Conventional wisdom suggests that for small data sets having substantial skew, one should attempt to determine the correct distributional form, if possible, and apply statistical methods appropriate for that distribution. Transformations such as the log or square root are often used. If an appropriate distributional form cannot be determined, a distribution-free procedure such as a rank transformation or a randomization test procedure can be used. To better appreciate the effect of such alternatives on both the type I error and power of detecting differences between treatment groups, simulation studies were conducted for responses having specific gamma G(r, Ø) and log-normal In(M, V) distributions. The gamma and log-normal distributions were selected so that they had the same first two moments. A simple two group design was assumed. The reference group always had an average disease level μ = 3.0 (μ = rø for gamma, μ = 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 statistical test procedures were compared for the gamma distributions. All test procedures were robust relative to the type I error. The UMP test based on a ratio of sample means produced the greatest power for all combinations of n, r and RT. The power losses associated with the randomization 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 absolute 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 scale was much larger, ranging from 5 per cent to 10 per cent smaller power 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 power for the shifted LOG (X + c) test increased monotonically to the asymptotic value of the ORE t-test. The same five test procedures based on 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 SQRT t-test. The loss in power varied between 2 per cent and 5 per cent for the RANK test. The RANK test performed considerably better than the t-test on the original scale. In contrast to the results for the gamma the power for the shifted LOG (X + c) test had its maximum for c = 0, and decreased monotonically to the asymptotic value of the ORIG t-test.
The results suggest that statistical inferences can be highly dependent on the distributional form and the scale of measurement of the response used in the statistical analysis.