Appendices S1-S5 and S7 from The coefficient of determination <i>R</i><sup>2</sup> and intra-class correlation coefficient from generalized linear mixed-effects models revisited and expanded

The coefficient of determination <i>R</i><sup>2</sup> quantifies the proportion of variance explained by a statistical model and is an important summary statistic of biological interest. However, estimating <i>R</i><sup>2</sup> for generalized linear mixed models (GLMMs) remains challenging. We have previously introduced a version of <i>R</i><sup>2</sup> that we called <i>R</i><sup>2</sup><sub>GLMM</sub> for Poisson and binomial GLMMs, but not for other distributional families. Similarly, we earlier discussed how to estimate intra-class correlation coefficients ICC using Poisson and binomial GLMMs. In this paper, we expand our methods to all other non-Gaussian distributions, in particular to negative binomial and gamma distributions that are commonly used for modelling 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 meaningful interpretation of GLMMs, while the delta method allows a general derivation of variance associated with non-Gaussian distributions. 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. However, our method can be used across disciplines and regardless of statistical environments.