Estimating sexual size dimorphism in fossil species from posterior probability densities

Abstract

Accurate characterization of sexual dimorphism is crucial in evolutionary biology because of its significance in understanding present and past adaptations involving reproductive and resource use strategies of species. However, inferring dimorphism in fossil assemblages is difficult, particularly with relatively low dimorphism. Commonly used methods of estimating dimorphism levels in fossils include the mean method, the binomial dimorphism index, and the coefficient of variation method. These methods have been reported to overestimate low levels of dimorphism, which is problematic when investigating issues such as canine size dimorphism in primates and its relation to reproductive strategies. Here, we introduce the posterior density peak (pdPeak) method that utilizes the Bayesian inference to provide posterior probability densities of dimorphism levels and within-sex variance. The highest posterior density point is termed the pdPeak. We investigated performance of the pdPeak method and made comparisons with the above-mentioned conventional methods via 1) computer-generated samples simulating a range of conditions and 2) application to canine crown-diameter datasets of extant known-sex anthropoids. Results showed that the pdPeak method is capable of unbiased estimates in a broader range of dimorphism levels than the other methods and uniquely provides reliable interval estimates. Although attention is required to its underestimation tendency when some of the distributional assumptions are violated, we demonstrate that the pdPeak method enables a more accurate dimorphism estimate at lower dimorphism levels than previously possible, which is important to illuminating human evolution.

Keywords: Bayesian estimate; fossils; human evolution; mixture analysis; sexual dimorphism.

Fig. 1.

Variation in shape of population distributions with a constant CV of 10%. (A–D) Hypothetical population (combined-sex) distributions with the same overall variation (CV of 10%) are plotted for several subgroup conditions. Log-normal distributions of males and females with a common wsxCV are mixed in equal proportions. Overall CV is fixed at 10% and the female mean at 10 mm. The wsxCV is set in four ways: 8%, 7%, 6%, and 5% (from left to right). Solid black curves indicate overall distributions, dashed curves are the latent within-sex distributions, red is for female, and blue is for male. Vertical dashed lines indicate within-sex means. The same distributions are shown in E–H segregated to subpopulations by the mean. Vertical solid lines indicate the presumed sex means of the mean method (MM). The true male mean/female mean (m/f ratio), as well as the MM ratio (mean of presumed males divided by that of the presumed females), is shown below each wsxCV condition. Note that the MM increasingly overestimates as the male and female distributions increasingly overlap (from right to left). The Rd values (see Application to Actual Cases), i.e., the distance between means in within-sex SD units, are 1.6, 2.2, 2.9, and 3.8, respectively in A, B, C, and D. Note also that, under the same overall CV, the true m/f ratio can vary substantially depending on wsxCV.

Fig. 2.

Distribution of wsxCVs of canine crown diameters in extant anthropoids. Blue histograms are for male and red for female. The upper canines (N = 29) and lower canines (N = 31) of both sexes are plotted together (N = 120). The CVs are those of species/subspecies samples with 15 or more specimens available for each sex. The correction suggested by Sokal and Braumann (28) was used in calculating the CVs. Data are from Plavcan (29) and Suwa et al. (21). See SI Appendix, SI Text and Dataset S1 for further details and source information. Out of 120 samples, a total of 29 samples have a wsxCV lower than 5%, and 20 samples have a wsxCV higher than 8%.

Fig. 3.

Simulation results under conditions concordant to assumptions. The pdPeak method is evaluated and compared with the MM, BDI, and the CVM. The CVM is that of Plavcan (15). Three conditions were set for sample size N: 10 (left plots), 15 (middle plots), and 30 (right plots). Two conditions were set for wsxCV: 5% (Upper) and 8% (Lower). This resulted in six combinations. Under each combination, the m/f ratio was set in 0.3 increments indicated by the markers, and 2,000 samples were generated for each m/f ratio in each condition. True m/f ratio is on the x axis, and estimated values are on the y axis. Solid lines indicate the mean, and dotted lines are the 5th- and 95th-percentile values of the 2,000 samples. The pdPeak was the only method capable of estimating m/f ratios between 1.1 and 1.2 with little bias when wsxCV was 8%. Note that, with regard to the pdPeak, MM, and BDI estimates, the wsxCV 8% results correspond to enlarged portions of the wsxCV 5% plots. This is because these methods are directly affected by the distance between the two means relative to within-sex variance. This can be expressed by alternative expressions of the same measure, the relative distance (Rd) (difference of means divided by female SD) or the Rm/f ratio (“m/f ratio minus 1” divided by wsxCV) (see Applications to Actual Cases). For example, with an m/f ratio of 1.30 and wsxCV of 8%, the Rm/f ratio is 3.75, which corresponds to an m/f ratio of ∼1.19 and a wsxCV of 5%. Thus, the wsxCV 8% plots largely correspond to the lower ∼60% subset (m/f ratio of 1.0 to 1.19) of the wsxCV 5% plots. This is not the case with the CVM, which depends on a regression relationship between the m/f ratio and combined-sex CV.

Fig. 4.

Simulation results under conditions disconcordant to assumptions. Simulation tests were conducted on background populations that deviate from assumptions in eight ways: two heteroscedastic conditions (Mσ > Fσ and Mσ < Fσ); four skewed distributions (both positive; both negative; male negative and female positive [tail to tail]; and male positive and female negative [head to head]); and two unbalanced sex ratios (M:F = 7:3 and 3:7). N = 15, wsxCV = 8% (average wsxCV used when heteroscedastic). Top Left reproduces the concordant condition to facilitate comparisons. See Fig. 3 for explanation of axes, lines, and symbols. The effects of deviation from assumptions are generally not conspicuous, except in the head-to-head and unbalanced sex ratio conditions.

Fig. 5.

Applications to extant anthropoid canine samples of known sex. The panels summarize results of the pdPeak and other methods applied to 110 extant anthropoid samples. The samples varied in size (N) from 5 to 89 and were separated into three categories. See SI Appendix, SI Text and Dataset S3 for sample and data information. Upper row shows the m/f ratio estimates (y axis) against the actual sample values (x axis). Note that the pdPeak estimates cluster around the y = x diagonal, especially when the sample size is large, whereas the other methods tend to overestimate when the m/f ratio is <1.2. The few cases of relatively large underestimation of the pdPeak estimate involve samples with large sample wsxCVs and, hence, with relatively larger uncertainties (see Applications to Actual Cases). Lower row shows the estimated and actual sample values of the quantity “m/f ratio minus 1” divided by sample wsxCV (male and female average), termed the Rm/f ratio (see Applications to Actual Cases). The y axis deviation from the y = x diagonal expresses difference between estimated and actual m/f ratios standardized by wsxCV. Lower row shows a pattern that conforms to what is expected from the simulation tests (Fig. 3); the pdPeak estimates are unbiased when the Rm/f ratio is >∼1.5, while MM and BDI overestimate when the Rm/f ratio is <∼2.5. The CVM is not shown in the Lower row because there are no general relationships between the CVM and the Rm/f ratio. The interrupted vertical lines indicate the 95% credible intervals of the pdPeak estimates. Note that the 95% credible intervals encompass the sample values in all cases. Note also that the pdPeak credible intervals broadly match the sample m/f ratio CIs of standard statistics (Dataset S3), but with a tendency to be wider (more conservative). From this, it is seen that the pdPeak credible intervals are reliable, even with small sample sizes.

Fig. 6.

The pdPeak method applied to the lower canine crown diameter of Ar. ramidus. Three runs were performed on the maximum canine diameter of different sample compositions. (A) Gona specimens (N = 5). (B) Middle Awash specimens (N = 6). (C) Both combined (N = 11). In each panel, Left shows the bivariate probability distribution of wsxCV (x axis) and the m/f ratio (y axis). Probability densities were obtained with the log-transformed data, and the axis labels were back-transformed to original scale (SI Appendix, SI Text). Dotted contour lines are the combined-sex CV levels. The red and yellow squares show distribution of the extant great ape and modern human reference data (SI Appendix, Table S1; the weighted wsxCVs of the males and females were used). In Right is the marginal posterior density plot of the logarithm of the m/f ratio, with a red diamond marker at its peak and dotted horizontal lines at the 68% and 95% credible interval levels.