CONTRAST Statement
- CONTRAST 'label' < fixed-effect values ...>
< | random-effect values ...> , ...< /
options > ;
The CONTRAST statement provides a mechanism for obtaining custom
hypothesis tests. It is patterned after the CONTRAST statement in
PROC GLM, although it has been extended to include random effects.
This enables you to select an appropriate inference space (McLean,
Sanders, and Stroup 1991).
You can test the hypothesis
,
where L' = (K' M') and
, in several inference spaces. The inference
space corresponds to the choice of M. When M = 0, your inferences apply to the entire population from which the
random effects are sampled; this is known as the broad
inference space. When all elements of M are nonzero, your
inferences apply only to the observed levels of the random effects.
This is known as the narrow inference space, and you can also
choose it by specifying all of the random effects as fixed. The GLM
procedure uses the narrow inference space. Finally, by zeroing
portions of M corresponding to selected main effects and
interactions, you can choose intermediate inference spaces.
The broad inference space is usually the most appropriate, and it is
used when you do not specify any random effects in the CONTRAST
statement.
In the CONTRAST statement,
- label
- identifies the contrast in the table. A label is
required for every contrast specified. Labels can be up
to 20 characters and must be enclosed in single quotes.
- fixed-effect
- identifies an effect that appears in the MODEL statement.
The keyword INTERCEPT can be used as an effect when an intercept is
fitted in the model. You do not need to include all effects
that are in the MODEL statement.
- random-effect
- identifies an effect that appears in the RANDOM statement.
The first random effect must follow a vertical bar (|); however,
random effects do not have to be specified.
- values
- are constants that are elements of the L matrix associated
with the fixed and random effects.
The rows of L' are specified in order and are separated by
commas. The rows of the K' component of L' are
specified on the left side of the vertical bars (|). These rows
test the fixed effects and are, therefore, checked for estimability.
The rows of the M' component of L' are specified on the
right side of the vertical bars. They test the random effects, and
no estimability checking is necessary.
If PROC MIXED finds the fixed-effects portion of the specified
contrast to be nonestimable (see the
SINGULAR= option),
then it displays "Non-est" for the contrast entries.
The following CONTRAST statement reproduces the F-test
for the effect A in the split-plot
example (see Example 41.1):
contrast 'A broad'
A 1 -1 0 A*B .5 .5 -.5 -.5 0 0 ,
A 1 0 -1 A*B .5 .5 0 0 -.5 -.5 / df=6;
Note that no random effects are specified in the preceding contrast;
thus, the inference space is broad. The resulting F-test has
two numerator degrees of freedom because L' has two rows.
The denominator degrees of freedom is, by default, the residual
degrees of freedom (9), but the DF= option changes the denominator
degrees of freedom to 6.
The following CONTRAST statement reproduces the F-test
for A when Block and A*Block
are considered fixed effects (the narrow inference space):
contrast 'A narrow'
A 1 -1 0
A*B .5 .5 -.5 -.5 0 0 |
A*Block .25 .25 .25 .25
-.25 -.25 -.25 -.25
0 0 0 0 ,
A 1 0 -1
A*B .5 .5 0 0 -.5 -.5 |
A*Block .25 .25 .25 .25
0 0 0 0
-.25 -.25 -.25 -.25 ;
The preceding contrast does not contain coefficients for B and
Block because they cancel out in estimated differences between
levels of A. Coefficients for B and Block are
necessary when estimating the mean of one of the levels of A
in the narrow inference space (see Example 41.1).
If the elements of L are not specified for an effect that
contains a specified effect, then the elements of the specified
effect are automatically "filled in" over the levels of the
higher-order effect. This feature is designed to preserve
estimability for cases when there are complex higher-order effects.
The coefficients for the higher-order effect are determined by
equitably distributing the coefficients of the lower-level effect as
in the construction of least squares means. In addition, if the
intercept is specified, it is distributed over all classification
effects that are not contained by any other specified effect. If an
effect is not specified and does not contain any specified effects,
then all of its coefficients in L are set to 0. You can
override this behavior by specifying coefficients for the
higher-order effect.
If too many values are specified for an effect, the extra ones are
ignored; if too few are specified, the remaining ones are set to 0.
If no random effects are specified, the vertical bar can be omitted;
otherwise, it must be present. If a SUBJECT effect is used in the
RANDOM statement, then the coefficients specified for the effects in
the RANDOM statement are equitably distributed across the levels of
the SUBJECT effect. You can use the E option to see exactly what
L matrix is used.
The SUBJECT and
GROUP options in the CONTRAST statement are useful
for the case when a
SUBJECT= or
GROUP= variable appears in the
RANDOM statement, and you want to contrast different subjects or
groups. By default, CONTRAST statement coefficients on random
effects are distributed equally across subjects and groups.
PROC MIXED handles missing level combinations of classification
variables similarly to the way PROC GLM does. Both procedures delete
fixed-effects parameters corresponding to missing levels in order to
preserve estimability. However, PROC MIXED does not delete missing
level combinations for random-effects parameters because linear
combinations of the random-effects parameters are always estimable.
These conventions can affect the way you specify your CONTRAST
coefficients.
The CONTRAST statement computes the statistic
![F = \frac{ [\hat{{\beta}} \ \hat{{\gamma}}
]'L'(L'\hat{C}L)^{-1}
L[\hat{{\beta}} \ \hat{{\gamma}}
]}
{{\rm rank}(L)}](images/mixeq31.gif)
and approximates its distribution with an F-distribution. In
this expression,
is an estimate of the
generalized inverse of the coefficient matrix in the mixed model
equations.
See the "Inference and Test Statistics" section for more information
on this F-statistic.
The numerator degrees of freedom in the F-approximation is
rank(L), and the denominator degrees of freedom is taken
from the "Tests of Fixed Effects" table and corresponds to the
final effect you list in the CONTRAST statement. You can change the
denominator degrees of freedom by using the DF= option.
You can specify the following options in the CONTRAST statement
after a slash (/).
- CHISQ
-
requests that
-tests be performed in addition to any
F-tests. A
-statistic equals its corresponding
F-statistic times the associate numerator degrees of freedom,
and this same degrees of freedom is used to compute the p-value
for the
-test. This p-value will always be less than
that for the F-test, as it effectively corresponds to an F-test
with infinite denominator degrees of freedom.
- DF=number
-
specifies the denominator degrees of freedom for the F-test.
The default is the denominator degrees of freedom taken from the
"Tests of Fixed Effects" table and corresponds to the
final effect you list in the CONTRAST statement.
- E
-
requests that the L matrix coefficients for the
contrast be displayed. For ODS
purposes, the label of this "L Matrix Coefficients"
table is "Coefficients".
- GROUP coeffs
- GRP coeffs
-
sets up random-effect contrasts between different groups
when a GROUP= variable appears in
the RANDOM statement. By default, CONTRAST statement
coefficients on random effects are distributed equally
across groups.
- SINGULAR=number
-
tunes the estimability checking. If v is a vector, define
ABS(v) to be the absolute value of the element of v
with the largest absolute value. If ABS(K'-K'T)
is greater than C*number for any row of K' in the
contrast, then K is declared nonestimable. Here T
is the Hermite form matrix (X'X)-X'X, and C is
ABS(K') except when it equals 0, and then C is 1. The value
for number must be between 0 and 1; the default is 1E-4.
- SUBJECT coeffs
- SUB coeffs
-
sets up random-effect contrasts between different subjects
when a SUBJECT= variable appears
on the RANDOM statement. By default, CONTRAST statement
coefficients on random effects are distributed equally
across subjects.
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.