SCALING, NORMALIZING, AND PER RATIO STANDARDS - AN ALLOMETRIC MODELING APPROACH

The practice of scaling or normalizing physiological variables (Y) by dividing the variable by an appropriate body size variable (X) to prod uce what is known as a ''per ratio standard'' (Y/X), has come under st rong criticism from various authors. These authors propose an alternat ive regression standard based on the linear regression of(Y) on (X) as the predictor variable. However, if linear regression is to be used t o adjust such physiological measurements (Y), the residual errors shou ld have a constant variance and, in order to carry out parametric test s of significance, be normally distributed. Unfortunately, since neith er of these assumptions appear to be satisfied for many physiological variables, e.g., maximum oxygen uptake, peak and mean power, an altern ative approach. is proposed of using allometric modeling where the con cept of a ratio is an integral part of the model form. These allometri c models naturally help to overcome the heteroscedasticity and skewnes s observed with per ratio variables. Furthermore, if per ratio standar ds are to be incorporated in regression models to predict other depend ent variables, the allometric or log-linear model form is shown to be more appropriate than linear models. By using multiple regression, sim ply by taking logarithms of the dependent variable and entering the lo garithmic transformed per ratio variables as separate independent vari ables, the resulting estimated log-linear multiple-regression model wi ll automatically provide the most appropriate per ratio standard to re flect the dependent variable, based on the proposed allometric model.