Discussion

Results for each taxon are summarized below and in Table 7.

Coefficients of variation yielded the initial estimates in determining the degree of variability between the intermembranous and endochondral bones of the species in this study. The majority of observations CVs are between 4 and 10, and 5 and 6 are the average values. Values lower than 4 usually mean that the sample size was not sufficient to show the variability. Table 1 shows that the CVs of the Paramylodon harlani and Nothrotheriops shastensis all fall under 10, which initially indicated that neither the intermembranous bones nor endochondral bones are highly variable. There is no consistent pattern of astragalus CVs being lower than patellar CVs, as would be predicted by our hypothesis. This is true even for the small inner and outer sesamoid bones, which might be expected to be the most variable in the skeleton (as they are in examples given below).

F-tests performed on Paramylodon harlani determined that variances between astragali and patellae widths and depths were unequal. The t-tests performed on Paramylodon harlani, as shown in Table 3 and Table 4, indicate that the means are all statistically different because the t-stat values fall outside of the range of t-critical values. Therefore, the null hypothesis that the means are equal is rejected. All p-values in Paramylodon harlani t-tests also fell below the 0.05 significance level; this result strongly rejects the null hypothesis. ANOVA results (as shown in Table 5) give f-values greater than f-critical values; this also rejects the null hypothesis of equal means and suggests a statistically significant difference. All results except the width dimensions indicate that the intermembranous bones are statistically different than the endochondral bones of this species. However, Levene's test (Table 6) indicated that none of the dimensions of the patellae were significantly more variable than those of the astragali.

CVs for the comparison of astragali and patellae of the Smilodon fatalis are all under 10, but the sesamoids have CVs of 11.71 and 12.11 (Table 1). All dimensions of measurement of the patellae and sesamoids (the intermembranous bones) had CVs greater than CVs of measurements of corresponding astragali, which is consistent with our hypothesis. T-test results reject the null hypothesis of equal means because all values of the t-stat fall outside of the range of t-critical values (Table 3 and Table 4). P-values also all fall below the 0.05 confidence interval; this strongly rejects the null hypothesis. In addition, because ANOVA yielded an f-value larger than the f-critical value in the comparison of astragalus to patella depth, the difference is considered statistically significant (Table 5). The same was true using Levene's test (Table 6). All results indicate that the endochondral bones are statistically different than the intermembranous bones in this species. In the case of Smilodon fatalis, intermembranous bones are consistently and significantly more variable than endochondral bones.

The CVs of Panthera atrox generated the most surprising results. CVs of patella length and depth were lower than those of astragalus length and depth. This implies that the patellae of this species are less variable than the astragali (Table 1). T-statistic values were outside of the range of t-critical value, which indicates that the means are statistically different, and the null hypothesis is rejected. Only the p-values of comparison of astragalus to patella depth falls below the significance level of 0.05, which strongly rejects the null hypothesis of equal means. However, p-values for comparison of astragalus to patella length and width are above the 0.05 significance level; this signifies that the results do not strongly reject the null hypothesis and therefore, may be due to a random sampling error. ANOVA results displayed an f-value larger than the f-critical value in the comparison of maximum depth. This indicates that the results are statistically significant, but contradict the prediction of our hypothesis. Levene's test, however, showed that none of these dimensions of astragali were statistically different from the corresponding dimensions of the patellae (Table 6).

The CVs of all dimensions of the patellae of Bison antiquus are higher than those of the astragalus, which agrees with our hypothesis (Table 1). In the f-tests (Table 2), the f values were always greater than the f-critical value, so the variances are not equal. In the t-tests assuming unequal variances, all three t values fell outside the critical range, so the means are statistically different. Thus, all three dimensions are consistent with our hypothesis and statistically significant. Using the modified Levene test (Table 6), the widths of patellae are not significantly more variable than those of the astragali, but the length and depth are significantly more variable.

The CV values of the horses are higher for the patella than for the astragalus in two dimensions (length and depth), consistent with our hypothesis of greater intermembranous bone variability, but not for the width measurements. The variances also show the same trend (Table 1). In the f-test (Table 2), two dimensions of the horse bones (length and depth) have unequal variances, but the third (width) has equal variances. The t-test (Table 3 and Table 4) shows that these differences are significant, as does the ANOVA (Table 5). Thus, the intermembranous bones of the horse are significantly more variable than the endochondral bones in two dimensions (length and depth), but not so in the third dimension (width). The modified Levene test (Table 6) gives the same result.

As mentioned above, the CVs of the patellae of C. hesternus are lower than those of the astragali for the length and depth measurements, but not for the width (Table 1). Using the f-test, the variances of all three dimensions were statistically equal, and the t-test (Table 3) shows that the means are also significantly different because they fall outside the range of critical values. In the ANOVA analysis (Table 5), the length dimension was significantly different, but not the width or depth dimensions. However, the modified Levene test (Table 6) shows that none of the dimensions of the patellae are significantly more variable than those of astragali. Thus, C. hesternus gives mixed results, with some that are consistent with our hypothesis, and others that are not.