A Diagnostic for Bias in Linear Mixed Model Estimators Induced by Dependence Between the Random Effects and the Corresponding Model Matrix

We explore how violations of the often-overlooked standard assumption that the random effects model matrix in a linear mixed model is fixed (and thus independent of the random effects vector) can lead to bias in estimators of estimable functions of the fixed effects. However, if the random effects of the original mixed model are instead also treated as fixed effects, or if the fixed and random effects model matrices are orthogonal with respect to the inverse of the error covariance matrix (with probability one), or if the random effects and the corresponding model matrix are independent, then these estimators are unbiased. The bias in the general case is quantified and compared to a randomized permutation distribution of the predicted random effects, producing an informative summary graphic for each estimator of interest. This is demonstrated through the examination of sporting outcomes used to estimate a home field advantage.