  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.
