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

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.

Tooth counts through growth in diapsid reptiles: implications for interpreting individual and size-related variation in the fossil record.

Tooth counts are commonly recorded in fossil diapsid reptiles and have been used for taxonomic and phylogenetic purposes under the assumption that differences in the number of teeth are largely explained by interspecific variation. Although phylogeny is almost certainly one of the greatest factors influencing tooth count, the relative role of intraspecific variation is difficult, and often impossible, to test in the fossil record given the sample sizes available to palaeontologists and, as such, is best investigated using extant models. Intraspecific variation (largely manifested as size-related or ontogenetic variation) in tooth counts has been examined in extant squamates (lizards and snakes) but is poorly understood in archosaurs (crocodylians and dinosaurs). Here, we document tooth count variation in two species of extant crocodylians (Alligator mississippiensis and Crocodylus porosus) as well as a large varanid lizard (Varanus komodoensis). We test the hypothesis that variation in tooth count is driven primarily by growth and thus predict significant correlations between tooth count and size, as well as differences in the frequency of deviation from the modal tooth count in the premaxilla, maxilla, and dentary. In addition to tooth counts, we also document tooth allometry in each species and compare these results with tooth count change through growth. Results reveal no correlation of tooth count with size in any element of any species examined here, with the exception of the premaxilla of C. porosus, which shows the loss of one tooth position. Based on the taxa examined here, we reject the hypothesis, as it is evident that variation in tooth count is not always significantly correlated with growth. However, growth trajectories of smaller reptilian taxa show increases in tooth counts and, although current samples are small, suggest potential correlates between tooth count trajectories and adult size. Nevertheless, interspecific variation in growth patterns underscores the importance of considering and understanding growth when constructing taxonomic and phylogenetic characters, in particular for fossil taxa where ontogenetic patterns are difficult to reconstruct.

Testing of the effect of missing data estimation and distribution in morphometric multivariate data analyses.

Missing data are an unavoidable problem in biological data sets and the performance of missing data deletion and estimation techniques in morphometric data sets is poorly understood. Here, a novel method is used to measure the introduced error of multiple techniques on a representative sample. A large sample of extant crocodilian skulls was measured and analyzed with principal component analysis (PCA). Twenty-three different proportions of missing data were introduced into the data set, estimated, analyzed, and compared with the original result using Procrustes superimposition. Previous work investigating the effects of missing data input missing values randomly, a non-biological phenomenon. Here, missing data were introduced into the data set using three methodologies: purely at random, as a function of the Euclidean distance between respective measurements (simulating anatomical regions), and as a function of the portion of the sample occupied by each taxon (simulating unequal missing data in rare taxa). Gower's distance was found to be the best performing non-estimation method, and Bayesian PCA the best performing estimation method. Specimens of the taxa with small sample sizes and those most morphologically disparate had the highest estimation error. Distribution of missing data had a significant effect on the estimation error for almost all methods and proportions. Taxonomically biased missing data tended to show similar trends to random, but with higher error rates. Anatomically biased missing data showed a much greater deviation from random than the taxonomic bias, and with magnitudes dependent on the estimation method.

Homology and architecture of the caudal basket of Pachycephalosauria (Dinosauria: Ornithischia): the first occurrence of myorhabdoi in Tetrapoda.

BACKGROUND: Associated postcranial skeletons of pachycephalosaurids, most notably those of Stegoceras and Homalocephale, reveal enigmatic osseous structures not present in other tetrapod clades. The homology and functional significance of these structures have remained elusive as they were originally interpreted to be abdominal ribs or gastralia, and more recently have been interpreted as de novo structures in the tail. PRINCIPAL FINDINGS: Analysis of these structures in nearly all pachycephalosaurid skeletons has facilitated a complete description of their architecture, and the establishment of patterns consistent with those of myorhabdoid ossifications--ossifications of the myoseptal tendons associated with myomeres. The presence and structure of myorhabdoid ossifications are well established for teleost fish, but this marks their first recognition within Tetrapoda. These elements are both structurally and histologically distinct from the deep, paraxial ossified tendon bundles of other ornithischian clades, although they may have performed a similar function in the stiffening of the tail. CONCLUSIONS/SIGNIFICANCE: These myorhabdoi are not de novo structures, but are instead ossifications (and therefore more amenable to fossilization) of the normally unossified plesiomorphic caudal myosepta of vertebrates. The ubiquitous ossification of these structures in pachycephalosaurids (all specimens preserving the tail also exhibit myorhabdoid ossifications) suggests it is a likely synapomorphic condition for Pachycephalosauria.

Cranial ontogeny in Stegoceras validum (Dinosauria: Pachycephalosauria): a quantitative model of pachycephalosaur dome growth and variation.

Historically, studies of pachycephalosaurs have recognized plesiomorphically flat-headed taxa and apomorphically domed taxa. More recently, it has been suggested that the expression of the frontoparietal dome is ontogenetic and derived from a flat-headed juvenile morphology. However, strong evidence to support this hypothesis has been lacking. Here we test this hypothesis in a large, stratigraphically constrained sample of specimens assigned to Stegoceras validum, the best known pachycephalosaur, using multiple independent lines of evidence including conserved morphology of ornamentation, landmark-based allometric analyses of frontoparietal shape, and cranial bone histology. New specimens show that the diagnostic ornamentation of the parietosquamosal bar is conserved throughout the size range of the sample, which links flat-headed specimens to domed S. validum. High-resolution CT scans of three frontoparietals reveal that vascularity decreases with size and document a pattern that is consistent with previously proposed histological changes during growth. Furthermore, aspects of dome shape and size are strongly correlated and indicative of ontogenetic growth. These results are complementary and strongly support the hypothesis that the sample represents a growth series of a single taxon. Cranial dome growth is positively allometric, proceeds from a flat-headed to a domed state, and confirms the synonymy of Ornatotholus browni as a juvenile Stegoceras. This dataset serves as the first detailed model of growth and variation in a pachycephalosaur. Flat-headed juveniles possess three characters (externally open cranial sutures, tuberculate dorsal surface texture, and open supratemporal fenestrae) that are reduced or eliminated during ontogeny. These characters also occur in putative flat-headed taxa, suggesting that they may also represent juveniles of domed taxa. However, open cranial sutures and supratemporal fenestrae are plesiomorphic within Ornithischia, and thus should be expected in the adult stage of a primitive pachycephalosaur. Additional lines of evidence will be needed to resolve the taxonomic validity of flat-headed pachycephalosaur taxa.