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 η². 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 η² 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).