Fixed and Random Effects into the Structural Model

Incorporating Fixed and Random Effects into the Structural Model 

One of the most basic questions in any mixed effects model analysis is which parameters should be treated as fixed and which are random. As repeatedly mentioned in the chapter on Linear Mixed Effects Models, an overparameterized random effects matrix can lead to inefficient estimation and poor estimates of the standard errors of the fixed effects, whereas too restrictive a random effects matrix may lead to invalid and biased estimation of the mean response profile (Altham, 1984). In a data rich situation where there are enough observations per subject to obtain individual parameter estimates, i.e., each subject can be fit individually using 

Another approach then is to first identify the fixed effects and then build the random effects matrix. However, there is an interaction between the random effects matrix and fixed effects such that the exclusion of a random effect may fail to identify a significant fixed effect. So what is an analyst to do A common strategy is to first treat all structural parameters in the model as independent random effects, i.e., to use a diagonal covariance matrix. Random effects with near zero variance are treated as fixed effects. Second, whenever possible, use an unstructured covariance matrix between those random effects identified in the first step as being important. If the unstructured covariance model does not minimize successfully, then treat the covariance as a simple matrix (no covariances between diagonal elements). Lastly, once the final model is selected, obtain the EBE (which are discussed later in the chapter) for the random effects and generate a scatter plot correlation matrix. EBEs that appear correlated should then have a covariance term included in the covariance, otherwise the covariance is set equal to zero. Keep in mind, however, that this approach is sequential in that A — B — C, but model building is not necessarily sequential. The process may be iterative such that the process may need to be modified based on the data and model. 

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