Journal of Applied Physiology
Vol. 82, No. 2, pp. 693-697, February 1997

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

ABSTRACT

INTRODUCTION

METHODS

RESULTS

DISCUSSION

FOOTNOTES

REFERENCES

ABSTRACT

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 (O_{2 peak}) 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 [Medline] ) 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 , where c is the general body size exponent and  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 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 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.

allometry; multiple regression; log-linear models

INTRODUCTION

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 modeling such relationships. Huxley's general allometric equation (11)

(1)

has been most often employed, where Y, m, and  represent the physiological dependent variable, body mass, and the multiplicative error term, respectively. When modeling the intraspecific relationship between maximal oxygen uptake (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) suggested that, in humans, this would inflate the derived mass exponent because of the disproportionate increase in metabolically active musculature within the sample, resulting in a higher 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

(2)

Stature functions as a continuous covariate and is assumed to accurately reflect "body size." It was proposed that the technique permits the identification of a mass exponent, free from the confounding influence of disproportionate proximal leg muscle mass with increasing 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-transformed dependent and independent variables

(3)

This method has been applied to three sets of data involving adults and children (13, 19, 28), with originally 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 method (18) depends on high collinearity between the two body size variables, paradoxically, this is its primary flaw. Two predictor variables are collinear with each other if the data vectors representing them lie on, or close to, the same line (22). If collinearity is severe, multiple allometric regression equations may be unstable and unreliable, and the exponents derived may be numerically inaccurate (4). It has been demonstrated that severe collinearity may even result in exponent sign changes, indicating a directional relationship contrary to the investigator's knowledgeable expectations (17). Clearly, such concerns are especially important when attempting to distinguish within 95% confidence limits between 0.67 and 0.75 mass exponents, and they reduce confidence in the interpretation of least squares multiple-regression models.

Berlin and Antman (5) identified three early warning signs for collinearity: large pairwise correlations between predictor variables, large changes in coefficients caused by the addition or deletion of other variables, and inflated SE values for coefficients. Mandel (16) stated that collinearity is the greatest problem encountered when using least squares regression models. It is curious, therefore, that despite the widespread use of multiple-regression techniques physiologists in the exercise sciences have paid little attention to the diagnosis and treatment of collinearity. McGiffen et al. (17) have urged that collinearity diagnostics be calculated and reported for all applications of multiple least squares regression models. Although Nevill's method (18) has been employed with apparent success in three previous studies, it is plausible that the restoration of the mass exponent to the "anticipated" 2/3 is a fortuitous statistical artifact resulting from collinearity confounds. The aim of this study was to critically examine Nevill's explanation of Kleiber's 3/4 mass exponent in modeling peak oxygen uptake (O_{2 peak}) in prepubertal, teenage, and adult males.

METHODS

Table 1. Physical and physiological characteristics of subjects Variable Prepubertal Teenage Adult n 26 26 23 Stature, m 1.44 ± 0.04  1.63 ± 0.09  1.77 ± 0.06  Mass, kg 34.3 ± 3.9  49.5 ± 8.9  76.7 ± 9.4   O2 peak, l/min 1.82 ± 0.22  2.6 ± 0.47  4.18 ± 0.5 Values are means ± SD; n, no. of subjects. O2 peak, peak O2 uptake.

Measurement of O_{2 peak}.

(4)

(5)

RESULTS

Testing for the commonality of slopes revealed no differences between groups (P > 0.05) for either body mass or stature. Multivariate log-linear regression revealed common slopes of 0.81 (SE = 0.08, 95% confidence interval 0.65-0.97; R² = 0.93, P < 0.05) for body mass and 2.33 (SE = 0.33, 95% confidence interval 1.68-2.98; R² = 0.91, P < 0.05) for stature for the three groups. Log-linear modeling including stature alongside mass as predictor variables (Eq. 5) reduced the mass exponent to 0.67 (SE = 0.12, 95% confidence interval 0.44-0.9) and the stature exponent to 0.66 (SE = 0.4, 95% confidence interval 0.13-1.46). Analysis of the standardized regression coefficients (beta weights) indicated that only the contribution of the mass exponent was significant (P < 0.05).

Pairwise comparisons between log-transformed stature and mass revealed a strong positive correlation (r = 0.96, P < 0.05). For Nevill's method (18) (Eq. 5), tolerance and VIF for stature and body mass were 0.08 and 11.5 and 0.07 and 13.7, respectively. The results of the principal components diagnostics are displayed in Table 2.

Table 2. PC collinearity diagnostics for modeling of O2 peak by group, body mass, and stature PC No. Variance Decomposition Proportions Eigenvalue CI Intercept Adult Prepubertal ln Mass ln Stature 1 3.6956 1.0 0.00008 0.06 0.004 0.00004 0.00023 2 1.01584 1.9 0.00001 0.08 0.103 0.00000 0.00003 3 0.28464 3.6 0.00025 0.31 0.213 0.00013 0.00099 4 0.00354 32.3* 0.10513 0.19 0.641* 0.00692 0.5588* 5 0.00038 98.8* 0.8945* 0.41 0.037 0.99292* 0.43987 * Condition indexes (CI) and variance decomposition proportions indicative of collinearity problems. PC, principal component.

DISCUSSION

The values attained for O_{2 peak} (l/min; Table 1) are consistent with previous treadmill-derived O_{2 peak} in similar samples (2, 28). The mean mass exponent common to all three groups of 0.81 approximates the 3/4 mass exponent identified by Kleiber (14) for basal metabolic rate in a range of mammals. Similar common mass exponents for O_{2 peak} have been reported previously for children and adults (2, 23, 28). These mass exponents appear higher than the 2/3 exponent anticipated from dimensionality theory and have led some investigators to postulate mechanisms to explain why theoretical principles have failed to provide an adequate account (18, 28). In the present study, however, it is noteworthy that the 95% confidence interval for the mass exponent includes both 2/3 and 3/4. It is, therefore, impossible to confidently reject dimensionality theory predictions. Many investigators fail to report confidence intervals for the derived mass exponents. This information is essential if meaningful interpretations are to be made. The confidence intervals for mass exponents for O_{2 peak} in human studies appear wider than those cited in investigations with the use of animals. Taylor et al. (26) reported a mean mass exponent of 0.79 (95% confidence interval, 0.75-0.83) in a range of wild mammals for treadmill-derived O_{2 peak}. This finding precludes the anticipated 2/3 exponent and indicates that O_{2 peak} scales similarly to basal metabolism.

If the true mass exponent for O_{2 peak} in humans is >2/3, it is possible that the body size ranges commonly studied are insufficient to detect it within 95% confidence limits. Kleiber (15) required a ninefold size ratio to distinguish between the 3/4 and 2/3 mass exponents. Interestingly, the mean stature exponent in the present study of 2.33 (95% confidence interval, 1.68-2.98) also conformed to dimensionality theory predictions (O_{2 peak}  mass^2/3  stature²).

Despite the relatively wide confidence intervals, it is clear that the mean mass exponent more closely approximates 3/4 than the anticipated 2/3. Nevill's method (18) of including stature as a continuous covariate to restore the mean mass exponent to 2/3 appears to have a sound physiological basis (1). The results of the collinearity diagnostics in the present study, however, indicate that the method may be numerically inaccurate, unreliable, and, therefore, invalid. Low tolerances and high VIFs were evident for both mass and stature predictor variables, with values exceeding published criteria for severe collinearity (10). Nevill's approach (18) assumes that stature is an effective proxy for mass to reflect the "general size" of the body. This assumption is supported by the strong pairwise correlation between the log-transformed mass and stature variables (r = 0.96, P < 0.05). Unfortunately, however, this also alerts the investigator to potential collinearity problems (5) and indicates variable redundancy. With stature and mass included in the same multiple log-linear regression (Eq. 5), the SE values for the exponents were inflated considerably, in comparison with those generated when modeling stature and mass separately (Eq. 4). Nevill's method (18) also reduced the mean stature exponent dramatically from 2.33 to 0.66. Hence, the three early warning signs for collinearity problems identified by Berlin and Antman (5) are all present. Moreover, it is noteworthy that the analysis of the beta weights revealed that the stature exponent no longer contributed to the prediction of O_{2 peak} (P > 0.05), even though stature has a strong positive bivariate correlation with the dependent variable. Inclusion of stature alone as a body size variable explained 91% of the variance in O_{2 peak}. The lack of significance for the stature exponent in Eq. 5 is due to its inflated variance and may lead to the erroneous conclusion that stature is not important in determining O_{2 peak}. In addition, Nevill's method (18) provides a stature exponent that is physiologically and theoretically implausible.

The mass exponent was reduced from the original mean of 0.81 to 0.67, exactly the value predicted from theoretical considerations. Without due consideration of collinearity confounds, this would, again, appear to support Nevill's method, adding a further study to those included in his meta-analysis (18). Due to variance inflation, however, the confidence interval for the "restored" 0.67 exponent was widened to 0.44-0.9, a range that includes 2/3, 3/4, and the original 0.81 exponent. Similarly, in the most recent of the three studies where the method has been applied (28), the confidence interval for the restored mean mass exponent of 0.71 (calculated from the SE provided) was 0.59-0.83. Notwithstanding the collinearity problems, the method is clearly unable to distinguish between theoretically predicted and empirically derived mass exponents within reasonable confidence limits.

The results of the collinearity diagnostics derived from the principal components analysis raise further doubts about the reliability and validity of Nevill's method (18). Severe collinearity was detected, with values exceeding the criterion defined by Belsey et al. (4). CI values of 32.3 and 98.8, and VDPs >50% for two coefficients, were obtained in principal components 4 and 5, respectively (Table 2).

The warning signs for collinearity, the preliminary diagnostics via tolerance and VIFs, and the specific principal components diagnostics, all indicate that little confidence can be placed in Nevill's method (18). The overlap between stature and mass is such that including both in the model (Eq. 5) is a somewhat redundant method of indicating body size. Among their suggestions for coping with collinearity, Berlin and Antman (5) include removing redundant variables from the model and reducing reliance on interpretation of coefficients for confounding variables. Both these suggestions have serious implications for Nevill's method (18). A further suggestion for coping with collinearity is to form a "summary" variable (5). With body size, this is a difficult task, as variables such as stature, mass, and surface area are expressed in different units and are highly interrelated. Indeed, the selection of an appropriate size variable has been described as the fundamental problem confronting all studies of allometry (12). As stated by Nevill (18), a fundamental assumption is that the physiological variable is influenced by active muscle mass, with body mass usually used as a proxy in the absence of muscle mass estimates. Although body mass and muscle mass are related (8), differences in body composition within samples can severely distort derived mass exponents (9). Where possible, therefore, measurements or estimates of involved musculature should be the scaling variable of choice (31).

Notwithstanding these considerations, with a large sample size and a body size range sufficient to overwhelm body composition variance (7), meaningful mass exponents may be derived, offering an advantage in practicality over sophisticated muscle mass measurement methods (27). The fact that the mean mass exponents reported in the literature frequently exceed theoretical predictions should not encourage a confirmatory bias in research. Recent attempts to account for apparent inconsistencies with theoretical predictions are a modern parallel to the process identified by Kleiber 50 years ago (14). The belief in the "surface law" for basal metabolism was so well established that empirical deviations were explained by particular conditions or measurement error. For example, when rabbits failed to comply with predicted daily heat production, their ear surface was removed from the model to restore the theoretical relationship (14).

The present study has demonstrated that Nevill's (18) explanation of Kleiber's 3/4 mass exponent is confounded by the severe collinearity problems detected in the multiple least squares regression model (Eq. 5). Empirical evidence in humans appears unable to distinguish between the 2/3 and 3/4 exponent for O_{2 peak}, as both are supported by the data. Further research, using large sample sizes selected for body size heterogeneity, is required to more fully elucidate the relationship between body size and O_{2 peak} in humans.

FOOTNOTES

Address for reprint requests: A. M. Batterham, Dept. of Exercise and Sport Science, Manchester Metropolitan Univ., Crewe and Alsager Faculty, Hassall Rd., Alsager ST7 2HL, UK (E-mail: A.Batterham{at}mmu.ac.uk).

Received 26 April 1996; accepted in final form 4 October 1996.

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