On the effect of ignoring correlation in the covariates when fitting linear mixed models

Abstract

We study in detail the effects of fitting the standard two-level linear mixed model with a single explanatory variable to clustered data. This model ignores clustering in the explanatory variable and we make explicit the effect of (the usually ignored) within-cluster correlation in the explanatory variable. This approach produces a number of unexpected findings. (i) Ignoring clustering in the explanatory variable affects estimators of both the regression and variance parameters and the effects are different for different estimators. (ii) Increasing the within cluster correlation of the explanatory variable introduces a second local maximum into the log-likelihood and reduced or restricted maximum likelihood (REML) criterion functions which eventually becomes the global maximum, producing a jump discontinuity (at different values) in the maximum likelihood (ML) and REML estimators of the parameters. (iii) Standard statistical software can return local rather than global ML and REML estimates in this very simple problem. (iv) Local ML and REML estimators may fit the data better than their global counterparts but, in these situations, ordinary least squares (OLS) may perform even better than the local estimators. We also establish central limit theorems hold for the ML and REML estimators of the parameters in misspecified linear mixed models which are of some independent interest.