Effect Size Estimation in Linear Mixed Models

Abstract In this note, we recollect some formulas and facts about linear mixed models in relation to Cohen’s effect size measure $$f^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>f</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> . It is shown how an estimate of the variance-covariance matrix for the estimated fixed effects parameter vector may serve to compute $$f^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>f</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> in 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 $$f^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>f</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> can be computed with the statistical software environment using . 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.