modeling in physiology Nevill’s explanation of Kleiber’s 0.75 mass exponent: an artifact of collinearity problems in least squares models? ALAN M. BATTERHAM, KEITH TOLFREY, AND KEITH P. GEORGE Department of Exercise and Sport Science, Crewe and Alsager Faculty, Manchester Metropolitan University, Alsager ST7 2HL, United Kingdom Batterham, Alan M., Keith Tolfrey, and Keith P. George. Nevill’s explanation of Kleiber’s 0.75 mass expo- nent: an artifact of collinearity problems in least squares models? J. Appl. Physiol. 82(2): 693–697, 1997.—Intraspe- cific allometric modeling (Y 5 a · mass b , where Y is the physiological dependent variable and a is the proportionality coefficient) of peak oxygen uptake (V ˙ O 2 peak ) has frequently revealed a mass exponent (b) greater than that predicted from dimensionality theory, approximating Kleiber’s 3/4 expo- nent 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 5 ln a 1 c · ln stature 1 b · ln mass 1 ln e, where c is the general body size exponent and e 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 modeling V ˙ O 2 peak 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 vari- ance decomposition proportions exceeding 50% for two regres- sion coefficients. The derived exponents may thus be numeri- cally inaccurate and unstable. In conclusion, the restoration of the mean mass exponent to the anticipated 2/3 may be a fortuitous statistical artifact. allometry; multiple regression; log-linear models RECENTLY , IN THE HUMAN SCIENCES, there has been a renewed interest in the influence of body size on selected physiological measurements. Several authors (20, 21, 29) have demonstrated the statistical and physiological validity of allometric equations in model- ing such relationships. Huxley’s general allometric equation (11) Y 5 a · m b · e (1) has been most often employed, where Y, m, and e represent the physiological dependent variable, body mass, and the multiplicative error term, respectively. When modeling the intraspecific relationship between maximal oxygen uptake (V ˙ O 2 max ; in l/min) and body mass, dimensionality theory predicts a mass exponent (b ) of 2/3 (3). A number of studies (2, 23, 28), however, have reported mean mass exponents greater than anticipated, often closer to the 0.75 exponent identified by Kleiber for basal metabolic rate (14). Nevill (18) proposed an explanation for these findings derived from Alexander et al. (1), who found that larger mammals have a greater proportion of proximal leg muscle mass in relation to their total body mass (leg muscle mass proportional to m 1.1 ). Nevill (18) sug- gested that, in humans, this would inflate the derived mass exponent because of the disproportionate in- crease in metabolically active musculature within the sample, resulting in a higher V ˙ O 2 max than anticipated from body mass. To accommodate this confound when modeling physiological variables in human subjects, Nevill argued that ‘‘stature’’ be entered together with body mass in a multiple allometric regression model Y 5 a · stature c · mass b · e (2) Stature functions as a continuous covariate and is assumed to accurately reflect ‘‘body size.’’ It was pro- posed that the technique permits the identification of a mass exponent, free from the confounding influence of disproportionate proximal leg muscle mass with increas- ing body size within the group. The proportionality coefficient (a) and the body size exponents (c and b ) are derived from multiple linear regression of log-trans- formed dependent and independent variables ln Y 5 ln a 1 c ln stature 1 b ln mass 1 ln e (3) This method has been applied to three sets of data involving adults and children (13, 19, 28), with origi- nally inflated mass exponents subsequently restored to the anticipated 2/3. As the theoretical basis for Nevill’s method (18) is derived from the work of Alexander et al. (1), it depends on the validity of stature as a proxy for body mass to accurately reflect ‘‘general body size’’ (6). Alexander et al. (1) did not provide data to evaluate the strength of the relationship between linear dimensions and mass in their sample. However, it can safely be assumed that, in humans, there is a strong relationship between stature and mass (3). Unfortunately, whereas Nevill’s 0161-7567/97 $5.00 Copyright r 1997 the American Physiological Society 693