A general and simple method for obtaining <i>R</i> ² from generalized linear mixed‐effects models

Abstract

Summary The use of both linear and generalized linear mixed‐effects models ( LMM s and GLMM s) has become popular not only in social and medical sciences, but also in biological sciences, especially in the field of ecology and evolution. Information criteria, such as Akaike Information Criterion ( AIC ), are usually presented as model comparison tools for mixed‐effects models. The presentation of ‘variance explained’ ( R 2 ) as a relevant summarizing statistic of mixed‐effects models, however, is rare, even though R 2 is routinely reported for linear models ( LM s) and also generalized linear models ( GLM s). R 2 has the extremely useful property of providing an absolute value for the goodness‐of‐fit of a model, which cannot be given by the information criteria. As a summary statistic that describes the amount of variance explained, R 2 can also be a quantity of biological interest. One reason for the under‐appreciation of R 2 for mixed‐effects models lies in the fact that R 2 can be defined in a number of ways. Furthermore, most definitions of R 2 for mixed‐effects have theoretical problems (e.g. decreased or negative R 2 values in larger models) and/or their use is hindered by practical difficulties (e.g. implementation). Here, we make a case for the importance of reporting R 2 for mixed‐effects models. We first provide the common definitions of R 2 for LM s and GLM s and discuss the key problems associated with calculating R 2 for mixed‐effects models. We then recommend a general and simple method for calculating two types of R 2 (marginal and conditional R 2 ) for both LMM s and GLMM s, which are less susceptible to common problems. This method is illustrated by examples and can be widely employed by researchers in any fields of research, regardless of software packages used for fitting mixed‐effects models. The proposed method has the potential to facilitate the presentation of R 2 for a wide range of circumstances.

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