Flexible approaches for estimating partial eta squared in mixed-effects models with crossed random factors

Mixed-effects models are frequently used in a variety of disciplines because they can appropriately specify multiple sources of variation. However, precisely because they distinguish between multiple sources of variation, it is difficult to specify a standardized effect size, such as eta(2). Several approaches to this problem have been proposed, but most do not address models with crossed random factors, and none allows for the range of data and models that researchers typically test. For example, no existing approach handles random slopes for a continuous predictor. We introduce several new, flexible approaches to estimating eta(2) in mixed-effect models with crossed random factors. We then conduct a simulation to assess new and old methods. We examine their respective strengths and weaknesses and offer recommendations for a simple approach based on the work of Snijders and Bosker (2011).

Automated summary

Mixed-effects models are commonly used in various disciplines, but it is challenging to specify a standardized effect size like eta(2) due to the distinction between multiple sources of variation. This paper introduces new, flexible approaches to estimating eta(2) in mixed-effect models with crossed random factors and conducts simulations to compare old and new methods. Recommendations for a simple approach based on previous work are provided after examining the strengths and weaknesses of the different methods.