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51-537: Mann–Whitney U {\displaystyle U} test (also called the Mann–Whitney–Wilcoxon ( MWW/MWU ), Wilcoxon rank-sum test , or Wilcoxon–Mann–Whitney test ) is a nonparametric statistical test of the null hypothesis that, for randomly selected values X and Y from two populations, the probability of X being greater than Y is equal to the probability of Y being greater than X . Nonparametric tests used on two dependent samples are

102-437: A K {\displaystyle K} -valued function of r {\displaystyle r} d {\displaystyle d} -dimensional variables. For each n ≥ r {\displaystyle n\geq r} the associated U-statistic f n : ( K d ) n → K {\displaystyle f_{n}\colon (K^{d})^{n}\to K}

153-403: A certain parameter is required. Suppose that a simple unbiased estimate can be constructed based on only a few observations: this defines the basic estimator based on a given number of observations. For example, a single observation is itself an unbiased estimate of the mean and a pair of observations can be used to derive an unbiased estimate of the variance. The U-statistic based on this estimator

204-401: A cost: in cases where a parametric test's assumptions are met, non-parametric tests have less statistical power . In other words, a larger sample size can be required to draw conclusions with the same degree of confidence. Non-parametric models differ from parametric models in that the model structure is not specified a priori but is instead determined from data. The term non-parametric

255-478: A finite population, where the defining property is termed ‘inheritance on the average’. Fisher's k -statistics and Tukey's polykays are examples of homogeneous polynomial U-statistics (Fisher, 1929; Tukey, 1950). For a simple random sample φ of size  n taken from a population of size  N , the U-statistic has the property that the average over sample values  ƒ n ( xφ )

306-428: A fixed size. The letter "U" stands for unbiased. In elementary statistics, U-statistics arise naturally in producing minimum-variance unbiased estimators . The theory of U-statistics allows a minimum-variance unbiased estimator to be derived from each unbiased estimator of an estimable parameter (alternatively, statistical functional ) for large classes of probability distributions . An estimable parameter

357-449: A hypothesis, for obvious reasons, is called parametric . Hypothesis (c) was of a different nature, as no parameter values are specified in the statement of the hypothesis; we might reasonably call such a hypothesis non-parametric . Hypothesis (d) is also non-parametric but, in addition, it does not even specify the underlying form of the distribution and may now be reasonably termed distribution-free . Notwithstanding these distinctions,

408-532: 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

459-406: A valid test. A very general formulation is to assume that: Under the general formulation, the test is only consistent when the following occurs under H 1 : 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., F 1 ( x ) = F 2 ( x + δ ) , we can interpret

510-440: 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

561-410: Is where n = n 1 + n 2 . 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 U 1 + U 2 = n 1 n 2 , the mean n 1 n 2 /2 used in the normal approximation is the mean of

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612-431: Is Order statistics , which are based on ordinal ranking of observations. The discussion following is taken from Kendall's Advanced Theory of Statistics . Statistical hypotheses concern the behavior of observable random variables.... For example, the hypothesis (a) that a normal distribution has a specified mean and variance is statistical; so is the hypothesis (b) that it has a given mean but unspecified variance; so

663-435: Is a measurable function of the population's cumulative probability distribution : For example, for every probability distribution, the population median is an estimable parameter. The theory of U-statistics applies to general classes of probability distributions. Many statistics originally derived for particular parametric families have been recognized as U-statistics for general distributions. In non-parametric statistics ,

714-483: 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 U 1 {\displaystyle U_{1}} or U 2 {\displaystyle U_{2}} ) and

765-425: Is calculated by dividing U by its maximum value for the given sample sizes, which is simply n 1 × n 2 . 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

816-452: Is defined as the average (across all combinatorial selections of the given size from the full set of observations) of the basic estimator applied to the sub-samples. Pranab K. Sen (1992) provides a review of the paper by Wassily Hoeffding (1948), which introduced U-statistics and set out the theory relating to them, and in doing so Sen outlines the importance U-statistics have in statistical theory. Sen says, “The impact of Hoeffding (1948)

867-418: Is defined as the smaller of: with The U statistic is related to the area under the receiver operating characteristic curve ( AUC ): Note that this is the same definition as the common language effect size , i.e. the probability that a classifier will rank a randomly chosen instance from the first group higher than a randomly chosen instance from the second group. Because of its probabilistic form,

918-573: Is defined to be the average of the values f ( x i 1 , … , x i r ) {\displaystyle f(x_{i_{1}},\dotsc ,x_{i_{r}})} over the set I r , n {\displaystyle I_{r,n}} of r {\displaystyle r} -tuples of indices from { 1 , 2 , … , n } {\displaystyle \{1,2,\dotsc ,n\}} with distinct entries. Formally, In particular, if f {\displaystyle f}

969-425: 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

1020-409: Is due to their more general nature, which may make them less susceptible to misuse and misunderstanding. Non-parametric methods can be considered a conservative choice, as they will work even when their assumptions are not met, whereas parametric methods can produce misleading results when their assumptions are violated. The wider applicability and increased robustness of non-parametric tests comes at

1071-428: Is exactly equal to the population value  ƒ N ( x ). Some examples: If f ( x ) = x {\displaystyle f(x)=x} the U-statistic f n ( x ) = x ¯ n = ( x 1 + ⋯ + x n ) / n {\displaystyle f_{n}(x)={\bar {x}}_{n}=(x_{1}+\cdots +x_{n})/n}

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1122-432: Is much more general than the corresponding parametric methods. In particular, they may be applied in situations where less is known about the application in question. Also, due to the reliance on fewer assumptions, non-parametric methods are more robust . Non-parametric methods are sometimes considered simpler to use and more robust than parametric methods, even when the assumptions of parametric methods are justified. This

1173-504: Is not meant to imply that such models completely lack parameters but that the number and nature of the parameters are flexible and not fixed in advance. Non-parametric (or distribution-free ) inferential statistical methods are mathematical procedures for statistical hypothesis testing which, unlike parametric statistics , make no assumptions about the probability distributions of the variables being assessed. The most frequently used tests include Early nonparametric statistics include

1224-563: Is overwhelming at the present time and is very likely to continue in the years to come.” Note that the theory of U-statistics is not limited to the case of independent and identically-distributed random variables or to scalar random-variables. The term U-statistic, due to Hoeffding (1948), is defined as follows. Let K {\displaystyle K} be either the real or complex numbers, and let f : ( K d ) r → K {\displaystyle f\colon (K^{d})^{r}\to K} be

1275-666: Is symmetric the above is simplified to where now J r , n {\displaystyle J_{r,n}} denotes the subset of I r , n {\displaystyle I_{r,n}} of increasing tuples. Each U-statistic f n {\displaystyle f_{n}} is necessarily a symmetric function . U-statistics are very natural in statistical work, particularly in Hoeffding's context of independent and identically distributed random variables , or more generally for exchangeable sequences , such as in simple random sampling from

1326-445: Is the hypothesis (c) that a distribution is of normal form with both mean and variance unspecified; finally, so is the hypothesis (d) that two unspecified continuous distributions are identical. It will have been noticed that in the examples (a) and (b) the distribution underlying the observations was taken to be of a certain form (the normal) and the hypothesis was concerned entirely with the value of one or both of its parameters. Such

1377-410: Is the same result as with the simple difference formula above. Nonparametric statistics The term "nonparametric statistics" has been defined imprecisely in the following two ways, among others: The first meaning of nonparametric involves techniques that do not rely on data belonging to any particular parametric family of probability distributions. These include, among others: An example

1428-942: Is the sample mean. If f ( x 1 , x 2 ) = | x 1 − x 2 | {\displaystyle f(x_{1},x_{2})=|x_{1}-x_{2}|} , the U-statistic is the mean pairwise deviation f n ( x 1 , … , x n ) = 2 / ( n ( n − 1 ) ) ∑ i > j | x i − x j | {\displaystyle f_{n}(x_{1},\ldots ,x_{n})=2/(n(n-1))\sum _{i>j}|x_{i}-x_{j}|} , defined for n ≥ 2 {\displaystyle n\geq 2} . If f ( x 1 , x 2 ) = ( x 1 − x 2 ) 2 / 2 {\displaystyle f(x_{1},x_{2})=(x_{1}-x_{2})^{2}/2} ,

1479-473: The U statistic can be generalized to a measure of a classifier's separation power for more than two classes: Where c is the number of classes, and the R k , ℓ term of AUC k , 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 . AUC k , k will always be zero but, unlike in

1530-830: 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. Let X 1 , … , X n 1 {\displaystyle X_{1},\ldots ,X_{n_{1}}} be group 1, an i.i.d. sample from X {\displaystyle X} , and Y 1 , … , Y n 2 {\displaystyle Y_{1},\ldots ,Y_{n_{2}}} be group 2, an i.i.d. sample from Y {\displaystyle Y} , and let both samples be independent of each other. The corresponding Mann–Whitney U statistic

1581-416: The median (13th century or earlier, use in estimation by Edward Wright , 1599; see Median § History ) and the sign test by John Arbuthnot (1710) in analyzing the human sex ratio at birth (see Sign test § History ). U statistic In statistical theory , a U-statistic is a class of statistics defined as the average over the application of a given function applied to all tuples of

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1632-414: The sign test and the Wilcoxon signed-rank test . Although Henry Mann and Donald Ransom 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

1683-892: The U-statistic is the sample variance f n ( x ) = ∑ ( x i − x ¯ n ) 2 / ( n − 1 ) {\displaystyle f_{n}(x)=\sum (x_{i}-{\bar {x}}_{n})^{2}/(n-1)} with divisor n − 1 {\displaystyle n-1} , defined for n ≥ 2 {\displaystyle n\geq 2} . The third k {\displaystyle k} -statistic k 3 , n ( x ) = ∑ ( x i − x ¯ n ) 3 n / ( ( n − 1 ) ( n − 2 ) ) {\displaystyle k_{3,n}(x)=\sum (x_{i}-{\bar {x}}_{n})^{3}n/((n-1)(n-2))} ,

1734-416: 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

1785-403: 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

1836-413: The finish line) is as follows, writing T for a tortoise and H for a hare: What is the value of U ? In reporting the results of a Mann–Whitney U test, it is important to state: 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

1887-408: The first sample and an observation in the second sample. Otherwise, if both the dispersions and shapes of the distribution of both samples differ, the Mann–Whitney U test fails a test of medians. It is possible to show examples where medians are numerically equal while the test rejects the null hypothesis with a small p-value. The Mann–Whitney U test / Wilcoxon rank-sum test is not the same as

1938-439: The first set. U for the other set is the converse (i.e.: U 2 {\displaystyle U_{2}} ). Method two: For larger samples: The maximum value of U is the product of the sample sizes for the two samples (i.e.: U i = n 1 n 2 {\displaystyle U_{i}=n_{1}n_{2}} ). In such a case, the "other" U would be 0. Suppose that Aesop

1989-568: 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%. The relationship between f and the Mann–Whitney U (specifically U 1 {\displaystyle U_{1}} ) is as follows: This is the same as the area under the curve (AUC) for the ROC curve . A statistic called ρ that is linearly related to U and widely used in studies of categorization ( discrimination learning involving concepts ), and elsewhere,

2040-415: The median tortoise beat the median hare, the hares collectively did better than the tortoises collectively. 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

2091-499: The normal distribution. m U and σ U 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 adjusted as follows: where the left side is simply the variance and the right side is the adjustment for ties, t k is the number of ties for the k th rank, and K is the total number of unique ranks with ties. A more computationally-efficient form with n 1 n 2 /12 factored out

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2142-410: 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.: U 1 {\displaystyle U_{1}} ) for

2193-494: The sample skewness defined for n ≥ 3 {\displaystyle n\geq 3} , is a U-statistic. The following case highlights an important point. If f ( x 1 , x 2 , x 3 ) {\displaystyle f(x_{1},x_{2},x_{3})} is the median of three values, f n ( x 1 , … , x n ) {\displaystyle f_{n}(x_{1},\ldots ,x_{n})}

2244-404: 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, U 2 is the smaller of the two, so U 2 = 10 . This formula then gives r = 1 – (2×10) / (10×10) = 0.80 , which

2295-508: The statistical literature now commonly applies the label "non-parametric" to test procedures that we have just termed "distribution-free", thereby losing a useful classification. The second meaning of non-parametric involves techniques that do not assume that the structure of a model is fixed. Typically, the model grows in size to accommodate the complexity of the data. In these techniques, individual variables are typically assumed to belong to parametric distributions, and assumptions about

2346-408: The sum of ranks in one of the samples, rather than U itself. The Mann–Whitney U test is included in most 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

2397-415: The test might run, However it would be rare to find such an extensive report in a document whose major topic was not statistical inference. For large samples, U is approximately normally distributed . In that case, the standardized value where m U and σ U are the mean and standard deviation of U , is approximately a standard normal deviate whose significance can be checked in tables of

2448-457: The theory of U-statistics is used to establish for statistical procedures (such as estimators and tests) and estimators relating to the asymptotic normality and to the variance (in finite samples) of such quantities. The theory has been used to study more general statistics as well as stochastic processes , such as random graphs . Suppose that a problem involves independent and identically-distributed random variables and that estimation of

2499-416: The two values of U . Therefore, the absolute value of the z -statistic calculated will be same whichever value of U is used. It is a widely recommended practice for scientists to report an effect size for an inferential test. The following measures are equivalent. 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,

2550-495: The two-class case, generally AUC k , ℓ ≠ AUC ℓ , k , which is why the M measure sums over all ( k , ℓ ) pairs, in effect using the average of AUC k , ℓ and AUC ℓ , k . The test involves the calculation of a statistic , usually called U , whose distribution under the null hypothesis is known: Alternatively, the null distribution can be approximated using permutation tests and Monte Carlo simulations. Some books tabulate statistics equivalent to U , such as

2601-559: The types of associations among variables are also made. These techniques include, among others: Non-parametric methods are widely used for studying populations that have a ranked order (such as movie reviews receiving one to five "stars"). The use of non-parametric methods may be necessary when data have a ranking but no clear numerical interpretation, such as when assessing preferences . In terms of levels of measurement , non-parametric methods result in ordinal data . As non-parametric methods make fewer assumptions, their applicability

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