SOME POWER CONSIDERATIONS WHEN DECIDING TO USE TRANSFORMATIONS

Conventional wisdom suggests that for small data sets having substanti al skew, one should attempt to determine the correct distributional fo rm, if possible, and apply statistical methods appropriate for that di stribution. Transformations such as the log or square root are often u sed. If an appropriate distributional form cannot be determined, a dis tribution-free procedure such as a rank transformation or a randomizat ion test procedure can be used. To better appreciate the effect of suc h alternatives on both the type I error and power of detecting differe nces between treatment groups, simulation studies were conducted for r esponses having specific gamma G(r, theta) and log-normal In(M, V) dis tributions. The gamma and log-normal distributions were selected so th at they had the same first two moments. A simple two group design was assumed. The reference group always had an average disease level mu = 3.0 (mu = r theta for gamma, mu = M for log-normal), and the treatment group always had means whose reductions ranged from 0 per cent to 50 per cent. The effect of distributional type and the degree of skewness was investigated by varying the population parameter values. Six stat istical test procedures were compared for the gamma distributions. All test procedures were robust relative to the type I error. The UMP tes t based on a ratio of sample means produced the greatest power for all combinations of n, r and R(T) The power losses associated with the ra ndomization test, the t-test on original scale, and the t-test on the square root scale were very small, (3 per cent to 6 per cent in absolu te value) for n = 10 and 15, and less than 2 per cent for group sizes of 25 or more. The power loss associated with the t-test on the log sc ale was much larger, ranging from 5 per cent to 10 per cent smaller po wer than the t-test on original scale. The Wilcoxon rank test produced similar results to that of the LOG t-test for small samples. The powe r for the shifted LOG (X + c) test increased monotonically to the asym ptotic value of the ORIG t-test. The same five test procedures based o n differences in sample means were then compared for the corresponding log-normal distributions. The UMP test, that is, LOG(X), produced the highest power. There was very little power lost for the SORT t-test. The loss in power varied between 2 per cent and 5 per cent for the RAN K test. The RANK test performed considerably better than the t-test on the original scale. In contrast to the results for the gamma the powe r for the shifted LOG (X + c) test had its maximum for c = 0, and decr eased monotonically to the asymptotic value of the ORIG t-test. The re sults suggest that statistical inferences can be highly dependent on t he distributional form and the scale of measurement of the response us ed in the statistical analysis.