An<i>R</i>²statistic for fixed effects in the generalized linear mixed model

Measuring the proportion of variance explained (R2) by a statistical model and the relative importance of specific predictors (semi-partial R2) can be essential considerations when building a parsimonious statistical model. The R2 statistic is a familiar summary of goodness-of-fit for normal linear models and has been extended in various ways to more general models. In particular, the generalized linear mixed model (GLMM) extends the normal linear model and is used to analyze correlated (hierarchical), non-normal data structures. Although various R2 statistics have been proposed, there is no consensus in statistical literature for the most sensible definition of R2 in this context. This research aims to build upon existing knowledge and definitions of R2 and to concisely define the statistic for the GLMM. Here, we derive a model and semi-partial R2 statistic for fixed (population) effects in the GLMM by utilizing the penalized quasi-likelihood estimation method based on linearization. We show that our proposed R2 statistic generalizes the widely used marginal R2 statistic introduced by Nakagawa and Schielzeth, demonstrate our statistics capability in model selection, show the utility of semi-partial R2 statistics in longitudinal data analysis, and provide software that computes the proposed R2 statistic along with semi-partial R2 for individual fixed effects. The software provided is adapted for both SAS and R programming languages.