Including Distorted Specimens in Allometric Studies: Linear Mixed Models Account for Deformation.

Allometry-patterns of relative change in body parts-is a staple for examining how clades exhibit scaling patterns representative of evolutionary constraint on phenotype, or quantifying patterns of ontogenetic growth within a species. Reconstructing allometries from ontogenetic series is one of the few methods available to reconstruct growth in fossil specimens. However, many fossil specimens are deformed (twisted, flattened, and displaced bones) during fossilization, changing their original morphology in unpredictable and sometimes undecipherable ways. To mitigate against post burial changes, paleontologists typically remove clearly distorted measurements from analyses. However, this can potentially remove evidence of individual variation and limits the number of samples amenable to study, which can negatively impact allometric reconstructions. Ordinary least squares (OLS) regression and major axis regression are common methods for estimating allometry, but they assume constant levels of residual variation across specimens, which is unlikely to be true when including both distorted and undistorted specimens. Alternatively, a generalized linear mixed model (GLMM) can attribute additional variation in a model (e.g., fixed or random effects). We performed a simulation study based on an empirical analysis of the extinct cynodont, , to test the efficacy of a GLMM on allometric data. We found that GLMMs estimate the allometry using a full dataset better than simply using only non-distorted data. We apply our approach on two empirical datasets, cranial measurements of actual specimens of ( = 16) and femoral measurements of the dinosaur ( = 26). Taken together, our study suggests that a GLMM is better able to reconstruct patterns of allometry over an OLS in datasets comprised of extinct forms and should be standard protocol for anyone using distorted specimens.