Mann–Whitney U test
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 nonparametric test of the null hypothesis that it is equally likely that a randomly selected value from one sample will be less than or greater than a randomly selected value from a second sample.
This test can be used to investigate whether two independent samples were selected from populations having the same distribution. A similar nonparametric test used on dependent samples is the Wilcoxon signed-rank test.
- 1 Assumptions and formal statement of hypotheses
- 2 Calculations
- 3 Properties
- 4 Examples
- 5 Normal approximation and tie correction
- 6 Effect sizes
- 7 Relation to other tests
- 8 History
- 9 Related test statistics
- 10 Example statement of results
- 11 Software Implementations
- 12 See also
- 13 Notes
- 14 References
- 15 External links
Assumptions and formal statement of hypotheses
Although 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.
A very general formulation is to assume that:
- All the observations from both groups are independent of each other,
- The responses are ordinal (i.e., one can at least say, of any two observations, which is the greater),
- Under the null hypothesis H0, the distributions of both populations are equal.
- The alternative hypothesis H1 is that the distributions are not equal.
Under the general formulation, the test is only consistent when the following occurs under H1:
- 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 · P(X = Y) ≠ 0.5.
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., F1(x) = F2(x + δ), we can interpret a significant Mann–Whitney U test as showing a difference in medians. 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. 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.
The Mann–Whitney U test / Wilcoxon rank-sum test is not the same as the Wilcoxon signed-rank test, although both are nonparametric and involve summation of ranks. The Mann–Whitney U test is applied to independent samples. The Wilcoxon signed-rank test is applied to matched or dependent samples.
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, approximation using the normal distribution is fairly good. Some books tabulate statistics equivalent to U, such as the sum of ranks in one of the samples, rather than U itself.
The Mann–Whitney 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 comparing two small sets of observations, a direct method is quick, and gives insight into the meaning of the U statistic, which corresponds to the number of wins out of all pairwise contests (see the tortoise and hare example under Examples below). For each observation in one set, count the number of times this first value wins over any observations in the other set (the other value loses if this first is larger). Count 0.5 for any ties. The sum of wins and ties is U (i.e.: ) for the first set. U for the other set is the converse (i.e.: ).
For larger samples:
- Assign numeric ranks to all the observations (put the observations from both groups to one set), beginning with 1 for the smallest value. Where there are groups of tied values, assign a rank equal to the midpoint of unadjusted rankings. E.g., the ranks of (3, 5, 5, 5, 5, 8) are (1, 3.5, 3.5, 3.5, 3.5, 6) (the unadjusted rank would be (1, 2, 3, 4, 5, 6)).
- Now, add up the ranks for the observations which came from sample 1. The sum of ranks in sample 2 is now determinate, since the sum of all the ranks equals N(N + 1)/2 where N is the total number of observations.
- U is then given by:
- where n1 is the sample size for sample 1, and R1 is the sum of the ranks in sample 1.
- Note that it doesn't matter which of the two samples 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
- 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
- U1 + U2 = n1n2.
- 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 (i.e.: ). In such a case, the "other" U would be 0.
Illustration of calculation methods
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 at once. The order in which they reach the finishing post (their rank order, from first to last crossing the finish line) 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, which means that U = 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, so U = 25. Note that the sum of these two values for U = 36, which is 6×6.
- Using the indirect method:
- rank the animals by the time they take to complete the course, so give the first animal home rank 12, the second rank 11, and so forth.
- the sum of the ranks achieved by the tortoises is 12 + 6 + 5 + 4 + 3 + 2 = 32.
- Therefore U = 32 − (6×7)/2 = 32 − 21 = 11 (same as method one).
- the sum of the ranks achieved by the hares is 11 + 10 + 9 + 8 + 7 + 1 = 46, leading to U = 46 − 21 = 25.
Illustration of object of test
A second example race illustrates the point that the Mann–Whitney U test does not test for inequality of medians, but rather for difference of distributions. Consider another hare and tortoise race, with 19 participants of each species, in which the outcomes are as follows, from first to last past the finishing post:
- H H H H H H H H H T T T T T T T T T T H H H H H H H H H H T T T T T T T T T
If we simply compared medians, we would conclude that the median time for tortoises is less than the median time for hares, because the median tortoise here (in bold) comes in at position 19, and thus actually beats the median hare (in bold), which comes in at position 20. However, the value of U is 100 (using the quick method of calculation described above, we see that each of 10 tortoises beats each of 10 hares, so U = 10×10). Consulting tables, or using the approximation below, we find that this U value gives significant evidence that hares tend to have lower completion times than tortoises (p < 0.05, two-tailed). Obviously these are extreme distributions that would be spotted easily, but in larger samples something similar could happen without it being so apparent. Notice that the problem here is not that the two distributions of ranks have different variances; they are mirror images of each other, so their variances are the same, but they have very different skewness.
Normal approximation and tie correction
where mU and σU are the mean and standard deviation of U, is approximately a standard normal deviate whose significance can be checked in tables of the normal distribution. mU and σU are given by
The formula for the standard deviation is more complicated in the presence of tied ranks. If there are ties in ranks, σ should be corrected as follows:
where n = n1 + n2, ti is the number of subjects sharing rank i, and k is the number of (distinct) ranks.
If the number of ties is small (and especially if there are no large tie bands) ties can be ignored when doing calculations by hand. The computer statistical packages will use the correctly adjusted formula as a matter of routine.
Note that since U1 + U2 = n1n2, the mean n1n2/2 used in the normal approximation is the mean of the two values of U. Therefore, the absolute value of the z statistic calculated will be same whichever value of U is used.
Proportion of concordance out of all pairs
The following three measures are equivalent.
Common language effect size
One method of reporting the effect size for the Mann–Whitney U test is with f, the common language effect size. As a sample statistic, the common language effect size is computed by forming all possible pairs between the two groups, then finding the proportion of pairs that support a direction (say, that items from group 1 are larger than items from group 2). To illustrate, in a study with a sample of ten hares and ten tortoises, the total number of ordered pairs is ten times ten or 100 pairs of hares and tortoises. Suppose the results show that the hare ran faster than the tortoise in 90 of the 100 sample pairs; in that case, the sample common language effect size is 90%. This sample value is an unbiased estimator of the population value, so the sample suggests that the best estimate of the common language effect size in the population is 90%.
The relationship between f and the Mann–Whitney U (specifically ) is as follows:
This is the same as the area under the curve (AUC) for the ROC curve below.
A statistic called ρ that is linearly related to U and widely used in studies of categorization (discrimination learning involving concepts), and elsewhere, is calculated by dividing U by its maximum value for the given sample sizes, which is simply n1×n2. ρ is thus a non-parametric measure of the overlap between two distributions; it can take values between 0 and 1, and it is an estimate of P(Y > X) + 0.5 P(Y = X), where X and Y are randomly chosen observations from the two distributions. Both extreme values represent complete separation of the distributions, while a ρ of 0.5 represents complete overlap. The usefulness of the ρ statistic can be seen in the case of the odd example used above, where two distributions that were significantly different on a Mann–Whitney U test nonetheless had nearly identical medians: the ρ value in this case is approximately 0.723 in favour of the hares, correctly reflecting the fact that even though the median tortoise beat the median hare, the hares collectively did better than the tortoises collectively.
Area-under-curve (AUC) statistic for ROC curves
Note that this is the same definition as the common language effect size from the section above.
Because of its probabilistic form, the U statistic can be generalised to a measure of a classifier's separation power for more than two classes:
Where c is the number of classes, and the Rk,l term of AUCk,l considers only the ranking of the items belonging to classes k and l (i.e., items belonging to all other classes are ignored) according to the classifier's estimates of the probability of those items belonging to class k. AUCk,k will always be zero but, unlike in the two-class case, generally AUCk,l ≠ AUCl,k, which is why the M measure sums over all (k,l) pairs, in effect using the average of AUCk,l and AUCl,k.
A method of reporting the effect size for the Mann–Whitney U test is with a measure of rank correlation known as the rank-biserial correlation. Edward Cureton introduced and named the measure. Like other correlational measures, the rank-biserial correlation can range from minus one to plus one, with a value of zero indicating no relationship.
There is a simple difference formula to compute the rank-biserial correlation from the common language effect size: the correlation is the difference between the proportion of pairs favorable to the hypothesis (f) minus its complement (i.e.: the proportion that is unfavorable (u)). This simple difference formula is just the difference of the common language effect size of each group, and is as follows:
For example, consider the example where hares run faster than tortoises in 90 of 100 pairs. The common language effect size is 90%, so the rank-biserial correlation is 90% minus 10%, and the rank-biserial r = 0.80.
An alternative formula for the rank-biserial can be used to calculate it from the Mann–Whitney U (either or ) and the sample sizes of each group:
This formula is useful when the data are not available, but when there is a published report, because U and the sample sizes are routinely reported. Using the example above with 90 pairs that favor the hares and 10 pairs that favor the tortoise, U2 is the smaller of the two, so U2 = 10. This formula then gives r = 1 – (2×10) / (10×10) = 0.80, which is the same result as with the simple difference formula above.
Relation to other tests
Comparison to Student's t-test
The Mann–Whitney U test tests a null hypothesis of that the probability that a randomly drawn observation from one group is larger than a randomly drawn observation from the other is equal to 0.5 against an alternative that this probability is not 0.5 (see Mann–Whitney U test#Assumptions and formal statement of hypotheses). In contrast, a t-test tests a null hypothesis of equal means in two groups against an alternative of unequal means. Hence, except in special cases, the Mann–Whitney U test and the t-test do not test the same hypotheses and should be compared with this in mind.
- Ordinal data
- The Mann–Whitney U test is preferable to the t-test when the data are ordinal but not interval scaled, in which case the spacing between adjacent values of the scale cannot be assumed to be constant.
- As it compares the sums of ranks, the Mann–Whitney U test is less likely than the t-test to spuriously indicate significance because of the presence of outliers. However, the Mann-Whitney U test may have worse type I error control when data are both heteroscedastic and non-normal.
- When normality holds, the Mann–Whitney U test has an (asymptotic) efficiency of 3/π or about 0.95 when compared to the t-test. For distributions sufficiently far from normal and for sufficiently large sample sizes, the Mann–Whitney U test is considerably more efficient than the t. This comparison in efficiency, however, should be interpreted with caution, as Mann-Whitney and the t-test do not test the same quantities. If, for example, a difference of group means is of primary interest, Mann-Whitney is not an appropriate test.
If one is only interested in stochastic ordering of the two populations (i.e., the concordance probability P(Y>X)), the Mann–Whitney U test can be used even if the shapes of the distributions are different. The concordance probability is exactly equal to the area under the receiver operating characteristic curve (ROC) that is often used in the context.
If one desires a simple shift interpretation, the Mann–Whitney U test should not be used when the distributions of the two samples are very different, as it can give erroneous interpretation of significant results. In that situation, the unequal variances version of the t-test may give more reliable results.
Similarly, some authors (e.g., Conover[full citation needed]) suggest transforming the data to ranks (if they are not already ranks) and then performing the t-test on the transformed data, the version of the t-test used depending on whether or not the population variances are suspected to be different. Rank transformations do not preserve variances, but variances are recomputed from samples after rank transformations.
See also Kolmogorov–Smirnov test.
The statistic appeared in a 1914 article by the German Gustav Deuchler (with a missing term in the variance).
As a one-sample statistic, the signed rank was proposed by Frank Wilcoxon in 1945, with some discussion of a two-sample variant for equal sample sizes, in a test of significance with a point null-hypothesis against its complementary alternative (that is, equal versus not equal).
A thorough analysis of the statistic, which included a recurrence allowing the computation of tail probabilities for arbitrary sample sizes and tables for sample sizes of eight or less appeared in the article by Henry Mann and his student Donald Ransom Whitney in 1947. This article discussed alternative hypotheses, including a stochastic ordering (where the cumulative distribution functions satisfied the pointwise inequality FX(t) < FY(t)). This paper also computed the first four moments and established the limiting normality of the statistic under the null hypothesis, so establishing that it is asymptotically distribution-free.
Related test statistics
The Mann–Whitney U test is related to a number of other non-parametric statistical procedures. For example, it is equivalent to Kendall's tau correlation coefficient if one of the variables is binary (that is, it can only take two values).
Example statement of results
In reporting the results of a Mann–Whitney U test, it is important to state:
- A measure of the central tendencies of the two groups (means or medians; since the Mann–Whitney U test is an ordinal test, medians are usually recommended)
- The value of U (perhaps with some measure of effect size, such as Common language effect size or Rank-biserial correlation).
- The sample sizes
- The significance level.
In practice some of this information may already have been supplied and common sense should be used in deciding whether to repeat it. A typical report might run,
- "Median latencies in groups E and C were 153 and 247 ms; the distributions in the two groups differed significantly (Mann–Whitney U = 10.5, n1 = n2 = 8, P < 0.05 two-tailed)."
A statement that does full justice to the statistical status of the test might run,
- "Outcomes of the two treatments were compared using the Wilcoxon–Mann–Whitney two-sample rank-sum test. The treatment effect (difference between treatments) was quantified using the Hodges–Lehmann (HL) estimator, which is consistent with the Wilcoxon test. This estimator (HLΔ) is the median of all possible differences in outcomes between a subject in group B and a subject in group A. A non-parametric 0.95 confidence interval for HLΔ accompanies these estimates as does ρ, an estimate of the probability that a randomly chosen subject from population B has a higher weight than a randomly chosen subject from population A. The median [quartiles] weight for subjects on treatment A and B respectively are 147 [121, 177] and 151 [130, 180] kg. Treatment A decreased weight by HLΔ = 5 kg (0.95 CL [2, 9] kg, 2P = 0.02, ρ = 0.58)."
However it would be rare to find so extended a report in a document whose major topic was not statistical inference.
In many software packages, the Mann–Whitney U test (of the hypothesis of equal distributions against appropriate alternatives) has been poorly documented. Some packages incorrectly treat ties or fail to document asymptotic techniques (e.g., correction for continuity). A 2000 review discussed some of the following packages:
- MATLAB has ranksum in its Statistics Toolbox.
- R's statistics base-package implements the test
wilcox.testin its "stats" package.
- SAS implements the test in its PROC NPAR1WAY procedure.
- Python (programming language) has an implementation of this test provided by SciPy
- SigmaStat (SPSS Inc., Chicago, IL)
- SYSTAT (SPSS Inc., Chicago, IL)
- Java (programming language) has an implementation of this test provided by Apache Commons
- JMP (SAS Institute Inc., Cary, NC)
- S-Plus (MathSoft, Inc., Seattle, WA)
- STATISTICA (StatSoft, Inc., Tulsa, OK)
- UNISTAT (Unistat Ltd, London)
- SPSS (SPSS Inc, Chicago)
- StatsDirect (StatsDirect Ltd, Manchester, UK) implements all common variants.
- Stata (Stata Corporation, College Station, TX) implements the test in its ranksum command.
- StatXact (Cytel Software Corporation, Cambridge, Massachusetts)
- PSPP implements the test in its WILCOXON function.
- Lepage test
- Cucconi test
- Kolmogorov–Smirnov test
- Wilcoxon signed-rank test
- Kruskal–Wallis one-way analysis of variance
- Mann, Henry B.; Whitney, Donald R. (1947). "On a Test of Whether one of Two Random Variables is Stochastically Larger than the Other". Annals of Mathematical Statistics. 18 (1): 50–60. doi:10.1214/aoms/1177730491. MR 0022058. Zbl 0041.26103.
- Fay, Michael P.; Proschan, Michael A. (2010). "Wilcoxon–Mann–Whitney or t-test? On assumptions for hypothesis tests and multiple interpretations of decision rules". Statistics Surveys. 4: 1–39. doi:10.1214/09-SS051. MR 2595125. PMC 2857732. PMID 20414472.
- , See Table 2.1 of Pratt (1964) "Robustness of Some Procedures for the Two-Sample Location Problem." Journal of the American Statistical Association. 59 (307): 655–680. If the two distributions are normal with the same mean but different variances, then Pr[ X>Y]=Pr[Y<X] but the size of the Mann-Whitney test can be larger than the nominal level. So we cannot define the null hypothesis as Pr[ X>Y]=Pr[Y<X] and get a valid test.
- Zar, Jerrold H. (1998). Biostatistical Analysis. New Jersey: Prentice Hall International, INC. p. 147. ISBN 978-0-13-082390-8.
- Wilkinson, Leland (1999). "Statistical methods in psychology journals: Guidelines and explanations". American Psychologist. 54 (8): 594–604. doi:10.1037/0003-066X.54.8.594.
- Nakagawa, Shinichi; Cuthill, Innes C (2007). "Effect size, confidence interval and statistical significance: a practical guide for biologists". Biological Reviews of the Cambridge Philosophical Society. 82 (4): 591–605. doi:10.1111/j.1469-185X.2007.00027.x. PMID 17944619.
- Kerby, D.S. (2014). "The simple difference formula: An approach to teaching nonparametric correlation." Comprehensive Psychology, volume 3, article 1. doi:10.2466/11.IT.3.1. link to full article
- McGraw, K.O.; Wong, J.J. (1992). "A common language effect size statistic". Psychological Bulletin. 111 (2): 361–365. doi:10.1037/0033-2909.111.2.361.
- Grissom RJ (1994). "Statistical analysis of ordinal categorical status after therapies". Journal of Consulting and Clinical Psychology. 62 (2): 281–284. doi:10.1037/0022-006X.62.2.281.
- Herrnstein, Richard J.; Loveland, Donald H.; Cable, Cynthia (1976). "Natural Concepts in Pigeons". Journal of Experimental Psychology: Animal Behavior Processes. 2 (4): 285–302. doi:10.1037/0097-7403.2.4.285.
- Hanley, James A.; McNeil, Barbara J. (1982). "The Meaning and Use of the Area under a Receiver Operating (ROC) Curve Characteristic". Radiology. 143 (1): 29–36. doi:10.1148/radiology.143.1.7063747. PMID 7063747.
- Mason, Simon J.; Graham, Nicholas E. (2002). "Areas beneath the relative operating characteristics (ROC) and relative operating levels (ROL) curves: Statistical significance and interpretation" (PDF). Quarterly Journal of the Royal Meteorological Society. 128 (584): 2145–2166. Bibcode:2002QJRMS.128.2145M. CiteSeerX 10.1.1.458.8392. doi:10.1256/003590002320603584.
- Hand, David J.; Till, Robert J. (2001). "A Simple Generalisation of the Area Under the ROC Curve for Multiple Class Classification Problems". Machine Learning. 45 (2): 171–186. doi:10.1023/A:1010920819831.
- Cureton, E.E. (1956). "Rank-biserial correlation". Psychometrika. 21 (3): 287–290. doi:10.1007/BF02289138.
- Wendt, H.W. (1972). "Dealing with a common problem in social science: A simplified rank-biserial coefficient of correlation based on the U statistic". European Journal of Social Psychology. 2 (4): 463–465. doi:10.1002/ejsp.2420020412.
- Motulsky, Harvey J.; Statistics Guide, San Diego, CA: GraphPad Software, 2007, p. 123
- Zimmerman, Donald W. (1998-01-01). "Invalidation of Parametric and Nonparametric Statistical Tests by Concurrent Violation of Two Assumptions". The Journal of Experimental Education. 67 (1): 55–68. doi:10.1080/00220979809598344. ISSN 0022-0973.
- Lehamnn, Erich L.; Elements of Large Sample Theory, Springer, 1999, p. 176
- Conover, William J.; Practical Nonparametric Statistics, John Wiley & Sons, 1980 (2nd Edition), pp. 225–226
- Lumley, Thomas; Diehr, Paula; Emerson, Scott; Chen, Lu (May 2002). "The Importance of the Normality Assumption in Large Public Health Data Sets". Annual Review of Public Health. 23 (1): 151–169. doi:10.1146/annurev.publhealth.23.100901.140546. ISSN 0163-7525.
- Conover, William J.; Iman, Ronald L. (1981). "Rank Transformations as a Bridge Between Parametric and Nonparametric Statistics". The American Statistician. 35 (3): 124–129. doi:10.2307/2683975. JSTOR 2683975.
- Kasuya, Eiiti (2001). "Mann–Whitney U test when variances are unequal". Animal Behaviour. 61 (6): 1247–1249. doi:10.1006/anbe.2001.1691.
- Kruskal, William H. (September 1957). "Historical Notes on the Wilcoxon Unpaired Two-Sample Test". Journal of the American Statistical Association. 52 (279): 356–360. doi:10.2307/2280906. JSTOR 2280906.
- Wilcoxon, Frank (1945). "Individual comparisons by ranking methods". Biometrics Bulletin. 1 (6): 80–83. doi:10.2307/3001968. hdl:10338.dmlcz/135688. JSTOR 3001968.
- Myles Hollander and Douglas A. Wolfe (1999). Nonparametric Statistical Methods (2 ed.). Wiley-Interscience. ISBN 978-0471190455.CS1 maint: uses authors parameter (link)
- Bergmann, Reinhard; Ludbrook, John; Spooren, Will P.J.M. (2000). "Different Outcomes of the Wilcoxon–Mann–Whitney Test from Different Statistics Packages". The American Statistician. 54 (1): 72–77. doi:10.1080/00031305.2000.10474513. JSTOR 2685616.
- "scipy.stats.mannwhitneyu". SciPy v0.16.0 Reference Guide. The Scipy community. 24 July 2015. Retrieved 11 September 2015.
scipy.stats.mannwhitneyu(x, y, use_continuity=True): Computes the Mann–Whitney rank test on samples x and y.
- Hettmansperger, T.P.; McKean, J.W. (1998). Robust nonparametric statistical methods. Kendall's Library of Statistics. 5 (First ed., rather than Taylor and Francis (2010) second ed.). London; New York: Edward Arnold; John Wiley and Sons, Inc. pp. xiv+467. ISBN 978-0-340-54937-7. MR 1604954.
- Corder, G.W.; Foreman, D.I. (2014). Nonparametric Statistics: A Step-by-Step Approach. Wiley. ISBN 978-1118840313.
- Hodges, J.L.; Lehmann, E.L. (1963). "Estimation of location based on ranks". Annals of Mathematical Statistics. 34 (2): 598–611. doi:10.1214/aoms/1177704172. JSTOR 2238406. MR 0152070. Zbl 0203.21105. PE euclid.aoms/1177704172.
- Kerby, D.S. (2014). The simple difference formula: An approach to teaching nonparametric correlation. Comprehensive Psychology, volume 3, article 1. doi:10.2466/11.IT.3.1. link to article
- Lehmann, Erich L. (2006). Nonparametrics: Statistical methods based on ranks. With the special assistance of H.J.M. D'Abrera (Reprinting of 1988 revision of 1975 Holden-Day ed.). New York: Springer. pp. xvi+463. ISBN 978-0-387-35212-1. MR 0395032.
- Oja, Hannu (2010). Multivariate nonparametric methods with R: An approach based on spatial signs and ranks. Lecture Notes in Statistics. 199. New York: Springer. pp. xiv+232. doi:10.1007/978-1-4419-0468-3. ISBN 978-1-4419-0467-6. MR 2598854.
- Sen, Pranab Kumar (December 1963). "On the estimation of relative potency in dilution(-direct) assays by distribution-free methods". Biometrics. 19 (4): 532–552. doi:10.2307/2527532. JSTOR 2527532. Zbl 0119.15604.
- Table of critical values of U (pdf)
- Interactive calculator for U and its significance
- Brief guide by experimental psychologist Karl L. Weunsch – Nonparametric effect size estimators (Copyright 2015 by Karl L. Weunsch)