Small sample sizes in the study of ontogenetic allometry; implications for palaeobiology

Abstract

Quantitative morphometric analyses, particularly ontogenetic allometry, are common methods used in quantifying shape, and changes therein, in both extinct and extant organisms. Due to incompleteness and the potential for restricted sample sizes in the fossil record, palaeobiological analyses of allometry may encounter higher rates of error. Differences in sample size between fossil and extant studies and any resulting effects on allometric analyses have not been thoroughly investigated, and a logical lower threshold to sample size is not clear. Here we show that studies based on fossil datasets have smaller sample sizes than those based on extant taxa. A similar pattern between vertebrates and invertebrates indicates this is not a problem unique to either group, but common to both. We investigate the relationship between sample size, ontogenetic allometric relationship and statistical power using an empirical dataset of skull measurements of modern Alligator mississippiensis. Across a variety of subsampling techniques, used to simulate different taphonomic and/or sampling effects, smaller sample sizes gave less reliable and more variable results, often with the result that allometric relationships will go undetected due to Type II error (failure to reject the null hypothesis). This may result in a false impression of fewer instances of positive/negative allometric growth in fossils compared to living organisms. These limitations are not restricted to fossil data and are equally applicable to allometric analyses of rare extant taxa. No mathematically derived minimum sample size for ontogenetic allometric studies is found; rather results of isometry (but not necessarily allometry) should not be viewed with confidence at small sample sizes.

Supplementary cranial description of the types of Edmontosaurus regalis (Ornithischia: Hadrosauridae), with comments on the phylogenetics and biogeography of Hadrosaurinae

“…kuukpikensis ). While we applaud the authors’ meticulousness, this approach is problematic, given the issues inherent with establishing a growth trajectory based on just a few specimens from one end of an ontogenetic spectrum [51]. …”

Positive allometry for exaggerated structures in the ceratopsian dinosaur Protoceratops andrewsi supports socio-sexual signaling

“…This measure has been shown by Dodson (1976) to change isometrically with other major features of the skull suggesting it is an appropriate metric for this purpose. Although there are obvious limitations with small datasets (Brown and Vavrek, 2015), the data set used here is larger than a number of previous assessments of allometry in fossil taxa and covers a greater range of animal sizes, and thus is sufficient given the limitations of the fossil record.…”

Scaling of major organs in hatchling female American alligators (Alligator mississippiensis)

“…The biphasic breakpoint observed for kidney and liver mass for young alligators appears unique in reptilian literature, but similar biphasic breakpoints have been demonstrated in other vertebrate groups. Our total sample size across all clutches ( N = 86) is robust, and likely to yield accurate allometric exponents, especially for our combined data across clutches (Figure ; Brown & Vavrek, ). The steeper scaling exponents for kidney and liver masses in small alligators may be indicative of the transition from yolk‐absorption to a protein‐based diet and development of fully mature cells.…”