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Special Communications: Methods

Ratio scaling of ˙VO2max penalizes women with larger percent body fat, not lean body mass


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Medicine & Science in Sports & Exercise: September 1996 - Volume 28 - Issue 9 - p 1204-1208
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There is abundant research evidence to suggest that expressing maximal oxygen uptake (˙VO2max) per unit of body mass (BM, i.e., ml·BM-1·min-1) actually fails to partial out the independent effect of BM(2-5,7-10,12,15). Such a BM-based ratio index is negatively correlated with BM(9) and therefore imposes a penalty on heavier subjects. An alternative index, free of the confounding effect of BM, is ml·BM-0.7·min-1(1-4,6,9). Such findings are congruent with the “theory of similarity” (TofS)(1) that posits that ˙VO2max is proportional to BM raised to the 2/3, or 0.67 power.

The exponent 0.67 is based on the theory that oxygen consumption in geometrically similar bodies varies with the rate of oxygen delivery that, in turn, varies with the cross-sectional areas of relevant blood vessels(1). Because cross-sectional area is proportional to the square of a body segment length, and a body segment length cubed varies directly with body mass, then cross-sectional area relates to body mass to the 0.67 power.

In the scaling of ˙VO2max by BM, the agreement between empirical findings and the TofS may, however, be coincidental. The TofS actually may apply best to lean body mass (LBM) because LBM is free of the confounding effect of fat mass (6). Lean body mass, an expression of the body's metabolically active tissue, is the only compartment of total BM that directly influences O2 uptake. Therefore, percent body fat could spuriously affect the magnitude of the exponent obtained empirically.

Allometric scaling (AS) is a technique in which a scaling convention can be developed for an outcome variable, y, to be expressed in terms of a scaling variable, x, in the form of y·x-a (e.g.,˙VO2max per BMa) such that the effects of the scaling variable are statistically partialed out (13). In short, AS solves for the value of a. This index, y·x-a, has a correlation of 0 with x, particularly desirable when correlating it with other variables. In such cases, the risk of spurious correlations is minimized by using AS (9,15). AS also can be used in a multivariate fashion to create performance variables that allow for unbiased comparisons between individuals who differ in more than one scaling variable, all of which are known to have an effect on the outcome variable(14).

To our knowledge, there has been only one investigation(6) that scaled ˙VO2max by LBM in young adults, with an exponent for LBM of 0.71. This finding suggested that scaling˙VO2max by BM or LBM would yield the same exponent of 0.7(1,4). One might then infer that body composition would not really affect the scaling exponent for BM. However, there were several serious design flaws in this study that make generalization of its findings problematic. First, men and women were combined into the same analysis without consideration for gender as a separate variable. Doing so created a bi-modal distribution (ascertained by our reanalysis of Dobeln's data) for ˙VO2max and LBM, clearly a violation of the assumption of normally distributed variables and a necessary condition for AS scaling(15). Second, there also may have been different slopes of the regression line for ˙VO2max regressed on BM in men and women, thus implying an interaction between gender and BM. This scenario would have resulted in one exponent that penalized lighter subjects of one gender and heavier subjects of the other gender. This should have been ascertained before calculating a universal exponent. In short, any AS analysis must consider group membership before calculating and interpreting exponents. Nevill et al. (9) and Vanderburgh et al.(13) elaborate further on this point. Third,˙VO2max was estimated by a submaximal heart rate response protocol, not by a criterion method such as indirect calorimetry. Finally, sample size was relatively small (33 men and 32 women). Because AS uses regression analysis to solve for the optimal scaling exponent a larger sample would provide more stable estimates of the exponent.

The present study was designed to overcome the limitations of the Dobeln study. Allometric scaling was used to scale ˙VO2max (assessed by indirect calorimetry) by BM and LBM (assessed by hydrostatic weighing) in a sizable sample of women to investigate the effects of body fatness on the magnitude of the exponent.


The subjects were 94 women volunteers (BM = 60.3 ± 8.4 kg, age = 27.4 ± 6.7 yr) who signed informed consent prior to testing. Subjects completed a medical history questionnaire and none had any contraindications for maximal exercise testing and hydrostatic weighing. Subject descriptives are shown in Table 1.


All subjects had ˙VO2max assessed via a continuous treadmill protocol with two variations depending on subjects' self-report of prior physical activity history and exercise training. For apparently untrained individuals, initial grade was 2.5%, speed was maintained at 6.0 mph, and elevation was increased by 2.5% every 2 min. For the other subjects, the protocol was identical, except the initial grade was 7.5%. Ventilation volume rate was assessed with a digital pneumotach, and expired air samples were analyzed with oxygen and carbon dioxide analyzers (Applied Electrochemistry, Pittsburgh, PA) calibrated before and after each treadmill test with reference gases verified by the micro-Scholander technique. Exercise heart rate was determined via radiotelemetry (Transkinetics, Canton, MA). Gas sampling began at approximately 70% of age-predicted maximum heart rate and was continued every 30 s. Strong verbal encouragement was given throughout the test. The criterion for ˙VO2max attainment was two of the following three: heart rate within 90% of age-predicted maximum (220-age), plateau of˙VO2 ≤ 2 ml·kg-1·min-1 per workload, or R ≥ 1.05.

Lean body mass was calculated by subtracting fat mass from body mass, the former calculated from body density determined from the Siri equation(11). Body density was determined from underwater weight, which was determined by 10 repeated trials with an average of the last three trials used as the criterion underwater weight score. Residual volume was determined via oxygen washout with subjects in a seated position out of the water. An average of two trials done within 5 min of each other was used as the subject's residual volume (r = 0.97, t = 0.72, P > 0.05).


Allometric scaling (AS) was the primary method of statistical analysis. Taking the log of both sides of the following relationship describing the best-fit curve of ˙VO2max (ml·min-1) vs BM:Equation 1,2

Equation 2 is an equation of a straight line, so the values of a, now the slope, and log b, the intercept, can be solved for via linear regression(10,12-15). Logarithmically transformed ˙VO2max values for each subject were regressed first on log BM, then log LBM in separate regression models. Normality of the distributions of log BM, log LBM, and log ˙VO2max was ascertained(the Statistica for Windows® Kolmogorov-Smirnov test for normality,P > 0.05 for each). The coefficients of log BM and log LBM in the two separate regression equations (the “B” values in the regression procedure) were the values of the exponent a for proper scaling of these variables. The standard error of the B values corresponded to the 68% confidence interval of a. Therefore, 1.96 times this value corresponded to the 95% confidence interval (an alpha level of 0.05). Vanderburgh and Mahar (13) provide additional discussion of AS procedures and considerations.


AS yielded the following exponents (±95% C.I.): BM: 0.61 ± 0.27, and LBM: 1.04 ± 0.26. These data suggest that conventional scaling of ˙VO2max in ml O2·BM-1·min-1 tends to penalize heavier women (Fig. 1) and that the ml O2·BM-0.61·min-1 convention eliminated this penalty (Fig. 2). The LBM exponent suggests that conventional scaling of ˙VO2max per unit of LBM (ml O2·LBM-1·min-1) does not penalize women with larger or smaller LBM (Fig. 3). Stated differently, this expression partials out the differential effect of LBM.

To our knowledge, this is the first investigation that has properly examined the effect of percent body fat on the scaling of ˙VO2max by BM using a relatively large sample of women, criterion methods for the assessment of ˙VO2max and LBM, and appropriate statistical design. In a similar study with 73 elderly men, Davies et al. (5) found similar results: BM exponent = 0.43, LBM exponent = 1.05, with LBM determined via dual energy x-ray absorptiometry. That this BM exponent was smaller than from our data is most likely due to the fact that the mean percent fat of the elderly men (27.7%) was higher than that of the women(24.2). Higher percent fat tends to further confound the relationship between body mass and ˙VO2max because a greater component of body mass is actually fat mass. In most cases, this confounding leads to a smaller BM exponent (5,13). The very close agreement between the two studies in LBM exponents provides preliminary evidence that the population exponent is probably closer to 1.0 for adult men and women. This, of course, needs further investigation, particularly with young men and elderly women.

The results of this investigation suggest that scaling by BM does in fact tend to penalize heavier women because the exponent 0.61 is less than 1. While the exponent's magnitude is in concert with prior investigations of this type regarding the scaling of BM (3,4,9), the TofS is not necessarily supported because the TofS actually applies best to BM free of the influence of fat mass, a rather unlikely occurrence in this or any similar population. In fact, fat mass and BM were moderately correlated in this sample (r = 0.48, P < 0.05). Therefore, the possibility that fat mass was spuriously affecting the ˙VO2max to BM relationship was certainly worth investigating.

That the LBM scaling exponent was 1.04, (and significantly different from 0.67, the exponent supported by the TofS), seems rather convenient because it supports the use of a ratio adjustment for ˙VO2max by LBM. However, its lack of congruence with the TofS presents an interesting dilemma, namely which convention of expressing ˙VO2max relative to body mass is most appropriate. The validity of the TofS, as applied to humans may be, however, suspect for a variety of reasons. First, the TofS applies to geometrically similar bodies. Even with fat mass partialed out, the overall trend for LBM differences between subjects in this study may have deviated from geometric similarity in the bone and muscle compartments that constitute LBM. This possibility could only be ascertained by more detailed analysis of different body compartments.

Second, and more importantly from a physiologic perspective, there may be other biological factors associated with increases in body size that affect the relationship between ˙VO2max and LBM that can lead to departures from the 0.67 exponent (3). Such factors might include hemodynamic characteristics, arteriovenous exchange, and cardiac function/dimensions, all of which, with respect to LBM, could change at rates different from that expected by the TofS. These possibilities can only be elucidated with similar AS analyses using large samples and somewhat more invasive assessment procedures.

Third, LBM includes active (those muscles used for physical work) and inactive tissue. Inactive tissue's relative proportion to total LBM may be different between large and small women, thus confounding the interpretation of the LBM exponent for all subjects. Stated another way, perhaps scaling by active LBM instead of total LBM would have led to an exponent more congruent with the TofS. Unfortunately, determining one's active LBM requires instrumentation and equipment not found in a majority (including ours) of exercise physiology laboratories.

Most interestingly, the present results indicate that expressing˙VO2max by the conventional body-mass adjusted ratio standard appears to impose a penalty only on those women who are heavier because of larger fat mass, and not larger LBM. Therefore, expression of˙VO2max in ml·BM-0.7·min-1, the convention proposed by others as a more fair index may in fact penalize lighter women because the exponent's deviation from 1 must be due to the relatively larger amount of fat mass in heavier women. If one takes the position that excess body fatness is undesirable from a health and performance perspective, then expressing ˙VO2max as ml·BM-1·min-1 appears to be a somewhat useful and even unbiased index of maximal oxygen uptake.

We should point out that there are statistical and health- or performance-related considerations for the interpretation of these data. From a statistical perspective, in expressing ˙VO2max so that the effect of BM is completely partialed out, one might use the ml·kg-0.61·min-1 convention. The health- or performance-related interpretation deals with how to best express˙VO2max adjusted for body size so that the resulting index imposes a penalty only on subjects with larger fat mass (e.g., ml·BM-1.04·min-1 or ml·BM-1·min-1). The choice of interpretation depends on one's objectives. For research purposes, in which statistical control may be most desired, the former may be more appropriate. The latter index is suggested for the clinician or the fitness professional who seeks an index more indicative of undesired health risk and/or performance liability.

We have used AS to show that although conventional ratio-scaling of maximal oxygen uptake tends to penalize heavier women, this penalty is imposed only on those who were heavier because of larger fat mass. From a health- and performance-related perspective, we conclude that expression of a young woman's ˙VO2max in ml·BM-1·min-1 is an appropriate expression of maximal oxygen uptake. The results also indicate that in contrast to previous findings (6), body composition does influence the scaling exponent of ˙VO2max by body size. This point should not be ignored in attempting to explain individual differences in ˙VO2max based on body size alone.

Figure 1-˙VO2max per unit of body mass (BM) vs BM. This negative relationship suggests that, as BM increases, ˙VO2max per unit of BM decreases.
Figure 1-˙VO2max per unit of body mass (BM) vs BM. This negative relationship suggests that, as BM increases, ˙VO2max per unit of BM decreases.
Figure 2-˙VO2max per unit of body mass (BM) raised to the 0.61 power vs BM. The near-zero relationship indicates that expressing˙VO2max in ml·BM-0.61·min-1 imposes no penalty based on BM.
Figure 2-˙VO2max per unit of body mass (BM) raised to the 0.61 power vs BM. The near-zero relationship indicates that expressing˙VO2max in ml·BM-0.61·min-1 imposes no penalty based on BM.
Figure 3-˙VO2max per unit of lean body mass (LBM) vs LBM. The near-zero relationship suggests that simple ratio scaling of˙VO2max by LBM imposes no penalty based on LBM.
Figure 3-˙VO2max per unit of lean body mass (LBM) vs LBM. The near-zero relationship suggests that simple ratio scaling of˙VO2max by LBM imposes no penalty based on LBM.


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©1996The American College of Sports Medicine