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Vol. 83, Issue 6, 2158-2166, December 1997
1 School of Social Sciences, 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.
polynomial models; regression diagnostics; scaling; weightlifting
INTRODUCTION 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 (14-16, 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 × massb × Allometric modeling is most often conducted via linear, least-squares
regression techniques applied to log-transformed dependent and
independent variables (ln y = ln
a + b
ln x + ln The specific aims of the current study were to
1) scale very-short-term 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, and
3) 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, very-short-term maximal human muscular function
is indicated by performances in elite, Olympic-style weightlifting
(snatch and clean-and-jerk 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 "jerk-drive"
phase of the clean-and-jerk increased in a nonlinear fashion across
body mass categories, ranging from 2,140 W in the 56-kg class to 4,786 W for a 110-kg 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
very-short-term 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, Olympic-style 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 clean-and-jerk) as a
function of body mass, using the log-transformed 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 two-thirds power [since muscle
strength (S) In 1984, Croucher (4) repeated Lietzke's log-linear models (13) for
the men's snatch and clean-and-jerk 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 log-log 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 90-kg 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 log-linearized 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 52-kg class and
then log transformed). The result was not a linear fit, but a
second-order polynomial, thus violating the allometric model. A similar
finding was reported by Tittel and Wutscherk (22) when modeling
two-event 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
strength-handicapping 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 using
Z scores (11) revealed an arithmetic
advantage for male lifters >120 kg. For women, the Sinclair model
demonstrated a much higher coefficient of variation, and
Z 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 log-linearized 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
In the exercise sciences, simple allometry
(y = axb) 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
"clean-and-jerk" 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 log-linearized allometric model (ln T = ln
a + b
ln M) yielded a common mass exponent
(b) of 0.47 (95% confidence
interval = 0.43-0.51, P < 0.01). However, size-related 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 second-order 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)M2
(R2 = 0.97) for men and T = (20.73 ± 24.14) + (5.662 ± 0.722)M (0.031 ± 0.005)M2
(R2 = 0.92) for women. We conclude that allometric scaling should be
applied only when all underlying model assumptions have been rigorously
evaluated.
) 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 (if
b 
1). A number of authors (16,
23, 26, 29-31) 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/massb.
Thus allometric modeling is held to permit meaningful intersubject or
intergroup comparisons free from the confounding influence of body
size.
). 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 log-linear 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 log-linear,
least-squares 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
log-transformed 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 ( or ln ) as a function
of the body size variable and checking for size-related 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), 2-mile run time (25), short-term peak and mean power output in a 30-s supramaximal arm ergometry test (12), and short-term peak and mean power output in a 30-s 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 log-linear allometric models.
muscle cross-sectional area height2 and mass height3, so S mass2/3]. Although no
summary statistics for the model were reported, it appears that
Lietzke's log-log plot (Fig. 1 in Ref. 13) resulted in a near-perfect
linear fit, with a slope of 0.6748, thus confirming the predictions
from dimensionality theory.
|
(1) |
|
(2) |
) was examined
via the Kolmogorov-Smirnov one-sample 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 log-linear regression model was primarily
investigated via detailed examination of the residuals. The ability of
the allometric model to provide a size-independent mass exponent was
evaluated by examining the specific relationship between the scaled
physiological variable (T/Mb)
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
Figure 1A is a scatter plot of two-event total vs. body mass in men and women. The apparent nonlinear, exponential nature of the plots suggested that a log-log 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 log-linearized general allometric equation (ln y = ln a + b ln x) yielded the following results
|
(3) |
|
(4) |
|
|
(5) |
, Eq. 2) vs. the predictor variable (ln M). If the
log-linear 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 log-linear allometric model thus appears to be wrongly specified. To confirm that
the log-log plots were not best represented by a straight-line fit, a
second-order polynomial was fitted to the log-transformed data (Fig.
3). The
R2 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 log-linear model
of the 1991 Men's World Records (sum of snatch and clean-and-jerk
lifts), Zatsiorsky plotted total weight lifted as a function of body
mass. The result was a straight line with a slope of 0.646 (R2 = 0.942). This mass exponent is close to the two-thirds value predicted
and confirmed by Lietzke (13). However, inspection of Zatsiorky's
log-log 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).
) resulting from allometric model
(Eq. 2) vs. predictor variable ln
body mass. Solid line, zero line.
Figure 4 displays the relationships between the allometrically scaled physiological variable (T/Mb) and body mass in men and women. Linear correlations revealed no relationship between body mass and T/M0.47 (P > 0.05). Visual inspection, however, indicated a quadratic curvature. Subsequent fitting of a second-order polynomial revealed significant relationships for men and women (R2 = 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 two-event 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/M0.58 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 strength-handicapping 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 size-independent mass exponent. Inherent nonlinear relationships in the data were revealed that in this specific case invalidate the allometric model. We strongly advise researchers using log-linearized 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 second-order polynomial
|
(6) |
|
(7) |
|
(8) |
|
(9) |
Inasmuch as the second-order polynomial is essentially a multiple linear regression model, the same assumptions apply as for the log-linearized allometric model. Diagnostic procedures were equivalent to those outlined previously. Model residuals were normally distributed (Kolmogorov-Smirnov test, P > 0.5) and homoscedastic (no correlation between residuals and predictor variables, P > 0.05). Figure 6 illustrates the two-event 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 two-event total was indeed unrelated to body mass (P > 0.9). Examination of the model residuals for the women (Fig. 6B) revealed no other nonlinear relationships. Visual inspection of the male data in Fig. 6A indicates a slight hint of quadratic curvature. However, attempts to fit a second-order polynomial to these residuals yielded a flat line. Hence, second-order polynomial modeling of male and female weightlifting performance data was successful in providing a size-independent scaling index. The present findings support those reported by Sinclair (18) and Tittel and Wutscherk (22). The latter study also found a second-order polynomial model when modeling the 1986 European Weightlifting Championships: T = 89.19 + 8.974M
0.036M2. 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, sample-specific 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 very-short-term maximal human muscular function in the present study is best described by second-order 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 third-order (cubic) polynomial provides as good a fit as the second-order 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 short-term, 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 gluteal-femoral 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 anabolic-androgenic 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 anabolic-androgenic 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 size-independent mass exponent for very-short-term maximal human muscular function indicated by elite Olympic-style weightlifting performance. Residual diagnostics from the allometric model revealed that the derived power function ratio (T/M0.47) 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 (E-mail: A.Batterham{at}tees.ac.uk).
Received 29 October 1996; accepted in final form 13 August 1997.
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ABSTRACT