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Relations between body mass, height, fat mass, and waist circumference in American and Korean men and women.

Dear Editor: 
 
Heymsfield et al. (1) have confirmed that the squaring of height in calculating BMI is well justified. For large samples of men and women—non-Hispanic whites, non-Hispanic blacks, Mexican Americans, and Asian Koreans, they separately related body mass [body weight (BW)], fat mass (FM), and waist circumference (WC) to heightβ, with β estimated by logarithmic regression. Some β values were adjusted for age and/or for percentage of fat (%fat) by including these with height as independent variables in the regressions. Here I distinguish the β values by subscripts so that, for example, βBW means β for BW. In most groups βBW (whether or not adjusted for age, but not for %fat) was lower in women than in men, as is generally true (2, 3). Here I integrate various ideas and quantitative evidence, extracting further information from the β values. The correlation coefficients, r, are all significant (P < 0.01). 
 
Adjustment for age was important. It increased βFM for all samples (Table 1, line 1), and presumably because BW includes FM, it increased βBW also, but to a lesser extent (line 2). Age had a striking influence in Korean women, probably because of generational differences in nutrition (1). 
 
 
 
TABLE 1 
 
Differences between some β values tabulated by Heymsfield et al.1 
 
 
 
 
Heymsfield et al. (1) discussed determinants of βBW in physical terms. In terms of statistics, βBW is determined by the SDs of log(BW) and log(height) and by r for log(BW) and log(height) (3). Thus, 
 
 
 
 
 
The value of r is generally more variable than the ratio of the SDs, explaining the correlation existing between βBW and r for BW and height (2, 3). Thus, βBW is determined by these 3 aspects of data scatter, with r dominating. As a separate issue, βBW must also depend on the unknown “true” (structural) relation between BW and height. Because BW and height are poorly correlated, the corresponding true value of βBW must be substantially higher than the values obtained by regression and probably close to 3, the power corresponding to isometry (2, 4). 
 
 
Heymsfield et al. (1) viewed BW as consisting of 2 components, one height-dependent component and one composed mainly of body fat and influenced by factors unrelated to height. This idea is backed up by mathematical modeling (3). The latter predicted that %fat should show some negative correlation with height, especially in women. This is relevant because the conventional theory underlying BMI requires there to be little such correlation (but see below). Heymsfield et al. (1) did find significant correlations between %fat and height. As they noted, β%fat may be calculated as (βFM – βBW). Table 1 (lines 3 and 4) shows β%fat to be negative for women (implying negative correlations). For men, adjustment for age made the 3 negative values positive. 
 
 
There was a strong link between βFM and βBW (1). With age adjustment, r is 0.98 with the regression equation being as follows: 
 
 
 
 
 
Modeling can produce a similar relation—with one component of FM increasing with height and another being independent of height. However, the model still needs quantitative development. 
 
 
Adjustment of βBW for age and %fat narrowed its range from 1.19–2.42 to 1.94–2.29 (1). One might suppose that adjustment for %fat could produce height exponents close to those for fat-free mass (FFM), but these were not recorded. Because there is no simple algebraic relation between βFFM and βBW adjusted for %fat, I calculated these for 4 other data sets; they differed by only −0.2 to +0.1 (i.e., rather little in this context). Because Heymsfield et al. (1) found βBW adjusted for both %fat and age to be ∼2 for all groups, the latter round number would seem to be about right for βFFM also. In terms of Equation 1 applied to FFM instead of to BW, the removal of variable adiposity as a source of data scatter has 2 effects that largely cancel out, namely increasing r (5) and making SDlog(FFM) smaller than SDlog(BW). An earlier study found only small differences between βBW and βFFM, both age-corrected (5). The closeness of βFFM to 2 provides a rationale for the BMI that does not depend on the usual assumption that %fat is uncorrelated with height (6). 
 
 
With regard to WC, which obviously correlates with BW and FM, Table 1 shows that the effects of age-adjustment on βWC correlated with those on βFM (r = 0.96; lines 1 and 5). The actual values of βWC and βBW (1) also correlated with each other, both with and without adjustment for age, and with adjusted and unadjusted values combined (r = 0.97, 0.94, and 0.93 respectively). Regression of βWC on βBW, both age-adjusted, gives the following equation for both sexes together: 
 
 
 
 
 
Other age-adjusted data (7) show a similar relation (with r = 0.88) (8): 
 
 
 
 
 
Regression of log(WC) on log(body volume/height) and log(age) has previously given regression coefficients for log(volume/height) of 0.62 for men and 0.61 for women (9). Taking their mean and assuming a body density of 1, the relation between βWC and βBW, adjusted for age, must accordingly be as follows: 
 
 
 
 
 
Over the relevant range, these 3 relations are almost identical. They may be largely understood through simple geometric modeling (9), especially with data scatter taken into account (8). The ratio WC2: height has been suggested as an adiposity index that tends to vary proportionally with BMI (8), but, at the expense of simplicity, the integer exponent might then be better replaced with a value closer to 1/0.615 (= 1.6). 
 
 
This brings us back to clinical utility, which is the main concern of this particular study by Heymsfield et al. (1). However, with basic underlying relations in mind, it is gratifying to find patterns emerging from their results. Here is a situation where evidence for differences between populations is also evidence, in the form of correlations, for underlying and informative similarities.