Otherwise, if both the dispersions and shapes of the distribution of both samples differ, the Mann-Whitney U test fails a test of medians. The Hodges–Lehmann estimate for this two-sample problem is the median of all possible differences between an observation in the first sample and an observation in the second sample. Under this location shift assumption, we can also interpret the Mann–Whitney U test as assessing whether the Hodges–Lehmann estimate of the difference in central tendency between the two populations differs from zero. Under more strict assumptions than the general formulation above, e.g., if the responses are assumed to be continuous and the alternative is restricted to a shift in location, i.e., F 1( x) = F 2( x + δ), we can interpret a significant Mann–Whitney U test as showing a difference in medians. The probability of an observation from population X exceeding an observation from population Y is different (larger, or smaller) than the probability of an observation from Y exceeding an observation from X i.e., P( X > Y) ≠ P( Y > X) or P( X > Y) + 0.5.
Under the general formulation, the test is only consistent when the following occurs under H 1: The alternative hypothesis H 1 is that the distributions are not equal.Under the null hypothesis H 0, the distributions of both populations are equal.The responses are at least ordinal (i.e., one can at least say, of any two observations, which is the greater),.All the observations from both groups are independent of each other,.Ī very general formulation is to assume that: 7.1.3 Area-under-curve (AUC) statistic for ROC curvesĪssumptions and formal statement of hypotheses Īlthough Mann and Whitney developed the Mann–Whitney U test under the assumption of continuous responses with the alternative hypothesis being that one distribution is stochastically greater than the other, there are many other ways to formulate the null and alternative hypotheses such that the Mann–Whitney U test will give a valid test.7.1 Proportion of concordance out of all pairs.6 Normal approximation and tie correction.5.1 Illustration of calculation methods.1 Assumptions and formal statement of hypotheses.
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