The GLM Procedure

CONTRAST Statement

  • CONTRAST ’label’ effect values <…effect values> </ options>;

The CONTRAST statement enables you to perform custom hypothesis tests by specifying an bold upper L vector or matrix for testing the univariate hypothesis bold upper L bold-italic beta equals 0 or the multivariate hypothesis bold upper L bold upper B bold upper M equals 0. Thus, to use this feature you must be familiar with the details of the model parameterization that PROC GLM uses. For more information, see the section Parameterization of PROC GLM Models. All of the elements of the bold upper L vector might be given, or if only certain portions of the bold upper L vector are given, the remaining elements are constructed by PROC GLM from the context (in a manner similar to rule 4 discussed in the section Construction of Least Squares Means).

There is no limit to the number of CONTRAST statements you can specify, but they must appear after the MODEL statement. In addition, if you use a CONTRAST statement and a MANOVA, REPEATED, or TEST statement, appropriate tests for contrasts are carried out as part of the MANOVA, REPEATED, or TEST analysis. If you use a CONTRAST statement and a RANDOM statement, the expected mean square of the contrast is displayed. As a result of these additional analyses, the CONTRAST statement must appear before the MANOVA, REPEATED, RANDOM, or TEST statement.

In the CONTRAST statement,

label

identifies the contrast on the output. A label is required for every contrast specified. Labels must be enclosed in quotes.

effect

identifies an effect that appears in the MODEL statement, or the INTERCEPT effect. The INTERCEPT effect can be used when an intercept is fitted in the model. You do not need to include all effects that are in the MODEL statement.

values

are constants that are elements of the bold upper L vector associated with the effect.

You can specify the following options in the CONTRAST statement after a slash (/).

E

displays the entire bold upper L vector. This option is useful in confirming the ordering of parameters for specifying bold upper L.

E=effect

specifies an error term, which must be one of the effects in the model. The procedure uses this effect as the denominator in F tests in univariate analysis. In addition, if you use a MANOVA or REPEATED statement, the procedure uses the effect specified by the E= option as the basis of the bold upper E matrix. By default, the procedure uses the overall residual or error mean square (MSE) as an error term.

ETYPE=n

specifies the type (1, 2, 3, or 4, corresponding to a Type I, II, III, or IV test, respectively) of the E= effect. If the E= option is specified and the ETYPE= option is not, the procedure uses the highest type computed in the analysis.

SINGULAR=number

tunes the estimability checking. If ABSleft-parenthesis bold upper L minus bold upper L bold upper H right-parenthesis greater-than upper C timesnumber for any row in the contrast, then bold upper L is declared nonestimable. bold upper H is the left-parenthesis bold upper X prime bold upper X right-parenthesis Superscript minus Baseline bold upper X prime bold upper X matrix, and C is ABSleft-parenthesis bold upper L right-parenthesis except for rows where bold upper L is zero, and then it is 1. The default value for the SINGULAR= option is 10 Superscript negative 4. Values for the SINGULAR= option must be between 0 and 1.

As stated previously, the CONTRAST statement enables you to perform custom hypothesis tests. If the hypothesis is testable in the univariate case, SS(upper H 0 colon bold upper L bold-italic beta equals 0) is computed as

left-parenthesis bold upper L bold b right-parenthesis prime left-parenthesis bold upper L left-parenthesis bold upper X prime bold upper X right-parenthesis Superscript minus Baseline bold upper L prime right-parenthesis Superscript negative 1 Baseline left-parenthesis bold upper L bold b right-parenthesis

where bold b equals left-parenthesis bold upper X prime bold upper X right-parenthesis Superscript minus Baseline bold upper X prime bold y. This is the sum of squares displayed on the analysis-of-variance table.

For multivariate testable hypotheses, the usual multivariate tests are performed using

bold upper H equals bold upper M prime left-parenthesis bold upper L bold upper B right-parenthesis prime left-parenthesis bold upper L left-parenthesis bold upper X prime bold upper X right-parenthesis Superscript minus Baseline bold upper L prime right-parenthesis Superscript negative 1 Baseline left-parenthesis bold upper L bold upper B right-parenthesis bold upper M

where bold upper B equals left-parenthesis bold upper X prime bold upper X right-parenthesis Superscript minus Baseline bold upper X prime bold upper Y and bold upper Y is the matrix of multivariate responses or dependent variables. The degrees of freedom associated with the hypothesis are equal to the row rank of bold upper L. The sum of squares computed in this situation is equivalent to the sum of squares computed using an bold upper L matrix with any row deleted that is a linear combination of previous rows.

Multiple-degrees-of-freedom hypotheses can be specified by separating the rows of the bold upper L matrix with commas.

For example, for the model 

proc glm;
   class A B;
   model Y=A B;
run;

with A at 5 levels and B at 2 levels, the parameter vector is

left-parenthesis mu alpha 1 alpha 2 alpha 3 alpha 4 alpha 5 beta 1 beta 2 right-parenthesis

To test the hypothesis that the pooled A linear and A quadratic effect is zero, you can use the following bold upper L matrix:

bold upper L equals Start 2 By 8 Matrix 1st Row 1st Column 0 2nd Column negative 2 3rd Column negative 1 4th Column 0 5th Column 1 6th Column 2 7th Column 0 8th Column 0 2nd Row 1st Column 0 2nd Column 2 3rd Column negative 1 4th Column negative 2 5th Column negative 1 6th Column 2 7th Column 0 8th Column 0 EndMatrix

The corresponding CONTRAST statement is 

contrast 'A LINEAR & QUADRATIC'
         a -2 -1  0  1  2,
         a  2 -1 -2 -1  2;

If the first level of A is a control level and you want a test of control versus others, you can use this statement: 

contrast 'CONTROL VS OTHERS'  a -1 0.25 0.25 0.25 0.25;

See the following discussion of the ESTIMATE statement and the section Specification of ESTIMATE Expressions for rules on specification, construction, distribution, and estimability in the CONTRAST statement.

Last updated: December 09, 2022