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TREATMENT
DESIGN
FOR
FERTILIZER USE EXPERIMENTATION
Foster B. Cady
Reggie J. Laird
CENTRO INTERNATIONAL DE MEJORAMIENTO DE MAIZ Y TRIO
INTERNATIONAL MAIZE AND WHEAT IMPROVEMENT CENTER
ton re 40 Apertedo Posr tl 6641 M1 # lle 6. D. F., MIlle*
CIMMYT Research Bulletin No. 26
4 14
I I'
Si CONTENTS
Page
ABSTRACT 1
BACKGROUND 2
A MODEL FOR FERTILIZER RESPONSE FUNCTIONS 4
PREDICTION ERRORS 8
S". Variance error 9
:" Bias error 10
I" '
SA combined variance and bias error 13
CRITERIA FOR COMPARING TREATMENT DESIGNS 14
: EMPIRICAL COMPARISON OF SELECTED DESIGNS 17
S: SUMMARY 25
' '. i
RESUME 26
S: RESUME 28
refereeRE NCES 29
I J .; REFERENCES 29
i 1;" .
'.; 29
'
TREATMENT DESIGN
FOR
FERTILIZER USE EXPERIMENTATION
Foster B. Cady and Reggie J. Laird 1
ABSTRACT
Ideally the experimenter would like to use treatment design to mini
mize both variance and bias problems in estimating yield from fertilizer
application. Selecting a treatment design for variance considerations will
not give a minimum bias design. Since different criteria of optimality lead
to different designs, the most appropriate criterion under a given set of
circumstances will be determined by the specific use that is made of the
data generated by the design, e.g., in fertilizer use studies the slope of
the estimated response surface near the economic optimal could be most
important.
For the quadratic polynomial estimating function and a square root
true function in two variables, treatment designs are compared empirically
using several variance criteria in addition to a volume concept of bias.
In general, bias is decreased by restricting the treatment combinations to
an area of the factor space not including the boundaries. Increased vari
ance of various estimated parameters may be controlled through replication
of the relatively small treatment designs needed for bias considerations.
1 Professor of Biological Statistics, Biometrics Unit, 337 Warren Hall, Cornell.
University, Ithaca, New York, 14850 and Soil Scientist. International Maize and Wheat
Improvement Center, Apdo. Postal 6641, Mexico 6, D. F. Mexico.
1
TREATMENT DESIGN
FOR
FERTILIZER USE EXPERIMENTATION
Foster B. Cady and Reggie J. Laird
Fertilizer experiments are conducted to obtain information on plant
response to applications of the essential nutrients. Experiments generally
consist of replicated fertilizer treatments where each treatment is a com
bination of rates of selected plant nutrients. A treatment design is a list
ing of treatments and may be visualized as points distributed in space.
For example, combinations of several rates of nitrogen and phosphorus
may be represented graphically as points in a twodimensional factor
space defined by axes corresponding to the nutrients.
Generally fertilizer experiments are limited to observing plant re
sponse to rates of the several nutrients that lie within discrete limits. For
example, in studying corn response to fertilization, information may be
desired for rates of nitrogen between the limits 0 and 300 kg/ha, rates
of phosphorus from 0 to 150 kg/ha, and rates of potassium from 0 to
200 kg/ha. These limits define the factor space within which observations
on corn response to applications of nitrogen, phosphorus, and potassium
are measured and the most meaningful information is thought to be in
cluded.
The central question in constructing a treatment design is how a
limited number of points should be distributed within a given factor space.
Literally an infinite number of rate combinations lie within the limits es
tablished for the different elements. However, the estimate of the plant
response to fertilization will depend, in part, on the selected rate combi
nations. The objective, then, of the investigator is to select that particular
treatment design which will best obtain the specific information he desires
on plant reaction to fertilization.
BACKGROUND
The selection of a treatment design presented no great problem in the
early years of fertilizer use research. The plant nutrient factors were com
monly varied, one at a time, with others held constant or not applied, and
the experimental treatments generally consisted of a small number of equal
ly or geometrically spaced fertilizer rates.
2
Following the publication by Yates (1937) of procedures for analyzing
factorial experiments, the ractorial arrangement of fertilizer treatments gradu
ally became accepted as the most adequate distribution of points within
a factor space. The treatments corresponding to a factorial consist of all
combinations of the several factor levels. Commonly, the levels of the fac
tors are evenly spaced so that the treatments are uniformly distributed
within the space defined by the rate limits selected for study. The factorial
arrangement of treatments permits the investigator to examine experimental
results for interaction in addition to simple linear, quadratic and higher
order main effects.
The total number of treatments required for factorial arrangements be
comes prohibitively large as the number of factors and levels increases. With
more than five levels of two factors (52) or three levels of three factors (33),
for example, the size of the area needed for a single block containing each
of the treatments often becomes so large that the precision of the experi
ment is greatly reduced. Fr certain factorials this deficiency can be elimi
nated by arranging the treatments in two or more blocks with certain higher
order interactions partially or completely confounded with block effects.
Large factorials have a second limitation in that the tlfChberof treatments
in a single replication may be excessively large in corpparison to the small
number of effects to be estimated. Fractional factoriTs 'offer one way of
reducing the total number of treatments without off ing. appreciably the
precision of certain estimates of fertilizer effects.
The postulate that factorial arrangement of treatments is most ade
quate for multifactor experiments was challenged in 1951 by Box and
Wilson. They assumedthe adequacy of the second degree polynomial for
representing the functional relationship, and proposed designs consisting
of a carefully selected number of treatments only moderately greater than
the number of effects to be estimated. A composite design for a three
factor experiment, for example, consists of only 15 treatments in comparison
with the 27 required by a 33 factorial. The distribution of treatment combi
nations in a central composite design for a threefactor experiment is shown
in Figure 8A.1 of Cochran and Cox (1957).
The question of optimal treatment design was explored further by
Box and Hunter in a paper published in 1957. They introduced the con
cept of a variance function for an experimental design and proposed that
the variance of a predicted yield should be constant at all points equi
distant from the center of the factor space. Designs having this property,
called rotatable designs, included a particular scaling or dispersion of the
treatments and a spherical factor space.
A different and basically intuitive approach to the selection of a de
sign with fewer treatments than the complete factorial arrangements
has been employed in recent years. Generally, the investigator begins
with a factorial arrangement and systematically eliminates treatment com
binations throughout the factor space. The number of possible partial
factorials corresponding to any given complete factorial arrangement is
large, and no objective criteria have been established to assure the selec
tion of the most efficient design for the specific objective at hand.
3
A MODI L FOR FERTILZER RESPONSE FUNCTIONS
The "goodness" with which the results of a fertilizer experiment ac
complish the objectives of an experimenter studying the relationship be
tween crop yield and level of fertilization depends largely on: (i) the as
sumed model that is used to express the results in the form of a yield
function and ats "closeness" to the true function, and (ii) the magnitude
of various random errors associated with doing the experiment.
Perhaps the first question which arises in considering the selection
of a statistical model for representing fertilizer use data is whether the
model should be discrete or continuous. If a discrete model is selected,
then comparisons are limited to treatment means and the investigator
chooses as optimal one of the combinations of fertilizer rates used as a
treatment. If a continuous model is selected, the yield data are used to
estimate the parameters corresponding to a specific mathematical func
tion, yields may be predicted for any levels of the input vairable, and
optimal fertilizer rates are calculated directly from the resulting function.
This may result in a predicted optimal fertilizer rate that is not a level
directly applied in the experiment.'
The interpretation given to earlier fertilizer use studies indicates that
researchers were generally thinking in terms of a discrete statistical model.
In 1909, however, Mitscherlich published a study in which he treated yield
as a continuous function of level of fertilization and suggested a "diminish
ing returns" model for the representation of experimental results. Over
if years investigators have continued to consider fertilizer use studies in
terms of both discrete and continuous models, although preference for
the latter has steadily increased. Basic studies on nutrient uptake and
plant growth tend to corroborate the thesis that the true fertilizer response
model is indeed continuous. However, since continuous models may as
sume an infinite variety of forms and laws governing plant growth have
only partially been elucidated, it has not been possible through deductive
processes to specify the true model characteristic of different production
conditions. Consequently, in his choice of a continuous model the inves
i;gaoor accepts the fact that the functional representation of his results
will be in error or biased to the extent that his assumed model differs
from the true model. If he decides to use a discrete model he is unable
to make inferences about rates of fertilization other than those actually
employed in the experiment and he can only hope to roughly approximate
'he true economic optimal level of fertilization.
As discussed by Anderson (1956) and Hildreth (1956) there are several
potential advantages in using a continuous model. The precision of the
estimated crop yields, produced at different fertilization levels, may be
substantially increased. The prediction equation provides a convenient
means for calculating the optimal rate of fertilization. The cumulative ex
perience of many years on fertilizer response, expressed in the form of
an equation, should eventually be useful in defining the true response
functions for specific productivity conditions.
Furthermore, fertilizer use researchers have developed a fairly reliable
picture of the general relationship between yield and rate of fertilization
for some crops. For example, it is fairly well established that the func
4
tionat relationship between the yield of corn grain and level of nitrogen
may be represented by a distorted sigmoidal curve similar to the one
shown in Figure 1. For most productivity conditions a reasonable assump
tion can be made as to the nature of the fertilizer response function.
Consequently, for studies designed to determine optimal levels of fertiliza
tion, investigators probably should select the continuous model that ap
pears most appropriate in the light of existing knowledge and take
measure to minimize the error due to the inadequacy of the model in
representing the true relationship.
The response curve shown in Figure 1 extends from the zero level of
nitrogen to an upper level well past that required for the maximum yield.
In practice, soil nutrient levels are always greater than zero and com
monly exceed a, the level corresponding to the inflection point on the
curve. Also, as discussed by Anderson (1957), for those cases where the
soil nutrient level is less than a, discarding observations on yield obtained
at the zero level of fertilization will lead to a diminishing returns type
function which can be more adequately represented by a second degree
polynomial. Consequently, in the selection of a representational model,
interest is generally restricted to that portion of the response curve that
lies to the right of a.
The relatively flat area beyond the maximum of the curve in Figure 1
also represents somewhat of a complication in the selection of a model.
However, in the study of economically optimal levels of fertilization, the
investigator is generally not concerned with the nature of the response
function past the maximum yield. Therefore it seems reasonable to limit
the range of interest to that lying between a and c.
The relationship between yield and nitrogen in the range from a to
c may be approximated by several algebraic functions. The two forms
which have been used most frequently in fertilizer use research are the
MitscherlichSpillman equation and the second order polynomial (Heady,
et al., 1961) The quadratic form of the polynomial has in it the advantage
that yields decline at nutrient levels greater than that corresponding to
the maximum. A second advantage of the quadratic polynomial model
is that its parameters are linear combinations of the observations and
may be readily estimated using the least squares procedure. No suitable
transformation to linear form is available for the MitscherlichSpillman
parameters, and they are usually estimated using a more laborious iterative
procedure. Thus, for finding optimal level of fertilization, many investigators
restrict the range of interest between a and c in Figure 1 and select a
polynomial as the estimating function.
The quadratic polynomial function with one factor may be expressed
algebraically as:
A
Y = bo + bX + b2X2
A
in which Y is the predicted yield, bo is the estimated yield without any
applied nutrient, b, is the estimated linear coefficient, b2 is the estimated
quadratic coefficient, and the independent variable X is the level of
fertilization. In practice the estimated coefficients can. be interpreted only
5
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0
CC
IM
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,
o g
o __l
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.21
Yield of Corn Grain
together as a pair. In fertilizer use research a diminishingg return' type
of response curve is characterized by a positive linear coefficient and a
negative quadratic coefficient. With this combination, the response curve
slopes upward for the first increments of X, but at a decreasing rate,
reaches a stationary or zero slope point at X equal to b,/b2 and then
slopes downward.
In addition to the quadratic polynomial, the square root model has
been a commonly used representation of fertilizer response data. This
function with one factor has the form
A s
Y = b0 + bX' + bX .
This transformation of the independent ariable is equivalent to ex
pressing the abscissa of the function in terms of VX iIn,s ic of X Th.js
for values of X less than one, the square roof function rises more rp:ijl,
than the quadratic; for vc..ues of X greater than one, the slope of the
square root function changes more slowly. hien the two forms are fitted
to the same set of fertilizer response data, the relotie magnitude of the
quadratic coefficient with respect to the linear coefficienr usually is much
greater for the square root than for the quadratic cqualion. The sI'Jre
root function has a larger slope than the quadralic at Io levels of ap
plication and becomes much flatter than the quadratic at high levels of
application. For the square root function to predict diminishing returns
with increasing levels of fertilization, b2 must be negaTrie. If the ilHs
are to increase for values of X less than the mrraxim u and dc.re:e
thereafter, then b, must also be positive. In this case the maximum .oLe
occurs at X = bi/4b
The quadratic and square root functions can be extended to q.
tions with two or more factors. A quadratic polynomial mzdel wvrn r...:
factors is often expressed as
A
Y = b, + b,X, + b2Xi + bX X bX + bsX,X2 .
The levels of the two factors or nutrients are denoted as X, and X: and
the estimated linear, pure quadratic and interaction coefficents are b,, b..
..., b respectively. Usually the signs of the linear terms are posiiie and
the pure quadratics negative. The estimated response surface has a
maximum and the calculation of an optimal level of fertilization is pruc
tical only if the pure quadratic terms are negative and the product of b,
and b4 exceeds bV/4.If the range of the observations on yield extends
beyond the level of fertilization corresponding to the maximum yield the
estimated response function will normally have a maximum. For expeirre.ts
in which the observations either fall short of or just reach the maximum
yield, the resulting estimated response function may not have a maximum
within the experimental range of the fertilizer variables.
The square root estimating function with two unknowns may be writ
ten in a form similar to the quadratic model. The efficiency of the quadratii
and square root functions in representing fertilizer response data has been
compared in several studies. Coefficients of determination (R2), indicating
7
the proportion of the total variability in yield which is accounted for by
the independent variables, have been reported for quadratic and square
root equations fitted to the same data by Heady and Dillon (1961). In
their work it appears that the square root prediction equation represents
the experimental observations somewhat more closely than the quadratic
equation, but differences in the size of the R2 coefficients are small.
Box and collaborators (1951, 1957), in their development of composite
designs, began with the assumption that the quadratic polynomial model
can be used to represent the response function within the region of experi
mentation. Thus, their treatment designs were derived specifically for use
in conjunction with a quadratic polynomial. This does not mean that experi
mental date generated with a composite design cannot be used to fit a
square root equation. It is true, however, that the moment requirements
for rotatable designs are not satisfied when the X variables are transformed
for the square root equation. Investigators who choose to employ composite
rototable designs usually use linear or quadratic polynomial models for re
presenting the response surface.
A variation of the polynomial model has been proposed by Nelder
(1966). His estimated model has the algebraic form
1 bo
 ++ b, + b2X.
A X
Y
Called the "inverse polynomial", it does not have symmetry about the maxi
mum. Also, only positive values of the response can be predicted when
the estimated model is used beyond the range of the data. However, in
the estimation procedure given by Nelder, it is implicitly assumed that the
variance is not constant for different levels of X. A thesis by Tejeda (1966)
is an example of applying Nelder's inverse polynomial.
In the following discussion of criteria for comparing treatment de
sirns it is assumed that the yield data will be used to estimate a quadratic
polynomial model. The selection of the quadratic equation as the estimating
function has been made for these reasons: (i) theoretical and empirical
evidence indicates that the quadratic is a reasonably adequate representa
tion of the fertilizer response function, if the region of interest is restricted
to the range between a and c in figure 1, (ii) computations are simplified
by using the quadratic equation, and (iii) economically optimal levels of
fertilizer can be easily calculated.
PREDICTION ERRORS
In fertilizer use studies, a functional relationship between yield and
the fertilizer variables) is envisioned and may be described by a mathe
matical model. The parameters of the model characterize the form of the
functional relationship. The data from an experiment are then used to
estimate the model parameters. Usually, the measure of "goodness" of
the estimated model is based on the difference between an observed re
sponse (Y) and the predicted response (Y) calculated from the model with
the estimated parameters and values of the fertilizer variabless. For ex
ample, when a straight line relationship between yield (Y) and applied
nitrogen (X) is assumed, the model is
Y = Po + 13,X + (1)
where the betas are the parameters for the intercept and the slope and E
is the random error. There is an unknown value of E for each value of Y
coming from a distribution assumed to have a mean of zero and variance
A
of o2. Each e is estimated by YY, the difference between the observed value
and that value predicted by the estimated model
A
Y = bo + b,X (2)
where bo and b, are the least squares estimators of 3o and /, respectively.
If the true model, i(X), is in fact equal to 3o + /PX, then these diff
erences are estimates of measures of the failure of the model to explain
the random error in the n data points. In this case,
1 ^
s2 = [Y (YY)2] (3)
n2
A
is an estimator of a2. That is, the only reason that Y is not equal to Y is
the random variability of the experiment caused by factors including soil,
environmental and measurement variability. The notation of X within the
parenthesis afer t1 is used to indicate that ) is a function of X.
However, if 11(X) is not equal to ,o + f/iX, then differences between
A
Y and Y are induced by a second type of error, called "bias error", in
addition to the random error. Bias error arises when the true model is not
A
equal to the expectation of the fitted model. With YY now including both
random and bias errors, we would expect expression (3) to estimate the
sum of 02 and a component which is due to the bias error. If the experi
menter has replicated at least one of the levels of the fertilizer variable,
an independent estimate of o2 can be calculated since differences in
replicated responses can be due only to random errors. This independent
estimate may then be compared with the mean square calculated from
the remaining sum of squares as an indication of bias error magnitude.
Variance error
Associated with a statistic arising from response function estimation
such as a linear combination of estimated parameters, e.g., the response
function, is its variance which we call variance error, having two distinct
components. One of these, the experimental error, o2, arises due to random
variation among experimental units or plots. In a completely randomized
experiment it is recognized as the failure of replicated plots of a given
treatment to yield the same. The estimated value, s2, when expressed on
8
9
the basis of the experimental unit, is an averaged squared deviation of
the yield of an individual plot from the treatment mean. It is influenced
by such things as variability in the plant, soil, management, and the way
in which the treatments are applied to the experimental units.
The second component of the variance error is somewhat more difficult
to visualize. It is a function of the distribution of the treatment combinations
in the factor space, and thus is influenced by the selection of the treatment
design. As an illustration, consider the problem of estimating the slope
of a function when it is accepted that the true relationship existing between
the two variables is linear. It can be shown that the most efficient way
to estimate the slope is to makeonehalf of the observations at each of
the two levels of the X variable corresponding to the extremes of the
factor space. This component of the variance error which is related to
the distribution of the treatment combinations appears in the regression
analysis as particular elements of the inverse matrix. The size of the
elements in the inverse matrix depends upon the number of treatment
combinations, the distribution of these treatments within the factor space,
and the number of times the set of treatments is replicated. The variance
error is the product of these two components.
Bias error
A
As shown in the first part of this section, the quantity, YY, contains
two sources of error if the true model is not equal to the expectation of
the estimated model. The difference between the true model and the
expectation of the estimated model,
A
TI(X) E(YX)), (4)
evaluated at a given point in the factor space, is called the bias of the
A
predicted value of Y. In general, the average bias of Y(X) is defined as
the difference (4) squared and integrated over a defined factor space.
This concept of average bias error was proposed and developed by Box
and Draper (1959).
To clarify the various kinds of "bias error" concepts, a formulation,
based on polynominal estimated and true models and using matrix
notation, is developed. It is convenient to express the true model at each
of the N design points as
10(X X,A, + Xf, (5)
The estimated model is shown as
(X) X, (6)
where the variables) in the estimated or fitted model are included in the
treatment design X, (N X p matrix where p is the number of estimated
parameters) and b, is a p X 1 column vector of the estirhated coefficients,
estimating .,. The variable(s) not included in the estimated model but
with nonzero coefficientes (1J are included in X1.
Expression (4) can be rewritten as
Ix?,, + xzJ E(XJ (7)
where a, is a set of x values of the p variables included in the estimated
model to predict Y for a given point in the factor space, i.e. x, is any
row vector not necessarily one of the rows of the X, matrix. The x values
for variables not in the estimated model for the same point in the factor
space are included in the row vector xlz
The expectation of xib, is not simply XiiI because the eApeclaton
has to consider the bias of the coefficients in the b, vector. The expec
tation of bt is gi + A46 where A = (XiXI)'X'Xi, as shown by Draper
A
and Smith (1966) and E(Y) = n(X). Therefore the bias of Y for a point
in the factor space is
fei;. + 5 3&) x,Ifl + A&] ,
or equivalently,
xe& x!Aa .
This is a criterion that can be readily considered for comparing al
ternative treatment designs because the alias matrix A is affected by :he
treatment design matrix Xi. If expression (8) is squared and integrated
over the factor space, the Box and Draper (1959) concept of average
A
bias in Y(X) can be written.
Another formulation of a bias concept in an estimated function in
volves the difference of the two terms in expression (8) evaluated at the
treatment design points. Now expression (8) can be written as the bias
vector
Xt XA (9)
This bias vector is associated with calculating lock of fit sum of squares
in analysis of variance tables for data from response surface experiments.
The squared bias term, a scalar, is the squared length of the bias vector
(9) and can be expressed as
O',X'XA2 X',XAI ,, (101
It is the part of the expectation of the sum of sauares for lack of fit in
an analysis of variance table due to Xg& + Xz~g being the true model
 10 
 11 
and Xlb the fitted or estimated model. It may be seen from (10) that
the magnitude of lack of fit may be influenced by the selection of the
treatment design, X,.
It is welt at .his point to digress momentarily and discuss the term
lack of fit, which is not rigorously defined in most textbooks. For example,
Cochran and Cox (19571, Chapter 8A, used the term in quotes. Draper
and Smith (1966) have five index references to lack of fit but without a
clear defiintion. Draper and Herzberger (1971) give a procedure for
dividing the lack of fit into two components in order to gain more insight
on the nature of the lack of fit. Generally speaking, lack of fit is asso
ciated with the failure of the fitted model to completely describe the
data. If a linear polynomial is fitted to data which are actually described
by a quadratic polynomial, then the failure of the straight line to show
the true shape of the function is attributable to lack of fit. By calculation,
lack of fit is a sum of squares based on the data and limited to the
discrete experimental points in the factor space. The residual sum of
squares, the difference between the total sum of squares and the sum
of squares associated with the fitted model, is partitioned into two parts.
The first is the experimental error sum of squares, assuming a designed
study, which may be calculated directly. Then the experimental error sum
of squares is subtracted from the residual sum of squares and the dif
ference called lack of fit. It cannot be calculated directly unless the
difference in the degrees of freedom associated with the fitted polynomial
model and the treatment degrees of freedom is associated with higher
order terms which are then added to the fitted model. The lack of fit
sum of squares is'then forced to be associated with the extra terms, i.e.,
the higher order terms are equated with the a vector of expression (10).
In this situation the assumed model that is used to force the lack of fit
sum of squares to be associated with higher order polynomial terms is
not necessarily the true model. Furthermore, the concept of average bias
A
error in Y involves integration over the entire factor space, not evaluation
at only the treatment design points as done in analysis of variance calcula
tions. Consequently, lack of fit as usually calculated in analysis of variance
tables, and average bias ,or bias defined y (10) where knowledge of
the true model i known, need not be, and usually aren't, equivalent.
However, if an experimenter tries to select between two alternative models,
then the model with the smallest lack of fit is probably the one with the
smallest average bias. An example is helpful in demonstrating the dif
ference. Consider a 3 X 3 factorial where the two factors are quantitative,
e.g., applied nitrogen and phosphorus, and a quadratic polynomial is
fitted to the data. If the true model is a quadratic polynomial, then the
lack of fit will be an estimate of experimental error. Assume that the
true model is a cubic polynomial. A bias exists but it has to be measured
using the data from the 3 X 3 factorial. Lack of fit is calculated by fitting
the two linear by quadratic terms and the quadratic by quadratic term.
In addition, the fitted model could do very well at estimating the true
model at the nine points in the factor space but the average bias could
be relatively large.
A combined variance and bias error
In the statistical literature, the variance and bias errors are com.
bined in a function called the mean squared error. Box and Draper (1959)
have used minimization of the mean squared error, which is the sum
A A
of the variance of Y(X) and the square of the bias of Y(X), as a criterion
for choosing a treatment design.
In the 1959 Box and Draper work, the combined error is standardized
by a2 on a per observation basis, involves a concept of coding or scal
ing as explained by Myers ('971), and is integrated over the whole
factor space. The mean squared error is defined as
J V+B
where V is the average variance of Y(X) and B is the average squared bias
A
of Y(X). It then seems reasonable to select a treatment design which would
minimized J. The advantages of this criterion are not only that J is not
affected by linear transformation of the independent variable and that it
considers both variance and bias but also that it is integrated over the
factor space, not limited to certain points within the factor space. As
an example, consider the situation where the experimenter fits
A
Y = bo + b,X
as his prediction model while the E(Y) or the true model is
E(Y) = TI(X) = fo + PX + 38,X
The average variance and bias have been worked out in detail by
Myers (1971) and if the design points are selected so that they are sym
metric about the center point,
V= 1+
3[111
NP,2
B =  l11 /32 + 4/45
52
where [11] is called by Box and Hunter (1957) the second moment of the
design, i.e., it is the sum of squares of X divided by N, the number of
observations.
The term (11] appears in both V and B. Because [11] is in the denomi
nator for the expression of V, variance can be decreased by increasing
the sum of squares of X for a given number of design points. For bias,
a particular value of (111 equal to 1/ is needed to minimize B. However,
a problem exists in trying to minimize J = V + B since the value [111
that will minimize J depends on the unknown parameter, 82, in the ex
pression for B. If f2 were relatively small, then values of 11] larger than
'/ would minimize J. For example, if P2 was equal to zero, then V would
 12 
 13 
be minimized by placing half the design points at each end of the range
of X, thereby making (111 as large as possible. On the other hand, if A
was relatively large, values of 1111 close to or equal to /3 would be best,
leading to designs with points between the extremes of the range of X.
Myers (1971) presents a table showing that, if the experimenter did
indeed know the values of Nfl/az, and could calculate the best second
moment for the treatment design to minimize J, the moment selected
would be cose to that design selected for only bias considerations. This
general rule of thumb holds through a range of V to B ratios up to
approximately a value of 6. An extension of this simple example to
more than one variable and more complex models is given by Myers
11971) with similar results. However, even if N31/o' were known, this
does not determine the best design, only its second moment. Other criteria
must be used to select the actual design (Thompson 1971).
CRITERIA FOR COMPARING TREATMENT DESIGNS
It should be clear from the previous discussion that an experimenter
has to consider many objectives when choosing a treatment design. Some
of these may be summarized as:,
1) interpretable data without extensive analysis, e.g., for at least
two levels of each factor, there should be three of the other
factors,
2) relatively small number of treatment combinations,
3) low variance of estimated coefficients and, if hypothesis testing
is important, independent estimates,
4) variance of the predicted values small over the central part of
the factor space,
5) variance of the estimated response function slopes small over the
central part of the factor space,
6) bias of the predicted values small over at least the central part
of the factor space,
7} measure of lack of fit,
8) inclusion of the "check" plot in the design, and
9) exclusion of the combinations of zero of one factor and high
level of another factor.
In order to compare two or more treatment designs certain restric
tions must be placed on the values which the X variables may assume.
Box and Wilson (1951) suggest the following convention for bringing
different designs to the same "size". They define the spread, S, of a
variable as the average deviation of that variable from its mean, cal
culated as the square root of the quantity
N
S, = (X XP/N
I
where X. is any value of the variable X, X is the mebn, and. N is the
number of observations. Two designs are considered to have comparable
size when the spread for each of the variables is the same.
The requirement imposed by the Box and Wilson (1951) work that
designs have the same spread in order to be comparable has been
questioned by Folks (1958) and Kempthorne (1965). They maintain that
it would seem more natural and appropriate for an experimenter to
simply define the factor space within which he expects to make inferences.
Designs are then evaluated over the same factor space.
The experimenter on the basis of a priori information decides that
the factor space should either extend to zero or should begin at some
level of fertilization above zero. He is also able to state in general terms
the upper rate limits which are to appear in the treatment design. The
upper limit of the factor space should generally extend slightly beyond
the fertilization level corresponding to the maximurr? ield. This is so
that the estimated response function will usually predict a maximum in
the region of experimentation. At the same time onri a small number
of yield observations should be made at fertilization, levels beyond the
maximum, as observations taken in this part of the' factor space may
increase the error due to bias in predicted yields in the region near the
optimum.
After the factor space has been decided, the treatment design prob
lem is how to place the treatment combinations in the factor space. In
order to select the best design among various alternatives, both bias
and variance criteria have to be considered.
Stigler (1971) has summarized the work on optimality of design and
has discussed several criteria which have been employed to compare
designs from the standpoint of minimal variance. Designs that minimize
the determinant of the covariance matrix of the estimated coefficients
or minimize the maximum variance of Y are discussed in detail by Kiefer
(1959). Equivalently, the determinant of X'X may be maximized as de
veloped by M. J. Box and Draper (1971). Perhaps the criterion which
has been viewed most favorably in past fertilizer use experimentation is
the minimization of the variance of the estimated response function
coefficients (b'sl. Box and associates (1957, 1959) developed the concept
of the variance function and considered the mean variance of the pre
dicted yield over the immediate region of interest as the appropriate
criterion to employ in comparing designs.
Certainly, as pointed out by Kempthorne (1965), the use of different
criteria of optimality leads to different designs. Also, it is equally clear,
that the most appropriate criterion under a given set of circumstances
will be determined by the specific use that is made of the data generated
by the design. In fertilizer use experimentation for the purpose of deter
mining economically optimum levels of fertilization, the investigator is
interested primarily in measuring the slope of the response function at
some point corresponding to the optimum. He is basically concerned in
measuring this slope with the greatest precision possible in accordance
with the facilities available. Consequently, for this type of research the
15 
 14 
minimization of the variance of the slope of the response function should
be a major variance criterion in comparing designs. Using this variance
criterion, Ott and Mendenhall (1972) give suggestions for onefactor
treatment designs. The procedure for estimating this variance for two
factors was developed by Fuller (1962).
If the variance of the slope is calculated for a large number of treat
ment combinations distributed over the experimental factor space, it be
comes apparent that the variance of the slope is minimal at the center
of the design and increases toward the limits. The question arises, there
fore, as to where the variance of the slope should be evaluated in the
comparison of treatment designs. Generally, at the time of selecting a
design, the investigator has reasonable assurance that the optimal level
will lie fairly close to the center of the design, but he has no information
as to the orientation of the optimum with respect to the center. Con
sequently, it wed seem reasonable to select a symmetrical space about
the center of the design as the region in which the optimum is expected
to occur and minimize the average or maximum variance of a slope
within this region.
Complete reliance on variance criteria is justified only if the true
model is known. In agronomic research however, the true model is not
known and bias error becomes an important consideration. Box and
Draper (1959, 1963) suggest mean squared error as a criterion for com
paoing designs. However, as observed previously, knowledge of the true
model is needed for calculating the mean squared error. Furthermore,
the design which is best for variance error will not be optimal for bias
error. Box and Draper circumvent this dilemma by recommending that a
design be selected for bias reasons alone.
Similar conclusions were reached by Draper and Lawrence (1967)
and Myers (1971) who conclude that the experimenter should choose
minimum bias designs unless it was known that variance was very im
portant relative to bias.
Thompson (1971), working with a class of minimum bias twofactor
designs and a square factor space, proposes secondary criteria of
moximizing the power to detect the inadequacy of the postulated model
and of minimizing the variance function of an estimated higher order
polynomial.
Another measure of bias has been proposed by Cady and Laird
(1969). With one independent variable, the bias measure is defined as
the area between the true and estimated models. With two independent
variables, the bias would be a volume. This concept of bias, easy to
understand and intuitively appealing, may be expressed as the absolute
difference integrated over the factor space, s:
IE(Y[X]) (X)JdX
It can be seen that several variance and bias criteria are available
for comparing designs. For fertilizer use experimentation with inadequate
knowledge of the" true model, priority will be given to bias considera
16 
tions in selecting a treatment design, i.e., the experimenter's control of
treatment design would then be used primarily for bias. Variance error
will be handled through experimental design, e.g., blocking and replica.
tion.
EMPIRICAL COMPARISON OF SELECTED DESIGNS
It is apparent from the previous sections that multiple criteria will
be necessary to select a treatment design because the experimenter
usually has more than one objective in a given experiment. If the in
dividual estimated regression coefficients are of interests, the design
minimizing the variances and covariances of the coefficients should be
considered. However, if the experimenter is using the estimated regres
sion equation for prediction, other criteria, including bias, become im
portant. In fact, in recent years, bias considerations have become the
dominant criterion in design selection.
Some quantitative results of treatment design on bias were obtained
in the 1969 work by Cady and Laird. For fertilizer use studies, it was
assumed that the nature of the relationship between yield and the in
dependent variables) is an increasing function at a decreasing rate, per
mitting a defined maximum and diminishing marginal products. Then,
the true model and any fitted models were selected from a class of equa
tions represented (in one variable) by
TI(X) = o + IXC + 8X2 
where .5 < c < 1, i.e., the two extremes are the square root and qua
dratic polynomial models.
For designs with a uniform spacing of treatments over the factor
space, the error due to bias decreases as the number of treatments in
creases. This decreasing function is not linear and the rate of decrease
in bias error per added treatment will become very small as the total
number of treatments becomes large. Bias error in subregions of the
factor space can be influenced by nonuniform spacing of the treatments.
For these asymmetrical designs, error due to bias is minimal in the region
of greatest concentration of treatments.
Escobar (1967) has compared factorials, partial factorials and com
posite designs by calculating the integrated variance and squared bias of
A
Y(X) when the fitted model was a quadratic polynomial in two variables,
X, and X2. The true model used was the quadratic polynomial plus two
specific cubic terms, X2X, and XiX2. The Escobar results show that the
composite designs are superior when bias is considered while the fac
torials and partial factorials have smaller variances. Designs minimizing
the generalized variance, rotatable designs, and designs based on regular
and irregular fractions of 2" factorials are compared using a quadratic
nolynomial model by Nalimov, et al. 11970).
A
Integrated bias and variance in Y(X) were the major considerations
in an empirical study comparing nine treatment designs in two factors
; 'i
 17 
as carried out out by Cady and Laird (1972). From the results, an investi
gator can be quided in the selection of a twofactor treatment design
for his specific objectives.
Response functions of the type
E(Y) + + ,0 + p,,X + 4 + fix
where .5 < c < 1.0 were considered. Reported by Cady and Laird (19721
were results from studies where c of the true function is 5, i.e., a square
root model, and the c of the fitted model is 1.0, a quadratic polynomial.
The betas of the square root model were chosen to give a representative
response function of corn to applied nitrogen (N) and phosphorus (P). The
true model where Y, N and P are in kilograms/hectare was:
E(Y) 3000 + 300.5/N + 374.4VP 15.10 N 23.33 P + 15.00\/NP.
The response surface is shown in Figure 2:0 where the ranges for N and
P were 0 to 320 and 0 to 240 kilograms/hectare respectively.
The nine designs studied are shown in Figure 3 and the fitted re
sponse surfaces (except for the central composite), using a quadratic poly
nomial as the estimating model, are given in Figure 2:1 through 9. Included
were:
1) 5 X 5 factorial, 25 treatments,
2) central composite, 9 treatments,
3) central composite modification by Myers, 9 treatments,
4) 3 X 3 factorial (rotated), 9 treatments,
5) 5 X 5 partial factorial, 13 treatments,
6) 5 X 5 partial factorial modification by Escobar, 13 treatments.
7) 7 X 7 partial factorial, 17 treatments,
8) 3 X 3 factorial, 9 treatments,
9) central composite modification by Thompson, 12 treatments.
The criteria for comparing the designs included:
1) bias residuals, liY),
2) index of integrated bias,
A A
3) index of variance of Y, V(Y),
4) index of variance of the derivative of the response to nitrogen,
V(dY/dN),
5) index of the variance of estimated regression coefficients.
The true responses were calculated from the true square root model
for the treatments in each design, quadratic equations were fitted to the
true responses, and predicted responses were then calculated from the
fitted quadratic equations. The bias residuals or differences between the
true response and the predicted response as summarized in Table 1 give
a quick but incomplete impression of the performance of each design.
The values in the table are the maximums and averages of the absolute
differences between the true yield from the square root model and the
F;
Figure 2.
The response surface. 0. for the square root (true) model used in the Cady and
Laird (1972) study, and the filed response surfaces using a quadratic polynomial
as the estimating model for the treatment designs: (1) 5 5 factorial. (3) central
composite modification by Myers, (4) 3 x 3 factorial (rotated), 5 x 5 partial factorial.
(8) 5 x 5 partial factorial modification by Escobar. (7) 7 x 7 partial factorial, (8)
3 x 3 factorial, and (9) central composite modification by Thompson.
 18 
..:.
%?i;
r
~~ r::::~: i~c~'*~
.:.:~
X
: ....
II '
xll
180
So
120i
o0
o 
240 
1oo 
O 
P 0
0
4
240
Iso
P 120 
so
0
o O I o I
M
I I
o SO ~ 340 DOS
n
o o
S0
a 0 a
121
predicted yield from the estimated quadratic polynomial, evaluated at the
various treatment combinations for each design. The central composite,
the 5 X 5 factorials, complete or partial, the 7 X 7 partial or the 3 X 3
rotated perform poorly relative to the designs modified by Myers, Escobar
or Thompson. Of surprising note are the small maximum and average
residuals of the 3 X 3 factorial. The weakness of looking at the residuals
calculated only at the treatment combinations becomes apparent. Eight
of the 9 treatment combinations of the 3 X 3 are on the boundaries of the
factor space. Hence the apparent good performance of the 3 X 3 might
be due to the estimating model fitting relatively well on the boundaries.
Hypothesized would be a poor fit across the remaining factor space.
A
TABLE 1. Bias Residuals, jY, kilograms/hectare
Number of
Design Design Points Maximum Average
1 25 1022 406
2 9 1199 508
3 9 301 135
4 9 996 442
5 13 882 333
6 13 222 84
7 17 766 350
8 9 326 204
9 12 372 161
The concept of integrated bias developed by Cady and Laird (1969)
was used to calculate the indices of integrated bias. Values for the indices
cf integrated bias in Table 2 were calculated by approximating the volume
between the true and estimated response surfaces for the central quarter
and the entire factor space. The central quarter is defined as the factor
space using ranges of 80 to 240 and 60 to 180 for N and P respectively.
The calculations are basically those described by Cady and Laird (1969)
using the intersections of a 30 X 30 grid placed over the factor space.
Firer sized grids were tried in preliminary calculations with differences
of less than 4%. Now the largest bias is given by the 3 X 3 factorials
followed by the 5 X 5 complete and partial factorials. The modified designs
have relatively low biases and the influence of number and distribution
of points is shown by the intermediate value of the 7 X 7 partial factorial.
Variance criteria were also considered. The designs are compared
in Table 2 by the contribution of the design matrix to the variance of a
predicted value, evaluated at 144 common points (intersections of a 12 X 12
grid) over the entire factor space. The maximums and averages given in
I I J I 2 3
0 0 WO 240 3
N
Figure 3.
The treatment designs compared in the Cady and Laird (1972) study: (1) 5 x 5
factorial. (2) central composite, (3) central composite modification by Myers.
(4) 3 x 3 factorial (rotated). (5) 5 x 5 partial factorial, (6) 5 x 5 partial fac
torial modification by Escobar, (7) 7 x 7 partial factorial. (8) 3 x 3 factorial, and
(9) central composite modification by Thompson.
 21 
M CDq C% . .C
0 CD C 0 0
ododood
to C M t C
TcD t' .n 'c CD
o  4 d 0
0
a
C
00
~ia
0
F;
Table 2 are based on values calculated by XX'XI'X.as given .by Draper
and Smith (1966) where X is a row vector of values for the columns in
the X matrix for a single observation. Also shown in the table are the
maximums of the central 36 points. As expected the 5 X 5 factorial has
the lowest averages. The Myers modification, the 3 X 3 rotated, and to a
lesser degree, the central composite, the Thompson and Escobar modififica
tions and the 7 X 7 partial factorial, have higher values reflecting the
scarcity of points on the boundaries. However, these latter designs perform
better if only the central quarter of the factor space is considered. In ad
dition, two or three replications of these designs will utilize approximately
the same number of plots as one replication of a 5 X 5 factorial. Increasing
A
replication will decrease the variance of Y. Consequently two replications
of the 5 X 5 partial factorial or three replications for the 3 x 3 factorial
will result in favorable variance values compared with one replication of
the 5 X 5. If emphasis is placed on the central quarter of the factor space,
then several designs compare very favorably with the 5 x 5 factorial,
including one of the modified designs with good bias properties, the Esco
bar modification .
In an economic evaluation of the estimated response surface, the
calculation of the optimal quantity of N involves the derivative of the re
sponse surface with respect to N. The calculated derivative is an estimated
parameter and consequently has a variance. Desired properties of a Ireat
ment design would include small values of the variances of the derivatives.
The values in Table 2 were calculated by the Fuller (1962) method. The
individual variances were evaluated at the intersections of a 5 X 5 grid
in the central quarter of the factor space. Again the superiority of the
5 X 5 factorial is seen. However, remembering that two or three replica
tions of the other designs can be used to lower the variances, the other
facto'ials, complete, partial and modified, do well.
TABLE 3. Diagonal Inverse Elements
I ?
a.
ei
CL
_ a
. 8
C
:2
(xlO9)
W4
(xl04)
P2z
(x10')
NP
1 .4508 .8015 .348 .1102 .4340
2 2.649 4.717 2.106 .6667 2.743
3 4.624 8.221 3713 1.173 4.802
4 3.645 6.481 1.907 .6028 10.85
5 .8021 1.426 .6390 .2019 .6382
6 1.488 2.645 1.202 .3800 1.238
7 2.298 4.086 2.112 .6675 .5603
8 .9440 1.678 .7629 .2411 .8181
9 2.540 4.516 2.178 .6883 1.615
 23 
1 CO 90 CD r
(x104) (xtO)
Design Nitrogen (N) Phosphorus (P)
If emphasis is placed on estimating ard testing the individual regres
sion coefficients, then low and uniform variances and covariances of the
estimated coefficients are desirable. The diagonal elements of the inverse
of X'X are used in the calculation. As seen previously X'X is the sum of
squares and cross products formed from the design matrix X. The diagonal
elements of the various inverses are given in Table 3. Of striking note
is the large value for the interaction term with a 3 X 3 rotated, reflecting
the 'absence of treatment points in the corners. As before, replication of
a treatment design will decrease the magnitude of the inverse elements.
As expected, the results in Tables13 confirm the inverse relationship
between the magnitudes of variance error and bias error associated with
the several designs. As argued earlier, however, bias error should be given
major importance in selecting a treatment design and variance error should
be controlled through experimental design, primarily through replication.
Looking again at Table 2 it is seen that the modifications by Myers, Escobar
and Thompson (designs 3. 6 and 9) have integrated bias values that are
about equal and well below the values for the other designs. The central
composite and the 7 x 7 partial factorial form a second group with higher
bias errors and the 5 X 5 factorial, the 3 X 3 rotated factorial and the
5 X 5 partial factorial comprise a third group with still larger bias errors.
The 3 X 3 factorial is in a group by itself with an integrated bias error
about three times that of the modified designs.
TABLE 4. The thirteen treatment combinations corresponding to a 5 X 5
partial factorial modification by Escobar, expressed both as coded values
and as actual levels of factor A with a range from 0 to 320 kg/ha and
factor B with a range from 0 to 240 kg/ha.
Coded Levels Actual Levels
Combination Factor A Factor B Factor A Factor B
1 .85 .85 24 18
2 0 .85 160 18
3 .85 .85 296 18
4 .40 .40 96 72
5 .40 .40 224 72
6 .85 0 24 120
7 0 0 160 120
8 .85 0 296 120
9 .40 .40 96 168
10 .40 .40 224 168
11 .85, .85 24 222
12 0 .85 160 222'
13 .85 .85 296 222
Although the bias errors of the Myers, Escobar and Thompson modifi
cations are about equal, the corresponding variance errors (Tables 2 and
3) are quite different. On the average the variance error for the 5 X 5
partial factorial modification by Escobar is only about one third that of the
central composite modification by Myers and twothirds that of the central
composite modification by Thompson.
Another advantage of the Escobar modification is that three levels
of each factor are studied at each of three levels of the other factor (Figure
3!. This characteristic of the design permits a rapid graphic evaluation of
the effects of the two factors. In research programs where a treatment de
sign is used at a number of sites and it is necessary to interpret the data
before doing a complete analysis, this spatial characteristic of the design
is very desirable.
The 5 X 5 factorial modification by Escobar, in terms of bias error,
variance error and spatial characteristics appears superior to the other
designs. The treatment combinations corresponding to the Escobar modifi
cation are given in Table 4 in terms of a range from 1 to +1 for each
factor .They may easily be transformed into actual levels as shown in the
last two columns in Table 4 once the range of each factor has been deter
mined. A combination of actual zero levels of both factors (a check plot)
may be added as a fourteenth combination if desired.
The selection of the modified 5 X 5 partial factorial has been based
on the premise that the experimenter has as an objective the estimation
of a response surface model. The 5 X 5 partial factorial would not be used
if estimation of one degree of freedom factorial'contrasts is desired. The
selection of the modified 5 X 5 partial factorial primarily resulted from
the empirical study and consequently is dependent on the parameter values
of the square root model used as the true model.
SUMMARY
In the planning of a twofactor study, a decision has to be made on
the choice of the levels of each factor that will be used in a combination
called a treatment. A factor space for two factors is defined as a two
dimensional area that encompasses all possible combinations of interest.
The axes will show the range of interest of each factor. Treatment design
construction involves the problem of selecting a relatively small number
of points in the factor space.
The best treatment design selected will depend on the specific objec
tives of the experimenter and his knowledge of the production system.
For example, it is well known in the single factor case that if the true re
lationship between the response variable and the treatment variable is a
straight line and if the objective is to minimize the variance of the esti
mated slope, then one should divide the total number of points equally
between the two extremes of the treatment variable range. However, this
would be a poor design if the true model was not a straight line and a
measure of the inadequacy of the estimated model is desired. In treatment
 24 
 25 
design then two general considerations are involved: variance of esti
mated parameters and bias, a measure of the differences between the true
model and the estimated model.
Past work has shown that designs which are good for minimizing
variance error are not the best for minimizing bias error. Unfortunately, in
order to study bias, the true model has to be known, a situation usually
not found in practice. Consequently, variance considerations have been
emphasized in the past and only more recent studies have shown the
importance of bias. In fact, the magnitude of bias is sufficiently large that
a recommendation has been made in fertilizer use studies to select a treat
ment design primarily for bias and to control variance error primarily
through replication (Cady and Laird, 1969).
The results are given of an empirical study comparing a number of
twofactor treatment designs including designs which theoretically have
been shown to be best in protecting against bias. A square root model
was selected for the true model and a quadratic polynomial used for the
estimated or fitted model.
In general the bias, or a measure of the integrated volume between
the estimated quadratic polynomial response surface and the true square
root response surface was decreased by restricting the treatment combina
tion to an area of the factor space not including the borders of the factor
space. This restriction led to larger deviations at the borders but over
most of the factor space, hopefully including that area where the economic
optimum will occur, the estimated quadratic polynomial more closely ap
proximated the true model. Not having treatment combinations at the
corners of the factor space increased the variance of estimated parameters,
particularly the variance of the estimated interaction coefficient. It was
concluded that, if a partial factorial which has less bias as compared with
the complete factorial is used, then extra replication may be included to
control the variance error and the total number of plots will be similar.
Three modified designs were best for minimizing bias. Among these
designs, the modified 5 X 5 partial factorial had lower variances for
several variance criteria and, in addition, permits a visual evaluation of
the response to the two factors at three levels of each of the other factors.
The modified 5 x 5 partial factorial is recommended as the best two
factor design among the nine alternative designs studied.
RESUME
Pour 6tablir le plan d'une 6tude bifactorielle. it faut prendre une
decision en ce qui concern le choix des niveaux de chaque facteur qui
seront utilis6s dans une combinaison nommbe traitement. L'espace factories
pour deux facteurs est d6fini par une surface bidimensionnelle qui englobe
toute combinaison possible. Les axes montreront t'intervalle d'int6r&t de
I~
 26 
chaque facteur. L'6laboration du plan de traitement impli4ue le rroblnme
de s6lectionner un petit nombre relatif de points dans F'space factoriel.
Le mailleur plan de traitement s6lectionn6 d6perdre des objectifs
sp6cifiques du chercheur et de sa connaissance du system de production.
Par example, dans le cas d'un seul facteur on sait bien que si la veritable
relation entire la variable dipendante et la variable du traitement est une
ligne droite, et si I'objectif est de r6duire au minimum la variance de Ia
pente estimbe, ii n'y aura qu'& diviser le nombre total de points 6galement
entire las deux extremes de l'intervalle de at variable du traitement. Ceci
sera, cependant, un plan incomplete si le vrai module n'a pas 6t4 une ligne
droite et si on veut une measure de I'insuffisance du module estim6. Ensuite,
dans le plan des traitements sont comprises deux considerations g6nerales:
at variance des parametres estimes et le biais, une measure des differences
entire le vrai modele et te module estime.
Le travail ant6rieur a d6montr6 que les bons plans pour diminuer
l'erreur de la variance ne sont pas les meilleure pour diminuer I'erreur du
biais. Malheureusement. pour 6tudier le biais, le module veritable doit 6tre
connu, une situation qui ne se rencontre pas dons le pratique. Par conse
quent, dans le pass les conditions de la variance talent souiignees et
seules des 6tudes plus r6centes ont soulign6 I'importance du biais. En effet,
la grandeur du biais esa tellement suffisamment important que dans tes
6tudes sur l'usage de fertilisants, il est recommande de selectionner en pri
mordial un plan de traitements par rapport au biais et pour contr6ler
I'erreur de la variance orincipalement au travers de r6p6titions. (Cady &
Laird, 1969).
Dans ce travail sont montr6s les r6sultats d'une 6tude empirique en
ce que sont compares divers plans de traitement bifactoriels, y compris cer
tains qui th6oriquement se sont montr6s les meilleurs pour se proteger des
biais. On a selectionn6 un module de racine carr6e comme le veritable mo
dele et on a utilis6 un polynomial quadratique comme le module estim6
ou just.
En general, le biais ou la measure d'un volume int6gri entre la surface
de r6ponse polynomiale quadratique et la surface de r6ponse de a racine
carr6e veritable, a et6 diminu6 de fa;on 6 restreindre ta combinaison de
traitement en une surface du facteur de I'espace qui ne comprenne pas les
limits de ce dit facteur. Cette restriction a amen6 a de meilleures devia
tions dans les limits; cependant, dons la majeure parties du facteur de
I'espace, o on espere inclure 'aire dan slaquelle se produit I'optimum
6conomique le polynomial quadratique estimr s'est rapproch6 plus pros
du veritable modele. Le fait de ne pas evoir de combinaisons de traitement
dans les angles du facteur de I'espace a augment la variance des para
m6tres estim6s, en particulier la variance du coefficient de ('interaction es
tim6e. On a conclu que si on utilise un facteur partial avec un biais plus
petit pour le compare avec le factriel complete, on pourra include des re
p6titons extra afin de contr6ler I'erreour de oa variance et obtenir un nom
bre similaire de parcelles qu'avec le factories complete.
Trois plans modifies ant 6t4 les meilleurs pour diminuer le biais a
son minimum. Entre ces plans, le factories partial de 5 x 5 a eu des variances
27 
mineures pour divers criteres de variance, et en plus, a permis une 6valua
tion visuelle de to r6ponse aux deux facteurs des trois niveaux de chacun
des autres facteurs. Le factories parties de 5 x 5 est recommand6 come
itant le meilleur plan bifactoriel des neuf plans 6tudi6s.
RESUME
En la planeaci6n de un studio bifactorial, hay que tomar una decision
con respect a k. escogencia de los niveles de cada factor que se usar6n
en una combinaci6n Ilamada un tratamiento. El espacio factorial para dos
factors se define como una 6rea bidimensional que abarca toda combi
naci6n possible tos eies mostrar6n el interval de interns de cada factor.
La elaboraci6n del disefio de tratamiento implica el problema de seleccio
nar un n6mero relative pequefio de puntos en el espacio factorial.
El mejor diseiio de tratamiento que se seleccione depender6 de los
objetivos especificos del investigator y de su conocimiento del sistema de
producci6n. Por ejemplo, en el caso de un solo factor se sabe bien que si
la verdadera relaci6n entire la variable dependiente y la variable del tra
tamiento es una linea recta, y si el objetivo es reducir al minimo la va
rianza de la pendiente estimada, habr6 que dividir el n6mero total de
puntos igualmente centre los dos extremes del intervalo de la variable
del tratamiento. Este seria, sin embargo, un disefio deficiente si el model
verdadero no fuese uno line recta y si se desea una medici6n de la inade
cuacidad del model estimado. Luego, en el disefio de los tratamientos
est6n involucradas dos consideraciones generates: la varianza de los par6
metros estimados y el sesgo, una media de las diferencias entire el mo
delo verdadero y el modelo estimado.
El trabajo anterior ha mostrado que los diseiios buenos para disminuir
el error de la varianza no son los mejores para disminuir el error del sesgo.
Desafortunadamente, para estudiar el sesgo, el model vexdadero debe
ser conocido, una situaci6n que ordinariamente no se encuentra en la
pr6ctica. Par consiguiente, en el pasado se han subrayado, las considera
clones de la varianza y s6o lo s studios m6s recientes han sefialado la
importancia del sesgo. En efecto, ia magnitude del sesgo es tan suficiente
mente grande que en los studios sabre uso de fertilizantes se ha recomen
dado seleccionar un diseio de tratamientos primordialmente con respect
al sesgo y para controlar el error de la varianza principalmente a trov6s
de repeticiones (Cady y Laird, 19691.
En este trabajo se muestron los resultados de un studio empirico en
el que se comparan various diseiios de tratamiento bifactorales, inclusive
algunos que te6ricamente han mostrado ser los mejores para proteger con
tra sesgos. Se seleccion6 un modelo de raiz cuadrada coma el model
verdadero y se us6 un polinomial cuadr6tico coma el modelo estimado o
ajustado.
En general, el sesgo o la media de un volume integrado entire la
superfiice de respuesta polinomial cuadr6tico estimada y to superficie de
respuesta de raiz cuadrada verdadera fue disminuido al restringir la
combinaci6n de tratamiento a una 6rea del factor de espacio que no in
cluia los limits de dicho factor. Esta restricci6n Ilev6 a mayores desvia
clones enlos limits; emperor, sobre ta mayor part, del factor de espacio
donde se espera incluir et 6rea en la cual ocurre el 6ptimo econ6mico
el polinomial cuadr6tico estimado se aproxim6 m6s de cerca al modelo
verdadero. El hecho de no tener combinaciones de tratamiento en las es
quinas del factor de espacio aument6 la varianza de los par6metros esti
mados, en particular ta varianza del coeficiente de interacci6n estimado.
Se concluy6 que, si se usa un factorial parcial que tenga menor sesgo al
compar6rsele con el factorial complete, se podr6n incluir repeticiones ex
tras a fin de controlar el error de la varianza y obtener un numero similar
de parcelas que con el factorial complete.
Tres disefios modificados fueron los mejores para disminuir al minimo
el sesgo. Entre estos disefios, el factorial parcial de 5 x 5 tuvo menores va
rianzas para various criterios de varianza y, adem6s, permit una evalua
ci6n visual de la respuesta a los dos foctores a tres niveles de cada uno
de los dem6s factors. El factorial parcial de 5 x 5 se recomeinda coma el
mejor disefo bifactorial entire los nueve disefos que se estudiaron.
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