## Abstract

The following is the abstract of the article discussed in the subsequent letter:

**Batterham, Alan M., Keith Tolfrey, and Keith P. George.** Nevill’s explanation of Kleiber’s 0.75 mass exponent: an artifact of collinearity problems in least squares models? *J. Appl. Physiol.* 82(2): 693–697, 1997.—Intraspecific allometric modeling (Y = a ⋅ mass^{b}, where *Y* is the physiological dependent variable and *a*is the proportionality coefficient) of peak oxygen uptake (V˙o
_{2peak}) has frequently revealed a mass exponent (*b*) greater than that predicted from dimensionality theory, approximating Kleiber’s 3/4 exponent for basal metabolic rate. Nevill (*J. Appl. Physiol. *77: 2870–2873, 1994) proposed an explanation and a method that restores the inflated exponent to the anticipated 2/3. In human subjects, the method involves the addition of “stature” as a continuous predictor variable in a multiple log-linear regression model: ln *Y* = ln *a* +*c* ⋅ ln stature + *b* ⋅ ln mass + ln ε, where *c* is the general body size exponent and ε is the error term. It is likely that serious collinearity confounds may adversely affect the reliability and validity of the model. The aim of this study was to critically examine Nevill’s method in modelingV˙o
_{2peak} in prepubertal, teenage, and adult men. A mean exponent of 0.81 (95% confidence interval, 0.65–0.97) was found when scaling by mass alone. Nevill’s method reduced the mean mass exponent to 0.67 (95% confidence interval, 0.44–0.9). However, variance inflation factors and tolerance for the log-transformed stature and mass variables exceeded published criteria for severe collinearity. Principal components analysis also diagnosed severe collinearity in two principal components, with condition indexes >30 and variance decomposition proportions exceeding 50% for two regression coefficients. The derived exponents may thus be numerically inaccurate and unstable. In conclusion, the restoration of the mean mass exponent to the anticipated 2/3 may be a fortuitous statistical artifact.

## Collinearity: A Function of the Sample Size, Range, and Similarity of Observations

*To the Editor*: I wish to express my surprise that the*Journal of Applied Physiology* decided to publish the paper by Batterham et al. (2) entitled “Nevill’s explanation of Kleiber’s 0.75 mass exponent: an artifact of collinearity problems in least squares models?”. Based on a sample size of *n* = 75 (prepubertal, teenage, and adult males), the authors examined how peak oxygen uptake (V˙o
_{2}; 1 min) might be scaled for differences in body size. Although the study confirms the work of Nevill (8) that an inflated mass exponent occurs (0.81) if mass alone is used to scale peakV˙o
_{2}, the authors are concerned that by including stature in addition to mass as a second covariate, the result may lead to collinearity.

The authors’ concerns are unfounded. There are numerous examples in the literature when both height and weight have been used to predict various dependent variables, e.g., body fat (1, 5, 7), blood pressure (6, 9), and the dependent variable of interest, peakV˙o
_{2} (3, 8, 10, and unpublished observations of Nevill and Holder). In all these papers, the contributions of height and weight were both significant, and no collinearity was reported. For example, when blood pressure was used as the dependent variable (see Refs. 6, 9), the contributions from both height and weight provided highly significant exponents (*P* < 0.0001) but with opposite signs, i.e., predicting a stature-adjusted weight index (approximately the body mass index ratio) as a measure of “overweight.”

There also exists overwhelming evidence that both height and weight are necessary when predicting peakV˙o
_{2}, i.e., taller subjects have significantly greater peakV˙o
_{2}, having already controlled for differences in body weight (3, 8, 10, and unpublished observations of Nevill and Holder). The sample sizes used in these studies were as follows; Baxter Jones et al. (3), *n* = 453 young elite athletes (231 men and 222 women); Welsman et al. (10), *n*= 156 school children (73 boys and 83 girls), and Nevill and Holder (unpublished observations), *n* = 1,732 adults, age 16 yr and over (852 men and 880 women). The strength of collinearity observed in Batterham et al. (2) (sample size *n* = 75) may be a function of the sample size and the range and similarity of observations (4), a problem more evident in Batterham et al. (2) than in the studies of Baxter-Jones et al. (3), Nevill (8), Welsman et al. (10), and Nevill and Holder (unpublished observations).

Nevill (8) suggested that the significant contribution of height in addition to body weight might be explained by a disproportionate increase in muscle mass with body size. However, based on the unpublished findings of Nevill and Holder, an alternative explanation now becomes apparent. The best model to predict maximalV˙o
_{2} incorporated a significant contribution from forced vital capacity (FVC; also known to be predominately stature related) as well as body mass, whereas the height term became redundant, no longer making a significant contribution to the regression model. Clearly, on the basis of these findings, the apparent advantage of being taller would appear to be more accurately explained by the subjects’ greater FVC. In effect, the significant contribution of the height term, found in the earlier studies (3, 8, 10), would appear to have been inadvertently providing an estimate of the subjects’ FVC. This was an unexpected result, although not entirely surprising. Taking the analogy with the motor car engine, we would expect a greater performance (maximum speed and power output) from an engine with a greater cubic capacity, so perhaps we might also expect a greater maximumV˙o
_{2} from a subject with a greater (forced) vital capacity.

- Copyright © 1997 the American Physiological Society

## REFERENCES

## REPLY

*To the Editor*: We are delighted to provide a rebuttal to Nevill’s letter. Nevill suggests that the collinearity reported in our paper (1-2) may be solely a function of a small sample size *n* and the range and similarity of observations. He attempts to bolster this argument by citing studies with larger sample sizes that allegedly provide evidence for the necessity of including both height and mass when predicting peak V˙o
_{2}. Interestingly, however, none of the cited studies report any appropriate collinearity diagnostics. If it could be demonstrated that collinearity was not present in the studies with larger sample sizes, then some support for Nevill’s position would be provided. Without this information, it is impossible for the reader to evaluate the validity of the obtained regression coefficients, which may well be very unstable and numerically inaccurate in the presence of severe collinearity (1-3). We are perplexed by Nevill’s apparent lack of concern regarding collinearity problems. In a 1994 study by Nevill and Holder (1-5) modeling maximumV˙o
_{2}, the authors’ state (p. 659): “in models which set out to distinguish between the effect of several influencing factors on a dependent variable, another desirable feature of the model is to obtain relatively low correlations between parameter estimates… . Not only are high correlations a nuisance at the inference stage, they can also cause instability in the numerical algorithms used to derive the parameter estimates.”

In a “recommendations” section of that paper, Nevill and Holder suggest the use of collinearity diagnostics of the type employed in our paper (1-2).

We must not, however, allow this debate to obscure the main issue, i.e., Nevill’s original explanation of Kleiber’s 0.75 mass exponent (1-4). In our reply to this article (1-2), we stated that this explanation was dependent on collinearity between height and mass within the sample. This was because the theoretical basis was derived from Alexander et al. (1-1), who found that proximal leg muscle mass was proportional to mass^{1.1}. Nevill attempted to account for any disproportionate increase in leg muscle mass with body mass increase, by inserting height as a proxy for body mass, to accurately reflect “general body size.” This was indeed a “conceptual leap,” as Alexander et al. did not report the relationship between mass and linear dimension in their sample of mammals. If, as Nevill argues, the collinearity found in our study was an artifact of the sample size and characteristics, then the explanation is flawed in any case. If there is no collinearity between height and mass, then height cannot be used as a surrogate of mass in the model—quod erat demonstrandum.

As Nevill’s explanation and method have been thoroughly debunked, he now appears to be attempting to smuggle in a “new” explanation, albeit with unpublished data, involving a significant contribution of FVC in the prediction of maximalV˙o
_{2}. We are not surprised that “the height term became redundant,” as this would be expected in the presence of severe collinearity. Interestingly, despite admitting in their unpublished manuscript that FVC is “predominantly stature related,” Nevill and Holder proceed to include height, FVC, mass, and a host of other variables in the regression model. It is revealing that, once again, no collinearity diagnostics are reported. We suspect that these diagnostics may invalidate the findings.

- Copyright © 1997 the American Physiological Society