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Special Communications: Statistics

Multivariate allometric scaling of men's world indoor rowing championship performance


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Medicine & Science in Sports & Exercise: May 1996 - Volume 28 - Issue 5 - p 626-630
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Allometric scaling (AS) is a measurement tool that yields a convention of expressing an outcome variable, y, (e.g., ˙VO2max in ml·min-1 or strength in kp) relative to a scaling variable, x(height, body mass, body surface area, etc.) that is free of the undue influence of the scaling variable: y·x-a. In other words, the independent effects of the scaling variable on the outcome variables are partialed out. Examples of AS applied to empirical data include˙VO2max per unit of body mass (kg): ml·kg-0.7·min-1(2-5,8,9,12); 3200-m run time (RT) per unit of lean body mass (LBM): RT·LBM-0.3(11); and grip strength (GS): GS·kg-0.5(10). Such conventions allow for comparisons between individuals that take into account differences in the outcome variable due solely to body size. These comparisons are preferred to simple ratio scaling(dividing the outcome by the scaling variable) as the latter has often been shown to be no better in controlling for the scaling variable's effect than no scaling at all (6-8,13).

AS has also been shown to be useful in correlational research where proper statistical control of the scaling variable is essential to avoid spurious correlations (13). Furthermore, others have promoted the use of AS to examine differential growth of ratio relationships, for example, the change in aerobic power and/or economy (˙VO2 per unit of body mass) during childhood (9).

AS also has a significant theoretical basis. Physiological markers such as˙VO2max and strength are known (based on models of geometric similarity) to be proportional to blood vessel and muscle cross-sectional area, respectively (1). Cross-sectional area (two dimensional), in turn, is related to mass (three dimensional) to the 2/3 power. Therefore, expressing these outcome variables per unit of mass (as is so often done in exercise science) actually “overadjusts” for the effects of body size, thereby penalizing those with larger body size.

In fact, if desired, one can even partial out the effect of more than one scaling variable. Nevill et al. (8) discussed the use of a multivariate allometric scaling (MAS) technique to control for not just one but two or more body size variables, x1 and x2. Such a convention, y·x1-a·x2-b, is particularly useful because it has zero correlation with x1 and x2, thus demonstrating its statistical control for both scaling variables. MAS has rarely been applied in exercise science but could have particular utility in situations where meticulous statistical control for scaling variables is paramount. Such a situation might be competitive sports.

In competitive sports there are many events that are not only skill and conditioning dependent but also body size dependent. Some include weight lifting, high jumping, discus or shot putting, and wrestling. For these that do not have weight classes, relatively small individuals cannot possibly be competitive at an elite level because of the potency of body size as a determinant in performance. AS, however, is a tool that allows for the“handicapping” of certain events such that no weight classes are needed and differences in body size are properly accounted for before evaluating performance.

The World Indoor Rowing Championships (WIRC) is such an event that could benefit from MAS. This competition is based solely on one event: the time to row a simulated 2500 m on a stationary rowing ergometer (Concept II, Model C, Morrisville, VT). This ergometer provides its accommodating resistance not by friction but by air resistance on a fan, thus simulating the curvilinear relationship between power and speed found on the water. In other words, as rowing power increases, rowing speed increases but is disproportionately less. As this 2500-m event is nonweight bearing, and one's body weight has no impact on drag (as it would in water) then one would expect that body size would be one of the determinants of indoor rowing performance. Furthermore, because this event lasts at least 7 min (the world record time is 7:10.7), aerobic power, or ˙VO2max, is likely to be another determinant.˙VO2max, however, is widely known to decrease with age and to be lower in women than men. Therefore, performance in the WIRC should be partially dependent on age, gender, and body size.

To this end, the WIRC has several age categories but only two weight classes: light and heavyweight for both the male and female divisions. The cut-off weights for these categories are 61.2 and 74.8 kg for females and males, respectively. This tends to penalize those just heavier or much lighter than the cut-offs. In fact, the winners in each category, particularly in the highly competitive “open” category, are invariably just at the cut-off (for the lightweights) and much heavier than the cut-off for the heavyweights. Stated differently, we submit that a 43.5-kg woman and a 77.1-kg man, regardless of conditioning and skill level, have virtually no chance of winning their respective weight divisions.

Dimensional analysis, applied to rowing time and its body-size related determinants, revealed that rowing speed cubed should be proportional to rowing power (the fan law principle) which, in turn, should be proportional to body mass to the 2/3 power (1). Combining these two relationships indicates that rowing speed relates to body mass to the 2/9 power, nearly 1/3. Since height also relates to body mass to the 1/3 power(1), then rowing time (the inverse of rowing speed) should relate approximately to the inverse of height. In other words, we used dimensional analysis to hypothesize that the proper scaling convention for rowing performance (7) relative to height (H) would be T·H. Because this rowing performance is related to ˙VO2max which, in turn, is related to age (A), we further hypothesized that MAS would reveal that T·H·A-b, such thatb was significantly larger than 0, would be the optimal scaling convention of height and age-adjusted WIRC rowing performance for males.

The purpose of this investigation, then, was to demonstrate the use of MAS as applied to competitive male rowers from the 1995 WIRC to develop a scaling convention of rowing performance, T·H-a·A-b. Such a convention would eliminate the need for age categories and would provide for much more precise control of body size then the current system of only the light and heavyweight divisions.


Subjects were 148 competitive male rowers from the 1995 WIRC. As such, all had extensive familiarity with the 2500-m rowing test on the Concept IIC stationary rowing ergometer. Informed consent was obtained from each subject. Subject rowing ability ranged from world/Olympic class to club or intercollegiate. Subject descriptive data are shown in Table 1.


For most subjects, body mass and height were measured before the rowing event, but in some cases just after. As we expected no significant weight change in an event that lasted no more than 10 min, the order of events was arguably not relevant. Body mass was recorded to the nearest 0.25 kg and height to the nearest 0.5 cm using a calibrated Healthometer scale. For both, subjects wore a shirt, shorts (in some cases, ankle-length lycra pants), and no shoes.

The rowing event was accomplished in heats, that is, each subject rowed in groups of five or more at a time. Room temperature was approximately 21°C. The object of the race was to simply row 2500 m as fast as possible. Race data for each subject: rowing speed (in minutes per 500 m), stroke rate, distance, and time were visible on the ergometer's electronic instrumentation itself as well as on a central console, which also showed each individual's relative position compared with others in the heat.

While we could not possibly calibrate each of the over 100 rowing ergometers used for this event, the reliability of the Concept IIC has been well accepted in the competitive rowing field. The very fact that the WIRC and many similar indoor rowing events worldwide, all using the Concept IIC rowing ergometer, attract world-class rowers from many countries is testament to the ergometer's apparent consistency of calibration. Experienced rowers, furthermore, are extremely familiar with their splits on the rowing ergometer and are very likely to detect small differences in calibration in machines. Finally, all the ergometers used for this WIRC were the new C model, making age of ergometer an unlikely contributor to substandard calibration.

Whether the ergometer is valid, that is, provides an accurate measure of the time one would row 2500 m on the water, is not relevant as water times are rarely, if ever, compared to indoor rowing times. Furthermore, the purpose of this investigation was to demonstrate the use of MAS as applied to a competitive sports event. Applications of the results to other non-WIRC-format rowing competitions is not as important as the understanding of how MAS can be used in competitive sporting events.


Multivariate allometric scaling is based on the assumption that the best-fit curve of the relationship between the outcome variable, y, an two or more scaling variables, x1, x2, etc., is described by the following function (in this case, using only two scaling variables): Eqn 1

where a, b, and c are constants. Rearranging the equation yields: Eqn 2

In this case,y·x1-a·x2-b proportional to 1, and therefore has zero correlation with eitherx1, or x2. The value of c is simply a constant multiplier. The purpose of MAS, then, is to solve for the values of a and b.

This is done by log-transforming equation 1 into:Eqn 3

Since logc is a constant it becomes the intercept of a linear function of the relationship between logy, logx1, and logx2. Multiple regression analysis can now be used, regressing logy on logx, and logx2 to solve for the values of a and b, which are, in fact, slopes. This analysis yields not only their respective values but their confidence intervals as well. In this investigation, substituting time to row 2500 m (T), height (H), and age (A) into equation 3, yields the following log-linear relationship: Eqn 4


Multiple regression analysis was applied to the logT, logH, and logA values of 148 male subjects. The resulting values of a and b(±SEE) were: -0.937 ± 0.12 (95% CI: 1.18 - 0.69) and 0.061± 0.01 (95% CI: 0.041 - 0.08), respectively. As -1 is within the 95% CI for a, the height exponent, we can conclude that -1.0 is an acceptable exponent for H. Since -1.0 is the exponent, then dividing T by H-1.0 is the same as multiplying by H1. This is rather convenient as raising any number to the first power is simply that number itself. Furthermore, since neither -1 nor 0 is within the 95% CI forb, the age exponent, then one can say that the optimal exponent for age is not 0 or 1. We reapplied MAS holding a constant at -1.0 and foundb to remain at 0.06. The resulting convention, T·H·A-0.06, is an index of rowing performance free of the confounding effects of H and A. Calculating this index for each male subject of the WIRC, then, creates a score that can now be compared between individuals, regardless of H or A.

We also applied AS in a univariate manner, adjusting T only for H in those male competitors less than 40 yr of age (N = 109). This was done because the T·H·A-0.06 convention, while statistically proper, may not be feasible (i.e., raising age to the -0.06 power cannot be easily computed by hand) for competitive situations due to the exponential function regarding age. Furthermore, approximately 70% of this sample was under the age of 40, providing us with a fairly large age-homogeneous sample. Lastly, as mentioned previously, dimensional analysis allowed us to hypothesize that the H exponent should be, in fact, -1.0. If this were true, then the T·H1 or T·H convention would be not only statistically correct but would be very feasible as well. This analysis revealed that the H exponent was -0.900 ± 0.12 (95% CI: 1.15 - 0.65), thus indicating that T·H is an index of rowing performance that partials out the confounding effect of height for competitive male rowers less than 40 yr of age.


MAS indicated that the T·H·A-0.06 convention of expressing WIRC performance partialed out the effects of age (A) and height(H) for this representative sample of male competitive rowers. This affords the opportunity to create norms against which other male rowers, from other WIRC-format competitions, can be compared. The means ± SD for the T·H·A-0.06 and T·H scores were 807.1 ± 51.4 s·m·yr-0.06, and 938.0 ± 46.2 s·m. Statistical analysis of both frequency distributions via Kolmogorov-Smirnov tests indicated that both were normal (P > 0.05 for both). Norms for these conventions are shown in Table 2.

The finding that the exponent for height was not different from 1.0 is congruent with the aforementioned dimensional analysis. The exponent for age indicated approximately a 0.1% improvement in rowing performance score (lower score is a better score) for every 1-yr increase in age, for constant T and H. This follows from the fact that the convention T·H·A-0.06 is handicapping increasing age to overcome the overall increase in T with age. Unlike the current WIRC age bracket system, however, the present use of MAS allows for comparisons of any male rower to another, regardless of age. This comparison is demonstrated in Table 3. As indicated, use of the T·H·A-0.06 convention dramatically changes overall race results. For example, subject C moves from first place (among all 148 rowers) to 23rd after adjustment for height (1.981 m) and age (17 yr). Subject B, on the other hand (42 yr old and 1.765 m tall) moves from 38th place overall to third.

The effect of age on rowing performance also suggests that adolescent rowers are likely to be slower than their post-pubescent counterparts, largely due to maturity-related differences. For this reason, rowers who were younger than 17 were excluded from this analysis.

Table 4 shows similar WIRC results changes with the univariate (AS) convention, scaling T by H (the T·H convention). Most notable of overall rank changes is subject A (1.71 m) who moves from 51st to first place overall. Considering that he is more than 26 cm shorter in height than subject D, who originally finished in 1st place, subject A's performance can be considered rather remarkable, and perhaps more deserving of first place.

Allometric scaling, whether univariate or multivariate, allows for comparisons of individuals both from within and outside the sample. One need only compute his individual adjusted score (T·H·A-0.06 or T·H) and use Table 2 for determining his percentile rank. This is neither possible nor feasible under any other type of adjustment techniques, such as analysis of covariance, which only allows comparison of adjusted group means, not individual scores, or the residuals methods (7), which is sample-specific and not generalizeable to the population (10).

Some cautions about the use of AS or MAS are warranted. First, sample sizes should be quite large (and, of course, representative of a particular population). In small samples (N < 30), outliers have a significant impact on the magnitude of the scaling exponent. Second, scaling conventions can only be compared to one another if the units are identical. For example, one cannot compare one subject whose performance is in the T·H·A-0.06 form with another whose performance is expressed as T·H, or even T·H·A-0.09 because the units of each convention are not exactly the same. Third, different populations should not be combined into the same MAS or AS procedure without first considering the grouping variable (gender, for example) as a separate independent predictor. Failure to account separately for gender often has a spurious effect on the magnitude of the scaling exponent. Vanderburgh et al.(10,12) detail more discussion about this point. Lastly, different populations should not be compared with each other without first verifying that their respective exponents are the same, or similar. This is analogous to ensuring that, in analysis of covariance, the slopes of the covariate-dependent variable regression lines are similar. If the exponents are not similar, comparison should not be done using AS. Nevill et al.(8) elaborate further on this issue.

We conclude that rowing performance according to the present standards of the WIRC is heavily biased toward those of larger height and, to some degree, younger age. Multivariate and/or univariate scaling can be used to adjust WIRC scores by height and age so that both can be statistically accounted for in determining overall race success. For evaluation of male competitive rowers of a wide range of ages, we recommend use of the T·H·A-0.06 convention. For evaluation of those older than 16 and less than 40, the T·H index is most feasible and appropriate. While actual use of the former may be a somewhat utopian expectation for WIRC-like competitions, the latter is as feasible as computing a simple ratio.


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©1996The American College of Sports Medicine