Abstract
This study examined the bivariate relationship between peak oxygen uptake (V˙o _{2 peak}; l/min) and body size in adult men (n = 1,314, age 17–66 yr), using both “simple” and “full” iterative nonlinear allometric models. The simple model was described byV˙o _{2 peak} =M^{b} (or FFM^{b}) exp(c SRPA) exp(a +d age) ε (whereM is body mass in kg; FFM is fatfree mass in kg; SRPA is selfreported physical activity; ε is a multiplicative error term; and exp indicates natural antilogarithms). The full model was described byV˙o _{2 peak} =M^{b} (or FFM^{b}) exp(c SRPA) exp(a +d age) +e (ε), wheree is a permittedYintercept term. TheM exponent obtained from simple allometry was 0.65 [95% confidence interval (CI), 0.59–0.71], suggestive of a curvilinear relationship constrained to pass through the origin. This “zeroYintercept” assumption was examined via the full allometric model, which revealed anM exponent of 1.00 (95% CI, 0.7–1.31), together with a positiveYintercept term (e) of 1.13 (95% CI, 0.54–1.73). The FFM exponents were not significantly different from unity in either the simple or full allometric models. It appears that the curvilinearity of the simple allometric model (using totalM) is fictitious and is due to the inappropriate forcing of the regression line through the origin. Utilizing FFM as the bodysize variable revealed a linear relationship between body size andV˙o _{2 peak}, irrespective of model choice. We conclude that the population mass exponent forV˙o _{2 peak} is close to unity.
 allometric scaling
 nonlinear regression
 loglinear modeling
 body size
 oxygen uptake
the relationship between body size and energy metabolism, at rest and during exercise, has interested researchers for over a century. In the exercise sciences, the past decade has seen a resurgence of interest in this problem. Several authors have suggested that allometric or power function modeling may be theoretically, physiologically, and statistically superior to alternative scaling models (2729, 35, 36). Huxley’s “simple” allometric equation (19), Y =aM^{b} ε, has been most often employed, whereY, M, and ε represent the physiological dependent variable, body mass, and a multiplicative error term, respectively. Feldman (15) stated that two empirical allometric laws for energy metabolism in adult animals operate simultaneously. These laws are based on theoretical speculations about mechanisms of heat loss (31, 32), biological similitude (4, 32), elastic similarity (26), or fractal geometry (37) and predict an intraspecific (withinspecies) mass exponent (b) of 2/3 and an interspecific (betweenspecies) value ofb = 3/4.
In humans, a wide variety of intraspecific mass exponents has been reported for maximal or peak oxygen uptake (V˙o _{2 peak}; in l/min). For example, Bergh et al. (9) reportedb values ranging from 0.47 to 0.86 for maximal oxygen uptake in seven different groups of adults. The weighted mean mass exponent of 0.71 closely approximated the anticipated 2/3 value. However, others have reported a mass exponent of 1.02 in adults, suggestive of a linear relationship between body size andV˙o _{2 peak} (30). Heil (17) argued that a mass exponent of 0.67 was appropriate in samples homogeneous with respect to age, training background, and body height, whereas a 0.75 value was more applicable to more heterogeneous samples. The lack of agreement regarding the proper value of the mass exponent has provoked some lively theoretical and methodological debate (7, 27).
Rogers et al. (30) suggested that the large variation in derived exponent values may be due to small sample sizes and/or bodysize homogeneity. In addition, failing to control, or account for, known covariates (e.g., age, physical activity status) could also confound empirical body sizephysiological function relationships (17). Furthermore, heterogeneity of body composition can spuriously affect the magnitude of the obtained mass exponents (35). Indeed, over 40 years ago Döbeln (14) stated that any scaling analysis in humans should be based ideally on body mass minus fat mass [i.e., fatfree mass (FFM)]. Estimated FFM has been found to be the best single predictor ofV˙o _{2 peak} in several studies (34, 35). Inasmuch as ∼95% of the oxygen passing through the lungs of a mammal exercising atV˙o _{2 peak} is bound for a “single sink” in the skeletal muscle mitochondria (25), it follows that estimated FFM may be a judicious choice as an indicator of body size.
Why derive a robust estimate of the population mass exponent forV˙o_{2 peak}?
It is apparent that the precise numerical value of the mass exponent for peak aerobic function has been the subject of much research and heated scientific debate. Deriving a robust estimate of the population mass exponent is clearly viewed as important. Most researchers in the exercise sciences attempt to use empirical, samplespecific mass exponents to 1) make interpretations of their biological significance and/or2) derive an index to remove the influence of body size from a dependent variable. Indeed, Feldman (15) stated that, in empirical allometry studies, it is the value of the mass exponent b that is of critical interest, inasmuch as it may provide “insight into the biological design of the class.” As Albrecht and Gelvin (2) argued, different numerical values for the mass exponent may result in different biological and theoretical explanations.
The importance of the precise numerical value of the mass exponent becomes apparent when the relationships among theory, data, and model are examined in the majority of the allometry literature. As SchmidtNielsen (32) stated, allometric equations are descriptive generalizations, or “empirical laws,” not biological laws. In most instances, biological, theoretical explanations have been “induced” from empirical data, modeled by using various forms of the general allometric equation. It has thus been argued that the allometric models are valuable inasmuch as they may reveal “principles and connections that otherwise remain obscure” (32). Clearly, if biological, theoretical expositions are to be developed on the basis of the empirically derived numerical value of mass exponents, a critical evaluation of the methods employed in allometry studies is warranted. The wide variety of obtained mass exponents for peak aerobic function reported in the literature underlines this need. Indeed, Rogers et al. (30) argued that, given this lack of consensus, it is difficult for researchers to determine which is the “most correct” mass exponent to use.
Critique of allometric models.
It has been suggested (17) that, given a large sample size and sufficient data to describe and account for known covariates, a robust mass exponent forV˙o _{2 peak} can be obtained. However, we contend that insufficient attention has been paid to the most appropriate statistical modeling procedures. The selection of an allometric modeling method is not a trivial issue. A robust estimation of the position of a line of best fit through a “data cloud” requires the critical evaluation of key assumptions underpinning the model chosen. Strauss (33) stated that, in any empirical study of allometry, the biologist is confronted with a confusing array of models and options. In the exercise sciences, allometric modeling ofV˙o _{2 peak} has been conducted exclusively via linear, least squares regression techniques applied to natural logtransformed dependent and independent variables (log V˙o _{2 peak} = loga + blog M + log ε). Although this approach simplifies the calculations greatly, it may introduce a systematic bias, resulting in a crude estimate of the mass exponent (33). The bias is due to the fact that the least squares technique minimizes the residual error of the linearized, loglog regression line, not that of the original untransformed data (38). Second, the equation of simple allometry,Y =aX^{b} ε, describes a curvilinear relationship that constrains the regression line to pass through the origin. Although intuitively appealing, this assumption has not been tested in the scaling ofV˙o _{2 peak} and has no empirical basis (2). Albrecht et al. (3), in an excellent methodological review, argued that the extrapolation of regressions through the origin may result in fictitious curvilinearity. Hence, if the true relationship betweenV˙o _{2 peak} and body mass were linear (b = 1), empirically derived mass exponents ofb < 1 (curvilinear) from simple allometry may be an artifact of forcing the regression line to curve at smaller values to reach the origin (1).
Because of the aforementioned limitations of simple, loglinear allometric scaling (log Y = loga + blog X) the validity of all previously reported mass exponents forV˙o _{2 peak} may be questionable. The purpose of the present study, therefore, was to determine a robust estimate of the population mass exponent (body mass and FFM) for V˙o _{2 peak} in a large sample of adult men, while controlling for some known covariates. First, we adopt an allometric model that applies an iterative nonlinear regression procedure to the original raw data (rather than the customary loglinearized method). Second, we tested the hypothesis that any obtained mass exponent ofb < 1, implying curvilinearity, would be due to imposing a “zeroYintercept” constraint. This simple experimental model involves manipulating the requirement for the regression line to pass through the origin and examining the effect on the obtained mass exponent (b). This is achieved via a “full” allometric model (13, 19), described by the equation Y =aX^{b} +c(ε), wherec is a permittedYintercept term.
METHODS
Subjects.
The 1,314 men selected for this study were employees at the National Aeronautics and Space Adminstration Johnson Space Center (Houston, TX). With the exception of astronauts, preemployment physical screening is not compulsory. Approximately 85% of the male workforce, however, choose to have a yearly health examination and a graded exercise stress test every 3 yr. At their annual physical examination, the men in this cohort were “apparently healthy,” asymptomatic, and free from cardiovascular disease. Additional exclusion criteria included the chronic use of cardiovascular medications and electrocardiographic abnormalities exhibited at rest or during the stress test. Written informed consent was obtained from all subjects before the procedures.
Test procedures.
The specific protocols employed in the present study are detailed elsewhere (21). Briefly, the subjects’ level of physical activity was assessed with the National Aeronautics and Space Adminstration selfreport physical activity (SRPA) scale (8). Each subject was asked to rate his physical activity during the previous month on an eightpoint scale (0–7). The scale’s lowest value of zero represents an almost completely sedentary lifestyle. The highest value of seven represents participation in a minimum of 3 h of regular, heavy aerobic physical activity each week.
Body fat percentage was estimated from the sum of chest, abdominal, and thigh skinfolds (23). Specific anatomic sites and procedures are described elsewhere (24). Total body mass (kg) and body fat percentage were used to partition body mass into fat mass and FFM compartments.V˙o _{2 peak} was determined during the Bruce treadmill exercise test protocol (10). A full outline of the equipment and test procedures is provided in another source (22). The criteria adopted for the attainment ofV˙o _{2 peak} were1) volitional exhaustion;2) a respiratory exchange ratio of ≥1.1; and 3) a peak exercise heart rate ≥90% of the agepredicted maximum.
Allometric modeling.
All analyses were carried out by using the SPSS (release 7.5 for Windows; SPSS, Chicago, IL) statistical software package. For comparative purposes both “simple” and full allometric models were derived, with separate models for body mass (M ) and FFM. Subject age and SRPA score were included in all models as covariates. Working in the arithmetic space defined by the original rawX and Y variables (3), we solved all model parameters by an iterative nonlinear protocol using the LevenbergMarquardt algorithm (16). In this procedure, small, successive corrections to regression parameter estimates are made until a global solution converges (on the basis of a cutoff criterion for the relative reduction in successive residual error of 1.0 E08). In this nonlinear regression program, the choice of initial values for the parameters can influence convergence. Where possible, realistic values should be selected close to the expected final solution as poor starting values can result in a locally, rather than globally, optimal solution (16). In the present study, therefore, linear multiple regression analysis of the following logtransformed simple allometric model was employed in an exploratory fashion to obtain starting values for the parameters in the subsequent, nonlinear simple allometric modeling
These values were used as initial values in the confirmatory, iterative nonlinear procedures described by the equations below. This exploratory method could not be applied to the full allometric model, however, inasmuch as the full model is not amenable to logarithmic transformation (3). Therefore, it was necessary to apply the same starting values (obtained from the simple, loglinear models) to the nonlinear full allometric models. The initial parameter value of the permitted Yintercept term (e) in the full model was defined as zero, to test the assumption of the simple allometric model that the regression line passes through the origin.
Simple allometric model.
The multivariable form of the general allometric modelY =aX^{b}
ε (19) is given by
In this model, body mass and FFM are variables assumed to relate in a nonlinear manner toV˙o _{2 peak}, whereas SRPA and age are linearly related to the dependent variable. The exp(c SRPA) exp(a +d age) term is thus the linear coefficient applied to M or FFM equivalent to the proportionality coefficienta that modifiesX in the general allometric equationY =aX^{b} ε (19).
Full allometric model.
Similarly, the multivariable form of the full allometric model, allowing for a Yintercept term, is given by
The full allometric model (Eq. 2 ) implies a curved line (unless b = 1) that does not pass through the origin (unless e = 0). It is intended to account for all statistical factors that describe the relationship between Y(V˙o _{2 peak}) andX (Mor FFM) (3). The full model reduces to the simple model (Eq. 1 ) if theYintercept term,e, is equal to zero. All regression parameters were calculated as point estimates, together with the associated estimates of uncertainty [95% confidence interval (CI)]. The 95% CI provides important information regarding the precision of the parameter estimates. In addition, it can be used to test for statistical significance (if desired) at the 0.05 alpha level. If the 95% CI for the coefficient does not cross zero, then that parameter makes a statistically significant contribution to the prediction ofV˙o _{2 peak}(P < 0.05). Several authorities, including the International Committee of Medical Journal Editors (20), have urged the reporting of CI in preference toP values, and this practice is adopted in the present study.
Regression diagnostics (5) revealed no violation of the assumptions of allometric modeling. There were no significant interactions among the model covariates, confirming the assumption of homogeneity of regression. In addition, KolmogorovSmirnoff onesample tests revealed that all allometric model residuals were normally distributed (P > 0.10).
RESULTS AND DISCUSSION
Table 1 presents the means, SDs, and range for the dependent variable and all model covariates. The subjects were heterogeneous for age, body mass, FFM, physical activity, andV˙o _{2 peak}. An ∼2.5fold body mass size ratio was evident across the sample (54.7–134.5 kg), with a twofold ratio for FFM (45.7–95.0 kg). A wide size range is vital to derive meaningful scaling expressions, and Calder (12) has urged that publication of size range should be mandatory in all allometric studies.
Table 2 provides the obtained regression coefficients for the simple allometric modeling ofV˙o _{2 peak} by body mass. Examination of the 95% CI reveals that all regression parameters make a statistically significant contribution to the modelR ^{2}(P < 0.05). This finding supports the results of Heil (17) and Jackson et al. (21) and confirms that both physical activity status and age are important covariates that must always be included in the model in studies of this type. The modelR ^{2} indicates that ∼58% of the sample variance inV˙o _{2 peak} is explained by the variables entered. The simple allometric model (Eq. 1 ) yielded a body mass exponent (b) of 0.65. Table 2 confirms that the 95% CI for this point estimate included the value of 2/3 anticipated from the intraspecific empirical law for mammals (18). This 2/3 exponent is consistent with the previous findings of Nevill and coworkers (29) using simple loglinear allometric modeling. Nevill et al. reported a mass exponent of 0.669, common to both men and women, in a study of 308 “recreationally active” young adults. Age and physical activity status were not included as covariates in this study as, presumably, the sample was homogeneous with respect to these variables. In a larger sample of 1,732 subjects, Nevill and Holder (28) obtained a b value of 0.66, with age and a dummy variable of “vigorous exercise” and “no vigorous exercise” included as covariates.
The 95% CI for the mass exponent from the simple allometric model (Table 2) indicates that the population mass exponent is significantly different from unity. The model implies that the relationship betweenV˙o _{2 peak} and body mass is curvilinear, with the regression line passing through the origin. The results of the full allometric model (Table3), however, suggest that this “curvilinearity” may be a fictitious artifact of an inappropriate modeling procedure. Table 3 displays a statistically significant, positive Yintercept term (e), together with a mass exponent (b) equal to one. The full model (Eq. 2 ) has thus reduced to a simple, linear model (Y =aX +c). Hence, it would seem that the curvilinearity (implied by b = 0.65) in the simple allometric model was caused by the forcing of the curve through the origin (13). Importantly, the relaxation of this constraint revealed a very different numerical value for the mass exponent, indicating that the critical, zeroYintercept assumption of the simple allometric model was untenable. This phenomenon is demonstrated graphically in Fig. 1, which depicts the bivariate relationship between body mass andV˙o _{2 peak}. For this analysis the dependent variable was first adjusted for the influence of model covariates. This was achieved by regressingV˙o _{2 peak} on age and SRPA and adding the resultant residuals to the meanV˙o _{2 peak} for the sample (3.08 l/min). Figure 1 illustrates clearly that, through the data cloud, the relationship between body mass andV˙o _{2 peak} is essentially linear. The curvilinearity of the simple allometric fit line results from an extrapolation of the curve well beyond the observed data. It appears that it is possible to force simple allometric regressions on bivariate data that are actually linear, as demonstrated previously by Albrecht et al. (3). Permitting aYintercept term revealed a different bivariate relationship between the Xand Y variables.
To our knowledge, no similar comparison of simple vs. full allometric models in the scaling ofV˙o _{2 peak} has been published. In the anthropological literature, Albrecht and Gelvin (2) reported that mass exponents were significantly affected by model choice (simple vs. full allometric) due to nonzeroYintercepts obtained in the full models. In the present study, the modelR ^{2} and SE of the estimate suggest that both the simple and full allometric models fit the data about equally well. However, the exponential termb differs significantly between the models because of the implicit assumption in simple allometry that theYintercepte = 0. If this assumption is relaxed, as with the full allometric model, then it is found that the bivariate relationship betweenV˙o _{2 peak} and body mass is described by a regression line that has aYintercepte > 0 and a mass exponentb = 1. As can be seen in Fig. 1, this means that the data are essentially linear within the range of observations, not curvilinear as implied by the mass exponentb = 0.65 from the simple allometric model.
We must emphasize that the suggestion of the addition of a constant to the simple allometric equation is not new. Over 60 years ago, Huxley (19) stated that the full model was the most complete and should be taken as the theoretical basis for analysis. Indeed, Albrecht et al. (3) argued that if the mass exponent (b) from the simple allometric model was considered biologically meaningful, then theb in the full model should be, also. The problem for the researcher is that, on the basis of our findings, the two models provide different mass exponents, due to the failure to satisfy a key assumption of the simple model. This occurred despite the fact that the two models displayed a similar degree of “predictive value,” in terms of explained variance inV˙o _{2 peak}and SE of the estimate. Hence, the mass exponent of 0.65 derived from simple allometry may lead to theoretical expositions such as the theory of geometric similarity (32). Conversely, exponentb in the full model implies thatV˙o _{2 peak} is linearly proportional to body mass. Interestingly, the 95% CI for the body mass exponent in the full model (Table 3), although wider than that in the simple model, did not include the anticipated intraspecific value of 2/3. It is noteworthy that several other variables (including pulmonary diffusing capacity, heart mass, blood volume, stroke volume, and skeletal muscle mass) in the chain of events involving the ability to take in, transport, and utilize oxygen also scale isometrically with body mass (albeit by using simple allometric equations) (25). Hence, the empirically derived mass exponentb = 1 from the full model may point to very different biological principles and connections from theb ≈ 2/3 from the simple model. We believe that, notwithstanding the equivalent predictive value of the two models, the more robust estimate of the population mass exponent is provided in this instance by the full allometric model. This assertion is based on the fact that a key assumption of the simple model was violated (0 Yintercept), whereas all regression assumptions were satisfied for the full allometric model. In the exercise sciences literature, there is no reported evidence of checks on the validity of the zeroYintercept assumption in simple allometric models.
The modeling of V˙o _{2 peak}by FFM provides additional insight. Table 4(simple allometry) displays an obtained FFM exponent of 0.97, with the 95% CI including unity, but precluding the value of 2/3. The modelR ^{2} indicates that the FFM model explained ∼7% more of the variance in the dependent variable than did the body mass models. This supports earlier contentions that FFM is a superior scaling variable forV˙o _{2 peak} (13, 34, 35). Full allometric modeling by FFM also revealed ab exponent not significantly different from unity (Table 5). Again, the 95% CI precludes the value of 2/3 for the mass exponent. The point estimate for the Yintercept term (e) is 0.4. However, the 95% CI for this parameter crosses zero, suggesting that for the population there is no statistically significantYintercept. Hence, the key zeroYintercept assumption of the simple allometric model has been satisfied. Both the full and simple allometric models, therefore, indicate that the bivariate relationship between V˙o _{2 peak} and FFM is essentially linear (b = 1) and passes through the origin (e = 0).
These findings are illustrated in Fig. 2, where it is clearly shown that the simple allometric and linear regression fit lines overlap throughout the range of observed data (in this analysis, V˙o _{2 peak}was adjusted for the effects of SRPA and age, as described previously). Both the full and simple allometric models have thus reduced to a simple, linear perratio standard model of the formY =aX. In modeling maximal oxygen uptake or V˙o _{2 peak}, exponent values not significantly different from unity for FFM have been reported previously in children (6), adult women (35), and elderly men (13). It is noteworthy that all of these studies (using simple, loglinear models) reported body mass exponents significantly less than one (implying curvilinearity), consistent with the findings of the simple allometric modeling in the present study. None of the studies, however, reported checks on the validity of all key model assumptions.
The differences between the reported exponents for body mass and FFM derived from simple allometry (Eq.1 ) may relate to heterogeneity of body composition across the sample. If there was no variance in body fat percentage, the derived exponent values should theoretically be exactly equivalent. In the present sample, however, a significant positive relationship was found between body mass and body fat percentage (Pearsonr = 0.58,P < 0.05). This suggests that, as body mass increases across the sample, the proportion of total body mass composed of FFM decreases. Inasmuch as only the metabolically active body cell mass is pertinent to the expression ofV˙o _{2 peak} (25, 35), this body composition heterogeneity may artificially deflate the obtained body mass exponents (35).
The positive Yintercept in the full allometric model (Eq. 2 , Table 3) for body mass may also relate to heterogeneity of body composition, in particular to the precise relationship between body mass and FFM in this sample. If the regression line is extrapolated to theYintercept, the suggestion is that someone of zero body mass would exhibit aV˙o _{2 peak} of 1.13 l/min (95% CI, 0.54–1.73). Recall that there was no statistically significant Yintercept in the full allometric modeling by FFM (Table 5). To explore this interesting phenomenon, we first examined the bivariate relationship between body mass and FFM in this sample. Theoretically, there should be a linear, proportional relationship between body mass and FFM if homogeneity of body composition is assumed. For example, if body fat percentage was constant at 15% across the sample, the relationship between body mass (X) and FFM (Y) would be described by a general linear model, Y = 0.85X + 0 (i.e., 0Yintercept). However, as stated, the present sample is highly heterogeneous for body composition. Figure3 illustrates the linear (Y =aX +c) and simple allometric (Y =aX^{b} , for comparison) relationships between body mass and FFM. The linear model (dashed line) reveals a significant positiveYintercept of 22.9 kg of FFM. The 95% CI for this parameter was 21.3–24.6 kg. The allometric regression (solid line) illustrates how the forcing of the curve through the origin causes the curvilinearity implied by the obtained body mass exponent of 0.65. Through the data cloud the regression lines overlap, and it is clear that the relationship is essentially linear.
The positive Yintercept from the linear model suggests that someone of zero body mass would exhibit FFM of 22.9 kg. This physiologically implausible phenomenon is due to the large variance in body fat percentage across the sample (range 3.9–39.3%, SD, 6% body fat). We next conducted a simulation of the influence of this heterogeneity of body composition on the results of the full allometric modeling by body mass (Eq.2 , Table 3). The 22.9 kg of FFM (theYintercept, Fig. 3) were entered into the simple allometric equation derived previously for FFM (Eq. 1 , Table 4). The simulation predicts that a hypothetical subject presenting with 22.9 kg of FFM would express a V˙o _{2 peak}of 1.15 l/min (95% CI, 0.41–1.89). This point estimate is almost exactly equivalent to the significant, positiveYintercept of 1.13 l/min obtained from the full allometric modeling by body mass (Table 3). Hence, it is possible that the physiologically implausible 1.13l/minYintercept in Table 3 (implying that someone of 0 body mass would still exhibit aV˙o _{2 peak} value) resulted, in part, from the specific relationship between body mass and FFM in this sample.
Taken together, our findings suggest that “ideally,” estimates of FFM should be secured to serve as the indicator of body size. The FFM models resulted in a larger coefficient of determination and a lower SE of the estimate in predictingV˙o _{2 peak}. When total body mass is used in scaling, the full allometric model may be preferable in situations where the zeroYintercept assumption does not hold. When FFM is used in scaling, the requirement of the regression line to pass through the origin was a valid assumption. This resulted in no significant difference between the FFM exponents derived from the simple (b = 0.97) and full (b = 1.1) allometric models.
The population body mass or FFM exponent for modelingV˙o _{2 peak} appears to be close to unity, implying a linear relationship. The 95% CIs do not include the anticipated value of 2/3. As stated at the beginning of this study, the intraspecific (withinspecies) allometric 2/3 law is an empirical law based on studies in mammals (15, 18). Theoretical speculations related to simple dimensionality theory or biological similitude, however, are frequently advanced (e.g., Refs. 4, 17, 27,29, 30). The conventional argument is thatV˙o _{2 peak} is a measure of work divided by time (aerobic power). Work is equal to the product of force (∝ L ^{2}, the square of the length dimension) and distance (∝L ^{1.0}). Work is therefore proportional toL ^{2} L ^{1.0} =L ^{3} ∝ M ^{1.0}. From Newtonian mechanics, time is assumed to be proportional to the length dimension (∝M ^{1/3}) (4, 17). Hence, V˙o _{2 peak}(metabolic work divided by time) is proportional toL ^{2} ∝M ^{2/3}. Despite intuitive appeal, however, this theory is overly simplistic and appears to have little or no foundation (15). Butler et al. (11) argued that energy metabolism is dependent on at least 12 dimensionally distinct factors. In a thorough and theoretically correct dimensional analysis of the relationship between metabolic rate and body mass, the authors concluded that the 2/3 mass exponent does not follow logically from dimensional reasoning. Indeed, an intrinsic 2/3 value could only be derived theoretically if just 1 pair of the 12 contributing variables (that does not contain a temperature dimension) were applied and held invariant as body mass varied. Other pairs of variables would result in completely different predicted mass exponents. Butler et al. calculated predicted intrinsic mass exponents of 1/5, 7/6, 2/3, and 5/4 for various pairs of invariant contributory variables. If energy metabolism depends on more than one pair of variables, in addition to mass, then simple dimensional theory cannot predict a mass exponent at all. Hence, the value of 2/3 is not a value anticipated from simple dimensionality theory.
The primary purpose of the present study was to determine a robust estimate of the mass exponent in the scaling ofV˙o _{2 peak}. However, it is noteworthy that the regression coefficient point estimates for the model covariates (SRPA and age) differ among the various analyses (Tables 25). These regression coefficients provide estimates of the independent influence of habitual physical activity and age onV˙o _{2 peak} in this crosssectional study. The more robust estimates for these effects can be derived from the models, including FFM as the indicator of general body size. Recall that these models (Tables 4 and 5) explained ∼7% more of the variance inV˙o _{2 peak} than the body mass models (Tables 2 and 3). The point estimates of the covariate regression coefficients for the simple allometric model (Table 4) indicate that a oneunit increase in the SRPA score is associated with a 4.5% increase inV˙o _{2 peak}[exp(0.044 ⋅ SRPA) = 1.045 exp(SRPA)]. The ageassociated decrease inV˙o _{2 peak} predicted from the simple FFM model (Table 4) is 0.7%/yr or 7%/decade [exp(−0.0072 ⋅ age) = 0.993 exp(age)]. Similarly, the full allometric model (Table 5) predicts that a oneunit increase in the SRPA score is associated with an increase in V˙o _{2 peak}of 5.2%, with an ageassociated decrease inV˙o _{2 peak} of 0.8%/yr or 8%/decade. Note that the 95% CIs for the covariate regression estimates in the simple and full allometric models overlap, implying that there is no statistically significant difference between them. This is due to the fact that there is no significantYintercept term (e) in the full allometric model; the simple and full FFM models are essentially equivalent.
The present study is subject to several limitations. First, the sample is exclusively male and delimited in age range (17–66 yr). We make no attempt to extrapolate the present findings beyond the male adult population within this age range. Indeed, we would recommend that our model be also crossvalidated on large, heterogeneous, and independent adult male samples. In the present study, crossvalidation within the same sample via various datasplitting algorithms was avoided, as this would have reduced the precision of the sample’s regression estimates. When distinguishing between mass exponent values of 2/3 and unity is attempted, for example, the sample size must be maximized to obtain an appropriately narrow (precise) 95% CI.
The limitations of the general methods employed to derive the physical activity and body composition estimates are well documented. Clearly, however, the adoption of criterion or “gold standard” methods for the above variables (such as doubly labeled water and hydrodensitometry) would have been impractical and prohibitively expensive in a crosssectional epidemiological study of this nature. The criteria adopted for the attainment ofV˙o _{2 peak} may represent a further limitation to our study. Volitional exhaustion, a respiratory exchange ratio of ≥1.1, and a peak exercise heart rate of ≥90% of agepredicted maximum are essentially “nonphysiological” indicators. Obtaining blood lactate data would have permitted us to establish a physiological state of exercise intensity. The lack of information pertaining to postexercise blood lactate concentrations is one reason the data are described as “peak” rather than “maximal” oxygen uptake. We are confident, however, that the values obtained are representative of very heavy exercise.
Despite the aforementioned limitations, the advantages in increased precision of regression parameter estimation, afforded by a large sample size, likely far outweigh any inherent measurement errors in the dependent variable or covariates. The present study was restricted to one species (intraspecific). An advantage of interspecific (betweenspecies) allometry is an increased range of body size (e.g., mouse to elephant). For example, this size range, coupled with the inclusion of data from a large number of animals, is held to overwhelm the variability inV˙o _{2 peak} because of sources other than body size and leads to more meaningful scaling expressions (12). To offset this advantage of interspecific allometry, we attempted to account for a proportion of the variability due to other sources by including important known covariates in the model. The sample was highly heterogeneous for physical activity status, age, and body composition. Including this information in the model covariates, together with a large sample somewhat heterogeneous for body mass (range from 54.7 to 134.5 kg), enabled us to distinguish (within 95% CI) between mass exponents of 2/3 and unity.
In summary, we have demonstrated empirically, by using the most complete (full) iterative nonlinear allometric models and controlling for key known covariates, that the population mass exponent forV˙o _{2 peak} is close to a value of one (linearity) for adult men. This mass exponent differs from that predicted by unfounded theoretical speculations about dimensional consistency (4, 17, 27). The obtainedb value of 1 contradicts the empirical 2/3 scaling law derived from simple, loglinear allometry. We have established that the curvilinearity implied by mass exponents <1 in studies employing simple allometric models may be fictitious, due to the inappropriate forcing of the regression line through the origin. If simple allometric models are employed, the key assumption that the regression line passes through the origin (Y = 0 whenX = 0) should be tested. We urge that full allometric models be applied in largesample scaling studies of this type to further examine the true relationship between body size and physiological function. If the empirically derived mass exponent value of 1.0 forV˙o _{2 peak} is found to be robust in new studies in diverse populations, then we are challenged to understand it from a physiological and/or biological perspective.
In addition, the application of a robust population mass exponent may be preferable in studies involving small, homogeneous samples, where an index must be derived to partition out the influence of body size from a dependent variable. In such studies, calculated samplespecific mass exponents may be confounded and inaccurate, because of lack of precision and wide CI (small n), and limited betweensubject variance in body size (range effects on regression estimates) (30). Moreover, attempts to apply nonlinear iterative full allometric models in small and/or homogeneous samples may be hampered by the failure of the model to converge cleanly with an optimal solution.
Footnotes

Address for reprint requests and other correspondence: A. M. Batterham, School of Social Sciences, Univ. of Teesside, Borough Road, Middlesbrough, TS1 3BA, UK (Email:A.Batterham{at}tees.ac.uk).

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 Copyright © 1999 the American Physiological Society