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00005768-199602000-0001600005768_1996_28_259_welsman_differences_2article< 114_0_12_7 >Medicine & Science in Sports & Exercise©1996The American College of Sports MedicineVolume 28(2)February 1996pp 259-265Scaling peak ˙VO2 for differences in body size[Special Communications: Methods]WELSMAN, JOANNE R.; ARMSTRONG, NEIL; NEVILL, ALAN M.; WINTER, EDWARD M.; KIRBY, BRIAN J.Children's Health and Exercise Research Centre, University of Exeter, Exeter, EX1 2LU, UNITED KINGDOMSubmitted for publication March 1993.Accepted for publication April 1995.This work was supported by grants from the British Heart Foundation and the University of Exeter.Current affiliations: Joanne R. Welsman and Neil Armstrong, Children's Health and Exercise Research Centre, University of Exeter, Heavitree Road, Exeter, EX1 2LU, U.K.; Brian J. Kirby, Postgraduate Medical School, University of Exeter, Barrack Road, Exeter, EX2 5DW, U.K.; Edward M. Winter, School of Humanities, Sport and Education, De Montfort University Bedford, Lansdowne Road, Bedford, MK40 2BZ, U.K.; Alan M. Nevill, School of Sport and Exercise Sciences, University of Birmingham, Birmingham, B15 2TT, U.K.Address for correspondence: Dr. Joanne Welsman, Children's Health and Exercise Research Centre, University of Exeter, Heavitree Road, Exeter, EX1 2LU, U.K.ABSTRACTThis paper examined the influence of different statistical modeling techniques on the interpretation of peak ˙VO2 data in groups of prepubertal, circumpubertal, and adult males (group 1M, N = 29; group 2M, N = 26; group 3M, N = 8) and females (group 1F,N = 33; group 2F, N = 34; group 3F, N = 16). Conventional comparisons of the simple per-body-mass ratio(ml·kg-1·min-1) revealed no significant differences between the three male groups (P < 0.05). In females, a decline in ˙VO2 between group 2F and 3F was observed (P< 0.05). Both linear and log-linear (allometric) models revealed significant increases across all three male groups for peak ˙VO2 adjusted for body mass (P < 0.05). In females these scaling models identified a significantly lower peak ˙VO2 in group 1F versus groups 2F and 3F (P < 0.05). Based upon the common mass exponent identified (b = 0.80, SE = 0.04), power function ratios(y·mass0.80) were generated and the logarithms of these compared. Again, results indicated a progressive increase in peak˙VO2 across groups 1M to 3M (P < 0.05) and an increase between groups 1F and 2F (P < 0.05). Incorporating stature into the allometric equation reduced the mass exponent to 0.71 (SE = 0.06) with the contribution of the stature exponent shown to be 0.44 (SE = 0.20). These results indicate that conventional ration standards do not adequately account for body size differences when investigating functional changes in peak˙VO2.The pediatric exercise physiologist is faced with the problem of how to interpret changes in physiological performance variables that occur during normal development or result from training. Comparisons between groups or between an individual's performance relative to that of peers are confounded by body size differences. Consequently, these differences must be partitioned out if physiological variables are to be studied meaningfully.Ratio ModelConventionally, such variables are expressed in ratio with body mass, e.g., peak ˙VO2 as ml·kg-1·min-1. In this way division of the physiological variable by body mass is assumed to“normalize” the data, i.e., remove the influence of body mass from subsequent analyses.Tanner (28) details the statistical limitations of this assumption and subsequent papers in the early 1970s corroborate these concerns (13,14). To summarize these briefly, the“ratio standard” assumes the mathematical relationship y = a·x between peak ˙VO2 and body mass, where a is the constant derived from the mean values of peak ˙VO2 and body mass, and the line passes through the origin. According to Tanner, the ratio model is only appropriate where the coefficient of variation (V) for the body size variable, e.g., peak ˙VO2, equals the Pearson product moment correlation coefficient (r) obtained between the two variables (28). The use of a ratio standard where this condition is not satisfied results in a distortion of the data with the magnitude of the distortion increasing as the discrepancy between the Vx/Vy and r increases(28,32). As a result, smaller individuals receive an arithmetic advantage while larger individuals are penalized. Furthermore, when the scaled value of peak ˙VO2(ml·kg-1·min-1) is correlated with body mass a negative coefficient is usually obtained, illustrating how the ratio standard fails to provide a dimensionless physiological performance variable by converting the positive correlation between body mass and peak ˙VO2 to a negative one (20).Linear Adjustment ModelWhere a linear relationship exists between the physiological performance variable and the body size variable, the use of a linear adjustment model to partition out body size differences has been recommended(28). Here the relationship between peak ˙VO2 and body mass is described by the linear regression equation y = a + b·x, where a is the intercept of the regression line on the y axis and b is the slope of the line.Where intergroup comparisons are required, the statistical procedure analysis of covariance (ANCOVA) is used to compare the slopes and intercepts of the regression lines generated for the specific groups under consideration and produce “adjusted means” for peak ˙VO2, i.e., means from which the influence of body mass has been partitioned out(27). While apparently providing a theoretically more appropriate method of scaling (28), linear adjustment models have also been shown to represent a better statistical fit in terms of a reduction in the residual error associated with the analysis compared with usual ratio techniques during the modeling of various peak performance measures (20,30).Several investigators have reworked traditionally scaled data using linear adjustment models and the results have challenged conventional interpretation. Eston et al. (10) demonstrate that differences in submaximal running economy between children and adults disappeared when a linear adjustment model was used to partial out the influence of body mass. Similarly, Winter et al. (33) report differences in maximal exercise performance between male and female subjects which were masked when power output data were expressed in ratio with lean leg volumes.Using a linear model to normalize peak ˙VO2 data in pre- and postpubertal males, Williams et al. (31) reveal a significantly higher adjusted peak ˙VO2 in postpubertal males compared with the younger subjects. Conversely, the ratio model supports the traditional interpretation of a consistent level of peak ˙VO2 during growth in males.Recently, gender differences in peak ˙VO2 related to fat free mass have been shown to disappear when linear scaling methods are applied(30), leading these authors to advocate strongly a general acceptance of the use of this type of model to interpret physiological performance.Log-Linear Adjustment ModelIn contrast, recent papers have highlighted the significant statistical limitations of the linear adjustment models(20,32). First, positive intercepts are common which suggest that a physiological response exists for zero body size; therefore, extrapolation beyond actual data points should be avoided. More importantly, the linear model assumes a constant (additive) error term, i.e., that the spread of scores around the regression line is constant throughout the range of y and x variables measured, a property known as homoscedasticity. This assumption is unlikely to be met when modeling human physiological performance variables in subjects who vary greatly in body size(20). Linear models might not therefore be appropriate even where the least squares regression line provides a better fit for data as reflected by a reduction in the residual sum-of-squares compared with the simple ratio model (20).Allometric analyses have a long history of use in the biological sciences for describing and interpreting size-related changes in physiological function(24) but apart from a few notable exceptions they have not been widely used in developmental exercise physiology. Allometric or power function equations describe curvilinear relationships between physiological variables which take the general form y = a·xb(24). Values of a and b are derived from linear regression on logarithmic transformations of data: ln y = ln a + b·ln x. The allometric modeling process allows group differences to be examined in two ways. First, when data are expressed in the log-linear form, groups may be compared using analysis of covariance in exactly the same way as for the linear adjustment model described above. Alternatively, following identification of the b exponent, power function ratios can be constructed by dividing the absolute value of the performance variable by mass raised to the appropriate exponent, i.e., y·m-b. Power function ratios for different groups may be compared using simple analysis of variance ort-test (20).Although the application of allometric models may produce only a modest improvement in statistical fit compared with a linear adjustment model, their use is justified for two reasons. First, they assume a multiplicative error term, which allows the model to control for spread in the data(heteroscedasticity) providing that the variables diverge at a constant rate(20). Second, the relationship is plausible as it goes through the origin.The numerical value of the b exponent in allometric equations describing physiological measures such as peak ˙VO2 has been the subject of some debate. According to the theory of geometric similarity, as human power output is proportional to stature2 or body mass0.67, peak˙VO2 should also be proportional to mass0.67,(4). However, based upon data obtained from a large variety of animal species, Kleiber (16) has argued for a mass exponent of b = 0.75. This value has been dismissed as statistical artifact (11) although others have attempted to provide theoretical support; for example, McMahon's theory of elastic similarity(18). The debate has not yet been resolved conclusively but at present, biological justification for a 0.75 power of mass has yet to be confirmed (12).Exponents identified for children have consistently exceeded either of these theoretical predictions with values usually close to b = 1.0(5,9,21). Faced with such findings, it is not surprising that others have concluded that there are no strong reasons for abandoning the conventional practice of expressing physiological variables in ratio with body mass (6,25).An explanation for these elevated values has recently been proposed by Nevill (19) based upon the findings of Alexander et al.(1), who demonstrate that larger mammals have a greater proportion of segmental muscle mass in relation to their total body mass, i.e., leg muscle mass is proportional to, mass1.1. Therefore, if the group of subjects under scrutiny exhibits a disproportionate increase in muscle mass, then the assumption of the allometric model that the proportion of muscle mass is common to all subjects will be violated. To accommodate this, Nevill (19) has suggested that when modeling measures such as peak ˙VO2, stature as well as body mass could be incorporated as a continuous covariate into the allometric equation, i.e., y = a·staturec·massb. By doing so, the independent contribution to peak ˙VO2 from body mass is separated from the contribution of the disproportionate increase in muscle mass with increasing size across the group.The aim of this paper was to explore how the application of these different scaling models influenced the interpretation of differences in peak˙VO2 among children, adolescents, and adults and to identify which model might provide the best statistical fit for this type of data.METHODSSubjectsDescriptive statistics of the subjects whose peak ˙VO2 data were used in the analyses are displayed in Table 1. Data for all groups were obtained at the Research Centre during 1991-1992 from untrained, volunteer subjects participating in various projects that required the determination of peak ˙VO2. All of the youngest children were classified as prepubertal; stage 1 for public hair and genitalia or breast development (29). Tanner ratings of maturity were not available for the teenagers but a maturity rating of 3 or 4 would be expected in children of these ages (3). In all subjects stature was measured to the nearest 0.01 m using an Harpenden stadiometer and body mass measured to the nearest 0.1 kg immediately prior to the exercise test using Avery beam balance scales. All testing procedures had received institutional ethical approval. Written, informed consent was obtained from all subjects and in the case of the prepubertal and circumpubertal children their parents/guardians as well.TABLE 1. Physical characteristics of subjects (mean ± SD).Test ProtocolAll peak ˙VO2 determinations were carried out on a motorized treadmill following identical procedures. After a 3-min warm-up jog at 7 km·h-1, the test protocol commenced at a belt speed of 8 km·h-1. Three-minute exercise bouts were separated by rest periods of 1 min. Belt speed was increased by 1 km·h-1 increments to 10 km·h-1 for the initial stages. Thereafter, the gradient was raised by 2.5% increments until the subject was unable to continue. In the absence of a plateau in ˙VO2 of less than 2 ml·kg-1·min-1 over the final stages of the test, the highest value of ˙VO2 recorded was accepted as peak˙VO2 if age-predicted maximum heart rate was recorded or heart rate failed to increase over the final stages, and respiratory exchange ratio was equal to, or greater than, 1.0. Peak values of heart rate and respiratory exchange ratio are displayed in Tables 2 and 3. Ventilation and expired O2 and CO2 were continuously monitored via a computerized on-line gas analysis system (Oxycon Sigma, Cranlea Ltd., Birmingham, U.K.). The system was calibrated according to the manufactures recommendations before each test using gases of known concentration. Heart rate was monitored throughout the test using a bipolar lead (Rigel, Morden, U.K.).TABLE 2. Mean peak heart rate, respiratory exchange ratio, peak˙VO2, and peak ˙VO2 adjusted for the covariate body mass from both linear and log-linear models (± SD) for males.TABLE 3. Mean peak heart rate, respiratory exchange ratio, peak˙VO2, and peak ˙VO2 adjusted for the covariate body mass from both linear and log-linear models (± SD) for females.AnalysesAll data were stored on a Dell 320LX personal computer, and statistical analyses were carried out using the SPSS-PC+ (SPSS Inc., Chicago, IL) statistics package. The following models were applied to the data to investigate differences in peak ˙VO2 across the subject groups. I. The conventional per-body-mass ratios were computed from absolute peak˙VO2 (l·min-1) divided by body mass. The mean values of peak ˙VO2 (ml·kg-1·min-1) for the six groups were compared using a 2 × 3 (gender by group) analysis of variance (ANOVA). II. The least squares linear relationships between peak ˙VO2 and body mass for the six groups were determined and compared using a 2 × 3 analysis of covariance (ANCOVA) with mass as the covariate. ANCOVA firstly examines whether the slopes of the regression lines are parallel, as equality of slopes across all groups must be satisfied for ANCOVA to proceed. Where this condition is satisfied, the regression lines are used to adjust the group means for differences in body mass, i.e., means for peak ˙VO2 from which the influence of body mass has been partitioned out. III. Subsequently, the allometric relationship between peak ˙VO2 and body mass was examined by computing a similar 2 × 3 ANCOVA but with linear regression applied to logarithmic transformations of peak˙VO2 and body mass data. This also provided the parameters a and b in the allometric relationship y = a·xb. IV. Power function ratios (˙VO2 ml·min-1·mass-b) were computed using the b exponent identified from the log-linear model above. The logarithms of the power function ratios were computed before comparison using ANOVA(20). V. To separate the independent contribution of body mass from the confounding influence of the contribution of the disproportionate increase in muscle mass with increasing body size anticipated in this group of subjects, log(stature) was included as a second covariate in the log-linear ANCOVA to predict peak˙VO2 described in (iii) above.After all ANOVA and ANCOVA analyses, where a significant F-ration was obtained the post-hoc Scheffé procedure was used to identify significantly different groups (minimum level of acceptanceP < 0.05).To examine the data for heteroscedasticity and identify which model, linear or allometric, provided the best statistical fit for the data, residuals(predicted - observed peak ˙VO2) from linear and log-linear models were converted to absolute values and correlated with the predictor variable, body mass.RESULTSMean values for peak ˙VO2 expressed in absolute terms(l·min-1) and in ratio with body mass(ml·kg-1·min-1) for males and females are summarized in Tables 2 and 3, respectively. ANOVA revealed that peak ˙VO2 (l·min-1) increased significantly across the age groups in both genders. At each age group, values for males were significantly higher than those for females.No significant differences in mass-related peak ˙VO2 were observed across the male groups (P > 0.05). In females, there was a significant decrease in mass-related peak ˙VO2 between group 2F and 3F (P < 0.05). Again, at each age group all comparisons between males and females were significantly different (P < 0.05).Significant correlation coefficients (P < 0.01) were obtained between peak ˙VO2 and body mass for all groups as follows: groups 1F, 2F, and 3F; r = 0.884, 0.849, and 0.722, respectively; groups 1M, 2M, and 3M; r = 0.866, 0.848, and 0.855, respectively. ANCOVA revealed that the slopes of the least squares linear regression lines were not significantly different across the six groups (b = 0.038, standard error {SE} = 0.002). However, significant gender and group differences were observed for the intercepts of the regression lines. Intercepts for the six groups were as follows: 1M = 0.443, 2M = 0.734, 3M = 1.226, 1F = 0.247, 2F = 0.392, 3F = 0.300. Adjusted means are presented in Tables 2 and 3. In contrast to the results of the conventional comparisons based upon per-body-mass ratios, adjusted means of peak ˙VO2 increased in males at each age group while in females a significant increase was observed between prepuberty(1F) and circumpuberty (2F).The ANCOVA on the log-transformed data produced the same outcome. Having first tested for commonality of exponents, a common slope of b = 0.80 (SE = 0.04) was obtained for the six groups. Intercepts for the groups were: 1M =-2.273, 2M = -2.162, 3M = -2.051, 1F = -2.393, 2F = -2.296, 3F = -2.325. A significant and progressive rise in peak ˙VO2 was noted for males with, again, a significant increase observed between groups 1F and 2F. The adjusted means from the log-linear model are presented inTables 2 and 3.The group mean values of the logarithms of the power function ratios computed using the common b exponent (peak ˙VO2 ml·kg-0.80·min-1) are displayed inTable 4. Results of the ANOVA were consistent with those from the linear and log-linear models. Significant increases in the power function ratio across all male groups were observed with a significantly lower value evident in group 1F compared with groups 2F and 3F.TABLE 4. Natural logarithms of power function ratios (in ml · kg-0.80 · min-1).Correlating the residuals from the linear-regression model with body mass produced a significant coefficient, r = 0.348 (N = 156, P< 0.01) (see Fig. 1). In contrast, a nonsignificant coefficient was obtained from the correlation of the residuals from the log-linear model with body mass, r = -0.013 (N = 156, P> 0.05) (see Fig. 2).Figure 1-Absolute residuals from the linear model vs body mass.Figure 2-Absolute residuals from the log-linear model vs log body mass.Log-linear modeling with both stature and mass as covariates produced exactly the same pattern of significant differences as the mass only model. However, the mass exponent was reduced to b = 0.71 (SE = 0.06) with the parameter for height shown to be c = 0.44 (SE = 0.20).DISCUSSIONThe values of peak ˙VO2 (l·min-1 and ml·kg-1·min-1) recorded in these subjects are consistent with previous treadmill peak ˙VO2 determinations in similarly aged subjects (2,3). As expected, absolute peak ˙VO2 increased with age in both males and females, largely reflecting the increase in body mass. To examine functional changes free from the influence of body mass, this study applied conventional and alternative statistical models for partitioning out body mass differences.The results of the standard comparisons of per-body-mass ratios support the frequently documented conclusions from both cross-sectional and longitudinal studies that peak ˙VO2 relative to body size remains consistent throughout growth and development in boys while there is a tendency for relative peak ˙VO2 to decline with growth in girls(2,3,15).The results from both the linear and log-linear adjustment models diverged markedly from this conventional interpretation. Functional increases in peak˙VO2 were observed from prepuberty to adulthood in males when more appropriate models were used to control for body mass differences. These findings are consistent with the indications from our preliminary study of pre and post-pubertal males (31). In females, peak˙VO2 appeared to improve between prepuberty and circumpuberty when linear or log-linear models were applied with levels then remaining consistent into adulthood rather than decreasing as usually observed.These findings were confirmed by the subsequent computation and comparison of the power function ratios based upon the b exponent identified from the log-linear modeling.The heteroscedasticity of the data was confirmed by the significant correlation found between the residuals from the linear model and body mass. This emphasizes the limitation of the linear adjustment model although the results produce an identical pattern of findings as the allometric model. The absence of correlation between residuals from the log-linear ANCOVA confirm the appropriate fit provided by this model.Although few published studies have reported mass exponents for peak˙VO2, particularly in untrained children and adults, there are sufficient data to demonstrate that peak ˙VO2 does not increase in direct proportion to body size, i.e., is not proportional to mass1.0. Bergh et al. (7) report exponents of 0.50 and 0.47 for trained and untrained adult males respectively and 0.74 for adult female runners. The mean exponent for all groups examined, including male and female athletes, was 0.71 (SE = 0.05). This is close to the value of 2/3 (0.67) predicted from the theory of geometric similarity (24). Nevill et al. (20) report mass exponents of 0.63 and 0.72 for peak ˙VO2 derived from large samples of men and women, respectively. Again the common exponent identified of 0.67 corresponded to the theoretically predicted value.The use of allometric analyses to model peak ˙VO2 in children and adolescents has consistently resulted in mass exponents well above the theoretical value. Cooper et al. (9) report a value of b= 1.01 for boys and girls aged 11-18 yr and therefore concluded that the ratio standard (ml·kg-1·min-1) would not change with increasing body size in children. A similar exponent of 1.02 is identified by Paterson et al. (21) in 11- to 15-yr-old trained boys. Other studies with untrained male subjects have reported exponents of 0.78(26) in 11- to 15-yr-old boys and 0.82 in 8- to 15-yr-old boys (5), values which closely approximate the value of 0.80 identified in the present study.A complete and satisfactory explanation as to why theoretical principles have failed to adequately describe the relationship between peak˙VO2 and body mass in young subjects does not appear to have been identified, although Cooper (8) has postulated that qualitative changes occur within muscles as part of the maturational process with the result that force generation for cross-sectional area increases; thus challenging an underlying assumption of the theory of geometric similarity that muscle stress is independent of body size. It is notable that the studies reporting mass exponents greater that 0.67 have included subjects ranging from prepuberty to postpuberty who thus represent a wide range of stature and mass. In such heterogeneous groups, simply modeling by body mass alone may be insufficient to partition out body size differences adequately; stature must also be considered (19). When modeling the peak power output/body mass relationship in a group of javelin throwers who differed considerably in body size, Nevill obtained a mass exponent of b = 0.76. With stature introduced as a second predictor variable in the allometric equation, a non significant stature parameter of 0.44 was obtained, with the mass exponent reduced to the theoretically predicted value of 0.66. The modeling of the data from the present study in this way produced results remarkably consistent with those demonstrated by Nevill, with the stature exponent shown to be 0.44 with the independent contribution from body mass reduced to b = 0.71, a value not significantly different from the theoretical value of 0.67. On the basis of these observations it may be postulated that the mass exponents observed in previous studies with children and adolsescents were elevated above the theoretical value as a result of stature not being included in the allometric model. For illustrative purposes, values of peak˙VO2 adjusted for mass0.67 are presented inTable 5 for the subjects included in this study.TABLE 5. Power function ratios calculated (ml · kg-0.67· min-1).While acknowledging the limitations of the cross-sectional design, the data presented here are consistent with the extant literature and indicate that when body size differences are controlled appropriately, significant growth-related increases in peak ˙VO2 are apparent that are masked when comparisons are based upon conventional ratio models. To what extent can these statistical findings be reconciled with the physiological changes taking place during maturation?In males, oxygen consuming muscle mass as a percentage of total body mass changes during growth from puberty to adulthood with a rapid period of muscle mass gain during the mid-teenage years (17). Similarly, other components of the oxygen transport chain continue to develop during growth such as hemoglobin concentration, particularly in males(22,23). Earlier maturation of females compared with males is well documented (29). In particular, female gains in the percentage of total body mass that is represented by muscle are not observed after the onset of puberty (17) as body fat stores increase.The results obtained in this study are consistent with these observations. Furthermore, they concur with the indications from investigations of changes in metabolic scope (the difference between resting and peak ˙VO2) that “the capacity to deliver O2 to working muscles improves during childhood years independent of gains in body size”(22).In conclusion, these data support previous findings(31) and suggest that the use of per-body-mass ratios has obscured our understanding of developmental changes in peak˙VO2. Using statistical techniques that theroretically are superior and can be shown to provide a better fit for the data to partition out body size, peak ˙VO2 is shown to increase progressively in males from prepuberty to adulthood and from prepuberty to circumpuberty in females. These findings appear to conform to the general pattern of maturation in the genders. The identification of a common mass exponent of 0.80 reinforces the argument that peak ˙VO2 does not increase in direct proportion to body mass and reflects the inability of simple ratio models to partition out body mass where functional changes in peak ˙VO2 with growth are under investigation. That the mass exponent identified is somewhat higher than the theoretically predicted value of 0.67 can be explained by the disproportionate increase in the subject's muscle mass with increasing body size. By also partitioning out stature the mass exponent was reduced to 0.71, and was not significantly different from 0.67. 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