Coefficient of determination R2 and intra class correlation coefficient ICC from generalized linear mixed effects models revisited and expanded
1. 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, but not for other distributional families. 2. Similarly, we earlier discussed how to estimate intra class correlation coefficients ICC (also known as repeatability in the field of ecology and evolution) using Poisson and binomial GLMMs, but not for other distributional families. ICC is related to R2 because they are both ratios of variance components. 3. In this article we expand our method to additional non Gaussian distributions, namely quasi Poisson, negative binomial and gamma GLMMs. However, in theory, our extension could be applied to any distribution and we include an explanatory calculation for the Tweedie distribution. 4. While expanding our approach, we highlight two useful concepts, Jensen9s inequality and the delta method, both of which help in understanding the properties of GLMMs. Jensen9s inequality has important implications for the interpretation 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. 5. We illustrate the implementation of our extension by worked examples in the R environment. However, our method can be used regardless of statistical packages and environments. We finish by referring to two alternative methods to our approach along with a cautionary note.