Analysis of covariance in the mixed model: higher-level, nonhomogeneous, and random regressions

The model generally considered in analysis of covariance has all levels of classification factors and interactions fixed, and also covariate regression coefficients fixed. Mixed models are more appropriate in most applications. A summary of estimation and hypothesis testing for analysis of covariance in the mixed model, including the case of random regression coefficients, is presented. Higher-level covariate regressions (i.e., regressions in which, for all levels of a factor or interaction, all observations on the same level have a common covariate value) are discussed. Nonestimability problems that result from defining such covariates at the levels of fixed effects are illustrated. The case of nonhomogeneous covariate regressions in the mixed model is considered in the context of interpreting predicted future differences among levels of a given factor or interaction. Nonhomogeneous regressions complicate interpretations only when they are associated with the contrast(s) of interest among fixed effects in the model. The question of whether the regressions are homogeneous is itself often of substantive interest. Different random regression coefficients associated with the levels of a random effect are also examined.