The GEE Procedure

A Type 3 analysis is similar to the Type 3 sums of squares used in PROC GLM, except that generalized score tests for Type 3 contrasts instead of Type 3 sums of squares are computed. Briefly, a Type 3 estimable function (contrast) for an effect is a linear function of the model parameters that involves the parameters of the effect and any interactions with that effect. A test of the hypothesis that the Type 3 contrast for a main effect is equal to 0 is intended to test the significance of the main effect in the presence of interactions. For more information about Type 3 estimable functions, see Chapter 53, The GLM Procedure, and Chapter 16, The Four Types of Estimable Functions. Also see Littell, Freund, and Spector (1991).

Boos (1992) and Rotnitzky and Jewell (1990) describe score tests applicable to testing in GEEs, where is a user-specified contrast matrix or a contrast for a Type 3 test of hypothesis.

Let be the regression parameters that result from solving the GEE under the restricted model , and let be the generalized estimating equation values at .

The generalized score statistic is

where is the model-based covariance estimate and is the empirical covariance estimate. The p-values for T are computed based on the chi-square distribution with r degrees of freedom, where r is the rank of .

A Type 3 analysis can consume considerable computation time because a constrained model is fitted for each effect. Wald statistics for Type 3 contrasts are computed if you specify the WALD option. Wald statistics for contrasts use less computation time than likelihood ratio statistics but might be less accurate indicators of the significance of the effect of interest. The Wald statistic for testing is defined by

where is the contrast matrix, are the GEE parameter estimates, and is the empirical covariance estimate. The asymptotic distribution of S is chi-square with r degrees of freedom, where r is the rank of .

The results of this type of analysis do not depend on the order in which the terms are specified in the MODEL statement. Type 3 analyses that use score statistics are not supported for nominal response data or weighted GEE methods. Type 3 analyses can be conducted using the Wald statistics for all the models that the GEE procedure supports.

The preceding development for Type 3 tests assumes that the rank of the empirical covariance matrix is not less than the row rank of the contrast matrix . When the rank of is less than the row rank of , estimability of the function is not sufficient to ensure that the chi-square test statistic has a unique value no matter what kind of generalized inverse is used to compute .

Although it is extremely rare, it is possible in practice that the uniqueness condition is not satisfied. For example, if the number of clusters is less than the number of nonsingular parameters in the model, then the matrix of coefficients for testing the overall null does not satisfy the uniqueness condition. If this condition is not satisfied, then the chi-square statistic for testing is not invariant to the choice of the -inverse of . This chi-square test is not recommended when the uniqueness condition is not satisfied. An alternative approach would be to increase the number of clusters or to find a parsimonious model so that the number of parameters is less than the number of clusters. When the rank of is less than the row rank of for a test, the procedure prints a note to the SAS log.