Using Mann-Whitney U test (Wilcoxon rank sum test), I am comparing two groups to see whether they are statistically different. Based on almost the same median and mean values between the two groups, I definitely thought that p-value would be very high. I am looking for a few rules of thumb of when to determine that my data is 'normal enough' to use a t-test vs. A Mann-Whitney U-test. From what I have read, most real world data sets are non-normal, and when sample sizes are large, tests including the Shaprio-Wilk will always reject the null hypothesis.
The Mann-Whitney (or Wilcoxon-Mann-Whitney) test is sometimes used for comparing the efficacy of two treatments in clinical trials. It is often presented as an alternative to a t test when the data are not normally distributed. Whereas a t test is a test of population means, the Mann-Whitney test is commonly regarded as a test of population medians. This is not strictly true, and treating it as such can lead to inadequate analysis of data.
Summary points
The Mann-Whitney test is used as an alternative to a t test when the data are not normally distributed
The test can detect differences in shape and spread as well as just differences in medians
Differences in population medians are often accompanied by equally important differences in shape
Researchers should describe the clinically important features of data and not just quote a P value
Use of Mann-Whitney test
The Mann-Whitney test is a test of both location and shape. Given two independent samples, it tests whether one variable tends to have values higher than the other. As Altman states, one form of the test statistic is an estimate of the probability that one variable is less than the other,1 although this statistic is not output by many statistical packages. In the case where the only distributional difference is a shift in location, this can indeed be described as a difference in medians. Hence, for example, the online help facility in Minitab 10.51 states that the Mann-Whitney test is “a two-sample rank test for the difference between two population medians . . . It assumes that the data are independent random samples from two populations that have the same shape.” Figure Figure11 shows two distributions for which this is the case. One distribution is shifted 0.75 units to the right: the medians differ by 0.75 units but the shapes are identical.
Two distributions with a difference in median but no difference in shape and spread
Theoretically, in large samples the Mann-Whitney test can detect differences in spread even when the medians are very similar. However, an alternative form of the test is better than the standard Mann-Whitney test for this purpose.2 The alternative test, however, is not very efficient when population medians are unequal and is not widely available in statistical packages.
Differences in population medians are often accompanied by other differences in spread and shape. Moreover, the difference in medians may not be the most striking or the most clinically important difference. It is important to look at distributional differences and discuss them. Figure Figure22 shows an example in which the median values are 0.65 and 1.14 units. The distribution with the larger median also has larger spread. The spread is shown clearly in figure figure33 , which shows box plots of samples of 25 drawn from these two distributions. (The P value from the Mann-Whitney test is 0.02.) If the difference is assumed to be merely a difference in medians other clinically important information could be ignored.
Two distributions with different medians and different shapes. The distribution with the larger median also has a greater spread
Box plots of samples of size 25 drawn from the distributions in figure figure2.2. Vertical lines indicate the medians and boxes the interquartile range
Methods
I examined the use of the Mann-Whitney test in papers published in the BMJ between September 1999 and August 2000. I did an online search of the electronic text of the journal using the keywords Wilcoxon, Mann, and Whitney. I identified five papers that had used the Mann-Whitney test but where, in my judgment, the information given suggested that there might be important distributional differences other than a shift in location. These are described briefly below.
Examples
Grande et al studied the impact on place of death of a hospital at home service for palliative care. The authors noted a significant difference among patients randomised to hospital at home care: “Patients in the hospital at home group who were admitted to the service survived significantly longer after referral than hospital at home patients who were not admitted (16 v 8 days).” There were 112 patients admitted to the service (median survival 16 days, interquartile range 5-42.5) and 73 patients who were not admitted (8, 3-18 days). The striking feature about these three sets of summary statistics is that each in the former group is about twice that for the second group. This suggests that the difference between the two distributions might not be just a shift of 8 days: the difference might be multiplicative, not additive—that is, patients who were admitted might survive twice as long as those who were not admitted.
Williams et al did a cost effectiveness study of open access follow up for inflammatory bowel disease. One of the measures was the total cost of secondary care, and this was compared for two groups: open access and routine visit. The mean (SD) cost was £582 (£807.94) for the 77 patients in the open access group and £611 (£475.47) for the 78 patients in the routine visit group. Although the mean is higher in the second group, the standard deviation is much higher in the first. There must, therefore, have been some very large values in the first group. Without further information it is difficult to be sure, but there seem to be distributional differences between the two groups. The choice of a Mann-Whitney test for these economic data has been criticised elsewhere. If total expenditure is the aspect of prime interest then a t test would have been more appropriate. If the interest lay in the distributions, it is unlikely that the medians alone would adequately have described the differences.
Lux et al studied responses of local research ethics committees. A conclusion was that “The required number of complete copies of protocols and documents . . . was significantly lower for the local committees that used a fast track system.” The 44 committees in the fast track group required a median of three copies (95% percentiles 2 and 13) compared with 11 (1 and 15) copies for the 55 committees in the standard group. Not only are the medians different, the distributions must also be different. About half of the fast track committees asked for two or three copies, whereas about half of the other committees asked for 11-15 copies. These differences, which the authors did not comment on, relate to shape as well as location of the distributions.
Macleod et al studied women with breast cancer from affluent and deprived areas. One of their conclusions is “The time between the date of the referral letter and the first clinic was one day shorter in women from affluent areas.” The median (interquartile range) time was 6 (1-13) days in the affluent area and 7 (4-20) days in the deprived area. Although the medians differ by one day, the summary statistics suggest that the data for the deprived group are more right skewed, and differences between the two groups might be much more pronounced for the higher waiting times. It would have been helpful to discuss this in the paper.
A similar feature is even more evident in data from a study of pain in blood glucose testing. A visual analogue scale was used to record pain at the ear or thumb. The authors report “The median pain score was 2 mm in the ear group and 8.5 mm in the thumb group . . . the difference in median pain score is small.” Although this is true, the box plots in the paper show that the spread of scores in the thumb group is much greater than for the ear group. In particular, at least three out of 30 people in the thumb group report a score that is at least twice the highest value in the ear group. Overall, values seem much higher in the thumb group. This is important because patients are likely to be more concerned with the worst pain they might experience than the median value.
Recommendations
Researchers should take care to describe their data and to be clear about the features that are most clinically important. They should use the statistical test that is most relevant for their hypotheses, and describe the features of the data that are likely to have caused a hypothesis to be rejected. As is always the case, it is not sufficient merely to report a P value. In the case of the Mann-Whitney test, differences in spread may sometimes be as clinically important as differences in medians, and these need to be made clear to the reader.
Footnotes
References
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 nonparametrictest 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.
Unlike the t-test it does not require the assumption of normal distributions. It is nearly as efficient as the t-test on normal distributions.
This test can be used to determine 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.
- 4Examples
- 6Effect sizes
- 7Relation to other tests
- 7.3Different distributions
- 9Related test statistics
Assumptions and formal statement of hypotheses[edit]
Although Mann and Whitney[1] 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.[2]
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.[3]
- 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.
Calculations[edit]
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.
Method one:
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 for the first set. U for the other set is the converse.
Method two:
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:[4]
- 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
Properties[edit]
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.
Examples[edit]
Illustration of calculation methods[edit]
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.
![Mann whitney u test sas Mann whitney u test sas](https://statistics.laerd.com/spss-tutorials/img/mwut/identical-shape-distributions.png)
Illustration of object of test[edit]
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 TH 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[edit]
For large samples, U is approximately normally distributed. In that case, the standardized value
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
- and
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.
Effect sizes[edit]
It is a widely recommended practice for scientists to report an effect size for an inferential test.[5][6]
Common language effect size[edit]
One method of reporting the effect size for the Mann–Whitney U test is with the common language effect size.[7][8] 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 hypothesis.[8] 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%.[9]
Rank-biserial correlation[edit]
A second 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.[10] 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 the proportion that is unfavorable (u). The value of f is the common language effect size. This simple difference formula is as follows:[7]
Stated another way, the correlation is the difference between the common language effect size and its complement:
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.
There is a formula to compute the rank-biserial from the Mann–Whitney U and the sample sizes of each group:[11]
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, U is the smaller of the two, so U = 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[edit]
Comparison to Student's t-test[edit]
The Mann–Whitney U test is more widely applicable than independent samplesStudent's t-test, and the question arises of which should be preferred.
- 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.
- Robustness
- As it compares the sums of ranks,[12] the Mann–Whitney U test is less likely than the t-test to spuriously indicate significance because of the presence of outliers, which implies the Mann–Whitney U test is more robust.[clarification needed][citation needed]
- Efficiency
- 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.[13] For distributions sufficiently far from normal and for sufficiently large sample sizes, the Mann–Whitney U test is considerably more efficient than the t.[14]
Overall, the robustness makes the Mann–Whitney U test more widely applicable than the t-test, and for large samples from the normal distribution, the efficiency loss compared to the t-test is only 5%, so one can recommend the Mann–Whitney U test as the default test for comparing interval or ordinal measurements with similar distributions.[citation needed]
![Mann whitney u test matlab Mann whitney u test matlab](http://www.numeracy-bank.net/images/mwu/two_boxplotsexample.jpg)
The relation between efficiency and power in concrete situations isn't trivial though. For small sample sizes one should investigate the power of the Mann–Whitney U test vs the t-test.
The Mann–Whitney U test will give very similar results to performing an ordinary parametric two-sample t-test on the rankings of the data.[15]
Area-under-curve (AUC) statistic for ROC curves[edit]
The U statistic is equivalent to the area under the receiver operating characteristic curve that can be readily calculated.[16][17]
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:[18]
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.
Different distributions[edit]
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.[citation needed]
Alternatives[edit]
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 erroneously significant results.[19] In that situation, the unequal variances version of the t-test may give more reliable results.
Alternatively, 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.
The Brown–Forsythe test has been suggested as an appropriate non-parametric equivalent to the F-test for equal variances.[citation needed]
See also Kolmogorov–Smirnov test.
History[edit]
The statistic appeared in a 1914 article[20] 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,[21] 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.[1] 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[edit]
Kendall's tau[edit]
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).[citation needed]
ρ statistic[edit]
A statistic called ρ that is linearly related to U and widely used in studies of categorization (discrimination learning involving concepts), and elsewhere,[22] 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.[citation needed]
Example statement of results[edit]
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
- 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.[23] 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.
Implementations[edit]
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:[24]
- MATLAB has ranksum in its Statistics Toolbox.
- R's statistics base-package implements the test
wilcox.test
in its 'stats' package. - SAS implements the test in its PROC NPAR1WAY procedure.
- Python (programming language) has an implementation of this test provided by SciPy[25]
- SigmaStat (SPSS Inc., Chicago, IL)
- SYSTAT (SPSS Inc., Chicago, IL)
- Java (programming language) has an implementation of this test provided by Apache Commons[26]
- 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.
See also[edit]
Notes[edit]
- ^ abMann, 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. MR0022058. Zbl0041.26103.
- ^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. MR2595125. PMC2857732. PMID20414472.
- ^[1], 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. ISBN978-0-13-082390-8.
- ^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. PMID17944619.
- ^ abKerby, 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
- ^ abMcGraw, 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.
- ^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
- ^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
- ^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. JSTOR2683975.
- ^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. PMID7063747.
- ^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. CiteSeerX10.1.1.458.8392. 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'(PDF). Machine Learning. 45 (2): 171–186. doi:10.1023/A:1010920819831.
- ^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. JSTOR2280906.
- ^Wilcoxon, Frank (1945). 'Individual comparisons by ranking methods'. Biometrics Bulletin. 1 (6): 80–83. doi:10.2307/3001968. JSTOR3001968.
- ^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.
- ^Myles Hollander and Douglas A. Wolfe (1999). Nonparametric Statistical Methods (2 ed.). Wiley-Interscience. ISBN978-0471190455.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. JSTOR2685616.
- ^'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.
- ^'org.apache.commons.math3.stat.inference.MannWhitneyUTest'.
References[edit]
- 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. ISBN978-0-340-54937-7. MR1604954.
- Corder, G.W.; Foreman, D.I. (2014). Nonparametric Statistics: A Step-by-Step Approach. Wiley. ISBN978-1118840313.
- 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. JSTOR2238406. MR0152070. Zbl0203.21105. PEeuclid.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. ISBN978-0-387-35212-1. MR0395032.
- 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. ISBN978-1-4419-0467-6. MR2598854.
- 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. JSTOR2527532. Zbl0119.15604.
External links[edit]
- 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)