Results

The f-statistic tests a given sample for normal probability distribution. Once all samples of this study were ranked, given a z-score (individual raw score minus population mean) and plotted on bivariate plots with regression lines, we determined that all samples were normally distributed. Because all of the distributions turned out to normal, coefficients of variation, t-tests, and ANOVA may be used if other standards are met.

As previously discussed, the f-test for equal variances is necessary in order to determine which type of t-test (assuming equal or unequal variances) to use, and whether or not standards for ANOVA are met. Results of the f-tests (Table 2) indicate that the comparison of variances of astragali to patellae of Paramylodon harlani length, Smilodon fatalis depth, Panthera atrox depth, Camelops hesternus length, width and depth, and Equus occidentalis width meet qualifications for further analysis through ANOVA and t-tests assuming equal variances because the f-values are smaller than the f-critical values for these samples. Thus, in the aforementioned measurement comparisons, the null hypothesis is accepted. In addition, the p-values for the Paramylodon harlani length, Panthera atrox depth, Equus occidentalis width, and Smilodon fatalis depth are greater than the 0.05 significance level.

In all other sample measurements (Paramylodon harlani width and depth, Smilodon fatalis length and width, Equus occidentalis length and depth, Bison antiquus length, width, and depth, and Panthera atrox length and width), f-values are larger than the f-critical values. This forces us to reject the null hypothesis of equal variances. Therefore, ANOVA is not applicable to these measurements, and only t-test assuming unequal variances may be used to further analyze this data. Also, p-values for all remaining samples are less than the 0.05 significance level. This provides strong evidence that the null hypothesis is not true for these samples.

In some cases (Equus occidentalis length and depth, Panthera atrox length and width), the variances (Table 1) of the astragalus (an endochondral bone) are smaller than those of the patella (a intermembranous bone), which is consistent with our hypothesis. In other cases (Paramylodon harlani width and depth, Bison antiquus length, width, and depth, and Smilodon fatalis length and width), the variances of the astragalus are larger than those of the patella, which is not consistent with our hypothesis.

Preliminary tests for equality of variances indicate that the variances between astragali and patellae in Paramylodon harlani length, Smilodon fatalis depth, Camelops hesternus length, width and depth, Equus occidentalis width, and Panthera atrox depth are statistically equal. Two-sample t-tests assuming equal variances were performed and results calculated (Table 3). All equal variance results except the B. antiquus depth and the C. hesternus width and depth have p-values that are less than the significance level of 0.05, and t-stat values fall outside of the range of t-critical values.

For the variances that were determined to be unequal (Paramylodon harlani width and depth, Smilodon fatalis length and width, Equus occidentalis length and depth, Bison antiquus length, width, and depth, and Panthera atrox length and width), two-sample t-tests assuming unequal variances were performed and results tabulated (Table 4). All results from t-tests assuming unequal variances fall outside of the range of t-critical values. All P-values for these taxa except for length and width in Panthera atrox and the depth measurement of Bison antiquus are less than the 0.05 significance level.

Analysis of variance was used to analyze how much of the variation among the observations is due to variation in each bone type influencing the character (measurement value) being studied. Most of the samples were determined to meet the standards for ANOVA: independent variables, normal distribution (calculated by f-stat), and equal variances (calculated by f-test). These samples, which include comparisons between the astragali and patellae of the Paramylodon harlani length, Smilodon fatalis depth, Camelops hesternus length, Equus occidentalis width, and Panthera atrox depth, yielded f-values higher than f-critical values in each case (Table 5), so the differences are significant. However, the depth and width dimensions of Camelops hesternus were below the f-critical value, so they are not significantly different.

As discussed above, the modified Levene test (Levene 1960; Plavcan and Cope 2001) is a popular method for testing for equal variability while using natural logs to correct for absolute size. Using ANOVA, we found that depth and length of bison and horse patellae were significantly more variable than their corresponding astragali, but not width (Table 6). None of the camel, sloth, or lion bones showed significantly more variability of patellae compared to astragali. However, all of the dimensions of Smilodon patellae were significantly more variable than those of the astragali.

As shown in Table 1, most of the coefficients of variation (20 out of 27) are higher for intermembranous bones than for their corresponding measurements of the endochondral bones. CVs from the samples of Panthera atrox are less than 10 with the exception of the maximum width of the astragalus, which has a CV of 11.95. However, the CVs of the endochondral bones (astragalus) are generally higher than those of the intermembranous bones (patella), in contrast to our expectations. Smilodon fatalis bones have CVs within the range of 5-13. The CVs of the endochondral bones of this species are all less than 10. Smilodon fatalis patellae CVs are also under 10. However, all three dimensions measurements of the carpal sesamoids in this species demonstrate CVs greater than 10. In addition, the CVs of both intermembranous bones (patella and sesamoids) are consistently higher than those of the astragalus, as expected in our hypothesis.

In both species of ground sloths (Paramylodon harlani and Nothrotheriops shastensis), all CVs are in the range of 3-10. The highest coefficient of variation in the order Xenarthra is the maximum depth of the outer sesamoid of digit 3 of the Paramylodon harlani with a value of 9.71. However, the CVs of the endochondral bones are in the same range as those of the intermembranous bones and not consistently lower. The sample size for Nothrotheriops shastensis is much smaller, but similar trends are observed. Mean, standard deviation, and coefficient of variation were not calculated for the Nothrotheriops shastensis patellae as only a few specimens were available.

In Camelops hesternus, the CVs were unusually high, with three values above 10 (Table 1). In contrast to our predictions, two out of three CVs (length and depth) of the astragalus were consistently higher than those of the patella. In Bison antiquus, the CVs of the astragalus were slightly lower than those of the patella, consistent with our initial hypothesis. The same is true of Equus occidentalis (Table 1).

Summarizing the CV data, 20 out of 27 of the comparisons of the CV of an intermembranous bone to the endochondral astragalus yield a higher CV for the intermembranous bone, in agreement with our prediction. Many of our taxa (Equus occidentalis, Bison antiquus, and Smilodon fatalis) show the predicted trend of greater CVs in intermembranous bones than in endochondral bones. The data from Paramylodon harlani are equivocal, with overlapping values of CVs for intermembranous and endochondral bones. But two of the taxa (Camelops hesternus and Panthera atrox) seem to have higher CVs in their endochondral bones (astragalus) than in their intermembranous bones.