The General Linear Mixed Effects Model

The problems inherent in the two-stage approach can be avoided by combining Eqs. (6.13) and (6.14) to yield 

It follows that conditional on the random effects, the expected value for the ith subject is 

Vd is the volume of distribution, and the negative of (3] is the elimination rate constant. A linear mixed effect model is appropriate for this dataset. In this case, Yj is a 3 x 1 vector of responses for each subject stacked on top of each other creating a Y vector with 12 rows. The other matrices are 

Equations (6.18) and (6.19) is referred to as the conditional model and implies that Y N(x(3 + zU, R). However, inference is not done on the conditional model but instead on the marginal distribution, which is obtained by integrating out the random effects. Under the marginal distribution, the expected value for the ith subject is 

the marginal model implies that Y N (x(3, zGzt+R). Inference is based on this marginal model unless the data are analyzed within a Bayesian framework. The major difference between the conditional model and marginal model is that the conditional model is conditioned on the random effects, whereas the marginal model does not depend on the random effects. So, the expected value for a subject is the population mean in the absence of further information (the random effects), and the variance for that subject is the total variance, not just the within-subject variance, without any further knowledge on that subject. 

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