If the assumptions are violated, T is compared with the wrong reference distribution, which may result in a deviation of the actual Type I error from the nominal significance level, in a loss of power relative to other tests developed for similar problems, or both. Where m X and m Y are the observed sample means, n X and n Y are the sample sizes of the two groups, and s is an estimate of the common standard deviation. If these assumptions are met and the null hypothesis of equal population means holds true, the test statistic T follows a t distribution with n X + n Y – 2 degrees of freedom: The test assumes independent sampling from normal distributions with equal variance. In Student’s t test, the expectations of two populations are compared. Altman, for example, stated that “most clinical trials yield data of this type, as do observational studies comparing different groups of subjects” (, p. Probably one of the most popular research questions is whether two independent samples differ from each other. criticized that contributors to medical journals often failed to examine and report that assumptions had been met when conducting Student’s t test. considered the use of parametric tests erroneous unless a test for normality had been conducted before. In this regard, violation of distributional assumptions has been identified as one of the most common problems: According to Olsen, a frequent error is to use statistical tests that assume a normal distribution on data that are actually skewed. Statistical tests have become more and more important in medical research, but many publications have been reported to contain serious statistical errors. The two-stage procedure might be considered incorrect from a formal perspective nevertheless, in the investigated examples, this procedure seemed to satisfactorily maintain the nominal significance level and had acceptable power properties. Interestingly, the overall Type I error rate and power of the entire two-stage procedure remained within acceptable limits. We discuss possible explanations for the observed results, the most important one being the selection mechanism due to the preliminary test. Preliminary testing for normality seriously altered the conditional Type I error rates of the subsequent main analysis for both parametric and nonparametric tests. Finally, we assessed the overall Type I error rate and the power of the two-stage procedure as a whole. By simulation, we separately estimated the conditional Type I error probabilities for the parametric and nonparametric part of the two-stage procedure. The two-sample t test was conducted if either both samples (Strategy I) or the collapsed set of residuals from both samples (Strategy II) had passed the preliminary Shapiro-Wilk test for normality otherwise, Mann-Whitney’s U test was conducted. MethodsĮqually sized samples were drawn from exponential, uniform, and normal distributions. The following two-stage procedure is widely accepted: If the preliminary test for normality is not significant, the t test is used if the preliminary test rejects the null hypothesis of normality, a nonparametric test is applied in the main analysis. Adequate control of the Type I error requires that the normality assumption holds, which is often examined by means of a preliminary Shapiro-Wilk test. Under the null hypothesis, the t test assumes that the two samples arise from the same normally distributed population with unknown variance. Select the category Probabilities and Distributions.Student’s two-sample t test is generally used for comparing the means of two independent samples, for example, two treatment arms. If the p-value is greater than 0.05, then the null hypothesis is not rejected.Īdditional information about assessing normality of data can be found in the JMP Learning Library. If the chosen alpha level is 0.05 and the p-value is less than 0.05, then the null hypothesis that the data are normally distributed is rejected. The Prob < W value listed in the output is the p-value. The null hypothesis for this test is that the data are normally distributed. The Shapiro-Wilk test for normality is available when using the Distribution platform to examine a continuous variable.

Shapiro wilk test