Scaling, normalizing, and 'per ratio' standards; an allometric modeling approach. Nevill, Alan. M., Roger. L. Holder. School of Sport and Exercise Sciences, University of Birmingham, UK, School of Mathematics and Statistics, University of Birmingham, UK,
APStracts 2:0199A, 1995.
The practice of scaling or normalizing physiological variables (Y), by dividing the variable by an 'appropriate' body size variable (X), to produce what is known as a 'per ratio' standard (Y.X-1), has come under strong criticism from various authors. These authors propose an alternative 'regression standard' based on the linear regression of (Y) upon (X) as the predictor variable. However, if linear regression is to be used to adjust such physiological measurements (Y), the residual errors should have a constant variance and, in order to carry out parametric tests of significance, be normally distributed. Unfortunately, since neither of these assumptions appear to be satisfied for many physiological variables, e.g. maximum oxygen uptake, peak and mean power, an alternative approach is proposed using allometric modeling where the concept of a ratio is an integral part of the model form. These allometric models naturally help to overcome the heteroscedasticity and skewness observed with 'per ratio' variables. Furthermore, if 'per ratio' standards are to be incorporated in regression models to predict other dependent variables, the allometric or log-linear model form is shown to be more appropriate than linear models. Using multiple regression, simply by taking logarithms of the dependent variable and entering the logarithmic transformed 'per ratio' variables as separate independent variables, the resulting estimated log-linear multiple regression model will automatically provide the most appropriate 'per ratio' standard to reflect the dependent variable, based on the proposed allometric model. 

Received 25 July 1994; accepted in final form 4 May 1995.
APS Manuscript Number A757-4.
Article publication pending Journal of Applied Physiology.
ISSN 1080-4757 Copyright 1995 The American Physiological Society.
Published in APStracts on 26 May 1995.