Full decomposition of phenotypic variance in dichotomous traits: New methods and key implications for behaviour, demography and evolutionary ecology

Abstract Numerous traits that define behaviour, development, life‐history and species ecology are expressed as dichotomous alternative phenotypes (e.g. morphological and behavioural polyphenisms, reproductive vs. non‐reproductive states, movement vs. philopatry). Accordingly, core objectives spanning behaviour, demography and evolutionary ecology are to decompose phenotypic variation in dichotomous traits into different environmental and/or genetic components, and thereby infer biological mechanisms and predict future dynamics. However, perhaps surprisingly, variance decompositions for dichotomous phenotypes have not been rigorously defined or comprehensively achieved. While analysts commonly use generalised linear mixed models (GLMMs) to estimate variance components on underlying, latent continuously distributed scales, conceptual and mathematical formalisations of back‐transformations onto the dichotomous scale of phenotypic expression have been lacking. These methodological limitations hinder full conceptualisation and evaluation of biological processes and impede cross‐system comparisons, but such challenges are rarely acknowledged. Accordingly, we first synthesise current approaches to modelling dichotomous phenotypic variation. Using the threshold model paradigm, we clarify the meaning of latent‐scale GLMM parameters. We then formulate and solve exact back‐transformation equations, linking the continuous latent scale (termed ‘liability’) to the dichotomous phenotypic scale. We use these derivations to devise a new methodology to systematically define and quantify all components of phenotypic variance that non‐linearly emerge from linear combinations of latent liability‐scale effects. We thereby provide an easy‐to‐implement calculation procedure (and R codes) for fully and accurately decomposing phenotypic variance in dichotomous traits using GLMM estimates. We highlight three main applications, with worked examples, to show how our novel back‐transformation methodology can address three core challenges in ecology and evolutionary biology: validly quantifying repeatability in binary behaviours, heterogeneity in individual contributions to population growth and genetic variances in dichotomous phenotypes. These applications illustrate how our full variance decompositions systematically quantify individual and/or genetic differences in phenotypic plasticity, revealing critical eco‐evolutionary processes. Overall, we provide new capability to comprehensively quantify and rationalise emerging variation in dichotomous phenotypic traits. By revealing key biological properties that were previously hidden and eliminating inaccuracies and misconceptions that are otherwise likely to be commonplace, we open broad opportunities to meaningfully dissect dichotomous phenotypic variation in ecological and evolutionary research.