Abstract
The coefficient of determination R2 quantifies the proportion of variance explained by a statistical model and is an important summary statistic of biological interest. However, estimating R2 for (generalized) linear mixed models (GLMMs) remains challenging. We have previously introduced a version of R2 that we called R2GLMM for Poisson and binomial GLMMs using biological examples, but not for other distributional families. Similarly, we earlier discussed how to estimate intra-class correlation coefficients ICC using only Poisson and binomial GLMMs. In this article, we expand our methods to all the other non-Gaussian distributions such as negative binomial and gamma distributions, which are common in biological data. While expanding our approach, we highlight two useful concepts for biologists, Jensen’s inequality and the delta method, both of which help us in understanding the properties of GLMMs. Jensen’s inequality has important implications for biologically more meaningful interpretation of GLMMs, while the delta method allows a general derivation of distribution-specific variances. We also discuss some special considerations for binomial GLMMs with binary or proportion data. We illustrate the implementation of our extension by worked examples from the field of ecology and evolution in the R environment although our method can be used regardless of statistical environments.
