samples cannot be distinguished from each other. Mathematically, the test statistic is U computed as follows: where if and is equal to 0 if . Another unique feature of this test is that Shieh (2006) et al. derived the probability distribution of under both the null and alternative hypotheses. When U both samples are larger than eight, a standardized statistic, termed may be approximated by U Z, a standard normal distribution under the null hypothesis: where and are the mean and standard deviation of the statistic under the null hypothesis, and U are both functions of only the known sample sizes and m n: When the size of both samples is less than eight, an exact empirical distribution of U under the null hypothesis should be computed from the ranks using formulae given by Mann and Whitney (1947). Bellera (2010) describes applications of this test to small samples of et al. unequal length. Type I error probability estimates from this test may be distorted if the variances of and differ significantly and corrections to the null hypothesis assumption of equal variances are not made (Kasuya, 2001). In contrast with conditions under the null hypothesis, determining the probability distribution associated with under the alternative hypothesis is much less straightforward U because the distribution of is unknown. Unless a Markov chain Monte Carlo method is applied U (Lee, 2014), one must define an alternative hypothesis based on a given distributional assumption (Blair and Higgins, 1980; Shieh 2006) or run Monte Carlo simulations (e.g. et al.,
samples cannot be distinguished from each other. Mathematically, the U test statistic is computed as follows: where if and is equal to 0 if . Another unique feature of this test is that Shieh et al. (2006) derived the probability distribution of U under both the null and alternative hypotheses. When both samples are larger than eight, a standardized U statistic, termed Z, may be approximated by a standard normal distribution under the null hypothesis: where and are the mean and standard deviation of the U statistic under the null hypothesis, and are both functions of only the known sample sizes m and n: When the size of both samples is less than eight, an exact empirical distribution of U under the null hypothesis should be computed from the ranks using formulae given by Mann and Whitney (1947). Bellera et al. (2010) describes applications of this test to small samples of unequal length. Type I error probability estimates from this test may be distorted if the variances of and differ significantly and corrections to the null hypothesis assumption of equal variances are not made (Kasuya, 2001). In contrast with conditions under the null hypothesis, determining the probability distribution associated with U under the alternative hypothesis is much less straightforward because the distribution of U is unknown. Unless a Markov chain Monte Carlo method is applied (Lee, 2014), one must define an alternative hypothesis based on a given distributional assumption (Blair and Higgins, 1980; Shieh et al., 2006) or run Monte Carlo simulations (e.g.