Abstract

This talk is concerned with regression models for the effect of covariates on correlated binary and correlated polytomous responses. In a generalized linear mixed model, correlations are induced by a random effect, additive on the logistic scale, so that the joint distribution p_x(y) obtained by integration depends on the covariate values x on the sampled units. The thrust of this talk is that the conventional formulation is inappropriate for most natural sampling schemes in which the sampled units inevitably arise from a random process. The conventional analysis incorrectly predicts parameter attenuation due to the random effect, thereby giving a misleading impression of the magnitude of treatment effects. The error in the conventional analysis is a subtle consequence of sampling bias that arises from random sampling of units. This talk will describe a non-standard but mathematically natural formulation in which the units are auto-generated by an explicit sampling plan. For a quota sample in which the x-configuration is pre-specified, the model distribution coincides with p_x(y) in the GLMM . However, if the sample units are selected at random, for example by simple random sampling from the available population, the conditional distribution p(y | x)$ is different from p_x(y). By contrast with conventional models, conditioning on x is not equivalent to stratification by x. The implications for likelihood computations and estimating equations will be discussed.