Collision between biological process and statistical analysis revealed by mean centring

Abstract

1. Animal ecologists often collect hierarchically structured data and analyse these with linear mixed-effects models. Specific complications arise when the effect sizes of covariates vary on multiple levels (e.g. within vs. among subjects). Mean centring of covariates within subjects offers a useful approach in such situations, but is not without problems. 2. A statistical model represents a hypothesis about the underlying biological process. Mean centring within clusters assumes that the lower level responses (e.g. within subjects) depend on the deviation from the subject mean (relative) rather than on the absolute scale of the covariate. This may or may not be biologically realistic. We show that mismatch between the nature of the generating (i.e. biological) process and the form of the statistical analysis produce major conceptual and operational challenges for empiricists. 3. We explored the consequences of mismatches by simulating data with three response-generating processes differing in the source of correlation between a covariate and the response. These data were then analysed by three different analysis equations. We asked how robustly different analysis equations estimate key parameters of interest and under which circumstances biases arise. 4. Mismatches between generating and analytical equations created several intractable problems for estimating key parameters. The most widely misestimated parameter was the among-subject variance in response. We found that no single analysis equation was robust in estimating all parameters generated by all equations. Importantly, even when response-generating and analysis equations matched mathematically, bias in some parameters arose when sampling across the range of the covariate was limited.

Most published selection gradients are underestimated: Why this is and how to fix it

“…2019; Westneat et al. 2020). Specifically, inflated estimates of among‐individual variance in traits can occur when environmental conditions eliciting reversible plasticity within‐individuals are themselves repeatable among individuals (Dingemanse et al.…”

Maternal allocation in relation to weather, predation and social factors in a colonial cooperative bird

“…We used bivariate models instead of a subject‐centring approach (Lejeune et al., 2016), to estimate female means as latent variables, and account for their uncertainty (credible intervals). This results in unbiased estimates of the within/between‐individual effects, which could otherwise be substantially affected by measurement error (Lüdtke et al., 2008; see also Westneat et al., 2020). We performed a Bayesian analysis using the MCMCglmm package (Hadfield, 2010) with the same fixed effect structure and transformations as for the univariate models.…”

Distinguishing within- from between-individual effects: How to use the within-individual centering method for quadratic patterns

“…In any case, when using the within-individual centering for non-linear relationships, the equation used should be chosen carefully in order to run meaningful analyses. In line with Westneat et al (2020), we stress that it is critical to have a clear understanding of the equation used to ensure that the assumptions made by a statistical model (using within-individual centering or not) match the biological process investigated. We hope that this manuscript will encourage the use of the within-individual centering method and promote its correct application for non-linear relationships.…”

“…It should be noted that extending equation 3 with random slopes for the linear and quadratic components of the within-individual effect would allow estimation of both varying rate and onset of senescence. However, those models, although theoretically sound, require much more data (van de Pol 2012; Westneat et al 2020).…”