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# Kolmogorov-Smirnov test vs Mann Whitney test for unpaired hypothesis testing?

I was telling a colleague that the best and most popular nonparametric alternative to the unpaired student t test is the Mann Whitney test. Ifigured that is probably what they use in clinical trials and other social science environment. But, he suggested an alternative the two-sample Kolmogorov-Smirnov (KS) test. I never heard of people using the KS test for this purpose. But, I wondered why not.

Can you help me figure out the relative strength and weaknesses of each test. Thanks in advance for any info.

Gaetan Lion

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You may find this text from the STATISTICA Electronic Manual helpful:

Mann-Whitney U Test

The Mann-Whitney U test assumes that the variable under consideration was measured on at least an ordinal (rank order) scale. The interpretation of the test is essentially identical to the interpretation of the result of a t-test for independent samples, except that the U test is computed based on rank sums rather than means. The U test is the most powerful (or sensitive) nonparametric alternative to the t-test for independent samples; in fact, in some instances it may offer even greater power to reject the null hypothesis than the t-test.

With samples larger than 20, the sampling distribution of the U statistic rapidly approaches the normal distribution (see Siegel, 1956). Hence, the U statistic (adjusted for ties) will be accompanied by a z value (normal distribution variate value), and the respective p-value.

Exact probabilities for small samples. For small to moderate sized samples, STATISTICA computes an exact probability associated with the respective U statistic. This probability is based on the enumeration of all possible values of U (unadjusted for ties), given the number of observations in the two samples (see Dinneen & Blakesley, 1973). Specifically, for small to moderate sized samples, the program will report (in the last column of the spreadsheet) the value 2 * p, where p is 1 minus the cumulative (one-sided) probability of the respective U statistic. To reiterate, the computations for this probability value are based on the assumption of no ties in the data (ranks). Note that this limitation usually leads to only a small underestimation of the statistical significance of the respective effects (see Siegel, 1956).

Kolmogorov-Smirnov test

The Kolmogorov-Smirnov test assesses the hypothesis that two samples were drawn from different populations. Unlike the parametric t-test for independent samples or the Mann-Whitney U test, which test for differences in the location of two samples (differences in means, differences in average ranks, respectively), the Kolmogorov-Smirnov test is also sensitive to differences in the general shapes of the distributions in the two samples (i.e., to differences in dispersion, skewness, etc.).
Thank you very much. This is a very good and educated answer.

Guy

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