Prof. Alan Welsh (Australian National University)

Linear mixed models are widely used in a range of application areas, including ecology and
environmental science. We study in detail the effects of fitting the two-level linear mixed model with
a single explanatory variable that is misspecified because it incorrectly ignores contextual effects. In
particular, 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) Incorrectly
omitting contextual effects affects estimators of both the regression and variance parameters not
just, as is currently thought, estimators of the regression 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 REML criterion functions which
eventually becomes the global maximum, producing a jump discontinuity (at different values) in the
maximum likelihood and REML estimators of the parameters. (iii) Standard statistical software such
as SAS, SPSS, STATA, lmer (fromlme4 in R) and GenStat often returns local rather than global
maximum likelihood and REML estimates in this very simple problem. (iv) Local maximum likelihood
and REML estimators may fit the data better than their global counterparts but, in these situations,
ordinary least squares may perform even better than the local estimators, albeit not as well as if we
fit the correct model. (Joint work with Hwan-Jin Yoon.)