE 25 11 Boote et al. (1996) referred to when criticizing it. One of the key areas of research has been the search for commonality of parameters for a given set of conditions, leading to evidence that certain parameters may be func tions of only one or two influences (e.g., genetics, soil type, water availability, etc.) Research by Scholtz (2002) on corn ( Zea mays L.) and ryegrass ( Lolum perenne L.) data sets concluded that the parameter b is a genetic based parameter, likely a function of plant species and possibly specific plant variety. Scholtz (2002) also tentat ively suggested that the product Ncl exp( b ), known as the secondary parameter Ncm in this dissertation (see Chapter 3), may also be a function of crop type, although some fluctuations from year to year were observed for perennials such as bermudagrass ( Cynodon dactylon (L.) Pers., Huneycutt et al., 1988), which he suggested may be due to annual fluctuations in region temperature or differences in the plantÂ’s initial quality at th e start of the season. Wilson ( 1995) also concluded from her dissertation research that there wa s evidence to suppor t the idea that b and Ncm were functions of crop type. If both b and Ncm are functions of crop type alone, it can be logically concluded that Ncl is also a function of crop type only. Wilson (1995) also noted that cn may be related to crop type, citi ng cases for ryegrass, bahiagrass ( Paspalum notatum Mills and Jones, 1996), and corn where each was grown at different locations, but for which cn was invariant with respect to loca tion and only varied with crop type. She noted, however, a study in which be rmudagrass did not exhibit the same cn when grown at different sites, and in other cases two different grasses grown at the same location had the same value for cn. The evidence so far, therefore, seems to suggest cn is likely not a function of crop type only. PAGE 26 12 Alternative Models and Approaches There are a myriad of m odels relating to crop beha vior. These range from incredibly simplistic linear equations to complex, computer driven, compartmental models with multiple subcomponents descri bing different aspects of the plant and environmental influences thereupon. Monteith (1996) quite humor ously points out the different philosophies from differe nt disciplines that have influenced the development of crop models. He states that physicists tend to take an extreme reductionist approach, employing the philosophy of OccamÂ’s razor, wh ile biologists tend to take the extremely overly complex approach, attempting to model a biological system in virtually infinite detail. Agronomists tend to adhere to the rela tively simplistic linear, quadratic, and sometimes higher order polynomials (3rd or 4th degree). A perfect example is the original analysis by Huneycutt et al. ( 1988) of the data from their forage experiment, which are the data subjected to analyses in this disse rtation. Dry matter yield response to applied N was classified as linear or quadratic, with some tendency to vary between the two from year to year, although some patterns were observed; e.g., broiler litter applied to nonirrigated bermudagrass tended to evoke a li near response in dry matter yield, while that of irrigated produced a quadratic re sponse (Huneycutt et al., 1988). Often, these models can be quite adequate within the range of data, and if one shares the philosophy that extrapolation of a mode l beyond data is never a good idea and that interpolation between data points should be done with ex treme caution, then there is no need for concern over the often il logical behavior these models exhi bit in such situ ations. In fact, Ratkowsky (1983) points out that if given the choice between several mo dels that fit data equally well, the one that is closest to a lin ear model is generally preferred. Linear and PAGE 27 13 polynomial models do have drawbacks, however, if attempts are being made to determine a physical or rational basis for crop res ponse, and they often lack broad-range applicability. Jones and Luyten (1998) outline the popular philosophy and techniques behind computer simulation in present day crop modeli ng. With this approach, it is common for a system (however it is defined) to be br oken into compartments whose behavior is described mathematically. The system is th en usually modeled by a set of first-order differential equations that are functions of time (temporal el ements). Distributed system modeling adds spatial elements to this type of analysis. Various numerical techniques (most often using computers) are employed to solve the resulting system of differential equations. Such analyses are subject to numeri cal errors. Jones and L uyten point out that the interactions of all subs ystems in biological systems are incompletely understood and incredibly complex, and so models that atte mpt to describe or predict these behaviors usually contain some level of empiricism, which is one reason calibration is usually required. Monteith (1996) reaffirms this id ea by noting that some of the submodels (or compartments) are based on Â“firm experime ntal evidenceÂ” while others are based on Â“arbitrary assumptionsÂ” (p 696). The Extended Logistic Model is in a way the median and perhaps even a potential intercessor between the two ex tremes of the simplistic mode ls that describe macroscale observations adequately but without much physical basis and the rather complex compartmental models that attempt to simu late, to varying degrees of success, the underlying processes resulting in such observations. It should be noted that many crop models or crop modeling software packages attempt to account for several environmental PAGE 28 14 effects, such as impact of insects, sunli ght, temperature, carbon dioxide concentration, etc. (Jones and Luyten, 1998). This goes far beyond the applied N effects that are the primary focus of the Extended Logistic Model in this study, and yet it is conceivable that some of the parameters in the Extended Logistic Model (most notably the linear parameter An) could potentially be tied to such m odels or their successors to account for such environmental effects. Monteith (1996) argues for a Â“balance betw een simplicity and complexityÂ” (p 696) in the vast field of crop modeling. He states that modelers need to aim for the right balan ce between (i) restricting algorithms to the minimum for a comprehensible model that allows current problems to be explored in the simplest possible wa y and (ii) making the structure of the model adaptable enough for more complex interactions to be introduced as new ideas and needs develop. (p 696) It can be argued the Extended Logistic Model was developed with this balanced approach in mind. It is relatively simple and broadly appl icable to a variety of situations, yet it is acknowledged that the physic al significance of all of its parameters is not well understood, and that as more information is as sessed, functions that describe or predict those parameters could be incorporated into the model with relative ease. Arkansas Bermudagrass and Tall Fescue Study The data analyzed in this study are de rived from a multiyear experiment with forage grasses conducted by Huneycutt et al. (1 988) at the Main E xperiment Station in Fayetteville, Arkansas. There were three factors considered in this study: crop type, fertilizer source (and application level), a nd irrigation treatment (irrigated versus nonirrigated). Its objective was to find yiel d responses to fertil izer treatments and irrigation treatments. Measurements taken included cumulative seasonal dry matter yield and crude protein concentrati on. For the analyses in this dissertation, crude protein was PAGE 29 15 converted to terms of N concen tration (or specific N), and to gether with yield data, this was used to calculate N uptake by the plant. All data, which were in English customary units in the original study, were conver ted to SI units for the analyses here. The two crops considered we re Tifton 44 bermudagrass ( Cynodon dactylon (L.) Pers.), a warm season grass with a harvest season from April through September; and Kenhy tall fescue ( Festuca arundinacea Shreb.), a cool season grass with a harvest season of September through August. Both are used as forage grasses for cattle. A third crop combination of tall fescue with clover was also included in the experiment, but it was not considered in the analyses here to maintain the focus on a single cropping system. Two types of fertilizer were used in th e study: commercial fertilizer and broiler litter. There were six treatment levels fo r commercial fertili zer, broken into subtreatments spread out over the season, as out lined in Table 2-1. The initial treatment each season included application of a 13-13-13 fertil izer. Additional N, P, and K were supplied as necessary to meet the experimentÂ’s levels by using ammonium nitrate, concentrated super-phosphate, and muriate of potash, respec tively. There were three treatment levels of broiler litter, applied in terms of total mass at 4.5, 9.0, and 13.5 Mg ha-1. A new batch of broiler litter was used each year, and the nutrient concentration varied as shown in Table 2-2. This resulted in different nutrient application rates each year, as outlined in Table 2-3. Unlike commercial fertilizer, all broi ler litter was applie d at the beginning of each season for each crop. It should be noted here that since the season ran from September through August of the following year its season is denoted by two years, e.g., 1981-1982. Thus in the tables, the year in parent heses denotes the beginning of the fescue PAGE 30 16 season, while that outside of th e parentheses denotes the end of the fescue season and the entire season for bermudagrass (which occurred within the same year). Irrigation treatments consisted of a nonirr igated treatment, in which the crops only received natural precipitation, and an irrigated treatment, in which crops received enough supplemental irrigation via perf orated-pipe sprinklers to br ing their weekly precipitation levels to a minimum of 3.8 cm for June through September of each season. The two grasses were established in separa te, contiguous plots on Captina silt loam soil with an initial (1980) pH of 6.2. Captina silt loam is of the taxonomic class fine-silty, siliceous, active, mesic Typic Fragiudults and used mainly in production of pasture and hay (United States Department of Agri culture [USDA], 2003). At the end of the experiment in 1985, the pH had shifted sligh tly, to 6.1 for the bermudagrass trials and 6.4 for the tall fescue trials. Each crop type wa s configured in a rando mized complete block design (RCBD) that included all fe rtilizer treatments (a zero pl ot, three levels of broiler litter, and six levels of comm ercial fertilizer) and three rep lications of each treatment. Two separate RCBDÂ’s were set up within ea ch crop, one irrigated and one nonirrigated. Plots were fertilized before in itial sprigging/planting for all trials. The analyses in this dissertation neglected the first year of data from the Arka nsas study with the idea that plots would have not been well established with in the first year. It should also be noted that while there were three replications for each treatment, only the average values of the three replications were reported by Huneycutt et al. (1988), and so it was these averaged values that were used in the data analysis in this dissertation. There are a few confounding points from th is experiment that could have had a potential impact on the results of the analyses included in this disse rtation. The Extended PAGE 31 17 Logistic Model as employed in this study is only used to describe N response and does not account for any response to P or K. Although a modified form of the model exists, termed the triple Extended Logistic Model, which can account for a ll three factors, it could not be applied here since N, P, and K effects were not experi mentally separated. In general, as N application increased, so did P and K application; thus increases in yield could have been partially due to increases in P and K application, not just N. While this was true for commercial fertilizer, the broi ler litter cases were even more confounded since nutrient content varied from year to year, and since there we re also micronutrients present that could have had positive or ne gative effects on plant response depending on their concentrations. The assumption in the an alyses here was that the response due to change in N application was far greater th an any response due to change in P or K application. This is generally tr ue, except in the cases where th ere is either a substantial P or K deficiency. It was assumed that the s upplemental P or K was sufficient to meet the needs of the plant at the given N applic ation rate, essentiall y making N the limiting nutrient, although no plant tissue analyses were available to clearly substantiate this claim. The authors made several interesting observations and conclusions from the Arkansas study. In general, yields declined over the years, especially in nonirrigated crops where stress due to dr ought in later years of the study likely impacted yields. Response to broiler litter was less pronounced than that to co mmercial fertilizer in both crops. Possible reasons given include incomp lete mineralization of the organic N in broiler litter (previous studies had suggested only 60% is mineralized after 300 days in the soil), upfront application of broiler litter versus spread out application of commercial PAGE 32 18 fertilizer over the season, and losses in avai lable N in the litter due to volatilization, leaching, or immobilization. The authors also note that the nutrient rati os in broiler litter differed significantly from those in the commercial fertilizer. PAGE 33 19 Table 2-1. Application schedule for commerci al fertilizer treatments (listed as N-P2O5K2O, each in kg ha-1) Time of Application Trial Treatment Beginning of season After 1st harvest After 2nd harvest After 3rd harvest Bermudagrass 112-112-135 56-112-67 56-0-67 --225-112-202 84-112-67 84-0-67 56-0-67 -337-112-270 112-112-101 84-0-84 84-0-84 56-0-0 449-146-337 140-146-112 112-0-90 112-0-67 84-0-67 562-180-404 168-180-135 140-0-101 140-0-101 112-0-67 674-213-472 225-213-135 168-0-135 168-0-135 112-0-67 Tall Fescue 112-112-135 56-112-67 56-0-67 --225-112-202 84-112-67 84-0-67 56-0-67 -337-112-270 112-112-101 84-0-101 84-0-67 56-0-0 449-146-337 140-146-112 140-0-90 84-0-67 84-0-67 562-180-404 168-180-135 168-0-135 112-0-67 112-0-67 674-213-472 225-213-135 225-0-135 112-0-101 112-0-101 Data adapted from Huneycutt et al. (1988) PAGE 34 20 Table 2-2. Broiler litter content analysis for 1982-1985 (in terms of kg nutrient Mg-1 litter) Year N P2O5 K2O (1981-)1982 45 37 31 (1982-)1983 37 29 42 (1983-)1984 33 23 30 1985 28 23 30 Data adapted from Huneycutt et al. (1988) Table 2-3. Equivalent N applicati on rates for broiler litter (kg ha-1) Year (1981-)1982 (1982-)1983 (1983-)1984 1985 Litter Rate (Mg/ha) 4.5 202 166 148 126 9.0 404 332 297 252 13.5 607 499 445 377 PAGE 35 21 CHAPTER 3 MATERIALS AND METHODS Analyses of data sets in this study incl ude linear regression, nonlinear regression, and several statistical analys es. This chapter begins with two methods for analyzing a single data set. This concept is then expanded to analyzing special Â“casesÂ” for multiple data sets in which some parameters are assumed common between data sets. Finally, a general set of equations is presented that can be used in all cases and for any number of data sets. Solutions for a Single Data Set: First Method Primary Parameters The five primary parameters of the l ogistic model can be considered as , , b c b An n n and clN These parameters do have a physical basis. nA is the maximum theoretical plant N uptake for the system. The intercept parameter for plant N uptake,nb relates to the initial condition of the soil; it can account for unm easured N that is initially present. The parameter nc is the N response coefficient, which accounts for some of the plantÂ’s response to applied N. The parameter b is the natural logarithm of the ratio of the upper theoretical N concentration limit, Ncm, to the lower theoretical plant N concentration limit, Ncl; it also relates nb to a corresponding secondary parameter in the yield model, termed b (presented later). PAGE 36 22 Secondary Parameters Useful secondary parameters include m nY K b A , ,and cmN At least three of these have significant physical meanings: Ais the maximum theoretical dry matter yield for the system; bis the intercept parameter for dry matter yield, related to initial soil conditions; and cmNis the maximum theoretical plant N concentration limit. Written in terms of the primary parameters, these are defined as b N A Acl n exp [3-1] b b bn [3-2] 1 exp b A Kn n [3-3] 1 exp 1 b N A Ycl n m [3-4] A A b N Nn cl cm exp [3-5] The logistic model can be broken into two parts: a linear portion and a nonlinear portion, as demonstrated by Overman and Scholtz (2002). Linear Model u m m n cN Y Y K N 1 [3-6] u n cl cl cN b A N N N 1 exp [3-7] Here cN is N concentration (or specific N, typically in units of g kg-1) and uN is N uptake (typically in units of kg ha-1). From this relationship and linear regression one can PAGE 37 23 determine the parameter clN as well as secondary parameters nK and mY Once nonlinear regression (presented later) is used to determinenA then b can also be determined from the above relationship. Nonlinear Model ) exp( 1 N c b A Nn n n u [3-8] Corresponding Model for Yield ) exp( 1 N c b A Yn [3-9] Linear Regression for Linear Model Parameters From least squares regression for intercept and slope, respectively: 2 2 2 u u c u u u c m n clN N n N N N N N Y K N [3-10] 2 21 1 exp u u c u c u m n clN N n N N N N n Y b A N [3-11] where n is the number of data point s in the data set Solving for mY and nK : c u c u u u mN N N N n N N n Y2 2 [3-12] c u c u c u u u c nN N N N n N N N N N K2 [3-13] Solving for b (including the parameter nA ): m n clY b A N1 1 exp [3-14] PAGE 38 24 cl m nN Y A b 1 exp [3-15] 1 exp cl m nN Y A b [3-16] 1 lncl m nN Y A b [3-17] Nonlinear Regression for Nonlinear Model Parameters This method is based on that outlined by Overman and Scholtz (2002). We want to minimize the error sum of squares, given by 2Âˆu uN N E [3-18] where uN is the true (data) value for N uptake and uN Âˆis the corresponding estimated (model) value for N uptake. Substituting the nonlinear model into this equation: 2) exp( 1 N c b A N En n n u [3-19] In order to optimize the three parameters, we must set each partial de rivative of the error sum of squares equal to zero: 0 nA E [3-20] 0 nb E [3-21] 0 nc E [3-22] Solving for the linear parameter nA from the first partial derivative: PAGE 39 25 0 exp 1 1 exp 1 2 N c b N c b A N A En n n n n u n [3-23] 0 exp 1 1 exp 1 exp 1 1 N c b N c b A N c b Nn n n n n n n u [3-24] N c b N N c b An n u n n nexp 1 exp 1 12 [3-25] 2exp 1 1 exp 1 N c b N c b N An n n n u n [3-26] Using the same method for the nonlinear parameters nb and nc however, only leads to implicit solutions: N c b N c b A N c b A N b En n n n n n n n u nexp exp 1 exp 1 2 02 [3-27] N c b N N c b A N c b A N c En n n n n n n n u nexp exp 1 exp 1 2 02 [3-28] 2exp 1 exp exp 1 2 0 N c b N c b N N c b A N A c En n n n n n n u n n [3-29] 3 2 2exp 1 exp 2 exp 1 exp 2 0 N c b N c b N A N c b N c b N N A c En n n n n n n n n u n n [3-30] PAGE 40 26 Since these solutions are implicit, the 2nd order Newton Raphson method is a suitable iterative method for finding values of nb and nc If reasonable initial guesses for nb and nc are made, nb and nc ', respectively, the partial derivatives with respect to nb and nc can be approximated for the Â“newÂ” estimates of nb and nc n nb b 'and n nc c respectively, as n c b n n n c b n c b n c c b b nc c b E b b E b E b En n n n n n n n n n ' 2 ' 2 2 ' ' [3-31] n c b n n c b n n c b n c c b b nc c E b b c E c E c En n n n n n n n n n ' 2 2 ' 2 ' ' [3-32] Since the value of both of these partials is zero at the minimum sum of squares, the following system of equations applies: n n n n n nc b n n c b n n n c b nb E c c b E b b E' ' 2 ' 2 2 [3-33] n n n n n nc b n n c b n n c b n nc E c c E b c b E' ' 2 2 ' 2 [3-34] Note the relationship n n n nc b n n c b n nc b E b c E' 2 ' 2 [3-35] has been employed here. Removing iterative subscripts for simplic ity and defining the following variables: n bb E Jn [3-36] n cc E Jn [3-37] PAGE 41 27 2 2n b bb E Hn n [3-38] 2 2 n c cc E Hn n [3-39] n n b c c bc b E H Hn n n n 2 [3-40] The system of equations can be simplified to the matrix form: n n n n n n n n n nc b n n c c c b c b b bJ J c b H H H H [3-41] Solving this system using CramerÂ’s Rule (Kolman and Shapiro, 1986): 2n n n n n n n n n n n n n n n n n n n n n n n n n nc b c c b b c b c c c b c c c b c b b b c c c c b b nH H H H J H J H H H H H J H J b [3-42] 2n n n n n n n n n n n n n n n n n n n n n n n n n nc b c c b b c b b b b c c c c b c b b b c c b b b b nH H H H J H J H H H H J H J H c [3-43] These values of nb and nc can then be used to update the previous Â“guessesÂ” for nband nciteratively until Eqs. [3-31] and [3-32] are equal to zero (or within a specified tolerance.) Solving for the second partial and cross derivatives: 3 2 2 2 2exp 1 exp 2 exp 1 exp 2N c b N c b A N c b N c b N A b b E Hn n n n n n n n n u n n n b bn n [3-44] PAGE 42 28 4 2 3 2 3 2 2exp 1 exp 3 exp 1 exp 2 exp 1 exp 2 exp 1 exp 2 N c b N c b N c b N c b A N c b N c b N N c b N c b N A Hn n n n n n n n n n n n n u n n n n u n b bn n [3-45] 3 2 2 2 2exp 1 exp 2 exp 1 exp 2 N c b N c b N A N c b N c b N N A c c E Hn n n n n n n n n u n n n c cn n [3-46] 4 2 2 3 2 2 3 2 2 2 2exp 1 exp 3 exp 1 exp 2 exp 1 exp 2 exp 1 exp 2 N c b N c b N N c b N c b N A N c b N c b N N N c b N c b N N A Hn n n n n n n n n n n n n u n n n n u n c cn n [3-47] 3 2 2 2exp 1 exp 2 exp 1 exp 2 N c b N c b A N c b N c b N A c c b E H Hn n n n n n n n n u n n n n b c c bn n n n [3-48] 4 2 3 2 3 2 2exp 1 exp 3 exp 1 exp 2 exp 1 exp 2 exp 1 exp 2 N c b N c b N N c b N c b N A N c b N c b N N N c b N c b N N A Hn n n n n n n n n n n n n u n n n n u n c bn n [3-49] Initial Estimates of Nonlinear Parameters (Linearization) Initial estimates of parame ters in the nonlinear model can be found by linearizing the model and performing linear regression. The following steps show how to linearize the model: ) exp( 1 N c b A Nn n n u [3-50] PAGE 43 29 ) exp( 1 N c b N An n u n [3-51] ) exp( 1 N c b N An n u n [3-52] N c b N An n u n 1 ln [3-53] n n u n ub N c N A N ln [3-54] To conduct this type of analys is, an initial guess of the nA parameter must be given and then the parameters nc and nb can be solved by simple linear regression. For the method employed in this analysis, 110% of the maximum value of uN for a given data set was used as a rough estimator for nA This is generally sufficient to generate reasonable initial estimates of nc and nb using the above linearization method. The solutions for nb and nc from linear regression, respectively, are: 2 2 2ln ln N N n N N A N N A N N N bu n u u n u n [3-55] 2 2ln ln N N n N A N N N A N N n cu n u u n u n [3-56] Iteration using the previously described nonlinear regression scheme can then be used to find optimized values for all three parameters. PAGE 44 30 Statistical Analyses fo r a Single Data Set Statistical measures of key interest in clude correlation coefficient, variance, covariance, and standard error of the parameters. The correlati on coefficient is defined as (Cornell and Berger, 1987): 2 1 2 2Âˆ 1 u u u uN N N N R [3-57] where uN is the actual N uptake data, uN Âˆis the estimator for N uptake based on the optimized logistic model, and uN is the average N uptake based on the data. Variance is defined as 2 1 2Âˆ 1 n i u uN N p n s [3-58] where n is the number of data points, and p is the number of parameters in the model (three, in this case, since only the nonlinear lo gistic model is being considered here). To find the covariance and standard error in the three parameters involved in nonlinear regression, three additional derivatives must first be found: 2 2n A AA E Hn n [3-59] n n A b b Ab A E H Hn n n n 2 [3-60] n n A c c Ac A E H Hn n n n 2 [3-61] Previously it was found from Eq. [3-23] that: PAGE 45 31 N c b N c b A N A En n n n n u nexp 1 1 exp 1 2 [3-62] Expanding and taking the second derivative: N c b N c b A N c b N A En n n n n n n u nexp 1 1 exp 1 2 exp 1 1 2 [3-63] 2exp 1 2 exp 1 1 2N c b A N c b N A En n n n n u n [3-64] 2 2 2exp 1 1 2N c b A E Hn n n A An n [3-65] Previously it was found from Eq. 3-27 that: 3 2 2exp 1 exp 2 exp 1 exp 2N c b N c b A N c b N c b N A b En n n n n n n n n u n n [3-66] Taking the derivative of this with respect to nA: 3 2 2exp 1 exp 4 exp 1 exp 2 N c b N c b A N c b N c b N b A E H Hn n n n n n n n n u n n A b b An n n n [3-67] Previously it was found from Eq. [3-30] that: 3 2 2exp 1 exp 2 exp 1 exp 2 N c b N c b N A N c b N c b N N A c En n n n n n n n n u n n [3-68] Taking the derivative of this with respect to nA : PAGE 46 32 3 2 2exp 1 exp 4 exp 1 exp 2 N c b N c b N A N c b N c b N N c A E H Hn n n n n n n n n u n n A c c An n n n [3-69] The resulting Hessian matrix for this model is (Overman and Scholtz, 2002): n n n n n n n n n n n n n n n n n nc c b c A c c b b b A b c A b A A AH H H H H H H H H H [3-70] To compute covariance and standard error, th e inverse of [3-70] must be obtained (see Appendix). The covariance for two differing parameters can then by found by (Overman and Scholtz, 2002): ] [ ) (1 2 abH s b a COV [3-71] where COV(a,b) is the covariance between parameter a and parameter b, and ] [1 abH is the corresponding element from the inverse of the Hessian matrix. For example, for the covariance between An and bn: ] [ ) (1 2 n nb A n nH s b A COV [3-72] The standard error is given by (Overman and Scholtz, 2002): 2 / 1 1 2) ( aaH s a SE [3-73] where SE ( a ) is the standard error of parameter a and Haa -1 is the corresponding element from the inverse of the Hessian matrix. Note for the linear portion of the model, th e correlation coefficient and variance can readily be calculated from the N concentration (rather than uptake) analogs to Eqs. [3-57] and [3-58], respectively. Because parameters in the linear portion are solved for directly in the method outlined above, standard erro r and covariance for the linear portion cannot PAGE 47 33 be assessed using this method (unless the co rresponding derivatives are taken). However, the actual method of regression used in this study, even for a single dataset, was actually based on the general equations presented near the end of this chapter, which are based on the method for multiple data sets presented la ter. These equations include the appropriate derivatives and cross derivatives to obtain covariance and standard error for both the nonlinear and linear portions of the model, so a complete set of sta tistical measures can be obtained. Solutions for a Single Data Set: Second Method Suppose the error sum of squares is redefi ned to include all three data measures (uN cN and Y ), as first proposed by Scholtz (20 02). The non-normalized version of this is: 2 2 2Âˆ Âˆ Âˆ Y Y N N N N Ec c u u [3-74] As a result of the new error sum of squares, only nonlinear regression is used to optimize all five primary parameters simultaneously rath er than the Â“splitÂ” of the model into linear and nonlinear sections as in the first method. Advantages of this a pproach include a more balanced influence of data and the ability to readily calculate sensible standard errors for all five primary parameters. Inserting the model into the new error sum of squares: 2 2 2) exp( 1 ) exp( 1 ) exp( 1 ) exp( 1 N c b A Y N c b A N c b A N N c b A N En n n n n c n n n u [3-75] Note here the model for cN Âˆis presented as a function if N only rather than Nu, as was used in the first method. The model for cN Âˆ in this form is simply the ratio of uN Âˆto Y Âˆ. PAGE 48 34 Normalization of this error sum of squares is sensible since it includes three different data types ( Nu, Nc, and Y ), which generally are of different magnitudes. The normalized version of the error su m of squares requires dividing the three elements of Eq. [3-75] through by An, cmN and A as shown: 2 2 2) exp( 1 ) exp( 1 ) exp( 1 ) exp( 1 A N c b A Y N N c b A N c b A N A N c b A N En cm n n n n c n n n n u [3-76] Simplifying further and rewriting in terms of primary parameters only: 2 2 2) exp( 1 1 ) exp( 1 ) exp( 1 ) exp( 1 1 N c b A Y N c b A N N c b A N N N c b A N En n n cm n n cm c n n n u [3-77] 2 2 2) exp( 1 1 exp ) exp( 1 ) exp( 1 exp ) exp( 1 1 N c b b A b YN N c b N c b b b N N N c b A N En n n cl n n n n cl c n n n u [3-78] Just as presented in the first method, to minimize the error sum of squares, the first derivatives with respect to all five basic parameters should be zero: 0 nA E [3-20] PAGE 49 35 0 nb E [3-21] 0 nc E [3-22] 0 b E [3-79] 0 clN E [3-80] This is the point where the second method beco mes rather complex. Compared to the first method, these first derivatives alone are qui te lengthy. Second de rivatives and cross derivatives becoming increasingly complex, and if commonality of parameters for multiple data sets is to be considered, this method becomes quite unwieldy. Thus, while the second method may be more ideal and unbi ased than the first method, it is less practical in application. For this reason the first method was us ed in this study. Solutions for Multiple Data Sets and Commonality of Parameters Of key interest in crop behavior analysis is the search for commonality of model parameters between different data sets. I ndividual parameters of highest concern are , b c bn n and clN nA is not considered in this st udy (except in the case of total commonality) because previous studies have shown it varies with water application and other environmental factors. Combinations of these parameters are also considered. For the cases presented below only two data se ts (denoted by Â“1Â” and Â“2Â” suffixes) are considered for simplicity. However, the poi nts presented can be easily expanded to analysis of several data sets simultaneously as was done in the pr ogram for this study. Indeed, the cases presented belo w can be grouped into a set of general equations (for any number of data sets). These more formal e quations better reflect the basis used for the PAGE 50 36 algorithms in the regression program, but the cases are presented as they are in this section for better clarity on how each is de rived. The corresponding general equations appear near the end of this chapter. Regres sion for the nonlinear porti on is presented first, followed by regression for the linear portion, sinc e the latter is partially dependent on the results of the former, as will be shown. Nonlinear Case 1: Common bn In general, the error sum of squares for the nonlinear portion for the two data sets would be: 2 22 Âˆ 2 1 Âˆ 1 u u u uN N N N E [3-81] For the case of common nb : 2 2) 2 2 exp( 1 2 2 ) 1 1 exp( 1 1 1 N c b A N N c b A N En n n u n n n u [3-82] Strictly speaking, it may be sensible to normalize this error sum of squares so that in the case that the scale of one data set is much greater than the other, it will not unduly bias the results. However, early attempts at using a normalized method in programming revealed computational instab ility. Therefore, the non-normalized approach was used, as is presented here. Since only one data type is involved, and data sets are usually within the same magnitude, the biasing effect of not normalizing was assumed to be minimal. Ratkowsky (1983) does not menti on normalizing when analyzing multiple sets of data in tests for commonality of parameters. Taking derivatives with respect to nA 1 andnA 2, setting equal to zero, and solving: 0 1 1 exp 1 1 1 1 exp 1 1 1 2 1 N c b N c b A N A En n n n n u n [3-83] PAGE 51 37 0 1 1 exp 1 1 1 1 exp 1 12 N c b A N c b Nn n n n n u [3-84] 1 1 exp 1 1 1 1 exp 1 1 12N c b N N c b An n u n n n [3-85] 21 1 exp 1 1 1 1 exp 1 1 1 N c b N c b N An n n n u n [3-86] Similarly, 22 2 exp 1 1 2 2 exp 1 2 2 N c b N c b N An n n n u n [3-87] Once again, only the nA parameters can be solved for directly. Other solutions are implicit only and so the nonlinear regression procedure illustrated previously must be applied to solve for the remaining parameters Taking the appropriate first, second, and cross derivatives: 2 2) 2 2 exp( 1 ) 2 2 exp( 2 ) 2 2 exp( 1 2 2 2 ) 1 1 exp( 1 ) 1 1 exp( 1 ) 1 1 exp( 1 1 1 2 N c b N c b A N c b A N N c b N c b A N c b A N b E Jn n n n n n n n u n n n n n n n n u n bn [3-88] 2 1) 1 1 exp( 1 ) 1 1 exp( 1 1 ) 1 1 exp( 1 1 1 2 1 N c b N c b N A N c b A N c E Jn n n n n n n n u n cn [3-89] PAGE 52 38 2 2) 2 2 exp( 1 ) 2 2 exp( 2 2 ) 1 2 exp( 1 2 2 2 2 N c b N c b N A N c b A N c E Jn n n n n n n n u n cn [3-90] 2 2 2 1 1) 1 1 exp( 1 1 2 1 N c b A E Hn n n A An n [3-91] 2 2 2 2 2) 2 2 exp( 1 1 2 2 N c b A E Hn n n A An n [3-92] 4 2 3 2 3 2 2 4 2 3 2 3 2 2 2 22 2 exp 1 2 2 exp 3 2 2 exp 1 2 2 exp 2 2 2 2 exp 1 2 2 exp 2 2 2 2 exp 1 2 2 exp 2 2 2 1 1 exp 1 1 1 exp 3 1 1 exp 1 1 1 exp 1 2 1 1 exp 1 1 1 exp 1 2 1 1 exp 1 1 1 exp 1 1 2 N c b N c b N c b N c b A N c b N c b N N c b N c b N A N c b N c b N c b N c b A N c b N c b N N c b N c b N A b E Hn n n n n n n n n n n n n u n n n n u n n n n n n n n n n n n n n u n n n n u n n b bn n [3-93] 4 2 2 3 2 2 3 2 2 2 2 2 2 1 11 1 exp 1 1 1 exp 1 3 1 1 exp 1 1 1 exp 1 1 2 1 1 exp 1 1 1 exp 1 1 2 1 1 exp 1 1 1 exp 1 1 1 2 1 N c b N c b N N c b N c b N A N c b N c b N N N c b N c b N N A c E Hn n n n n n n n n n n n n u n n n n u n n c cn n [3-94] PAGE 53 39 4 2 2 3 2 2 3 2 2 2 2 2 2 2 22 2 exp 1 2 2 exp 2 3 2 2 exp 1 2 2 exp 2 2 2 2 2 exp 1 2 2 exp 2 2 2 2 2 exp 1 2 2 exp 2 2 2 2 2 N c b N c b N N c b N c b N A N c b N c b N N N c b N c b N N A c E Hn n n n n n n n n n n n n u n n n n u n n c cn n [3-95] 0 1 2 2 12 1 2 2 2 1 n n A A n n A AA A E H A A E Hn n n n [3-96] 3 2 2 1 2 11 1 exp 1 1 1 exp 1 4 1 1 exp 1 1 1 exp 1 2 1 1 N c b N c b A N c b N c b N A b E H b A E Hn n n n n n n n n u n n A b n n b An n n n [3-97] 3 2 2 2 2 22 2 exp 1 2 2 exp 2 4 2 2 exp 1 2 2 exp 2 2 2 2 N c b N c b A N c b N c b N A b E H b A E Hn n n n n n n n n u n n A b n n b An n n n [3-98] 3 2 2 1 1 2 1 11 1 exp 1 1 1 exp 1 1 4 1 1 exp 1 1 1 exp 1 1 2 1 1 1 1 N c b N c b N A N c b N c b N N A c E H c A E Hn n n n n n n n n u n n A c n n c An n n n [3-99] 0 2 1 1 22 2 1 2 1 2 n n A c n n c AA c E H c A E Hn n n n [3-100] 0 1 2 2 12 1 2 2 2 1 n n A c n n c AA c E H c A E Hn n n n [3-101] PAGE 54 40 3 2 2 2 2 2 2 22 2 exp 1 2 2 exp 2 2 4 2 2 exp 1 2 2 exp 2 2 2 2 2 2 2 N c b N c b N A N c b N c b N N A c E H c A E Hn n n n n n n n n u n n A c n n c An n n n [3-102] 4 2 3 2 3 2 2 2 1 2 11 1 exp 1 1 1 exp 1 3 1 1 exp 1 1 1 exp 1 1 2 1 1 exp 1 1 1 exp 1 1 2 1 1 exp 1 1 1 exp 1 1 1 2 1 1 N c b N c b N N c b N c b N A N c b N c b N N N c b N c b N N A b c E H c b E Hn n n n n n n n n n n n n u n n n n u n n n b c n n c bn n n n [3-103] 4 2 3 2 3 2 2 2 2 2 22 2 exp 1 2 2 exp 2 3 2 2 exp 1 2 2 exp 2 2 2 2 2 exp 1 2 2 exp 2 2 2 2 2 exp 1 2 2 exp 2 2 2 2 2 2 N c b N c b N N c b N c b N A N c b N c b N N N c b N c b N N A b c E H c b E Hn n n n n n n n n n n n n u n n n n u n n n b c n n c bn n n n [3-104] 0 1 2 2 12 1 2 2 2 1 n n c c n n c cc c E H c c E Hn n n n [3-105] In order to solve for the nonlinear paramete rs, the Newton Raphson procedure outlined in the first method for solutions of a single da ta set can be employed here. Using similar notation as in that section, in this case the following system of equations applies: n n n n n n n n n n n nc c b n n c c b n n n c c b n n n c c b nb E c c b E c c b E b b E' 2 1 ' 2 1 2 2 1 2 2 1 2 22 2 1 1 [3-106] PAGE 55 41 n n n n n n n n n n n nc c b n n c c b n n n c c b n n c c b n nc E c c c E c c E b b c E' 2 1 ' 2 1 2 2 1 2 2 2 1 21 2 2 1 1 1 1 [3-107] n n n n n n n n n n n nc c b n n c c b n n c c b n n n c c b n nc E c c E c c c E b b c E' 2 1 ' 2 1 2 2 2 1 2 2 1 22 2 2 1 1 2 2 [3-108] Replacing these derivatives with the Â“ J Â” and Â“ H Â” notation for first and second partial derivatives (respectively) used previously, this system of equations in matrix form becomes: n n n n n n n n n n n n n n n n n n n n nc c b n n n c c c c b c c c c c b c c b c b b bJ J J c c b H H H H H H H H H2 1 2 2 1 2 2 2 1 1 1 1 2 12 1 [3-109] The extension of this matrix system in the case for more than two data sets should be fairly obvious. For each additional data set, one additional row and column would be added to the Hessian matrix, and one additi onal element would be added to each of the remaining matrices. For example, for three data sets, there would be one additional row, column, and element to reflect the inclusion of c3n and c3n. Since a matrix inversion routine is already necessary when determining standard error, the method of inversion is convenient to use to solve this system: n n n n n n n n n n n n n n n n n n n n nc c b c c c c b c c c c c b c c b c b b b n n nJ J J H H H H H H H H H c c b2 1 1 2 2 1 2 2 2 1 1 1 1 2 12 1 [3-110] PAGE 56 42 This solution gives the Â“ Â” values which can then be used to update parameter values and iteratively solve for the parameters until all first partial derivatives are within a specified tolerance of zero, as presented in the first solution for a single data set The Hessian matrix for standard error analysis in this case is: n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n nc c c c b c A c A c c c c c b c A c A c c b c b b b A b A b c A c A b A A A A A c A c A b A A A A AH H H H H H H H H H H H H H H H H H H H H H H H H2 2 1 2 2 2 2 1 2 2 1 1 1 1 2 1 1 1 2 1 2 1 2 2 1 2 2 2 2 1 2 2 1 1 1 1 2 1 1 1H [3-111] This matrix can be readily simplified by noting the cross derivatives between two differing data sets must be zero: n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n nc c b c A c c c b c A c c b c b b b A b A b c A b A A A c A b A A AH H H H H H H H H H H H H H H H H2 2 2 2 2 1 1 1 1 1 2 1 2 1 2 2 2 2 2 1 1 1 1 10 0 0 0 0 0 0 0 H [3-112] Once the inverse of this matrix is obtained, the equations presented earlier for covariance and standard error (Eqs. [3-71] and [3-73], respectively) can be used. It should be noted, however, that the variance us ed now includes both sets of data: 2 2 2) 2 2 exp( 1 2 2 ) 1 1 exp( 1 1 1 1 N c b A N N c b A N p n sn n n u n n n u [3-113] where n is now the total number of data points in both data sets, and p is the total number of parameters (five in this specific case). PAGE 57 43 Nonlinear Case 2: Common cn The error sum of squares for this case is: 2 2) 2 1 exp( 1 2 2 ) 1 1 exp( 1 1 1 N c b A N N c b A N En n n u n n n u [3-114] Using the same approach as in the previous case: 0 1 1 exp 1 1 1 1 exp 1 1 1 2 1 N c b N c b A N A En n n n n u n [3-115] 21 1 exp 1 1 1 1 exp 1 1 1 N c b N c b N An n n n u n [3-116] 22 2 exp 1 1 2 2 exp 1 2 2 N c b N c b N An n n n u n [3-117] Taking the remaining first, second, and cross derivatives: 2 1) 1 1 exp( 1 ) 1 1 exp( 1 ) 1 1 exp( 1 1 1 2 1 N c b N c b A N c b A N b E Jn n n n n n n n u n bn [3-118] 2 2) 2 2 exp( 1 ) 2 2 exp( 2 ) 2 2 exp( 1 2 2 2 2 N c b N c b A N c b A N b E Jn n n n n n n n u n bn [3-119] 2 2) 2 2 exp( 1 ) 2 2 exp( 2 2 ) 1 2 exp( 1 2 2 2 ) 1 1 exp( 1 ) 1 1 exp( 1 1 ) 1 1 exp( 1 1 1 2 N c b N c b N A N c b A N N c b N c b N A N c b A N c E Jn n n n n n n n u n n n n n n n n u n cn [3-120] PAGE 58 44 2 2 2 1 1) 1 1 exp( 1 1 2 1 N c b A E Hn n n A An n [3-121] 2 2 2 2 2) 2 2 exp( 1 1 2 2 N c b A E Hn n n A An n [3-122] 4 2 3 2 3 2 2 2 2 1 11 1 exp 1 1 1 exp 3 1 1 exp 1 1 1 exp 1 2 1 1 exp 1 1 1 exp 1 2 1 1 exp 1 1 1 exp 1 1 2 1 N c b N c b N c b N c b A N c b N c b N N c b N c b N A b E Hn n n n n n n n n n n n n u n n n n u n n b bn n [3-123] 4 2 3 2 3 2 2 2 2 2 22 2 exp 1 2 2 exp 3 2 2 exp 1 2 2 exp 2 2 2 2 exp 1 2 2 exp 2 2 2 2 exp 1 2 2 exp 2 2 2 2 N c b N c b N c b N c b A N c b N c b N N c b N c b N A b E Hn n n n n n n n n n n n n u n n n n u n n b bn n [3-124] 4 2 2 3 2 2 3 2 2 2 2 4 2 2 3 2 2 3 2 2 2 2 2 22 2 exp 1 2 2 exp 2 3 2 2 exp 1 2 2 exp 2 2 2 2 2 exp 1 2 2 exp 2 2 2 2 2 exp 1 2 2 exp 2 2 2 2 1 1 exp 1 1 1 exp 1 3 1 1 exp 1 1 1 exp 1 1 2 1 1 exp 1 1 1 exp 1 1 2 1 1 exp 1 1 1 exp 1 1 1 2 N c b N c b N N c b N c b N A N c b N c b N N N c b N c b N N A N c b N c b N N c b N c b N A N c b N c b N N N c b N c b N N A c E Hn n n n n n n n n n n n n u n n n n u n n n n n n n n n n n n n n u n n n n u n n c cn n [3-125] 0 1 2 2 12 1 2 2 2 1 n n A A n n A AA A E H A A E Hn n n n [3-126] PAGE 59 45 3 2 2 1 1 2 1 11 1 exp 1 1 1 exp 1 4 1 1 exp 1 1 1 exp 1 2 1 1 1 1 N c b N c b A N c b N c b N A b E H b A E Hn n n n n n n n n u n n A b n n b An n n n [3-127] 0 1 2 2 12 1 2 2 2 1 n n A b n n b AA b E H b A E Hn n n n [3-128] 3 2 2 1 2 11 1 exp 1 1 1 exp 1 1 4 1 1 exp 1 1 1 exp 1 1 2 1 1 N c b N c b N A N c b N c b N N A c E H c A E Hn n n n n n n n n u n n A c n n c An n n n[3-129] 0 2 1 1 22 2 1 2 1 2 n n A b n n b AA b E H b A E Hn n n n [3-130] 3 2 2 2 2 2 2 22 2 exp 1 2 2 exp 2 4 2 2 exp 1 2 2 exp 2 2 2 2 2 2 N c b N c b A N c b N c b N A b E H b A E Hn n n n n n n n n u n n A b n n b An n n n [3-131] 3 2 2 2 2 22 2 exp 1 2 2 exp 2 2 4 2 2 exp 1 2 2 exp 2 2 2 2 2 N c b N c b N A N c b N c b N N A c E H c A E Hn n n n n n n n n u n n A c n n c An n n n [3-132] 0 1 2 2 12 1 2 2 2 1 n n b b n n b bb b E H b b E Hn n n n [3-133] PAGE 60 46 4 2 3 2 3 2 2 2 1 2 11 1 exp 1 1 1 exp 1 3 1 1 exp 1 1 1 exp 1 1 2 1 1 exp 1 1 1 exp 1 1 2 1 1 exp 1 1 1 exp 1 1 1 2 1 1 N c b N c b N N c b N c b N A N c b N c b N N N c b N c b N N A b c E H c b E Hn n n n n n n n n n n n n u n n n n u n n n b c n n c bn n n n [3-134] 4 2 3 2 3 2 2 2 2 2 22 2 exp 1 2 2 exp 2 3 2 2 exp 1 2 2 exp 2 2 2 2 2 exp 1 2 2 exp 2 2 2 2 2 exp 1 2 2 exp 2 2 2 2 2 2 N c b N c b N N c b N c b N A N c b N c b N N N c b N c b N N A b c E H c b E Hn n n n n n n n n n n n n u n n n n u n n n b c n n c bn n n n [3-135] Using an approach very similar to Case 1, the solution to th e corresponding system of equations for this case is: n n n n n n n n n n n n n n n n n n n n nc b b c c b c b c c b b b b b c b b b b b n n nJ J J H H H H H H H H H c b b2 1 1 2 1 2 2 2 1 2 1 2 1 1 12 1 [3-136] The extension of this matrix system in the case for more than two data sets would be very similar to such an extension in Case 1 (ex cept, of course, additi onal elements would be the result of additional Â“nbx Â” parameters rather than Â“ncx Â” parameters). The Hessian matrix for standard e rror analysis in this case is: n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n nc c b c b c A c A c c b b b b b A b A b c b b b b b A b A b c A b A b A A A A A c A b A b A A A A AH H H H H H H H H H H H H H H H H H H H H H H H H2 1 2 1 2 2 2 1 2 2 2 1 2 1 2 1 1 1 2 1 1 1 2 2 2 1 2 2 2 1 2 1 2 1 1 1 2 1 1 1H [3-137] PAGE 61 47 This matrix can be readily simplified by no ting that the cross derivatives between two differing data sets must be zero: n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n nc c b c b c A c A c c b b b A b c b b b A b c A b A A A c A b A A AH H H H H H H H H H H H H H H H H2 1 2 1 2 2 2 2 2 1 1 1 1 1 2 2 2 2 2 1 1 1 1 10 0 0 0 0 0 0 0H [3-138] Once the inverse of this matrix is obtained, the equations presented earlier for covariance and standard error (E qs. [3-71] and [3-73], respectively) can be used. Again, however, the variance now includes both data sets: 2 2 2) 2 2 exp( 1 2 2 ) 1 1 exp( 1 1 1 1 N c b A N N c b A N p n sn n n u n n n u [3-139] where n is the total number of data points in both data sets, and p is the total number of parameters (five in this specific case). Nonlinear Case 3: Common bn and cn The error sum of squares for this case is: 2 2) 2 exp( 1 2 2 ) 1 exp( 1 1 1 N c b A N N c b A N En n n u n n n u [3-140] Using the same approach as in the previous case: 0 1 exp 1 1 1 exp 1 1 1 2 1 N c b N c b A N A En n n n n u n [3-141] PAGE 62 48 21 exp 1 1 1 exp 1 1 1 N c b N c b N An n n n u n [3-142] 22 exp 1 1 2 exp 1 2 2 N c b N c b N An n n n u n [3-143] Taking the remaining first, second, and cross derivatives: 2 2) 2 exp( 1 ) 2 exp( 2 ) 2 exp( 1 2 2 2 ) 1 exp( 1 ) 1 exp( 1 ) 1 exp( 1 1 1 2 N c b N c b A N c b A N N c b N c b A N c b A N b E Jn n n n n n n n u n n n n n n n n u n bn [3-144] 2 2) 2 exp( 1 ) 2 exp( 2 2 ) 1 exp( 1 2 2 2 ) 1 exp( 1 ) 1 exp( 1 1 ) 1 exp( 1 1 1 2 N c b N c b N A N c b A N N c b N c b N A N c b A N c E Jn n n n n n n n u n n n n n n n n u n cn [3-145] 2 2 2 1 1) 1 exp( 1 1 2 1 N c b A E Hn n n A An n [3-146] 2 2 2 2 2) 2 exp( 1 1 2 2 N c b A E Hn n n A An n [3-147] PAGE 63 49 4 2 3 2 3 2 2 4 2 3 2 3 2 2 2 22 exp 1 2 exp 3 2 exp 1 2 exp 2 2 2 exp 1 2 exp 2 2 2 exp 1 2 exp 2 2 2 1 exp 1 1 exp 3 1 exp 1 1 exp 1 2 1 exp 1 1 exp 1 2 1 exp 1 1 exp 1 1 2 N c b N c b N c b N c b A N c b N c b N N c b N c b N A N c b N c b N c b N c b A N c b N c b N N c b N c b N A b E Hn n n n n n n n n n n n n u n n n n u n n n n n n n n n n n n n n u n n n n u n n b bn n [3-148] 4 2 2 3 2 2 3 2 2 2 2 4 2 2 3 2 2 3 2 2 2 2 2 22 exp 1 2 exp 2 3 2 exp 1 2 exp 2 2 2 2 exp 1 2 exp 2 2 2 2 exp 1 2 exp 2 2 2 2 1 exp 1 1 exp 1 3 1 exp 1 1 exp 1 1 2 1 exp 1 1 exp 1 1 2 1 exp 1 1 exp 1 1 1 2 N c b N c b N N c b N c b N A N c b N c b N N N c b N c b N N A N c b N c b N N c b N c b N A N c b N c b N N N c b N c b N N A c E Hn n n n n n n n n n n n n u n n n n u n n n n n n n n n n n n n n u n n n n u n n c cn n [3-149] 0 1 2 2 12 1 2 2 2 1 n n A A n n A AA A E H A A E Hn n n n [3-150] 3 2 2 1 2 11 exp 1 1 exp 1 4 1 exp 1 1 exp 1 2 1 1 N c b N c b A N c b N c b N A b E H b A E Hn n n n n n n n n u n n A b n n b An n n n [3-151] 3 2 2 1 2 11 exp 1 1 exp 1 1 4 1 exp 1 1 exp 1 1 2 1 1 N c b N c b N A N c b N c b N N A c E H c A E Hn n n n n n n n n u n n A c n n c An n n n [3-152] PAGE 64 50 3 2 2 2 2 22 exp 1 2 exp 2 4 2 exp 1 2 exp 2 2 2 2 N c b N c b A N c b N c b N A b E H b A E Hn n n n n n n n n u n n A b n n b An n n n [3-153] 3 2 2 2 2 22 exp 1 2 exp 2 2 4 2 exp 1 2 exp 2 2 2 2 2 N c b N c b N A N c b N c b N N A c E H c A E Hn n n n n n n n n u n n A c n n c An n n n [3-154] 4 2 3 2 3 2 2 4 2 3 2 3 2 2 2 22 exp 1 2 exp 2 3 2 exp 1 2 exp 2 2 2 2 exp 1 2 exp 2 2 2 2 exp 1 2 exp 2 2 2 2 1 exp 1 1 exp 1 3 1 exp 1 1 exp 1 1 2 1 exp 1 1 exp 1 1 2 1 exp 1 1 exp 1 1 1 2 N c b N c b N N c b N c b N A N c b N c b N N N c b N c b N N A N c b N c b N N c b N c b N A N c b N c b N N N c b N c b N N A b c E H c b E Hn n n n n n n n n n n n n u n n n n u n n n n n n n n n n n n n n u n n n n u n n n b c n n c bn n n n[3-155] Using an approach similar to Case 1, the solution to the co rresponding system of equations for the iterative Â“ Â” values for this case is: n n n n n n n n n nc b c c b c c b b b n nJ J H H H H c b1 [3-156] Note that symbolically this is the same as the first met hod for the solution for a single data set. For conformity with Case 1 and Case 2, it is presented here in the Â“invertedÂ” solution form rather than the CramerÂ’s Rule solution presented in that section. The extension of this case to more than two da ta sets would not alter this final system PAGE 65 51 symbolically, but additional elements would be present in the unde rlying derivatives since for each new data set, a new element would be added to the error sum of squares equation. The Hessian matrix for standard e rror analysis in this case is: n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n nc c b c A c A c c b b b A b A b c A b A A A A A c A b A A A A AH H H H H H H H H H H H H H H H2 1 1 2 1 2 2 2 2 1 2 1 1 2 1 1 1H [3-157] This matrix can be simplified by noting that the cross derivatives between two differing data sets must be zero: n n n n n n n n n n n n n n n n n n n n n n n n n n n nc c b c A c A c c b b b A b A b c A b A A A c A b A A AH H H H H H H H H H H H H H2 1 2 1 2 2 2 2 1 1 1 10 0 H [3-158] Once the inverse of this matrix is obtained, the equations presented earlier for covariance and standard error (E qs. [3-71] and [3-73], respectively) can be used. Again, the variance now accounts for both sets of data: 2 2 2) 2 exp( 1 2 2 ) 1 exp( 1 1 1 1 N c b A N N c b A N p n sn n n u n n n u [3-159] where n is the total number of data points in both data sets, and p is the total number of parameters (four in this specific case). PAGE 66 52 Nonlinear Case 4: Common An, bn, and cn The error sum of squares for this case is: 2 2) 2 exp( 1 2 ) 1 exp( 1 1 N c b A N N c b A N En n n u n n n u [3-160] Using the same approach as in the previous case: 0 2 exp 1 1 2 exp 1 1 2 1 exp 1 1 1 1 exp 1 1 2 N c b N c b A N N c b N c b A N A En n n n n u n n n n n u n [3-161] 2 22 exp 1 1 1 exp 1 1 2 exp 1 2 1 exp 1 1 N c b N c b N c b N N c b N An n n n n n u n n u n [3-162] Taking the remaining first, second, and cross derivatives: 2 2) 2 exp( 1 ) 2 exp( ) 2 exp( 1 2 2 ) 1 exp( 1 ) 1 exp( ) 1 exp( 1 1 2 N c b N c b A N c b A N N c b N c b A N c b A N b E Jn n n n n n n n u n n n n n n n n u n bn [3-163] 2 2) 2 exp( 1 ) 2 exp( 2 ) 1 exp( 1 2 2 ) 1 exp( 1 ) 1 exp( 1 ) 1 exp( 1 1 2 N c b N c b N A N c b A N N c b N c b N A N c b A N c E Jn n n n n n n n u n n n n n n n n u n cn [3-164] 2 2 2 2) 2 exp( 1 1 2 ) 1 exp( 1 1 2 N c b N c b A E Hn n n n n A An n [3-165] PAGE 67 53 4 2 3 2 3 2 2 4 2 3 2 3 2 2 2 22 exp 1 2 exp 3 2 exp 1 2 exp 2 2 exp 1 2 exp 2 2 2 exp 1 2 exp 2 2 1 exp 1 1 exp 3 1 exp 1 1 exp 2 1 exp 1 1 exp 1 2 1 exp 1 1 exp 1 2 N c b N c b N c b N c b A N c b N c b N N c b N c b N A N c b N c b N c b N c b A N c b N c b N N c b N c b N A b E Hn n n n n n n n n n n n n u n n n n u n n n n n n n n n n n n n n u n n n n u n n b bn n [3-166] 4 2 2 3 2 2 3 2 2 2 2 4 2 2 3 2 2 3 2 2 2 2 2 22 exp 1 2 exp 2 3 2 exp 1 2 exp 2 2 2 exp 1 2 exp 2 2 2 2 exp 1 2 exp 2 2 2 1 exp 1 1 exp 1 3 1 exp 1 1 exp 1 2 1 exp 1 1 exp 1 1 2 1 exp 1 1 exp 1 1 2 N c b N c b N N c b N c b N A N c b N c b N N N c b N c b N N A N c b N c b N N c b N c b N A N c b N c b N N N c b N c b N N A c E Hn n n n n n n n n n n n n u n n n n u n n n n n n n n n n n n n n u n n n n u n n c cn n [3-167] 3 2 3 2 2 22 exp 1 2 exp 4 2 exp 1 2 exp 2 2 1 exp 1 1 exp 4 1 exp 1 1 exp 1 2 N c b N c b A N c b N c b N N c b N c b A N c b N c b N A b E H b A E Hn n n n n n n n n u n n n n n n n n n u n n A b n n b An n n n [3-168] 3 2 3 2 2 22 exp 1 2 exp 2 4 2 exp 1 2 exp 2 2 2 1 exp 1 1 exp 1 4 1 exp 1 1 exp 1 1 2 N c b N c b N A N c b N c b N N N c b N c b N A N c b N c b N N A c E H c A E Hn n n n n n n n n u n n n n n n n n n u n n A c n n c An n n n [3-169] PAGE 68 54 4 2 3 2 3 2 2 4 2 3 2 3 2 2 2 22 exp 1 2 exp 2 3 2 exp 1 2 exp 2 2 2 exp 1 2 exp 2 2 2 2 exp 1 2 exp 2 2 2 1 exp 1 1 exp 1 3 1 exp 1 1 exp 1 2 1 exp 1 1 exp 1 1 2 1 exp 1 1 exp 1 1 2 N c b N c b N N c b N c b N A N c b N c b N N N c b N c b N N A N c b N c b N N c b N c b N A N c b N c b N N N c b N c b N N A b c E H c b E Hn n n n n n n n n n n n n u n n n n u n n n n n n n n n n n n n n u n n n n u n n n b c n n c bn n n n [3-170] Since nA can still be solved for directly as long as nb and nc are known, the solution to the corresponding sy stem of equations for the iterative Â“ Â” values is symbolically the same as Case 3: n n n n n n n n n nc b c c c b c b b b n nJ J H H H H c b1 [3-156] As in Case 3, the extension of this case to more than two data sets would not alter this final equation symbolically, but would add el ements to the error sum of squares and the corresponding derivatives. The Hessian matrix for standard error analysis in this case is the same symbolically as that in the case of a single data set: n n n n n n n n n n n n n n n n n nc c b c A c c b b b A b c A b A A AH H H H H H H H HH [3-70] Once the inverse of this matrix is obtained, the equations presented earlier for covariance and standard error (E qs. [3-71] and [3-73], respectively) can be used. Again, the variance includes both sets of data: PAGE 69 55 2 2 2) 2 exp( 1 2 ) 1 exp( 1 1 1 N c b A N N c b A N p n sn n n u n n n u [3-171] where n is the total number of data points in both data sets, and p is the total number of parameters (three in this case). Linear Case A: Common b Recall the linear portion of the model is: 1 exp b A N N N Nn u cl cl c [3-172] The corresponding error sum of squares is: 2 21 exp Âˆ b A N N N N N N En u cl cl c c c [3-173] For the case of two data sets (Â“1Â” an d Â“2Â”) the error sum of squares is: 2 22 Âˆ 2 1 Âˆ 1c c c cN N N N E [3-174] 2 21 exp 2 2 2 2 2 1 exp 1 1 1 1 1 b A N N N N b A N N N N En u cl cl c n u cl cl c [3-175] Note here it is assumed the nA Â’s of the two data sets differ. For the purpose of this analysis, the values of nA are assumed Â“knownÂ”. The values of nA used here would be those found in the previous nonlinear regressi on. Therefore, dependi ng on the case, they may be the same or different, but here the more general case of unique values is presented. Simplifying back to primary parameters: PAGE 70 56 Again, the derivatives with respect to th e unknown parameters should be set to zero in order to minimize the error sum of squares. Rearranging the error sum of squares and then taking the derivative with respect to b : 2 22 2 2 exp 2 2 2 2 2 1 1 1 exp 1 1 1 1 1n u cl n u cl cl c n u cl n u cl cl cA N N b A N N N N A N N b A N N N N E [3-176] b A N N A N N b A N N N N b A N N A N N b A N N N N b En u cl n u cl n u cl cl c n u cl n u cl n u cl cl cexp 2 2 2 2 2 2 exp 2 2 2 2 2 2 exp 1 1 1 1 1 1 exp 1 1 1 1 1 2 [3-177] Setting equal to zero, rea rranging, and solving for b : n u cl n u cl cl c n u cl n u cl cl c n u cl n u clA N N A N N N N A N N A N N N N A N N A N N b b b E 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 2 2 2 1 1 1 exp exp 2 02 2 [3-178] PAGE 71 57 n u cl n u cl cl c n u cl n u cl cl c n u cl n u clA N N A N N N N A N N A N N N N A N N A N N b 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 2 2 2 1 1 1 exp2 2 [3-179] 2 22 2 2 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 expn u cl n u cl n u cl n u cl cl c n u cl n u cl cl cA N N A N N A N N A N N N N A N N A N N N N b [3-180] 2 22 2 2 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 lnn u cl n u cl n u cl n u cl cl c n u cl n u cl cl cA N N A N N A N N A N N N N A N N A N N N N b [3-181] Taking the derivatives for the two remaining parameters: 1 exp 1 1 1 1 1 exp 1 1 1 1 1 1 2 1 b A N A N b A N A N N N N En u n u n u n u cl c cl [3-182] Setting equal to zero, rea rranging, and solving for clN 1: PAGE 72 58 1 exp 1 1 1 1 1 2 1 exp 1 1 1 1 1 2 0 12b A N A N N b A N A N N N En u n u c n u n u cl cl [3-183] 21 exp 1 1 1 1 1 exp 1 1 1 1 1 1 b A N A N b A N A N N Nn u n u n u n u c cl [3-184] Similarly: 1 exp 2 2 2 2 1 exp 2 2 2 2 2 2 2 2 b A N A N b A N A N N N N En u n u n u n u cl c cl [3-185] 21 exp 2 2 2 2 1 exp 2 2 2 2 2 2 b A N A N b A N A N N Nn u n u n u n u c cl [3-186] Since all parameters can be solved for e xplicitly, each parameter can be updated by iterating through the sequence of equations. In other words,clN 1and clN 2 can be updated each time using Eqs. [3-184] and [3-186], respectively, and these new values of clN 1and clN 2 can then be used in Eq. [3-181] to update b which can then be used to update clN 1and clN 2again. This process is repeated until a ll first partial derivatives are within a specified tolerance of zero. The extension to cases of more than two data sets should be fairly obvious. The corresponding error sum of squares and the de rivatives thereof would simply have an PAGE 73 59 additional element for each data set, and an additional equation for each clNx would be formed. Standard error analysis will require the second derivatives and cross derivatives. The first derivative with respect to b written slightly differently here, is: n u cl n u cl cl c n u cl n u cl cl c n u cl n u clA N N A N N N N A N N A N N N N b A N N A N N b b E 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 exp 2 2 2 2 1 1 1 exp 22 2 2 [3-187] Taking the second derivative: n u cl n u cl cl c n u cl n u cl cl c n u cl n u clA N N A N N N N A N N A N N N N b A N N A N N b b E 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 exp 2 2 2 2 1 1 1 exp 42 2 2 2 2 [3-188] The first derivate with respect to clN 1, rearranged here, is: 1 exp 1 1 1 1 1 2 1 exp 1 1 1 1 1 2 12b A N A N N b A N A N N N En u n u c n u n u cl cl [3-189] Taking the second derivative: PAGE 74 60 2 2 21 exp 1 1 1 1 2 1 b A N A N N En u n u cl [3-190] Similarly for : 2clN 2 2 21 exp 2 2 2 2 2 2 b A N A N N En u n u cl [3-191] Taking the cross derivative of b and N 1cl: 2 2 21 exp 1 1 1 1 2 1 1 b A N A N b N b E b N En u n u cl cl [3-192] Completing the derivative: b A N b A N A N b N En u n u n u clexp 1 1 1 exp 1 1 1 1 4 12 [3-193] Similarly: b A N b A N A N b N En u n u n u clexp 2 2 1 exp 2 2 2 2 4 22 [3-194] The Hessian matrix for standard e rror analysis in this case is: cl cl cl cl cl cl cl cl cl cl cl clN N N N b N N N N N b N bN bN b bH H H H H H H H H2 2 2 1 2 2 1 1 1 1 2 1H [3-195] This matrix can be simplified a bit by noti ng the cross derivatives between two differing data sets must be zero: cl cl cl cl cl cl cl clN N b N N N b N bN bN b bH H H H H H H2 2 2 1 1 1 2 10 0H [3-196] PAGE 75 61 Once the inverse of this matrix is obtained, the equations presented earlier for covariance and standard error (Eqs. [3-71] and [3-73], respective ly) for the nonlinear portion can now be used for the linear portion as well. It shou ld be noted, however, that a modified form of the variance must be used, based on the normalized error sum of squares for the linear portion: 2 2 21 exp 2 2 2 2 2 1 exp 1 1 1 1 1 1 b A N N N N b A N N N N p n sn u cl cl c n u cl cl c [3-197] where n is now the total number of data points in both data sets, and p is the total number of parameters (three in this specific case). Linear Case B: Common Ncl The normalized error sum of squares for this case is: 2 22 2 2 exp 2 2 2 1 1 1 exp 1 1 1n u cl n u cl cl c n u cl n u cl cl cA N N b A N N N N A N N b A N N N N E [3-198] Taking the derivative with respect to 1b : 1 exp 1 1 1 1 1 exp 1 1 1 2 1b A N N A N N b A N N N N b En u cl n u cl n u cl cl c [3-199] Setting equal to zero and solving: PAGE 76 62 1 exp 1 1 1 1 1 exp 1 1 1 2 0 1 b A N N A N N b A N N N N b En u cl n u cl n u cl cl c [3-200] n u cl n u cl cl c n u clA N N A N N N N A N N b 1 1 1 1 1 1 1 1 exp2 [3-201] 21 1 1 1 1 1 1 ln 1n u cl n u cl n u cl cl cA N N A N N A N N N N b [3-202] Similarly for 2b : 22 2 2 2 2 2 2 ln 2n u cl n u cl n u cl cl cA N N A N N A N N N N b [3-203] Taking the derivative with respect to clN: 1 2 exp 2 2 2 2 1 2 exp 2 2 2 2 2 2 1 1 exp 1 1 1 1 1 1 exp 1 1 1 1 1 2 b A N A N b A N A N N N b A N A N b A N A N N N N En u n u n u n u cl c n u n u n u n u cl c cl [3-204] Setting to zero, rearra nging and solving for clN : PAGE 77 63 1 2 exp 2 2 2 2 2 1 1 exp 1 1 1 1 1 2 1 2 exp 2 2 2 2 1 1 exp 1 1 1 1 2 02 2b A N A N N b A N A N N b A N A N b A N A N N N En u n u c n u n u c n u n u n u n u cl cl [3-205] 2 21 2 exp 2 2 2 2 1 1 exp 1 1 1 1 1 2 exp 2 2 2 2 2 1 1 exp 1 1 1 1 1 b A N A N b A N A N b A N A N N b A N A N N Nn u n u n u n u n u n u c n u n u c cl [3-206] Since all parameters can be solved for e xplicitly, each parameter can be updated by iterating through the sequence of equations until all of the fi rst derivatives are within a selected tolerance of zero. The extension to cases of mo re than two data sets should be fairly obvious. The corresponding error sum of squares would si mply have an additional element for each data set, and an additional equation for each clbx would be formed. Standard error and covariance analysis will require the second derivatives and cross derivatives. The first derivative with respect to clN expanded out here, is: PAGE 78 64 1 2 exp 2 2 2 2 2 1 1 exp 1 1 1 1 1 2 1 2 exp 2 2 2 2 1 1 exp 1 1 1 1 22 2b A N A N N b A N A N N b A N A N b A N A N N N En u n u c n u n u c n u n u n u n u cl cl [3-207] The second derivative is: 2 2 2 21 2 exp 2 2 2 2 1 1 exp 1 1 1 1 2 b A N A N b A N A N N En u n u n u n u cl [3-208] The second derivative with respect to 1b is: n u cl n u cl cl c n u clA N N A N N N N b A N N b b E 1 1 1 1 1 1 exp 2 1 1 1 exp 4 12 2 2 2 [3-209] Similarly, the second deri vative with respect to 2 b is: n u cl n u cl cl c n u clA N N A N N N N b A N N b b E 2 2 2 2 2 2 exp 2 2 2 2 exp 4 22 2 2 2 [3-210] The cross derivative between clN and 1 b is: 1 exp 1 1 1 1 exp 1 1 1 1 4 12b A N b A N A N b N En u n u n u cl [3-211] PAGE 79 65 Similarly, the cross derivative between clN and 2 b is: 2 exp 2 2 1 2 exp 2 2 2 2 4 22b A N b A N A N b N En u n u n u cl [3-212] The Hessian matrix for standard e rror analysis in this case is: cl cl cl cl cl clN N b N b N N b b b b b N b b b b bH H H H H H H H H2 1 2 2 2 1 2 1 2 1 1 1H [3-213] This matrix can be simplified a bit by noti ng the cross derivatives between two differing data sets must be zero: cl cl cl cl cl clN N b N b N N b b b N b b bH H H H H H H2 1 2 2 2 1 1 10 0H [3-214] Once the inverse of this matrix is obtained, the equations presented earlier for covariance and standard error (Eqs. [3-71] and [3-73], respectively) can be used. The variance used in this case is: 2 2 21 2 exp 2 2 2 1 1 exp 1 1 1 1 b A N N N N b A N N N N p n sn u cl cl c n u cl cl c [3-215] where n is the total number of data points in both data sets, and p is the total number of parameters (three in this specific case). PAGE 80 66 Linear Case C: Common b and Ncl The error sum of squares for this case is: 2 21 exp 2 2 2 1 exp 1 1 1 b A N N N N b A N N N N En u cl cl c n u cl cl c [3-216] Taking the derivative with respect to b : b A N N A N N b A N N N N b A N N A N N b A N N N N b En u cl n u cl n u cl cl c n u cl n u cl n u cl cl cexp 2 2 2 2 exp 2 2 2 2 exp 1 1 1 1 exp 1 1 1 2 [3-217] Setting equal to zero and solving: 2 22 2 1 1 2 2 2 2 2 1 1 1 1 1 lnn u cl n u cl n u cl n u cl cl c n u cl n u cl cl cA N N A N N A N N A N N N N A N N A N N N N b [3-218] Taking the derivative with respect to clN : PAGE 81 67 1 exp 2 2 2 2 1 exp 2 2 2 2 2 2 1 exp 1 1 1 1 1 exp 1 1 1 1 1 2 b A N A N b A N A N N N b A N A N b A N A N N N N En u n u n u n u cl c n u n u n u n u cl c cl [3-219] Setting to zero and solving: 2 21 exp 2 2 2 2 1 exp 1 1 1 1 1 exp 2 2 2 2 2 1 exp 1 1 1 1 1 b A N A N b A N A N b A N A N N b A N A N N Nn u n u n u n u n u n u c n u n u c cl [3-220] The extension to more than two data sets w ould simply involve additional elements in the above equations corresponding to additional datasets (cNx ,uNx andnAx ). Standard error analysis will require the second derivatives and cross derivatives. The second derivative with respect to b is: n u cl n u cl cl c n u cl n u cl cl c n u cl n u clA N N A N N N N A N N A N N N N b A N N A N N b b E 2 2 2 2 2 2 1 1 1 1 1 1 exp 2 2 2 1 1 exp 42 2 2 2 2 [3-221] The second derivative with respect to clN is: PAGE 82 68 2 2 2 21 exp 2 2 2 2 1 exp 1 1 1 1 2 b A N A N b A N A N N En u n u n u n u cl [3-222] The cross derivative between clN and b is: b A N b A N A N b A N b A N A N b N En u n u n u n u n u n u clexp 2 2 1 exp 2 2 2 2 4 exp 1 1 1 exp 1 1 1 1 42 [3-223] The Hessian matrix for standard e rror analysis in this case is: cl cl cl clN N b N bN b bH H H HH [3-224] Once the inverse of this matrix is obtained, the equations presented earlier for covariance and standard error (Eqs. [3-71] and [3-73], respectively) can be used. The variance for this case is: 2 2 21 exp 2 2 2 1 exp 1 1 1 1 b A N N N N b A N N N N p n sn u cl cl c n u cl cl c [3-225] where n is the total number of data points in both data sets, and p is the total number of parameters (two in this case). Generalized Equations for All Analyses Mainly for programming purposes, a set of general equations can be derived to encompass all of the above cases. Then, with only small addendums, these equations can be used to fit any case. The resulting equatio ns are presented in this section. Little PAGE 83 69 description or derivation is shown since these are merely ge neralizations of the patterns apparent in previously presented cases. It s hould be noted that th ese general equations also apply to cases involving a single data set. Indeed, because of slight discrepancies observed between the non-normalized and norma lized approaches when analyzing data, the normalized approach was used even when an alyzing a single data set for consistency. This had the added benefit that standard erro r analysis could easily be extended to the linear portion for a single data set without having to derive any more equations. Variable Definitions Let i denote the current data set and j the current data point. Let nds equal the number of data sets in the analysis and ndp ( i ) the number of data points in data set i Let N ( i,j ) denote the N uptake value for data point j within data set i Similarly, let Nu( i,j ) denote the N uptake va lue for data point j within data set i Similarly, let Nc( i,j ) denote the N concentration value for data point j within data set i Let An( i ) denote the value of An for data set i Similarly let bn( i ), cn( i ), b ( i ), and Ncl( i ) denote the values of bn, cn, b and Ncl, respectively, for data set i Let ENu and ENc denote the total normalized error sum of squares for the nonlinear portion (based on N uptake) a nd linear portion (based on N concentration), respectively. The Â“ J Â” and Â“ H Â” notation used previously will also be used here to represent first and second (or cro ss) derivatives, respectively. The suffix ( i ) will denote that the derivative applies specifically to data set i General Equations for Nonlinear Portion For all cases the following equations apply: nds i i ndp j n n n u Nuj i N i c i b i A j i N E1 ) ( 1 2)) ( ) ( ) ( exp( 1 ) ( ) ( [3-226] PAGE 84 70 ) ( 1 2 ) ( 1) ( ) ( ) ( exp 1 1 ) ( ) ( ) ( exp 1 ) ( ) (i ndp j n n i ndp j n n u nj i N i c i b j i N i c i b j i N i A [3-227] ) ( 1 2) ( ) ( ) ( exp( 1 )) ( ) ( ) ( exp( ) ( )) ( ) ( ) ( exp( 1 ) ( ) ( 2 ) ( ) (i ndp j n n n n n n n n u n bj i N i c i b j i N i c i b i A j i N i c i b i A j i N i b E i Jn [3-228] ) ( 1 2)) ( ) ( ) ( exp( 1 )) ( ) ( ) ( exp( ) ( ) ( )) ( ) ( ) ( exp( 1 ) ( ) ( 2 ) ( ) (i ndp j n n n n n n n n u n cj i N i c i b j i N i c i b j i N i A j i N i c i b i A j i N i c E i Jn [3-229] ) ( 1 2 2 2) ( ) ( ) ( exp( 1 1 2 ) ( ) (i ndp j n n n A Aj i N i c i b i A E i i Hn n [3-230] ) ( 1 4 2 3 2 ) ( 1 3 2 2 2 2) ( ) ( ) ( exp 1 ) ( ) ( ) ( exp 3 ) ( ) ( ) ( exp 1 ) ( ) ( ) ( exp ) ( 2 ) ( ) ( ) ( exp 1 ) ( ) ( ) ( exp ) ( 2 ) ( ) ( ) ( exp 1 ) ( ) ( ) ( exp ) ( ) ( 2 ) ( ) (i ndp j n n n n n n n n n i ndp j n n n n u n n n n u n n b bj i N i c i b j i N i c i b j i N i c i b j i N i c i b i A j i N i c i b j i N i c i b i N j i N i c i b j i N i c i b j i N i A i b E i i Hn n [3-231] PAGE 85 71 ) ( 1 4 2 2 3 2 2 ) ( 1 3 2 2 2 2 2 2) ( ) ( ) ( exp 1 ) ( ) ( ) ( exp ) ( 3 ) ( ) ( ) ( exp 1 ) ( ) ( ) ( exp ) ( ) ( 2 ) ( ) ( ) ( exp 1 ) ( ) ( ) ( exp ) ( ) ( 2 ) ( ) ( ) ( exp 1 1 ) ( ) ( exp ) ( ) ( ) ( 2 ) ( ) (i ndp j n n n n n n n n n i ndp j n n n n u n n n n u n n c cj i N i c i b j i N i c i b j i N j i N i c i b j i N i c i b j i N i A j i N i c i b j i N i c i b j i N j i N j i N i c i b N i c i b j i N j i N i A i c E i i Hn n [3-232] ) ( 1 3 ) ( 1 2 2 2) ( ) ( ) ( exp 1 ) ( ) ( ) ( exp ) ( 4 ) ( ) ( ) ( exp 1 ) ( ) ( ) ( exp ) ( 2 ) ( ) ( ) ( ) ( ) ( ) (i ndp j n n n n n i ndp j n n n n u n n A b n n b Aj i N i c i b j i N i c i b i A j i N i c i b j i N i c i b j i N i A i b E i i H i b i A E i i Hn n n n [3-233] ) ( 1 3 ) ( 1 2 2 2) ( ) ( ) ( exp 1 ) ( ) ( ) ( exp ) ( ) ( 4 ) ( ) ( ) ( exp 1 ) ( ) ( ) ( exp ) ( ) ( 2 ) ( ) ( ) ( ) ( ) ( ) (i ndp j n n n n n i ndp j n n n n u n n A c n n c Aj i N i c i b j i N i c i b j i N i A j i N i c i b j i N i c i b j i N j i N i A i c E i i H i c i A E i i Hn n n n [3-234] PAGE 86 72 ) ( 1 4 2 3 2 ) ( 1 3 2 2 2 2) ( ) ( ) ( exp 1 ) ( ) ( ) ( exp ) ( 3 ) ( ) ( ) ( exp 1 ) ( ) ( ) ( exp ) ( ) ( 2 ) ( ) ( ) ( exp 1 ) ( ) ( ) ( exp ) ( ) ( 2 ) ( ) ( ) ( exp 1 ) ( ) ( ) ( exp ) ( ) ( ) ( 2 ) ( ) ( ) ( ) ( ) ( ) (i ndp j n n n n n n n n n i ndp j n n n n u n n n n u n n n b c n n c bj i N i c i b j i N i c i b j i N j i N i c i b j i N i c i b j i N i A j i N i c i b j i N i c i b j i N j i N j i N i c i b j i N i c i b j i N j i N i A i b i c E i i H i c i b E i i Hn n n n [3-235] For common bn, each bn( i ) would be assigned the same value bn, and the following equations would also apply: nds i b bi J Jn n1) ( [3-236] nds i b b b bi i H Hn n n n1) ( [3-237] For common cn, each cn( i ) would be assigned the same value cn, and the following equations would also apply: nds i c ci J Jn n1) ( [3-238] nds i c c c ci i H Hn n n n1) ( [3-239] Note Eqs. [3-236] through [3-239] also apply for both common bn and cn, as does the following equation. nds i c b b c c bi i H H Hn n n n n n1) ( [3-240] For common An, bn, and cn, all previous equations would a pply, as well as the following: PAGE 87 73 nds i i ndp j n n nds i i ndp j n n u nj i N i c i b j i N i c i b j i N A1 ) ( 1 2 1 ) ( 1) ( ) ( ) ( exp 1 1 ) ( ) ( ) ( exp 1 ) ( [3-241] nds i A A A Ai i H Hn n n n1) ( [3-242] nds i b A A b b Ai i H H Hn n n n n n1) ( [3-243] nds i c A A c c Ai i H H Hn n n n n n1) ( [3-244] General Equations for Linear Portion nds i i ndp j n u cl cl c Nci b i A j i N i N i N j i N E1 ) ( 1 21 ) ( exp ) ( ) ( ) ( ) ( ) ( [3-245] ) ( 1) ( exp ) ( ) ( ) ( ) ( ) ( ) ( ) ( exp ) ( ) ( ) ( ) ( ) ( 2 ) (i ndp j n u cl n u cl n u cl cl ci b i A j i N i N i A j i N i N i b i A j i N i N i N j i N i b E [3-246] ) ( 1 ) ( 1 2 2 2 2) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( exp 2 ) ( ) ( ) ( ) ( exp 4 ) ( ) (i ndp j n u cl n u cl cl c i ndp j n u cl b bi A j i N i N i A j i N i N i N j i N i b i A j i N i N i b i b E i i H [3-247] PAGE 88 74 ) ( 11 ) ( exp ) ( ) ( ) ( ) ( 1 ) ( exp ) ( ) ( ) ( ) ( ) ( ) ( 2 ) (i ndp j n u n u n u n u cl c cli b i A j i N i A j i N i b i A j i N i A j i N i N j i N i N E [3-248] ) ( 1 2 2 21 ) ( exp ) ( ) ( ) ( ) ( 2 ) ( ) (i ndp j n u n u cl N Ni b i A j i N i A j i N i N E i i Hcl cl [3-249] ) ( 1 2 2) ( exp ) ( ) ( 1 ) ( exp ) ( ) ( ) ( ) ( 4 ) ( ) ( ) ( ) ( ) ( ) (i ndp j n u n u n u cl b N cl bNi b i A j i N i b i A j i N i A j i N i b i N E i i H i N i b E i i Hcl cl [3-250] For cases with independent b Â’s, the following would apply ) ( 1 2 ) ( 1) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ln ) (i ndp j n u cl i ndp j n u cl n u cl cl ci A j i N i N i A j i N i N i A j i N i N i N j i N i b [3-251] For cases with independent NclÂ’s, the following would apply: ) ( 1 2 ) ( 11 ) ( exp ) ( ) ( ) ( ) ( 1 ) ( exp ) ( ) ( ) ( ) ( ) ( ) (i ndp j n u n u i ndp j n u n u c cli b i A j i N i A j i N i b i A j i N i A j i N j i N i N [3-252] For common b each b ( i ) would be assigned the same value b and the following equations would also apply: nds ii b E b E1) ( [3-253] PAGE 89 75 nds i b b b bi i H H1) ( [3-254] nds i i ndp j n u cl nds i i ndp j n u cl n u cl cl ci A j i N i N i A j i N i N i A j i N i N i N j i N b1 ) ( 1 2 1 ) ( 1) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ln [3-255] For common Ncl, each Ncl( i ) would be assigned the same value Ncl, and the following equations would also apply: nds i cl cli N E N E1) ( [3-256] nds i N N N Ni i H Hcl cl cl cl1) ( [3-257] nds i i ndp j n u n u nds i i ndp j n u n u c cli b i A j i N i A j i N i b i A j i N i A j i N j i N N1 ) ( 1 2 1 ) ( 11 ) ( exp ) ( ) ( ) ( ) ( 1 ) ( exp ) ( ) ( ) ( ) ( ) ( [3-258] For common b and Ncl, all of the above equations would apply as well as the following: nds i bN bNi i H Hcl cl1) ( [3-259] Analysis of Variance (ANOVA) for Co mmonality of Parameters (F-test) In this study, several data sets are analyzed in multiple ways, both independently and with various parameters held in common fo r a selection of data sets. To test whether or not parameters should be c onsidered common, an F-test is employed similar to that outlined in Ratkowsky (1983). For the purposes of this study, the F-te st was based on a normalized residual sum of squares that included all data ( Nu, Nc, and Y ). This way the fit of different models (with PAGE 90 76 varying common parameters) to the complete data set could be compared. Normalization was employed since Nu, Nc, and Y are often of different orde rs of magnitude. The general form for the normalized residual sum of squares, RSS using the conventions presented in the previous section is: nds i n i ndp j cl cl i ndp j c c n i ndp j u ui A j i Y j i Y i b i N i b i N j i N j i N i A j i N j i N RSS1 2 ) ( 1 2 2 2 2 2 ) ( 1 2 2 ) ( 1 2) ( ) ( Âˆ ) ( ) ( exp ) ( ) ( exp ) ( ) ( Âˆ ) ( ) ( ) ( Âˆ ) ( [3-260] where uN Âˆ, cN Âˆ, and Y Âˆ are model-calculated values (as functions of applied N only): )) ( ) ( ) ( exp( 1 ) ( ) ( Âˆ j i N i c i b i A j i Nn n n u [3-261] )) ( ) ( ) ( exp( 1 ))) ( ) ( ) ( ) ( exp( 1 ))( ( exp( ) ( ) ( Âˆ j i N i c i b j i N i c i b i b i b i N j i Nn n n n cl c [3-262] ))) ( ) ( ) ( ) ( exp( 1 ))( ( exp( ) ( ) ( ) ( Âˆ j i N i c i b i b i b i N i A j i Yn n cl n [3-263] Note that the value for the cN Âˆ here is merely the ratio of uN Âˆ to Y Âˆ. The general form of the F-test is presented below. Let A denote a test case in which no comm on parameters are assumed between data sets; i.e., all parameters are individual. Let B denote a test in which one or more of the parameters are common. Let pA and pB denote the number of parameters estimated in each test case A and B respectively. Let dfA and dfB denote the degrees of freedom in each test case A and B respectively. Let n denote the total number of data points in the analysis (from all data sets in cluded in the analysis, counting Nu, Nc, and Y points PAGE 91 77 separately). Let RSSA and RSSB denote the residual sum of squares in each test case A and B respectively. Let F denote the calculated F-test st atistic (or variance ratio) and F the critical value for F at significance level With these definitions, the following equations apply: A Ap n df [3-264] B Bp n df [3-265] A A A B A Bdf RSS df df RSS RSS F / / [3-266] Once F is calculated, it is compared to F( dfB-dfA, dfA, ) (which is obtained from the appropriate F distribution tables). If the following holds true: F F [3-267] then the null hypothesis is re jected for a test of level (Wackerly et al., 2002). The null hypothesis, in all cases, is that the para meters assumed to be common are so; the converse of this hypothesis is th at all parameters are unique. Therefore, if Eq. [3-267] holds true, it cannot be assumed that the parameters in question are common at the selected test level PAGE 92 78 CHAPTER 4 RESULTS AND DISCUSSION In this chapter, the strategy and results of analysis are discussed in detail, including evidence of commonality of parameters. Numerical results are listed in the tables at the end of the chapter, followed by figures used to exemplify some results. Broad conclusions comparing the results of this study to other studies with respect to the Extended Logistic Model are reserved, for the most part, for the next chapter. Analysis Strategy Admittedly, this analysis was in some ways daunting because of the large amount of data and the complexity of the experiment. Indeed, altogether there were some 28 data sets with a total of 140 indivi dual data points (or 420 if the three data types of N uptake, N concentration, and dry matter yield are counted separately) spread across multiple years. These data were characterized based on combinations of thr ee different factors: crop type (bermudagrass or tall fescue), irrigation treatment (irrigated or nonirrigated), and fertilizer treatment (commercial fertilizer or broiler litter application). Clearly, the potential number of combinati ons of analyses was massive. In the first round of analyses, each data se t was analyzed individually to obtain its own optimized set of parameters and sta ndalone statistics. In the second round of analyses, data were isolated based on crop type and fertilizer treatment. Irrigation treatment was considered both separately and together. Analyses were not conducted within single years but rather across all years. Th is was done partially because data from a single year could be extremely limited, espe cially in the case of broiler litter where PAGE 93 79 there were only four applica tion rates (including zero). This was also done to simplify analyses somewhat, and because the interest lies primarily in comm onality of parameters that extends beyond a single season. In this second round of analyses, a group of data was first analyzed with all parameters independent (the most general mode l) in order to establish a baseline residual sum of squares for use in the F-test. After this analysis, termed Mode (0), there were 15 additional modes that could potentially be c onsidered, split into four levels. Mode descriptions can be found in Table 4-1. In th e first level (1), one of the parameters was assumed invariant and the othe rs allowed to vary between data sets. Parameters whose invariance could not be rejected in this first level were th en grouped together into the varying combinations of levels (2)-(4) and an alyzed further to determine whether two or more parameters could be considered simulta neously invariant between data sets. Note this round also included special cases for comme rcial fertilizer where data of a given crop type and a given N application rate were averaged over the years into two averaged data sets of nonirrigated and irrigated data. This was possible only for co mmercial fertilizer because while N application rates for commerc ial fertilizer were the same scheme each year, N application rates for br oiler litter varied from year to year because broiler litter was applied based on schemes of total tonnage, not N rate. The results for the second round were then perused to identify possible parameters that could be common on even broader scales (for example, across both crop types or across both types of fertili zer). The groupings chosen for the third round were based on these results, but often additional parameters were tested for commonality even if PAGE 94 80 commonality was not suggested by the sec ond round, since such a test was very convenient anyway. Overview of Results Round One The results of first round analyses can be seen in Table 4-2 th rough Table 4-9. Note that covariance is not included in these tables. This is partially for brevity. In the vast majority of cases, covariance between paramete rs was very small, often very close to zero. However, in cases where standard erro rs were relatively high (note in a few cases the standard error was more than 50%!), a significant covariance between parameters was observed. The effect the standard errors have on the resulting model equations is discussed and illustrated in figures for all fi ve parameters in the Â“Sensitivity AnalysisÂ” section of the Appendix. Examples of results from each data ca tegory can be seen in Figure 4-1 through Figure 4-20. As noted in these figures, they include the best fit example and the poorest fit example from each group of data. The averaged data values for irrigated and nonirrigated data with commer cial fertilizer are also am ong these figures. The figures demonstrate that, even with the Â“worstÂ” fits the model describes th e data consistently well. It is also difficult to deny that on an i ndividual data set scale, the Extended Logistic Model describes the data extremely well if the nonlinear correlation coefficient, R is used as the chief measure of goodness of fit. In only five out of 34 cases did RNc (the correlation coefficient for the N concentrati on portion of the model) fall below 0.90. Granted, two of these R values were exceptionally low (0.57 and 0.68, from the tall fescue with broiler litt er data of 1982-3). The RNu value (the correlation coefficient for the PAGE 95 81 N uptake portion of the model) never fell below 0.97 and in the vast majority of cases was 0.99 or greater. Only once was the RY value (the correlation coefficient for the dry matter yield model) less than 0.97, a nd even then it was better than 0.95. What is not surprising here is that the N uptake model had the highest R values. This can be attributed, at leas t in part, to the fact that three of the five parameters, An bn and cn, are optimized solely against the N uptake data, and so in this sense the regression scheme gives extra weight to properly fitting N uptake at the potential expense of fit to the other two data types. What is perhaps more noteworthy is the extremely good fit to dr y matter yield data, even though these data are not directly invol ved in the regression scheme. This evidence further solidifies the concept of the strong interconnectedness of N uptake, N concentration, and yield in response to applied N. For a preliminary search for patterns and potential commonality among parameters, each parameter was arithmetically averaged across various groups of data, as shown in Table 4-10. The simple standard deviation was also taken for the parameters in the groups, as shown in the corresponding Table 4-11 Note this should not be confused with standard errors calculated in regression analysis. Following is a discussion of each of the parameters. While the linear parameter An was not the focus of this study (because it is known to vary in response to several different environmental inputs), some patterns were observed in the averaging procedure describe d above. This was a rather simplistic and qualitative approach, however, since no statis tical tests of significant difference were PAGE 96 82 performed for An. More focus is given on An in this section than the other four parameters since it was not considered in the round two and round thr ee analyses and discussions. First, it is rather clear there is a differe nce in response due to crop type: the average An value for all bermudagrass was 404 kg ha-1, while for all tall fescue it was 337 kg ha-1, about 17% lower than for bermudagrass, although both had high standard deviations. This difference is more apparent if only the commercial fertilizer treatments are considered (note that broiler litter trea tments had far fewer treatment levels, four versus seven, and much higher standard errors in An overall than commercial fertilizer treatments). Bermudagrass treated with commercial fertilizer had a mean An of 490 kg ha-1, while tall fescue with comm ercial fertilizer had a mean An of 342 kg ha-1, 30% lower than for bermudagrass; however, there wa s a very large standard deviation of 36% for tall fescue. There is also evidence of differences in An due to fertilizer source. The average An for all commercial fertilizer treatments was 398 kg ha-1, and that for all broiler litter treatments was 324 kg ha-1, about 19% lower than that fo r commercial fertilizer, although both had high standard deviations as well. This apparent diffe rence is even greater if only bermudagrass is considered (490 kg ha-1 for commercial fertilizer and 318 kg ha-1 for broiler litter), but it is less apparent if only tall fescue is considered (342 kg ha-1 for commercial fertilizer and 331 kg ha-1 for broiler litter, well with in the standard deviations of each other). Evidence of differences in An due to water availability is also present. The average An for all irrigated tr eatments was 406 kg ha-1, and that for all nonirrigated treatments was 345 kg ha-1, about 15% lower than that for irriga ted treatments, but these are within PAGE 97 83 standard deviations of each ot her. If only commercial fertil izer treatments are considered, differences are more apparent: 528 kg ha-1 for irrigated bermudagrass versus 452 kg ha-1 for nonirrigated bermudagrass, and 433 kg ha-1 for irrigated tall fescue versus 251 kg ha-1 for nonirrigated tall fescue, alt hough the latter did have a very high standard deviation of 45%. Overall, there appear to be possible differences in bn due to both irrigation and crop type. Effects on bn due to fertilizer source are less apparent. Although there is some overlap when considering standard devi ations, from the averages alone for both bermudagrass and tall fescue, bn tended to be more of a function of fertilizer source in an irrigated situation than in a nonirrigated situati on, where the values of bn were about the same within each crop. Taking st andard deviations into account, bn was much more variable in tall fescue than in bermudagrass. The most marked difference in cn was due to fertilizer source, with a value of 0.00836 ha kg-1 for commercial fertilizer and 0.00625 ha kg-1 for broiler litter, although both exhibited standard deviat ions of 32%. In many cases, cn was rather highly variable, especially with tall fescue, in which very high standard deviations were often observed for this parameter. The parameter b was most highly variable in br oiler litter groups, and especially so when tall fescue was the crop. With such high standard deviations, there was no obvious factor that influenced b most strongly. Irrigation treatment, fertilizer source, and crop type all could have played potential roles in its value. There is strong evidence that Ncl is likely a function of cr op type for the most part, with an average value of 13.2 g kg-1 for all bermudagrass and 18.2 g kg-1 for all tall PAGE 98 84 fescue, each with very low standard deviati ons (6% and 9%, respectively) compared to other parameters. Most differen ces due to fertilizer source were easily within standard deviations of each other, although there was so me evidence of small differences due to irrigation. Round Two The results of the second round of anal yses can be found in Table 4-12 through Table 4-25. Each of these tables includes the value of parameters assumed invariant (according to mode) and the resulting F-values from comparison of the variances of each mode to the most general one, mode (0). Cr itical F-values (obtai ned from an F-value calculator similar to an F di stribution textbook table) at the = 0.10 level of significance (or 90% confidence) are listed in the tables as well, each being a function of the number of degrees of freedom in the most genera l model, and the difference in degrees of freedom of the general model from the mode being tested. Recall from Chapter 3 that if the calculated value of F falls below the critical value of F, then the null hypothesis cannot be rejected at the stated confidence level. Thus in this analysis, if the calculated Fvalue fell below the critical F, then the null hypothesis (that th e selected invariant parameters are truly common) could not be rejected with 90% confidence, and so commonality of the stated parameters could be assumed. An F-value above the critical Fvalue meant one could reject the null hypothesis with 90% confidence and so commonality could not be assumed. The 90% leve l was chosen in this study because it is considered standard and is more rigorous than higher levels of 95% or 99%. This may seem counter-intuitive at first thought, but in essence it is easier to reject a null hypothesis with 90% confiden ce than with 95% or 99% confidence, thus, it is harder to PAGE 99 85 statistically support commonality (the null hypo thesis) at 90% versus higher levels. In other words, if the null hypot hesis is supported (i.e., cannot be rejected) at 90%, it will also be supported at 95% and 99% confid ence levels. More discussion on the F-test, including anomalous observations, is presented in the next major section of this chapter. The tables for round two all include the four possible versions of mode (1) (holding each parameter common one at a time). If a parameter failed this initial test for commonality, it was not considered in further combinations with other parameters for the data set grouping in question. An exception to this is Table 4-12, in which additional combinations were included to illustrate the poi nt that even if a parameter fails the initial test (as cn did in this case), it may not fail when combined with other parameters (note cn paired with either b or Ncl passed the commonality test). This is likely because one of the parameters that passes the commonality test alone can in some circumstances Â“compensateÂ” for the one that fails the test so that both appear common when paired together. This is the primary reason why if a parameter failed a commonality test on its own, it was not considered further. To clar ify, the concurrent commonality pairings in Table 4-12 that include cn are not considered valid and are not considered when discussing commonality pairs from this point forward. In a few cases where a large number of pairings were tested, some pairs that failed the commonality test were not included in the table for brevity. Out of the 14 data groupings analyzed in round two, common bnÂ’s within a group occurred three times; common cnÂ’s occurred six times; common b Â’s occurred eight times; and common NclÂ’s occurred 12 times. Concurrent commonality occurred far less often, however. There was only one instance in which b and Ncl were concurrently PAGE 100 86 common, and only once were cn, b and Ncl concurrently common. The parameters bn and b were concurrently common twice, as were the pairs bn, Ncl and cn, b The pair bn, Ncl was concurrently common five times. Round Three Results for round three can be found in Table 4-26 throughTable 4-34. The groupings chosen for these analyses were ba sed in large part on the potential broader commonalities suggested by examining results from round two. In the nine analyses of round three, common bnÂ’s were observed twice, as were common cnÂ’s. Common b Â’s were observed three times, and common NclÂ’s were observed five times. Concurrent common pairs bn, b ; cn, b ; and cn, Ncl were observed once each. The concurrent common pair bn, Ncl was observed twice. Discussion of Commonality of Parameters Validity of the F-test The rare occurrence of negative F-values, a theoretical impossibility, in regression results for some groupings of data reveals a statistical quandary. This anomaly can be explained, but it brings into que stion the validity of the F-te st as it was employed in this study to test for common ality of parameters. The F-test, when used to test general models against more simplified ones, assumes that in each model all paramete rs are perfectly optimized, generating the lowest residual sum of squares possible using that model. Th is was the case in analyses performed by Ratkowsky (1983), which served as the basis fo r the approach used in this study. Even with perfect, least squares op timization of parameters, Ratk owsky (1983) points out that tests for invariance of parameters in non linear models are only approximate due to PAGE 101 87 intrinsic nonlinearity in each model, which can cause bias in the difference between the residual sum of squares, which is the basis of the F-test as used to test for parameter invariance. Perfect least squares optimization, moreove r, was technically not the case in the somewhat simplified regression approach us ed in this study, further bringing into question the validity of perfor ming an F-test based on the parameters estimates obtained. The regression approach used in this study (recall the Â“first methodÂ” in the previous chapter) first found values of An bn and cn based solely on N uptake data. The values of Ncl and b were then found using th e resulting value(s) of An and the N concentration (or specific N) and N uptake data. However, the re sidual sum of squares us ed for F-values in this study was based on the sum of normalized error sum of squares for all three data types: N uptake, N concentration, and dr y matter yield. To perfectly optimize all parameters for this residual sum of square s, the Â“second methodÂ” of regression (discussed in the previous chapter) woul d have to be employed, which was deemed impractical and prone to computationa l error compared to the first method. Strictly speaking, with th e method of regression employe d in this study, the only rigorous F-test that could be legitimately pe rformed would be on the first part of the regression using the N uptake data only; thus only commonality of An bn and cn could be tested, and the residual sum of squares would be based on the N uptake data only. This is because this was the only instance in this study in which parameters were truly optimized. The remaining two parameters depe nd, at least partiall y, on the resulting values of the first three (since An is carried over). Moreover, yield data were never directly considered in this regression method, although it was implicit in the N PAGE 102 88 concentration and N uptake data pairs. Finally normalization was not used in regression, but it was used in calculating the residual sum of squares, for reasons discussed in the previous chapter. Therefore, this met hod does not truly optimize any of the five parameters for the entire normalized data se t of all three data types, except under ideal conditions. It was assumed, however, that if the model fit the data reasonably well (e.g., if the results had R values very close to unity), then the resulting parameters were very close to truly optimized values, or at the least the rela tive differences in variances were very close to those if the parameters had been perfectly optimized. This assumption, however, was not rigorously tested. But with this, the resul ting F-values would have been very close to the F-values resulting from truly optimized pa rameters, and thus the F-test would be a reasonably valid test for comparing different commonality schemes. However, in cases where the model did a particularly Â“badÂ” j ob of describing the data (such as the N concentration model for 1982-3 tall fescue with broiler litter), it is perfectly conceivable that the resulting parameters and variances differed significantly from their perfectly optimized counterparts. This could have resu lted in significantly inaccurate F-values and, on occasion, in even nonsensical negative ones, which occurred when a simplified model with fewer parameters actually had a lower residual sum of squares than the more general model. It should also be noted that in the expe riment upon which this study is based, the unfertilized Â“checkÂ” treatment was not repeated for both broiler litter and commercial fertilizer experiments; only one check trea tment was conducted and counted as the zero N rate value for both fertilizer trials. This was accounted for in the degrees of freedom PAGE 103 89 calculation in analyses that included both broiler litter and commercial fertilizer by counting these Â“repeatedÂ” data points only once. Commonalities Suggested by the F-test The clearest occurrence of a co mmon parameter is that of Ncl, which passed individual commonality tests in 17 out of 23 data groupings. This is followed somewhat distantly by b (11 out of 23), cn (8 out of 23), and bn (5 out of 23). A summary of individual commonalities is listed in Table 4-35. There is strong evidence that Ncl is a function of crop type and reasonably good evidence it is a function of crop type only. This is most appare nt by the absence of Ncl commonality in Table 4-35 in the two tests wh ere both crops were gr ouped together; note this is the only instance of absence of Ncl commonality in the tests listed in the table. Evidence that it is a function of crop t ype only is also suggested by the test group containing all bermudagrass, in which Ncl, with a value of 13.1 g kg-1, was the only parameter that passed the commonality test. All tall fescue, however, did not pass the test, and there was a difference between the two common Ncl values of all irrigated tall fescue versus all nonirrigated tall fescue, 18.6 g kg-1 and 17.4 g kg-1, respectively. This difference, however, was only about 7%, and all tall fescue with commercial fertilizer (both irrigated and nonirrigated) did pass the commonality test, with an Ncl value of 17.5 g kg-1. It is possible the Â“oddÂ” results of the 19823 tall fescue with broiler litter impacted an overall test for Ncl for tall fescue. Evidence that any factor in this experiment other than crop type significantly impacts Ncl is quite limited. It is concluded, therefore, that Ncl is apparently independent of water availability and fertilizer source. PAGE 104 90 The factors that influence b are less apparent. It is po ssible it is unaffected by crop type and is affected by water availability, as suggested by the two test groups that included both crops in Table 4-35, in which b was the only parameter found common in each of the tests, with a value of 0.447 for the irrigated group and 0.589 for the nonirrigated group. This is cha llenged, however, by the separate commercial fertilizer test groups for bermudagrass and tall fescue (com bined irrigated and nonirrigated), both of which had common b Â’s that differed from each other (0.573 for bermudagrass and 0.465 for tall fescue, a difference of about 23%). From this latter test, water availability appears unimportant and instead b appeared to be a functi on of crop type. Separating commercial fertilizer irrigated from commer cial fertilizer nonirrigated groups of a specific crop type, however, resulted in common b Â’s within each group but differing between groups (0.404 for irrigated tall fe scue, 0.529 for nonirrigate d tall fescue, 0.493 for irrigated bermudagrass, and 0.639 for nonirrigated bermudagrass). The results for broiler litter were even less conclu sive. Overall results suggest that b may be affected by all three factors: crop type, water availability, and fertilizer source. Excluding the common value of cn for nonirrigated bermudagrass (0.514 ha kg-1), in all other cases where a common value of cn was observed, its value was within about 5% of the average value 0.00766 ha kg-1 (more precisely +6.5%, -3.3%). Common cn was observed twice as often in irrigated groups (4) compared to nonirrigated (2). Also, in the averaged data for each crop (consisting of two groups, irrigated and nonirrigated), a common cn was observed. Only two of the eight common cnÂ’s occurred in broiler litter groups. Based on these results, it is unlikely cn is a strong function of crop type, but there is some evidence that both fertilizer source and water availability can affect its value. PAGE 105 91 Conclusions concerning bn are perhaps the most unclear of the four parameters. Curiously, a common bn occurred in bermudagrass only in nonirrigated groups, while with tall fescue, a common bn occurred only in irrigated grou ps. A look at the individual test results (Table 4-2 through Table 4-9) reveals that in ev ery case for bermudagrass, the nonirrigated value of bn was greater than the irrigated value. Such a pattern was not observed with fescue, however, and the values of bn for fescue data sets tended to vary greatly. Evidence tend s to suggest that bn is influenced by all th ree factors: crop type, water availability, and fertilizer source, w ith water availability as possibly the most influential of the three. Note the above comments on commonali ty are confounded by the fact that concurrent commonality was rare. So in most cases where more than one parameter passed a single commonality test it was unclear which parame ter of the four to Â“chooseÂ” as being truly invariant. It is apparent that based on F-tests alone, conclusive evidence of commonality of multiple paramete rs among different data sets is limited in scope and certainty. Beyond the F-test: Commonality Based on Goodness of Fit The F-test is probably the best, most rigor ous test for commonality of parameters. However, from a more practical, engin eering standpoint, dismissing the idea of commonality based on the F-test alone may be in undue haste. A simplified model (with invariant parameters) may indeed produce re sults significantly diffe rent from a more general one, but this may not warrant its dismi ssal if it still does a reasonably good job of describing the data (i.e., if the correlation coefficients remain fa irly high). Ratkowsky (1983) does caution, however, against being misled on the goodness of a model based on examination of the residual sum of squares alone (as is done in when calculating PAGE 106 92 correlation coefficients) in the case of relativ ely small data sets. The applicability of a model to several data sets is an important f actor in such a consideration. These points are further discussed and illustra ted in the next chapter. PAGE 107 93 Table 4-1. Mode descriptions. Mode Common parameters (0) -(1a) bn (1b) cn (1c) b (1d) Ncl (2a) bn, cn (2b) bn, b (2c) bn, Ncl (2d) cn, b (2e) cn, Ncl (2f) b Ncl (3a) bn, cn, b (3b) bn, cn, Ncl (3c) bn, b Ncl (3d) cn, b Ncl (4) bn, cn b Ncl Table 4-2. Optimized parameters for bermudagrass with commercial fertilizer. Year Water An, Kg ha-1 bn cn, ha kg-1 b Ncl, g kg-1 1982 I 449 1.58 0.00827 0.552 13.2 1982 N 513 1.90 0.00723 0.630 13.2 1983 I 542 1.87 0.00839 0.444 14.2 1983 N 413 1.99 0.00809 0.569 13.0 1984 I 587 1.58 0.00566 0.488 13.6 1984 N 401 1.89 0.00863 0.727 11.7 1985 I 534 2.03 0.00714 0.494 13.5 1985 N 480 2.47 0.00886 0.637 12.6 Avg I 519 1.75 0.00735 0.481 13.7 Avg N 451 2.05 0.00812 0.642 12.6 Avg I/N 485 1.88 0.00768 0.566 13.1 Notes: N = nonirrigated, I=irri gated, I/N = average of bot h irrigated a nd nonirrigated data; Â“AvgÂ” means data at a given N applic ation rate were averaged over all years PAGE 108 94 Table 4-3. Statistics for optimized para meters for bermudagrass with commercial fertilizer. Year Water RNu RNc RY SE ( An) SE ( bn) SE ( cn) SE (b ) SE ( Ncl) 1982 I 0.9948 0.98240.98342.9 8.4 10.0 2.6 1.2 1982 N 0.9984 0.99610.99422.3 4.1 5.7 1.4 0.7 1983 I 0.9970 0.93450.99722.2 6.1 7.0 3.6 1.3 1983 N 0.9904 0.97740.99174.9 10.6 13.8 1.1 0.5 1984 I 0.9840 0.85540.986310.9 11.0 20.5 4.7 1.7 1984 N 0.9891 0.98930.97954.2 11.5 12.8 1.3 0.8 1985 I 0.9906 0.94910.98666.5 9.5 14.6 3.2 1.2 1985 N 0.9923 0.95320.98664.4 10.4 12.6 3.5 1.8 Avg I 0.9952 0.98760.99453.6 7.2 9.8 1.1 0.4 Avg N 0.9967 0.99080.99432.8 6.3 7.8 1.0 0.5 Avg I/N 0.9971 0.99150.99502.7 5.7 7.4 0.9 0.4 Note: SE Â’s, standard errors, are in percent (%) of the parameter value. Table 4-4. Optimized parameters fo r bermudagrass with broiler litter. Year Water An, kg ha-1 bn cn, ha kg-1 b Ncl, g kg-1 1982 I 284 1.27 0.00677 0.153 14.5 1982 N 428 1.77 0.00350 0.447 13.5 1983 I 429 1.75 0.00679 0.430 13.5 1983 N 370 2.05 0.00603 0.473 13.3 1984 I 273 1.39 0.01016 0.409 13.5 1984 N 303 2.19 0.00767 0.605 11.9 1985 I 240 1.84 0.00892 0.302 13.5 1985 N 215 2.32 0.00620 0.463 11.8 Table 4-5. Statistics for optimized parame ters for bermudagrass with broiler litter. Year Water RNu RNc RY SE ( An) SE ( bn) SE ( cn) SE (b ) SE ( Ncl) 1982 I 0.9991 0.87470.9970 3.0 6.3 8.0 11.0 1.2 1982 N 0.9931 0.99350.9869 52.5 29.1 35.3 3.9 0.9 1983 I 0.999997 0.99720.999830.3 0.3 0.5 1.5 0.4 1983 N 0.9885 0.92420.9900 38.8 17.7 37.6 5.3 1.5 1984 I 0.9932 0.99410.9889 6.7 18.9 20.0 3.0 1.0 1984 N 0.9998 0.90450.9859 3.6 1.9 4.3 9.5 3.8 1985 I 0.9933 0.98180.9868 18.7 13.2 26.2 6.1 1.1 1985 N 0.9983 0.99330.9968 35.6 13.3 18.1 2.7 0.5 PAGE 109 95 Table 4-6. Optimized parameters for ta ll fescue with commercial fertilizer. Year Water An,k ha-1 bn cn, ha kg-1 b Ncl, g kg-1 1981-2 I 454 1.46 0.00759 0.423 18.4 1981-2 N 370 1.65 0.00800 0.458 18.7 1982-3 I 397 1.88 0.00743 0.457 17.3 1982-3 N 239 1.52 0.00659 0.581 15.7 1983-4 I 449 1.54 0.00789 0.323 18.5 1983-4 N 145 1.17 0.01726 0.582 15.9 Avg I 435 1.59 0.00749 0.397 18.1 Avg N 248 1.36 0.00779 0.517 17.2 Avg I/N 342 1.49 0.00752 0.456 17.7 Table 4-7. Statistics for optimized parameters for tall fescue with commercial fertilizer. Year Water RNu RNc RY SE ( An) SE ( bn) SE ( cn) SE (b ) SE ( Ncl) 1981-2 I 0.9995 0.96700.99760.9 2.5 2.8 2.7 0.9 1981-2 N 0.9954 0.93030.98342.8 7.6 8.9 4.4 1.7 1982-3 I 0.9879 0.87860.98535.2 12.0 14.4 5.1 1.8 1982-3 N 0.9750 0.95140.95478.1 16.3 21.0 3.0 1.4 1983-4 I 0.9970 0.96920.99662.1 6.2 7.1 2.5 0.6 1983-4 N 0.9846 0.97030.97682.1 20.2 14.1 1.8 1.0 Avg I 0.9970 0.95550.99572.3 6.0 7.0 3.1 1.0 Avg N 0.9965 0.98010.98752.2 6.8 7.6 2.2 1.0 Avg I/N 0.9981 0.97840.99491.8 4.8 5.5 2.2 0.8 Table 4-8. Optimized parameters fo r tall fescue with broiler litter. Year Water An, kg ha-1 bn cn, ha kg-1 b Ncl, g kg-1 1981-2 I 388 1.22 0.00354 0.277 20.3 1981-2 N 708 2.41 0.00297 0.680 20.0 1982-3 I 248 0.97 0.00521 0.074 20.5 1982-3 N 126 1.00 0.00656 0.175 17.5 1983-4 I 399 1.46 0.00665 0.244 18.8 1983-4 N 119 0.80 0.00647 0.427 16.2 PAGE 110 96 Table 4-9. Statistics for optimized parame ters for tall fescue with broiler litter. Year Water RNu RNc RY SE ( An) SE ( bn) SE ( cn) SE (b ) SE ( Ncl) 1981-2 I 0.9976 0.9686 0.9964 17.6 15.6 19.0 4.0 0.6 1981-2 N 0.9990 0.9254 0.9917 52.6 21.8 15.9 7.6 1.9 1982-3 I 0.9897 0.5688 0.9800 21.3 26.6 35.9 30.5 1.4 1982-3 N 0.9960 0.6805 0.9988 7.3 13.9 17.0 15.5 1.9 1983-4 I 0.99997 0.9978 0.99986 1.2 1.0 1.8 1.4 0.2 1983-4 N 0.99996 0.9561 0.9949 0.9 1.7 1.9 4.1 1.3 Table 4-10. Arithmetic average of para meters over select data groups Data Group Description An bn cn b Ncl Irrigated bermuda, commercial fertilizer 528 1.77 0.00737 0.495 13.6 Nonirrigated bermuda, commercial fertilizer 452 2.06 0.00820 0.641 12.6 Irrigated bermuda, broiler litter 309 1.56 0.00816 0.324 13.8 Nonirrigated bermuda, broiler litter 329 2.08 0.00585 0.497 12.6 Irrigated fescue, commercial fertilizer 433 1.63 0.00764 0.401 18.1 Nonirrigated fescue, commercial fertilizer 251 1.45 0.01062 0.540 16.8 Irrigated fescue, broiler litter 345 1.22 0.00513 0.198 19.9 Nonirrigated fescue, broiler litter 318 1.40 0.00533 0.427 17.9 Bermuda, commercial fertilizer 490 1.91 0.00778 0.568 13.1 Bermuda, broiler litter 319 1.82 0.00701 0.410 13.2 Fescue, commercial fertilizer 342 1.54 0.00913 0.471 17.4 Fescue, broiler litter 331 1.31 0.00523 0.313 18.9 Irrigated bermuda 418 1.66 0.00776 0.409 13.7 Nonirrigated bermuda 390 2.07 0.00703 0.569 12.6 Irrigated fescue 389 1.42 0.00639 0.300 19.0 Nonirrigated fescue 285 1.43 0.00798 0.484 17.3 All commercial fertilizer 427 1.75 0.00836 0.526 15.0 All broiler litter 324 1.60 0.00625 0.369 15.6 All irrigated 406 1.56 0.00717 0.362 16.0 All nonirrigated 345 1.80 0.00743 0.532 14.6 All bermuda 404 1.87 0.00739 0.489 13.2 All fescue 337 1.42 0.00718 0.392 18.2 All data 375 1.68 0.00730 0.447 15.3 PAGE 111 97 Table 4-11. Standard deviations of pa rameters over select data groups Data Group Description SD ( An) SD ( bn) SD ( cn) SD (b ) SD ( Ncl) Irrigated bermuda, commercial fertilizer 11 13 17 9 3 Nonirrigated bermuda, commercial fertilizer 12 13 9 10 5 Irrigated bermuda, broiler litter 26 18 20 39 4 Nonirrigated bermuda, broiler litter 28 11 30 15 7 Irrigated fescue, commercial fertilizer 7 14 3 17 4 Nonirrigated fescue, commercial fertilizer 45 17 55 13 10 Irrigated fescue, broiler litter 24 20 30 55 5 Nonirrigated fescue, broiler litter 106 63 38 59 11 Bermuda, commercial fertilizer 13 15 14 16 6 Bermuda, broiler litter 25 20 29 32 7 Fescue, commercial fertilizer 36 15 44 21 8 Fescue, broiler litter 67 45 31 69 9 Irrigated bermuda 32 15 19 31 3 Nonirrigated bermuda 24 11 25 18 6 Irrigated fescue 19 22 27 46 6 Nonirrigated fescue 80 41 61 37 10 All commercial fertilizer 28 18 32 20 16 All broiler litter 46 33 32 47 20 All irrigated 27 19 23 38 18 All nonirrigated 48 29 44 26 18 All bermuda 28 17 22 28 6 All fescue 51 31 50 46 9 All data 38 26 35 36 18 Note: SD Â’s, standard deviations, are in percen t (%) of the average parameter value. PAGE 112 98 Table 4-12. Analysis of variance for para meters for irrigated bermudagrass with commercial fertilizer. Mode Common parameters No. Parameters Estimated Degrees of freedom Residual Sum of Squares F Critical F (90%) (0) -20 64 0.1400 (1a) bn = 1.77 17 67 0.1600 (1a)-(0) 3 0.0200 3.05 2.17 (1b) cn = 0.00736 17 67 0.1565 (1b)-(0) 3 0.0165 2.51 2.17 (1c) b = 0.493 17 67 0.1443 (1c)-(0) 3 0.0043 0.66 2.17 (1d) Ncl = 13.6 17 67 0.1432 (1d)-(0) 3 0.0032 0.49 2.17 (2d) cn = 0.00736 b = 0.489 14 70 0.1621 (2d)-(0) 6 0.0221 1.68 1.87 (2e) cn = 0.00736 Ncl = 13.6 14 70 0.1595 (2e)-(0) 6 0.0195 1.49 1.87 (2f) b = 0.497 Ncl = 13.6 14 70 0.1464 (2f)-(0) 6 0.0064 0.49 1.87 (3d) cn = 0.00736 b = 0.491 Ncl = 13.6 11 73 0.1798 (3d)-(0) 9 0.0398 2.02 1.73 (4) bn = 1.75 cn = 0.00736 b = 0.486 Ncl = 13.7 8 76 0.2301 (4)-(0) 12 0.0901 3.43 1.65 Note: This table includes extra analyses fo r illustration only, as discussed in the Â“Overview of ResultsÂ” section of this chapter. PAGE 113 99 Table 4-13. Analysis of variance for para meters for nonirrigated bermudagrass with commercial fertilizer. Mode Common parameters No. Parameters Estimated Degrees of freedom Residual Sum of Squares F Critical F (90%) (0) -20 64 0.1332 (1a) bn = 2.07 17 67 0.1514 (1a)-(0) 3 0.0182 2.91 2.17 (1b) cn = 0.00816 17 67 0.1386 (1b)-(0) 3 0.0054 0.86 2.17 (1c) b = 0.639 17 67 0.1378 (1c)-(0) 3 0.0046 0.74 2.17 (1d) Ncl = 12.6 17 67 0.1371 (1d)-(0) 3 0.0039 0.62 2.17 (3d) cn = 0.00816 b = 0.638 Ncl = 12.6 11 73 0.1538 (3d)-(0) 9 0.0206 1.10 1.73 Table 4-14. Analysis of variance for para meters for bermudagrass with commercial fertilizer (both irrigated and nonirrigated). Mode Common parameters No. Parameters Estimated Degrees of freedom Residual Sum of Squares F Critical F (90%) (0) -40 128 0.2732 (1a) bn = 1.90 33 135 0.3332 (1a)-(0) 7 0.0600 4.01 1.76 (1b) cn = 0.00773 33 135 0.3003 (1b)-(0) 7 0.0271 1.81 1.76 (1c) b = 0.573 33 135 0.2993 (1c)-(0) 7 0.0261 1.75 1.76 (1d) Ncl = 13.1 33 135 0.2867 (1d)-(0) 7 0.0135 0.90 1.76 (2f) b = 0.571 Ncl = 13.1 26 142 0.3438 (2f)-(0) 14 0.0706 2.36 1.56 PAGE 114 100 Table 4-15. Analysis of variance for paramete rs for average of bermudagrass data with commercial fertilizer (split into tw o groups: nonirrigated and irrigated). Mode Common parameters No. Parameters Estimated Degrees of freedom Residual Sum of Squares F Critical F (90%) (0) -10 32 0.02919 (1a) bn = 1.88 9 33 0.03499 (1a)-(0) 1 0.0058 6.36 2.86 (1b) cn = 0.00770 9 33 0.03022 (1b)-(0) 1 0.00103 1.13 2.86 (1c) b = 0.568 9 33 0.03459 (1c)-(0) 1 0.0054 5.92 2.86 (1d) Ncl = 13.1 9 33 0.03108 (1d)-(0) 1 0.00189 2.07 2.86 (2e) cn = 0.00770 Ncl = 13.1 8 34 0.03196 (2e)-(0) 2 0.00277 1.52 2.48 Table 4-16. Analysis of variance for paramete rs for irrigated bermudagrass with broiler litter. Mode Common parameters No. Parameters Estimated Degrees of freedom Residual Sum of Squares F Critical F (90%) (0) -20 28 0.02811 (1a) bn = 1.62 17 31 0.04934 (1a)-(0) 3 0.02123 7.05 2.29 (1b) cn = 0.00767 17 31 0.03269 (1b)-(0) 3 0.00458 1.52 2.29 (1c) b = 0.331 17 31 0.04041 (1c)-(0) 3 0.0123 4.08 2.29 (1d) Ncl = 13.7 17 31 0.02937 (1d)-(0) 3 0.00126 0.42 2.29 (2e) cn = 0.00767 Ncl = 13.7 14 34 0.03401 (2e)-(1) 6 0.00593 0.98 2.00 PAGE 115 101 Table 4-17. Analysis of variance for para meters for nonirrigated bermudagrass with broiler litter. Mode Common parameters No. Parameters Estimated Degrees of freedom Residual Sum of Squares F Critical F (90%) (0) -20 28 0.04273 (1a) bn = 2.11 17 31 0.04443 (1a)-(0) 3 0.0017 0.37 2.29 (1b) cn = 0.00532 17 31 0.04696 (1b)-(0) 3 0.00423 0.92 2.29 (1c) b = 0.514 17 31 0.04443 (1c)-(0) 3 0.0017 0.37 2.29 (1d) Ncl = 12.6 17 31 0.04850 (1d)-(0) 3 0.00577 1.26 2.29 (4) bn = 1.94 cn = 0.00601 b = 0.492 Ncl = 12.4 8 40 0.1438 (4)-(0) 12 0.1011 5.52 1.79 (3a) bn = 1.94 cn = 0.00601 b = 0.475 11 37 0.09027 (3a)-(0) 9 0.04754 3.46 1.87 (3b) bn = 1.94 cn = 0.00601 Ncl = 12.6 11 37 0.08714 (3b)-(0) 9 0.04441 3.23 1.87 (3c) bn = 2.11 b = 0.483 Ncl = 12.8 11 37 0.1270 (3c)-(0) 9 0.08427 6.14 1.87 (3d) cn = 0.00532 b = 0.480 Ncl = 13.0 11 37 0.1177 (3d)-(0) 9 0.0750 5.46 1.87 (2b) bn = 2.11 b = 0.524 14 34 0.04569 (2b)-(0) 6 0.00296 0.32 2.00 (2c) bn = 2.11 Ncl = 12.6 14 34 0.04777 (2c)-(0) 6 0.00504 0.55 2.00 (2e) cn = 0.00532 Ncl = 12.6 14 34 0.05239 (2e)-(0) 6 0.00966 1.05 2.00 PAGE 116 102 Table 4-18. Analysis of variance for paramete rs for bermudagrass with broiler litter (both irrigated and nonirrigated). Mode Common parameters No. Parameters Estimated Degrees of freedom Residual Sum of Squares F Critical F (90%) (0) -40 56 0.07084 (1a) bn = 1.79 33 63 0.1331 (1a)-(0) 7 0.0623 7.04 1.83 (1b) cn = 0.00674 33 63 0.1073 (1b)-(0) 7 0.0365 4.12 1.83 (1c) b = 0.407 33 63 0.09836 (1c)-(0) 7 0.02752 3.11 1.83 (1d) Ncl = 13.1 33 63 0.08635 (1d)-(0) 7 0.01551 1.75 1.83 PAGE 117 103 Table 4-19. Analysis of variance for parameters for irrigated tall fescue with commercial fertilizer. Mode Common parameters No. Parameters Estimated Degrees of freedom Residual Sum of Squares F Critical F (90%) (0) -15 48 0.06978 (1a) bn = 1.61 13 50 0.07606 (1a)-(0) 2 0.00628 2.16 2.42 (1b) cn = 0.00765 13 50 0.07119 (1b)-(0) 2 0.00141 0.49 2.42 (1c) b = 0.404 13 50 0.07606 (1c)-(0) 2 0.00628 2.16 2.42 (1d) Ncl = 18.0 13 50 0.07229 (1d)-(0) 2 0.00251 0.86 2.42 (4) bn = 1.58 cn = 0.00751 b = 0.402 Ncl = 18.0 7 56 0.1299 (4)-(0) 8 0.06012 5.17 1.80 (2a) bn = 1.58 cn = 0.00751 11 52 0.09488 (2a)-(0) 4 0.02510 4.32 2.07 (2b) bn = 1.61 b = 0.404 11 52 0.08187 (2b)-(0) 4 0.01209 2.08 2.07 (2c) bn = 1.61 Ncl = 18.0 11 52 0.07768 (2c)-(0) 4 0.0079 1.36 2.07 (2d) cn = 0.00765 b = 0.404 11 52 0.07707 (2d)-(0) 4 0.00729 1.25 2.07 (2e) cn = 0.00765 Ncl = 18.0 11 52 0.07367 (2e)-(1) 4 0.00389 0.67 2.07 (2f) b = 0.490 Ncl = 17.2 11 52 0.1072 (2f)-(0) 4 0.0374 6.43 2.07 PAGE 118 104 Table 4-20. Analysis of variance for parame ters for nonirrigated tall fescue with commercial fertilizer. Mode Common parameters No. Parameters Estimated Degrees of freedom Residual Sum of Squares F Critical F (90%) (0) -15 48 0.1499 (1a) bn = 1.59 13 50 0.1683 (1a)-(0) 2 0.0184 2.95 2.42 (1b) cn = 0.00803 13 50 0.2459 (1b)-(0) 2 0.0960 15.4 2.42 (1c) b = 0.529 13 50 0.1527 (1c)-(0) 2 0.0028 0.45 2.42 (1d) Ncl = 16.9 13 50 0.1577 (1d)-(0) 2 0.0078 1.25 2.42 (2f) b = 0.527 Ncl = 16.9 11 52 0.1786 (2f)-(0) 4 0.0287 2.30 2.07 Table 4-21. Analysis of variance for parameters for tall fescue with commercial fertilizer (both irrigated and nonirrigated). Mode Common parameters No. Parameters Estimated Degrees of freedom Residual Sum of Squares F Critical F (90%) (0) -30 96 0.2196 (1a) bn = 1.60 25 101 0.2456 (1a)-(0) 5 0.0260 2.27 1.91 (1b) cn = 0.00775 25 101 0.3201 (1b)-(0) 5 0.1005 8.79 1.91 (1c) b = 0.465 25 101 0.2410 (1c)-(0) 5 0.0214 1.87 1.91 (1d) Ncl = 17.5 25 101 0.2350 (1d)-(0) 5 0.0154 1.35 1.91 (2f) b = 0.473 Ncl = 17.4 20 106 0.3205 (2f)-(1) 10 0.1009 4.41 1.67 PAGE 119 105 Table 4-22. Analysis of variance for paramete rs for average of tall fescue data with commercial fertilizer (split into tw o groups: nonirrigated and irrigated). Mode Common parameters No. Parameters Estimated Degrees of freedom Residual Sum of Squares F Critical F (90%) (0) -10 32 0.03189 (1a) bn = 1.53 9 33 0.03486 (1a)-(0) 1 0.00297 2.98 2.87 (1b) cn = 0.00756 9 33 0.03259 (1b)-(0) 1 0.00070 0.70 2.87 (1c) b = 0.456 9 33 0.03532 (1c)-(0) 1 0.00343 3.44 2.87 (1d) Ncl = 17.7 9 33 0.03290 (1d)-(0) 1 0.00101 1.01 2.87 (2e) cn = 0.00756 Ncl = 17.7 8 34 0.03358 (2e)-(0) 2 0.00169 0.85 2.48 PAGE 120 106 Table 4-23. Analysis of variance for parameters for irrigated tall fescue with broiler litter. Mode Common parameters No. Parameters Estimated Degrees of freedom Residual Sum of Squares F Critical F (90%) (0) -15 21 0.01580 (1a) bn = 1.41 13 23 0.01252 (1a)-(0) 2 -0.00328 -2.18* 2.58 (1b) cn = 0.00506 13 23 0.02389 (1b)-(0) 2 0.00809 5.38 2.58 (1c) b = 0.200 13 23 0.01916 (1c)-(0) 2 0.00336 2.23 2.58 (1d) Ncl = 19.7 13 23 0.01789 (1d)-(0) 2 0.00209 1.39 2.58 (3c) bn = 1.41 b = 0.185 Ncl = 20.3 6 30 0.04341 (3c)-(0) 9 0.02761 4.08 1.95 (2b) bn = 1.41 b = 0.236 11 25 0.1372 (2b)-(0) 4 -0.00208 -0.69* 2.23 (2c) bn = 1.41 Ncl = 19.7 11 25 0.01427 (2c)-(0) 4 -0.00153 -0.51 2.23 (2f) b = 0.160 Ncl = 20.2 11 25 0.05279 (2f)-(0) 4 0.03699 12.3 2.23 *These anomalies are discussed in the Â“Valid ity of the F-testÂ” section of this chapter Table 4-24. Analysis of variance for parameters for nonirrigated tall fescue with broiler litter. Mode Common parameters No. Parameters Estimated Degrees of freedom Residual Sum of Squares F Critical F (90%) (0) -15 21 0.01528 (1a) bn = 1.58 13 23 0.1336 (1a)-(0) 2 0.1183 81.29 2.58 (1b) cn = 0.00365 13 23 0.02800 (1b)-(0) 2 0.01272 8.74 2.58 (1c) b = 0.379 13 23 0.02911 (1c)-(0) 2 0.1383 9.50 2.58 (1d) Ncl = 18.3 13 23 0.02351 (1d)-(0) 2 0.00823 5.66 2.58 PAGE 121 107 Table 4-25. Analysis of variance for paramete rs for tall fescue with broiler litter (both irrigated and nonirrigated). Mode Common parameters No. Parameters Estimated Degrees of freedom Residual Sum of Squares F Critical F (90%) (0) -30 42 0.03108 (1a) bn = 1.50 25 47 0.06900 (1a)-(0) 5 0.03792 10.25 1.99 (1b) cn = 0.00458 25 47 0.05215 (1b)-(0) 5 0.02107 5.69 1.99 (1c) b = 0.273 25 47 0.05372 (1c)-(0) 5 0.02264 6.12 1.99 (1d) Ncl = 19.0 25 47 0.04851 (1d)-(0) 5 0.01743 4.71 1.99 Table 4-26. Analysis of variance for se lect parameters for all bermudagrass. Mode Common parameters No. Parameters Estimated Degrees of freedom Residual Sum of Squares F Critical F (90%) (0) -80 176 0.3440 (1b) cn = 0.00761 65 191 0.4284 (1b)-(0) 15 0.0844 2.88 1.53 (1c) b = 0.531 65 191 0.4308 (1c)-(0) 15 0.0868 2.96 1.53 (1d) Ncl = 13.1 65 191 0.3731 (1d)-(0) 15 0.0291 0.99 1.53 Table 4-27. Analysis of vari ance for select parameters fo r all irrigated bermudagrass. Mode Common parameters No. Parameters Estimated Degrees of freedom Residual Sum of Squares F Critical F (90%) (0) -40 88 0.1681 (1a) bn = 1.75 33 95 0.2179 (1a)-(0) 7 0.0498 3.72 1.79 (1b) cn = 0.00741 33 95 0.1892 (1b)-(0) 7 0.0211 1.58 1.79 (1c) b = 0.445 33 95 0.2040 (1c)-(0) 7 0.0359 2.68 1.79 (1d) Ncl = 13.7 33 95 0.1726 (1d)-(0) 7 0.0045 0.34 1.79 (2e) cn = 0.00741 Ncl = 13.7 26 102 0.1934 (2e)-(0) 14 0.0253 0.95 1.58 PAGE 122 108 Table 4-28. Analysis of varian ce for select parameters for all nonirrigated bermudagrass. Mode Common parameters No. Parameters Estimated Degrees of freedom Residual Sum of Squares F Critical F (90%) (0) -40 88 0.1759 (1a) bn = 2.07 33 95 0.1952 (1a)-(0) 7 0.0193 1.38 1.79 (1b) cn = 0.00786 33 95 0.2425 (1b)-(0) 7 0.0666 4.76 1.79 (1c) b = 0.612 33 95 0.1900 (1c)-(0) 7 0.0141 1.01 1.79 (1d) Ncl = 12.6 33 95 0.1858 (1d)-(0) 7 0.0099 0.71 1.79 (3c) bn = 2.07 b = 0.650 Ncl = 12.3 19 109 0.3763 (3c)-(0) 21 0.2004 4.77 1.50 (2b) bn = 2.07 b = 0.613 26 102 0.2128 (2b)-(0) 14 0.0369 1.32 1.58 (2c) bn = 2.07 Ncl = 12.6 26 102 0.2051 (2c)-(0) 14 0.0292 1.04 1.58 (2f) b = 0.668 Ncl = 12.1 26 102 0.2775 (2f)-(0) 14 0.1016 3.63 1.58 Table 4-29. Analysis of variance for select parameters for all fescue. Mode Common parameters No. Parameters Estimated Degrees of freedom Residual Sum of Squares F Critical F (90%) (0) -60 132 0.2507 (1a) bn = 1.58 49 143 0.3889 (1a)-(0) 11 0.1382 6.62 1.62 (1b) cn = 0.00722 49 143 0.4530 (1b)-(0) 11 0.2023 9.68 1.62 (1c) b = 0.423 49 143 0.3244 (1c)-(0) 11 0.0737 3.53 1.62 (1d) Ncl = 18.0 49 143 0.2905 (1d)-(0) 11 0.0392 1.88 1.62 PAGE 123 109 Table 4-30. Analysis of variance for se lect parameters for irrigated fescue. Mode Common parameters No. Parameters Estimated Degrees of freedom Residual Sum of Squares F Critical F (90%) (0) -30 66 0.08559 (1a) bn = 1.57 25 71 0.08785 (1a)-(0) 5 0.00226 0.35 1.94 (1b) cn = 0.00725 25 71 0.1271 (1b)-(0) 5 0.04151 6.40 1.94 (1c) b = 0.354 25 71 0.1128 (1c)-(0) 5 0.0272 4.19 1.94 (1d) Ncl = 18.6 25 71 0.09750 (1d)-(0) 5 0.01191 1.84 1.94 (2c) bn = 1.57 Ncl = 18.6 20 76 0.09663 (2c)-(0) 10 0.01104 0.85 1.70 Table 4-31. Analysis of variance for sele ct parameters for nonirrigated fescue. Mode Common parameters No. Parameters Estimated Degrees of freedom Residual Sum of Squares F Critical F (90%) (0) -30 66 0.1651 (1a) bn = 1.59 25 71 0.3041 (1a)-(0) 5 0.1390 11.11 1.94 (1b) cn = 0.00713 25 71 0.3260 (1b)-(0) 5 0.1609 12.86 1.94 (1c) b = 0.500 25 71 0.1933 (1c)-(0) 5 0.0282 2.25 1.94 (1d) Ncl = 17.4 25 71 0.1822 (1d)-(0) 5 0.0171 1.37 1.94 PAGE 124 110 Table 4-32. Analysis of variance for select parameters for all irrigated samples. Mode Common parameters No. Parameters Estimated Degrees of freedom Residual Sum of Squares F Critical F (90%) (0) -70 154 0.2537 (1a) bn = 1.70 57 167 0.3095 (1a)-(0) 13 0.0558 2.61 1.57 (1b) cn = 0.00735 57 167 0.3189 (1b)-(0) 13 0.0652 3.04 1.57 (1c) b = 0.399 57 167 0.3239 (1c)-(0) 13 0.0702 3.28 1.57 (1d) Ncl = 15.5 57 167 0.4425 (1d)-(0) 13 0.1888 8.82 1.57 Table 4-33. Analysis of vari ance for select parameters for all irrigated samples with commercial fertilizer. Mode Common parameters No. Parameters Estimated Degrees of freedom Residual Sum of Squares F Critical F (90%) (0) -35 112 0.2098 (1a) bn = 1.72 29 118 0.2441 (1a)-(0) 6 0.0343 3.05 1.83 (1b) cn = 0.00747 29 118 0.2266 (1b)-(0) 6 0.0168 1.49 1.83 (1c) b = 0.447 29 118 0.2260 (1c)-(0) 6 0.0162 1.44 1.83 (1d) Ncl = 15.4 29 118 0.2967 (1d)-(0) 6 0.0869 7.73 1.83 (2d) cn = 0.00747 b = 0.445 23 124 0.2423 (2d)-(0) 12 0.0325 1.45 1.61 PAGE 125 111 Table 4-34. Analysis of varian ce for select parameters for all nonirrigated samples with commercial fertilizer. Mode Common parameters No. Parameters Estimated Degrees of freedom Residual Sum of Squares F Critical F (90%) (0) -35 112 0.2830 (1a) bn = 1.99 29 118 0.3850 (1a)-(0) 6 0.1020 6.73 1.83 (1b) cn = 0.00814 29 118 0.3842 (1b)-(0) 6 0.1012 6.68 1.83 (1c) b = 0.589 29 118 0.2941 (1c)-(0) 6 0.0111 0.73 1.83 (1d) Ncl = 14.1 29 118 0.3488 (1d)-(0) 6 0.0658 4.34 1.83 Table 4-35. Summary of re sults for single parameter commonality F-test Data Group Description bn cn b Ncl All bermudagrass 13.1 Both crops, irrigated, commercial fertilizer 0.00747 0.447 Both crops, nonirrigated, comme rcial fertilizer 0.589 Irrigated bermudagrass 0.00741 13.7 Nonirrigated bermudagrass 2.07 0.612 12.6 Irrigated tall fescue 1.57 18.6 Nonirrigated tall fescue 17.4 Bermudagrass, commercial fertilizer 0.573 13.1 Irrigated bermudagrass, commercial fertilizer 0.493 13.6 Nonirrigated bermudagrass, commercial fertilizer 0.00816 0.639 12.6 Bermudagrass, broiler litter 13.1 Irrigated bermudagrass, broiler litter 0.00767 13.7 Nonirrigated bermudagrass, broi ler litter 2.11 0.00532 0.514 12.6 Tall fescue, commercial fertilizer 0.465 17.5 Irrigated tall fescue, commercial fertilizer 1.61 0.00765 0.404 18.0 Nonirrigated tall fescue, comme rcial fertilizer 0.529 16.9 Irrigated tall fescue, br oiler litter 1.41 0.200 19.7 Bermudagrass, commercial fertilizer, across year average 0.00770 13.1 Tall fescue, commercial fertilizer, across year average 0.00756 17.7 PAGE 126 112 N, kg ha-1 0100200300400500600700800 Y, Mg ha-1 0 10 20 Nu, kg ha-1 0 200 400 c, g kg-1 0 10 20 30 Data Figure 4-1. Dry matter yield (Y), N uptake (Nu), and N concentration (Nc) versus N application rate (N) fo r 1982 nonirrigated bermud agrass with commercial fertilizer.This was the data set with th e best overall model fit in this category of data. Model lines are drawn using Eqs. [3-261] through [3-263] based on parameters listed in Table 4-2. PAGE 127 113 Nu, kg ha-1 0100200300400500600 Y, Mg ha-1 0 10 20 Nc g kg-1 0 10 20 30 Data Figure 4-2. Phase plot for 1982 nonirrigated be rmudagrass with commercial fertilizer. Model lines are drawn using Eq. [3-7 ] and Eq. [2-2] based on parameters listed in Table 4-2. PAGE 128 114 N, kg ha-1 0100200300400500600700800 Y, Mg ha-1 0 10 20 Nu, kg ha-1 0 200 400 c, g kg-1 0 10 20 30 Data Figure 4-3. Dry matter yield, N uptake, and N concentration ve rsus N application rate for 1984 irrigated bermudagrass with commerci al fertilizer. This was the data set with the poorest overall model fit in this category of data. Model lines are drawn using Eqs. [3-261] through [3-263] based on parameters listed in Table 4-2. PAGE 129 115 Nu, kg ha-1 0100200300400500600 Y, Mg ha-1 0 10 20 Nc g kg-1 0 10 20 30 Data Figure 4-4. Phase plot for 1984 irrigated berm udagrass with commercial fertilizer. Model lines are drawn using Eq. [3-7] and Eq [2-2] based on parameters listed in Table 4-2. PAGE 130 116 N, kg ha-1 0100200300400500600700800 Y, Mg ha-1 0 10 20 Nu, kg ha-1 0 200 400 c, g kg-1 0 10 20 30 Irrigated Nonirrigated Figure 4-5. Dry matter yield, N uptake, and N concentration ve rsus N application rate for the average of all irrigated and nonirr igated bermudagrass with commercial fertilizer. Model lines are drawn usin g Eqs. [3-261] through [3-263] based on parameters listed in Table 4-2. PAGE 131 117 Nu, kg ha-1 0100200300400500600 Y, Mg ha-1 0 10 20 Nc g kg-1 0 10 20 30 Irrigated Nonirrigated Figure 4-6. Phase plot for average of all irrigated and nonirrigated bermudagrass with commercial fertilizer. Model lines are drawn using Eq. [3-7] and Eq. [2-2] based on parameters listed in Table 4-2. PAGE 132 118 N, kg ha-1 0100200300400500600700800 Y, Mg ha-1 0 10 20 Nu, kg ha-1 0 200 400 c, g kg-1 0 10 20 30 Data Figure 4-7. Dry matter yield, N uptake, and N concentration vers us N application rate for 1983 irrigated bermudagrass with broiler li tter. This was the data set with the best overall model fit in this category of data. Model lines are drawn using Eqs. [3-261] through [3-263] based on parameters listed in Table 4-4. PAGE 133 119 Nu, kg ha-1 0100200300400500600 Y, Mg ha-1 0 10 20 Nc g kg-1 0 10 20 30 Data Figure 4-8. Phase plot for 1983 irrigated bermud agrass with broiler li tter. Model lines are drawn using Eq. [3-7] and Eq. [2-2] base d on parameters listed in Table 4-4. PAGE 134 120 N, kg ha-1 0100200300400500600700800 Y, Mg ha-1 0 10 20 Nu, kg ha-1 0 200 400 c, g kg-1 0 10 20 30 Data Figure 4-9. Dry matter yield, N uptake, and N concentration vers us N application rate for 1982 irrigated bermudagrass with broiler li tter. This was the data set with the poorest overall model fit in this category of data. M odel lines are drawn using Eqs. [3-261] through [3-263] based on parameters listed in Table 4-4. PAGE 135 121 Nu, kg ha-1 0100200300400500600 Y, Mg ha-1 0 10 20 Nc g kg-1 0 10 20 30 Data Figure 4-10. Phase plot for 1982 irrigated berm udagrass with broiler litter. Model lines are drawn using Eq. [3-7] and Eq. [2-2] based on parameters listed in Table 44. PAGE 136 122 N, kg ha-1 0100200300400500600700800 Y, Mg ha-1 0 10 20 Nu, kg ha-1 0 200 400 c, g kg-1 0 10 20 30 Data Figure 4-11. Dry matter yield, N uptake, and N concentration versus N application rate for 1981-2 irrigated tall fescue with comm ercial fertilizer. This was the data set with the best overall model fit in this category of data. Model lines are drawn using Eqs. [3-261] through [3-263] based on parameters listed in Table 4-6. PAGE 137 123 Nu, kg ha-1 0100200300400500600 Y, Mg ha-1 0 10 20 Nc g kg-1 0 10 20 30 Data Figure 4-12. Phase plot for 19812 irrigated tall fescue with commercial fertilizer. Model lines are drawn using Eq. [3-7] and Eq [2-2] based on parameters listed in Table 4-6. PAGE 138 124 N, kg ha-1 0100200300400500600700800 Y, Mg ha-1 0 10 20 Nu, kg ha-1 0 200 400 c, g kg-1 0 10 20 30 Data Figure 4-13. Dry matter yield, N uptake, and N concentration versus N application rate for 1982-3 irrigated tall fescue with comm ercial fertilizer. This was the data set with the poorest overall m odel fit in this category of data. Model lines are drawn using Eqs. [3-261] through [3-263] based on parameters listed in Table 4-6. PAGE 139 125 Nu, kg ha-1 0100200300400500600 Y, Mg ha-1 0 10 20 Nc g kg-1 0 10 20 30 Data Figure 4-14. Phase plot for 19823 irrigated tall fescue with commercial fertilizer. Model lines are drawn using Eq. [3-7] and Eq [2-2] based on parameters listed in Table 4-6. PAGE 140 126 N, kg ha-1 0100200300400500600700800 Y, Mg ha-1 0 10 20 Nu, kg ha-1 0 200 400 c, g kg-1 0 10 20 30 Irrigated Nonirrigated Figure 4-15. Dry matter yield, N uptake, and N concentration versus N application rate for the average of all irrigated and nonirrigated tall fescue with commercial fertilizer. Model lines are drawn usin g Eqs. [3-261] through [3-263] based on parameters listed in Table 4-6. PAGE 141 127 Nu, kg ha-1 0100200300400500600 Y, Mg ha-1 0 10 20 Nc g kg-1 0 10 20 30 Irrigated Nonirrigated Figure 4-16. Phase plot for average of all ir rigated and nonirrigated tall fescue with commercial fertilizer. Model lines are drawn using Eq. [3-7] and Eq. [2-2] based on parameters listed in Table 4-6. PAGE 142 128 N, kg ha-1 0100200300400500600700800 Y, Mg ha-1 0 10 20 Nu, kg ha-1 0 200 400 c, g kg-1 0 10 20 30 Data Figure 4-17. Dry matter yield, N uptake, and N concentration versus N application rate for 1983-4 irrigated tall fescue with broile r litter. This was the data set with the best overall model fit in this categor y of data. Model lines are drawn using Eqs. [3-261] through [3-263] based on parameters listed in Table 4-8. PAGE 143 129 Nu, kg ha-1 0100200300400500600 Y, Mg ha-1 0 10 20 Nc g kg-1 0 10 20 30 Data Figure 4-18. Phase plot for 1983-4 irrigated tall fescue with br oiler litter. Model lines are drawn using Eq. [3-7] and Eq. [2-2] base d on parameters listed in Table 4-8. PAGE 144 130 N, kg ha-1 0100200300400500600700800 Y, Mg ha-1 0 10 20 Nu, kg ha-1 0 200 400 c, g kg-1 0 10 20 30 Data Figure 4-19. Dry matter yield, N uptake, and N concentration versus N application rate for 1982-3 irrigated tall fescue with broile r litter. This was the data set with the poorest overall model fit in this cat egory of data. Model lines are drawn using Eqs. [3-261] through [3-263] based on parameters listed in Table 4-8. PAGE 145 131 Nu, kg ha-1 0100200300400500600 Y, Mg ha-1 0 10 20 Nc g kg-1 0 10 20 30 Data Figure 4-20. Phase plot for 1982-3 irrigated tall fescue with br oiler litter. Model lines are drawn using Eq. [3-7] and Eq. [2-2] base d on parameters listed in Table 4-8. PAGE 146 132 CHAPTER 5 SUMMARY AND CONCLUSIONS Summary of Research In this study, the Extended Logistic M odel, which describes crop response to applied N, was applied to data from a previ ous forage study with th ree factors: fertilizer treatment (application level and source, from eith er commercial fertilizer or broiler litter), irrigation treatment (supplemental irrigation or no supplemental irrigation), and crop type (bermudagrass or tall fescue). The Extended Lo gistic Model consists of five parameters: An, the maximum theoretical plant N upt ake for the system being analyzed; bn, the intercept parameter for plant N uptake in response to applied N; cn, the N response coefficient; b the logarithm of the ratio of the upper theoretical plant N concentration limit to the lower theoretical limit for plant N concentration, the latter of which is the fifth parameter, Ncl. A computer program was deve loped based on a new regression scheme that split the Extended Logistic Model into two compon ents, one a nonlinear relationship between N uptake and applied N, and the other a line ar relationship between N concentration and N uptake. Parameters were optimized using this nonlinear/linear regression scheme, and key statistics were calculated, including standard error, covariance, correlation coefficients, and a total normali zed residual sum of squares. The Extended Logistic Model was applied to each data set from the forage study to obtain an optimized set of parameters for that specific data set. Data sets were then grouped together in various conf igurations in an effort to observe possible invariance of PAGE 147 133 some of the five parameters under a given se t of conditions, with the hope that certain parameters could be isolated as likely functions of one or two factors (e.g., crop type) only. Key Observations and Conclusions 1. The analyses of individual data sets in this study add to the already vast amount of evidence that the Extended Logistic Mode l is a consistently good descriptor of crop response to applied N in terms of dry matte r yield, N uptake by the plant, and plant N concentration. This conclusion is reache d without consideration of the possible invariance of some parameters within the m odel, since a modelÂ’s validity cannot be based on parameter invariance alone. 2. The new regression scheme and co rresponding software developed in conjunction with this research has proven to be a very useful tool in finding good estimates of the parameters in the Extended Logistic Model, both for individual data set analysis and multiple data set analysis in which one or more parameters are assumed invariant. The regression scheme was shown to be computationally stable, even when analyzing 16 data sets simultaneously and iterating until all first derivatives (as discussed in Chapter 3) are very close to zero (10-6 was the specific toleran ce used for all analyses in this study). 3. Many of the statistical analyses base d on the new regression scheme, however, are not well founded and are, at best, approximations of resu lts that would be obtained with the more complex nonlinear regressi on scheme involving a normalized sum of squares from all three data types (dry matter yield, N uptake, and N concentration), as presented in Chapter 3, which would hypothe tically solve for all five parameters simultaneously. Problems with statistical calcu lations include standard errors for each PAGE 148 134 parameter, which are based only on the data t ypes used to optimize the given parameter. For example, the standard error of An is based only on regression results involving N uptake data. This does not take into account uncertainty resulting from its involvement in the N concentration and dry matter yield portio ns of the model. Therefore, its standard error would likely differ from a standard error calculated using the aforementioned alternate method. Perhaps most troublesom e are problems with the residual sum of squares as used in the F-test to test for parameter invariance. Parameters using this regression scheme are, at best, only good appr oximations of parameters that would be obtained by a true least square s regression involving all three data types. The resulting error sum of squares, therefor e, is likely rarely the true minimum sum of squares that would be obtained by an alternate, more rigo rous method. This results in potentially significantly inaccurate F-values, bringing into question the validity of using of such values in an F-test. 4. In many cases, the linear parameter An was the most variable of the five. Evidence from this study suggests it is affected by all three of the f actors considered in this study: crop type, fertilizer source, and wa ter availability. This agrees with previous evidence that An is a strong function of environmental inputs. 5. Evidence also suggests that bn is influenced by all three factors considered in this study, perhaps with water availa bility being the most influe ntial. There is no apparent pattern, however, to the value of bn found in the analysis of data here. Of the four parameters considered for commonality in this study ( bn, cn, b and Ncl), bn failed commonality tests most often. Hypothetically, bn should be influenced the most by initial conditions, including initial fertilizer levels in the soil. Since these initial conditions are PAGE 149 135 likely strong functions of the hi story of a plot, they would lik ely vary from year to year and site to site, and so the lack of a pattern for bn is not surprising, es pecially considering all grouped analyses consisted of da ta from all years of the study. 6. Evidence suggests that cn is likely not a strong func tion of crop type, but it may be impacted by both fertilizer source and wate r availability, with perhaps slightly more impact by the former. However, the variability of cn when considering individual analyses was considerable, and so patterns were unclear. In the vast majority of groupings where a common cn was found, its value hovered within about 5% of 0.00766 ha kg-1. These results add to the evidence that cn is likely not a functi on of crop type only, conflicting with evidence from some studies cited by Wils on (1995) but agreeing with others she cited in which cn was not a clear function of crop type. At best, it can be concluded that cn is not a function of crop type only, which agrees with the mixed results of previous studies. Hypoth etically, one would expect cn to vary with more than just crop type, since it is a measure of a cropÂ’s respons e to the applied N. Indeed, if fertilizer sources vary, as they did in this experiment, then it is conceivable cn would vary to indicate the difference in availa bility of the applied N to th e plant. For example, studies have shown that broiler litter breaks down relatively slowly and so its N is likely less available than that of commerc ial fertilizer; the conclusions of Huneycutt et al. (1988) agree with this. Therefore, one would expect, overall, for the cn value to be greater for commercial fertilizer than for broiler litter sources, holding all other factors constant. There is some evidence of this from this study, such as the average value of 0.00836 ha kg-1 for the complete commercial fertilizer group versus 0.00625 ha kg-1 for the complete broiler litter group, but in that specific cas e the high standard deviations (both 32%) place PAGE 150 136 these values within statistical reach of each other. Theref ore, with the high general variability of cn results are inconclusive as to whether there is a clear effect of fertilizer source on cn, but the idea remains plausible. 7. No firm conclusions could be drawn regarding the parameter b While some tests suggested it to be a func tion of crop type and not water availability, others suggested just the opposite. Evidence was furthe r confounded by the high variability of b observed in broiler litter groups. Theref ore the results in this study cannot corroborate conclusions by Scholtz (2002) and Wilson (1995) that b is a genetic parameter only. 8. The strongest evidence for co mmonality in this study was for Ncl. Results suggest that Ncl is a strong function of crop type and possibly crop type only, making it a solely genetic parameter. The simple averages and standard deviations of the individual results revealed a marked difference in Ncl between the two crops with relatively low standard deviations (compare d to any other parameter). Fr om commonality tests, there was some evidence, albeit far limited compared to that for crop type, that water availability slightly affected the value of Ncl. Previous studies by Scholtz (2002) and Wilson (1995) had suggested the related parameter Ncm (equal to Ncl expb ) is a strong function of crop type, which combin ed with their conclusion that b is a function of crop type (not corroborated in this study), suggests Ncl was a function of crop type in these studies as well. 9. It should be noted that all comments and conclusions from this study regarding commonality (of any of the parameters) are limite d to the three factors considered in this experiment: fertilizer source, crop type, and irrigation treatment. There could be no conclusive evidence as to whether soil type, regional temperature, or other effects due to PAGE 151 137 location affected these parameters since all tests were conducted on the same soil at the same location. However, research by Scho ltz (2002) on data fro m several different locations suggested b and Ncm (and therefore Ncl) were invariant to location. 10. Data groupings rarely passed te sts for concurrent commonality among parameters (i.e., in tests where more than one parameter was assumed invariant simultaneously, such a combination rarely pa ssed the F-test, even if individual tests suggested commonality among parameters when considered separately). This was a confounding factor in deciding which parameters were Â“trulyÂ” invariant. However, even if a concurrent commonality group failed the Ftest, this did not necessarily mean the assumption of concurrent commonality resulte d in a poor model in terms of goodness of fit. For example, all bermuda grass with chemical fertilizer failed the F-test for any concurrent commonalities, even though b and Ncl individually passed the test and cn almost passed the test. However, regre ssion was done on this data group assuming concurrent invariance of cn, b and Ncl. The resulting parameters are shown in Table 5-1, and their respective statistics are shown in Ta ble 5-2. The resulting models and data are shown in Figure 5-1 through Figure 5-4. As can be seen in Table 5-2 and the figures, even with all three parameters simultaneously i nvariant, the model still fits the data very well with all correlation coeffici ents greater than 0.9, with the exception of 1984 irrigated bermudagrass, for which the model for N concentration is rather poor with an R of 0.66. A similar exercise was conducted for all tall fe scue with commercial fertilizer, the results of which are shown in Table 5-3 and Table 5-4. These results can be visualized in Figure 5-5 through Figure 5-7. While th ere are four out of 18 correla tion coefficients less than 0.9 in this group, the overall f it of the model to the data is still very good. This illustration PAGE 152 138 suggests that if the overall goal is reliable engineering estimates for crop response, then the restrictions on commonality sugges ted by the F-test alone may be overly conservative. Future Work There are several ar eas with regard to the Exte nded Logistic Model and the research in this dissertation that can be expanded on and fu rther explored. The regression method used in this research proved qu ite adequate for obtaining good parameter estimates, but its statistical basis was questiona ble. The statistical difficulties associated with the Extended Logistic ModelÂ—mainly because it is a model that takes into consideration three different data typesÂ—are not trivial and more research into the best way to perform regression and useful statisti cal calculations is certainly warranted. The nonlinear regression rou tine based on a total normalized sum of squares (the Â“second methodÂ” in Chapter 3) at first glance seems the most robust, but wr iting a stable program than can handle such intens e calculations is no small task There may be even more appropriate methods yet to be considered. Certainly more data should be analyzed as it becomes available to continue the search for patterns that may aid in furthe r developing the Extended Logistic Model or obtaining more insightful knowledge about th e nature of its parameters. Analyses involving data from previous stud ies are often criticized. Some argue that it is an abuse of data, since the original purpose of the experiment had nothing to do with the model at the time it was conducted. Some argue that Â“old da taÂ” are not reliable, th at Â“better dataÂ” can be obtained today. Such statements are nav e and disrespectful, in the opinion of this author, who greatly admires th e talent and devotion it takes for agronomists and other experimentalists to conduct such massive and detailed experiments. Certainly, it would PAGE 153 139 be very desirable to conduct new experime nts geared more specifically toward the information needed to develop the Extended Logistic Model. Such experiments could include measurements of additional factors, su ch as solar radiation or evapotranspiration, that may lend insight into findi ng functions that determine so me of the parameters, such as An. While this is not an unrealistic goal, it certainly would be no trivial task, and it would be more difficult to find funding for su ch well managed, multi-year experiments in present day academic environments that often push for fast results. The software developed in conjunction with this research is the first of its kind, with a relatively user-friendly, visual inte rface and the ability to examine multiple cases of commonality of parameters on multiple data sets quickly. Yet this first version is still in its infancy compared to its potential. More file type options for input and output could improve its efficiency for use with other program s, such as spreadsheets or databases. An intrinsic graphing system for a convenient, vi sual view of results would be useful. An intrinsic library with respect to common para meters based on crop type could also be a possibility as further knowledge about parame ter behavior is obtai ned. This software could also be expanded to use new regression methods or to incorporate other types of models, such as the Expanded Growth Model that models crop growth with time over a single season, which can be linked to the Extended Logist ic Model as discussed by Scholtz (2002). Closing Remarks The program developed in conjunction with th is research was affectionately termed DAEDALUS. This was done not to indicate a ny sort of catchy acr onym, but rather to reflect a philosophy. In Greek mythology, Daedalus was the father of Icarus, and both were imprisoned in the Labyrinth on the island of Crete by King Minos. In order to PAGE 154 140 escape, Daedalus constructed a pair of wings consisting of feathers held together with wax for both him and his son. Despite warni ngs from Daedalus not to fly too low (to avoid the water) nor too high (to a void the sun), Icarus in the th rill of flight flew too close to the sun, which melted the wax of his wings, sending him to his death below. How this relates to crop modeling, and speci fically the Extended Logistic Model, is quite likely less than obvious! The philosophy is this: the incredible complexity of nature should always be respected, and any attempts to understand even the smallest aspects of it, while a worthy effort, should always be proceeded upon with caution and humility. The research behind the Extended Logistic Model has been vast, but it has always been a relatively cautious search for gene ral, consistent patterns across many crops, locations, and circumstances. As was stated in Chapter 2, the model has been criticized for its lack of predictive ability. Many mode ls that do attempt to predict yield are based on multiple subcomponents that attempt to simu late some of the myriad of processes within a plant and at its in terface with its environment. Many of these subcomponents are based on quite simplistic assumptions, and the po wer of their predictive ability is, at the least, questionable. Currently, and quite likely for some time to come, predictive ability is not the focus of the research involving the Extended Logistic Model. Rather, it is a continued, inductive, Â“top downÂ” search for pa tterns and insight into a plantÂ’s behavior. Anything further would quite lik ely be in undue haste, but th is does not mean the patterns already observed and equations developed so far are not extraordinary achievements. PAGE 155 141 Table 5-1. Parameters for bermudagrass w ith commercial fertilizer assuming invariance of cn, b and Ncl across all data sets in this group. Year Water An, kg ha-1 bn cn, ha kg-1 b Ncl, g kg-1 1982 I 456 1.52 0.00773 0.565 13.1 1982 N 503 1.97 0.00773 0.565 13.1 1983 I 553 1.78 0.00773 0.565 13.1 1983 N 418 1.93 0.00773 0.565 13.1 1984 I 522 1.78 0.00773 0.565 13.1 1984 N 411 1.76 0.00773 0.565 13.1 1985 I 520 2.11 0.00773 0.565 13.1 1985 N 499 2.26 0.00773 0.565 13.1 Notes: N = nonirrigated, I=irrigated Table 5-2. Statistics for parameters for be rmudagrass with commercial fertilizer assuming invariance of cn, b and Ncl across all data sets in this group. Year Water RNu RNc RY SE ( An) SE ( bn) SE ( cn) SE (b ) SE ( Ncl) 1982 I 0.9945 0.98270.98162.8 8.1 4.2 1.2 0.5 1982 N 0.9981 0.95820.98383.0 6.2 4.2 1.2 0.5 1983 I 0.9965 0.91050.99232.6 6.1 4.2 1.2 0.5 1983 N 0.9902 0.97790.99023.5 7.3 4.2 1.2 0.5 1984 I 0.9787 0.66150.97152.8 6.6 4.2 1.2 0.5 1984 N 0.9881 0.95410.97113.3 7.9 4.2 1.2 0.5 1985 I 0.9902 0.91140.98513.1 5.8 4.2 1.2 0.5 1985 N 0.9912 0.93710.98003.3 5.6 4.2 1.2 0.5 Note: SE Â’s, standard errors, are in percent (%) of the parameter value. PAGE 156 142 Table 5-3. Parameters for tall fescue with commercial fertilizer assuming invariance of cn, b and Ncl across all data sets in this group. Year Water An, kg ha-1 bn cn, ha kg-1 b Ncl, g kg-1 1981-2 I 452 1.47 0.00775 0.476 17.4 1981-2 N 372 1.62 0.00775 0.476 17.4 1982-3 I 393 1.94 0.00775 0.476 17.4 1982-3 N 228 1.66 0.00775 0.476 17.4 1983-4 I 451 1.53 0.00775 0.476 17.4 1983-4 N 154 0.50 0.00775 0.476 17.4 Table 5-4. Statistics for parameters for tall fescue with commercial fertilizer assuming invariance of cn, b and Ncl across all data sets in this group. Year Water RNu RNc RY SE ( An) SE ( bn) SE ( cn) SE (b ) SE ( Ncl) 1981-2 I 0.9995 0.9490 0.9980 1.9 5.8 4.2 1.7 0.7 1981-2 N 0.9953 0.8336 0.9690 2.4 6.3 4.2 1.7 0.7 1982-3 I 0.9878 0.8575 0.9825 2.5 5.5 4.2 1.7 0.7 1982-3 N 0.9729 0.9125 0.9447 3.7 9.9 4.2 1.7 0.7 1983-4 I 0.9970 0.5893 0.9755 2.0 5.7 4.2 1.7 0.7 1983-4 N 0.9347 0.9110 0.8816 4.0 41.3 4.2 1.7 0.7 PAGE 157 143 N, kg ha-1 0100200300400500600700800 Y, Mg ha-1 0 10 20 Nu, kg ha-1 0 200 400 c, g kg-1 0 10 20 30 Irrigated Nonirrigated Figure 5-1. Dry matter yield, N uptake, and N concentration ve rsus N application rate for 1982 bermudagrass with commercial fertilizer using common cn, b and Ncl for all bermudagrass with commercial fertilizer (1982-1985). Model lines are drawn using Eqs. [3-261] through [3-263] based on parameters listed in Table 5-1. PAGE 158 144 N, kg ha-1 0100200300400500600700800 Y, Mg ha-1 0 10 20 Nu, kg ha-1 0 200 400 c, g kg-1 0 10 20 30 Irrigated Nonirrigated Figure 5-2. Dry matter yield, N uptake, and N concentration ve rsus N application rate for 1983 bermudagrass with commercial fertilizer using common cn, b and Ncl for all bermudagrass with commercial fertilizer (1982-1985). Model lines are drawn using Eqs. [3-261] through [3-263] based on parameters listed in Table 5-1. PAGE 159 145 N, kg ha-1 0100200300400500600700800 Y, Mg ha-1 0 10 20 Nu, kg ha-1 0 200 400 c, g kg-1 0 10 20 30 Irrigated Nonirrigated Figure 5-3. Dry matter yield, N uptake, and N concentration ve rsus N application rate for 1984 bermudagrass with commercial fertilizer using common cn, b and Ncl for all bermudagrass with commercial fertilizer (1982-1985). Model lines are drawn using Eqs. [3-261] through [3-263] based on parameters listed in Table 5-1. PAGE 160 146 N, kg ha-1 0100200300400500600700800 Y, Mg ha-1 0 10 20 Nu, kg ha-1 0 200 400 c, g kg-1 0 10 20 30 Irrigated Nonirrigated Figure 5-4. Dry matter yield, N uptake, and N concentration ve rsus N application rate for 1985 bermudagrass with commercial fertilizer using common cn, b and Ncl for all bermudagrass with commercial fertilizer (1982-1985). Model lines are drawn using Eqs. [3-261] through [3-263] based on parameters listed in Table 5-1. PAGE 161 147 N, kg ha-1 0100200300400500600700800 Y, Mg ha-1 0 10 20 Nu, kg ha-1 0 200 400 c, g kg-1 0 10 20 30 40 Irrigated Nonirrigated Figure 5-5. Dry matter yield, N uptake, and N concentration ve rsus N application rate for 1981-2 tall fescue with commer cial fertilizer using common cn, b and Ncl for all tall fescue with commercial fe rtilizer (1981-1984). Model lines are drawn using Eqs. [3-261] through [3-263] based on parameters listed in Table 5-3. PAGE 162 148 N, kg ha-1 0100200300400500600700800 Y, Mg ha-1 0 10 20 Nu, kg ha-1 0 200 400 c, g kg-1 0 10 20 30 40 Irrigated Nonirrigated Figure 5-6. Dry matter yield, N uptake, and N concentration ve rsus N application rate for 1982-3 tall fescue with commer cial fertilizer using common cn, b and Ncl for all tall fescue with commercial fe rtilizer (1981-1984). Model lines are drawn using Eqs. [3-261] through [3-263] based on parameters listed in Table 5-3. PAGE 163 149 N, kg ha-1 0100200300400500600700800 Y, Mg ha-1 0 10 20 Nu, kg ha-1 0 200 400 c, g kg-1 0 10 20 30 40 Irrigated Nonirrigated Figure 5-7. Dry matter yield, N uptake, and N concentration ve rsus N application rate for 1983-4 tall fescue with commer cial fertilizer using common cn, b and Ncl for all tall fescue with commercial fe rtilizer (1981-1984). Model lines are drawn using Eqs. [3-261] through [3-263] based on parameters listed in Table 5-3. PAGE 164 150 APPENDIX SUPPLEMENTAL DISCUSSIONS Matrix Inversion There are multiple possible strategies for inverting matrices. The so-called inversion by determinants method may be mo re efficient for 3x3 matrices (Bronson, 1989), such as those involved in the singl e data set example outlined previously. However, the program used in this research was designed to analyze multiple data sets simultaneously in order to test for common factors between data sets. In such cases the resulting matrices can be much larger than 3x3, and so inversion by determinants is not the most efficient method. Therefore an algorithm involving elementary row operations and a partial pivoting strategy (to help pr event rounding error) based on the methods presented by Bronson (1989) was employed instead, since this is generally more efficient in larger matrices. It is assumed for the purpos es of this research that all matrices are nonsingular, meaning they do have inverses. This inversion method is presented below for a generalized 3x3 matrix, given by i h g f e d c b aA [A-1] First, A and the identity matrix I are used to form the partitioned matrix 1 0 0 0 1 0 0 0 1 i h g f e d c b aI A [A-2] PAGE 165 151 Through a series of row operations, th e ultimate goal is to transform I Ainto the form r q p o n m l k j 1 0 0 0 1 0 0 0 1B I [A-3] where now B is the inverse of A. The series of operations required to reach this conclusion depend on the actual values on A; i.e., there is no general formula that can be shown here for a solution using this method. Rather, there is a general algorithm for processing I A to reach the form B I. This is presented in some detail belo w and can be considered the Â“pseudo-codeÂ” for what was used in the regression program, although here it is pr esented in terms the 3x3 matrix A. The matrix I A needs to be transformed into row-echelon form using elementary row operations, which include 1) interchangi ng any two rows, 2) multiplying elements of any entire row by a scalar that is not zer o, and 3) adding the elements in one row (multiplied by a scalar) to the corresponding elements in another row. These row operations must be performed on the enti re row (and thus will begin transforming I as well as A). To be in row-echelon form, I A must be transformed to the form ' ' ' ' 1 0 0 1 0 ' 1 r q p o n m l k j c b aB' A' [A-4] The first step in attaining row-echelon form (including a partial pivot procedure to prevent significant round-off error) is to find the element in the first column of I A with PAGE 166 152 the largest absolute value and interchange its row with the first row (if different from the first row). This element is known as the pivot element. The second step is to multiply the each el ement of the first row by the inverse of the current pivot element to ensure the fina l value of the pivot element becomes unity (e.g., if the pivot element has a value a then multiply each element of the entire row by 1/ a ). This completes the work on the first row, which should therefore be left alone in subsequent iterations. The third step is to convert the remaining elements in the first column to zero if they are not so already. Suppose ther e remains a row with nonzero element g in the first column. To convert this to zero, add each element in this row to Â– g times the corresponding element in the fi rst row. This step is repe ated for all remaining rows containing nonzero elements in the first column so that, once completed, there is only the value 1 in the first row, first column positi on and zeroes in the first column position of the remaining rows. These three steps are then applied to th e second row, second column and then the third row, third column (and so on for highe r order matrices) until the row-echelon form illustrated in Eq. [A-4] is obt ained. It should be re-itera ted that these operations are applied to the entire partitioned matrix I A. Once row-echelon form is obtained, all th at remains is to make the necessary operations to transform the elements ' b aand 'cto zero. In this example, assuming these values were not zero this would be accomp lished by first adding to each element of the second row 'c times each corresponding element of the third row, which would eliminate 'c. Second, 'b times the third row added to the first row would eliminate 'b. PAGE 167 153 Finally 'a times the (new) second row added to the first row would eliminate 'a, so that the final form illustrated in Eq. [A-3] is reached, and B, the inverse of A, is obtained. Sensitivity Analysis To demonstrate the effect of standard errors on parameters, a hypothetical Extended Logistic Model was created using Â“typical valuesÂ” for the five parameters: An = 400 kg ha-1, bn = 1.7, cn = 0.007 ha kg-1, b = 0.5, and Ncl = 15 g kg-1. Seventeen points were then calculated for applied N values ranging from zero to 800 kg ha-1 in increments of 50 kg ha-1. Each parameter was then varied, one at a time, +20% and then -20% to simulate the effect of a 20% standard erro r. The percent change of each simulation point (in terms of the original, unmodified values ) was then calculated, and then the average change over each data type (dry matter yi eld, N uptake, and N concentration) was calculated. The results can be visuali zed in Figure A-1 through Figure A-5. The effect of varying the linear parameter An can be seen in Figure A-1. An is essentially a scaling parameter present in the numerator of both the N uptake equation and dry matter yield equation, which explai ns why altering it has no effect on the N concentration model, as seen in the figure. Also as a result of th is, obviously increasing An by 20% will increase all N uptake model values and all dry matter yield model values by 20% as well; similarly, a decrease of 20% w ill decrease all values in those two models by 20%. While its relative influence is constant over the model, since it is a scaling factor, the effect An has in terms of absolute magnitude is, of course, higher at increasing levels of applied N. The effect of varying the parameter bn can be seen in Figure A-2. This parameter is an intercept parameter, affecting the y-inte rcept of both the yield model and the N uptake PAGE 168 154 model, as can be seen in the figure. Its infl uence on these two models is at a maximum in terms of relative magnitude at zero applied N and attenuates with increasing applied N, having little effect as the models appr oach their plateau. A +20% change in bn decreased model values for N uptake by an average of 11%, decreased N concentration model values by an average of 2%, and decreased yi eld model values by an average of 9%. A -20% change in bn increased model values for N uptake by an average of 12%, increased N concentration model values by an average of 2%, and increased yield model values by an average of 9%. The effect of varying the parameter cn can be seen in Figure A-3. This parameter, the N response coefficient, affects the slope of the logistic model and so its impact is most noticeable in the N uptake and yield mode ls near the midpoint of the models, where the slopes are at their maximum. Its effect on the N concentration model is similar. A +20% change in cn increased model values of the N uptake, N concentration, and yield models by an average of 8%, 2%, and 6%, respectively. A +20% change in cn decreased model values of the N uptake, N concentrati on, and yield models by an average of 11%, 3%, and 8%, respectively. The effect of varying the parameter b can be seen in Figure A-4. Since it is not part of the N uptake model, a change in b has no effect on this model. It influences the scaling factor A (a secondary parameter that is a function of An, b and Ncl) for the yield model, thus altering b has a partial inverse sc aling effect on the yield model as well as a shifting effect, since it also influences the yintercept for this model. Its effect on the N concentration model is also a matter of both s caling and shifting, as seen in the figure. A +20% change in b increased model values for N concentration by an average of 8% and PAGE 169 155 decreased model values for yield by an average of 7%. A -20% change in b decreased model values for N concentration by an aver age of 7% and increased model values for yield by an average of 8%. The effect of varying the parameter Ncl can be seen in Figure A-5. Ncl, the theoretical lower limit for plant N concentration, does not influence the N uptake model since it is not part of this equation. It serves as a dire ct scaling factor for the N concentration model, and an i nverse scaling factor for the yi eld model. A +20% change in Ncl increased each model value for N concen tration by 20% and decreased each model value for yield by about 17%. A -20% change in Ncl decreased model values for N concentration by 20% and increased model values for yield by 25%. PAGE 170 156 N, kg ha-1 0100200300400500600700800 Y, Mg ha-1 0 10 Nu, kg ha-1 0 200 400 Unmodified Model Model with 1.2An Model with 0.8An c, g kg-1 0 10 20 30 Figure A-1. Model dry matter yi eld, N uptake, and N concentration versus N application rate demonstrating the effect on the m odels of a change of 20% in the parameter An. PAGE 171 157 N, kg ha-1 0100200300400500600700800 Y, Mg ha-1 0 10 Nu, kg ha-1 0 200 400 Unmodified Model Model with 1.2bn Model with 0.8bn c, g kg-1 0 10 20 30 Figure A-2. Model dry matter yi eld, N uptake, and N concentration versus N application rate demonstrating the effect on the m odels of a change of 20% in the parameter bn. PAGE 172 158 N, kg ha-1 0100200300400500600700800 Y, Mg ha-1 0 10 Nu, kg ha-1 0 200 400 Unmodified Model Model with 1.2cn Model with 0.8cn c, g kg-1 0 10 20 30 Figure A-3. Model dry matter yi eld, N uptake, and N concentration versus N application rate demonstrating the effect on the m odels of a change of 20% in the parameter cn. PAGE 173 159 N, kg ha-1 0100200300400500600700800 Y, Mg ha-1 0 10 Nu, kg ha-1 0 200 400 Unmodified Model Model with 1.2 b Model with 0.8 b c, g kg-1 0 10 20 30 Figure A-4. Model dry matter yi eld, N uptake, and N concentration versus N application rate demonstrating the effect on the m odels of a change of 20% in the parameter b PAGE 174 160 N, kg ha-1 0100200300400500600700800 Y, Mg ha-1 0 10 Nu, kg ha-1 0 200 400 Unmodified Model Model with 1.2Ncl Model with 0.8Ncl c, g kg-1 0 10 20 30 Figure A-5. Model dry matter yi eld, N uptake, and N concentration versus N application rate demonstrating the effect on the m odels of a change of 20% in the parameter Ncl. PAGE 175 161 LIST OF REFERENCES Boote, K.J., J.W. Jones, and N.B. Pickering. 1996. Potential uses and limitations of crop models. Agronomy Journal 88: 704-716. Bronson, R. 1989. SchaumÂ’s Outline of Theory and Problems of Matrix Operations McGraw-Hill. New York, NY. Cornell, J. A., and R. D. Berger. 1987. Factor s that influence the value of the coefficient of determination in simple linea r and nonlinear regression models. Phytophathology 77: 63-70. Huneycutt, H. J., C. P. West, and J. M. Phillips. 1988. Responses of Bermudagrass, Tall Fescue and Tall Fescue-Clover to Bro iler Litter and Commercial Fertilizer Arkansas Agricultural Experiment Station, University of Arkansas Division of Agriculture. Fayetteville, AR. Jones, J.W., and J.C. Luyten. 1998. Simulation of Biological Processes. Agricultural Systems Modeling and Simulation Marcel Dekker, Inc. New York, NY. Kolman, B., and A. Shapiro. 1986. Algebra for College Students 2nd edition. Academic Press, Inc. Orlando, FL. Mills, H.A. and J. B. Jones, Jr. 1996. Plant Analysis Handbook II Micromacro Publishing, Inc. Athens, GA. Monteith, J. L. 1996. The quest for balance in crop modeling. Agronomy Journal 88: 695-697. Overman, A. R., and R. V. Scholtz. 2002. Mathematical Models of Crop Growth and Yield. Marcel Dekker, Inc. New York, NY. Overman, A. R., S. R. Wilkinson, and D. M. Wilson. 1994. An extended model of forage grass reponse to applied nitrogen. Agronomy Journal 86: 617-620. Pearl, R., L. J. Reed, and J. F. Kish. 1940. The logistic curve and the census count of 1940. Science 92: 486-488. Ratkowsky, D. A. 1983. Nonlinear Regression Modeling: A Unified Practical Approach Marcel Dekker, Inc. New York, NY. PAGE 176 162 Russell, E. J. 1937. Soil Conditions and Plant Growth 7th edition. Longmans, Green and Co. London, England. Scholtz, R. V. 2002. Mathematical Modeling of Agronomic Crops: Analysis of Nutrient Removal and Dry Matter Accumulation Ph.D. Dissertation. University of Florida. Gainesville, FL. Soil Science Society of America. 1997. Glossary of Soil Science Terms Soil Science Society of America. Madison, WI. United States Department of Agriculture [USDA]. 2003. Captina Series Internet Site: http://ortho.ftw.nrcs.usda. gov/osd/dat/C/CAPTINA.html United States Department of Agriculture. Washington, D. C. Accessed: February 20, 2004. Wackerly, D. D., W. Mendenhall, and R. L. Scheaffer. 2002. Mathematical Statistics with Applications 6th edition. Duxbury. Pacific Grove, CA. Wilson, D. M. 1995. Estimation of Dry Matter and Nitr ogen Removal by the Logistic Equation Ph.D. Dissertation. University of Florida. Gainesville, FL. PAGE 177 163 BIOGRAPHICAL SKETCH Kelly Hans Brock was born in Thomasvill e, GA, in 1975. He grew up on a small farm in Monticello, FL, and graduated high school in 1993 from Aucilla Christian Academy as valedictorian of hi s class. Kelly entered the Univ ersity of Florida in 1993 as a National Merit Scholar and honors student. He completed his B.S. in agricultural and biological engineering in May 1999 with a focu s in agrisystems engineering and a minor in chemistry, graduating with highest honors and a Four-Year Scholar Honorable Mention. After being awarded a National Sc ience Foundation fellowship, Kelly entered a Ph.D. direct program at the University of Florida in the Fall of 1999 in agricultural and biological engineering, focusing on natural resources engineering. |