Thinking

Ruff’s Surface Law: Evidence from the Holocene

In the previous dispatch we argued that throughout the Holocene, stature was not correlated with latitude and mean temperatures, but BMI was. Consequently, the gradient of stature along latitude in modern data encodes information on global polarization in everyday living standards. We can thus interpret the cross-sectional variation in modern stature as reflecting the differential health transition in the mid-twentieth century. Conversely, the interpretation of cross-sectional variation in BMI is confounded by the fact that BMI is a linear function of pelvic bone width, and pelvic bone width encodes information on the very long run morphological adaptation of geographically-clustered populations to the macroclimate in accordance with Ruff’s surface law. In what follows we introduce a new anthropometric measure that captures this morphological adaptation. And we refine our empirical strategy in light of our lack of precise knowledge of the antiquity of the fossils in the Goldman Osteometrics Dataset.

Ruff’s surface law asserts that because body heat is a function of body mass (or volume) and the rate of heat dissipation is a function of surface area, in colder climes, surface area ought to be minimized relative to volume; conversely, in warmer climes, surface area ought to maximized relative to volume. We operationalize this observation by constructing a new metric that we label Ruff’s ratio defined as the ratio of surface area in squared centimeters (SA) exponentiated to the power 1.5, to body mass in kilograms (BM),

\textup{Ruff's ratio}:=\frac{\text{SA}^{1.5}}{\text{BM}},

where we have chosen the power of the numerator to equalize the dimensions of the two quantities. If Ruff’s surface law holds for human beings, Ruff’s ratio should be a strong correlate of latitude. Ruff argued that the surface area of the human body is well-approximated by the area of the curved surface of a cylinder with diameter equal to pelvic bone width and height equal to stature. We therefore estimate the surface area via the formula,

\text{SA}:=\pi\cdot BW\cdot ST,

where BW is pelvic bone width and ST is stature; both measured in centimeters. Table 1 presents summary statistics for our three sex-combined variables.

Table 1. Summary statistics. 
Stature (cm) BMI Ruff’s ratio
Mean 158.7 22.0 27.0
SD 8.698 1.709 0.778
CV 0.055 0.078 0.029
Source: Goldman Osteometric Dataset, author’s computations.

Note that the coefficient of variation of Ruff’s ratio is extremely low. Since there is such little variation in this metric, small differences in Ruff’s ratio between populations can be expected to be correspondingly significant.

Before we examine the empirical correlations it must be observed that the location of the fossils is not always a good indication of population geohistory. Specifically, it may well be the case that New World populations were not fully adapted to the macroclimate of the location where the fossil was recovered. We will therefore report both full sample estimates and those obtained by restricting the sample to the Old World. Also, Spearman’s rank correlation is a more robust measure in the presence of heteroskedasticity than the standard Pearson correlation coefficient; as will be clear from the stability of the estimates. For the same reason we report robust standard errors for our gradient estimates.

Table 2 reports Spearman’s rank correlation coefficients for latitude. The results are consistent with our hypothesis. Moreover, Ruff’s ratio is a stronger correlate of latitude than BMI in accordance with Ruff’s surface law. Spearman’s rho for Ruff’s ratio is 25% higher than the same for BMI in the full sample. Note also the extraordinary stability of the coefficient for Ruff’s ratio vis-à-vis BMI when we restrict the sample to the Old World. While Spearman’s correlation coefficient for BMI jumps by 15% when we restrict the sample to the Old World (consistent with our interpretation of the New World data points being noiser), the coefficient for Ruff’s ratio is unchanged to the third decimal point. Ruff’s ratio is therefore a less noisy measure of climatic adaptation than BMI.

Table 2. Spearman’s rank correlation coefficients for latitude.
Stature BMI Ruff’s ratio
Full sample (N=45) 0.211 0.493 0.616
pVal 0.164 0.001 0.000
Old World (N=27) 0.218 0.567 0.616
pVal 0.274 0.003 0.001
Source: Goldman Osteometric Dataset, author’s computations.

Table 3 reports Pearson’s correlation coefficients. Stature is now a (just barely) significant correlate of latitude in the full sample. But not after we restrict the sample to the Old World. Also, consistent with our interpretation, BMI is a more significant correlate of latitude once we restrict the sample to the Old World. Meanwhile, Ruff’s ratio remains unchanged to the second decimal. Note that Pearson’s correlation coefficient is less reliable than Spearman’s in the presence of heteroskedasticity, which probably explains why it shows Ruff’s ratio to be only a weakly stronger correlate of latitude than BMI.

Table 3. Pearson’s correlation coefficients for latitude.
Stature BMI Ruff’s ratio
Full sample (N=45) 0.298 0.516 0.548
pVal 0.047 0.000 0.000
Old World (N=27) 0.331 0.544 0.554
pVal 0.092 0.004 0.003
Source: Goldman Osteometric Dataset, author’s computations.

Table 4 displays our OLS gradient estimates and robust standard errors. Stature again just barely clears the threshold for significance in the full sample, while falling into insignificance in the restricted sample. Meanwhile, latitude is strongly priced into BMI and Ruff’s ratio. And the gradient estimate exhibit absolute stability when we restrict the sample. Latitude explains roughly 10 percent of the variation in stature, 25-30 percent of the variation in BMI, and 30 percent of the variation in Ruff’s ratio. Note also the extraordinary stability of the parameter estimates for our last two variables when we restrict the sample to the Old World.

Table 4. Gradients of latitude. 
Stature (cm) BMI Ruff’s ratio
Full sample (N=45)
Gradient 0.083 0.047 0.020
Robust std error 0.049 0.014 0.006
pVal 0.048 0.001 0.001
R^2 0.089 0.267 0.300
Old World sample (N=27)
Gradient 0.094 0.047 0.021
Robust std error 0.058 0.015 0.007
pVal 0.057 0.002 0.002
R^2 0.109 0.296 0.307
Source: Goldman Osteometric Dataset, author’s computations.

Again it is very hard to reject our hypothesis. The evidence marshaled in the present and previous dispatch are consistent with the hypothesis that climatic information is encoded in body shape rather than stature. We have seen thus far that both BMI and Ruff’s ratio are strong correlates of latitude. The Spearman coefficients in Table 1 suggest that the latter is a stronger correlate of latitude than BMI in accordance with Ruff’s surface law. But the evidence from least squares is more equivocal in this regard. In order to test this more carefully, we run a horse race between our three variables. Specifically, we run “reverse regressions” with latitude as the dependent variable and see “who kills whom” among our three regressors. Table 5 displays our estimates.

Table 5. Reverse multivariate OLS regressions.
Dependent variable is latitude. All regressors have been standardized to have mean 0 and variance 1.
N=44 Estimate Robust Std Error pVal
Model 1 Const 33.72 2.22 0.00
Stature 3.05 2.44 0.11
BMI 8.02 2.43 0.00
Model 2 Const 33.67 2.21 0.00
Stature 0.58 2.84 0.42
Ruff’s ratio 9.44 2.59 0.00
Model 3 Const 33.65 2.18 0.00
BMI 4.43 3.44 0.10
Ruff’s ratio 6.53 3.21 0.02
Model 4 Const 33.67 2.16 0.00
Stature 0.89 2.62 0.37
BMI 4.51 3.42 0.10
Ruff’s ratio 5.93 3.26 0.04
Source: Goldman Osteometric Dataset, author’s computations.

A very consistent picture emerges from the multivariate regression estimates displayed in Table 5. First, BMI kills stature (Model 1). Second, Ruff’s ratio kills stature even more thoroughly (Model 2). Compare the coefficient and standard error of stature in Model 1 and 2. The gradient estimate for stature falls from three to one-half and the pValue jumps from 0.11 to 0.42. Third, and most importantly, Ruff’s ratio kills BMI (Model 3). This is also true of the “kitchen sink” regression (Model 4). In sum, neither stature nor BMI is a statistically significant correlate of latitude once we control for Ruff’s ratio. This is very strong evidence indeed in favor of Ruff’s surface law.

The following two graphs display the variation in Ruff’s ratio across Holocene populations labeled by their modern location.

Ruff_ratio_boxplot.png

Figure 2. Boxplot for Ruff’s ratio. Source: Goldman Osteometric Dataset, author’s computations.

Ruff_ratio_scatter.png

Figure 2. Latitude and Ruff’s ratio. New World observations in red; Old World in blue. Source: Goldman Osteometric Dataset, author’s computations.

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