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Estimating Fixed and Random Effects in the Mixed Model

ML and REML methods provide estimates of and , which are denoted and , respectively. To obtain estimates of and predicted values of , the standard method is to solve the *mixed model equations* (Henderson, 1984):

The solutions can also be written as

and have connections with empirical Bayes estimators (Laird and Ware, 1982; Carlin and Louis, 1996). Note that the are random variables and not parameters (unknown constants) in the model. Technically, determining values for from the data is thus a prediction task, whereas determining values for is an estimation task.

The mixed model equations are extended normal equations. The preceding expression assumes that is nonsingular. For the extreme case where the eigenvalues of are very large, contributes very little to the equations and is close to what it would be if actually contained fixed-effects parameters. On the other hand, when the eigenvalues of are very small, dominates the equations and is close to 0. For intermediate cases, can be viewed as shrinking the fixed-effects estimates of toward 0 (Robinson, 1991).

If is singular, then the mixed model equations are modified (Henderson, 1984) as follows:

Denote the generalized inverses of the nonsingular and singular forms of the mixed model equations by and , respectively. In the nonsingular case, the solution estimates the random effects directly. But in the singular case, the estimates of random effects are achieved through a back-transformation
where is the solution to the modified mixed model equations. Similarly, while in the nonsingular case itself is the estimated covariance matrix for , in the singular case the covariance estimate for is given by where

An example of when the singular form of the equations is necessary is when a variance component estimate falls on the boundary
constraint of 0.