Alternatives to the t-Test

When the normality assumption does not hold, a nonparametric alternative to the $t$-test can often have better statistical power.

Learning Objective

Explain how Wilcoxon Rank Sum tests are applied to data distributions

Key Points

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.
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.
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 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.

Terms

central limit theorem
The theorem that states: If the sum of independent identically distributed random variables has a finite variance, then it will be (approximately) normally distributed.
Wilcoxon Rank Sum test
A non-parametric test of the null hypothesis that two populations are the same against an alternative hypothesis, especially that a particular population tends to have larger values than the other.
Wilcoxon signed-rank test
A nonparametric statistical hypothesis test used when comparing two related samples, matched samples, or repeated measurements on a single sample to assess whether their population mean ranks differ (i.e., it is a paired difference test).

Full Text

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 $\chi^2$ 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 $\chi^2$ distribution. If the data are substantially non-normal and the sample size is small, the $t$-test can give misleading results. 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.

Slutsky's theorem extends some properties of algebraic operations on convergent sequences of real numbers to sequences of random variables. The theorem was named after Eugen Slutsky. The statement is as follows:

Let $\{X_n\}$, $\{Y_n\}$ be sequences of scalar/vector/matrix random elements. If $X_n$ converges in distribution to a random element $X$, and $Y$ converges in probability to a constant $c$, then:

$\displaystyle{ X_n + Y_n \overset{a}{\rightarrow} X + c\\ Y_nX_n \overset{d}{\rightarrow} cX\\ Y_n^{-1}X_n \overset{d}{\rightarrow} c^{-1} X }$

where $\overset{d}{\rightarrow}$ denotes convergence in distribution.

When the normality assumption does not hold, a nonparametric 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.

One-way analysis of variance generalizes the two-sample $t$-test when the data belong to more than two groups.

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