On the effect of using ratios in the analysis of variance

In experimental designs used by a number of disciplines, raw data are transformed into ratios and statistical analyses are performed on the ratio data. Although the designs themselves may be appropriate, the mathematical properties induced by ratio transformations can produce a loss of sensitivity in statistical tests. Since there are times when such designs must be used, it would be desirable to characterize the circumstances under which the statistical analysis of ratio data is or is not appropriate. To provide such information, a computer simulation approach was used to generate bivariate normal observations X and Y which were analysed using: (1) an analysis of variance ignoring the covariate; (2) an analysis of covariance; (3) an analysis of variance on the ratio Y/X. Comparisons were made between the three models by accumulating the number of rejections (1−β) on critical F-values for each model. Results taken from over a million analyses are discussed for a wide range of specific treatment effects and known correlations between the independent variable X and the dependent variable Y.