![]() Currently, these two statistics are only described for binomial and Poisson GLMMs. However, at least one important issue seems to remain. The descriptions were initially limited to random-intercept GLMMs, but have later been extended to random-slope GLMMs, widening the applicability of these statistics (see also ). These previous articles featured generalized linear mixed-effects models (GLMMs) as the most versatile engine for estimating R 2 and ICC (specifically and ICC GLMM). We have reviewed methods for estimating R 2 and ICC in the past, with a particular focus on non-Gaussian response variables in the context of biological data. In the field of ecology and evolution, a type of ICC is often referred to as repeatability R, where the grouping factor is often individuals that have been phenotyped repeatedly. ![]() The intra-class correlation coefficient (ICC) is a related statistic that quantifies the proportion of variance explained by a grouping (random) factor in multilevel/hierarchical data. In this context, it is not surprising that the coefficient of determination R 2 is a commonly reported statistic, because it represents the proportion of variance explained by a linear model. One of the main purposes of linear modelling is to understand the sources of variation in biological data. However, our method can be used across disciplines and regardless of statistical environments. We illustrate the implementation of our extension by worked examples from the field of ecology and evolution in the R environment. We also discuss some special considerations for binomial GLMMs with binary or proportion data. Jensen's inequality has important implications for biologically meaningful interpretation of GLMMs, whereas the delta method allows a general derivation of variance associated with non-Gaussian distributions. 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. In this paper, we generalize our methods to all other non-Gaussian distributions, in particular to negative binomial and gamma distributions that are commonly used for modelling biological data. Similarly, we earlier discussed how to estimate intra-class correlation coefficients (ICCs) using Poisson and binomial GLMMs. We have previously introduced a version of R 2 that we called for Poisson and binomial GLMMs, but not for other distributional families. However, estimating R 2 for generalized linear mixed models (GLMMs) remains challenging. The coefficient of determination R 2 quantifies the proportion of variance explained by a statistical model and is an important summary statistic of biological interest.
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