Clarifying and generalizing R² measures for generalized linear mixed models

R² measures are widely used to quantify effect size, but their interpretation is less straightforward for generalized linear models (GLMs) and generalized linear mixed models (GLMMs) than for ordinary least squares regression. This thesis clarifies the interpretation of R² measures for GLMs and GLMMs by distinguishing two broad families: measures that quantify proportion reduction in model misfit when comparing a null model to the full model, and measures that decompose latent-scale variance into explained and unexplained portions. The first family, which I term proportion-reduction-in-misfit measures, includes Cox-Snell, Nagelkerke, and deviance-ratio R², and the second family, which I term latent-variance-decomposition measures, includes McKelvey-Zavoina R² for single-level GLMs and Nakagawa-Schielzeth marginal and conditional R² for GLMMs. In this thesis, I help clarify and demonstrate that these two approaches answer different questions and have different properties. The thesis further proposes an expanded variance-decomposition framework for GLMMs that separates latent-scale variance into level-specific portions, and distinguishes contributions from level-1 vs. level-2 predictors and random intercept vs. random slope heterogeneity. I also developed a custom R function for logistic GLMMs fitted with lme4, and demonstrate its use with an empirical example. I conclude by providing concrete recommendations for researchers interested in using R² for GLM and/or GLMM, and discuss avenues for future research.