Mann-Whitney U
Encyclopedia
In statistics
, the Mann–Whitney U test (also called the Mann–Whitney–Wilcoxon (MWW) or Wilcoxon rank-sum test) is a non-parametric
statistical hypothesis test for assessing whether one of two samples
of independent observations tends to have larger values than the other. It is one of the most well-known non-parametric significance tests. It was proposed initially by Gustav Deuchler in 1914 (with a missing term in the variance) and later independently by Frank Wilcoxon
in 1945, for equal sample sizes, and extended to arbitrary sample sizes and in other ways by Henry Mann
and his student Donald Ransom Whitney in 1947.
than the other, there are many other ways to formulate the null
and alternative hypotheses such that the MWW test will give a valid test.
A very general formulation is to assume that:
Under more strict assumptions than those 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 MWW test as showing a difference in medians. Under this location shift assumption, we can also interpret the MWW 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.
, 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, rather than U itself.
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.
Method one:
First, arrange all the observations into a single ranked series. That is, rank all the observations without regard to which sample they are in.
For small samples a direct method is recommended. It is very quick, and gives an insight into the meaning of the U statistic.
Method two:
For larger samples, a formula can be used:
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:
What is the value of U?
The median tortoise here comes in at position 19, and thus actually beats the median hare, which comes in at position 20.
However, the value of U (for hares) is 100
(9 Hares beaten by (x) 0 tortoises) + (10 hares beaten by (x) 10 tortoises) = 0 + 100 = 100
Value of U(for tortoises) is 261
(10 tortoises beaten by 9 hares) + (9 tortoises beaten by 19 hares) = 90 + 171 = 261
Consulting tables, or using the approximation below, shows that this U value gives significant evidence that hares tend to do better than tortoises (p < 0.05, two-tailed). Obviously this is an extreme distribution that would be spotted easily, but in a larger sample something similar could happen without it being so apparent. Notice that the problem here is not that the two distributions of ranks have different variance
s; they are mirror images of each other, so their variances are the same, but they have very different skewness
.
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; the full formula is given in the text books referenced below. However, 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 = n1 n2, the mean n1 n2/2 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.
, and the question arises of which should be preferred.
Ordinal data: U remains the logical choice when the data are ordinal but not interval scaled, so that the spacing between adjacent values cannot be assumed to be constant.
Robustness: As it compares the sums of ranks, the Mann–Whitney test is less likely than the t-test to spuriously indicate significance because of the presence of outlier
s – i.e. Mann–Whitney is more robust
.
Efficiency: When normality holds, MWW has an (asymptotic) efficiency
of
or about 0.95 when compared to the t test. For distributions sufficiently far from normal and for sufficiently large sample sizes, the MWW can be considerably more efficient than the t.
Overall, the robustness makes the MWW 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 MWW as the default test for comparing interval or ordinal measurements with similar distributions.
The relation between efficiency
and power
in concrete situations isn't trivial though. For small sample sizes one should investigate the power of the MWW vs t.
MWW will give very similar results to performing an ordinary parametric two-sample t test on the rankings of the data.
curve that can be readily calculated.
curve (ROC) that is often used in the context.
If one desires a simple shift interpretation, the U test should not be used when the distributions of the two samples are very different, as it can give erroneously significant results.
Alternatively, some authors (e.g. Conover) 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.
correlation coefficient if one of the variables is binary (that is, it can only take two values).
involving concept
s) 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. This statistic was first proposed by Richard Herrnstein
(see Herrnstein et al., 1976). 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 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.
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,
A statement that does full justice to the statistical status of the test might run,
However it would be rare to find so extended a report in a document whose major topic was not statistical inference.
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, the Mann–Whitney U test (also called the Mann–Whitney–Wilcoxon (MWW) or Wilcoxon rank-sum test) is a non-parametric
Non-parametric statistics
In statistics, the term non-parametric statistics has at least two different meanings:The first meaning of non-parametric covers techniques that do not rely on data belonging to any particular distribution. These include, among others:...
statistical hypothesis test for assessing whether one of two samples
Sampling (statistics)
In statistics and survey methodology, sampling is concerned with the selection of a subset of individuals from within a population to estimate characteristics of the whole population....
of independent observations tends to have larger values than the other. It is one of the most well-known non-parametric significance tests. It was proposed initially by Gustav Deuchler in 1914 (with a missing term in the variance) and later independently by Frank Wilcoxon
Frank Wilcoxon
Frank Wilcoxon was a chemist and statistician, known for the development of several statistical tests....
in 1945, for equal sample sizes, and extended to arbitrary sample sizes and in other ways by Henry Mann
Henry Mann
Henry Berthold Mann was a professor of mathematics and statistics at Ohio State University. Mann proved the Schnirelmann-Landau conjecture in number theory, and as a result earned the 1946 Cole Prize. He and his student developed the U-statistic of nonparametric statistics...
and his student Donald Ransom Whitney in 1947.
Assumptions and formal statement of hypotheses
Although Mann and Whitney developed the MWW test under the assumption of continuous responses with the alternative hypothesis being that one distribution is stochastically greaterStochastic dominance
Stochastic dominance is a form of stochastic ordering. The term is used in decision theory and decision analysis to refer to situations where one gamble can be ranked as superior to another gamble. It is based on preferences regarding outcomes...
than the other, there are many other ways to formulate the null
Null hypothesis
The practice of science involves formulating and testing hypotheses, assertions that are capable of being proven false using a test of observed data. The null hypothesis typically corresponds to a general or default position...
and alternative hypotheses such that the MWW test will give a valid test.
A very general formulation is to assume that:
- All the observations from both groups are independentStatistical independenceIn probability theory, to say that two events are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other occurs...
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 the distributions of both groups are equal, so that the probability of an observation from one population (X) exceeding an observation from the second population (Y) equals the probability of an observation from Y exceeding an observation from X, that is, there is a symmetry between populations with respect to probability of random drawing of a larger observation.
- Under the alternative hypothesis the probability of an observation from one population (X) exceeding an observation from the second population (Y) (after exclusion of ties) is not equal to 0.5. The alternative may also be stated in terms of a one-sided test, for example: P(X > Y) + 0.5 P(X = Y) > 0.5.
Under more strict assumptions than those 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 MWW test as showing a difference in medians. Under this location shift assumption, we can also interpret the MWW 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
Median
In probability theory and statistics, a median is described as the numerical value separating the higher half of a sample, a population, or a probability distribution, from the lower half. The median of a finite list of numbers can be found by arranging all the observations from lowest value to...
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 statisticStatistic
A statistic is a single measure of some attribute of a sample . It is calculated by applying a function to the values of the items comprising the sample which are known together as a set of data.More formally, statistical theory defines a statistic as a function of a sample where the function...
, usually called U, whose distribution under the null hypothesis
Null hypothesis
The practice of science involves formulating and testing hypotheses, assertions that are capable of being proven false using a test of observed data. The null hypothesis typically corresponds to a general or default position...
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, rather than U itself.
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.
Method one:
First, arrange all the observations into a single ranked series. That is, rank all the observations without regard to which sample they are in.
For small samples a direct method is recommended. It is very quick, and gives an insight into the meaning of the U statistic.
- 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."
- Taking each observation in sample 1, count the number of observations in sample 2 that have a smaller rank (count a half for any that are equal to it). The sum of these counts is U.
Method two:
For larger samples, a formula can be used:
- 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.
- 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 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
- 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
- Knowing that R1 + R2 = N(N + 1)/2 and N = n1 + n2 , and doing some algebra, we find that the sum is
Properties
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.Illustration of calculation methods
Suppose that AesopAesop
Aesop was a Greek writer credited with a number of popular fables. Older spellings of his name have included Esop and Isope. Although his existence remains uncertain and no writings by him survive, numerous tales credited to him were gathered across the centuries and in many languages in a...
is dissatisfied with his classic experiment
The Tortoise and the Hare
The Tortoise and the Hare is a fable attributed to Aesop and is number 226 in the Perry Index. The story concerns a hare who ridicules a slow-moving tortoise and is challenged by him to a race. The hare soon leaves the tortoise behind and, confident of winning, decides to take a nap midway through...
in which one tortoise
Tortoise
Tortoises are a family of land-dwelling reptiles of the order of turtles . Like their marine cousins, the sea turtles, tortoises are shielded from predators by a shell. The top part of the shell is the carapace, the underside is the plastron, and the two are connected by the bridge. The tortoise...
was found to beat one hare
Hare
Hares and jackrabbits are leporids belonging to the genus Lepus. Hares less than one year old are called leverets. Four species commonly known as types of hare are classified outside of Lepus: the hispid hare , and three species known as red rock hares .Hares are very fast-moving...
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 is beaten by, getting 0, 5, 5, 5, 5, 5, which means U = 25. Alternatively, we could take each hare in turn, and count the number of tortoises it is beaten by. In this case, we get 1, 1, 1, 1, 1, 6. So U = 6 + 1 + 1 + 1 + 1 + 1 = 11. 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.
Illustration of object of test
A second example illustrates the point that the Mann–Whitney does not test for equality of medians. Consider another hare and tortoise race, with 19 participants of each species, in which the outcomes are as follows:- 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
The median tortoise here comes in at position 19, and thus actually beats the median hare, which comes in at position 20.
However, the value of U (for hares) is 100
(9 Hares beaten by (x) 0 tortoises) + (10 hares beaten by (x) 10 tortoises) = 0 + 100 = 100
Value of U(for tortoises) is 261
(10 tortoises beaten by 9 hares) + (9 tortoises beaten by 19 hares) = 90 + 171 = 261
Consulting tables, or using the approximation below, shows that this U value gives significant evidence that hares tend to do better than tortoises (p < 0.05, two-tailed). Obviously this is an extreme distribution that would be spotted easily, but in a larger sample something similar could happen without it being so apparent. Notice that the problem here is not that the two distributions of ranks have different variance
Variance
In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...
s; they are mirror images of each other, so their variances are the same, but they have very different skewness
Skewness
In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. The skewness value can be positive or negative, or even undefined...
.
Normal approximation
For large samples, U is approximately normally distributed. In that case, the standardized valuewhere 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; the full formula is given in the text books referenced below. However, 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 = n1 n2, the mean n1 n2/2 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.
Comparison to Student's t-test
The U test is useful in the same situations as the independent samples Student's t-testStudent's t-test
A t-test is any statistical hypothesis test in which the test statistic follows a Student's t distribution if the null hypothesis is supported. It is most commonly applied when the test statistic would follow a normal distribution if the value of a scaling term in the test statistic were known...
, and the question arises of which should be preferred.
Ordinal data: U remains the logical choice when the data are ordinal but not interval scaled, so that the spacing between adjacent values cannot be assumed to be constant.
Robustness: As it compares the sums of ranks, the Mann–Whitney test is less likely than the t-test to spuriously indicate significance because of the presence of outlier
Outlier
In statistics, an outlier is an observation that is numerically distant from the rest of the data. Grubbs defined an outlier as: An outlying observation, or outlier, is one that appears to deviate markedly from other members of the sample in which it occurs....
s – i.e. Mann–Whitney is more robust
Robust statistics
Robust statistics provides an alternative approach to classical statistical methods. The motivation is to produce estimators that are not unduly affected by small departures from model assumptions.- Introduction :...
.
Efficiency: When normality holds, MWW has an (asymptotic) efficiency
Efficiency (statistics)
In statistics, an efficient estimator is an estimator that estimates the quantity of interest in some “best possible” manner. The notion of “best possible” relies upon the choice of a particular loss function — the function which quantifies the relative degree of undesirability of estimation errors...
of
or about 0.95 when compared to the t test. For distributions sufficiently far from normal and for sufficiently large sample sizes, the MWW can be considerably more efficient than the t.Overall, the robustness makes the MWW 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 MWW as the default test for comparing interval or ordinal measurements with similar distributions.
The relation between efficiency
Efficiency (statistics)
In statistics, an efficient estimator is an estimator that estimates the quantity of interest in some “best possible” manner. The notion of “best possible” relies upon the choice of a particular loss function — the function which quantifies the relative degree of undesirability of estimation errors...
and power
Statistical power
The power of a statistical test is the probability that the test will reject the null hypothesis when the null hypothesis is actually false . The power is in general a function of the possible distributions, often determined by a parameter, under the alternative hypothesis...
in concrete situations isn't trivial though. For small sample sizes one should investigate the power of the MWW vs t.
MWW will give very similar results to performing an ordinary parametric two-sample t test on the rankings of the data.
Area-under-curve (AUC) statistic for ROC curves
The U statistic is equivalent to the area under the receiver operating characteristicReceiver operating characteristic
In signal detection theory, a receiver operating characteristic , or simply ROC curve, is a graphical plot of the sensitivity, or true positive rate, vs. false positive rate , for a binary classifier system as its discrimination threshold is varied...
curve that can be readily calculated.
Different distributions
If one is only interested in stochastic ordering of the two populations (i.e., the concordance probability P(Y > X)), the 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 characteristicReceiver operating characteristic
In signal detection theory, a receiver operating characteristic , or simply ROC curve, is a graphical plot of the sensitivity, or true positive rate, vs. false positive rate , for a binary classifier system as its discrimination threshold is varied...
curve (ROC) that is often used in the context.
If one desires a simple shift interpretation, the U test should not be used when the distributions of the two samples are very different, as it can give erroneously significant results.
Alternatives
In that situation, the unequal variances version of the t test is likely to give more reliable results, but only if normality holds.Alternatively, some authors (e.g. Conover) 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.
Kendall's τ
The U test is related to a number of other non-parametric statistical procedures. For example, it is equivalent to Kendall's τKendall tau rank correlation coefficient
In statistics, the Kendall rank correlation coefficient, commonly referred to as Kendall's tau coefficient, is a statistic used to measure the association between two measured quantities...
correlation coefficient if one of the variables is binary (that is, it can only take two values).
ρ statistic
A statistic called ρ that is linearly related to U and widely used in studies of categorization (discrimination learningDiscrimination learning
In psychology, discrimination learning is the process by which animals or people learn to make different responses to different stimuli. It was a classic topic in the psychology of learning from the 1920s to the 1970s, and was particularly investigated within:...
involving concept
Concept
The word concept is used in ordinary language as well as in almost all academic disciplines. Particularly in philosophy, psychology and cognitive sciences the term is much used and much discussed. WordNet defines concept: "conception, construct ". However, the meaning of the term concept is much...
s) 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. This statistic was first proposed by Richard Herrnstein
Richard Herrnstein
Richard J. Herrnstein was an American researcher in animal learning in the Skinnerian tradition. He was one of the founders of quantitative analysis of behavior....
(see Herrnstein et al., 1976). 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 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.
Example statement of results
In reporting the results of a Mann–Whitney test, it is important to state:- A measure of the central tendencies of the two groups (means or medians; since the Mann–Whitney 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 (ref. 5 below). 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
- Online implementation using javascript
- ALGLIB includes implementation of the Mann–Whitney U test in C++, C#, Delphi, Visual Basic, etc.
- RR (programming language)R is a programming language and software environment for statistical computing and graphics. The R language is widely used among statisticians for developing statistical software, and R is widely used for statistical software development and data analysis....
includes an implementation of the test (there referred to as the Wilcoxon two-sample test) aswilcox.test(and in cases of ties in the sample:wilcox.exactin theexactRankTestspackage, or use theexact=FALSEoption). - StataStataStata is a general-purpose statistical software package created in 1985 by StataCorp. It is used by many businesses and academic institutions around the world...
includes implementation of Wilcoxon-Mann-Whitney rank-sum test with ranksum command. - SciPySciPySciPy is an open source library of algorithms and mathematical tools for the Python programming language.SciPy contains modules for optimization, linear algebra, integration, interpolation, special functions, FFT, signal and image processing, ODE solvers and other tasks common in science and...
has themannwhitneyufunction in thestatsmodule. - MATLABMATLABMATLAB is a numerical computing environment and fourth-generation programming language. Developed by MathWorks, MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages,...
implements the test with functionranksumin the statistics toolbox. - MathematicaMathematicaMathematica is a computational software program used in scientific, engineering, and mathematical fields and other areas of technical computing...
implements the function asMannWhitneyTest.
See also
- Kolmogorov–Smirnov test
- Wilcoxon signed-rank testWilcoxon signed-rank testThe Wilcoxon signed-rank test is a non-parametric statistical hypothesis test used when comparing two related samples or repeated measurements on a single sample to assess whether their population mean ranks differ The Wilcoxon signed-rank test is a non-parametric statistical hypothesis test used...
- Kruskal–Wallis one-way analysis of variance
External links
- Table of critical values of U (pdf)
- Discussion and table of critical values for the original Wilcoxon Rank-Sum Test, which uses a slightly different test statistic (pdf)
- Interactive calculator for U and its significance
- Mann, Henry Berthold (biography at Ohio State UniversityOhio State UniversityThe Ohio State University, commonly referred to as Ohio State, is a public research university located in Columbus, Ohio. It was originally founded in 1870 as a land-grant university and is currently the third largest university campus in the United States...
)