Statistical Properties of Ratios. I. Empirical Results

Abstract

Atchley, W. R., C. T. Gaskins, and D. Anderson (Departments of Biological Sciences, Animal Sciences, and Biomedical Engineering and Computer Medicine, Texas Tech University, Lubbock, Texas 79409) 1976. Statistical properties of ratios. I. Empirical results. Syst. Zool. 25:137–148.—Results are presented of an empirical analysis on the statistical consequences of compounding ratios of continuous variables. Three commonly employed relationships among the ratio variables Y and Z were examined including 1) Y = X1/X2, Z = X3/X2; 2) Y = X1/X2, Z = X1; and 3) Y = X1/X2, Z = X2. Simulation studies with minimum sample sizes of 25,000 indicated large and systematic changes in both the structure and the underlying distributions of data when ratios and proportions were compounded between continuous variables. Ratio variables are skewed to the right and leptokurtic and the nonnormality is increased when magnitude of the denominator coefficient of variation is increased. Further, there is a pronounced increase in spurious correlation between variables when ratios are compounded and the magnitude of this spurious correlation is a function of the size of the denominator coefficient of variation. This spurious correlation may increase from r = 0.0 between the original raw variables to r = 0.99 between the derived ratio variables. Multivariate statistical procedures such as principal components analysis are greatly affected when the data upon which the analyses are based include ratios or proportions. In this case, there is often an inflation of the first eigenvalue together with large changes in the magnitude and direction of the coefficients on the various principal components. Several common applications of ratios in biological research are discussed. Contrary to a widely held belief, it is shown that in the scaling of data, ratios do not remove the effect of the scaling variables but rather increase the correlation between the ratio variable and the original scaling variable.