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performing a conventional analysis of variance
of the drought indices and testing the significance
of treatment and block effects. A high
association between drought indices and treatments
was taken to mean that the definition
of a given treament should include not only
the direct effect of the applied treatment on
yield, but also the indirect effect of the moisture
stress that is associated with the treatment.
In such a case, the drought indices
should be adjusted for treatments and only
the remaining variation among the drought
indices used as a covariate. Drought indices
should also be adjusted for blocks, in case of
a significant association bewteen indices and
blocks.
In summary, it is seen that the evaluation
of the drought index, for use as a covariate,
requires three operations: (1) an estimate of
me.asurement error, (2) comparison of the total
variability With measurement error by means
of an F test, and (3) analysis of variance of
the drought indices and tests of significance
of treatment and block effects. When either
or both of these effects is significant, it is
necessary to adjust the drought indices for
the indicated effects. This procedure is applicable
in the evaluation of other production
factors for use as a covariate in the analysis
of yield data.
Evaluation of the Predictive Ability
of Regression Models
Fertilizer recommendations for farmers are
commonly developed using information generated
in a large number of well-conducted
field trials. These trials are distributed over
time and geographical area to sample a wide
range of values among important factors affecting
yields. Variables are described carefully
for 'each experiment at the plot or site
level, and the results are combined into a
prediction function, using mUltiple linear regression
procedures.
The number of potential variables to be
included in the prediction equation is large
as it includes applied fertilizer variables, site
variables, and interaction variables. Currently
most investigators select the variables for the
prediction equation by means of a stepwise
procedure. One such approach introduces
potential variables into the equation in an
order determined by their correlation with the
yield or response variable. A second stepwise
procedure begins with all the potential
predictor variables and eliminates some of
them, one at a time, until a satisfactory prediction
equation is obtained,
These procedures are known by the terms
forward selection and backward elimination,
respectively. The criterion used in the selection
or elimination of a variable is the change
in the residual sum of squares, calculated by
using the differences between the observed
and predicted observations.
Recent CIMMYT studies have shown that
the full model (the equation containing all
potential predictor variables) performs well
when used in predicting yields for combinations
of the site variables measured at the
experimental sites. However, the full model
performs poorly when used to predict yields
for combinations which are not the specific
combinations measured in the study. This
holds true even though the combinations are
within the range of values studied. The backward
elimination model and, to a lesser extent
the forward selection model, provide about
the same general results.
These irregularities led to CIMMYT's development
of a new criterion for the selection
of predictor variables in the general yield
function. Called the predictive sum of squares,
it is based on the performance of the estimated
equation for predicting observations
not included in the estimation of the predictor
variable coefficients. Calculation is similar to
the commonly used residual sum of squares,
as both are the sum of squares of the deviations
between the observed and predicted
observations.
A predictor variable selection procedure,
based on the predictive sum of squares criterion,
involves the following steps:
The total number of fertilizer trials is randomly
divided into a small number of groups.
Setting aside one of the groups, the prediction
equations is estimated for each of the
potential predictor variables; the yields predicted;
and the predictive sum of squares
calculated for the omitted group. The procedure
is then repeated, with each of the groups
set aside in turn. At the end of the cycle, the
total predictive sum of squares is calculated
by summing over the omitted groups, and
the predictor with the smallest value is 'selected.
The entire procedure is then repeated
in a stepwise manner for the one-by-one selection
of several predictor variables.
As is commonly known, the residual sum
of squares decreases as more predictors are
added to the general yield function and asymptotically
approaches zero as the number of
variables approaches the number of observations.
With the usual significance levels of
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