Abstract
Batterham, Alan M., and Keith P. George. Allometric modeling does not determine a dimensionless power function ratio for maximal muscular function. J. Appl. Physiol. 83(6): 2158–2166, 1997.—In the exercise sciences, simple allometry (y =ax^{b} ) is rapidly becoming the method of choice for scaling physiological and human performance data for differences in body size. The purpose of this study is to detail the specific regression diagnostics required to validate such models. The sum (T, in kg) of the “snatch” and “cleanandjerk” lifts of the medalists from the 1995 Men’s and Women’s World Weightlifting Championships was modeled as a function of body mass (M, in kg). A loglinearized allometric model (ln T = lna + bln M) yielded a common mass exponent (b) of 0.47 (95% confidence interval = 0.43–0.51, P < 0.01). However, sizerelated patterned deviations in the residuals were evident, indicating that the allometric model was poorly specified and that the mass exponent was not size independent. Model respecification revealed that secondorder polynomials provided the best fit, supporting previous modeling of weightlifting data (R. G. Sinclair.Can. J. Appl. Sport Sci. 10: 94–98, 1985). The model parameters (means ± SE) were T = (21.48 ± 16.55) + (6.119 ± 0.359)M − (0.022 ± 0.002)M^{2}(R ^{2} = 0.97) for men and T = (−20.73 ± 24.14) + (5.662 ± 0.722)M − (0.031 ± 0.005)M^{2}(R ^{2} = 0.92) for women. We conclude that allometric scaling should be applied only when all underlying model assumptions have been rigorously evaluated.
 polynomial models
 regression diagnostics
 scaling
 weightlifting
recently in the exercise sciences, there has been a resurgence of interest in the importance of body size as a potentially confounding influence in studies of physiological function. In several articles (1416, 30, 31) the utility of the class of models often referred to as “allometric or power function models” has been advanced. These authors have suggested that allometric modeling (general equation, y =a × mass^{b}× ε) may be theoretically, physiologically, and statistically superior to alternative methods of scaling physiological variables for differences in body size. Briefly, the assumption of a multiplicative error term in allometric models is believed to overcome the heteroscedasticity and skewness frequently observed with linear, per ratio standard modeling. Moreover, the true relationship between the body size (independent) variable and the physiological or performance (dependent) variable may be curvilinear, as implied by the general allometric equation (ifb ≠ 1). A number of authors (16,23, 26, 2931) reported that allometry is often more successful than other approaches in providing a dimensionless dependent variable via construction of a power function ratio standard:y/mass^{b}. Thus allometric modeling is held to permit meaningful intersubject or intergroup comparisons free from the confounding influence of body size.
Allometric modeling is most often conducted via linear, leastsquares regression techniques applied to logtransformed dependent and independent variables (ln y = lna + bln x + ln ε). Hence, the general assumptions of linear regression apply to the model. Nevill (15) criticized previous research for failing to check and confirm the assumptions of homoscedasticity (constant error variance) and normal distribution of residuals in linear or loglinear modeling of physiological data sets. We contend, however, that, particularly in the exercise and sport science literature, insufficient attention has been paid to perhaps the most important assumption of loglinear, leastsquares models, i.e., that the regression model is correctly specified. Clearly, the model must adequately describe the relationship between the independent and dependent variables, or parameter estimation and hypothesis testing rests on a false premise (8). For the logtransformed allometric model, there should be a strong, linear relationship between the independent and dependent variables. However, the bivariate or multivariate correlation coefficient is a poor criterion for evaluating whether a model is correctly specified (19). Certainly, a high and statistically significant linear correlation coefficient can be obtained, despite an inherent nonlinearity in the data (8). Indeed, several authorities, including Gould (9), Albrecht et al. (1), and Strauss (20), issued cautionary warnings about the inappropriate “forcing” of particular regression models on the data at hand. Investigation of potential model misspecification is best conducted by plotting the model residuals (ε or ln ε) as a function of the body size variable and checking for sizerelated distributional patterns that would violate the model assumptions (1, 19). To our knowledge, these recommendations and cautionary warnings have not been adequately heeded in the exercise and sport science literature, where allometric modeling of physiological or human performance data is becoming increasingly prevalent. Recent examples of variables scaled for body size differences using allometry include grip strength (26), peak or maximal oxygen uptake (2, 23), indoor rowing ergometer performance (24), 2mile run time (25), shortterm peak and mean power output in a 30s supramaximal arm ergometry test (12), and shortterm peak and mean power output in a 30s supramaximal sprint treadmill test (16). It appears that allometric scaling is rapidly becoming the method of choice for partitioning out the effects of body size on physiological or human performance data sets. Indeed, Winter (30, p. 678) has called for the “preferential use of allometric modeling,” and Vanderburgh et al. (26, p. 83) encouraged future research to apply allometric scaling to many physiological, human performance, and anthropometric variables. If allometric models are to be applied more widely, it is imperative that they be correctly specified for the data set in question. Our primary intention is therefore to detail the appropriate diagnostic procedures required to test the assumptions underlying loglinear allometric models.
The specific aims of the current study were to1) scale veryshortterm maximal human muscular function in men and women for differences in body mass via a generalized allometric model,2) employ a range of regression diagnostics to test for potential model misspecification, and3) evaluate the success of the allometric model in providing a dimensionless physiological index (i.e., an index that is unrelated to body mass). In the event of an incorrectly or inadequately specified allometric model, a further aim was to revise or respecify the model using alternative modeling procedures to better account for the data at hand.
In the current study, veryshortterm maximal human muscular function is indicated by performances in elite, Olympicstyle weightlifting (snatch and cleanandjerk events). Garhammer and Takano (7) state that the major lifting forces are applied to the bar for ∼800 ms in the snatch and the clean and for 200 ms in the jerk. Weightlifting performances therefore rely almost exclusively on anaerobic energy systems. It has been conclusively demonstrated that weightlifting performance is strongly and positively correlated with power output (6). Garhammer (6) reported that power output in the “jerkdrive” phase of the cleanandjerk increased in a nonlinear fashion across body mass categories, ranging from 2,140 W in the 56kg class to 4,786 W for a 110kg lifter. The power outputs of the heavier lifters in Garhammer’s study exceed published maximal estimates for external human power output during brief exertions (28). Clearly, performance in elite weightlifting represents the maximum of achievable veryshortterm human muscular function. Furthermore, because weightlifting competition is organized into weight categories (from 54 to >108 kg in men and from 46 to >83 kg in women), a wide body size range is guaranteed, together with a wide range in the physiologically dependent variable. Calder (3) stated that a wide body size range is crucial in allometry to derive meaningful scaling expressions.
The problem of how best to model the relationship between body size and elite, Olympicstyle weightlifting performance has received considerable attention in the literature (4, 10, 11, 13, 18, 22). Vorobjev (27) assumed the relationship to be linear, although it seems certain that this may apply only in a very restricted part of the size range. The progressively larger body size of the heavier weight classes is associated with a lower performance total than would be predicted from simple, linear proportionality (18). This nonlinear relationship was first documented in 1956 by Lietzke (13), in modeling men’s world record totals in three lifts (press, snatch, and cleanandjerk) as a function of body mass, using the logtransformed general allometric equation. Lietzke chose only to include lifters up to 90 kg in the model, because in 1956 all athletes >90 kg competed in a single “heavyweight” class. Because only the somatotypical feature “muscle mass” causally influences maximal muscular strength (22), an assumption of the allometric model is that the proportion of muscle mass is common to all lifters within the sample. Clearly, the wide range of body masses in this class (at the time from just >90 kg to >150 kg) may have distorted this homogeneity with respect to body shape, structure, and composition. Lietzke’s hypothesis, drawn from simple dimensionality theory (9), was that world record total would be proportional to mass raised to the twothirds power [since muscle strength (S) ∝ muscle crosssectional area ∝ height^{2} and mass ∝ height^{3}, so S ∝ mass^{2/3}]. Although no summary statistics for the model were reported, it appears that Lietzke’s loglog plot (Fig. 1 in Ref. 13) resulted in a nearperfect linear fit, with a slope of 0.6748, thus confirming the predictions from dimensionality theory.
In 1984, Croucher (4) repeated Lietzke’s loglinear models (13) for the men’s snatch and cleanandjerk events (the press was dropped from competition after the 1972 Munich Olympic Games). Croucher’s model included additional body weight classes at 52, 100, 110, and >110 kg (a body mass of 145 kg was ascribed to the “unlimited” body weight class). The loglog plots revealed a straight line with a slope of 0.58 for both lifts, considerably lower than the exponent reported by Lietzke (13) and representative of “negative allometry” (9) (mass exponent lower than “isometry,” i.e., smaller than expected from maintenance of geometrical similarity with size increase). It is possible that this was due to the inclusion of three weight classes above Lietzke’s 90kg limit, at which homogeneity of body composition may begin to become distorted.
Allometric models of weightlifting performance have gained widespread acceptance (22, 32). However, neither Lietzke (13) nor Croucher (4) reported the results of any diagnostic procedures to support the assumptions underpinning their models. Estimates of uncertainty for the derived exponents are not stated. Moreover, no plots of the model residuals are provided to detect potential patterned deviations related to body mass.
Sinclair (18) was the first to expose the loglinearized allometric equation as a specious model for weightlifting performance data. Sinclair plotted the men’s world record in each class as a function of body mass (with both variables normalized about the 52kg class and then log transformed). The result was not a linear fit, but a secondorder polynomial, thus violating the allometric model. A similar finding was reported by Tittel and Wutscherk (22) when modeling twoevent weightlifting totals from the 1986 European Championships as a function of body mass. Sinclair avoided the problem of the breakdown of homogeneity of body structure and composition in the unlimited class by deriving the maximum body mass statistically. This was achieved by an extension of the method of least squares, the details of which are provided in the original source (18). The Sinclair model ultimately provides a table of coefficients that can be used for interindividual comparisons of weightlifting performance and has been the scaling approach endorsed by the International Weightlifting Federation since 1979. In a review of a variety of strengthhandicapping models by Hester et al. (10), the Sinclair model was found to be the best overall model for men’s weightlifting, demonstrating the lowest coefficient of variation. However, an analysis of the Sinclair model residuals usingZ scores (11) revealed an arithmetic advantage for male lifters >120 kg. For women, the Sinclair model demonstrated a much higher coefficient of variation, andZ scores revealed a consistent biasing, favoring the lighter classes and penalizing the heavier classes. Hence, Hester et al. concluded that the definitive scaling formula has probably not been developed.
As the current study focuses on one very specific feature of human muscular function, we must emphasize that it is not our intention to generalize the specific findings to allometric modeling of muscular function more broadly. In addition, the aim is not to provide a model with which to scale other weightlifting performance data sets for differences in body size. Rather, as loglinearized allometric models are becoming increasingly popular in the exercise sciences, elite weightlifting performance data represent a useful vehicle with which to illustrate the regression diagnostics essential to their validation.
METHODS
Performance and body size data.
Data from the 9th Women’s and 67th Men’s World Weightlifting Championships (1995) were employed (data obtained from the International Weightlifting Federation Scientific and Research Committee). Veryshortterm maximal human muscular function (the dependent variable) was represented by “twoevent total” lifted (in kg), i.e., the sum of the best lifts in the snatch and the cleanandjerk events. Actual body mass (in kg) recorded at the official competition “weighin” defined the independent body size variable.
Subjects.
Subjects were selected from each of the 10 body mass classes for men and 9 classes for women. The top three performers (medalists) in each category were selected to decrease the possibility of extreme influential outliers biasing the model. The inclusion of data from all lifters in the competition would have increased sample size and statistical power. However, the sample was restricted to the “most elite” lifters in an attempt to secure a population relatively homogeneous with respect to factors that may confound the true relationship between body size and muscular function (including technical ability, genetic predisposition, training background, motivation, nutritional status). Twoevent totals (means ± SD in kg) were 207 ± 22 (range 165–240) for women and 364 ± 50 (range 272.5–442.5) for men. Actual body mass (in kg) recorded at weighin was 64.7 ± 14.8 (range 44.8–99.7) for women and 83.5 ± 25.1 (range 53.6–158.1) for men.
Allometric modeling.
All analyses were carried out using the SPSS 6.0 for Windows (SPSS, Chicago, IL) statistical package. The allometric relationships between twoevent total and body mass were derived via log transformations of the absolute data. Normality of the logtransformed variables was confirmed via the KolmogorovSmirnov onesample test (P > 0.1). The general curvilinear allometric equation, y =ax^{b}
, can be linearized by taking natural logarithms of both sides: lny = lna + bln x. The exponentb is simply the slope of the loglog plot, and a is derived from the antilog of the yintercept. Initially, separate loglinear regressions were conducted for men and women. Commonality of slopes of the relationship between body mass and twoevent total between men and women was tested by including gender (coded “0” for women and “1” for men), together with a genderbyln body mass interaction term, in a multiple, loglinearized regression model
Regression diagnostics.
Normality of the distribution of the residuals (ln ε) was examined via the KolmogorovSmirnov onesample test. The assumption of homoscedasticity was checked via the correlation between the absolute residual and the independent body size variable (ln M). A significant correlation would indicate heteroscedasticity, with error variance not constant throughout the range of observations. The assumption of a correctly specified loglinear regression model was primarily investigated via detailed examination of the residuals. The ability of the allometric model to provide a sizeindependent mass exponent was evaluated by examining the specific relationship between the scaled physiological variable (T/M^{b}) and body mass. There should be no relationship if the power function ratio is free from the confounding influence of body size.
RESULTS AND DISCUSSION
Figure1 A is a scatter plot of twoevent total vs. body mass in men and women. The apparent nonlinear, exponential nature of the plots suggested that a loglog transformation may provide a good linear fit and thus that the allometric model may be appropriate.
Allometric model.
Separate modeling of the male and female data using the loglinearized general allometric equation (ln y = lna + bln x) yielded the following results
As reported, homogeneity of regression was found, with no significant difference between the male and female mass exponents. This represents an interesting and significant finding, which to our knowledge has not been reported previously. In their 1990 review article, Hester et al. (10) stated that fundamental differences exist in the strengthbody weight relationship between men and women. In contrast, the present findings indicate that, although the elevations of the slopes of ln T as a function of ln M differ by gender, the gradients are not different. Because the male and female mass exponents are remarkably similar, the remaining analysis utilizes the common mass exponent of 0.47.
The derived mass exponent for men in the current study is lower than those reported by Lietzke (13) and Croucher (4). However, no estimates of uncertainty were provided by these authors to permit an evaluation of whether the 95% confidence interval for the mass exponent overlaps that reported for men in the present study of 0.42–0.54. Certainly, Lietzke’s (0.6748) and Croucher’s (0.58) point estimates are outside these limits. Differences in weight categories modeled between the studies of Lietzke and Croucher have already been described, which may explain the mass exponents derived. In the current study the range of male weight classes (54 to >108 kg) was similar to that in Croucher’s 1984 study. One notable difference was that Croucher assumed a body mass of 145 kg for the “unlimited,” superheavyweight class, whereas we had access to the actual body masses recorded at competition weighin. For the top three finishers, these were 158.1, 133.9, and 121.3 kg. However, remodeling our data using a value of 145 kg for the superheavyweight class revealed no significant alteration to the derived mass exponent. The explanation for the lower mass exponent in the present findings is thus unclear but is possibly related to changes over time in the weightlifting performancebody weight relationship. Interestingly, remodeling our male data by omitting weight categories beyond the 91kg limit resulted in a mass exponent of 0.68, which is virtually identical to Lietzke’s result. This finding supports the previous explanation of the differences between Lietzke’s and Croucher’s mass exponents. It seems that one assumption of the allometric model, that muscle mass represents a constant proportion of body mass within the sample, may be violated in the heavier weight categories. Unfortunately, body composition data are not available to aid in the evaluation of this hypothesis.
Regression diagnostics.
The KolmogorovSmirnov test confirmed the assumption of normally distributed residuals (P > 0.5). The lack of correlation between the absolute residuals and ln M (r = 0.08,P > 0.05) suggested that the model errors displayed homoscedasticity. Hence, the diagnostics recommended by Nevill (15) have been employed, and the assumptions are satisfied. These findings, together with the high reported coefficients of determination for the male and female loglog plots (Eqs. 3 and 4 ), lend support to the validity of the allometric model. Figure 2, however, illustrates the plot of the raw residuals (ln ε, Eq.2 ) vs. the predictor variable (ln M). If the loglinear model is correctly specified, the residuals should be randomly scattered about zero with constant variance. The residuals in Fig. 2 display systematic variations, with mainly negative residuals at low and high values of ln M and mainly positive residuals at intermediate body sizes. This pattern of residuals is strongly suggestive of a nonlinearity in the data, despite the aforementioned high, significant, linear correlation coefficients. The loglinear allometric model thus appears to be wrongly specified. To confirm that the loglog plots were not best represented by a straightline fit, a secondorder polynomial was fitted to the logtransformed data (Fig.3). TheR ^{2} values revealed that the quadratic fit was able to account for an additional 8 and 7% of the variance for men and women, respectively, compared with the linear relationship. The simple, general allometric model (Eq. 2 ) may thus be inappropriate for this data set. Conversely, Zatsiorsky (32) recently argued that the allometric model was indeed valid for describing the relationship between body mass and weightlifting performance. In a loglinear model of the 1991 Men’s World Records (sum of snatch and cleanandjerk lifts), Zatsiorsky plotted total weight lifted as a function of body mass. The result was a straight line with a slope of 0.646 (R ^{2} = 0.942). This mass exponent is close to the twothirds value predicted and confirmed by Lietzke (13). However, inspection of Zatsiorky’s loglog plot (32, p. 70, Fig. 3.7) reveals a definite nonlinearity in the data. This is analogous to the relationship displayed in Fig. 3, confirming that strong, significant linear relationships can be obtained, despite a poorly specified regression model (8).
Figure 4 displays the relationships between the allometrically scaled physiological variable (T/M^{b}) and body mass in men and women. Linear correlations revealed no relationship between body mass and T/M^{0.47}(P > 0.05). Visual inspection, however, indicated a quadratic curvature. Subsequent fitting of a secondorder polynomial revealed significant relationships for men and women (R ^{2 }= 0.69 and 0.46 for men and women, respectively,P < 0.05). This finding indicates that the derived mean mass exponent is not size independent. Rather, the power function ratio standard constructed from the allometric mass exponent penalizes small and large lifters and systematically favors lifters of intermediate body size. The allometric model is therefore invalid and inappropriate for making dimensionless interclass comparisons of twoevent lifting total. To our knowledge, there is little evidence in the allometric modeling literature within the exercise sciences of such rigorous examination of the specific relationships between the scaled physiological variable and body size. In 1984, Croucher (4) concluded that weightlifting performance totals scaled by the power function ratio T/M^{0.58} were lower in the smaller and larger weight categories and that the “stronger” men were those in the middle weight divisions. On the basis of the findings in the present study, we contend that the larger and smaller men in Croucher’s analysis were likely suffering from the penalty of statistical artifact because of a poorly specified regression model. A similar finding was reported by Hester et al. (10) in their 1990 review of strengthhandicapping formulas, where analysis of the Croucher model residuals via Z scores indicated a consistent favoring of lifters in the middle of the mass distribution.
Using rigorous diagnostic procedures, we have clearly demonstrated that simple, linear correlations are a poor criterion with which to confirm a sizeindependent mass exponent. Inherent nonlinear relationships in the data were revealed that in this specific case invalidate the allometric model. We strongly advise researchers using loglinearized allometric scaling or any other model to conduct a detailed examination of the model residuals to ensure that the data fit the assumptions. This echoes the cautionary warnings given by Smith (19) that are only gradually disseminating to the exercise sciences.
Model respecification.
The results of the diagnostics applied to the allometric model strongly indicate that a secondorder polynomial
In the current study, fitting of separate secondorder polynomial models for men and women
Figure 5 illustrates the bestfit secondorder polynomial models for men and women. Visual comparison to the power function fits in Fig. 1 Breveals the superiority of the polynomial model, particularly if the data points representing the heaviest weight category are inspected. This underlines the point made previously that inclusion of the heavier lifters presents particular problems. It may be argued that because of the apparent breakdown of homogeneity of body shape, structure, and composition in the heaviest classes these lifters should not be included in the model. Our decision to include all classes was based on three criteria. First, the aim of our study is to demonstrate the diagnostic procedures required when allometric modeling is applied to physiological or performance data sets. In the exercise sciences, this almost always involves scaling the data for the whole sample. Of course, we acknowledge that, in the absence of estimates of fatfree mass, or preferably muscle mass, variance in body composition within the sample may distort the relationships derived. Second, Hunter et al. (11) suggested that strengthhandicapping models should evaluate the strengthbody weight relationship by including the full spectrum of body weights. Third, authorities in the biological sciences, including Gould (9), have cautioned against modeling restricted parts of the total size range, because spurious findings may result.
Inasmuch as the secondorder polynomial is essentially a multiple linear regression model, the same assumptions apply as for the loglinearized allometric model. Diagnostic procedures were equivalent to those outlined previously. Model residuals were normally distributed (KolmogorovSmirnov test, P > 0.5) and homoscedastic (no correlation between residuals and predictor variables, P > 0.05). Figure6 illustrates the twoevent total scaled for body size via the above quadratic models (Eqs.8 and 9 ) vs. body mass in men and women, respectively. Linear correlations revealed that the scaled twoevent total was indeed unrelated to body mass (P > 0.9). Examination of the model residuals for the women (Fig. 6 B) revealed no other nonlinear relationships. Visual inspection of the male data in Fig. 6 A indicates a slight hint of quadratic curvature. However, attempts to fit a secondorder polynomial to these residuals yielded a flat line. Hence, secondorder polynomial modeling of male and female weightlifting performance data was successful in providing a sizeindependent scaling index. The present findings support those reported by Sinclair (18) and Tittel and Wutscherk (22). The latter study also found a secondorder polynomial model when modeling the 1986 European Weightlifting Championships: T = 89.19 + 8.974M − 0.036M^{2}. Clearly, this set of coefficients is markedly different from those reported in the current study. This is to be expected, inasmuch as the model generated is intended to be sample specific and thus cannot be generalized to other weightlifting performance data. Hence, the present model lacks the predictive power and generalizability of the Sinclair model but is likely to be a more accurate scaling method for this particular data set. In the exercise sciences, samplespecific scaling approaches are usually required, because, for most variables and population groups, a truly robust and valid formula has not been demonstrated. For example, even for peak or maximal oxygen uptake, where most of the research efforts in scaling have been concentrated, a vast array of allometric mass exponents have been reported for particular populations (17). Hence, extreme caution must be exercised in generalizing scaling models derived from specific samples to other samples.
That veryshortterm maximal human muscular function in the present study is best described by secondorder polynomial models suggests a body mass limit beyond which maximal muscular function does not improve and may even begin to deteriorate. However, a caveat is essential here, in that remodeling of our data by a thirdorder (cubic) polynomial provides as good a fit as the secondorder polynomial and demonstrates an upturn at the end of the body mass range. Nevertheless, adult men >108 kg and women >83 kg are far less numerous in the population. Moreover, beyond a certain body mass any small increases in positive factors (including muscle mass) may be outweighed by negative effects of the larger body size. Because weightlifting is a shortterm, explosive power event, the heavier athlete must rapidly overcome the inertia of her/his own body mass as well as the loaded bar. In addition, weightlifting technique may be adversely affected, as excessive adipose tissue deposition, particularly in the abdominal and/or glutealfemoral region, may affect the correct vertical trajectory of the loaded bar. These potential negative influences of large body masses are underscored by the belief that there may be an upper limit to lean body mass in humans. Forbes (5) estimated that the upper limits for lean body mass are ∼100 kg in men and 60 kg in women. Hence, beyond body masses of ∼110–120 kg for men and 70–80 kg for women, increases in body mass may primarily represent gains in fat mass. Because only the somatotypical variable muscle mass causally influences maximal muscular function, it is possible that little or no advantage may be gained by increasing size beyond these limits. A caveat must be issued here, however, with respect to potential use of anabolicandrogenic steroids and other anabolic agents in this sample. Clearly, such use may influence any upper limits to lean body mass in humans. Furthermore, various patterns of anabolicandrogenic steroid use across body mass categories or between genders may have distorted the relationships between body mass and maximal muscular function presented here. Unfortunately, no information regarding use of anabolic agents was available in this sample.
The present study has clearly demonstrated the failure of allometric modeling to provide a sizeindependent mass exponent for veryshortterm maximal human muscular function indicated by elite Olympicstyle weightlifting performance. Residual diagnostics from the allometric model revealed that the derived power function ratio (T/M^{0.47}) systematically favored subjects of intermediate body size and penalized smaller and larger subjects. If allometry is to be widely used to model physiological or human performance data to partition out the influence of body size, all underlying model assumptions must be rigorously checked and satisfied. We echo the cautionary warnings from the biological sciences (1, 19) by suggesting that particular attention be paid to evaluation of whether the model is correctly specified. This can be achieved by a detailed examination of the model residuals. In the event of model misspecification, the model should be revised or respecified to increase its theoretical, physiological, and statistical validity. It is hoped that this more rigorous approach will prevent any potential indiscriminate application of allometric scaling.
Footnotes

Address for reprint requests: A. M. Batterham, School of Social Sciences, Centre for Sport Science, University of Teesside, Borough Rd., Middlesbrough, Cleveland TS1 3BA, UK (Email:A.Batterham{at}tees.ac.uk).
 Copyright © 1997 the American Physiological Society