Inference for the Random Effects

Although in practice one is usually primarily interested in estimating the parameters in the marginal linear mixed-effects model (the fixed effects β and the variance components in D and in all ∑_i), it is often useful to calculate estimates for the random effects b _i as well, since they reflect how much the subject-specific profiles deviate from the overall average profile. Such estimates can then be interpreted as residuals which may be helpful for detecting special profiles (i.e., outlying individuals) or groups of individuals evolving differently in time. Also, estimates for the random effects are needed whenever interest is in prediction of subject-specific evolutions (see Section 7.5).

As indicated in Section 5.1, it is then no longer sufficient to assume that the marginal distribution of the responses Y _i is given by model (5.1), because it does not imply that the variability in the data can be explained by random effects. In this section, we will therefore explicitly assume that the hierarchical model (3.8) is appropriate. Since random effects represent a natural heterogeneity between the subjects, this assumption will often be justified for data where the between-subjects variability is large in comparison to the within-subject variability.