Scaling, normalizing, and per ratio standards: an allometric modeling approach

Abstract

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), has come under strong criticism from various authors. These authors propose an alternative regression standard based on the linear regression of (Y) on (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 of 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. By 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.