Current events Charles A. Lockwood Randomization procedures and sexual & Brian G. Richmond dimorphism in Australopithecus Doctoral Program in Anthropological afarensis Sciences, State University of New York, Stony Brook, NewYork 11794-4364, U.S.A. William L. Jungers Department of Anatomical Sciences, State University of New York, Stony Brook, New York 11794-8081, U.S.A. William H. Kimbel Institute of Human Origins, 1288 Ninth St, Berkeley, California 94710, U.S.A. Journal of Human Evolution (1996) 31, 537–548 Randomization procedures have recently been used in a variety of applications in the field of paleoanthropology, centered on the assessment of taxonomic composition and sexual dimorphism of fossil samples (e.g., Richmond & Jungers, 1995; Kramer et al., 1995; Cope & Lacy, 1995; Grine et al., 1996). In particular, Richmond & Jungers (1995) used the exact randomization method to assess the probability that the range of size and shape variation in specimens attributed to Australopithecus afarensis could be found in comparative samples of extant hominoids. To compare size, the exact randomization method involves computing all possible pairwise ratios of size variables in the comparative sample and determining what percentage of these ratios exceed the maximum ratio found in the fossil sample. Based on the available sample of mandibles, proximal femora and humeri attributed to A. afarensis, Richmond & Jungers (1995) concluded that the size range of variation for each element was so rarely exceeded by ratios between same-species pairs of extant hominoids that, if the Hadar remains are accepted to represent a single species, a degree of sexual dimorphism at least as great as that of the most dimorphic living apes (gorillas and orang-utans) is probable. While exact randomization is an appropriate procedure to determine the relative affinities between two fossil specimens, there are potential problems with focusing exclusively on the extremes of a sample without taking into account the presence of intermediate specimens. One alternative is to compare the total distribution of all pairwise comparisons among fossils with those among reference samples. However, because fossil specimens are differentially preserved, there is clearly value in finding a method with which the extremes of size in a given sample can be used to test the null hypothesis of a single species. Bootstrapping is a technique for estimating standard errors and confidence limits that involves resampling with replacement from the population (Efron & Tibshirani, 1993). Resampling from reference samples at sample sizes equal to that of the fossil sample is a form of bootstrapping that may be suitable for comparing extremes or, in some cases, statistics of variation such as the coefficient of variation (CV). Given the assumptions of independent and random sampling in the fossil record, the random samples created using this method approximate potential ‘‘fossil samples’’ for each extant hominoid species. 0047–2484/96/120537+12 $25.00/0 ? 1996 Academic Press Limited 538 . ET AL. Using this more conservative, and probably more appropriate, resampling procedure, the present study compares size variation in A. afarensis with that of the same reference samples used by Richmond & Jungers (1995). We ask the question, what is the probability of sampling a set of n individuals from an extant hominoid species whose size variation is greater than that present in a sample of n individuals of A. afarensis? In other words, we acknowledge that more than two individuals contribute to the pattern of variation in the fossil assemblage (cf. Kramer, 1993; Cope & Lacy, 1995). The corresponding null hypotheses are that for each hominoid reference taxon and for each skeletal element, the size variation in A. afarensis does not exceed that of the reference group. Rejection of the null hypothesis for each group would support an alternative hypothesis that multiple species are included in the Hadar sample. Rejection of the null hypothesis for some but not all groups would support the hypothesis of a high degree of sexual dimorphism in A. afarensis relative to those groups whose size variation is less. Methods The mandibular corpus, the proximal femur and the humerus are used here to examine sexual dimorphism. The measurements of these bones and the reference samples of modern hominoids are described in Table 1 of Richmond & Jungers (1995). In brief, the reference samples are from multiple populations or subspecies of Homo sapiens, Gorilla gorilla, Pan troglodytes, and Pongo pygmaeus. Total sample sizes for each element in each hominoid species range from 48–50 for the mandible and humerus, and from 31–41 for the proximal femur. Corpus breadth and height at P4/M1 are measured for mandibles. Eight measurements are used to describe the proximal femur. The size variable for each individual mandible or proximal femur is the geometric mean, which equals the nth root of the product of n measurements. The geometric mean, therefore, condenses the size information from a number of variables into a single dimension (Mosimann, 1970; Jungers et al., 1995). For comparisons of humeri, the size variable is simply total humeral length. Before we address questions of probability, it is useful and appropriate to establish some estimate of average sexual dimorphism in A. afarensis for the elements being considered. This is done herein only for the mandibles of A. afarensis, as the sample size and attendant accuracy are higher than for the other elements. To estimate the average degree of sexual dimorphism in the mandible, indices of sexual dimorphism (ISDs) are first calculated for the extant hominoids. The ISD is the ratio of male to female mean values for geometric means. If the CVs of geometric means for the mandibular corpora are correlated with the ISDs for our reference samples of extant hominoids, it may be postulated that the CV for the Hadar sample can be directly compared with those of extant hominoids in order to ascertain the degree of sexual dimorphism in A. afarensis. In addition, a mean index of sexual dimorphism may be estimated from the CV by a reduced major axis regression analysis analogous to those used by Fleagle et al. (1980), Kay (1982) and Leutenegger & Shell (1987) to estimate degrees of sexual dimorphism in the canines of extinct anthropoid species without a priori knowledge of specimen gender. Plavcan (1994) suggests that this method may overestimate the degree of sexual dimorphism present in a species with low sexual dimorphism. This is a fault common to most estimators of sexual dimorphism (see also Josephson et al., 1996), and we regard the resampling procedures described below as another opportunity to test the validity of the CV-based estimate of sexual dimorphism. A. AFARENSIS 539 The method of hypothesis testing used here is a resampling method commonly referred to as bootstrapping. Bootstrapping was developed to calculate standards errors for unconven- tional statistics, although the term is also applied to general methods of resampling to determine the validity of a result from a single sample (Efron, 1979; Efron & Tibshirani, 1993; Manly, 1991; Sokal & Rohlf, 1995). We use bootstrapping to simulate random samples of extant taxa comparable in sample size with those of the fossil record. By assessing the probability of finding a certain degree of size variation in a modern taxon, it is possible to determine the validity of directly comparing statistics of size variation. The assumption that these random and independent samples represent a legitimate set of comparisons for the fossil record may be criticized on various grounds. For reasons related to the socioecological structure of the group being sampled and the type of fossil deposition, a fossil sample may be biased toward one sex or the other or toward one end of the body size range. However, because there is no a priori reason to suspect a specific bias at Hadar, we have chosen not to explore these alternatives here. Another consideration is that the time depth represented in some fossil assemblages could artificially increase the apparent range of variation. As suggested by Richmond & Jungers (1995), the inclusion of multiple subspecies or populations in the extant reference samples may address this problem in part. For each skeletal element, 1000 random samples of geometric means are selected with replacement from each of the extant hominoid groups. This is a sufficient number of replications to detect significance at the P=0·05 level for all but borderline cases (Manly, 1991; Efron & Tibshirani, 1993). Two statistics of variation are used: the max/min ratio and the CV. The bootstrap procedure is performed for pairs of size variables (for max/min ratios) and for sample sizes equal to the number of specimens in the hypodigm of A. afarensis for each element (for max/min ratios and CVs). Pairwise sampling of max/min ratios should produce results similar to the exact randomization procedure used by Richmond & Jungers (1995). Using larger sample sizes in the bootstrapping analysis should lead to larger ranges of variation in the extant hominoids (Cope & Lacy, 1995). The maximum sample size for fossil specimens is 17 for the mandible, five for the proximal femur, and three for the humerus. The sample of 17 adult mandibles whose corpus height and breadth can be measured at P4/M1 includes eight undescribed specimens. The measurements for two of these, A.L. 444-2b and A.L. 417-1a, were also used by Richmond & Jungers (1995), but six other undescribed specimens have been added for the current study. The largest mandibular corpus is that of A.L. 438-1g (undescribed), and the smallest is that of A.L. 207-13. For the proximal femur, the sample size of five contains those specimens of A. afarensis for which absolute size in the dimensions used here can be compared, although the full suite of measurements may not be available. These are A.L.