Alternatives To The t-test For Location Problems
The t-test provides an exact test for the equality of the means of two normal populations with unknown, but equal, variances. (The Welch's t-test is a nearly exact test for the case where the data are normal but the variances may differ.) For moderately large samples and a one tailed test, the t is relatively robust to moderate violations of the normality assumption.
For exactness, the t-test and Z-test require normality of the sample means, and the t-test additionally requires that the sample variance follows a scaled χ2 distribution, and that the sample mean and sample variance be statistically independent. Normality of the individual data values is not required if these conditions are met. By the central limit theorem, sample means of moderately large samples are often well-approximated by a normal distribution even if the data are not normally distributed. For non-normal data, the distribution of the sample variance may deviate substantially from a χ2 distribution. However, if the sample size is large, Slutsky's theorem implies that the distribution of the sample variance has little effect on the distribution of the test statistic. If the data are substantially non-normal and the sample size is small, the t-test can give misleading results. See Location test for Gaussian scale mixture distributions for some theory related to one particular family of non-normal distributions.
When the normality assumption does not hold, a non-parametric alternative to the t-test can often have better statistical power. For example, for two independent samples when the data distributions are asymmetric (that is, the distributions are skewed) or the distributions have large tails, then the Wilcoxon Rank Sum test (also known as the Mann-Whitney U test) can have three to four times higher power than the t-test. The nonparametric counterpart to the paired samples t test is the Wilcoxon signed-rank test for paired samples. For a discussion on choosing between the t and nonparametric alternatives, see Sawilowsky.
One-way analysis of variance generalizes the two-sample t-test when the data belong to more than two groups.
Read more about this topic: Student's t-test
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