Tests of Homogeneity for Generalized Linear Models

Abstract

We propose two tests for testing homogeneity among clustered data adjusting for the effects of covariates. The first is a score test for a generalized linear model with random effect, in which the distribution of the response variable given the random effect is entirely defined. In contrast to the likelihood ratio test, however, the score test does not require estimation of the parameters of a mixed-effects model nor specification of the mixing distribution. The second test is proposed in the framework of the generalized estimating equation (GEE) approach. In deriving this test, we need only the specification of the marginal expectation and variance of the response variable and the fourth moment for the overdispersion term, whereas for deriving the score test for mixed effects models, the entire conditional distribution must be specified. We demonstrate that the two tests are identical when the covariance matrix assumed in the GEE approach is that of the random-effects model. In both approaches, the test statistic can be decomposed into a pairwise correlation statistic and a statistic of overdispersion. We performed a simulation study to compare the power of the score test and of the test based on their pairwise correlation statistic only, and also to compare their type I errors in cases where data present overdispersion not due to the clustering studied. On the basis of these results, we recommend using the pairwise correlation statistic, which is more robust than the complete statistic to overdispersion not due to the clustering studied.

Key Words:

• Clustered data
• Generalized estimating equations
• Intraclass correlation
• Random effects model
• Score test