title: 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.
authors.affiliation: Royal Tyrrell Museum of Palaeontology , Drumheller, Alberta , Canada.
authors.forename: Caleb Marshall
authors.initials: CM
authors.lastname: Brown
authors.affiliation: Department of Natural History, Royal Ontario Museum , Toronto, Ontario , Canada.
authors.forename: Matthew J
authors.initials: MJ
authors.lastname: Vavrek
keywords: Alligator; Error; Fossil; Growth; Isometry; Morphometrics; Palaeontology; Sample size; Shape; Size