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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 (Ftest) ..............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 Ftest.. ....................................................... .......... ................. 86 Commonalities Suggested by the Ftest ...................................... ............... 89 Beyond the Ftest: 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 21 Application schedule for commercial fertilizer treatments (listed as NP205K20, each in k g h a1) ..................................................................... 19 22 Broiler litter content analysis for 19821985 (in terms of kg nutrient Mg1 litter)...20 23 Equivalent N application rates for broiler litter (kg ha1)..................................20 41 M ode descriptions. ........................................... ... .... ........ ......... 93 42 Optimized parameters for bermudagrass with commercial fertilizer....................93 43 Statistics for optimized parameters for bermudagrass with commercial fertilizer...94 44 Optimized parameters for bermudagrass with broiler litter. ...................................94 45 Statistics for optimized parameters for bermudagrass with broiler litter ...............94 46 Optimized parameters for tall fescue with commercial fertilizer.............................95 47 Statistics for optimized parameters for tall fescue with commercial fertilizer.........95 48 Optimized parameters for tall fescue with broiler litter. ......................................95 49 Statistics for optimized parameters for tall fescue with broiler litter.....................96 410 Arithmetic average of parameters over select data groups.................. ............96 411 Standard deviations of parameters over select data groups............... ................ 97 412 Analysis of variance for parameters for irrigated bermudagrass with commercial fertilize er. .......................................................... ................ 9 8 413 Analysis of variance for parameters for nonirrigated bermudagrass with com m ercial fertilizer. ....................................... .......................... 99 414 Analysis of variance for parameters for bermudagrass with commercial fertilizer (both irrigated and nonirrigated). ........................................ ......................... 99 415 Analysis of variance for parameters for average of bermudagrass data with commercial fertilizer (split into two groups: nonirrigated and irrigated).............100 416 Analysis of variance for parameters for irrigated bermudagrass with broiler litter. 100 417 Analysis of variance for parameters for nonirrigated bermudagrass with broiler litter. .............................................................................. 10 1 418 Analysis of variance for parameters for bermudagrass with broiler litter (both irrigated and nonirrigated)..................... ........ ............................. 102 419 Analysis of variance for parameters for irrigated tall fescue with commercial fertilizer. .......................................................................... 103 420 Analysis of variance for parameters for nonirrigated tall fescue with commercial fertilizer. .......................................................................... 104 421 Analysis of variance for parameters for tall fescue with commercial fertilizer (both irrigated and nonirrigated)..................... ........ ............................. 104 422 Analysis of variance for parameters for average of tall fescue data with commercial fertilizer (split into two groups: nonirrigated and irrigated). ...............................105 423 Analysis of variance for parameters for irrigated tall fescue with broiler litter.....106 424 Analysis of variance for parameters for nonirrigated tall fescue with broiler litter. 106 425 Analysis of variance for parameters for tall fescue with broiler litter (both irrigated and nonirrigated). ........................................... ... .... ................. 107 426 Analysis of variance for select parameters for all bermudagrass...........................107 427 Analysis of variance for select parameters for all irrigated bermudagrass. ...........107 428 Analysis of variance for select parameters for all nonirrigated bermudagrass. .....108 429 Analysis of variance for select parameters for all fescue..................................... 108 430 Analysis of variance for select parameters for irrigated fescue ...........................109 431 Analysis of variance for select parameters for nonirrigated fescue. ......................109 432 Analysis of variance for select parameters for all irrigated samples....................110 433 Analysis of variance for select parameters for all irrigated samples with commercial fertilizer. ................................................................ ..... .......... 110 434 Analysis of variance for select parameters for all nonirrigated samples with com m ercial fertilizer. .............................................. ..... .. .... .............. .. 111 435 Summary of results for single parameter commonality Ftest ...........................111 51 Parameters for bermudagrass with commercial fertilizer assuming invariance of cn, Ab, and Nc1 across all data sets in this group. ................................. ............... 141 52 Statistics for parameters for bermudagrass with commercial fertilizer assuming invariance of cn, Ab, and Nc1 across all data sets in this group. .............................141 53 Parameters for tall fescue with commercial fertilizer assuming invariance of cn, Ab, and Nc~ across all data sets in this group.............. .............................................. 142 54 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 41 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 42 Phase plot for 1982 nonirrigated bermudagrass with commercial fertilizer ..........113 43 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 44 Phase plot for 1984 irrigated bermudagrass with commercial fertilizer ..............115 45 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 46 Phase plot for average of all irrigated and nonirrigated bermudagrass with comm ercial fertilizer.. .............................................. ...... ...... .. .......... .. 117 47 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 48 Phase plot for 1983 irrigated bermudagrass with broiler litter.............................119 49 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.............. .............................................. 120 410 Phase plot for 1982 irrigated bermudagrass with broiler litter..............................121 411 Dry matter yield, N uptake, and N concentration versus N application rate for 19812 irrigated tall fescue with commercial fertilizer. This was the data set with the best overall model fit in this category of data. ..............................................122 412 Phase plot for 19812 irrigated tall fescue with commercial fertilizer................. 123 413 Dry matter yield, N uptake, and N concentration versus N application rate for 19823 irrigated tall fescue with commercial fertilizer. This was the data set with the poorest overall model fit in this category of data. ............... .......... .........124 414 Phase plot for 19823 irrigated tall fescue with commercial fertilizer................. 125 415 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. ...126 416 Phase plot for average of all irrigated and nonirrigated tall fescue with commercial fertilize e r. ............................ .. ................ ................ ................ ............. ........ 12 7 417 Dry matter yield, N uptake, and N concentration versus N application rate for 19834 irrigated tall fescue with broiler litter. This was the data set with the best overall model fit in this category of data.... ...................................128 418 Phase plot for 19834 irrigated tall fescue with broiler litter ..............................129 419 Dry matter yield, N uptake, and N concentration versus N application rate for 19823 irrigated tall fescue with broiler litter. This was the data set with the poorest overall model fit in this category of data.... ...................................130 420 Phase plot for 19823 irrigated tall fescue with broiler litter.............................131 51 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 (19821985). .............. .... ..............143 52 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 (19821985). .............. .... ..............144 53 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 (19821985). .............. .... ..............145 54 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 (19821985). .............. .... ..............146 55 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 (19811984).....................................................147 56 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 (19811984).....................................................148 57 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 (19811984).....................................................149 A1 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 A2 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 A3 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 A4 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 A5 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 cropbased 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 typeas has been found in previous studiesand 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 aquiferbased 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 plantsoilwater 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 Ftest 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 42 through Table 49, as well as Table 412 through Table 434, are presented to cover in great detail the results of each analysis. The most important and useful tables in Chapter 4, however, are Table 410, Table 411, and Table 435, 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 Sshape 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 VerhulstPearl 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, MorganMercerFlodin, and Weibull distributionbased 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 [21] 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= [22] 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 socalled 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 (noninfinite) 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 broadbased 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 2425). 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 broadrange 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 firstorder 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 21. The initial treatment each season included application of a 131313 fertilizer. Additional N, P, and K were supplied as necessary to meet the experiment's levels by using ammonium nitrate, concentrated superphosphate, 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 22. This resulted in different nutrient application rates each year, as outlined in Table 23. 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., 19811982. 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 perforatedpipe 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 finesilty, 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 21. 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 NP205 After 3rd harvest 112112135 225112202 337112270 449146337 562180404 674213472 112112135 225112202 337112270 449146337 562180404 674213472 5611267 8411267 112112101 140146112 168180135 225213135 5611267 8411267 112112101 140146112 168180135 225213135 Data adapted from Huneycutt et al. (1988) Trial Bermuda grass Tall Fescue 56067 84067 84084 112090 1400101 1680135 56067 84067 840101 140090 1680135 2250135 56067 84084 112067 1400101 1680135 56067 84067 84067 112067 1120101 5600 84067 112067 112067 5600 84067 112067 1120101 Table 22. Broiler litter content analysis for 19821985 (in terms of kg nutrient Mg1 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 23. Equivalent N application rates for broiler litter (kg ha1) 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 [31] N,, exp(Ab) b = b, Ab [32] A KAn [33] Sexp(Ab)1 An 1 [ m Nj exp(b) 1 [34] Ncm No, exp(Ab) A [35] 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, [36] No = N, + N [exp(Ab)l [37] Here No is N concentration (or specific N, typically in units of g kg1) and N, is N uptake (typically in units of kg ha1). 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= [38] 1 + exp(b c,,N) Corresponding Model for Yield Y = [39] 1 + exp(b cN) Linear Regression for Linear Model Parameters From least squares regression for intercept and slope, respectively: N = (,N)( ) (N) (2 )) [310] Y n NZ N N X )2 N l=1 n N Ne)W NzXv S[exp(Ab) 1] 1 n)) [311] 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 [312] n(_ (N.NcN)) X N)[) Nc t N.2) N, N 2(N, N, K = ( [313] n( (NNj)) (INJXZNC) Solving for Ab (including the parameter A,): Nc' [exp(Ab)l]= 1 [314] A, Y, [exp(Ab)1]= [315] Y.Nl A exp(Ab)= A +l [316] Ab = In I" +1 [317] \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 [318] 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. ," [319] 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 [320] OAn dE =0 [321] bb dE = 0 [322] 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 [32 OA LL 1+ exp(b cN) 1+ exp(bn cN) 1 I F[N 1 + exp(bn cN) [ L )]N [32 A 1 A rr1 + exp(bn cN) 1+ exp(b cN), 0 AnZ[[1+exp(b c N)]2 ] 1+exp(b$ cN)] [32 An = [32 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(bc 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(bcn N)] 2' Nexp(b cAN) [1 + exp(b, cN) ]2 An [ [+ exp(bn cnN)]3 3] 4] 5] 6] [327] [328] [329] [330] 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' , ] [332] [333] [334] 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 [335] [336] [337] 02E H b [338] n 82E Hn 2 [339] a2E Hb. = H = [340] The system of equations can be simplified to the matrix form: Hbnbn Hbnn ]bn L Jbn [341] 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[342] 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. [331] and [332] are equal to zero (or within a specified tolerance.) Solving for the second partial and cross derivatives: 02E Hbb Hbnbn 2 2 abn [344] 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 [346] 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 [347] +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 [349] 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 = [350] 1 + exp(b, cN) S=l+exp(b cnN) [351] Nn 1 1 exp(b, cN) [352] Nn In A1= b cN [353] N. In AN N =c.Nb [354] 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 ) lnN A)' N b 2)2 [355] l n N2 _~ln) n I n.A N N In A NN n 2 ( [356] 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 [357] 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 " ) [358] np 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 [359] 82E HA = Hbn =Ab [360] 32E H Ac = H CA [361] A c nAn OAnoc Previously it was found from Eq. [323] that: E = 2 N 1 [362] 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)[363] + 2 y A n ] 1 S_ 1+ exp(b cnN) 1 + exp(b cN) ] OE= 2 f [[N. +2y [[ An [364] OA, I+ exp(b, c N) Z [I+ exp(b c Nc)]2 H2 = 2 [365] A 2En [1+ exp(bn cN)]2 Previously it was found from Eq. 327 that: E 4 N. exp(b cnN) 2 exp(b cnN) b, + exp(b cn)]2 [ + exp(bn cN)] [366] Taking the derivative of this with respect to A : 82E H Anbn = HbnAn , = 2 [N +exp(b cN) 4An exp(bn cN) [367] N. [1+ exp(bn c N)]2 [1 + exp(b c N)]3 Previously it was found from Eq. [330] that: OE F2A NNexp(b cnN) +2A Nexp(b cnN) [368] 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 [369] = 2 NNexp(b cN) +4A Nexp(b c cN) 369] 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 [370] H cnAn H CA H cnn To compute covariance and standard error, the inverse of [370] 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] [371] 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] [372] The standard error is given by (Overman and Scholtz, 2002): SE(a) =(s 2H )1/2 [373] 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. [357] and [358], 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 nonnormalized version of this is: E= [N ^J. + [N+N +[Y2 [374] 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(bcN) 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. [375] 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. [376] .N) [377] c N) [378] [320] dE =0 [321] Obn dE = 0 [322] acc dE S= 0 [379] 8Ab dE = 0 [380] 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, [381] For the case of common b : Al A2 E =ANl ", + N2, "A2, [382] 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 nonnormalized 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 [383] ,A1 [ 1+ exp(b cl,% N1l)L + exp(b clc Nl) jNlu +y ]p,[l1" ] =O 0 [384] 1 + exp(b cl Nl) [I + exp(b, cl N1)]2 Al 1 )]2 =j[ %" [385] [1 + exp(b cl N1)]2 1 + exp(b clN) [385] A r+ exp(bn cl ) [386] Al,= [386] [1 + exp(b, clN1l)]2 Similarly, 2, 1+ exp(b c2n N2) A2 = [387] 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 N1F77 Jbn = l+ exp(b, c1,N1) (l+exp(b cl N1))2 [388] 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 [389] N2 A2,, OE !_ 2 1+ exp(b c2. N1) Jc2, 2C [390] J" Oc2, A2 N2exp(b c2 N2) L( + exp(b c2 N2))2 HAI Al 2 = 21 [391] H a A1n ( + exp(b cl N1))2 HA2 A2 2' 1 [392] 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 [393] +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 + [394] [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 [395] 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 [396] H2E 22E H1,b" Ab 1 2 bbnAln b2 EbA1 A b [397] 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) [39 Z [1 + exp(bn c2,,N2)]2 1[ + exp(b c2,N2)]3 d2E d2E H = H,  dAl1 c l "C' cc AlA1 [399] Nl,,Nlexp(bn clN1) N+4AI I exp(b cl,,N1) [399] L [1+exp(bn clNl)]2 1 [1 +exp(b, cl N1l)] d2E d2E H A2cl  H 2 0 [3100] n2, c" 8A2 acl ,'"A2" cl, 2A2 82E 82E H Al H = 2 0 [3101] A," 8Al c2 c2 1 c2 Al 82E 82E H A2 c2, A2 c 2nA2n Hc2 ,A2 N2,N2exp(b c2 N2) [3102] [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 [3103] 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 [3104] [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 [3105] 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 [3107] 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' [3108] 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 [3109] 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 [3110] 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 [31 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 [31 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. [371] and [373], 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= [3113] np 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  _] [3114] = "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) [3115] A1 = + 2 " =0 [3116] OA [l + exp(bl c1 c N)] l+ exp(bl2 cN. ) A2 = [3117] y1 A1 + exp(b2n cnN2)] [3117] 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(blcN1))2 [3118] Jb" 81 O 1 + exp(bl, cN1) ](I + exp(bl cn N1))2 aE Jb2, 2 Ob2n 2 N2 A2n A2 exp(b2 cN2) [3119] 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 [3120] 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 [3123] 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 [3124] 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, cN1)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 [3125] H2E H 2 E A "A'" dA 2,A = A2 ,1 [3121] [3122] S2E H AI, A2 A A2 [3126] H 2E H 2E HAl,,abl n bA l b n bln Aln bl Al [3127] 2! Nl,, exp(bl cN1) 4A exp(bl cN1) [3127l) [1 + exp(bl, clN)]2 41 [I + exp(bl,, cNl)] HbE H0E H b2 HbAI =b2 E1 0 [3128] b2" Al Ob2 n b2 nAl 2E H a Al"c" 8Al Sc 02E H c"Al" Bc A41 [3129] 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 [3130] 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 [3131] cN2)]  c N2)] 2E H A2 . Sc,,8A2,, S N2,,N2 exp(b2n cN2)' 1 [l+ exp(b2n cN2)]2 + 4A2 N2exp(b2ncnN2) [l1+ exp(b2n cN2)]3J [3132] a2E H b b2 b2nbl" 8b2,8bl, b 2E H 2 O2, oblnb2, [3133] 82E 2E H =H bl"c" Obl Cn c"bl" nOcbl1 = 2A N1, Nlexp(bl cNl) 2N1l N[exp(bl cnN1)]2 [3134] [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 [3135] 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 [3136] &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 [3137] 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,, [3138] 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. [371] and [373], respectively) can be used. Again, however, the variance now includes both data sets: Aln 2 1 1+ exp(bl, cN1) [ s = [3139] +np 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 [3140] 1 1 + exp(bn cN1) I 1 + exp(b cnN2) Using the same approach as in the previous case: E =21 Nl C 10 [3141] MAn 1+ exp(bn c l)N1 1 + exp(b cN1) [ NI. 1+ exp(bn cN1)[3 Al = c [3142] 1 [ + exp(b cN1)]2 N 2 , A2 xpb 2)] [3143] 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 [3144] + 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 [3146] A A A12 21 (I + exp(b, cN1))2 H 2E =2 + [3147] 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 cnN2)]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 [3148]  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 cNN2)]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 [3152] 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, [3149] [3150] [3151] 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 [3155] 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 ] [3156] 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 [3157] 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 [3158] 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. [371] and [373], respectively) can be used. Again, the variance now accounts for both sets of data: 2 Al 2 1 N 1 + exp(b cN1) s [3159] 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" [3160] 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[3161] +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 [3162] 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) [3163] +2A A2 exp(b, cN2) aZL =2 1+ exp(b, cN2) L(1 + exp(b cN2))2 ] F FE A1 A N12exp(b cN1) [3164] (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 [3165] 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 [3166] 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 [3167] [3168] [3169] 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 [3156] &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 [370] HcnAn HcAbn Hcncn Once the inverse of this matrix is obtained, the equations presented earlier for covariance and standard error (Eqs. [371] and [373], respectively) can be used. Again, the variance includes both sets of data: 2 1 1 1+ exp(bn cNNl) [3171] np 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] [3172] An The corresponding error sum of squares is: E =Z[N N,' Z N N~'N N [exp(Ab) ] [3173] A, For the case of two data sets ("1" and "2") the error sum of squares is: E= [Nl ~, + [N2c N2c [3174] E=Z N1 N 1, Nl l E= NlN Nli N "iNiu [exp(Ab)1] Al 12 [3175] + 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 [3176] 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 [3178] A2,, 1 A2, Setting equal to zero, rearranging, and solving for Ab: 0 SAb N N1 N[3178] = 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,, [3179] + 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] [3182] 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 [3183] + 1[N,[Nl_ NI +2N N1C NI. exp(Ab) Al Al Z[FNl1u Ni, lexp0xb N1lL = LI A2 [3184] N1, N1, exp(Ab)l Al A1, Similarly: SN2C [N2 N2, exp(Ab) O2E +IA2n A2n = 2Y [3185] A2n A2n 2 N2 N2 exp(Ab)1 N2,, z[N2 exp 2 [3186] 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. [3184] and [3186], respectively, and these new values of Nl~, and N2,, can then be used in Eq. [3181] 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 [3187] =2exp2LAb) 1 +N A2 [3187] 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.rIN +NH 2 OAb2 Al A2 SNNl Nl 1 ~ 1 [3188] 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 [3189] +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, [3196] [3190] N2 N exp(Ab) I A2 I [3191] NI exp(Ab) Al 12 [3192] [3193] [3194] [3195] Once the inverse of this matrix is obtained, the equations presented earlier for covariance and standard error (Eqs. [371] and [373], 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 = [3197] S+ N2 N N21 N2 [exp(Ab) l] np 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_ NlNc NNl exp(Abl)+ NcINI 1 2 [3198] + N2 N NcN2 exp(Ab2)+ Nc N2 A2 A2 Taking the derivative with respect to Abl: Nlc N, NcINI exp(Abl)+ N1N1l OE A' n Al1 A 2Y A [3199] Setting equal to zero and solving: NIc N NNI" 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 [3200] Nc N1,A Nj1 N1, A,1L Al, [3201] NANI, Al [3202] [3203] =2 +2Y [3204] 1, exp(Abl)1 A Al N2 N2 : NNd + A exp(Ab2)12 + N2,N2 AS 2 exp(Ab2)1 +AI IA, NA2, [3205] Nl Nu N exp(Abl)1 A A, A + IN2c N" N2 exp(Ab2) i[h A2n A2 Nc I I 2 n 2A2 [3206] 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 [3207] +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 [3208] 2exp+ (Ab N N exp(Ab2) 1 A2l A2 The second derivative with respect to Ab A is: =4exp2 (Ab2l)  8Abl2 A\ Abl2 A12 [3209] 2exp(Abl) N1Nc t [I b [A2, Al,, Similarly, the second derivative with respect to Ab2 is: =4exp2(Ab2) AZb22 'L^ A2 LA2[3210] 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 [3211] 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 [3212] cN,8OAb2 IA2 A2, A2,, The Hessian matrix for standard error analysis in this case is: HAblAbl HAblAb2 HAb1lNe H= HAb2Abl HAb2Ab2 HAb2Nc [3213] 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 [3214] HNcAbl HNcAb2 HN,,N Once the inverse of this matrix is obtained, the equations presented earlier for covariance and standard error (Eqs. [371] and [373], 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 [3215] np + 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 [3216] + 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 ) [3217] 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 Ncc+ NN2 FNcN27 Ab = 1n A2 [3218] 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, [3219] 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 [32 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, [32A, L [N1, N1,, + N N1,, NN1, [32 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 [3222] 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, [3223] 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 [3224] HNcAb HNciNc Once the inverse of this matrix is obtained, the equations presented earlier for covariance and standard error (Eqs. [371] and [373], respectively) can be used. The variance for this case is: N1 Nc NNl [exp(Ab)1] 2 1 Al s = [3225] np + 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 nonnormalized 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) [3226] 11 J1 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)) [3230] [3231] 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) [3227] [3228] [3229] 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 [3232] j))]2 [3233] [3234] 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 (ic (i)N(i, j))]2 = 2An(i) ] [3235] 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) [3236] l=1 nds Hbb, = Hbb (ii) [3237] 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) [3238] i=1 nds H,,, = (i,i) [3239] 1=1 Note Eqs. [3236] through [3239] also apply for both common bn and cs, as does the following equation. nds Hb,,c = Hcb = aHbcn (i,i) [3240] =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) 11 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)] [3241] [3242] [3243] [3244] [3245] [3246] [3247] .( ). 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(() JLJL 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 [3252] 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) [3253] OE 8N,, (i) [3248] (2E) /N', (i> [3249] 02E aNc, (i)8Ab(i) 1][N(i) exp(Ab(i)) An (i)0 [3250] Ab(i) = In N (i)N (i, j) An (i) [3251] nds HAbAb = Hbb(i,i) [3254] Sn A A(i) Ab = In 2 [3255] 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 ^= [3256] ON1 ,, 11 N,, (i) nds HN, = HN,, (i,i) [3257] 1 1 nds ndp(i) (, j NA (i,) N. (i, j) exp(( 1 I N(ij)A (i A (i) N 1 = [3258] 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 (,) [3259] 1=1 Analysis of Variance (ANOVA) for Commonality of Parameters (Ftest) 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 Ftest is employed similar to that outlined in Ratkowsky (1983). For the purposes of this study, the Ftest 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 + J1 dSS A,2 (i) N2 (i)exp (Ab(i)) RSS = [3260] S NC (i) exp (Ab(i)) [Y(i,j) Y(i,j)1 + 1 A2 (i) where N,, N9, and Y are modelcalculated values (as functions of applied N only): Aj(i) N (i, j) =Ai) [3261] 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) [3262] 1 + exp(b (i) c,, (i)N(i, j)) Y(i,j) =i) [3263] 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 Ftest 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 Ftest 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 [3264] df, = n p [3265] F = (RSS RSSA)/(df dfA [3266] RSSA /dfA Once F is calculated, it is compared to Fo(dfBdfA, dfA, a) (which is obtained from the appropriate F distribution tables). If the following holds true: F > F, [3267] 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. [3267] 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 Ftest. 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 41. 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 42 through Table 49. 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 41 through Figure 420. 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 19823). 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 410. The simple standard deviation was also taken for the parameters in the groups, as shown in the corresponding Table 411. 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 ha1, 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 ha1, while tall fescue with commercial fertilizer had a mean An of 342 kg ha1, 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 ha1, 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 ha1 for commercial fertilizer and 318 kg ha1 for broiler litter), but it is less apparent if only tall fescue is considered (342 kg ha1 for commercial fertilizer and 331 kg ha1 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 ha1, and that for all nonirrigated treatments was 345 kg hal, 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 ha1 for irrigated bermudagrass versus 452 kg ha1 for nonirrigated bermudagrass, and 433 kg ha1 for irrigated tall fescue versus 251 kg ha1 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 kg1 for commercial fertilizer and 0.00625 ha kg1 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 kg1 for all bermudagrass and 18.2 g kg1 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 412 through Table 425. Each of these tables includes the value of parameters assumed invariant (according to mode) and the resulting Fvalues from comparison of the variances of each mode to the most general one, mode (0). Critical Fvalues (obtained from an Fvalue 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 Fvalue 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 counterintuitive 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 Ftest, 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 412, 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 412 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 426 throughTable 434. 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 Ftest The rare occurrence of negative Fvalues, 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 Ftest as it was employed in this study to test for commonality of parameters. The Ftest, 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 