R2s for Correlated Data: Phylogenetic Models, LMMs, and GLMMs

Many researchers want to report an R 2 to measure the variance explained by a model. When the model includes correlation among data, such as phylogenetic models and mixed models, defining an R 2 faces two conceptual problems. (i) It is unclear how to measure the variance explained by predictor (independent) variables when the model contains covariances. (ii) Researchers may want the R 2 to include the variance explained by the covariances by asking questions such as “How much of the variance is explained by phylogeny?” Here, I investigate three R 2 s for phylogenetic and mixed models. A least-squares R 2 ls is an extension of the ordinary least-squares R 2 that weights residuals by variances and covariances estimated by the model; it is closely related to R 2 glmm proposed by Nakagawa & Schielzeth (2013). The conditional expectation R 2 ce is based on “predicting” each residual from the remaining residuals of the fitted model. The likelihood ratio R 2 lr was first used by Cragg & Uhler (1970) for logistic regression, and here is used with the standardization proposed by Nagelkerke (1991). These three R 2 s are formulated as partial R 2 s, making it possible to compare the contributions of mean components (regression coefficients in phylogenetic models and fixed effects in mixed models) and variance components (phylogenetic correlations and random effects) to the fit of models. The properties of the R 2 s for phylogenetic models were assessed using simulations for continuous and binary response data (phylogenetic generalized least squares and phylogenetic logistic regression). Because the R 2 s are designed broadly for any model for correlated data, the R 2 s were also compared for LMMs and GLMMs. R 2 ls , R 2 ce , and R 2 lr all have good performance, and each has advantages and disadvantages for different applications. These R 2 s are computed in the R package rr2 ( https://github.com/arives/rr2 ). [Binomial regression, coefficient of determination, non-independent residuals, phylogenetic model, pseudo-likelihood]