Linear hypotheses for are expressed in matrix form as
where is a matrix of coefficients for the linear hypotheses, and is a vector of constants. The vector of regression coefficients includes slope parameters as well as intercept parameters. The Wald chi-square statistic for testing is computed as
where is the estimated covariance matrix. Under , has an asymptotic chi-square distribution with r degrees of freedom, where r is the rank of .
For models that use the less-than-full-rank parameterization (as specified by the PARAM=GLM option in the CLASS statement), a Type 3 test of an effect of interest is a test of the Type III estimable functions that are defined for that effect. When there are no missing cells, the Type 3 test of a main effect corresponds to testing the hypotheses of equal marginal means. For more information about Type III estimable functions, see Chapter 44: The GLM Procedure, and Chapter 15: The Four Types of Estimable Functions. Also see Littell, Freund, and Spector (1991).
For models that use a full-rank parameterization, all parameters are estimable when there are no missing cells; so it is unnecessary to define estimable functions. The Type 3 test of an effect of interest is the joint test that the parameters associated with that effect are zero. For a model that uses effects parameterization (as specified by the PARAM=EFFECT option in the CLASS statement), testing a main effect is equivalent to testing the equality of marginal means. For a model that uses reference parameterization (as specified by the PARAM=REF option in the CLASS statement), the Type 3 test is a test of the equality of cell means at the reference level of the other model effects. For more information about the coding scheme and the associated interpretation of results, see Muller and Fetterman (2002, Chapter 14).
For a model without an interaction term, the Type 3 tests of main effects are the same regardless of the type of parameterization that is used. For a model that contains an interaction term and no missing cells, the Type 3 test of a component main effect is the same under GLM parameterization and effect parameterization, because both test the equality of cell means. But this differs from reference parameterization, which tests the equality of cell means at the reference level of the other component main effect. If some cells are missing, you can obtain meaningful Type 3 tests only by testing a Type III estimable function, so in this case you should use GLM parameterization.
The results of a Type 3 analysis do not depend on the order in which the terms are specified in the MODEL statement.