Nonparametric Statistics

Nonparametric statistics refer to methods of measurement that do not rely on assumptions that the data are drawn from a specific distribution. Non-parametric statistical methods have been widely used in various kinds of research designs to make statistical inferences. In practice, when the normality assumption on the measurements is not satisfied, parametric statistical methods might provide misleading results. In contrast, nonparametric methods make much less stringent distributional assumptions on the measurements. They are valid methods regardless of the underlying distributions of the observations. Because of this attractive advantage, ever since the first introduction of non-parametric tests in the last century, many different types of nonparametric tests have been developed to analyze various types of experimental designs. Such designs encompass one-sample design, two-sample design, randomized-block design, two-way factorial design, repeated measurements design, and high-way layout. The observations in each experimental condition could be equal or unequal. The targeted inferences include the comparisons of treatment effects, the existence of interaction effects, ordered inferences of the effects, and multiple comparisons of the effects. All these methods share the same feature that instead of using the actual observed measurements, they used the ranked values to form the statistics. By discarding the actual measurements, the methods gain the robustness to the underlying distributions and the potential contamination of outliers. This gain of robustness is only at the price of losing a relatively small amount of efficiency. In this entry, a brief review of the existing nonparametric methods is provided to facilitate the application of them in practical settings.

Tests for One or Multiple Populations

The Mann–Whitney–Wilcoxon (MWW) test is a nonparametric test to determine whether two samples of observations are drawn from the same distribution. The null hypothesis specifies that the two probability distributions are identical. It is one of the best-known nonparametric tests. It was first proposed by Frank Wilcoxon in 1945 for equal sample sizes, and it was later extended to arbitrary sample sizes by Henry B. Mann and Donald R. Whitney in 1947. To obtain the statistic, the observations are first ranked without regard to which sample they are in. Then for samples 1 and 2, the sum of ranks R1 and R2, respectively, are computed. The statistic takes the form of

where N1,N2 denotes the samples sizes. For small samples, the distribution of the statistic is tabulated. However, for sample sizes greater than 20, the statistic can be normalized into

The significance of the normalized statistic z can be assessed using the standard normal table. As a test to compare two populations, MWW is in spirit very similar to the parametric two-sample t test. In comparison, the parametric t test is more powerful if the data are drawn from normal distribution. In contrast, if the distributional assumption is violated in practice, then MWW is more powerful than its parametric counterpart. In terms of efficiency, under the normality assumption, the efficiency of MWW test is 95% of that of t test, which implies that to achieve the same power, the t test will need 5% less data points than the MWW test. Under other non-normal, especially heavy-tailed distributions, the efficiency of MWW could be much higher.

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