MATERIALS AND METHODS

Twelve specimens of Protypotherium preserving postcranial bones were examined at The Field Museum, Chicago (FMNH) (Table 1). Due to the lack of a complete Protypotherium skeleton in the collections, measurements from 10 specimens were combined (averaged) to provide a complete dataset for the multivariate analyses described below. Only one of these specimens had been identified to species (FMNH PM 13235, P. australe), but no differences in postcranial morphology have been described among Protypotherium species that would be expected to affect these analyses. Functional indices were calculated before averaging and were only recorded for specimens in which all necessary variables could be measured (i.e., indices were not calculated from mean values).

The body mass of Protypotherium has been estimated at 5-10 kg based on limb bone scaling relationships (Elissamburu 2004; Anderson and Croft 2006; Anderson 2007). Owing to the lack of extant notoungulates, Protypotherium was compared to a variety of mammals of similar size from FMNH and Cleveland Museum of Natural History (CMNH) zoology collections including: artiodactyls (10 genera), carnivorans (11 genera), caviomorph rodents (12 genera), hyraxes (3 genera), and lagomorphs (3 genera) (Appendix). One to three adults of each genus were measured, depending on the number of available skeletons; in total, 80 individuals were studied. For most genera (28/29), all individuals were of a single species. Each genus was placed into one of five locomotor categories based on its primary mode of locomotion (Appendix): (1) arboreal (climbers plus scansorial mammals); (2) semifossorial (non-subterranean diggers); (3) generalized (terrestrial, non-cursorial non-diggers); (4) bounding (quadrupedal jumpers, leapers, bounders, and hoppers); and (5) cursorial (quadrupedal runners).

For fossil and modern specimens, 11 measurements were taken to the nearest tenth of a millimeter using digital calipers (Figure 2; Table 2). These particular measurements were chosen because they encompass the chief lever arms and muscle insertion areas important for locomotor performance. The data were not log transformed because the species under examination span a relatively small range of body sizes (one order of magnitude). From these measurements, eight indices were calculated that describe characteristics of bones (and muscles) related to limb function (cf. Hildebrand 1985; Carrano 1997, 1999; Vizcaíno et al. 1999; Vizcaíno and Milne 2002; Elissamburu and Vizcaíno 2004; Shockey et al. 2007) (Table 3). Such indices are useful for minimizing the effects of body size and for grouping species into broad locomotor categories, but care must be taken when comparing such indices among phylogenetically disparate species; distantly related mammals of similar locomotor habit do not necessarily have equivalent index values (Garland and Janis 1993). As described by Carrano (1999), because these indices (ratios) express unique mechanical relationships and do not have a common denominator, they likely do not suffer from the statistical shortcomings that affect ratio data in other types of analyses. All indices in this study are expressed as percents (i.e., * 100).

The locomotor habit of Protypotherium was assessed using qualitative attributes of the postcranial skeleton and quantitative multivariate analyses of: (1) linear postcranial measurements; and (2) functional indices computed from those measurements. Multivariate analyses included principal components analysis (PCA) and discriminant function analysis (DFA), both executed using SPSS 11.0 statistical software.

PCA reduces a large number of variables into a smaller number of factors (principal components; PCs) that retain most of the variance observed in the original dataset; the smaller number of factors (relative to variables) facilitates comparisons of similarity among species. PCA is generally employed as an exploratory technique in ecomorphological studies, permitting extinct species to be compared with extant ones using only two or three axes (e.g., Spencer 1995; Woodman 1995; Gingerich 2003, 2005; Van Valkenburgh et al. 2003; Andersson 2005; Woodman and Croft 2005; Weisbecker and Warton 2006; Shockey et al. 2007). Interpreting PCs (axes) requires examining loadings (i.e., relative contribution) of variables. Because all variables usually load highly on PC1, it is often interpreted as summarizing variation due to body size. Subsequent PCs may summarize mostly taxonomic or ecomorphological variation, depending on which variables load highly and how species are distributed along the PC axis.

DFA attempts to sort cases into pre-determined groups using variables believed to be correlated with group membership. In ecomorphological studies, variables are generally osteological (i.e., tooth or bone measurements) and groups are generally behavioral (e.g., locomotor habit, diet). DFA differs from PCA in that the discriminant functions (DFs, analogous to PCs) preferentially weight variables based on how well they discriminate among groups; such discriminatory power is assessed using variable means and variances for each group. Variables with low variances may therefore be important in DFA and relatively unimportant in PCA, if they are good at discriminating between groups. As in PCA, a score for each species can be calculated for each DF. If discrimination among groups is good, the scores of species in one group will differ significantly from those in other groups on at least one DF axis. This is usually evident in discriminant plots; with good discrimination, little overlap exists between groups, and group centroids are clearly separated from one another. In reality, a DFA is rarely able to separate all cases; the percentage of cases classified correctly is a general measure of its effectiveness.

Once DFs have been generated using a 'training' sample (e.g., extant mammals), scores can be calculated for unknowns (e.g., extinct mammals) and used to infer group membership. The probability that an unknown belongs to a particular group (the 'posterior probability') is inversely proportional to the distance in discriminant space between the unknown and the group's centroid. The unknown therefore has the greatest probability of pertaining to the group with the closest centroid (at least when prior probabilities – the a priori probabilities of group membership – are equal). Based on the dispersion of group members around the centroid in discriminant space, a 'conditional probability' is also calculated; this is the probability that an unknown pertains to the group, given the characteristics of other group members. If the unknown falls well within the cloud of other group members in discriminant space, the conditional probability is high; if the unknown falls well outside the cloud, the conditional probability is low. In a general sense, the conditional probability describes the degree to which the unknown resembles other members of that group. It is only computed for the group to which the unknown most likely pertains. In contrast, posterior probabilities are computed for all groups. The position of the unknown relative to other cases (and groups) can be assessed visually using DF plots, which are analogous to PC plots.

In this study, significance levels of DFA variables and functions were assessed using Wilks' Lambda (p < 0.05). Prior probabilities for group membership were considered equal (i.e., it was assumed equally likely that Protypotherium could pertain to any locomotor group).