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The coefficient of determination R2 and intra-class correlation coefficient from generalized linear mixed-effects models revisited and expanded.

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 [Formula: see text] for Poisson and binomial GLMMs, but not for other distributional families. Similarly, we earlier discussed how to estimate intra-class correlation coefficients (ICCs) using Poisson and binomial 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. 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, whereas 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.

SUBMITTER: Nakagawa S

PROVIDER: S-EPMC5636267 | biostudies-literature | 2017 Sep

REPOSITORIES: biostudies-literature

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The coefficient of determination R2 and intra-class correlation coefficient from generalized linear mixed-effects models revisited and expanded.

Nakagawa Shinichi S Johnson Paul C D PCD Schielzeth Holger H

Journal of the Royal Society, Interface 20170913 134

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 [Formula: see text] for Poisson and binomial GLMMs, but not for other distributional families. Similarly, we earlier discussed h ...[more]

PMID: 28904005

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Project description:BackgroundThe intraclass correlation coefficient (ICC) is widely used in biomedical research to assess the reproducibility of measurements between raters, labs, technicians, or devices. For example, in an inter-rater reliability study, a high ICC value means that noise variability (between-raters and within-raters) is small relative to variability from patient to patient. A confidence interval or Bayesian credible interval for the ICC is a commonly reported summary. Such intervals can be constructed employing either frequentist or Bayesian methodologies.MethodsThis study examines the performance of three different methods for constructing an interval in a two-way, crossed, random effects model without interaction: the Generalized Confidence Interval method (GCI), the Modified Large Sample method (MLS), and a Bayesian method based on a noninformative prior distribution (NIB). Guidance is provided on interval construction method selection based on study design, sample size, and normality of the data. We compare the coverage probabilities and widths of the different interval methods.ResultsWe show that, for the two-way, crossed, random effects model without interaction, care is needed in interval method selection because the interval estimates do not always have properties that the user expects. While different methods generally perform well when there are a large number of levels of each factor, large differences between the methods emerge when the number of one or more factors is limited. In addition, all methods are shown to lack robustness to certain hard-to-detect violations of normality when the sample size is limited.ConclusionsDecision rules and software programs for interval construction are provided for practical implementation in the two-way, crossed, random effects model without interaction. All interval methods perform similarly when the data are normal and there are sufficient numbers of levels of each factor. The MLS and GCI methods outperform the NIB when one of the factors has a limited number of levels and the data are normally distributed or nearly normally distributed. None of the methods work well if the number of levels of a factor are limited and data are markedly non-normal. The software programs are implemented in the popular R language.

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The coefficient of determination R2 and intra-class correlation coefficient from generalized linear mixed-effects models revisited and expanded.

Publications

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

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