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

Abstract

Many researchers want to report an R2 to measure the variance explained by a model. When the model includes correlation among data, such as phylogenetic models and mixed models, defining an R2 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 R2 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 R2s for phylogenetic and mixed models. A least-squares R2ls is an extension of the ordinary least-squares R2 that weights residuals by variances and covariances estimated by the model; it is closely related to R2glmm proposed by Nakagawa & Schielzeth (2013). The conditional expectation R2ce is based on \"predicting\" each residual from the remaining residuals of the fitted model. The likelihood ratio R2lr was first used by Cragg & Uhler (1970) for logistic regression, and here is used with the standardization proposed by Nagelkerke (1991). These three R2s are formulated as partial R2s, 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 R2s for phylogenetic models were assessed using simulations for continuous and binary response data (phylogenetic generalized least squares and phylogenetic logistic regression). Because the R2s are designed broadly for any model for correlated data, the R2s were also compared for LMMs and GLMMs. R2ls, R2ce, and R2lr all have good performance, and each has advantages and disadvantages for different applications. These R2s are computed in the R package rr2 (https://github.com/arives/rr2). [Binomial regression, coefficient of determination, non-independent residuals, phylogenetic model, pseudo-likelihood]