Methodological Procedures in the
Economic Analysis of
Fertilizer Use Data
r
i p
PARTICIPANTS IN THE SYMPOSIUM FROM WHICH THIS PRESENTATION HAS BEEN DEVELOPED
Front sitting, L to R: D. D. Mason, E. L. Baum, John T. Pesek, Frank F. Bell, and Leland G. Allbaugh.
Rear Standing, L to R: Roger C. Woodworth, Clifford G. Hildreth, W. G. Brown, Earl R. Swanson, Donald D.
Ibach, Glenn L. Johnson, Earl O. Heady, Travis P. Hignett, R. L. Anderson, W. L. Parks,
John Blackmore, and Earl W. Kehrberg.
hUCIB \. ~I'. E
iElilsPil ~1C.`~"
Methodological Procedures in the
Economic Analysis of
Fertilizer Use Data
E. L. BAUM
Agricultural Economist,
Tennessee Valley Authority
EARL O. HEADY
Professor of Economics,
Iowa State College
Edited by Economic Consultant,
Tennessee Valley Authority
JOHN BLACKMORE
Agricultural Economist, FAO,
The United Nations
Formerly, Chief, Agricultural
Economics Branch, Tennessee
Valley Authority
THE IOWA STATE COLLEGE PRESS Ames, Iowa, U.S.A.
1956 by The Iowa State College Press.
All rights reserved.
Library of Congress Catalog Card Number: 567375
LITHOPRINTED IN THE UNITED STATES OF AMERICA BY
CUSHINGMALLOY. INC., ANN ARBOR. MICHIGAN. 1956
Foreword
Economic considerations in fertilizer use are only now beginning to
get the attention of research workers. Problems of fertilizer produc
tion and use have been studied for many years, but for the most part
attention has been directed primarily at the technical aspects of the
problems. As the fertilizer industry continues to adopt new processes
to produce improved products, there will be a continued need for more
technical studies. In addition, however, there is a tremendous need for
research on the economic aspects of fertilizer use.
Fertilizer is a major item of expense on many farms and an increas
ingly important factor of production on many other farms in the United
States. Farmers are interested in knowing how much and what kinds of
fertilizer to use to maximize their profits. They also seek information
on how to buy their fertilizer needs at the least cost. Agricultural ex
tension workers and others, who provide production planning advice to
farmers operating under widely varying conditions, recognize the need
for more and better information on the economic aspects of fertilizer
use. Research workers thus are being called on to conduct the research
necessary to answer agronomiceconomic questions basic to the develop
ment of practical fertilizer recommendations.
As an aid to these research workers, TVA sponsored a symposium
in June 1955, bringing together a group of economists, agronomists, and
statisticians. The papers which they presented have served as the basis
for this book. The objective of this book is the same as that of the sym
posium to present the most recent information and techniques bearing
upon some of the important questions involved in studies of the econom
ics of fertilizer use, thus facilitating the development of needed research.
LELAND G. ALLBAUGH, Director
Division of Agricultural Relations
Tennessee Valley Authority
Knoxville, Tennessee
September, 1955
Preface
A technological revolution has occurred in American agriculture
over the past quarter of a century. However, the role of chemical fer
tilizers in this change has largely escaped public notice. The public is
generally aware that fewer Americans now are engaged in farming, but
are producing more agricultural products; it knows that since 1930,
American farm income has risen from depression depths to levels un
attained in any other period in our history, or by farmers in any other
part of the world. Common is the knowledge that mechanization and im
provements in crops and livestock have contributed to these changes.
The impacts of the tractor and of hybrid corn are well known. However,
relatively few persons are aware of the phenomenal increases in ferti
lizer use associated with the increases in production and income.
Since the mid1930's, fertilizer consumption in the United States has
risen from 6 million to 23 million tons. In the Midwest, the increase in
fertilizer use has been particularly striking, being 78fold in Illinois
and 130fold in Iowa. In Tennessee, where fertilizers have been in com
mon use for many years, the increase has been 6fold during the period
19341954.
Another striking change in fertilizer use has been the rapid rise in
average analysis. In the 35year period 19001934, the plant food con
tent of American fertilizer increased by only four units. However, in
the 19year period 19341952, the increase amounted to nearly nine
units. This improvement in average analysis is a reflection of changes
in fertilizer production technology. First the United States Department
of Agriculture and then the Tennessee Valley Authority began programs
of research on fertilizer production problems. United States Depart
ment of Agriculture laboratories at Beltsville, Maryland, and the Ten
nessee Valley Authority laboratories and pilot plants at Muscle Shoals,
Alabama, made important contributions to fertilizer production technol
ogy. In addition, fertilizer firms expanded their investments in research.
The result of all these research efforts has been an expanding fertilizer
industry in which obsolete plants and processes are being replaced by
facilities to produce better fertilizers at lower costs to farmers. De
mand and use of fertilizer has grown similarly, and the trend will con
tinue upward given economic stability and further research and educa
tion in the production and use of chemical fertilizers.
The need for research on fertilizers and fertilization at a time when
the Nation's warehouses are filled with stored food items and when pro
duction controls are in use may be questioned. However, the ultimate
economic goals of a society are never reached by placing restraints on
imagination and ingenuity in research. Moreover, the research with
PREFACE
which this book deals is directed to problems for which solutions are
to be sought five years or more in the future. With a rapidly growing
population, and one which is increasing in urbanization, farm technolo
gies need to be improved still further in order to increase farm output
in the decades ahead. Then, too, a more efficient farm industry with
more production from fewer resources is in line with national economic
growth. Further strides in efficiency allow food production with a mini
mum of resources, so that the Nation can produce more of those semi
luxury and other goods which characterize a wealthy society.
Research on fertilizers and crop response to fertilizer use is greatly
needed in other parts of the world, even more so than in the United
States. Fortunately, the United Nations and the United States, with their
technical assistance programs, have assisted in maintaining the increase
of the world's food supply at a level, equal to the need. This is not to say
the problem of hunger has been completely solved. Without being famine
stricken, there still are millions of people who are chronically hungry 
people who are alive but who are so poorly nourished that they lack the
energy to work efficiently. More important, their physiological status
prevents full development of the capacities of the human resource and
the personal satisfactions which accompany such developments.
Hungry people fall easy prey to diseases, not only diseases of the
body but also diseases of the mind, robbing man's faith in his ability to
govern himself. If all mankind is to be provided with enough to eat,
there is need for further increases in crop output. In many parts of the
world, chemical fertilizers are a primary need to this end.
In recent years there have been rapid advances in the chemical and
engineering aspects of fertilizer production. Agronomic research has
been greatly improved; data from many experiments carried on more
than five years previously are now obsolete. Still, as leading agrono
mists recognize, there is need for further improvement and expansion
in agronomic research on fertilizer use.
Until very recently, economists have given little attention to ferti
lizer use as an area for empirical research. Still, nearly every elemen
tary economics text uses fertilizer examples to illustrate the principle
of diminishing returns. Because of lack of data, however, these exam
ples have been based on hypothetical cases. While students may assume
that research workers have thoroughly explored the production relation
ships between fertilizer use and crop use, such is not the case.
The early work of Mitscherlich and Spillman serves as a landmark
on fertilizer response curves. While it appears strange that Spillman's
work was not extended by any significant research on the fertilizer re
sponse economics until recently, there are sound reasons for this phe
nomenon. First, until quite recently, relatively few agricultural econo
mists had enough training in mathematics and statistical techniques to
use this type of analysis. Emphasis on econometrics in the graduate
training has provided a larger number of economists with the requisite
training. Second, there has been an overspecialization in agricultural
research. Specialization can be an aid to efficiency, but in many state
PREFACE
agricultural experiment stations, specialists have isolated themselves
from each other by walls of administration and communication. In
many instances, the barriers between agronomists and farm economists,
for example, have grown too great for productive research in the physi
cal response and economic use of fertilizer. Recent developments in
interdepartmental cooperation promise to remove this barrier.
However, even though agronomists and economists realize a need to
work together, there often is a formidable barrier in their conceptuali
zation of the problem. Economists are inclined to look at a problem of
fertilizer use as a production problem in which there is a functional re
lationship between input and output, with the relationship generally non
linear in nature. On the other hand, some agronomists in designing
their research have conceptualized their problem as one of comparison
of discrete phenomena. Although these concepts are not necessarily in
consistent, they increase the difficulty of interdisciplinary cooperation.
Economists find it difficult to understand why agronomists have not
pushed their rate trials higher; why they are so insistent on numerous
replications; and why they have avoided multivariable experiments.
Agronomists, on the other hand, have been dismayed by the complex
terminology and models employed by economists to describe ideas
which otherwise seem essentially simple.
The framework for carrying on fertilizer research, particularly that
to be used in farm decisionmaking, appears to be on the verge of rapid
change. Acceptance of the concept of the farm production unit in terms
of the economist's model of a "firm" gives the new perspective to the
role of physical research. Recommendations on all production practices
and enterprises must fit together in an economic sense if the farm is to
maximize returns. The farm operator, not the production specialist or
economist, should make the choice of the types and combinations of pro
duction factors or practices to be employed, as well as the types and
amounts of products to be produced. He must make these selections in
terms of his capital, ability to stand risks, and family's goal as a con
suming unit. The production specialist, or the economist, cannot supply
a single "best answer" to a production problem if the farm is considered
a firmhousehold combination. The farmer must be given data from
which to fashion a plan to fit his own particular circumstances. Accord
ingly, the data from research may need to take special forms, such as
that explained in the chapters which follow.
Farm production economists, because of their concern with the farm
as a whole and the economic aspects of planning, are proving to be use
ful collaborators in many kinds of production experiments. Then, too,
farm production research workers are gradually abandoning the practice
of giving the results of completed research to economists with the re
quest that they "analyze" the economic results of data which already
take on a "predetermined" form. Production scientists are increasingly
seeking the assistance of economists along with statisticians, in initia
tion of design of experiments and in statistical analyses which conform
to the economic models used in decisionmaking.
PREFACE
This changing trend towards greater interdepartmental cooperation
and increased use of economic models in fertilizer research was the
basis for planning the Tennessee Valley Authority sponsored symposium
on methodological procedures in the economic analysis of fertilizer use
data. If economists are to contribute effectively to such research, they
should have a broad understanding of agronomic research and of statis
tical methodologies, as well as the economic principles and methods of
economic analysis. The symposium and this book were planned with
these needs in mind.
Part I of this book includes statements on the overall methodologi
cal problems involved in estimating and using fertilizer response func
tions. It indicates the practical uses which can be made of improved
inputoutput data, the fundamental economic relationships in fertilizer
responses, how these can be applied in considering prices and the capi
tal situations of farmers, and how research workers from different dis
ciplines can work together on designing and initiating fertilizer research.
Part II deals with fundamental statistical problems involved in designing
experiments and estimating functions of fertilizer response. It consid
ers designs in relation to analytical models and statistical efficiency,
alternative algebraic forms of functions as these relate to alternative
designs and predictions, and discrete and continuous models in relation
to both experimental design and farmer recommendations.
Part III relates to the agronomic problems of conducting experiments
from which production functions can be designed. It considers the size
and type of the experiment in relation to the resources and personnel
available. It also includes detailed discussions of soil, moisture, cul
tural practices, and other variables as they relate to conducting experi
ments. The feasibility of using soil test data is discussed and examples
of fitting standard curves to existing agronomic data are included. Part
IV deals with the application of improved response data. It indicates
the type of data needed in farm and home planning programs. Examples
are included, showing how budgeting and linear programming can be
used to relate fertilizer to the whole farm business. Finally, simple
monographs are used to illustrate how complex estimates can be trans
formed to provide simple calculations for the extension worker or
farmer.
Part V presents important trends in fertilizer use and costs. It in
dicates developments which have taken place in the relative price, pro
duction, and use of particular plant nutrients. It traces developments
in the source and processes for nutrients. Finally, it outlines some of
the prospective trends and problems in fertilizer use.
A debt of gratitude is owed particularly to those in the Tennessee
Valley Authority's management who made possible this symposium and
its reporting in this book, and to the Iowa State College Press, through
which publication was effected. Appreciation is extended to Lois R.
Carr, Helen P. Long, and Mary L. Robinette, Division of Agricultural
Relations, Tennessee Valley Authority, for their fine cooperation in pre
paring the manuscript for publication.
PREFACE xi
The editors believe that the information presented in this book will
contribute materially to the improvement and expansion of research in
the economics of fertilizer use.
E. L. BAUM
Tennessee Valley Authority
Knoxville, Tennessee
EARL O. HEADY
Iowa State College
Ames, Iowa
JOHN BLACKMORE
FAO, United Nations
Rome, Italy
September, 1955
Table of Contents
PART I: OVERALL METHODOLOGICAL CONSIDERATIONS
1. Methodological Problems in Fertilizer Use ........ 3
Earl 0. Heady, Iowa State College
2. Interdisciplinary Considerations in Designing Experiments
to Study the Profitability of Fertilizer Use . . ... 22
Glenn L. Johnson, Michigan State University
PART II: FUNDAMENTAL DESIGN AND
PREDICTION PROBLEMS
3. A Comparison of Discrete and Continuous Models in
Agricultural Production Analysis . . . . . ... 39
R. L. Anderson, North Carolina State College
4. Discrete Models With Qualitative Restrictions . . ... 62
Clifford G. Hildreth, North Carolina State College
5. Functional Models and Experimental Designs for
Characterizing Response Curves and Surfaces . . 76
David D. Mason, North Carolina State College
PART III: AGRONOMIC AND RELATED CONSIDERATIONS IN
EXPERIMENTS AND FITTING FUNCTIONS TO
EXISTING DATA
6. Agronomic Problems in Securing Fertilizer Response Data
Desirable for Economic Analysis . . . . ... 101
John T. Pesek, Iowa State College
7. Methodological Problems in Agronomic Research
Involving Fertilizer and Moisture Variables ...... .113
W. L. Parks, University of Tennessee
8. Some Problems Involved in Fitting Production Functions
to Data Recorded by Soil Testing Laboratories ..... 134
Earl W. Kehrberg, Purdue University
TABLE OF CONTENTS
9. Evaluating Response to Fertilizer Using Standard
Yield Curves .......................142
D. B. Ibach, United States Department of Agriculture
PART IV: APPLICATION OF DATA
10. Practical Applications of Fertilizer Production Functions . 151
William G. Brown, Iowa State College
11. Organizing Fertilizer InputOutput Data in Farm
Planning ......................... 158
Roger C. Woodworth, University of Georgia
12. Selecting Fertilizer Programs by Activity Analysis . . 171
Earl R. Swanson, University of Illinois
13. Fertilization in Relation to Conservation Farming and
Allocation of Resources Within the Farm . . . .... 188
Earl O. Heady, Iowa State College
PART V: TRENDS IN USE AND MANUFACTURE OF
FERTILIZER
14. Our Changing Fertilizer Technology . . . . ... 203
T. P. Hignett, Tennessee Valley Authority
INDEX ............................... 215
List of Figures
Figure
Number Title
1.1.Possible effect of weather variations on the distribution
of fertilizer production functions . . . . . . 9
1.2.Production surface and yield isoquants for nutrients
& 1.3. which are technical complements .... . . . .13
1.4.Yield resource curves for nitrogen with P205 fixed at
different levels ....................... 14
1.5.Predicted yield surface for corn . . . . .... 15
1.6.Predicted yield isoquants for corn . . . . ... 16
1.7.Isoclines showing equal nutrient ratios in relation to
yield isoquants ............ ... ...... .18
1.8.Isoclines for corn and alfalfa showing convergence to
& 1.9. maximum yield ....................... 19
2.1.Schematic presentation of continuous corn experiment,
Michigan State University, 1953 . . . . ... .. 35
3.1.The Box and Wilson composite design for estimating
quadratic surfaces ..................... 49
4.1.Cornnitrogen price map . . . . . . .... 74
5.1.Yield contours (bushels per acre, corn) for 4 x 4 factorial
experiment ......... ........ ....... 88
5.2.Yield contours, 4 x 4 factorial experiment with corn.
(Data are in bushels per acre). X's on linear scale . 89
5.3.Yield contours, 4 x 4 factorial experiment with corn,
for square root transformation of X's . . . .... 89
5.4."Value" contours, after adjustment for cost of fertilizer
and stand. Assumed constants: Corn = $1.40 per bushel;
Nitrogen = 18 cents per pound; Stand = $1.00 per 1000
plants per acre ........................ 90
5.5.Design configuration and treatment means for multifactor
potato yield experiment (yields are pounds U. S. No. l's,
per 2 row, 25foot plot) ................... 93
LIST OF FIGURES
5.6.Yield contours for U. S. No. 1 potatoes, pounds per plot,
for variations in Xi (N) and X2 (P20), with Xs (K20) held
at +1 (50 pounds per acre) ................. 94
5.7.Yield contours of lettuce tops (gms. dry wt.) as affected
by additions of Cu and Fe to nutrient solutions containing
Fe+2 NIl + NO; and the middle level of Mo. Observa
tional points and yields are underlined. The point at the
center of the contours is the predicted maximum yield. .. 95
7.1.Volume composition of a Maury silt loam soil . . . .114
7.2.Curves showing the relation between the soil moisture
tension and the moisture content of the soil . . ... 116
7.3.Clay particle surrounded by ion swarm . . . ... 118
7.4.Plant root in contact with soil particles . . . ... 119
7.5.Yield of grain and water use by corn with different irriga
tion treatments. L. S. D. (5% level) for yield = 21
bu/acre . . .. . . . .. .. . . . .. 123
7.6.Effect of nitrogen and moisture variables on the
composition of a meadow . . . . . . .... 124
7.7.Effect of moisture tension on leaf elongation . . . 126
7.8.Effects of nitrogen and moisture on the yield of grain
sorghum .. .. .. .. ..... ... .... ....127
7.9.Effect of nitrogen, moisture, and spacing on the yield of
grain sorghum ..... .................... .128
7.10.The effect of nitrogen on forage yield and water utilization
of Coastal Bermudagrass . . . . . . ... 129
8.1.Correlation of independent variables . . . . . .. 139
10.1.Yield isoclines and isoquants for corn on IdaMonona
soil, Iowa. Optimum rates are indicated by dashed lines
representing the nitrogencorn price ratio . . . ... 153
10.2.Yield isoclines and isoquants for alfalfa on Webster soil,
Iowa. Optimum rates are indicated by dashed lines
representing the phosphorusalfalfa price ratio . . 156
11.1.Corn yield response to nitrogen for three soil types . 159
11.2.Corn yield response from nitrogen for three fertility
situations on Hayesville clay loam . . . . . .. 161
11.3.Returns above nitrogen cost from the use of nitrogen for
corn on low fertility Hayesville clay loam . . . ... 162
11.4.Effect of management on response to fertilizer . . 163
LIST OF FIGURES xvii
11.5.Corn yield response from nitrogen on low fertility
Hayesville clay loam for different years . . . . 164
11.6.Returns from the use of nitrogen on corn for
low fertility Hayesville clay loam . . . . . .. 165
11.7.Relationship between different soils and yield of
cotton and corn when fertilizer and other production
factors are used in "optimum" amounts . . ... .167
13.1.Net income predicted for typical 160acre farm on
IdaMonona soils of western Iowa . . . . .... .192
13.2.Net income on typical farm using variable discount rates .194
13.3.Use of additional fertilizer to reduce the income gap
on farm shown in figures 13.1 and 13.2 . . ... .195
14.1.Annual U. S. consumption of primary plant nutrients
in fertilizer .. .. . .. .. . . . . .. .. .205
List of Tables
Table
Number Title
2.1. Rates of Fertilizer Application, Continuous Corn
Experiment, Brookston Soil, Michigan, 1953 . . ... 35
4.1. Corn Fertilization Data . . . . . . . .... 69
4.2. Analysis of Variance for Equation . . .. ........ .. 70
4.3. Estimates of Coefficients in Equation . . . . ... 70
4.4. Estimated Responses to Nitrogen for Coastal Soil and
Two Types of Weather ................... 72
5.1. Observed and Predicted Yields by Three Functions
for Corn Yields, 1952 .................... 81
5.2. Yield of Oats in Pot Experiments with Varied
Phosphate Dressings and Varied Water . . . . ... 83
5.3. Number of Constants To Be Fitted for Equations of
Varying Degree ....................... . 85
5.4. Yields, Bushels per Acre, As Influenced by Variation
in Plant Stand and N Levels . . . . . . ... 86
5.5. Analysis of Variance of 4 x 4 Factorial Experiment . .. 87
5.6. Rates and Coded Values Used in Potato Fertility
Experiment ................... ...... 91
5.7. Treatments, and Treatment Means for Three
Replications, Potato Fertility . . . . . . .... 91
5.8. Regression Coefficients and Their Standard Errors,
and the Analysis of Variance for Second Degree
Surface for Data in Table 5.7 . . . . . . ... 92
5.9. Analysis of Variance of Yield . . . . . . ... 94
6.1. Comparison of the Variation in Three Experiments as
Determined by Deviations from Regression, and
Among Plots Treated Alike . . . . . . . ... 109
7.1. Volume Percent Composition and Bulk Density of
Four Tennessee Valley Soils . . . . .... . 114
LIST OF TABLES
9.1. Comparison of Results from Graphic and Mathematical
Solutions for LeastSquares Fit When Applied to
Three 12Rate Experiments Involving Nitrogen on
Irrigated Corn .................... . 144
9.2. Sums of Squared Residuals Explained by Exponential
and Quadratic SquareRoot Equations as Applied to
Three 9 x 9 Partial Factorial Experiments . . . ..145
9.3. Sums of Squares Reported from Calculated Yields ... .146
10.1. Bushels of Corn per Acre for Varying Levels of Fertilizer
on Calcareous Ida Silt Loam Soil in Western Iowa in 1952 .152
11.1. Corn Yield Response to Nitrogen on Hayesville Clay
Loam When 60 Pounds P205 and 60 Pounds K20
Are Applied per Acre .................... 160
11.2. Soil Characteristics Fred Nichols' Farm of
160.6 Acres in Catoosa County . . . . . .... .168
11.3. Response to Fertilizer Corn on Hayesville Clay Loam,
Georgia, 610% Slope, Soil Test P2Q, Low, K20 Medium .169
12.1. Lime and Fertilizer Requirements for "BuildUp" and
"Maintenance" (Pounds per Rotation Acre) Muscatine
Silt Loam, Starting with "Low" Phosphate Test a
ThreeTon Limestone Requirement, and Adequate
Potassium . . ......................174
12.2. Total Capital Outlays for Lime and Fertilizer Required
for Various Phosphate "BuildUp" Programs . . . 177
12.3. Optimum Cropping and Livestock Systems for Various
Situations on a 200Acre Muscatine Silt Loam Farm. . .180
13.1. Description of Processes or Activities . . . . .. 196
13.2. Optimum Plans Under the Various Capital Situations . 198
14.1. U. S. Fertilizer Consumption and Composition ..... .204
14.2. Adjusted Wholesale Price, Bulk, F.O.B. Works or Ports,
1955, Dollars per Unit of N, P205 or K20 . . . . 206
14.3. Estimated Consumption and Adjusted Wholesale Prices
of Fertilizer Nitrogen, Phosphate, and Potash by Sources 207
PART I
OverAll Methodological
Considerations
> Practical Uses
> Predicting Production Functions
> Economic Interpretation of Data
> Interdisciplinary Cooperation
EARL O. HEADY
Iowa State College
Chapter 1
Methodological Problems in
Fertilizer Use
HE central methodological problem in fertilizer use on a single
crop is prediction of the mathematical form and the probability
distribution of the response function. This is a task, of course,
for various soils, crops, and climatic situations. However, there are
other methodological problems which are auxiliary to this central prob
lem. They include: (a) the design of experiments to allow efficient pre
diction of the response function, and (b) the estimating procedure for
predicting the surface and optimum use of nutrients. Since the last two
problems are being given detailed treatment in other chapters, this
chapter will focus on the fundamental and basic problems which relate
to estimating the response functions.
Practical Importance of Knowledge in Response Functions
Although this chapter has the main objective of treating methodologi
cal problems in fertilizer economics, some of the practical or applied
aspects of these fundamental considerations need to be pointed out.
First, greater knowledge of simple, singlevariable response functions
can encourage greater use of fertilizer. The slope of the response func
tion represents the incremental or marginal yield due to small increases
in fertilizer use. The farmer with limited capital needs this information
in determining how much fertilizer to apply. Knowledge represented by
a response function is more useful than knowledge represented by the
mean yield increase of one or two fertilizer (level) treatments.
Suppose a farmer with limited capital can earn $2.50 return on funds
spent for other lines of his business (such as tractor fuel, mule feed,
crop seed, or hog supplement). He is given information showing that
one discrete level of fertilization, 30 pounds of nitrogen, will increase
oat yield by 17 bushels. With oats at 70 cents per bushel and nitrogen
application costing 18 cents per pound, the total return is $11.90 and
the total cost is $5.40, a net of $6.50. However, the return per dollar
spent on fertilizer ($11.90 + $5.40) is only $2.20, and the farmer will
allocate his scarce funds where he can get $2.50.
Suppose, however, that the farmer is given even three points from
a response function showing: the first 10 pounds of N has a marginal
yield of 10 bushels; the second 10 pounds has a marginal yield of 5
bushels; and the third 10 pounds has 2 bushels marginal yield. With a
EARL O. HEADY
unit costing $1.80, the first 10 pounds returns $3.89 per dollar invested
in fertilizer, and the second returns $1.95. Hence, since the farmer
can realize only $2.50 elsewhere in his business, he now is encouraged
to invest in at least 10 pounds of N. With more detailed knowledge of
the response function, he may even invest in 15 pounds. Obviously then,
knowledge of the response function, coupled with information on the eco
nomics of fertilizer use, can encourage a greater investment in this re
source on that great majority of farms with limited capital. (See Chap
ter 11 for indications of use of these notions in farm planning.)
Knowledge of the response function is equally important for the
farmer who considers his crop in the environment of unlimited capital.
This is the case of tobacco producers; it is becoming the case of many
other farmers. It is known that the optimum or most profitable level
of fertilization for these farmers is defined by equation 1 where the
term to the left of the equality
dY Pf
(1 ) = Pf
(1) dF Py
is the marginal yield or response and the term to the right is the price
ratio (price per unit of fertilizer divided by the price per unit of yield).
The marginal yield is the derivative of yield in respect to nutrient; it is
the slope of the response function for any particular input level. This is
the type of information basic for making recommendations to farmers
who seek to maximize profits in a decisionmaking environment of un
limited capital.
It is obvious that the most profitable level of fertilization changes
as the term to the right of the equality changes. (Likewise the optimum
level of fertilization will change for the limitedcapital farmer previ
ously cited, as the price of crop yield, fertilizer, or any other product
or resource for his farm changes.) How much change needs to be made
in fertilizer use, as prices change, again depends on the slope of the re
sponse function. If the slope changes only slightly over a wide range of
fertilizer inputs, the loss (profit depression) from not shifting rates
can be great;' if the slope changes greatly over a small input range, the
farmer may lose but little in not adjusting his rates to price change.
Finally, greater knowledge of the response curve is needed as an
aid in farm planning and linear programming, to allow improved predic
tions of how and where fertilizer fits into the program of the farm as a
whole. If numerous points are known for the response curve, each sug
gested level of fertilization can be treated as an activity or investment
opportunity. The optimum level of fertilization relative to (a) other in
vestment alternatives (activities), and (b) complete farm organization
can then be predicted. Data in a form for this purpose will generally
encourage use of more fertilizer. The reason has been suggested
'This statement applies particularly where the previous price ratio was equal to a deriv
ative of the function high (low) on the response function and the new price ratio is equal to a
derivative low (high) on the curve.
METHODOLOGICAL PROBLEMS IN FERTILIZER USE
already: Knowledge of high marginal returns for small fertilizer inputs
can specify use of this resource, even by the farmer with very limited
funds. This knowledge also will indicate how far in the use of fertilizer
the farmer with more funds can profitably go.
The farmer is the only one who can make the decision as to the most
profitable quantity of fertilizer to use. Optimum quantity is determined
partly by the response function for his particular soil, tempered as it is
by previous soil management, weather, insects and pests, and other va
riables which are both endogenous and exogenous to his decisionmaking
environment. But aside from the purely physical and biological varia
bles of the fertilizer production function, the optimum quantity is as
much a function of present nutrient and future (crop) price ratios as it
is of the response ratios. Since prices, and even yields, are held with
uncertainty, the fertilizer recommendation must conform to the farmer's
uncertainty or riskbearing ability which includes (a) his equity position;
(b) his psychological makeup; and (c) other phenomena which cause him
to temper the quantity and kinds of the resources which he employs.
Refined estimates of the fertilizer response function can help provide
the basic data needed to guide these decisions which are unique to each
farm.
Knowledge of multivariable response functions also has great prac
tical implications. Anyone knowing the basic principles of production
recognizes immediately that the production coefficient for, and the re
turn from, any one input category is a function of the amount and kind
of other input categories with which it is combined. The economic po
tential in, and limits of, any one resource can be determined only by
studies which consider numerous input categories as variables. These
variables may include different fertilizer nutrients, seeding rates, seed
varieties, irrigation, and various other technologies. A fertilizer rate
study may show a much lower response curve for one nutrient, if it is
varied alone, than if it is varied along with another nutrient. Similarly,
a multivariable response study may be applied productively when a
new crop variety, which has a great yieldboosting effect, is discovered.
In much of the Midwest higheryielding varieties have little effect un
less used with sufficient fertilizer nutrients. A simple singlevariable
response study may fail to "lift the lid on yield potential," under new
varieties or other developments in technology. Finally, knowledge of
isoclines from multivariable studies provides a practical guide in
fertilizer manufacture.
Methodological Problems in SingleVariable Functions
A few practical applications of fundamental fertilizer research have
been presented above because (a) the practical problems and their solu
tions are the main goals of fundamental research and methodological
considerations, and (b) fundamental research can result in a greater
and more efficient use of fertilizer if it provides refinements for ob
taining more practical recommendations for the individual farmers.
(Practicality is characterized by recognition of the variables peculiar
EARL O. HEADY
to each farm, including capital, equity position, risk considerations,
and other economic variables, as well as physical and biological varia
bles such as the crop and variety, alternative nutrients, soil conditions,
etc.)
In discussing practical applications first, the cart has been put be
fore the horse. The remainder of this chapter will deal with the funda
mental science or methodological considerations in this instance, the
"horse."
Form of SingleVariable Function
For research on simple response functions with a singlevariable
nutrient, for a particular soil and management system, there are two
basic methodological problems, viz., (a) the appropriate algebraic form
of the response function, and (b) the betweenyear variability in the pro
duction function.
As far as this writer knows and as pointed out by Mason in Chapter
5, there is no biological proof that the fertilizer response function con
forms universally to a particular algebraic form of equation. It is likely
that the bestfitting form of the fertilizer production function varies by
crop, year, soil, or other variables. One algebraic form which has been
popular over time with research workers has been the Mitscherlich
Spillman type of function. One form of this function is equation 2,
(2) Y = m arF .
(Another form is shown in equation 2 of Chapter 5.) This function em
ploys specific assumptions about the nature of the response curve: (a)
It assumes that the elasticity of response is less than 1.0 over all ranges
of fertilizer applications, a condition likely to be encountered in most
situations but one which need not hold true universally (some experi
ments at particular locations show a short range of increasing returns).
(b) It assumes that fertilization rates never become so great as to cause
negative marginal products (i.e., declining total yields), since yield be
comes asymptotic to the limit m. (c) It assumes the condition of equa
tion 3,
AY AY AY
(3) 2 7 n_
namely, that the ratios of successive increments to total yield over all
fertilizer inputs are equal. Lastly, (d) the function assumes that where
two nutrients are involved, the maximum yield per acre can be attained
with a large number of nutrient combinations (i.e., it does not allow the
isoclines to converge at the point of maximum yield).
A function which also forces particular assumptions into the predic
tions, although these are considerably different from the Mitscherlich
equation, is the CobbDouglas function, listed as equation 4. It does not
METHODOLOGICAL PROBLEMS IN FERTILIZER USE
(4) Y = aFb
assume that the ratios of marginal yields are equal. However, it does
assume that the percentage increase in yield is constant and equal to b
for all increments of fertilizer. This assumption, illustrated in equation
5 below, may be as realistic as the parallel assumption of the Mitscher
lich equation.
Y Y Y
(5) Y Y
F AF V F
F F F
The CobbDouglas equation allows the yield to increase at either a
diminishing, constant, or increasing rate, although the response curve
can be represented by only one of these and never by a combination. If
total yield increases at a diminishing rate, the function assumes nega
tive marginal products and, therefore, that total yield becomes asymp
totic to some limit.
Somewhat more flexible functions are the simple quadratic and square
root forms indicated respectively as equations 6 and 7 below:
(6) Y = a + bF cF2
(7) Y=a+bVFcF.
These equations do not force certain of the elasticity and marginal ratio
restraints of the previous equations. Also, they allow the total yield to
reach a maximum, followed by negative marginal yields. Equation 6
may apply particularly where a maximum is reached with relatively
low fertilization level; equation 7 may apply where marginal yields
change rapidly over low fertilization levels but "straighten out" for
higher levels, if no other practices or inputs are limitational. But again
these functions may have no unique biological base. Is there a unique
biological base for response functions?
The research worker makes a biological (and at this stage of knowl
edge, a subjective) assumption when selecting a particular function.
Methodological effort should be devoted to proving either that (a) biolog
ical responses do follow particular mathematical forms, or that (b)
there is no unique algebraic response function for all situations. The
hypothesis followed is that the latter will most likely prove correct.
While fundamental greenhouse research may prove the first to have
some validity, objective statistical tests may be used to specify which
function is most appropriate under field conditions. This methodologi
cal problem merits further attention, since every fertilizer recommen
dation to farmers implies knowledge of the mathematical nature of the
response function. Greater knowledge of the response form is needed
for most efficient designs. If the mathematical form is known to be a
EARL O. HEADY
quadratic equation, a Box design may be most efficient (see page 48,
Chapter 3). However, another design may be more efficient if the math
ematical form proves to be logarithmic or exponential.
Distribution of Response Functions
Conventionally, fertilizer recommendations are made as if the re
sponse or regression coefficients were singlevalued. It would be con
venient if farmers' decisions could be made in this framework of cer
tainty in respect to both prices and yield increments. Unfortunately
this is not true. A methodological problem arises in providing response
information which recognizes that risk and/or uncertainty must be in
corporated into farmers' decisions: The farmer is not faced with a
single response function but with a distribution of response functions.
He recognizes this situation and makes his decisions accordingly. In
corporation of riskuncertainty and probability concepts into fertilizer
research and recommendations would aid him in these decisions.
The problem can be brought into focus by viewing fertilizer response
in the manner of the generalized production function represented by
equation 8. Yield response (Y) is represented as a function of
(8) Y = f (FI F2 . F, X,, X2 . X Z, Z2 . Zn)
fertilizer nutrients F1 through Fn and other types of inputs (practices
represented by X, through Xn and Z 1 through Zn). The last two cate
gories of inputs (Xi and Zi) are denoted by soil type, nutrients already
in the soil, seed variety, cultural practices, number of cultivations,
seeding rate, moisture of particular weeks, temperature at critical
times, and other variables (resource inputs) which affect yield. In this
case a single bar follows Fi, denoting that nutrient F, alone is the input
in the production function which is variable or which can be controlled.
All variables between the single and double bars, F, through Xn, are
endogenous to the decisionmaking environment, (can be controlled by
the farmer or decisionmaker) but are held fixed for the particular pro
duction period (i.e., crop year). These represent seeding rates, number
of cultivations, application of particular nutrients in fixed levels, etc.
To the right of the double bar are variables, such as weather, which are
exogenous to the decisionmaking framework and cannot be controlled
by the farmer. These exogenous variables vary within and between
seasons. Hence, the response curve for the single variable F, will
take on a different height and slope with each change in the exogenous
variables. The result is a distribution of response functions such as
shown in figure 1.1. The most likely hypothesis is that the response
functions are normally distributed. There have been suggestions, how
ever, that this is not the case, at least over a period of a few years (the
span usually relevant in a farmer's decisions). In case the response
curves are not normally distributed, the mean may be represented by
the dotted line in figure 1.1 and is above the mode (the "most probable"
METHODOLOGICAL PROBLEMS IN FERTILIZER USE
FIGURES AFTER
.05 CURVE REPRESENT
ASSUMED PROB
.10 ABILITIES. 0
S IS MEAN RESPONSE
FUNCTION.
.20
M
Z!.40
.10
.\05
QUANTITY OF FERTILIZER
Fig. 1.1 Possible effect of weather variations on
the distribution of fertilizer production functions.
curve of any one year). (The mean might also fall below the mode, de
pending on the skewness of the distribution.)
How should the farmer make decisions when the response curve
varies between years? Even though the distribution of functions might
be established (and hence conform with Knight's (2)2 risk concept), the
curve of any particular year represents uncertainty. The answer de
pends on the individual farmer and his ability to bear risk as character
ized by his capital, his equity position, and his aversion for risk. If he
is a conservative individual with little capital and a low equity, he may
wish to take few or no chances. In this case he may, in effect, count on
the lowest possible response function and apply fertilizer accordingly.
Using this type of "uncertainty precaution" (discount system), he feels
assured that the probability is in favor of outcomes better than expected,
and that there is slight chance of outcomes worse than predicted.3 Un
doubtedly, this type of uncertainty precaution causes farmers to use
fertilizer in quantities smaller than conventionally recommended.
The farmer in a better capital position and with less risk aversion
may make decisions on the basis of model response expectations. He
2Numbers in parentheses which appear in sentences refer to reference citations listed at
the end of each chapter.
3Regardless of the decision and the outcome, the farmer is always faced with the possi
bility of two kinds of errors. First, he may assume "the best" and act accordingly. If he is
wrong, he may be penalized by a depression of profits greater than if he had anticipated
"the worst." Secondly, he can assume "the worst" and act accordingly. If he is wrong, his
profits will be less than if he had used an alternative expectation and planned for "the best."
EARL O. HEADY
wishes the greatest probability of success in expectations and plans.
He will, of course, never be 100 per cent correct. He will apply too
little fertilizer for maximum profits in good years and too much in
poor years.
Data on the distribution of the production function are lacking in
most locations. To fill this gap in the farmer's decisionmaking envi
ronment, time sequences of fertilizer experiments are needed, with all
endogenous variables (soil, seeding rate, previous management, etc.)
held constant over a period of years. The exogenous variables then
would be reflected in the distribution of functions, which would be use
ful in recommendations to, and decisions by, farmers. There is some
preliminary indication that farmers believe the fertilizer response to
"reflect the best yield to be expected" and, therefore, that deviations
from this quantity are likely in the direction of lower yields.4
Information is needed to show whether the fertilizer functions are
normally distributed and to indicate to farmers that "better incomes"
are just as probable as "lower outcomes." But most important, this
type of information would provide the decisionmaking basis for farmers
who must use different plans because of variations in their ability to as
sume uncertainty. Table 11.3 (page 169) provides some insight into the
need for variability data for farm planning.
Carryover and Alternative Rates in Succeeding Years
Under the research needs outlined above, level of fertilization would
be a variable handled similarly in a series of years. The focus here is
on the distribution of functions, due to weather and other variations,
without regard to: (a) carryover effects or (b) the results of alterna
tive fertilization rates in succeeding years. However, both of the latter
are needed if fertilizer is to become a resource used to its full economic
potential.
Leaching is great in parts of the Southeast and carryover response
is unimportant in economic decisions. In some localities, however,
carryover responses are important. Information on these residuals
can increase the quantity of fertilizer used. With carryover effects in
years following the one of application, the optimum level of fertilization
can be determined by equating the discounted value of marginal responses
with the discounted value of marginal costs of each fertilizer increment.
The value of the marginal response for any fertilizer input (i.e., the jth
input) then becomes, as shown in equation 9, the sum of the marginal
response values
i1 R.
1
(9) Vj.= E 
i=n
4This statement is based on a survey of farmers' expectations being conducted by the
writer.
METHODOLOGICAL PROBLEMS IN FERTILIZER USE
divided by the discount coefficient. For example, suppose the third in
crement of fertilizer gives a response of 8 bushels in the first year, 4
bushels in the second year, and 2 bushels in the third year. The price
of the crop is $1 and the farmer's discount rate is 10 per cent. With
discounting for yearly periods, the present value of the sequences of
yield response is:
$8 + $4 $2
(1 +.10) (1 + .10)2 (1 + .10)3 =$12.08
Without knowledge of residual responses, the firstyear discounted
marginal value of the third input is only $7.27. Obviously, then, more
fertilizer will be used where residual effects exist and are made known
to farmers. Knowledge of residual effects can reduce uncertainty con
siderations if the farmer knows that even though weather of the first
year is bad, probabilities are high for getting a large residual effect in
following years. He then will not be so timid about using fertilizer.
Finally, residual response functions allow farmers to discount fer
tilizer returns to fit their own particular capital and uncertainty situa
tions. The magnitude of the discount rate should differ with each farmer.
On the one hand, it will be a function of the alternative returns on capital
in other parts of the farm business; the beginning farmer may discount
at 40 per cent while the wealthy, established farmer may discount at 4
per cent. On the other hand, the magnitude of the discount rate will be
a function of the subjective price and yield uncertainty in the farmer's
mind. By supplying information on time sequences of yield responses,
the research worker aids the farmer in using the fertilizer to fit his
own unique circumstances.
A final phase of time should be mentioned. It is the effect of rate of
fertilizer application in previous years on the response function in sub
sequent years. How much difference is there in the response function
for corn this year on fields which received respectively 20, 40, 60, and
80 pounds of nitrogen last year?
Nature of the Production Surface
In order to be systematic, we have discussed singlevariable func
tions or curves first. In following this procedure, the cart is placed
before the horse. The reason is that one cannot know which singleva
riable curve is the appropriate one to predict unless he knows or as
sumes something about the response surface itself. Hence, he turns to
the concepts and methodological problems involved in production func
tions involving two or more variables. Of course, what has been said
about appropriate biological or algebraic forms of functions, about the
distribution of the fertilizer response function, and other time consid
erations also applies to functions involving two or more variables.
When more than two nutrients can be variable for a single crop, two
economic problems are involved: (a) the leastcost combination of nu
trients for any given yield level, and (b) the most profitable level of
EARL O. HEADY
fertilization, considering the nutrient combinations, which yields the
lowest cost for each yield level. These decisions must be made by both
the farmer with limited capital and the farmer with unlimited capital.
If he has unlimited capital, then the optimum level of fertilization and
the optimum combination of nutrients are simultaneously attained when
the partial derivatives for both nutrients are equated with the crop/nu
trient price ratio for each.
Using data for an Iowa corn experiment (1), for example, we have
the twovariable response functions in equation 10. Using prices of
$1.40 per bushel for corn, 18 cents per pound for nitrogen, and 12 cents
per pound for phosphorus,
(10) Y = 5.68.316N.417P + 6.35V + 8.52V\/ + .341 VN,
the partial derivatives to equal the price ratios in equations 11 and 12
are set. From these, one solves for the quantities of the two nutrients
in equations 13 and 14. Given this particular function, the optimum
level of fertilization and combination of nutrients include 142.5 pounds
of N and 156.5 pounds of P205.
S.C 3.1756 .1705 v .18
(11) 316 + + 1
aN 77 1.40
(1) C = .417 4.2578 .1705V .12
OP(12 VP 1.40
(13) N = 142.48 Ibs.
(14) P = 156.45 lbs.
Even if the farmer has limited capital and cannot push fertilization
to the point that the value of the last increment of yield is just equal to
the cost of the last increment of fertilizer, he still needs to know the
leastcost combination of nutrients for the particular yield to be attained.
The leastcost combination is determined by equating the marginal rate
of substitution of the two nutrients (the derivative of one nutrient in re
spect to the other with yield considered constant at a specific level) with
the nutrient price ratio. Using the response function of equation 10,
equation 15 is obtained, which defines the marginal rate of substitution
between N and P2 05. Setting this equation of substitution rates to equal
(15) dN .8348 V/PN + 8.5155 VN+ .3410N .12
(') dP .6323 \PN + 6.3512 /P"+ .3410P .18
the P price ratio of .12 it is determined that for a 50bushel yield, the
S~18'
least'cost nutrient combination includes 11i.8 pounds of N and 24.3 pounds
METHODOLOGICAL PROBLEMS IN FERTILIZER USE
of P2 Os; for a 100bushel yield, the leastcost fertilizer ratio includes
79.3 pounds of N and 101.6 pounds of P2 O.5
The Nature of Yield Isoquants and Fertilizer Isoclines
The question of nutrient substitutability is now raised and, hence,
the nature of the fertilizer production surface. Some concepts assume
that nutrients are not substitutes in attaining a given crop yield. Liebig's
classical Law of the Minimum assumed, for example, that the fertilizer
yield surface reduces to a "knife's edge" as shown in figure 1.2. Higher
yields can be attained only if higher rates of fertilization follow some
limitational nutrient ratio. This also is the assumption employed in the
socalled practical information which pictures crop production in the
vein of a barrel, wherein yield cannot be raised above the shortest stave,
namely, a particular fertilizer nutrient.
CL
0 L
POUNDS OF P205 POUNDS OF P205
Figs. 1.2 and 1.3 Production surface and yield isoquants for
nutrients which are technical complements.
Now, for every yield surface, there is a corresponding map of yield
isoquants or contours.8 For the Liebig response surface, the yield iso
quants take the form suggested in figure 1.3. Both nutrients are limita
tional in the sense that increasing one alone (a) neither reduces the
sIn addition to knowing the leastcost nutrient ratio for a specified yield, the farmer with
limited capital needs to use this information to determine the return per dollar invested in
fertilizer as compared to other alternatives. This information will aid him in determining
how much to invest in fertilizer.
If the yield response for two nutrients is pictured as a surface or "hill" on a 3dimen
sional diagram, it can be reproduced in 2dimensional form just as a hill is reproduced by
the soils expert on a topographical map, as a set or family of contours. Each contour rep
resents a given yield level and the points on it represent the various nutrient combinations
which allow attainment of this specified yield level. The yield contour, showing all possible
combinations of nutrients allowing its attainment, is termed a yield isoquant (equal quantity).
EARL O. HEADY
amount of the other required to produce the given yield, or (b) increases
the level of yield. This is denoted by the fact that the isoquant forms a
180degree angle. However, if it is assumed that addition of one nutri
ent, without change in the other, causes toxic or other effects reducing
total yield, the isoquants reduce to a single point consistent with the
corner of the angles in figure 1.3.
However, a strict Liebig type of production surface is the exception
rather than the rule. Otherwise agronomists would not have (or have
been able to have) successfully conducted a relatively large number of
singlenutrient experiments. Perhaps it is true that such distinct nu
trients as nitrogen, P2 O0, or K20 do not substitute in the chemical
processes of the plant (although close substitution may hold true for ele
ments such as Na and K). However, availability of one nutrient may af
fect the ability of the plant to utilize other nutrients. Hence, in any case
where variation of one nutrient, with another fixed at specific levels as
in figure 1.4, results in different response curves, substitution does take
place in the sense that different nutrient combinations can be used to
attain a given yield. For example, if a 10bushel response is attained
with 20 pounds of N and 120 pounds of P2 Os, with 60 pounds of N and 90
pounds of PzOs, or with 120 pounds of N and 40 pounds of P20s, the
given response can be attained with various nutrient combinations. It
may be stated that nutrients are substitutes, at least at the level of farm
decisionmaking.7
SP205 = 120
W P205 = 80
z
0 P605 40
wi fd at d t l .
I I I
I I
I I I I
20 60 120 200
POUNDS NITROGEN
Fig. 1.4 Yield resource curves for nitrogen
with P20s fixed at different levels.
'These statements need, of course, to be conditioned in terms of plant composition and
quality.
METHODOLOGICAL PROBLEMS IN FERTILIZER USE
a I I
0 80 160 240 320
POUNDS NITROGEN PER ACRE
Fig. 1.5 Predicted yield surface for corn. Source: Pesek, Heady,
and Brown, Iowa Agr. Exp. Sta. Bul. 424.
The response surface for many crops and soils is more likely to
parallel that shown in figure 1.5 for N and P20s5 on corn in western
Iowa, or some modification of it (1). The corresponding family of yield
isoquants is shown in figure 1.6. At high levels, the isoquants bend
sharply to a purely vertical position at the "upper" end and to a purely
horizontal position at the "lower" end. At these points of infinite and
zero slope, respectively, the nutrients actually do become limitational
or technical complements in the sense of Liebig; increase of one nutrient
alone, at the vertical and horizontal points of the curves, will not result
in reduction of the amount of the other nutrient, with yield remaining at
the specified level or addition to the total yield. (Yield may actually be
reduced if one nutrient is increased while the other is held constant at
the level indicated at the points of infinite or zero slope.) However, be
tween the two points of complementarity, the curves have a negative
slope, denoting that they are substitutes in the sense that addition of one
nutrient reduces the quantity of the other nutrient required to attain
EARL O. HEADY
I I I I I I
250
FIGURES ARE YIELDS
IN BUSHELS
200
0
0 I *50
150
120
S100 0
\ 100
50
80
060
0 ,I I I lI
0 50 100 150 200 250 300
POUNDS OF NITROGEN
Fig. 1.6 Predicted yield isoquants for corn (from Fig. 1.5).
(maintain) the given yield.8 Furthermore, the curvature or slope of the
isoquant changes, denoting that increasing quantities of the nutrient
being added are necessary to offset constant decrements of the nutrient
being replaced.
An important methodological problem in fertilizer research is that
of obtaining more information on the slope and degree of curvature of
the yield isoquants. If the slope changes only slightly and its length be
tween the points of complementarity (i.e., the vertical point on the
"upper" end and the horizontal point on the "lower" end) is great, the
nutrients can be classed as "good" substitutes (i.e., "poor" complements).
If the curvature is sharp (i.e., the slope changes rapidly) and the range
between complementary points is narrow, the nutrients are poor substi
tutes (i.e., "good" complements). Now it is just as important to know
that nutrients are "good" substitutes as it is to know that they are "good"
complements. Perhaps too much research and too many recommenda
tions have supposed that nutrients are only good complements. Given
the meager knowledge which exists, the specialist making recommenda
tions can suggest specific nutrient ratios with less burden on his
'For other alternatives in fertilizer production surfaces and isoquant maps, see (1).
METHODOLOGICAL PROBLEMS IN FERTILIZER USE
conscience (and less profit depression to the farmer if the expert is not
entirely correct) if he knows that substitution is "good" over a wide
range.
If the slope of the isoquant is relatively constant over most of its
range, and if this slope does not deviate greatly from the magnitude of
the price ratio, a large number of nutrient combinations give costs and
profits of fertilization which are quite similar. Here, again, the expert
making fertilizer recommendations need not let his conscience be both
ered greatly if he recommends a particular ratio such as 20200 rather
than a 10200. However, if the curvature changes greatly between the
complementary points and if the slope at either one or both ends devi
ates considerably from the magnitude of the price ratio, the expert
needs to give particular heed to his recommendations on nutrient ratios.
He will want to consider price ratios; he will want to consider the ef
fects of nutrient prices on the optimum nutrient combination and the op
timum fertilization level. The optimum nutrient combination will change
with yield level, if the slopes of the yield isoquants differ greatly as
successively higher yields are attained. Under these conditions, the
recommendation on nutrient ratios should differ between farmers (a)
who have funds for only low fertilization levels, and farmers (b) who
have unlimited capital and can use higher fertilization ratios. Similarly,
if slopes between isoquants change greatly with higher yields, the nutri
ent ratio will need to be changed as the price of the product changes
(and higher or lower yield levels are profitable), even if the nutrient
price ratio remains unchanged. The extent to which these facets of
economics need to be incorporated into fertilizer recommendations de
pends on the nature of the production surfaces and isoquant maps. While
they are fundamental science aspects of agronomic phenomena, knowl
edge is still too meager to determine where, and the extent to which,
these considerations become important.
Fertilizer Isoclines
The slopes of isoquants change (i.e., the marginal rate of substitution
between nutrients) as higher yields are attained. However, slope or sub
stitution rate changes must be defined in a particular manner. They
must be in reference to a fixed ratio of nutrients such as that illustrated
in figure 1.7. The straight lines, A and B, passing through the origin,
denote that nutrients are held in fixed ratios at higher fertilization
levels. Changes in slopes or substitution rates on successive isoquants,
in relation to needs for different nutrient ratios at varying yield levels,
are measured at the point of intersection of the fixed ratio lines and the
yield isoquants. If the slope of the isoquants were identical at all points
where they are intersected by a fixed ratio line, the same fertilizer mix
would be optimum for all yield levels. If the slope changes along a fixed
ratio line, the nutrient ratio which is optimum for one yield level is not
also optimum for another yield level.
A concept with perhaps greater application and more fundamental
EARL O. HEADY
NUTRIENT RATIOs10100
0= NUTRIENT RATIOI0200
POUNDS P205
Fig. 1.7 Isoclines showing equal nutrient ratios
in relation to yield isoquants.
importance than the fixed ratio line is the fertilizer yield isocline. An
isocline map exists for every fertilizer production surface. An isocline
is a line connecting all points of equal slopes or substitution rates on a
family of isoquants. In other words, it connects all nutrient combina
tions which have the same substitution rates for the various yield levels.
There is a different isocline for each possible nutrient substitution rate.
Of course, if the fertilizer production surface is of the Liebig knife
edge type illustrated in figure 1.2, the map reduces to a single isocline,
denoting a zero substitution rate.
The isocline is also an expansion path, showing the leastcost and
highestprofit combination of nutrients to use as higher yield levels are
attained under a given price ratio for nutrients. In other words, it indi
cates whether the same nutrient ratio should be recommended and used
regardless of the yield to be attained. Chapter 10 illustrates practical
uses of this concept. Isoclines can be straight lines, such as A and B
in figure 1.7. In this case they become identical with a fixed ratio line
and the leastcost nutrient ratio will be the same for all yield levels.
The expert need not inquire about the yield level to be attained when he
makes his recommendation. However, an isocline map composed en
tirely of straight lines (fixed ratios) is very unlikely and perhaps impos
sible. Under maps of this nature, the isoclines would never converge
but, instead, would spread farther apart at higher yield levels. There
fore, straightline isoclines would indicate no limit to total yield level.
Limits in total production exist only if the isoclines converge to the
point of maximum yield and, therefore, are curved rather than straight
(see Chapter 6 for other details on this point).
Isocline maps may take on many different forms. Little is known
about them, and their nature can be established only by basic research.
All isoclines for a given production surface may be bent in the same
METHODOLOGICAL PROBLEMS IN FERTILIZER USE
CORN ISOCLINES
F ISOCLINE FOR 1.5
SUBSTITUTION RATE
ALFALFA ISOCLINES
ISOCLINES FOR SUB
STITUTION RATE OF
1.50 
1.0, /
0
240
U.
120 MAXIMUM YIELD //
MAXIMUM
YIELD
0 120 240 360 0 120 240
POUNDS OF N POUNDS OF K20
Figs. 1.8 and 1.9 Isoclines for corn and alfalfa showing convergence
to maximum yield.
direction and none may be linear. Alternatively, one may be nearly
straight while those above and below it bend in opposite directions. Dif
ferent isocline maps, based on research in Iowa (1), are shown in fig
ures 1.8 and 1.9. The two for corn, covering likely limits in price ratios
for nutrients, are quite straight, with a slope relatively close to 1:1, de
noting that recommendations of a constant nutrient combination may not
deviate far from leastcost ratios for all yield levels. (Cognizance of
the slight curvature in recommendations might cause more bother than
savings in cost would merit.) In the case of the alfalfa data, however,
the relevant isoclines bend rather sharply, suggesting that the least
cost nutrient ratio for one yield level may differ considerably from that
for another yield level.
Two isoclines can be called ridgeliness" (see figure 10.1, page 153).
They correspond to all points in figure 1.5, where the slope of the sur
face changes from positive to negative (i.e., the tops of the ridges denot
ing zero marginal responses). The ridgelines denote the points on suc
cessive yield isoquants where the nutrient substitution rate becomes
zero. Since they denote technical complementarity of nutrients, they
might appropriately be given the term "Liebig lines" because these are
the limitational conditions which Liebig had in mind in his law of the
minimum. The ridgelines (Liebig lines) converge, along with the other
isoclines, at the point of maximum yield where nutrient substitution
also is impossible.9
If (a) the ridgelines are not far apart, (b) the isoclines within their
boundary are fairly straight, and (c) the yield isoquants for a particular
'The isoquant at the point of maximum yield reduces to a single point.
EARL O. HEADY
yield have only a slight curvature, with slopes not too different from
the nutrient price ratio; several nutrient ratios, within the boundaries
of the ridgelines, will give costs which are only slightly different (al
though only one will denote the leastcost ratio). If (a) the ridgelines
are "sprung far apart," (b) the isoclines "bend sharply," and (c) the
isoquants "curve greatly" away from the price ratios, the saving from
changing nutrient ratios along an isocline can be quite considerable.
Only basic research can indicate the frequency and extent of different
isocline maps. The situation likely varies with soil, crop, year, and
other variables.
Information of this nature not only has methodological importance
but also practical significance. Therefore, the full economic potential
of fertilizer use will be uncovered only by multivariable response re
search. This is true since, as production economics logic has long
suggested, the productivity of any one resource always depends on the
level of input for other resources. While much of the logic is illustrated
with two variables, analysis should be extended to variables which in
clude other nutrients, seeding rates, moisture, quantities of nutrients
already in the soil, soil type, and others. In other words, one should
view the production function in the generalized form of equation 8. It
is not inconceivable that soil typing and classification might be rela
tive to the fertilizer production function. For example, with other inputs
specified, economic distinction need not be made between soils where
marginal response for parallel fertilizer inputs are the same. While
they may be complex, steps to incorporate this concept into fertilizer
research might obviate the need for considering experiments at isolated
locations and in particular years as unrelated facts.
At the outset it was stated that the paramount methodological prob
lem was that of the mathematical form of the fertilizer production func
tion. Experimental designs and estimating procedures are auxiliary
problems to it but at the same time are the foundation tools for estab
lishing the mathematical characteristics of the function, at a given point
in time and over time. To what extent is replication necessary when in
terest is in prediction of the response curve or function and the standard
error which attaches to it, rather than the mean differences between
treatment? Supposing that yield distributions are heteroscedastic in
respect to variance; under what conditions would recommendations
differ among regression lines predicted with nonreplicated treatments
and means of treatments based on replications? What experimental de
signs allow both statistical and economic efficiency in estimate of com
plete surfaces, including isoquants, isoclines, and ridgelines? Is it
unlikely that responses for different fertilizer inputs follow in the man
ner of a continuous function, and that other estimating procedures are
necessary? There are hypotheses in respect to the answers of some of
these questions; however, lack of time and space prevents the unraveling
of their logic.
METHODOLOGICAL PROBLEMS IN FERTILIZER USE 21
References Cited
1. HEADY, E. O., PESEK, J. T., and BROWN, W. G., 1955. Crop response sur
faces and economic optima in fertilizer use. Iowa Agr. Exp. Sta. Res. Bul. 424.
2. KNIGHT, F., 1933. Risk and Uncertainty. London School of Economics, Chap. 7.
GLENN L. JOHNSON
Michigan State University
Chapter 2
Interdisciplinary Considerations in
Designing Experiments To Study the
Profitability of Fertilizer Use
MORE interdisciplinary cooperation among agronomists, statis
ticians, and economists is an important need in agricultural re
search. Fertilization research should be looked at from an
agriculturist's viewpoint rather than from the confined viewpoints of
the farm management specialist, the soils specialist, the marketing
specialist, the mathematical statistician, or the specialist in legumi
nous nitrogen fixation.
The Economics of Designing Experiments
Economics is concerned with the use of scarce resources in attaining
multiple objectives. Experimental designs involving interdisciplinary
research involve economic considerations. In designing interdepartmen
tal experiments, some of the objectives pursued are in conflict; other
objectives are complementary, i.e., attainment of one objective may
make it easier to attain another. Such conflicts and complementarities
occur both within and between the sets of objectives commonly of inter
est to agronomists, economists, and statisticians.
The job of agriculturists in designing an experiment is to approach
the "best combination" of objectives in designing a particular fertiliza
tion experiment. The best combination of objectives should recognize
any existing complementarity. Of course, the best combination of objec
tives depends on the relative costs of attaining the objectives. Mention
of the cost of attaining objectives calls attention to the relationships
among research resources and attainment of research objectives.
Pairs of Resources May Be Substitutes or Complements
If substitution is "near perfect," the designer should use the cheaper
of the two resources in designing his study; i.e., if two identical fields
are available, one for $400 an acre and the other for $350 an acre, he
should use the latter. At the other extreme, pairs of resources may
complement or contribute to the productivity of each other. For in
stance, an agronomist and a statistician working together may design
an experiment which is superior to the product of either working alone.
Their effort is then complementary. If two resources are perfect com
plements in the sense that they are unproductive used alone, or in only
DESIGNING EXPERIMENTS TO STUDY PROFITABILITY 23
one proportion, the designers should take full advantage of this comple
mentarity. The two research resources should be used in the one pro
portion.
The difficult problems in selecting research resources arise, how
ever, when resources are neither perfect complements nor perfect sub
stitutes but are, instead, complements over wide ranges and substitutes
over narrower ranges. In this case, the designer has to match the
added costs of and returns from using another unit of one resource
against the added costs of and returns from using another unit of an al
ternative resource. If a unit of one resource is more productive rela
tive to its costs than another, it is logical to expand its use relative to
the other. When research funds for a given experiment are limited, the
best experimental design is one which yields equal additional returns
for equal additional expenditures on the resources subject to the de
signer's control. If, as is very unlikely, there are unlimited funds to
support the experiment, the best experimental design is one which yields
additional returns equal to additional costs for all resources subject to
the designer's control.
The designer should also ask himself whether (a) any part, or all,
of any of the fixed resources could be disposed of (by sale or transfer
to another experiment) at a net return in excess of what it would produce
in the experiment under consideration, and whether (b) more of any of
the fixed resources can be acquired at a net cost below what it would
produce in the experiment. If the answer to either of these two ques
tions is "yes" for a particular resource, the designer should cause the
resource to become variable and adjust its use according to the rules
previously considered.
In experiments on the economics of fertilization, a high degree of
complementarity exists among the services of agronomists, statisticians,
and economists. In most fertilizer experiments, agronomic (both in
soils and in crops) and statistical training are complementary. And, if
the experimental results are to be interpreted economically, the serv
ices of an economist complement those of the agronomist and the statis
tician. Thus, with the exception of a highly technical fertilization experi
ment intended to yield technical information for noneconomic application,
most fertilization experiments can advantageously employ the services
of agronomists, statisticians, and economists.
Reconciliation of Objectives
Agronomists, statisticians, and economists, as a result of their dif
ferent training, comprehend and prefer to pursue objectives which are
sometimes conflicting. Also, because research workers are specialists
in different organizations or different parts of a given organization,
their preferences and objectives may differ still further. These differ
ent objectives and preferences have to be reconciled and aggregated into
group choices in designing cooperative experiments.
Generally speaking, the reconciliation and aggregation process is a
bargaining one, with weights assigned to individual and institutional
GLENN L. JOHNSON
preferences on various bases such as, (a) the amount of resources con
tributed by the different organizations, (b) the professional repute of
the individuals, (c) the democratic procedure of one vote per participant,
or (d) the principle of "greasing the wheel which squeaks the loudest."
If it were possible to price the objectives separately and produce re
search on some sort of a free enterprise basis, a free price system
might be used instead of a bargaining process in making these decisions.
Similarly, consensus or deference to recognized authority would make
it unnecessary to use bargaining processes in arriving at these design
decisions. But administrative authority is not well enough informed to
make these decisions; uniformly recognized professional authorities do
not exist and differences, not consensus, as to preferences are the rule,
not the exception. Thus, the bargaining process seems inevitable in the
committee meetings, Kaffeeklatsches, seminars, and informal coopera
tive arrangements in which experiments are designed.
The problem is not one of eliminating bargaining decisions in design
ing experiments. Instead, it is one of improved bargaining leading to
design decisions. Such decisions can be improved first by appealing for
agricultural statesmanship, rather than by encouraging competition
among departments of institutions or among institutions. Agricultural
research statesmanship, rather than destructive competition or per
sonal position and sprfeigeaif olg TndividUtials or illadvised loyalty'to
o6fedfscipline among those serving agriculture, will lead to cooperative
research which solves the problems of agriculture. A second important
Wy'dif'improvmng decisions on&experimenta i design is to increase the
knowledge of the designer (whether an individual or a committee) about
(a) the nature and importance of objectives held by different research
organizations, different disciplines, and different individuals, (b) the
nature of different research resources and their usefulness in attaining
the objectives listed in (a), and (c) research techniques or methods of
value in using the resources considered in (b), to attain the objectives
considered in (a).
In the remainder of this chapter, fertilization experiments in general
will first be considered. Following this, special problems of making
economic interpretation of data secured from fertilizer experiments
will be considered along with the desirable characteristics of experi
mental data from the standpoint of economic analysis. Finally, a re
cently designed Michigan experiment will be reviewed. This outline
will permit emphasis of two principle methods available for improving
decisions on experimental design. They are (a) use of agricultural
statesmanship, and (b) use of more knowledge about objectives, research
resources, and research methods.
Specification of Function for Investigation
Most fertilization experiments involve investigation of a set of func
tional relationships such as that represented by equation 8 in Chapter 1.
This generalized function is, of course, too complex and extensive
to be handled with the intellectual and physical resources of any research
DESIGNING EXPERIMENTS TO STUDY PROFITABILITY 25
organization. Hence, the first step is to restrict the general area of
investigation to a manageable size or number of input categories. This
is commonly done in two ways. First, autonomous subfunctions within
the function are isolated for study. he word autonomnous._here means
that outcomes within the subfunction are not influenced by events in the
remainder of the function. choices amonga~li.i'iiv~` sGfunctions de
p on the preferences of individuals and agencies and upon the com
parative productivity of research resources in such alternatives. If, as
is generally the case, such autonomous subfunctions are still too large
to work with, controls have to be imposed on certain of the variables to
limit further the realm of inquiry. Here the conflicting ends are "gen
erality" and "accuracy." For given resources, the study can cover a
larger subfunction with a low degree of accuracy or a smaller subfunc
tion with greater accuracy. The designer must decide how much of one
he is willing to sacrifice in order to get the other.
To illustrate the above two steps, consider the problem of setting up
a fertilizer experiment within a generalized function, including all pos
sible products, inputs, and associated technologies. This function could
be cut down to, e.g., a corn, oats, and clover rotation which can be pre
sumed to be independent of other rotations. This, however, would still
require a very large experiment. There is almost an infinity of inputs
to consider land with all its variation, labor, nitrogen, different
sources of phosphorus, potash, machinery, different technologies, vari
eties of oats, cultural practices, etc. If an attempt were made to study
all of these factors at once, the resources required for the project
would be spread very thinly, and only very inaccurate results (i.e., those
with great variance) would be secured.
Restriction of scope can be attained by the imposition of controls,
both selective and experimental. Here, many individual and organiza
tional preferences must be considered. One agronomist may be particu
larly interested in corn over the cornbelt, while a cooperating colleague
may be endeavoring to become a national authority on planting and fer
tilization practices for small grains. The experiment station director
may know that agricultural leaders favor investigation of corn fertiliza
tion on a soil type within one state. Hence, all of these kinds of prefer
ences and others, along with the conflict between generality and accu
racy, enter into the series of negotiations leading to the final choice.
The final choice might involve, for example, (a) restricting the ex
periment to a rotation on the given soil type to include: (i) given varie
ties of corn, oats, and clover, (ii) given cultural practices, and (iii)
given levels of available K2 O, and (b) restricting the experiment further
to N and P2 Os as the primary variable inputs to be studied in application
to corn only.
This last step would narrow the realm of inquiry to a consideration
of only the following subfunction:
(1) Yc = f'(N, P2 0 s oats, clover, KO, Xf . Xn) +u .
GLENN L. JOHNSON
This function reads as follows: the yield, Yc, of a given variety of corn
is a function of the amount of N and P2 05 applied to corn grown in a
COCL rotation with K20 and other inputs Xf. . Xn (such as, soil
type, oats variety, clover varieties, cultural practices, etc.), fixed at
specified conditions, or levels.
Unexplained Residuals
The introducedd in equation 1 stands for variations of actual
yields from the functional relationship specified in l() above. In prac
teliThe u's are always, partially, functions of more or less uncon
trolled and unstudied variables, such as lack of uniformity in soil types,
variations in weather, and disease or insect infestation. So long as the
u's behave substantially as though they are randomly and independently
distributed with respect to the studevariaestheyanbeaveraged
outiwitstatistlcalTramcur or instance themefliodEff'Teast
squares may beapplied'to secure es inmates of equation 1 which mini
mize the sum of the squared deviations in the Yc's. This procedure is
appropriatesolongas theu'scan.beinterpreted as due to errors in
measuring the Yc's,, oras random stochastic movements in the function,
either with or without antecedent causes.
Another practical requirement is that the u's be small enough for
the estimats_.QfXe..ta.bjusble._ At this point the objectives of the stat
iscian nd agronomist may come in conflict. Trained in estimating
procedures, and perhaps charged by the experiment station director
with responsibility for the statistical accuracy of estimates based on the
data produced by the experiment, the statistician desires accuracy. Or
dinarily, the agronomist does too, but not at the expense of what he may
consider undue restriction of his work and expensive randomization
and control procedures.
In investigating equation 1, the statistical conditions required with
respect to the u's may be secured, in part at least, by (a) procedures
which reduce errors in measuring X d c, (b) controls on nonstudied
inputs and factors,_and (c) procedures designed to rarnimzlhe inci
denice otfunstudied and uncontrolled variables in the experiment and,
hencrr fethe u's generated biHevariables. Examples of the first
set of procedures are doublechecking and the measurement of nutrients
in the soil as well as those applied. The imposition of controls was
illustrated above. Plot layouts to randomize the distribution of soil
differences between plots are a common example of the third set of pro
cedures. Decisions on such procedures must be made early in the ex
periment. As an earlier step, the total amount of resources to be de
voted to the experiment has to be determined and allocated among such
competing ends as: number of plots, measurement accuracy, search
for uniform fields, etc. After the number of plots is determined, its
use in producing accuracy versus generality must be determined.
DESIGNING EXPERIMENTS TO STUDY PROFITABILITY 27
Desirable Characteristics of Experiments
for Economic Interpretation
To this point, the discussion has been general. It applies to purely
agronomic experiments as well as to experiments to be interpreted
economically. Experimental data to be used in agronomic analysis,
however, may or may not possess certain desirable characteristics for
economic interpretation. It is important that the nature of characteris
tics which are desirable for economic analysis be known before fertili
zation experiments expected to yield data of economic significance are
designed. The nature of these desirable characteristics can be seen
most clearly by examining the uses which an economist may wish to
make of the data.
The first required modification of concepts used to this point, if
economic analysis is to be carried out, is the introduction of input
prices, Pxj, and output prices, Pyi, to produce a profit equation of the
form:
(2) g(Py Y1 .... PynYn:X1 Px .... XmPxm)= T
When narrowed down to manageable size, as previously done by isola
tion of an autonomous subfunction and imposition of selective and ex
perimental controls, the following type of subfunction is secured:
(3) g'(YcPc, NPn, P2 5 PP2o, oats, clover, KO2, X, . X.) = 7r .
Application of maximization procedures (as taught in any elementary
calculus course) to equation 3 or portions thereof, permits location of
such economic optima as the quantity of Y to produce maximum profit
and the leastcost combination of N and P2 Os to use in producing that
amount of Y.
Corresponding applications also permit determination of how these
optima shift with price changes. The laws of growth, of the minimum,
or of diminishing returns (which are highly interrelated and are inves
tigated by agronomists and economists alike) tend to assure the second
order conditions necessary to locate these optima. The most important
economic optima tend to occur on the function where the
a Y
S>10 are decreasing.
9aX
As an example, when P2 Q is constant, d7r defines the most
dN
profitable amount of nitrogen to use with the constant amount of P2 O,.
Under ordinary competitive conditions dr dYc Py Pn. Thus, an
dN dN
estimate of which is the slope of equation 3 in the YCN dimension,
dN
GLENN L. JOHNSON
is important to the economist attempting to ascertain the most profitable
amount of input N to use.
Suppose, however, that the economist's interest is somewhat more
complex. He may desire to find the best (most profitable) combination
of N and P2zO in producing a given amount of Yc. The condition
oYc
ON n defines the leastcost combination of N and P, O0 to use
OYc Pp2o5
OP20,
in producing the amount of Yc under consideration. As the
0 c P P. and ar acpyc p
N ON P 020s 4P205 Pz05
are the slopes of equation 3 in the YcN and YCP2Os dimensions, respec
tively, also, slopes are crucial to the economist attempting to ascertain
the most profitable (leastcost) combination of N and P2 05 to use in ob
taining a given yield (Yc = a constant) of corn. These steps parallel
those of equations 11 through 14 in Chapter 1.
If the economist is considering the problem of a farmer with a given
amount of money to spend on N and P2Os then, instead of fixing Yc,
0Y
C
the relationship ON Pn is solved simultaneously with
OYC
P0 05
OP, Os
PnN + P 205 P2 05 = C (the amount of money which can be spent on N
and P2 Os), to determine N and P2 O5. These values for N and P2 O can,
in turn, be substituted in equation 3 to determine Ye.
In both this and the previous instance involving Yc = a constant, the
productivity of N may depend on the amount of P2 05 present (and vice
QY
versa) and the study should be designed so that the estimates of c
ON
and OY, can reflect such relationships.
OP2 0s
When the economist desires to determine the most profitable amounts
of N and P2 Os to use and of Yc to produce, he sets 07r and 07r
SN 0P2Os
equal to zero, and solves simultaneously for N and P2 0s. Having se
cured N and P2 O0 in this manner, he then substitutes them in equation
3 and solves for Yc. Alternatively, the optimum combination of N and
P2 Os and the optimum level of Yc can be solved in the manner of equa
tions 11 through 14 in Chapter 1. As in the previous cases, Oar and
6N
DESIGNING EXPERIMENTS TO STUDY PROFITABILITY 29
Oir involve estimates of aYc and OY the slopes of equation
OP2 05 N a0P205
3 in the YeN and YcP2 Os dimensions, as the crucial values to be deter
mined from the fertilization experiment.
Consideration of more complex subproduction functions involving
more than two inputs reveals that, in each instance, Ye turns out to
OXj
be crucial in estimating the most profitable quantities of Yc to produce,
and of the inputs Xj. The same is true if Yc is fixed, or if the money
which can be spent on the variable inputs is limited.
If the subfunction being investigated involves two products, Yc and
YL (corn and a legume) with the amount of Yc produced affecting the
productivity of resources used in producing YL (and vice versa), these
influences should be measured and reflected in the estimates of the OYi
In such subfunctions, an additional problem of determining the most
profitable combination of Yc and YL exists. Yc and YL are in the most
profitable combination and amounts when the following equations hold
simultaneously:
(4a) = 0
(4a) .ON(Yc)
(4b) aOP (Y) =
(4c) OL)
(4d) 0r =
OP2 Os (YL)
where ON(Yc) stands for a change in the amount of N used in producing
Yc, as contrasted to a change in N used in producing YL, which is writ
ten ON(YL), or a change in P2z used in producing Yc, which is written
OP2 O (Yc). After solution of (4a), (4b), (4c), and (4d) for N(Yc), N(YL),
P2 Os(Yc), and P205 (YL), these values can be substituted into equation 3
to determine the most profitable amounts of Yc and YL to produce.
The above example involving two outputs Y, and YL, and two inputs
N and P2 Q is easily generalized to "n" outputs and "m" inputs. In this
generalized form, the same conclusion holds, i.e., the crucial estimates
required to determine high profit points, least cost combinations of
inputs, and high profit combination of outputs are the estimates of
OY1
1Y such estimates to reflect interactions among the Y as well as
xj (Yi)
among the Xj.
The economist's strong preference for accurate estimates of the
GLENN L. JOHNSON
SYi aYi
() where a are positive and decreasing may come into
sharp conflict with the interests of agronomists in the early stages of
interdepartmental negotiations on the design of fertilization experi
ments. The agronomist, after many earlier negotiations with statisti
cians, has a strong preference for accuracy in estimations yields for'
some combination of fertilizer nutrients; the economist has, for reasons
expressed above, a strong preference for accuracy in estimates of
ayi
SThe agronomist is led to seek replications at points on the
SX ((Yi) _________
surface while the economist is led to seek less replication and more
"spread" ofthepbserations over the surface. These two objectives,
While competitive over a narrow range, are also quite complementary
over wider ranges since the standard error of estimate for yields is a
aY,
X (Y fact, an experimental
component of the standard error for In fact, an experimental
Yj
design yielding low standard errors for can be made
to yield as low or even a lower standard error of? estimate for Yi than
aY
ayi
one in which the standard error of is high. When the agrono
0x.Y,
mist sees these complementarities and opportunities for cooperation,
it is a relatively short step toward agreement and the presentation of
a unified research proposal backed by personnel from both areas of
work.
Alternative Agronomic Objectives
and Linear Programming Determinations
Other objectives of agronomists, while not always complementary
with those of economists, are seldom in sharp conflict. This is espe
cially true if the need for full use of fixed research resources is con
sidered, as well as the need for economy in the use of variable, or "out
of pocket," research resources. For instance, fertilizer placement and
tillage practices can be tested in subseries within a design with only a
small increase in variable costs and probably no increase in fixed or
overhead costs.
Another consideration involving slopes of function should be men
tioned here. Some persons argue that economic interpretations of fer
tilization data can be made on a comparative budget and/or on a linear
aY.
1
programming basis which does not require estimates of the
from continuous production functions. This is, of course, true. In such
procedures profits are computed for each discretely estimated point on
the relevant subproduction function for which an estimate of yield is
available. Comparison of profits among such points permits the
DESIGNING EXPERIMENTS TO STUDY PROFITABILITY 31
economist to determine the most profitable among them, as discrete
opportunities. While these procedures do not make direct use of
aYi
SXj(Yi) estimates, they locate the "best" point by comparing finite dif
ference between points. The smaller these differences, the more accu
rately the "best" point can be located. Thus, regardless of whether or
not the economic analysis is to be based on point estimates or on esti
mates of derivatives from continuous functions, experimental observa
tions should yield information on a multiplicity of points on that area of
the surface where the derivatives are positive and decreasing.
Another point of similarity should be noted in the data requirements
of economic analyses based on point versus continuous function esti
mates. In both instances, the "best" amounts of the different fertilizers
to use vary with prices of the inputs and of the output. These variations
occur in areas of the function where decreasing increments in yields
result from equal successive increments in the variable inputs. This
mutual characteristic of the different methods of economic interpreta
tion further increases the desirability of having yield information over
large areas of the surface, or on a multiplicity of points on the surface.
Thus, we note again that the same complementarity which exists be
tween the agronomist's desire for a low standard error of estimate for
SYi
yields and the economist's desire for a low standard error of
also exists between the desires of (a) the budgeter or linear program
mer on the one hand, and (b) the continuous function analyst on the other.
Economists carrying out continuous function analyses sometimes
are devotees to certain functions. For instance, prior knowledge that
one will predict a CobbDouglas, Spillman, or linear function creates
the desire for special designs; i.e., a CobbDouglas analyst may want
to avoid all zero rates of application since the log 0 = o. However,
because of the current lack of knowledge of which function best fits the
data, it appears desirable to avoid designs which confine the analysis to
a particular function, unless resource limitations restrict the analyst
to one of the simpler functions.
Methods of Attaining Desirable Characteristics
for Economic Analysis
The objectives outlined above are attained in designing experiments
by:
A. Ascertaining on the basis of existing information the range of com
0Y.
binations of Xj's for which the >O and decreasing and con
aX i(Yj)
centrating experimental observations on these combinations.
B. Securing observations for a sufficient number of combinations in the
area defined in (A) to give the economist flexibility in selecting
GLENN L. JOHNSON
functional forms if he elects to use continuous functions or, if he
elects not to use continuous functions, confidence that he has data on
sufficient alternatives to make the relevant discrete comparisons.
It should be recognized that while this may reduce the number of rep
lications which can be made with given resources for any one com
bination there are complementarities between the desire of accuracy
iY.
in Yi estimates and accuracy in  estimates. This requirement
insures that data on the interactions among the Xj's will be available.
C. Allocating experimental observations among the possible combinations
of the Xj in such a way as to minimize the linear correlations among
terms whose coefficients are likely to be estimated; i.e., if
Yc = A + biX1+ beXLX1 + b3X2 + b4X + bs X is likely to be fitted,
an experimental design which minimizes (with due consideration to
the cost of minimization) the linear correlations among X, and XX2
or between X, and X2, or X, and X etc., is desirable. Minimiza
tion of the intercorrelations among the variables whose coefficients
reto be estimated reduces the standard erFroFsFi the estimated
coefficients.
D. 7Allocating experimental observations among the possible combina
tions ol the Xj's in such a way as to increase the standard deviation
of the terms whose coefficients are likely to be estimated; i.e., if
Y = a X 2b" is to be estimated linearly in the logarithms, then
clog Y, alog X, and olog X2 should be kept large or, alternatively,
if Y = a + biXi + b2X2Xi + bsX3 is to be estimated, then oX1,.XX2,
and Ox, should be kept large.
E. Controlling or measuring the influence of the YL's on each other's
functional relationships.ith.the.Xi. This can be done if allbut one
of the Yi is held constant or, if more than one of the Yi is to be stud
ied, by (a) measuring the byproducts of each Yi studied and the in
fluence of these byproducts on the production of the other Yi, or
(b) by simply letting the separate functions for the Yi reflect the
levels at which the other Yi are produced. If byproducts and/or
"bylosses" involving humus, biologically fixed nitrogen, soil, nutri
ent removal, soil structure, erosion, etc., can be measured and in
corporated into the functions, this is probably the preferable solution.
Simply letting the separate functions for the Yi reflect the levels at
which the other Yi are produced may cause estimates of the produc
tivity of one or more of the applied nutrients to reflect either by
product losses or gains.
F. Maximizing, with available resources and in view of direct and op
portunity costs, the number of observations made.
There are at least two important sets of interrelationships to be
kept in mind in using the above methods. First, the objectives, both
economic and noneconomic, being sought are in some instances compet
itive or conflicting while, in other instances, they are complementary
DESIGNING EXPERIMENTS TO STUDY PROFITABILITY 33
with attainment of either or both of two objectives making it easier to
attain the other. Second, the methods of agronomists, statisticians,
and economists are in some instances competitive but are in many
instances complementary.
Agronomic Methodologies
However, considering the interrelationships among objectives of
economists and agronomists in some detail, certain agronomic meth
odological developments should be mentioned. The mechanization of
plot work is extremely important in lessening some of the competitive
aspects of objectives of experimental designs. In effect, agronomists
are substituting especially adapted or constructed machines for much
of the labor previously used in handweighing and measuring fertilizers
and in handharvesting and measuring the crops produced. Use of such
equipment calls for larger lanes and turning areas. Thus, these new
technologies make it "profitable" to substitute both capital and land for
labor in the research process.
This substitution tends to increase the overhead or fixed cost of an
experiment but reduces the perunit costs of adding plots to the design.
It also makes possible an increase in the number of experimental ob
servations. The increase in observations involves only a small increase
in cost, with the advantage of spreading the fixed cost over more plots.
Thus, designs are becoming increasingly possible whereby the agrono
mists can supply economists with the kinds of data needed for economic
interpretations.
The worksimplification methods developed at Michigan State Uni
versity can be mentioned as examples of techniques which make more
elaborate experiments possible. One device is a fertilization attach
ment for corn planters, a mechanism both accurate enough for experi
mental work and for reducing the fertilizer cleaning work in moving
from one plot to another. Another device is a onerow mounted corn
picker which makes it possible to pick one row without knocking down
adjacent rows. Accurate calibration of fertilizer drills also makes it
possible to vary rates of application from plot to plot without hand
measurement and weighing. Also, an accurately calibrated fertilizer
drill on a garden tractor makes it possible to sidedress corn rapidly
and efficiently. While machine work may be somewhat less accurate
than hand work (though this is debatable if reliable labor is hard to get),
reduced costs make it possible to offset these inaccuracies (if they
exist) with more and larger plots. So promising are these developments
that many experimental procedures need a thorough worksimplification
study. The accuracy of machine work needs an equally thorough statis
tical evaluation.
An Example
The reconciliation process in designing an experiment for studying
the economics of fertilization can be well illustrated with an example
GLENN L. JOHNSON
from Michigan. For some years there has been a rather close coopera
tion between members of the Agricultural Economics staff and the staff
of the Soils Department. Also, there has been a fair interchange of
graduate students, as well as a number of seminars and informal ses
sions. Thus, personnel involved have known and understood each other
and, in general, there exists an environment favorable to agricultural
statesmanship.
After some preliminary meetings, a decision was made to develop a
joint project between the two departments to study the economics of fer
tilizing some of the major Michigan crops. Six people from the various
departments actively designed the experiment. Statisticians, while not
project members, were consulted and used, both directly and indirectly.
Decisions had to be made on: (a) crops to be fertilized, (b) range of
fertilizer nutrients to be studied, and (c) soil types to be studied. The
problem had to be confined to portions of an autonomous subfunction in
order to make the problem manageable.
Preliminary discussions of objectives of the two departments and of
the Michigan farmers tentatively indicated that three subprojects should
be developed. The first of these was concerned with a corn, oats, wheat,
and alfalfabrome rotation on Miami silt loam, one of south central
Michigan's upland soils. Another subproject dealt with corn under con
tinuous cultivation on the Brookston series. The third dealt with the
fertilization of pasturage on one of the pasture soils of north central
Michigan.
Further consideration of the relative importance of these three stud
ies and of the cost of doing experimental work at the different locations
considerably modified the tentative conclusions. For instance, the pas
ture experiment was dropped because it was too far away from the cam
pus to be conducted economically, and the pasture fertilization problem
was less important to Michigan farmers than further strengthening of
the continuous corn experiment. Also, it was decided to carry out the
corn, oats, wheat, alfalfabrome rotation on a soil in the Fox series be
cause of the difficulty of getting a sufficiently homogeneous field of
Miami soil. It was found that, after preliminary soil tests, the contin
uous corn experiments on Brookston would have to be moved to a more
northern county from the county in which it was originally planned to
locate them. The farmers in the original area had already fertilized
the soil to such a high level that the response to fertilizer would be of
little significance for economic analysis.
The continuous corn experiment on Brookston soil will be considered
below. In this experiment, it was decided that each of the three nutri
ents would be applied at seven different levels including the zero rate
of application. It was judged by the agronomists involved that these
aY,
rates would fall mainly in the area where >O, and decreasing.
The seven levels are presented in table 2.1.
DESIGNING EXPERIMENTS TO STUDY PROFITABILITY 35
TABLE 2.1. Rates of Fertilizer Application, Continuous Corn Experiment,
Brookston Soil, Michigan, 1953
Rate
Nutrient 0 1 2 3 4 5 6
N 0 20 40 80 160 240 320
K20 0 20 40 80 160 240 320
P2Os 0 40 80 160 320 480 640
It was also possible to incorporate into the design, work of special
interest to the agronomists (Fig. 2.1). Thus, the plots running up the
main diagonal were replicated and split into two parts. On onehalf of
each plot in one replication, a different method of fertilizer placement
was employed. This made it possible for one agronomist involved to
gain certain information in which he was particularly interested. It
should also be noted that all the plots were large enough to be split in
subsequent years to absorb similar supplementary projects having to
do with, e.g., type of fertilizer, variety of corn, planting rates, and va
rious other cultural practices. Soil tests were made for each plot to
enable both economists and agronomists to study the effects of differ
ence in soil fertility on yields as well as accumulation of fertilizer
residuals.
P205
0 123 3 4
5 6110 1
P205
2 3 4 5 6
P205
0 1 2 3
4 5 6
P205
0 1 2 3 4
0 1 I 1 1 1 1 1 I I I I I
I I I 2 1 1 2 1 2 I 1 I I I I I I
2 I i i i I,
31 I I :4
5 11 1 I I I I
6 2 11 2 2 I I I I I I I I:l
The over printed
large numbers
refer to the
rates at which N
is app led.
corn experiment,
Michigan State University, 1953.
0
2
K20 3
4
5
6
Fig. 2.1 Schematic presentation of continuous
2 I I 2 I 2 jI I I I 2 J I A 2. I 2
,12 1 21 2 I 1 1 2 1 1 1L 2 1 2
I I I I I I 2 I I I
GLENN L. JOHNSON
A 7 x 7 x 7 experiment involves 343 different plots, if none of the
plots were replicated. Several members of the committee had interests
in replication of certain of the plots. For instance, the economists were
interested in replication of the O, O, O plots for a number of reasons
one encounters in fitting various alternative functions. The agronomists
were also interested in having a replicated 3 x 3 x 3 factorial. After
provision was made for 11 repetitions of the O, O, O plot and a repli
cated 3 x 3 x 3 factorial, it was obvious that project resources were in
adequate (even after cancellation of the pasture experiment) to permit
separate plots for each of the 343 cells in the design. It was decided
therefore, that the observations which could be afforded would be scat
tered throughout the sample spaotn dadeviatns
for the three utrientlarge ndto minimize t correlations
effcinaon e fertiizer nutrients applied.
Plans were made to control unstudied variabIes a to randomize
the influences of those which could not be controlled. Controls were
imposed in selection of the field and parts thereof as well as in selection
of workers and equipment. Within the portion of the field selected by
our soil classification expert as one being homogeneous, plot locations
were randomized.
At this point a member of the Soils Department took active participa
tion in the project and indicated to the economists that there were advan
tages of work simplification procedures in research. Hence, the num
ber of plots were expanded somewhat. Some of the extra plots were
scattered over the surface to be estimated. Others, however, were used
to secure more information about the relationships between yields and
each fertilizer nutrient considered separately with zero amounts of the
other nutrients applied. The distribution of plots, while probably not
ideal for fitting a given function, would give considerable flexibility in
selecting functions for analytical purposes. The last requirement ap
pears advantageous in view of certain modifications, which were devel
oped at Michigan State University, in fitting modified CobbDouglas
functions which are asymmetric and nonconstant, and have elasticities
capable of reflecting more than one stage of a production function.
It is not claimed that the ultimate in experimental design has been
secured. It is felt, however, that a moderately good job has been done
in taking into account the various objectives of economists and agrono
mists. Experimental designs were used which reflect, rather satisfac
torily, group choices (i.e., recognizing the wants, preferences, and ob
jectives of the people and organizations concerned). The economists
are pleased; the agronomists feel they will secure more than ample re
turns for their investment in the project. And both the experiment sta
tion administrators and the National Fertilizer Association administra
tors were favorable to financing the project. It has been shown that
when representatives from various fields of work join forces and agree
on a mutually advantageous research program to serve agriculture, such
a program receives high priority in the minds of administrators charged
with using research resources efficiently.
PART II
Fundamental Design and
Prediction Problems
> Alternative Designs
> Appropriate Functions
> Continuous Functions and Discrete Models
> Estimational Procedures
R. L. ANDERSON
North Carolina State College
Chapter 3
A Comparison of
Discrete and Continuous Models in
Agricultural Production Analysis
Types of Experimental Procedures
Historical Review
IN a recent review article (1), this writer traced the development of
multifactor experimental procedures. A brief resume' of this de
velopment seems desirable at this time: In the first multifactor
experiments, a single factor was varied at a time. For example, with
five factors, one might plan 5f experiments, in which each of the factors
in turn was used at levels while the other four factors were held at
some starting level. Fisher (14) and Yates (25) encouraged the use of
complete factorials and developed a large number of special designs in
volving them. In a complete factorial, all combinations of the factor
levels are used, e.g., Ifor the above experiment. These designs were
developed for experiments in which the experimental error could not be
neglected. In order to estimate the magnitude of this error in each ex
periment, the experiment had to be repeated several times, e.g., r.
These factorial designs were formed largely for useful field experiments
in which sequential experimentation would be less than the laboratory
experiments, and the factors were often of the discrete type, e.g., varie
ties or rations.
Because of the large number of factor combinations required in
many field experiments, it was felt that some form of incomplete block
design was needed to reduce the experimental error. This resulted in
the socalled confounded designs, e.g., with 2k, 3k, 3 x 2k, 3k x 2, 4k
designs. These are described by Yates (26). More complicated fac
torial designs have been constructed by Nair (21, 22), Bose (4), Finney
(13), and Li (20), among others.
When physical scientists and engineers became interested in multi
factor experiments, they found that complete and confounded factorials
required too many experimental units, especially since the experimental
errors were often much lower than in field experiments. One method of
reducing the number of experimental units was to use higher order in
teraction effects to estimate the error and hence avoid repetition of the
design. Fisher (14) and Cornish (9) described the analysis of the singly
replicated unconfounded factorial design and used the higher order
40 R. L. ANDERSON
interactions for this purpose. Jeffreys (17) and Kempthorne (18) have
advance justifications for this approach. Then Finney (11, 12), Plackett
and Burman (23), Kempthorne (18), Rao (24), and Davies and Hay (10)
developed the fractional replication designs, based on using parts of the
confounded designs. Yates (25) and Hotelling (16) had already men
tioned the use of such designs.
Some General Considerations of Factorial Experiments
The results of multifactor experiments are usually summarized in
various two and moreway tables of means and an analysis of vari
ance. For example, let us assume there are two factors (A and B), one
with p and the other with q groups, each of the pq classes having r sam
ples. Some characteristic, such as yield, is measured for each of the
pqr samples. The results are summarized in a (p x q) table of class
means (Yij) with the corresponding (p + q) border means (Ai and Bj).
B
1 2 ... q
S11Y Vi ... Yq A,
2 tY1 Y22 2q A2
A .
P Yp, Yp * Ypq Ap
Bi B2 ... Bq Y
For example, the border means for A represent averages over all
Bgroups. There are two circumstances under one or both of which
these Ameans are of importance:
1. Differences between Bgroups are the same for all Agroups,
i.e., there is no AB interaction.
2. The experimenter desires to make inferences regarding A only
when averaged over these particular Bgroups.
If item 1 is true, one can set up the following model to represent the
yield for a given sample:
(1) Y = (mean) + (A effect) + (B effect) + (error).
The A and B effects are estimated by computing the deviations of
group means from the general mean, e.g.,
Al effect = A1 Y.
The errors are assumed to be normally and independently distributed
COMPARISON OF DISCRETE AND CONTINUOUS MODELS 41
with zero means and same variances, a2. In this case, analysis of vari
ance is:
Source of Variation d. f. S.S. M. S.
A p1 SSA MSA
B q1 SSB MSB
Residual (p1)(q1) SSI MSI
Error (r1) pq SSW s2
In the above analysis of variance, SSA = qr 2 A2 pqr Y2 and SSB
= pr B pqr Y2. The residual sum of squares measures the failure
of the A and B effects to be additive, i.e., presence of AB interaction.
It is computed as:
22
r Yfij pqr SSA SSB.
The error variance, a is estimated from the variability within classes.
The mean squares are all computed by dividing the sums of squares by
the corresponding degrees of freedom. One can test for the existence
of interaction by use of F = MSI/s2. Presumably, if this is significant,
inferences about A effects must be confined to averages over these q B
groups. Otherwise one should consider the general model:
(2) Y = (class mean) + (error).
Then each of the pq classes is considered separately and the simple
analysis of variance is:
Source of Variation d. f. S. S. M. S.
Treatments pq1 SST MST
Error (r1) pq SSW s2
SST = r S Y2 pqr Y2
The same procedures can be followed for more than two classifi
cation variables. In this it is advisable to look at the individual contri
butions to the interaction: AB, AC, BC ... ABC ... In many cases it
is even possible to subdivide SSA, for example, into pertinent single de
gree of freedom contracts; hence, SS(AB) can also be subdivided. This
subdivision of SSI is useful in detecting particular aspects of nonaddi
tivity which may be concealed in blanket tests of MSI/s2. For more ex
act discussion of these problems, see Chapter 20 of Anderson and
Bancroft (2).
R. L. ANDERSON
Extension of Factorial Experimentation to Continuous Variables
In the past, even though the factors could be varied continuously,
most analyses of experimental data have followed the same procedures
as for discrete classifications. For example, if one had an experiment
to study the effect of nitrogen (n) and potash (k) on the yield of corn, one
might consider a simple 2 x 2 experiment with four treatment combi
nations: low n and k (00); low n and high k (02); high n and low k (20);
and high n and high k (22).1 Suppose each treatment were randomly as
signed to r plots. The usual summary procedure would be to form the
fourtreatment totals and means in 2 x 2 tables. The totals are indi
cated as follows:
Potash
low high
low (00) (02) No
Nitrogen
high (20) (22) N2
Ko K2 G
The border totals are indicated by capital letters, with G for the grand
total.
If one were unwilling to make any assumptions about the compara
bility of the four treatments, he would look only at the fourcell mean
(cell totals divided by r) and use model 2 and the accompanying analy
sis.
If the experimenter feels that the effect of increased n or k is the
same regardless of the level of the other element, he would use an
adaptation of model 1 as follows:
(1') Y = (mean) + (n effect) + (k effect) t (error) ,
where the + sign refers to high level plots and the to low level plots.
For example, the average or expected yield for a plot receiving high n
and low k is:
(mean) + (n effect) (k effect) .
The n effect, for example, represents the expected increase in yield due
to high n over the average of high and low n, and is estimated by
N2 N= N
4r 2r
The analysis of variance is the same as for model 1. The residual
can be used to test the adequacy of the additive model 1', i.e., test for
the existence of an (NK) interaction. If this residual is significant, the
10 is used for the low level and 2 is used for the high level, so that 1 may be introduced
as a middle level.
COMPARISON OF DISCRETE AND CONTINUOUS MODELS 43
Source of Sum of Squares =
Variation Mean Square
N (N, N0)2
4r
K (K2 Ko)2
4r
Residual [(00)(02)(20)+(22)] 2
4r
effect of increased n is not the same for low and for high k (and vice
versa). Hence it is necessary to interpret each cell mean separately,
i.e., use model 2.
Continuing this aping of the models for discrete factors, the follow
ing general model has been constructed for the 2 x 2 experiment:
(3) Y = (mean) (n effect) t (k effect) t (nk interaction effect)+ (error),
where the interaction effect receives a plus sign for the (0,0) and (2,2)
plots and a minus sign for the (0,2) and (2,0) plots. For example, the
expected yield for a plot receiving high n and low k is:
(mean) + (n effect) (k effect) (nk interaction effect) .
The interaction effect is estimated by:
(00) (02) (20) + (22)
4r
If the response surface can be approximated by a simple mathemati
cal function, it seems more logical to estimate the parameters of this
function instead of main effects and interactions. In the present ex
ample, consider the following continuous model:
(4) Y = 9o + PiX1 + f2X2 + 12X1X2 + (error).
Xi and X2 represent the respective levels of nitrogen and potash as
deviations from the mean level in the experiment (X = 1 for low and
X = +1 for high level); fPo is the expected yield for n and k midway be
tween the amounts applied in the experiment (Xi = X2 = 0); fI and p2 are
linear effects of added n and k; 312 is the interaction parameter. Using
model 4, the cell totals (of r plots each) have these expectations:
low k(X2=1) high k (Xz=l) Total
low n (X1= 1) r(o1 22) r(.o 1+f2 12) 2r(fo i)
high n (X1=l) r(3o+9132Pz12) r(0o+P1+02+f12) 2r(jo+j1)
Total 2r(po32) 2r(f3o+9f2) 4r o
R. L. ANDERSON
The estimators of the 3's in equation 4 are:
Parameter Estimator
Al bi No
1 bl = K024
1.2 b_ = (00)(02)(20H)22)
4r
go bo = G/4r = Y.
The variance of each estimator is o2/4r. Note that these estimators
are the same as for the effects of model 3. j3, for example, measures
the average difference in yield per unit change in n for these two k
treatments, i.e., the change in Y for a unit change in X,, neglecting
interaction.2 Also the analysis of variance produces the same three
orthogonal sums of squares for treatments, using either models 3 or 4.
Hence it appears that models 3 and 4 are identical. However, there
is a very important difference. Model 3 makes no assumption regard
ing the shape of the response surface, but model 4 implies a definite
continuity of response; hence, one would feel free to use the results of
model 4 to interpolate between the actual levels used in the experiment.
If he did use model 3 for this purpose, he would actually be assuming
the continuous model 4. One is often tempted to extrapolate the results
beyond the levels used in the experiment; such extrapolation assumes
the same response surface holds beyond the experimental levels. In
other words, one uses model 2 or 1 if he does not wish to assume a
quadratic response surface, but uses model 4 if experience or theory
indicates such a surface would be satisfactory.
If the design is spread out so that the low and high levels differ by
2d units (instead of 2), bi will have a denominator of 4rd and b,2 a de
nominator of 4rd2. Hence the variance of bi is reduced by a factor of d2
and b12 by a factor of d. The only reason for not using extremely di
vergent levels is that the response surface may have a different shape
at extremely large or small fertilizer applications.
If the continuous model 4 is used, it seems unreasonable to include a
quadratic term involving XX2 without also including terms involving X.
and X4. The shape of a response surface such as model 4 is rather gro
tesque. In other words one would be more likely to consider the follow
ing general quadratic model:
(5) Y = o0 + I1X1 + fX2 + 312X1X2 + IX2 + j2Z2Xi + (error) .
"It should be clear that the difference between low and high levels is a twounit change,
e.g., if low is 50 pounds per plot and high is 100 pounds per plot, a unit change is 25 pounds
and fl and 0, measure the linear effects of 25pound increases.
COMPARISON OF DISCRETE AND CONTINUOUS MODELS 45
If model 5 is the true continuous model, the expectations of the cell
totals are:
low k (X2=l) high k (X2=l) Total
high n(X,=l) r(A+g +gi+f2t ) r(4+ + 1+6i+ 2+8,) 2r(P+8+4+4a)
Total 22ro4+1+,) 2r(o+4+,+8a) 4r(A o+x,+ ,)
It turns out that the estimates of 31, z2, and 112 are the same as for
model 4 and are not mixed up with the quadratic terms (3,i and 322), i.e.,
they are unbiased estimates. However, there is no method of estimating
A%, 311, or 322, since their sum is estimated by Y. Hence this analysis
indicates that it is safe if only statements are made regarding the
treatments used in the experiment and no attempt is made to predict the
results for other fertilizer levels.
Of course the solution to the above dilemma is to add other levels of
n and k. The traditional design to estimate quadratic effects is the 3 x 3
complete factorial with the three levels of n and k equally spaced.3 As
suming the middle values of n and k are the averages of the low and
high levels used in the 2 x 2 experiment, i.e., if the low and high appli
cations were 50 and 100 pounds per plot, the middle application would be
75 pounds per plot. In the factorial setup, the levels are designated as
0, 1, and 2 with X = 1, 0, 1, respectively. Henceforth, factor combi
nation will be designated by the levels used, e.g., (1, 1). Assuming r
plots per cell and using model 5, the expectations for the (1, 1),
(1, 1), (1, 1) and (1, 1) totals would be as before. The expectations
for the other five class totals and the border totals would be:
(1, 0) r(fo A + 3u)
(0, 1) r(go 32 + 322)
(0, 0) r fo
(0, 1) r(3o + f2 + 922)
(1, 0) r(1o + 83 + in)
No 3r(3o 01 + 111) + 2r 122
N, 3r 3o + 2r 922
N, 3r(go + P3 + 13) + 2r 122
Ko 3r(go 92 + 122) + 2r 0,
K, 3r go + 2r 03n
K2 3r(po + 2 + 22) + 2r n11
G 9r 3o + 6r (311 + 32
SEqual spacing enables one to analyze linear and quadratic components in a simple man
ner, but it is not an essential, or even the most efficient, method of spacing.
R. L. ANDERSON
The following estimators and variances are obtained:
Variance of
Parameter Estimator Estimator
b1 bi =(N2No)/6r o2/6r
2 b2 = (K2Ko)/6r a2/6r
912 b2 = [(,) (1,1) (,) + (l,l)/4r 2/4r
11 bi = (N22Ni+No)/6r a2/2r
022 b22 = (K22K1+Ko)/6r a2/2r
go bo = [5(Ni+K1)(No+N2+Ko+K2)]/18r 5 a2/9r
Note that b,, b2, and bz1 are the same as before; also, if the levels are
(d,O,d), the variances for the linear coefficients are again reduced by
a factor of d2 and for the quadratic and interaction coefficients by a
factor of d4.
The analysis of variance
quadratic component):
is as follows (( stands for linear and q for
Effect d.f. M. S.
N( 1 (N2No2/6r
Kf 1 (KK)2/6r
NfK 1 [(1, )( 11) (1,1)]2/4r
Nq 1 (N2N,+N) /18r
Kq 1 (K22Ki+K2/18r
Residual 3 [SST SS(NC + K + ... + Kq)]/3
Error 9(r1) s2 = SSW/9(r1)
The residual mean square can be used to test for the adequacy of the
model. If the 3 x 3 complete factorial is used, it turns out that these
three degrees of freedom can be subdivided into three orthogonal com
ponents, which measure NgK, NqKf, and Nq Kq interaction effects
[8122XIX2 + 2 112X1X2 +,1122X'X2 is added to model 5].
Once again a factorial model similar to model 3 can be constructed
with the same linear and quadratic effects as in model 5. However,
there seems little reason for estimating such effects unless one is will
ing to assume a quadratic response surface. If he does not wish to as
sume a quadratic response surface, he has two possible factorial
models:
1. Model 2 with nine treatments
COMPARISON OF DISCRETE AND CONTINUOUS MODELS 47
2. Model 1' with two effects for each factor: above and below the
middle application or referred to either the high or low appli
cation. 4
The analysis based on model 1' would include a sum of squares attriut
able to the interactions, giving a test of the adequacy of the model.
These remarks hold for any number of factors and levels per factor.
If there is a mixture of classification variables (e.g., varieties) and
continuous variables, a combined factorial and continuous model can be
set up and analyzed in a manner analogous to covariance. This would
assume that the parameters for the continuous variables were the same
for each discrete classification; a test of this hypothesis can also be
constructed.
The Use of Blocking Methods to Reduce Experimental Error
The use of blocking methods in the previous discussion has not been
considered because they only complicate the presentation without alter
ing any of the conclusions. However, one must consider the blocking
procedure if there is confounding. Unfortunately, the procedures used
in constructing such designs have been based on confounding certain
parts of the higher order interactions which are not related to higher
degree components. For example, the socalled I and J parts of the NK
interaction in a 3 x 3 experiment do not pertain to any one of the four
degree components, NpNJ, NfKq, NqKf, or NqKq. One would prefer a
design which minimized the confounding on NfKf.
A bulletin now in press by Binet, Leslie, Weiner, and Anderson (3)
presents the confounding patterns in terms of degree components. This
bulletin should be of use in three ways:
1. It presents shortcut methods of analyzing these confounded
experiments when degree components are of interest.
2. Several new confounded designs are presented.
3. It presents the confounding patterns for various designs, so the
reader can select the design which will be best for his problem.
To illustrate the procedures, suppose the nine treatments in the 3x3
experiment were put in 3 blocks of 3 plots each. One such arrangement
would be (the treatments refer to levels, and Bi are block totals):
Block
1 2 3
(1,1) (1,1) (0,1)
(1,0) (0,0) (1,0)
(0,1) (1,1) (1,1)
Bi B2 B,
'Cf. Anderson and Bancroft (2), Section 20.5.
R. L. ANDERSON
If b'j represents the mean of the jth block, then two block contrasts
are formed:
2C, = b3 b' and 6c2 = bI 2bI + b .
The least squares equations for the two block and four NK effects are:
Ci NTK q NqKf C2 NfK f NqKq Yield Sum
6 6 6 0 0 0 B3B,
6 12 0 0 0 0 (NVKq)
6 0 12 0 0 0 (NqKf)
0 0 0 18 6 18 B, 2B2+ B3
0 0 0 6 4 0 (NEKE)
0 0 0 18 0 36 (NqKq)
The yield sum for NfKq, for example, is:
[(1,1) 2(1,0) + (1,I)] [(1,1)2(1,0) + (1,1)].
The usual procedure in analyzing these results would be to assume
the block contrasts and NfK were the only real effects. This leaves
only one contrast for testing the model, since there are only four de
grees of freedom in the above six equations. The method of analysis
proposed in the bulletin is the abbreviated Doolittle method, which is
also discussed in detail by Anderson and Bancroft (2). Obviously there
is no estimate of error from this experiment. If such an estimate is
needed, another replicate should be used, preferably one which has a
different confounding pattern, as indicated in the bulletin.
For experiments with many factors, it is often possible to estimate
the pertinent contrasts by use of fractional designs.
Special Designs To Estimate Parameters of Response Surfaces
The material by Binet et al. (3) furnishes a method of using existing
confounded factorial designs to estimate the important degree compo
nents. However, for most experiments in which the experimenter has
evidence that a smooth response surface is suitable, he should consider
designs especially constructed to estimate the parameters of this sur
face and not to estimate class means for a classification model. Box (5)
developed some general design principles for estimating the parameters
of planar surfaces.
Box and Wilson (8) proposed a new design for estimating quadratic
surfaces which gives more information on the quadratic effects and less
on the highdegree effects. Their composite design would push the (0,1),
(0,1), (1,0), and (1,0) points a units from the center of the design as
indicated in figure 3.1.
COMPARISON OF DISCRETE AND CONTINUOUS MODELS 49
(:C 0)
C I, I)
* 3 X 3 DESIGN
X COMPOSITE DESIGN
Fig. 3.1 The Box and Wilson composite design for
estimating quadratic surfaces.
If a = 2, the expectations of the totals for the four altered cells are:
(2,0) r (o, 20 + 401)
(0,2) r (io 2(2 + 443,)
(0,2) r (3o +2/, + 4P,,)
(2,0) r (9, + 2Pi + 41,)
In this case one cannot analyze the results as for a 3 x 3 table, be
cause it is an incomplete 5 x 5 factorial experiment. Here one must
use the general leastsquares approach. The matrix for the normal
equations is:
Coefficients of Estimators Right hand side
Equation bo b b b b12 b1 b22
bo 9r 0 0 0 12r 12r G
bi 0 12r 0 0 0 0 g, = SX1Y
b2 0 0 12r 0 0 0 g, = SX2Y
b12 0 0 0 4r 0 0 gl2 = SXX 2Y
bl1 12r 0 0 0 36r 4r g,1 = SX2Y
b22 12r 0 0 0 4r 36r g22 = SX2Y
I, I)
: Oz
(I, I)
(, 0)
(1,I)
, (0,<)
R. L. ANDERSON
In the preceding, for example, g, = (1,1) + 2(2,0) + (1,1) (1,1)
2(2,0)(1,1), where (1,1), etc., stand for class totals. The solutions
and variancess of the estimators are:
Variance of Estimator (X a2r)
3x3
Parameter Estimator Composite 3 x 3 (Adjusted)
g1/12r 1/12 1/6 1/12
~2 g2/12r 1/12 1/6 1/12
z12 g12/4r 1/4 1/4 1/16
Pi, (30gn + 18g22 64G)/384r 5/64 1/2 1/8
P22 (30g22 + 18gn 64G)/384r 5/64 1/2 1/8
Po (10G 3gn 3g22)/18r 5/9 5/9 5/9
One gets the impression that there is a tremendous reduction in
variances of estimators by use of the composite design instead of the
3 x 3 factorial. However, most of this gain is the natural result of us
ing a wider range of X's; the incompleteness of the factorial in the
composite design is not responsible for all the gain. This was indicated
for the 2 x 2 experiment. One could adjust the coordinates of the 3 x 3
design so that the spread is the same as for the composite design. The
variance of the coordinates for the latter (with a = 2) is [2(4) + 4(1)
+ 3(0)]/9 = 4/3. Let the new coordinates for the 3 x 3 design be (d,0,d),
so that the variance of these coordinates is 2d2/3 = 4/3; or d = 2.
Hence, the variances of linear terms are reduced by 1/2 and of quad
ratic terms by 1/4. Therefore, the composite design has improved the
quadratic estimators at the expense of the interaction one. Box and
Wilson (8) show that this is desirable in estimating the optimal factor
combination.
Another criterion of the relative efficiency of two different designs
in estimating the parameters of a response surface would be the amount
of information used to estimate the high degree coefficients, which are
assumed to be unimportant.
Box and Hunter (7) have advanced another principle of a good
surfacefitting design; it should be rotatable; i.e., the accuracy of the
estimates of the parameters should not depend on the orientation of the
design with respect to the true surface itself. They have constructed
several incomplete factorial designs which meet this requirement.
Mason discusses in Chapter 5 some recent experiments in which the
composite designs have been used.
'These are obtained by inverting the lefthand matrix. The abbreviated Doolittle or
squareroot method is usually used, although special pattern matrices can be used.
COMPARISON OF DISCRETE AND CONTINUOUS MODELS 51
Sequential Experimentation
Much of the impetus for the BoxWilson paper (8) came from a need
to develop sequential procedures for determining optimal factor combi
nations. Various procedures have been summarized in Anderson's re
view article (1). Since then, Box (6) has published an extensive dis
cussion of the entire problem. Although the use of these sequential
methods may be somewhat limited in fertilizer experiments because of
the length of time needed to obtain results, it probably would be desira
ble to develop a more systematic procedure of utilizing past experience
in designing future experiments.
Better methods are needed to pool data from a series of experi
ments. Researchers should be encouraged to spend more time on these
problems.
Some Special Comparisons of Discrete and Continuous Models
Comparison of Discrete Model 2 and Quadratic Model 5
Using 3 x 3 Design
1. The quadratic model is correct. In this case the estimated aver
age yield for plots receiving X, units of N and X2 units of K (measured
from the mean level) is:
= bo + biX + b2X2 + b12X1X2 + bX2 = b22X.
A
In order to obtain the sampling variance of Y, it requires the variances
of the estimators given previously and the covariances. All of these
could be obtained by inverting the matrix of sums of squares and prod
ucts of the regression variables in the normal equations. This matrix
is as follows:
bo b, b2 bl2 b11 b22
bo 9r 0 0 0 6r 6r
bi 0 6r 0 0 0 0
b2 0 0 6r 0 0 0
b12 0 0 0 4r 0 0
bui 6r 0 0 0 6r 4r
b22 6r 0 0 0 4r 6r
Since bo, b1u, and b12 are the only correlated variables, consider them
separately in a 3 x 3 matrix A, which when multiplied by its inverse C
is the identity matrix.
9r 6r 6r C1 C2 C2 1 0 0
A 6r 6r 4r C C2 C3 C4 = 0 1 0
6r 4r 6r C2 4 Cs 0 1
R. L. ANDERSON
There are only four different elements of C. These can be determined
quite simply as follows:
9r C1 + 12r C2 = 1 C 2 = 1/3r; C0 = 5/9r
6r C0 + 10r C2 =0
6r C2 + 6r Cs + 4r C = 1 4 = 1/2r 3C2 = 0
9r C2 + 6r Cs + 6r C4 = 0 C3 = 1/6r C2 = 1/2r
Hence the matrix of variances and covariances of the b's is:
5/9 0 0 0 1/3 1/3
0 1/6 0 0 0 0
2 0 0 1/6 0 0 0
r 0 0 0 1/4 0 0
1/3 0 0 0 1/2 0
1/3 0 0 0 0 1/2
A
The variance of Y is:
2A o2
a2(Y) = [5/9 + 1/6 (X2 + X ) + 1/4 (XeXl)+ 1/2 (XI + X')
2/3 (X2 + Xi)]
2
= [5/9 + 1/2 (X1+ X4 X' X.) + 1/4 X ] .
If the discrete model is used, every mean will have a sampling vari
ance of T2/r. For even the most divergent points ( 1, 1),
A
2(Y) = 29 a2/36r
which is less than a2/r. Hence, if the quadratic model is correct, even
the yields at the experimental points are estimated more accurately
from the regression model instead of the simple average yield at that
point. Of course Y is even more accurate for the other five points.
The same conclusions hold for comparing two mean yields. The
largest variance using Y is the comparison of (1,1) and 1,1), which is
5 a2/3r, as compared to 2 a2/r for model 2. Many of the comparisons
using Y have much lower variances than this.
The results might be even more favorable if another design were
used.
2. The quadratic model is biased. Suppose the true model is model
5 plus 3 X). In this case Y is too small by 3 when X, = 1 and too large
by 3 when X = 1. Some mean differences would be biased by 20,
others by 0, and others not at all. However, the estimates using
COMPARISON OF DISCRETE AND CONTINUOUS MODELS 53
model 2 would be unbiased. The problem of whether to use the biased
estimates depends on a comparison of the suspected magnitude of the
bias and the variances mentioned above. This problem may be even
more serious if the form of the response equation is radically different
from the quadratic, e.g., if. it is exponential or logistic.
Returning to the bias of 3 X], it should be mentioned that at least one
of the other P's will also be biased if this term is not considered in the
estimation procedure (when j/ 0); for example:
E (b) = p3+/3 .
3 is called an alias of 01. Box and Wilson (8) consider possible aliases
in evaluating various designs. It is possible to construct designs so
that possible aliases will not have much effect on the estimates. This
may be one of the chief reasons why agricultural experimenters have
not considered continuous models. Hildreth (15) has considered an esti
mation procedure which is built on model 2, but uses certain inequality
restrictions on the production function. The estimation procedure used
by Hildreth is discussed in Chapter 4.
PseudoInteractions in Some Factorial Experiments
The tendency to follow the mechanical procedure of analyzing fac
torial experiments in terms of main effects and interactions can result
in serious loss of information, often of a misleading nature. As an ex
ample, consider an experiment involving two levels of nitrogen (coded
n = 1 and 1) and two different cover crops to be plowed under. Suppose
C, supplies no nitrogen to the soil, whereas C2 supplies 2 units of n
(coded n = 1 and 1). In addition, the two crops supply other unspecified
nutrients. Assume that the yield is a quadratic function of ni plus some
additive amount due to the unspecified nutrients in the soil and furnished
by the two crops: 3o y for C1 and p3o+ y for C2(y may be positive or
negative). Hence the model is:
Y = Ao + fln + 1nn2 v y + (error)
where y is added for C2 plots and subtracted for C1 plots.6 The expected
class and border total yields are:
Crop 1 Crop 2 Total
n = r(3o201+40uy) r(3o + y) 2r(Po00+2011)
n = 1 r( o ) r(P,+2P1+431++ y) 2r(Po+3++2A,)
2r(AoA+2a13 V) 2r(Po+Aj+2ax+ y) 4r(0o+2 11)
'The center of the system is now one unit more than the average of the two nitrogen
levels.
R. L. ANDERSON
The estimators and their variances are:
Parameter Estimator Variance
S(N Nt)/4r 2/4r
S [(1,1)(1,2)(1,1)+(,2)] /8r 2 /16r
S [(1,2)(1,1)] /2r a2/2r
o [(1,2)+(1,1)] /2r /2r
Compare these results with those obtained by use of traditional fac
torial methods.
Effect Yield E(Yield) E(MS)
Nitrogen NiNI 4rpI 4rpf+ 2
Crop C2C1 4r( v +P) 4r( y +J3)2 + 2
Nx C (1,1)(1,2)(1,1)+(l,2)8rj31 16rpI,+ a 2
An N x C interaction is indicated if there is a quadratic effect of nitro
gen; also the crop effect will be mixed up with the linear effect of
nitrogen (this is satisfactory if one only wants to test for differences in
yields and not to determine basic causes of such differences). But a
major criticism is a failure to provide a method of estimating the quad
ratic effect of nitrogen. The N x C interaction effect is the least
squares estimate of 1j3, but this fact is concealed in a routine factorial
analysis of variance.
This is a very simple illustration of the need for more basic models
in discussing responses to treatments. Classification models may con
ceal basic response patterns. One might consider this problem when
three instead of two levels of n were used. In this case the factorial
estimate of 31, probably would be inefficient, because of neglect of the
information from the N x C interaction.
Yates (26) presents a 23 experiment with 4 replications, the factors
being N, K, and D (dung). Levels were none and some, the latter being
0.45 cwt. N per acre, 1.12 cwt. K20 per acre, and 8 tons of D per acre.
Assume that this amount of dung supplies the same as the "some" of n
and k, plus "some" other nutrients (called d). Code these data with 1
for none and +1 for some. Hence, the values of the variables for the
various plots are:7
'A unit of nitrogen is 0.225 cwt., of potash is 0.56 cwt., and of dung is 4 tons: the center
is at 0.45 N, 1.12 K and 4 D.
COMPARISON OF DISCRETE AND CONTINUOUS MODELS
N K D n k d Yield of 4 plots
0 0 0 2 2 1 425
1 0 0 0 2 1 426
0 1 0 2 0 1 1118
1 1 0 0 0 1 1203
0 0 1 0 0 1 1283
1 0 1 2 0 1 1396
0 1 1 0 2 1 1673
1 1 1 2 2 1 1807
Assume a quadratic equation in n and k,
Hence:
with d appearing linearly.
(6) Y = Po+An+32k+jj3n2+322k2+f312nk+93d+ (error) .
Because this experiment was not designed to estimate quadratic effects,
it turns out that if a complete quadratic model was used with P33d2,
013nd, and 323kd included, the following pairs of coefficients could not be
separated: 3o and P33; t ij and 13s; and /22 and 323. In other words the
constant and d2, n2, and nd and k2 and kd are aliases. It is assumed
here that d is essentially a residual variable, which is unlikely to have
any effect and especially not a quadratic one; however, one cannot be
sure which of two aliases is responsible for an effect.
The matrix for the least squares equations for model 6 is:
bo bi b2 biu b22 bl2 b3 Yield Sum
32 0 0 64 64 32 0 9,331
64 32 0 0 0 32 3,320
64 0 0 0 32 5,258
256 128 128 0 18,984
256 128 0 17,324
128 0 8,928
32 2,987
The forward solution of the abbreviated Doolittle method is as
follows:
R. L. ANDERSON
b11 b22 b12 b3
64 64 32 0
Yield
9331
UO 1 0 0 2 2 1 1 9331/32
64 32 0 0 0 32 3320
bl1 1/2 0 0 0 1/2 3320/64
48 0 0 0 16 3598
b2 1 0 0 0 1/3 3598/48
128 0 64 0 322
bi 1 0 1/2 0 322/128
128 64 0 1338
b22 1 1/2 0 1338/128
32 0 105
b12 1 0 105/32
32/3 383/3
b3 1 383/3
The variancecovariance matrix and the estimates are:
bo bl b2 b1 b22 b12 b3 Estimates
16 0 0 4 4 4 0 1 bo 310.75
4 0 0 0 0 4
4 0 0 0 4
2 1 2 0
2 2 0
4 0
12
bi 10.41**
b2 70.97**
bni .88
b22 12.09**
b12 3.28
bo 11.97*
Since the error variance in the experiment was 347.01 (with 21 de
grees of freedom), r2/128 is estimated by 2.71. This is multiplied by
the diagonal terms to obtain the estimated variances for the estimates.
All linear terms and the quadratic term for k are significant (b3 barely
so at the 5% level) while b12 is about the same size as its standard
error. The sum of squares can be compared with those of Yates as
follows:
~;
COMPARISON OF DISCRETE AND CONTINUOUS MODELS 57
Effect Yates Here8
Nf 3,465.3 172,225.0
K& 161,170.0 269,700.1
Nq and N'De 810.0 810.0
Kq and KfDf 13,986.3 13,986.3
NfKf 344.5 344.5
Df 278,817.8 1,528.0
Nf K Df 124.0 124.0
This is only an illustrative example, however, and should serve as
an example of the procedure. There may be some questions concerning
the use of the coded values. These are put in so that the estimators
will be as nearly uncorrelated as possible; this enables one to better
evaluate the usefulness of various predictors in the model. Box and
Wilson (8) generally follow this procedure.
Problem of Adjustment for Available Nutrients With Continuous Models
One of the major needs in the determination of fertilizer response
surfaces is a method of adjusting for nutrients available in the soil be
fore the experiment is started. In a single experiment, it is usually
assumed that the variation in basic levels is random, with the average
level being taken account of by the constant term. If there are no es
sential differences between the basic levels in the plots for each of the
treatments, the results of the experiment can be used to indicate treat
ment contrasts. However, if a continuous model such as the quadratic
model 5 is used, the experimenter should be careful about extending the
results to plots with different available nutrients.
If the effect of the available nutrients is to merely increase the
actual levels of X, the results can be converted to a prediction equation
in terms of the available plus added nutrients. In order to simplify the
results, consider a quadratic prediction equation for an experiment in
volving only one nutrient,
(7) E (X) = go + lX + j3X2
where X is the added amount of the nutrient. The actual amount (avail
able plus added) of the nutrient in an experiment is designated as
N=X+d(X=Nd). Then:
(8) E (N) = (3o 3id + Pi3d2) + (P1 293nd) N + PAN 2
Now try to apply the results of this experiment to a farm. The
8These are not adjusted sums of squares; i.e., Nj is not adjusted for Kf or Df; K is not
adjusted for DJ; and Nq and Kq not for NjK f. Note the Nq = Yates' NfDf and the Kq = Yates'
KJDf, as indicated above.
R. L. ANDERSON
predicted yield if X is applied is E(X). Suppose the value of d for this
farm is do (N = X + do); then the expected yield when X is added should
be:
F(X) = (P3o Old + 3n&d2) + (P1 20,,d) (X + do) + f(3X + d 2
= [(3o 01(d do) + 31n (d do)2] + [0i 211 (d do)] X + pIX2
The bias in using E (X) instead of F (X) is:
(9) E (X) F (X) = (d do) (pl + 2AX) (d do)2 11n .
One might suppose that even though the predicted yield is biased, at
least the difference between the predicted yields for two different levels
of farm application would be unbiased. Even this is not true. The bias
in the predicted increase in yield for an application of X, instead of Xi
is 203u (X2 Xi) (d do), which will be negative for X2 > X1 and d > do,
since B1 is expected to be negative; hence, one would tend to under
estimate the effect of added nutrients if the available nutrients at the
farm are less than at the experimental plots.
These problems become further aggravated when one attempts to
combine the results of experiments at two locations with different
values of d. Suppose d = di for one location and d = d2 for a second lo
cation, but the same rates of application are used in each experiment,
e.g., X = 1, 0, 1. If a quadratic model is used, the experimental model
E(N) is:
(10) E (N) = Po* + P1* N + 1* N2
where Po*, Pi* and 031* can be found from model 8 above. The values of
the 3* are assumed to be the same for each experiment (neglecting
other nutrients in this discussion); however, the values of go and pi in
model 7 are not the same. Let P3l and Pi' represent the values of 3i in
experiments 1 and 2, respectively.
Then solving for the p's in terms of the 3*'s, yields:
'o = ot + dlPt + d2 11* and W' = P0 + dI* + d~2 1
P0 = o3 + 2d13n1* and P" = Pf + 2d21 *
On the basis of the above results, the experimenter would make one
of two incorrect decisions if he did not take account of the inequality of
the available nutrients for the two experiments:
1. He would conclude that the true response pattern was different at
the two localities and publish two predition equations, each of which
represents an inefficient use of the data in estimating the basic para
meters. This may prevent the savings in extension work which over
all recommendations entail. However, the biases mentioned above are
less likely to be so important, because the experimenter realizes his
prediction equation is different for different locations.
2. If the experimental error is large compared with (d2 d1), he
might conclude that the differences in the estimates of the P's was a
COMPARISON OF DISCRETE AND CONTINUOUS MODELS 59
chance difference, and use average ('s for his prediction equation.
This would produce the biased results mentioned above. However, the
important point here is that the estimates of the parameters are quite
inefficient because the large spread in N over both experiments is neg
lected. There is uncertainty as to which incorrect procedure is worse,
since this is a matter of weighing the extra costs of a wide variety of
recommendations against the inefficiencies and biases of overall
recommendations.
To illustrate the fact that one can obtain more information regarding
the response surface by combining the two experiments, suppose only
two levels of X (X = 1 and 1) are used for each experiment but the
available coded levels are di = 1 and d2 = 1. If single estimates are
made for each experiment, no estimator of 0i, will be available; hence,
if (31 is not zero, the separate estimators of the linear coefficient will
be biased. However, in this case, the pooled estimator of pi will be un
biased because di + d2 = 0. Also, in this case, the objective is to com
pare the response surfaces in terms of the total nutrients (X + d). The
number of plots for each level of N and the estimators of P* and their
expected values when 31f / 0:
N
Experiment 2 0 2 b* E(bf)
1 r r (NoN.2)/2r pt 2Pl
2 r r (NNo)/2r P* + 2,f*
In both experiments, a2 (bt) = a2 /2r. The pooled estimate of t* is un
biased and has a2 (bZ) = a2 /4r.
If a combined analysis is made, biT = (N2 N0 + N2)/8r, where No
is the sum of the yields of the 2 rplots with N = 0; a2(bl*) = 2/16r. In
this case b* = (N2 N2)/4r with a 2(b) = a2 /8r; note that this variance
is onehalf the pooled variance. Even if the experimenter wants to as
sume different values of P3 in each experiment because of unequal
amounts of other nutrients, he obtains the same estimate of A( from
the combined data, and the above pooled estimate of *T.
If a more complicated model is considered, such as an exponential
or logistic model, the experimenter will probably find that the inclusion
of the available nutrients in the model is just as important. It may be
that one of the reasons for obtaining such unrealistic production func
tions from combined data is the failure to adjust for the available nutri
ents. Also, this may account for the divergent shape of combined re
sponse surfaces when various mathematical forms are used. Someone
might make studies similar to these for the more complicated pro
duction models.
If one can obtain more efficient and more nearly unbiased estimates
by adjusting for available nutrients in several experiments in a com
bined analysis, why is this not done more often? In many cases, the
answer may be lack of knowledge of how to make even the simple
R. L. ANDERSON
combined analyses. However, the real answer may be generally more
complimentary to experimenters:
1. Statisticians have not developed easy and efficient estimation pro
cedures for the more complicated models.
2. Procedures for determining available nutrients are not too well de
veloped.
3. It is often difficult to calibrate available and applied nutrients.
4. Even though only a few nutrients are added in the experiment, ad
mustments must be made for all available nutrients. This may result
in a much more complicated analysis.
5. Research has not been well coordinated. As a result, computations
may be complicated and total levels may not be spread out very much
in the various experiments.
6. Adjustments for weather factors are also needed, especially when
combining data from several years. Cropweather and soilweather
relationships are even more poorly known than are cropnutrient
relationships.
Much of the computing difficulty will probably be relieved as more
use of electronic computers is made. Hence, it should be recommended
that coordination of efforts in the direction of setting up realistic
models and measuring and calibrating available nutrients is needed.
References Cited
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Analysis of confounded factorial experiments in single replications. N.C. Agr.
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factorial designs in industrial research. Biometrics 6:23349.
COMPARISON OF DISCRETE AND CONTINUOUS MODELS 61
References Cited
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12. 1946. Recent developments in the design of field experiments.
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16. HOTELLING, H., 1944. Some improvements in weighing and other experimen
tal techniques. Annals of Math. Stat. 15:297306.
17. JEFFREYS, H., 1939. Theory of Probability. Oxford University Press, New
York.
18. KEMPTHORNE, O., 1947. A simple approach to confounding and fractional
replication in factorial experiments. Biometrika. 34:25572.
19. 1952. The Design and Analysis of Experiments. Wiley, New York.
20. LI, JEROME C. R., 1944. Design and statistical analysis of some confounded
factorial experiments. Iowa Agr. Exp. Sta. Bul. 333.
21. NAIR, K. R., 1938. On a method of getting confounded arrangements in the
general symmetrical type of experiments. Sankhya 4:12138.
22. 1940. Balanced confounded arrangements for the 5n type of experi
ments. Sankhya 5:5770.
23. PLACKETT, R. L. and BURMAN, J. P., 1946. The design of optimum multi
factorial experiments. Biometrika 33:30525.
24. RAO, C. R., 1947. Factorial experiments derivable from combinatorial ar
rangements of arrays. Jour. Roy. Stat. Soc. Suppl. 9:12839.
25. YATES, F., 1935. Complex experiments. Jour. Roy. Stat. Soc. Suppl.
2:181247.
26. 1937. The design and analysis of factorial experiments. Imperial
Bureau of Soil Science Technical Communication No. 35, England.
CLIFFORD G. HILDRETH
North Carolina State College
Chapter 4
Discrete Models With
Qualitative Restrictions
IN statistical analyses, as in many other human endeavors, the prod
uct of a particular undertaking is closely related to the input. At the
final stage of a statistical application, what one puts in are some ob
servations and a specification; what one gets out are some statistical
inferences, i.e., estimates, tests, and/or optimal decisions. Ways in
which good observations contribute to useful inferences are generally
well understood and are quite properly stressed in most applied statis
tics courses. The possible contributions, positive or negative, of alter
native specifications are not as easily understood and, for many prob
lems, have not been adequately explored by statistical theorists.
Specifications
Since the rationale for the procedure to be outlined and illustrated
depends entirely on considerations of specification, a few general re
marks on these matters may be helpful. First, a statistical specifica
tion is defined as the complete set of assumptions which are accepted
as a basis for a particular statistical investigation. Another way of put
ting this is to say that a specification includes all statements about the
underlying statistical population which the investigator accepts a priori.
Specification and model are nearly synonymous terms. According
to a fairly well accepted usage (1, 6) observed here, the model is the
class of all statistical populations which are consistent with the specifi
cation, i.e., which satisfy the a priori assumptions.' For most of the
discussion the terms will be used interchangeably.
In general, an investigator's situation is such that if he adds assump
tions to his specification (narrows his model), the prospective accuracy
of his inferences is increased, provided the added assumptions are re
alistic. However, if the assumptions are unrealistic, biased inferences
will generally result. Thus, a researcher should clearly use in his
specification all of the relevant a priori information that he is sure is
realistic.2 In doubtful cases, the investigator may be helped by
'This statement would require some modification in contexts in which one needs to dis
tinguish the statistical population from the theoretical structure which explains it. Such in
stances have arisen mainly in economics and psychology and need not be taken into account
in the following discussion.
2Sometimes a researcher may ignore potentially useful a priori information to simplify
computations. This possibility is left aside to keep from diverting the discussion.
DISCRETE MODELS WITH QUALITATIVE RESTRICTIONS 63
theoretical research indicating the extent to which a particular added
assumption may improve the inferences to be drawn and, on the other
hand, the biases to which particular errors in the assumption will lead.
If the possible biases are large relative to potential gains, a doubtful
assumption should, of course, be rejected. If the prospects are reversed,
a doubtful assumption might well be utilized. Considerable reliance upon
the judgment of the investigator is unavoidable in all but the most routine
applications, and good judgment combined with technical skill is what
makes a good applied statistician.
That the contribution of a priori information differs from one prob
lem to another may be observed by considering estimates obtained from
a random sample from a normal population. If the investigator is pri
marily interested in a good estimate of the population variance, he may
improve his estimate by specifying the population mean a priori, if it is
known. This specification will substantially improve his estimate of the
variance, if he has only a few observations, but will only be a slight im
provement if he has many observations. Thus, if he (a) has a fair a pri
ori notion of the mean but does not know it exactly, and (b) has a small
number of observations, he might very well use his best a priori value
for the mean; otherwise he may neglect his a priori notion.3 On the
other hand, if the investigator is primarily interested in estimating the
mean, a priori knowledge of the variance is not of any help.
Clearly the difficult case is the one in which an uncertain assumption
(a) may improve the analysis significantly if correct, and (b) damage it
badly if incorrect.4 A thorough knowledge of the field of application
should help the research worker to judge the likelihood of bias. Some
times a test of significance can be developed as an additional aid to
judgment. However, this precaution has often been pointed out; i.e., to
test an assumption and then use it (if not rejected) as part of the specifi
cation on which subsequent estimates and tests are based complicates
the interpretation of the traditional probability statements that are later
made about test statistics or confidence regions. While this statement
is undeniable, it should not seriously inhibit use of preliminary tests.
The basic difficulty is not that a preliminary test is performed but that
the investigator is under pressure to utilize an uncertain assumption.
Proceeding without attest does not remedy this basic difficulty.
The particular specification problem with which we shall be con
cerned is that of formulating appropriate assumptions about the form of
a response surface. For convenience, a certain observable response,
y, depends upon the magnitudes of certain observable, and sometimes
controllable, variables, z,, z . zk, and certain unobservable varia
bles whose net effect may be approximately represented by a random
variable, u. The unobserved variables may be partly controllable, es
pecially in carefully conducted experiments. The assignment of the z's
3If his a priori information could be put in the form of a distribution function for the pop
ulation mean, and the weight function for various possible errors in the estimate of the va
riance were taken into account, this could be handled as a statistical decision problem.
4A simple but suggestive example has been presented by Leonid Hurwicz (4).
CLIFFORD G. HILDRETH
may be randomized to assure that u is independent of the z's, and con
ditions can sometimes be held sufficiently stable from one observation
to another that u will have a small variance.
Form of Equation
Familiar statistical procedures give the investigator two types of
alternatives. He may assume a priori that an equation of a certain
known form will represent the surface to a close approximation and use
the observations to estimate several unknown parameters in the equation.
Alternatively, he may forego the assumption as to form and regard each
distinct combination of the z's as a different treatment, unrelated to the
others in his statistical model. These alternatives correspond to the
continuous and discrete models discussed by Anderson in Chapter 3.
To use a discrete model it is necessary to make some specifications
about the form of the function. Also, assumptions must be made about
the way in which the random component, the u, enters. It is usually
found desirable to make some assumptions about the interactions of the
z's. There are, of course, an infinite number of models for each type of
interaction from which an investigator might choose.
Advantages and Disadvantages of Continuous Models
Continuous models offer several potential advantages. There may
be a substantial gain in efficiency in having a small number of parame
ters to estimate and in estimating response at a particular point (a par
ticular combination of the z's) from all of the observations rather than
just the observations at that point. The estimated equation provides a
convenient means of interpolation and limited extrapolation. Further
more, the form of the relation, once it is well established, may have
interesting theoretical implications.
The principal disadvantage of continuous models lies in the biases
which may accrue if an inappropriate form is used, and the difficulty of
designing a satisfactory test of the appropriateness of a particular as
sumption regarding the form. It is particularly disconcerting that, in
many instances in which several alternative assumptions have been in
vestigated, alternative fitted equations have resulted which differ little
in terms of conventional statistical criteria, such as multiple correla
tion coefficients or F tests of the deviation, but differ much in their
economic implications (cf. 5, 9). It is also worth noting that bias due to
inappropriate form does not decrease as sample size increases,5 whereas
inefficiencies in discrete or formfree methods become less important
in large samples. In many contexts the convenience of interpolation of
fered by a continuous function may not be very important. Frequently
the discrete alternatives analyzed will be sufficiently numerous to
'In general, bias will decrease if the range of the observations is increased along with
sample size and, of course, can also be decreased by changing the assumed form as dis
crepancies become apparent.
DISCRETE MODELS WITH QUALITATIVE RESTRICTIONS 65
determine an optimal decision to the degree of accuracy permitted by
the data. In addition, when results of analyses are put to practice, there
will always be relevant discrepancies between the conditions underlying
the analysis and the conditions faced in commerical production on farms.
Some judgment will of necessity be exercised at this stage; interpolation
may be as effective as using a predetermined formula.
As noted earlier, there are many situations in which choosing a spec
ification involves delicate judgment and a thorough knowledge of the par
ticular field of application. Where judgment plays a large part, two dif
ferent researchers may use somewhat different models and procedures
without any existing way of labeling one, right, and the other, wrong.
Instead of seeking "the" way to proceed in such instances, mathematical
statisticians might better try to give the applied worker the means for
employing any of a variety of models and procedures, thus enlarging the
area over which judgment can be exercised.
Situations sometimes arise, for production economics analysis as
elsewhere, in which the investigator does not find either the continuous
or traditional discrete type of model to be ideal. He may feel that no
particular form of function has been sufficiently well established in his
area to give reasonable assurance against bias in a continuous model.
He may have rather firm notions about some properties underlying the
relation. These properties are ignored if he treats distinct input com
binations as unrelated treatments. An economist might, for instance,
strongly believe that a particular production function is characterized
by diminishing returns; that a certain demand equation is homogeneous;
that a certain supply curve slopes upward. To the extent that he knows
these properties exist, it is wasteful to analyze statistical results that
are inconsistent with them. For such situations it might be useful to
have procedures enabling the researcher to include in his specification
such qualitative properties as seem sufficiently well established, with
out forcing him to specify his relation as completely as a continuous
model requires.
A Discrete Model
A possible approach is to formulate discrete models which include
the appropriate qualitative restrictions and to work out appropriate sta
tistical procedures for these models. Appropriate procedures can be
found for a variety of such models. In an article by Hildreth (2), pro
cedures were developed for obtaining estimates of points on a production
surface under the assumption that inputs are subject to diminishing re
turns.6 The work is now being extended and, while it is highly incom
plete, a sketch of accomplishments may serve to suggest possibilities
of the approach and the kinds of problems, mostly unsolved, which are
encountered in using it.
"This exposition is marred by the inclusion of a hastily attempted generalization which
can be shown to be false. A correction may be found in the December 1955 issue of the Jour.
Amer. Stat. Assn. Fortunately, the false generalization does not affect the main result or
the applications which have been developed.
CLIFFORD G. HILDRETH
The extensions have been worked out jointly by the author and A. P.
Stemberger. They will be more fully reported in Stemberger's doctoral
thesis. The data come from experiments on the response of corn yields
to nitrogen, conducted by Krantz and Chandler (7).
The model initially employed was of the following form:
(1) ynt= P (Zn) + unt n = 1,2 . N
t =1,2 ... Tn
where the Nobserved levels of nitrogen have been arranged in ascend
ing order and z is the pounds per acre in the nth level (zn+l > zn).
ynt is the observed yield for the tth trial (observation) with application
zn. Tn is the number of plots to which zn pounds have been applied.
Unt is a random disturbance assumed to be independent of zn.
The algebraic form of the production or response function, p (z), is
regarded as unknown except that successive increments of z are as
sumed to increase y at a nonincreasing rate. In other words p (z) is
d2 <
concave, or d = O if the derivative exists. With only this assumption
regarding form it is not generally possible to estimate the response to
levels of nitrogen other than those (N in number) for which observations
are available.7
Since there is no loss of generality in taking E (unt) = 0, the follow
ing may be written:
(2) ?in = E (ynt) = (n).
The assumption of diminishing returns then requires:
(3) +2 ?7n+1 < 'n+ 7 n = 1,2... N2.
n+2 n+1 Z+ Z
n+1 n
Regarding the f7n as the magnitudes to be estimated, the application of
the method of maximum likelihood (if the Unt are normally distributed)
or the method of least squares leads to the problem of finding estimates,
n', which minimize the sum of squares:
N Tn A
(4) Q = Z Z (ynt ln2
n=l t=l
'It is possible to estimate upper and lower bounds for all z such that z,
bounds can be estimated for z > zNor z < zz.
DISCRETE MODELS WITH QUALITATIVE RESTRICTIONS 67
when the restrictions, equation 3 above, believed to hold for the popula
tion parameters, are also required to hold for the estimates.
Thus the estimation problem is one of minimizing a positive definite
quadratic form subject to constraints in the form of inequalities. Prob
lems like this have been studied in activity analysis and in game theory.
With the aid of a theorem by Kuhn and Tucker (8), it was possible to
develop an iterative procedure for obtaining the required estimates.8
The use of this procedure to obtain yield estimates from the Krantz
Chandler data is described in the article mentioned previously. At the
time of the estimates, only data pertaining to "good" weather and one
type of soil were available. When access to the complete data was ob
tained, it was found that numerous other observations were available
covering weather experience classified into three main categories: good,
fair, and dry. Also, several soil types were available which could be
placed in three fairly homogeneous classes: Piedmont, Coastal, and
Drained Coastal.
The problem of using all of the data in a unified analysis was similar
to problems sometimes encountered in combining data from different
experiments. The model was modified to allow for soil and weather ef
fects and could then be indicated:
(5) Yijnt = a i + Yj + rn + Uijnt
1 = 1,2,3
j = 1,2,3
n=1,2 ... 12
t = 1,2 ... Tijn
where:
the tth yield observed on soil i with weather j and
Yijnt = nitrogen level n.
a = a general constant
Pi = the contribution to yield of soil i
vj = the contribution to yield of weather j
the contribution to yield of applying z pounds of
n nitrogen
uijnt = a random disturbance
the number of observations with soil i, weather j,
Tijnt = and level of nitrogen n.
The twelve levels of nitrogen were in 20pound intervals from 0 to 220,
inclusive.
8The computing procedure developed may also be used to solve a number of nonlinear
programming problems, including some involving monopoly and risk elements.
CLIFFORD G. HILDRETH
Interaction Among Soil, Water, and Fertilizer
The model indicated by 5 assumes no interaction among soil, weather,
and nitrogen effects. With observed yield as the dependent variable,
this would mean, for instance, that dry weather should cut yield the same
number of bushels on heavily fertilized plots as on lightly fertilized plots,
and similarly for other effects. This assumption is not entirely plausi
ble. A somewhat more promising possibility is the assumption that a
change in weather has the same percentage effect on plots with various
combinations of soil and fertilizer. To modify these assumptions re
garding interaction, log yijnt is substituted for Yijnt in equation 5.
For convenient future reference, write:
(6) Yijnt = a + Pi + Vj + +n + uijnt
where Yijnt = log Yijnt and other symbols have meanings similar to
their meanings in equation 5, except that the constants are now logs of
factors in an expression for observed yield. Equation 6 is equivalent to
t = A + i Yj + + ijnt
(7) Yijnt = A
where A is the base of the system of logarithms used.
For several reasons it seemed desirable to initially analyze both
equations 5 and 6 without imposing restrictions on the 17n. Before doing
this it seemed reasonable to test the interaction assumption in equation
5. The restrictions on the 77n in equation 6 which would express dimin
ishing marginal productivity are nonlinear; direct estimation of the co
efficients of equation 6, subject to restrictions, would be even more dif
ficult. While the interaction assumption implicit in equation 6 seems
more plausible a priori than that in equation 5, it still seemed desirable
to test this assumption before deciding what other analyses might be
worthwhile. The data on which the analyses are based are given in
table 4.1.
The tests for interaction confirmed the a priori belief that equal per
centage effects were more plausible than equal absolute effects. The
test showed significant interaction in equation 5 at the 0.01 level,9
whereas the test applied to equation 6 shows no significant interaction,
as can be seen in table 4.2. Accordingly, further analysis was confined
to equation 6. The estimates of coefficients for equation 6 are given in
table 4.3.
All of the indicated F ratios are significant at the 0.001 level, except
for interaction which is not significant at the 0.05 level. In testing for
'For equation 5, the interaction mean square was 364.08, within cells mean square was
189.24, giving an F of 1.92. Degrees of freedom are 39 and 182 as in equation 6. The
assistance of R. L. Anderson in performing these tests is gratefully acknowledged.
DISCRETE MODELS WITH QUALITATIVE RESTRICTIONS 69
TABLE 4.1 Corn Fertilization Data
Levels of Nitrogen in Pounds
0 20 40 60 80 100 120 140 160 180 200 220
A Piedmont Soil
(Dry) (Dry) (Dry) (Dry) (Dry) (Dry) (Dry) (Dry)
18.0 29.9 44.5 50.4 50.4 51.0 52.1 51.1
33.3 41.4 51.1 62.3 60.2
35.6 48.5 37.0 37.7 39.2
31.7 42.0 54.9 54.5 57.8
63.8 67.1 74.5
29.6 40.4 50.3 50.4 54.4 51.0 56.8 51.1
B Drained Coastal Soil
(Dry) (Dry) (Dry)
S 27.6 50.2 61.4
(Fair) (Fair) (Fair)
66.0 86.5 88.9
(Good) (Good) (Good) (Good) (Good) (Good) (Good)
50.1 39.8 62.9 77.7 89.0 102.8 86.6
80.8 63.5 78.1 114.3 90.0
22.2 57.0 111.5 123.2
84.4 110.2 123.9
70.3 86.7
90.6 99.4
102.4 114.5
50.1 56.8 61.1 91.5 101.6 102.8 103.5
Cell means
Cell means
Cell means
Cell means
C Coastal Soil
(Dry) (Dry) (Dry) (Dry)
53.4 43.9 52.5 59.0
60.5 61.3 40.0
50.6 52.7 66.4
68.2 80.9 62.6
40.9 62.2
(Good)
90.7
90.7
(Dry) (Dry) (Dry)
45.5 59.2 41.5
68.6 63.0 81.2
58.5 75.3 70.7
50.7 61.8 70.7
64.8
53.9 54.7 60.2 55.4 59.0 57.6 64.8 66.0
(Fair) (Fair) (Fair) (Fair) (Far) (Fair) (Fair) (Pair) (Fair)
9.9 43.4 39.1 52.2 60.3 81.0 72.0 83.6 85.5
31.3 24.9 66.0 59.5 74.1 88.2
24.2 64.3 81.8 78.8
15.2 74.0 74.8
12.9
18.7 43.4 32.0 64.1 69.1 81.0 75.0 85.9 85.5
(Good) (Good) (Goo) (Goo) (Good) (Good) (Good) (d) (G ) Good)
19.8 35.7 40.2 59.0 59.8 77.2 80.8 86.9 81.5
63.9 50.1 96.9 77.3 102.2 86.7 73.0 117.1
22.8 56.0 52.1 58.5 81.7 83.6 117.0 102.3
51.7 42.0 85.1 34.0 107.1 98.3 100.5 114.3
18.8 31.8 63.6 50.1 80.0 108.3 115.5 101.0
17.4 27.3 44.9 49.5 69.0 92.1 98.3
2.8 42.2 49.1 62.2 94.6 83.9 97.8
13.3 35.3 79.0 90.6 104.9
14.6 68.6 96.3 70.2
24.4 88.1 107.0
11.6 59.0 102.5
20.8 94.4 78.1
19.1 90.2 68.7
7.2 79.5 69.8
16.6 62.4
6.9
8.3
9.2
7.7
25.8
19.3 40.1 61.7 55.8 82.4 90.8 89.2 86.9 98.6
S (Fair) 
81.0
85.8
(Good) (Good) (Good)
115.7 72.9 107.8
95.1 108.0 116.5
110.3 74.7
102.7
120.9
103.9
98.2
128.6
109.4 90.5 99.7
(Dry) (Dry) (Dry) (Dry)
9.4 21.0 37.8
21.3 44.6 52.6
4.3 22.1 50.6
14.4 29.6 42.0
15.2 31.6 56.0
3.7 56.9
20.4
17.2
8.4
"3.5
14.7
39.5
14.1 29.8 49.3
Cell means
Treatment means 19.4 41.9 54.7 55.2 74.8 71.6 78.4 71.3 97.6 89.5 75.9 81.7
Soil means Weather means General means
y1.. 47.7 y.L 44.5 y... 61.7
y2.. 81.7 y2. 55.0
3.. 60.1 y.3. 72.3
 
CLIFFORD G. HILDRETH
TABLE 4.2. Analysis of Variance for Equation 6
Source of Variation d.f. S.S. M.S. F. Ratio
Mean 1 691.358 
Regression 15 17.882 1.192 47.11
Soils (2) (.364) .182 7.28
Weather (2) (1.136) .568 22.45
Nitrogen (11) (15.438) 1.403 55.45
Error 221 5.598 .0253 
Interaction (39) (1.023) .0261 1.04
Within cells (182) (4.576) .0250
Total 237 714.839 
TABLE 4.3. Estimates of Coefficients in Equation 6
Estimated Standard
Coefficient Estimated Error of Antilog of
Symbol Interpretation Coefficient Coefficient Coefficient
General constant
Piedmont soil
Drained Coastal soil
Coastal soil
Dry Wx
Fair Wx
Good Wx
0* Nitrogen
20# Nitrogen
40f Nitrogen
60* Nitrogen
80f Nitrogen
100# Nitrogen
120# Nitrogen
140# Nitrogen
160# Nitrogen
180* Nitrogen
200# Nitrogen
220# Nitrogen
1.2408
.12090
.08394
0
.17836
.02589
0
0
.38941
.48193
.56049
.63435
.67796
.67381
.66649
.71981
.75580
.72706
.73117
.0270
.0397
.0333
0
.0270
.0344
0
0
.0415
.0470
.0449
.0370
.0521
.0382
.0552
.0467
.0507
.0618
.0617
17.41
1.321
1.213
1.000
.663
.942
1.000
1.000
2.451
3.034
3.635
4.309
4.764
4.719
4.640
5.246
5.699
5.334
5.385
I
DISCRETE MODELS WITH QUALITATIVE RESTRICTIONS 71
interaction, the within cells sum of squares was placed in the denomi
nator. It has 182 degrees of freedom because only 55 of the 72 cells
have any observations from which to estimate cell means. The error
mean square has been used as the denominator for the other F ratios.
This has been done so that both the estimates and the tests, other than
the test for interaction itself, would be based on the same specifications.
The adjusted R2 is 0.72.
The estimates of coefficients are given in table 4.3, along with esti
mated standard errors and antilogs. The coefficients of equation 6 are
not unique. The meaning of the equation would be unchanged if a con
stant were added to a, and the same constant were subtracted from all
of the 0i, or all of the y., or all of the r7n. This makes it possible to
select arbitrary values for one coefficient for each type of effect. B3,
73 and r7/ were set equal to zero.
The antilogs indicate how estimated yields change from one soil
weather combination to another. In going from Coastal soil to Piedmont,
32.1 percent was added to the estimated yield regardless of weather and
nitrogen; in going from good weather and Coastal soil to fair weather and
drained Coastal soil, 14.3 percent was added (1.213 x .942 1 = .143), etc.
It was desirable to obtain an estimate of the nitrogen effects subject
to the diminishing returns restrictions. This estimate was complicated
by the fact that cell frequencies were highly disproportionate and by the
fact that the restrictions on the log of yield are nonlinear. The first dif
ficulty is perhaps not too serious. Since the restrictions apply only to
the nitrogen effects and since interaction is not significant, it seems a
reasonable conjecture that imposing the restrictions would affect the
soil and weather coefficients very little. The estimates of these coeffi
cients are, in any case, unbiased but would be somewhat more efficient
if estimated subject to the restrictions.
One might proceed by correcting the original observations on logs of
yield by the estimated soils and weather effects and then reestimate the
nitrogen effects, treating these corrected values as observations. This
procedure would go quite smoothly except for the second complication 
the nonlinearity of the restrictions on log of yield. While the estimates
subject to nonlinear inequalities can be developed, time has not been
available, and therefore the author will not speculate as to how much
the computations would be increased.10 An approximation to the results
'"It appears that quadratic restrictions would suffice for this problem.
Let y = '(x), Y = log y
dY dy
dx dx
'Y d 2 ,dy 2 dY2
W ' Yy y d Y().
Since y is positive
< o< + ( )2 < 0.
Thus, imposing the condition on the right is equivalent to imposing the condition on the
left. While this relation only holds exactly at a point, its interval analogue will be suffi
ciently close for practical purposes and this will involve quadratic restrictions on the
treatment effects in the log form.
CLIFFORD G. HILDRETH
that would be obtained under this procedure can be found by converting
the corrected logs of yields back to yields and proceeding as in the
original problem cited.
AA
(8) y*k= 10 (Yijnt i" Yj)
where k runs from 1 to Kn, and Knis the number of observations at
the nth level of nitrogen (Kn = T.. ). Then, choose estimates
77 to minimize the sum of squares.
n
(9) Q* = S S (y*k n)2 subject to the restrictions
n k
in+2 fn+l
z z
n+2 n+1l
71n+1 ?n
z n=1,2 ... N2.
zn+
n+1 n
This procedure is not quite consistent with the assumptions implicit in
equation 6 since the sums of squares of deviations are minimized in
yields rather than in logs of yields. However, a comparison of restricted
and unrestricted estimates in table 4.4 confirms that the error is not
large. Estimates are presented for good and dry weather and Coastal
soil. To obtain the estimate, either restricted or unrestricted, for any
other soilweather class and for any level of nitrogen, one could
TABLE 4.4. Estimates Responses to Nitrogen for Coastal Soil and Two Types of
Weather
Nitrogen Level Good Weather Estimates Dry Weather Estimates
(inPounds) Unrestricted Restricted Unrestricted Restricted
0 17.41 21.75 11.54 14.42
20 42.68 44.30 28.30 29.37
40 52.81 54.91 35.01 36.41
60 63.11 65.30 41.84 43.29
80 75.01 75.70 49.73 50.19
100 82.95 80.85 55.00 53.60
120 82.15 84.78 54.47 56.21
140 80.78 88.70 53.56 58.81
160 91.33 92.63 60.55 61.41
180 99.22 96.56 65.78 64.02
200 92.88 95.95 61.58 63.61
220 93.80 95.34 62.19 63.21
DISCRETE MODELS WITH QUALITATIVE RESTRICTIONS 73
multiply the estimate from table 4.4 for good weather by the product of
the antilogs, from table 4.3, of the coefficients for the desired soil
weather combination.
To become a generally useful tool, estimation subject to qualitative
restrictions needs to be developed in several directions. Better proce
dures for handling transformation of variables are needed. It would be
useful to have confidence regions and tests which take account of the
restrictions." As more variables are restricted, improved computa
tional procedures will be needed.
Even when these developments take place, the procedures should be
regarded as supplementing rather than supplanting existing techniques.
There will still be the advantages of efficiency and convenience attached
to continuous models when the appropriateness of a particular algebraic
form can be rigorously established. However, criteria for goodness of
fit are needed that take account of the implications to be drawn from
fitted relations.
Certain other improvements in statistical capabilities are needed
irrespective of the type of model chosen. In crop production studies,
more effective procedures are needed for incorporating data on the ini
tial condition of the soil into models and for relating response to specific
observable weather variables.
There is one additional topic that should be mentioned, viz., the
drawing of economic implications from our results. After estimating
a continuous production surface for an economic unit, the natural pro
cedure is to form a net revenue function with prices of inputs and out
puts appearing as variables. This can be maximized with respect to
inputs and outputs yielding the optimal quantity as a function of all of
the prices. When the economic unit is a firm, these equations are the
individual firm's supply and demand functions. More generally, these
might be designated as the optimal decision relations.
When the analysis takes the form of estimation of response to a set
of discrete alternatives, the natural analogue to the functions described
above is a construction of a price map (3). If all possible prices of in
puts and outputs are considered as points in a multidimensional Euclid
ean space, then the price map is a partitioning of this price space into
regions which correspond to the production alternatives in such a way
that a particular alternative (or combination of alternatives in extended
analyses) is optimal whenever the actual price combination falls inside
the corresponding region. A price map corresponding to the restricted
estimates in table 4.4 is shown in figure 4.1. The procedure for deter
mining regions is the same as that used for cotton fertilization data in
the reference cited previously. Crosses show the price combinations
which actually prevailed (on the average) in North Carolina in the indi
cated years.
"It should be recognized that conventional tests which ignore the restrictions are unbiased
even when the restrictions are known to hold. Utilizing the restrictions would generally in
crease the power of our tests.
CLIFFORD G. HILDRETH
1.60  1944
X 1943 1945 X 1950
x 193
1.20
ISO
180
X 1942 100
.80 ^ _11
X 1941 .
X 19P l 9' 0
.080
80
0  
0
PRICE OF NITROGEN (LB.)
Fig. 4.1 Cornnitrogen price map.
Experimenters have increasingly accepted the desirability of taking
statistical considerations into account, in planning their investigations,
and of examining the statistical implications of their results. It now
appears that a good start is being made toward assigning economic con
siderations and implications of their proper role. Actually, a set of
optimal decision relations or its discrete counterpart, a price map,
might well be regarded as just as necessary to a complete report of an
investigation as the analysis of variance table.
X Ig 8
9
*cu
DISCRETE MODELS WITH QUALITATIVE RESTRICTIONS 75
References Cited
1. HAAVELMO, T., 1944. The probability approach in econometrics, Econometrica
Suppl. 12:3147.
2. HILDRETH, C., 1954. Point estimates of ordinates of concave functions. Jour.
Amer. Stat. Assn. 49:598619.
3. 1955. Economic implications of some cotton fertilizer experiments.
Econometrica 23:8898.
4. HURWICZ, L., 1951. Some specification problems and applications to econo
metric models. (Abstract.) Econometrica 19:343.
5. JOHNSON, P. R., 1953. Alternative functions for analyzing a fertilizeryield
relationship. Jour. Farm. Econ. 35:51921.
6. KOOPMANS, T. C., 1949. Identification problems in economic model construc
tion. Econometrica 17:5263.
7. KRANTZ, B. A., and CHANDLER, W. V., 1954. Fertilize corn for higher
yields. N. C. Agr. Exp. Sta. Bul. 366 (revised).
8. KUHN, H. W., and TUCKER, A. W., 1951. Nonlinear programming. Proc.
2nd Berkeley Symposium on Math. Stat. and Probability. University of Cali
fornia Press.
9. PRAIS, S. J., 1953. Nonlinear estimation of the Engel curves. Rev. Econ.
Stu. 20:102.
DAVID D. MASON
North Carolina State College
Chapter 5
Functional Models and Experimental
Designs for Characterizing Response
Curves and Surfaces
HE yield of a particular crop is a function of many possible factors,
as has been pointed out in Chapters 1 and 2. The climate, variety,
management practices, and soil factors are, in fact, broad catego
ries which in themselves contain a number of subfactors, each of which
may be modifying or limiting. This chapter is concerned primarily with
the functional relationship between yield and a portion of the soil factor,
that relating to the nutrient status of the soil.
Background
Even a superficial examination of the numbers and types of factors
affecting crop yield will reveal that any function completely describing
the relationship would be extremely complex. It is small wonder that
widely different hypotheses have been developed and supported, since
one may find almost any pattern of response, varying from strong posi
tive linear relations to strong negative linear relations. From a statis
tical standpoint, the failure of hypotheses, purporting to have general
application, to agree arises from failure of the experimenters to ade
quately sample the population to which inferences are made.
Russell (25) gives an excellent review of the historical development
of the concepts of plant nutrition, and of the attempts to obtain rational
explanations of various phenomena. Liebig, with his first publication in
1840 and subsequent papers and books on the subject, together with his
heavy ridicule of the efforts of his predecessors and contemporaries,
contributed much, particularly in the way of stimulating controversy
and subsequent research. His law of the minimum, which he stated as
"by the deficiency or absence of one necessary constituent, all the others
being present, the soil is rendered barren for all those crops to the life
of which that one constituent is indispensable," is perhaps his best re
membered contribution.
The field experiment approach to the problems of plant nutrition and
response initiated by Boussingault (about 1834) and Lawes and Gilbert
in 1843 furnished positive evidence of the response of crops to natural
and artificial manures. However, Russell reports that the controversy
regarding the use of "chemical manures" went on for many years before
their general acceptance was indicated. Even today a remnant of this
controversy is evidenced by the "organic gardening" school of thought.
76
MODELS AND DESIGNS FOR SURFACES
Mitscherlich's contributions, beginning in the first decade of this
century, marked the first major attempt to formulate a general func
tional model. His experiments were made with plants grown in sand
cultures supplied with "excess" of all nutrients excepting the one under
investigation. His expression is commonly known by the descriptive
term, "law of diminishing returns," and has the mathematical properties
outlined in Chapter 1. Mitscherlich's work, like Liebig's, produced con
troversy and has both ardent supporters and critics. His function, to
gether with modifications and contributions by other workers, will be
given more quantitative expression in the following section. Spillman
(26) later, but independently, developed the same function (in the alge
braic form of equation 2 in Chapter 1) and extended the methodology to
computation of economically optimum rates of fertilization. Spillman,
as did Mitscherlich, suggested optimum experimental designs for ob
taining data necessary for the estimation of the parameters of the model.
Anderson has adequately outlined, in Chapter 3, the procedural de
velopments from the standpoint of the statistical approach of developing
empirical polynomial functions to characterize the response. The de
velopment of the factorial experiment and appropriate methods of statis
tical analysis led to the definition and characterization of interaction
between factors (also called complementarity, or joint effects). This,
in turn, has led to the geometrical concept of a response surface as the
realistic expression of the contribution of two or more nutrients to yield.
With the increased interest of production economists in the applica
tion of quantitative methods in the past several years, several papers
have been concerned with the choice of a proper functional model for the
characterization of inputoutput relationship in plant growth. Johnson
(17), Heady (11), McPherson (18), Ibach and Mendum (16), Paschal (22),
Hutton (14), Hutton and Elderkin (15), and Heady, Pesek, and Brown (12)
have set forth, in varying degrees, bases of comparison and procedures
for evaluation.
Functional Models for SingleVariable Response Curves
Two general approaches have been used in developing mathematical
expressions for the relationship between the amounts of the various fac
tors present, and the amounts of plant growth. They are:
1. Attempts to define a model which expresses basic laws of plant
behaviour, and fitting the experimental data to this more or less
rigid model.
2. The experimental data are studied by statistical methods and an
empirical polynomial equation of "best fit" is developed, with no
assumption or hypothesis as to the underlying causes.
The first approach is logically and intuitively more appealing. It has
its counterpart in the simple physical and chemical systems where de
terministic models are common and useful. However, even the simplest
of biological systems is relatively complex, and together with errors of
DAVID D. MASON
technique and measurement, exact relationships are to be viewed
askance. Some of the more common functional models for which some
biological justification has been claimed are first considered in the fol
lowing paragraph.
The Mitscherlich Function
Expressing quantitatively the statement that the increase of crop
produced by unit increment of the lacking factors is proportional to the
decrement from the maximum, one has:
dy
(1) = (A y)c
where y is the yield obtained when x = the amount of the factor present,
A is the maximum yield obtainable if the factor were present in excess,
this being computed from the equation, and c is a constant. Upon inte
gration, and assuming that y = O when x = 0,
(2) y = A(1 e' ).
Mitscherlich maintained that the "c" values in his expression were
constant for a given nutrient over different crops and growing conditions.
Most of the early controversy about his work centered around his hy
pothesis concerning the "c" values. The workers subsequently men
tioned as using the Mitscherlichtype equation have assumed that "c" is
a parameter to be estimated from the data. This function is expressed
in other algebraic forms by Spillman (26) and Stevens (27), and has been
widely used by many workers. Ibach and Mendum (16) have detailed in
structions for computations, together with examples, using the Spillman
form. Monroe (19), PimentelGomez (24), and Stevens (27) give simpli
fied least squares procedures for estimation of parameters for solution,
when the X levels are equally spaced. Also, standard errors may be
computed for the estimated parameters.
Prior to the comparatively recent publication of the three references
mentioned above, and a paper by Hartley (10), least squares estimates
involved such heavy labor that they were seldom made. An interesting
example of the application of the Mitscherlich model is given by Crow
ther and Yates (6), in summarizing all published results of oneyear
fertilizer experiments conducted in Great Britain and the northern
European countries since 1900, in order to formulate a wartime ferti
lizer distribution policy. Economic analyses, in terms of setting out
optimum rates for maximum profit, were made of the data.
One of the other early criticisms of Mitscherlich's equation was that
no allowance was made for possible yield depression by harmful ex
cesses of the factor. Mitscherlich, after extensive study of his experi
mental data, introduced a modification of the following form to allow
for such depressions:
MODELS AND DESIGNS FOR SURFACES
(3) y = A(1 10C)10kX2,
with the constant "k" being called the "factor of injury." He felt that
this would apply mainly to the response of grain crops to nitrogen. He
provided estimates of "k" for several crops.
The Logistic Function
The logistic is the commonly used function for fitting growth curves
in biological populations, and may be expressed in the form:
(4) yt=
1 + bat
l+be
where a, b, and k are parameters to be estimated from the observed
data, and yt is the value of the growth character studied at point of time,
t. For yield response models, x, for increment of fertilizer, would be
substituted for t.
This curve has a lower asymptote of O, and an upper asymptote at k,
k
and the point of inflection is at y = a point midway between the two
asymptotes. Thus, we have the familiar Sshaped or sigmoid curve.
Such a model would be useful to characterize the initial "lag" that may
occur when the amount of the factor in the soil is very low, and small
increments are applied in the low range. In the usual situation this ini
tial lag is not observable. Nair (21) gives an extensive discussion of the
logistic function together with methods and illustrations of fitting.
The Power Function (CobbDouglas)
The power function,
(5) Y = a Xb,
has been employed as the model in various economic investigations. In
this equation, Y is the yield, a and b are constants, with X as the level
of the factor. The equation may be written in the linear form as
(6) Log Y = Log A + b. log X.
Hutton et al. (15) discuss the general characteristics of the CobbDouglas
function, and suggest methods of analysis. Heady (11) and McPherson
(18) also describe the various characteristics of this function and modi
fied forms of the power function. If b>O, as would be the case in the
yield response curve, y continues to increase as X increases.
DAVID D. MASON
The Polynomial
The terms in a polynomial equation may vary from one to n1 where
n is the number of levels of the factor X. In the single variable case,
the number of terms and the degree of the equation are normally (but
not necessarily) parallel. The first degree (or linear) equation de
scribes a straight line, while the second degree (or quadratic) describes
a monotonic curve. The degree less one indicates the number of times
the curve may change direction. The usual forms are:
Linear : Y = o + X
Quadratic: Y = 0o + PiX + n,,X2
Cubic : Y = go + iXX + +X2 + 11X3
General : Y = fo + ,X + ...... 3 x"(n1)
..(n1)
The X may be transformed to the square root, logarithm, reciprocal,
or other form, with the same general process of fitting applied. Meth
ods of fitting such curves are straightforward. Discussion of fitting
procedures, with examples, is given by Anderson and Bancroft (2) and
other texts.
Discussion of Application of Exponential,
Power, and Polynomial Models
The functions mentioned above are only a few of the better known of
a large number of possible functions. Within the polynomial class alone
an almost infinite number of possibilities exist. The problem, therefore,
of choosing the "best" function is not soluble from a simple set of rules.
By the use of least squares procedures the value of the constants for the
equations may be computed. These procedures give the "best" fit for
the particular form of functional model, in the sense of describing a
curve from which the mean of the squares of the deviations of the indi
vidual points from that curve are a minimum.
It cannot be claimed that any of the functions represent fundamental
biological laws of growth, although one may rationalize the form of a
particular function in a particular situation. One procedure of choosing
the "best" function, mentioned by Heady (11) and by Hutton et al. (15),
is to examine possible applicable functions, and select the one that best
fits the data. A useful procedure, where data are being examined from
a replicated experiment (more than one observation at each increment),
is to examine the size of the "lack of fit" term, as given in the analysis
of variance. The following data, from Veits, Nelson, and Crawford (28),
serves to illustrate the procedure.
If the lack of fit term is of the same order of magnitude as the ex
perimental error, then the function is characterizing the data adequately.
A significant lack of fit mean square indicates that the model is inade
quate to describe the functional relationship.
