Effect Size Estimation in Linear Mixed Models

In this note, we recollect some formulas and facts about linear mixed models in relation to Cohen’s effect size measure f2. It is shown how an estimate of the variance-covariance matrix for the estimated fixed effects parameter vector may serve to compute f2in the presence of random effects. The advantage of the purported approach lies in the fact that any variance-covariance estimate already available may be applied. This also illuminates the circumstance that an actual computed effect size value necessarily depends on the employed linear mixed model estimation procedures. To demonstrate possible applications, it is shown how f2can be computed with the statistical software environment R using lme4 . Based on an artificially generated data set, the possible impact of the inclusion of fixed and/or random effects is exemplarily discussed. In addition, the application to several random variables is demonstrated with a publicly available dat set.