Scaling of the Humerofemoral and Intermembral Indices
The relationship between body mass and humerus length is remarkably consistent among modern humans and African apes (Jungers, 1994). Humerus length divided by the cube root of body mass does not distinguish among gorillas, chimpanzees, bonobos, African Pygmies and largerbodied humans; rephrased in a slightly different manner, the interspecific loglog scaling relationship between these two variables is highly significant and isometric. This perhaps surprising result indicates that humerus length itself can be used as a valuable surrogate for body mass or overall body size. I take advantage of this empirical finding to examine the scaling relationship between the humerofemoral and intermembral indices and humerus length (much like Lovejoy [1993] did with femur length) in order to evaluate the hypothesis that these interlimb indices are necessarily inversely correlated with "size" in humans.
Figure 9.3 is a bivariate plot of the humerofemoral index on humerus length in the total sample of 314 modern humans. Figure 9.4 is a similar plot of the intermembral index on humerus length. In both plots, a nonparametric Loess line (tension = 0.5) is provided to guide the eye through a scatter
Fig. 9.5 Bivariate lnln plot of femur length (mm) on humerus length (mm). The parametric correlation coefficient is 0.9, and the reduced major axis (RMA) slope is 1.033. The 95% confidence interval includes the value of 1.0, indicating thereby an isometric scaling relationship (as was demonstrated in a different way in Fig. 9.3).
Fig. 9.3 Bivariate plot of the humerofemoral index against humerus length (used here as a size surrogate, in mm). A Loess line is provided to guide the eye through the scatter of points. The relationship is not statistically significant as judged by a Spearman rank order correlation (rho). This indicates that that the humerofemoral index does not increase as size decreases (i.e., the relationship is isometric).
Fig. 9.4 Bivariate plot of the intermembral index against humerus length (in mm). The same conventions apply here as in Fig. 9.3. The intermembral index does not change in a predictable manner with size.
Fig. 9.5 Bivariate lnln plot of femur length (mm) on humerus length (mm). The parametric correlation coefficient is 0.9, and the reduced major axis (RMA) slope is 1.033. The 95% confidence interval includes the value of 1.0, indicating thereby an isometric scaling relationship (as was demonstrated in a different way in Fig. 9.3).
For those readers more familiar with the loglog regression approach to scaling, it is easy to demonstrate the equivalency of the two methods. Figure 9.5 is a bivariate plot of ln(femur length) on ln(humerus length) for the total sample. The correlation is 0.9 and the reduced major axis estimate of the slope is 1.033. The full equation is ln(FL) = 1.033 x ln(HL) + 0.14351
Fig. 9.4 Bivariate plot of the intermembral index against humerus length (in mm). The same conventions apply here as in Fig. 9.3. The intermembral index does not change in a predictable manner with size.
of points that are statistically uncorrelated (Spearman's rho = 0.095 in both). This lack of correlation between a shape variable (the index) and a size variable (humerus length) is what Mosimann and James (1979) define as "isometry." That is, shape fails to change in a predictable fashion with size. The Loess fit might suggest that both indices actually increase with humerus length initially and then change very little if at all afterwards. The important takehome message from this finding is that there is no increase in either index as size decreases. The predicted negative allometry between each index and size simply does not obtain for the total sample.
Permutationbased 95% confidence intervals on the slope are 0.981 and 1.086. In other words, the relationship is best described again as isometric, just as it was using the Mosimann and James methodology. The expectation of negative allometry derived from the sizerequired hypotheses is firmly rejected regardless of how one approaches the problem.
One might still wonder if the predicted negative scaling relationships might exist within more homogeneous human subsamples, especially among the smallest modern humans. I evaluate this possibility by calculating correlations between each index and humerus length within each of the seven ethnic samples, and the results are summarized in Table 9.2. In most groups, the correlations are nonsignificant (4/7 for the humerofemoral index, 5/7 for the intermembral index), indicating an essentially isometric relationship in which the index does not change in a predictable direction. Equally important, when there is a significant correlation, it is in the wrong direction for the allometric scenarios that predict higher indices in smaller humans. For example, both indices in the Andaman Islanders and Khoesan (among some of the smallest people on earth) are significantly positive; i.e., the indices increase with size rather than decrease. Accordingly, if one were to predict humerofemoral indices for LB1 and AL2881
Humerofemoral 
Intermembral index  
index and humerus 
and humerus  
Group 
length (rho) 
length (rho) 
African Pygmies 
0.073 
0.143 
Andaman Islanders 
0.613* 
0.674* 
Khoesan 
0.746* 
0.658* 
Zulu 
0.141 
0.196 
African Americans 
0.194 
0.223 
Sami 
0.330** 
0.244 
Inuit 
0.062 
0.019 
Total sample 
0.095 
0.095 
from the Andaman Islanders and Khoesan, the relevant equations would be:
Humerofemoral Index = 0.090 x Humerus Length
Humerofemoral Index = 0.089 x Humerus Length
The Andaman Islander equation predicts an index of 67.6 for LB1 and 67.2 for A.L. 2881; the Khoesan equation predicts 65.6 and 65.3, respectively. These values are obviously poor estimates of the known values (Table 9.1).
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