Describing posterior distributions of variance components: Problems and the use of null distributions to aid interpretation

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Joel L Pick, Claudia Kasper, Hassen Allegue , Niels J Dingemanse, Ned A Dochtermann, Kate L Laskowski, Marcos R Lima, Holger Schielzeth, David F Westneat, Jonathan Wright, Yimen G Araya-Ajoy


Assessing the biological relevance of variance components estimated using MCMC-based mixed-effects models is not straightforward. Variance estimates are constrained to be greater than zero and their posterior distributions are often asymmetric. Different measures of central tendency for these distributions can therefore be very different, and credible intervals cannot overlap zero, making it difficult to assess the the size and statistical support for among-group variance. This is often done through visual inspection of the whole posterior distribution, and so relies on subjective decisions for interpretation. We use simulations to demonstrate the difficulties of summarising the posterior distributions of variance estimates from MCMC-based models. We compare commonly used summary statistics of posterior distributions of variance components showing that the posterior median is predominantly the least biased. We also describe different methods for generating null distributions (i.e. a distribution of effect sizes that would be obtained if there was no among-group variance) that can be used to aid in the interpretation of variance estimates. We further show how null distributions could be used to derive a p-value that provides complimentary information to the commonly presented measures of central tendency and uncertainty and also facilitates the implementation of power analyses within an MCMC framework.



Ecology and Evolutionary Biology, Life Sciences


hierarchical modelsvariance, null distribution, permutation, simulations, squidSim, Hierarchical Models, variance


Published: 2022-11-26 11:39

Last Updated: 2023-02-17 12:18

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CC-BY Attribution-NonCommercial 4.0 International

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Data and Code Availability Statement:
All code and generated data for the simulated examples are available at

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