Mann-Whitney Test for Independent Samples

Basic Concepts

The Mann-Whitney U test is essentially an alternative form of the Wilcoxon Rank-Sum test for independent samples and is completely equivalent.

Define the following test statistics for samples 1 and 2 where n₁ is the size of sample 1 and n₂ is the size of sample 2, and R₁ is the adjusted rank-sum for sample 1 and R₂ is the adjusted rank-sum of sample 2. It doesn’t matter which sample is bigger.

As for the Wilcoxon version of the test, if the observed value of U is < U_crit then the test is significant (at the α level), i.e. we reject the null hypothesis. The values of U_crit for α = .05 (two-tailed) are given in the Mann-Whitney Tables.

Example using the table of critical values

Example 1: Repeat Example 1 of the Wilcoxon Rank Sum Test using the Mann-Whitney U test.

Figure 1 – Mann-Whitney U Test

Since R₁ = 117.5 and R₂ = 158.5, we can calculate U₁ and U₂ to get U = 39.5. Next, we look up in the Mann-Whitney Tables for n₁ = 12 and n₂ = 11 to get U_crit = 33. Since 33 < 39.5, we cannot reject the null hypothesis at α = .05 level of significance.

Properties

Property 1:

Property 2: For n₁ and n₂ large enough the U statistic is approximately normal N(μ, σ²) where

Observation: Click here for proofs of Properties 1 and 2.

Property 3: Where there are a number of ties, the following revised version of the variance gives better results:

where n = n₁ + n₂, t varies over the set of tied ranks and f_t is the number of times (i.e. frequency) the rank t appears. An equivalent formula is

Continuity Correction

A further complication is that it is often desirable to account for the fact that we are approximating a discrete distribution via a continuous one by applying a continuity correction. This is done by using a z-score of

instead of the same formula without the .5 continuity correction factor.

Example using a normal approximation

Example 2: Repeat Example 2 of the Wilcoxon Rank Sum Test using the Mann-Whitney U test.

Figure 2 shows the results of the one-tailed test (without using a ties correction). Column W displays the formulas used in column T.

Figure 2 – Mann-Whitney U test using normal approximation

As can be seen in cell T19, the p-value for the one-tail test is the same as that found in Wilcoxon Example 2 using the Wilcoxon rank-sum test. Once again we reject the null hypothesis and conclude that non-smokers live significantly longer.

Effect Size

The effect size for the data using the Mann-Whitney test can be calculated in the same manner as for the Wilcoxon rank-sum test, namely

and the result will be the same, which for Example 2 is r = .31, as shown in cell T21 of Figure 2. This is a medium-sized effect.

Here, .1 is a small effect, .3 is a medium effect, and .5 is considered to be a large effect.

Rank-biserial Correlation

There is another measure of effect size, namely

p represents the probability that a score randomly generated from population A will be bigger than a score randomly generated from population B, where A and B are the populations corresponding to the two samples and A corresponds to the sample with the higher value. The higher this value is the larger the effect.

The difference between p and its complement 1–p provides a standard effect size r called the rank-serial correlation. Thus

where μ is the mean of the normal approximation. For Example 2, the rank-serial correlation is r = 1 – 486/760 = .36, which is a medium-sized effect.

Worksheet Functions

Real Statistics Functions: The Real Statistics Resource Pack provides the following functions:

MANN(R1, R2) = U for the samples contained in R1 and R2

MANN(R1, n) = U for the sample contained in the first n columns of R1 and the sample consisting of the remaining columns in R1. If the second argument is omitted it defaults to 1.

MWTEST(R1, R2, tails, ties, cont) = p-value of the Mann-Whitney U test for the samples contained in R1 and R2 using the normal approximation. tails = 1 or 2 (default). If ties = TRUE (default) the ties correction factor is applied. If cont = TRUE (default) a continuity correction is applied.

Any empty or non-numeric cells in R1 or R2 are ignored.

Observations

For Example 2, we can use the Real Statistics MANN function to arrive at the value of 486 for U shown in cell T9 of Figure 2, namely =MANN(J6:M15,N6:Q15) = 486. Similarly, the p-value of 0.003081 in cell T19 can be calculated by =MWTEST(J6:M15,N6:Q15,1,FALSE,TRUE).

Note that the z-score and the effect size r can be calculated using the Real Statistics function MWTEST as follows:

z-score = NORM.S.INV(MWTEST(R1, R2))

r = NORM.S.INV(MWTEST(R1, R2))/SQRT(COUNT(R1)+COUNT(R2))

The results of analysis for Example 2 can be summarized as follows: The life expectancy of non-smokers (Mdn = 76.5) is significantly higher than that of smokers (Mdn = 70.5), U = 486, z = -2.74, p = .0038 < .05, r = .31, based on a one-tailed test Mann-Whitney test with continuity correction, but no correction for ties.

Of course, you can also use a two-tailed test with a ties correction, as we demonstrate shortly.

Worksheet Array Function

Real Statistics Function: The Real Statistics Resource Pack provides the following function that returns output consisting of the U-stat, z-stat, r effect size, and the three types of p-values (the normal approximation, exact test, and simulation).

MW_TEST(R1, R2, lab, tails, ties, cont, exact, iter): returns a column array with the output described above for the samples contained in R1 and R2. tails = 1 or 2 (default). For the normal approximation, if ties = TRUE (default) the ties correction factor is applied; if cont = TRUE (default) a continuity correction is applied; when exact = TRUE (default FALSE) then the p-value of the exact test is output and if iter ≠ 0 then the p-value of the simulation version of the test is output where the simulation consists of iter samples (default 10,000). If lab = TRUE (default FALSE) then an extra column of labels is appended to the output.

Any empty or non-numeric elements in R1 or R2 are ignored. See Mann-Whitney Exact Test and Mann-Whitney Simulation for more information about the exact test and simulation p-values.

Figure 3 displays the output from =MW_TEST(A6:A17,B6:B17,TRUE) for Example 2.

Figure 3 – Output from MW_TEST

Even if the argument is set to FALSE, the p-value of the exact test will be produced provided both samples have fewer than 800 elements and the smaller sample has at most 300 elements.

Data Analysis Tool

Real Statistics Data Analysis Tool: The Real Statistics Resource Pack also provides a data analysis tool that performs the Mann-Whitney test for independent samples, automatically calculating the medians, rank sums, U test statistic, z-score, p-values, and effect size r.

For example, to perform the analysis in Example 1, press Ctrl-m and choose the T Test and Non-parametric Equivalents data analysis tool from the menu that appears (or from the Misc tab if using the Multipage user interface). The dialog box shown in Figure 4 now appears.

Figure 4 – Dialog box for Real Statistics Mann-Whitney Test

Enter A5:B17 as the Input Range 1 (alternatively, insert A5:A17 in Input Range 1 and B5:B17 in Input Range 2), click on Column headings included with data, choose the Two independent samples and Non-parametric options, and click on the OK button. Keep the default of 0 for Hypothetical Mean/Median and .05 for Alpha (although these values are not used) For this version of the test, we check the Use continuity correction, Include exact test, and Include table lookup options, but leave the Use ties correction option unchecked.

The output is shown in Figure 5.

Figure 5 – Mann-Whitney test data analysis tool output

Note that both the one-tail and two-tail tests are displayed. Also, three versions of the test are shown: the test using the normal approximation (range E17:F17), the test using the exact test (range E18:F18), and the simulation test (range E19:F19). The fact that the “Yates” continuity correction factor is used is noted in cell F15.

Ties correction version

If we check the Use Ties correction option in Figure 4 we would obtain the output shown in Figure 6.

Figure 6 – Mann-Whitney test with ties correction

In this case, the ties correction of Property 3 is applied to the normal approximation. As you can see there is very little difference between the outputs shown in Figures 5 and 6.

Note too that the ties correction (as well as the continuity correction) only applies to the normal approximation. The ties and continuity corrections are not applied to the exact and simulation versions of the test. The difference in the simulation p-values (row 19) in Figures 5 and 6 is due to the randomness of the simulations and not the ties correction.

Rank-serial correlation version

If the Use rank-serial correlation option were checked in Figure 4, then cell K16 in Figure 6 would contain the formula =1-K12/K13 with value .4015. Cell L16 would contain the value “rank-serial”.

Ties correction worksheet function

Real Statistics Function: The Real Statistics Pack provides the following function to calculate the ties correction used in the data analysis tool.

TiesCorrection(R1, R2, type) = ties correction value for the data in R1 and optionally R2, where type = 0: one sample, type = 1: paired sample and type = 2: independent samples

For the Mann-Whitney test type = 2.  The ties correction is used in the calculation of the standard deviation (cell U15 of Figure 6) as follows

=SQRT(K13*(K6+L6+1)/6*(1-TiesCorrection(A6:A17,B6:B17,2)/ ((K6+L6)^3-K6-L6)))

Exact Test

Click here for a description of the exact version of the Mann-Whitney Test using the permutation function.

Simulation

Click here for a description of how to use simulation to determine the p-value for the Mann-Whitney test. This approach takes ties into account.

Confidence Interval of the Median

Click here for a description of how to calculate a confidence interval of the median based on the Mann-Whitney Test.

Statistical Power and Sample Size

Click here for a description of how to calculate the statistical power or minimum sample size required for the Mann-Whitney Test.

Examples Workbook

Click here to download the Excel workbook with the examples described on this webpage.

References

Howell, D. C. (2010) Statistical methods for psychology (7^th ed.). Wadsworth, Cengage Learning.
https://labs.la.utexas.edu/gilden/files/2016/05/Statistics-Text.pdf

Zar. J. H. (2010) Biostatistical analysis 5^th Ed. Pearson
https://bayesmath.com/wp-content/uploads/2021/05/Jerrold-H.-Zar-Biostatistical-Analysis-5th-Edition-Prentice-Hall-2009.pdf

Kerby, D. S. (2014). The simple difference formula: an approach to teaching nonparametric correlation. Comprehensive Psychology.
https://scirp.org/(S(351jmbntv-nsjt1aadkposzje))/reference/referencespapers.aspx?referenceid=3598009