U-Mann-Whitney - Wilcoxon Rank-Sum Table

Probabilities relate to the distribution of W_A, the rank sum for group A, when H_o : A = B is true. The tabulated value for the lower tail is the largest value of w_A for which Pr(W_A ≤ w_A) ≤ prob. The tabulated value for the upper tail is the smallest value of w_A for which pr(WA ≥ wA) ≤ prob. For sample sizes (n_A & n_A) larger than 12 use Normal Approximation and the Standard Normal Table to calculate critical values.

Lower Tail nA nB 0.005 0.01 0.025 0.05 0.10 0.20 4 4 . . 10 11 13 14 4 5 . 10 11 12 14 15 4 6 10 11 12 13 15 17 4 7 10 11 13 14 16 18 4 8 11 12 14 15 17 20 4 9 11 13 14 16 19 21 4 10 12 13 15 17 20 23 4 11 12 14 16 18 21 24 4 12 13 15 17 19 22 26 5 5 15 16 17 19 20 22 5 6 16 17 18 20 22 24 5 7 16 18 20 21 23 26 5 8 17 19 21 23 25 28 5 9 18 20 22 24 27 30 5 10 19 21 23 26 28 32 5 11 20 22 24 27 30 34 5 12 21 23 26 28 32 36 6 6 23 24 26 28 30 33 6 7 24 25 27 29 32 35 6 8 25 27 29 31 34 37 6 9 26 28 31 33 36 40 6 10 27 29 32 35 38 42 6 11 28 30 34 37 40 44 6 12 30 32 35 38 42 47 7 7 32 34 36 39 41 45 7 8 34 35 38 41 44 48 7 9 35 37 40 43 46 50 7 10 37 39 42 45 49 53 7 11 38 40 44 47 51 56 7 12 40 42 46 49 54 59 8 8 43 45 49 51 55 59 8 9 45 47 51 54 58 62 8 10 47 49 53 56 60 65 8 11 49 51 55 59 63 69 8 12 51 53 58 62 66 72 9 9 56 59 62 66 70 75 9 10 58 61 65 69 73 78 9 11 61 63 68 72 76 82 9 12 63 66 71 75 80 86 10 10 71 74 78 82 87 93 10 11 73 77 81 86 91 97 10 12 76 79 84 89 94 101 11 11 87 91 96 100 106 112 11 12 90 94 99 104 110 117 12 12 105 109 115 120 127 134

Upper Tail nA nB 0.20 0.10 0.05 0.025 0.01 0.005 4 4 22 23 25 26 . . 4 5 25 26 28 29 30 . 4 6 27 29 31 32 33 34 4 7 30 32 34 35 37 38 4 8 32 35 37 38 40 41 4 9 35 37 40 42 43 45 4 10 37 40 43 45 47 48 4 11 40 43 46 48 50 52 4 12 42 46 49 51 53 55 5 5 33 35 36 38 39 40 5 6 36 38 40 42 43 44 5 7 39 42 44 45 47 49 5 8 42 45 47 49 51 53 5 9 45 48 51 53 55 57 5 10 48 52 54 57 59 61 5 11 51 55 58 61 63 65 5 12 54 58 62 64 67 69 6 6 45 48 50 52 54 55 6 7 49 52 55 57 59 60 6 8 53 56 59 61 63 65 6 9 56 60 63 65 68 70 6 10 60 64 67 70 73 75 6 11 64 68 71 74 78 80 6 12 67 72 76 79 82 84 7 7 60 64 66 69 71 73 7 8 64 68 71 74 77 78 7 9 69 73 76 79 82 84 7 10 73 77 81 84 87 89 7 11 77 82 86 89 93 95 7 12 81 86 91 94 98 100 8 8 77 81 85 87 91 93 8 9 82 86 90 93 97 99 8 10 87 92 96 99 103 105 8 11 91 97 101 105 109 111 8 12 96 102 106 110 115 117 9 9 96 101 105 109 112 115 9 10 102 107 111 115 119 122 9 11 107 113 117 121 126 128 9 12 112 118 123 127 132 135 10 10 117 123 128 132 136 139 10 11 123 129 134 139 143 147 10 12 129 136 141 146 151 154 11 11 141 147 153 157 162 166 11 12 147 154 160 165 170 174 12 12 166 173 180 185 191 195

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Mann-Whitney U

In statistics, the Mann-Whitney U test (also called the Mann-Whitney-Wilcoxon (MWW), Wilcoxon rank-sum test, or Wilcoxon-Mann-Whitney test) is a non-parametric test for assessing whether two samples of observations come from the same distribution. It is one of the best-known non-parametric significance tests. It was proposed initially by Wilcoxon (1945), for equal sample sizes, and extended to arbitrary sample sizes and in other ways by Mann and Whitney (1947). MWW is virtually identical to performing an ordinary parametric two-sample t test on the data after ranking over the combined samples.

Contents 1 Formal statement of object of test 2 Calculations 3 Example 4 Approximation 5 Relation to other tests 5.1 Comparison to Student's t-test 5.2 Different distributions 5.2.1 Alternatives 5.3 Kendall's τ 5.4 ρ statistic 6 Example statement of results 7 See also 8 External links 9 References 10 Implementations

[edit] Formal statement of object of test

It is commonly stated that the MWW test tests for differences in medians but this is not strictly true. The null hypothesis in the Mann-Whitney test is that the two samples are drawn from a single population, and therefore that their probability distributions are equal. It requires the two samples to be independent, and the observations to be ordinal or continuous measurements, i.e. one can at least say, of any two observations, which is the greater. In a less general formulation, the test may be thought of as testing the null hypothesis that the probability of an observation from one population exceeding an observation from the second population is 0.5. This formulation requires the additional assumption that the distributions of the two populations are identical except for possibly a shift (i.e. f₁(x) = f₂(x + δ)). Another alternative interpretation is that the test assesses whether the Hodges-Lehmann estimate of the difference in central tendency between the two populations is zero. 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.

Calculations

The test involves the calculation of a statistic, usually called U, whose distribution under the null hypothesis is known. In the case of small samples, the distribution is tabulated, but for sample sizes above ~20 there is a good approximation using the normal distribution. Some books tabulate statistics equivalent to U, such as the sum of ranks in one of the samples.

The U test is included in most modern statistical packages. It is also easily calculated by hand, especially for small samples. There are two ways of doing this.

For small samples a direct method is recommended. It is very quick, and gives an insight into the meaning of the U statistic.

1. Choose the sample for which the ranks seem to be smaller (The only reason to do this is to make computation easier). Call this "sample 1," and call the other sample "sample 2."

2. Taking each observation in sample 2, count the number of observations in sample 1 that are smaller than it (count a half for any that are equal to it).

3. The total of these counts is U.

For larger samples, a formula can be used:

1. Arrange all the observations into a single ranked series. That is, rank all the observations without regard to which sample they are in.

2. Add up the ranks for the observations which came from sample 1. The sum of ranks in sample 2 follows by calculation, since the sum of all the ranks equals N(N + 1)/2 where N is the total number of observations.

3. "U" is then given by:

where n₁ is the two sample size for sample 1, and R₁ is the sum of the ranks in sample 1.

Note that there is no specification as to which sample is considered sample 1. An equally valid formula for U is

The smaller value of U₁ and U₂ is the one used when consulting significance tables. The sum of the two values is given by

Knowing that R₁ + R₂ = N(N + 1)/2 and N = n₁ + n₂ , and doing some algebra, we find that the sum is

The maximum value of U is the product of the sample sizes for the two samples. In such a case, the "other" U would be 0.

Example

Suppose that Aesop is dissatisfied with his classic experiment in which one tortoise was found to beat one hare in a race, and decides to carry out a significance test to discover whether the results could be extended to tortoises and hares in general. He collects a sample of 6 tortoises and 6 hares, and makes them all run his race. The order in which they reach the finishing post (their rank order) is as follows, writing T for a tortoise and H for a hare:

T H H H H H T T T T T H

What is the value of U?

• Using the direct method, we take each tortoise in turn, and count the number of hares it beats, getting 6, 1, 1, 1, 1, 1. So U = 6 + 1 + 1 + 1 + 1 + 1 = 11. Alternatively, we could take each hare in turn, and count the number of tortoises it beats. In this case, we get 5, 5, 5, 5, 5, 0, which means U = 25. Note that the sum of these two values for "U" is 36, which is 6 × 6.

• Using the indirect method:

the sum of the ranks achieved by the tortoises is 1 + 7 + 8 + 9 + 10 + 11 = 46.

Therefore U = 46 − 6×7/2 = 46 − 21 = 25.

the sum of the ranks achieved by the hares is 2 + 3 + 4 + 5 + 6 + 12 = 32, leading to U = 32 - 21 = 11.

Approximation

For large samples, the normal approximation:

can be used, where z is a standard normal deviate whose significance can be checked in tables of the normal distribution. m_U and σ_U are the mean and standard deviation of U if the null hypothesis is true, and are given by

All the formulae here are made more complicated in the presence of tied ranks, but if the number of these is small (and especially if there are no large tie bands) these can be ignored when doing calculations by hand. The computer statistical packages will use them as a matter of routine.

Note that since U₁ + U₂ = n₁ n₂, the mean n₁ n₂/2 used in the normal approximation is the mean of the two values of U. Therefore, you can use U and get the same result, the only difference being between a left-tailed test and a right-tailed test.

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