2083:"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 weight for subjects on treatment A and B respectively are 147 and 151 kg. Treatment A decreased weight by HLΔ = 5 kg (0.95 CL kg,
2955:, 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%.
8421:
8407:
22:
8445:
8433:
3984:
test can have inflated type I error rates even in large samples (especially if the variances of two populations are unequal and the sample sizes are different), a problem the better alternatives solve. As a result, it has been suggested to use one of the alternatives (specifically the
Brunner–Munzel
1899:
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,
1185:
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
2613:
4096:
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:
2830:
4530:
59 (307): 655–680. If the two distributions are normal with the same mean but different variances, then Pr = Pr but the size of the Mann–Whitney test can be larger than the nominal level. So we cannot define the null hypothesis as Pr = Pr and get a valid
3154:
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.
758:
1260:
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
3596:. 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.
3171:
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.
1707:
4298:
with a point null-hypothesis against its complementary alternative (that is, equal versus not equal). However, he only tabulated a few points for the equal-sample size case in that paper (though in a later paper he gave larger tables).
2858:
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.
3450:
2394:
2409:
4020:-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.
4302:
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
3508:
of a randomly drawn observation from one group is the same as the probability distribution of a randomly drawn observation from the other group against an alternative that those distributions are not equal (see
3222:
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
3175:
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
2660:
1000:
906:
1522:
1389:
5843:. Kendall's Library of Statistics. Vol. 5 (First ed., rather than Taylor and Francis (2010) second ed.). London; New York: Edward Arnold; John Wiley and Sons, Inc. pp. xiv+467.
4344:). 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.
2181:
4275:
2290:
559:
3054:
4684:
Mason, S. J., Graham, N. E. (2002). "Areas beneath the relative operating characteristics (ROC) and relative operating levels (ROL) curves: Statistical significance and interpretation".
3869:
3978:
3804:
524:
447:
309:
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.,
1864:
3909:
804:
3675:
365:
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.
3709:
3288:
3261:
3217:
2991:
1553:
1246:
1215:
450:
2040:
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,
915:, 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.
5715:
1276:
Now, add up the ranks for the observations which came from sample 1. The sum of ranks in sample 2 is now determined, since the sum of all the ranks equals
544:
471:
86:
4001:
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
6002:. 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.
5643:
Bergmann, Reinhard; Ludbrook, John; Spooren, Will P.J.M. (2000). "Different
Outcomes of the Wilcoxon–Mann–Whitney Test from Different Statistics Packages".
3296:
1040:(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
806:
being the sum of the ranks in groups 1 and 2, after pooling all samples in one set (see below) and where the smallest value obtains rank 1 and so on.
5987:
5035:
2608:{\displaystyle \sigma _{\text{ties}}={\sqrt {{n_{1}n_{2}(n_{1}+n_{2}+1) \over 12}-{n_{1}n_{2}\sum _{k=1}^{K}(t_{k}^{3}-t_{k}) \over 12n(n-1)}}},\,}
4911:
4882:
4373:
2299:
5230:
Zimmerman, Donald W. (1998-01-01). "Invalidation of
Parametric and Nonparametric Statistical Tests by Concurrent Violation of Two Assumptions".
4952:
Nakagawa, Shinichi; Cuthill, Innes C (2007). "Effect size, confidence interval and statistical significance: a practical guide for biologists".
7542:
3517:
tests a null hypothesis of equal means in two groups against an alternative of unequal means. Hence, except in special cases, the Mann–Whitney
3459:
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,
8047:
5081:
3150:
statistic can be seen in the case of the odd example used above, where two distributions that were significantly different on a Mann–Whitney
3142:
are randomly chosen observations from the two distributions. Both extreme values represent complete separation of the distributions, while a
1916:
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
828:
8197:
5186:
Wendt, H.W. (1972). "Dealing with a common problem in social science: A simplified rank-biserial coefficient of correlation based on the
3114:
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
2825:{\displaystyle \sigma _{\text{ties}}={\sqrt {{n_{1}n_{2} \over 12}\left((n+1)-{\sum _{k=1}^{K}(t_{k}^{3}-t_{k}) \over n(n-1)}\right)}},}
1926:. 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
7821:
6462:
4012:
Similarly, some authors (e.g., Conover) suggest transforming the data to ranks (if they are not already ranks) and then performing the
1955:
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.
4219:
928:
7595:
4081:
2044:"Median latencies in groups E and C were 153 and 247 ms; the distributions in the two groups differed significantly (Mann–Whitney
8034:
837:
4672:
3184:)). This simple difference formula is just the difference of the common language effect size of each group, and is as follows:
1441:
1308:
6046:
6007:
5848:
5441:
5406:
4776:
6457:
6157:
7061:
6209:
3540:
but not interval scaled, in which case the spacing between adjacent values of the scale cannot be assumed to be constant.
2127:
2102:
However it would be rare to find such an extensive report in a document whose major topic was not statistical inference.
173:
6119:
5875:
4848:
753:{\displaystyle U_{1}=n_{1}n_{2}+{\tfrac {n_{1}(n_{1}+1)}{2}}-R_{1},U_{2}=n_{1}n_{2}+{\tfrac {n_{2}(n_{2}+1)}{2}}-R_{2}}
5703:
scipy.stats.mannwhitneyu(x, y, use_continuity=True): Computes the Mann–Whitney rank test on samples x and y.
2399:
The formula for the standard deviation is more complicated in the presence of tied ranks. If there are ties in ranks,
2232:
342:
test as showing a difference in medians. Under this location shift assumption, we can also interpret the Mann–Whitney
8476:
7844:
7736:
5895:
4414:
4235:
2208:, is approximately a standard normal deviate whose significance can be checked in tables of the normal distribution.
822:
5686:
8449:
8022:
7896:
6067:(December 1963). "On the estimation of relative potency in dilution(-direct) assays by distribution-free methods".
4312:
2999:
1181:
For comparing two small sets of observations, a direct method is quick, and gives insight into the meaning of the
388:
test is applied to independent samples. The
Wilcoxon signed-rank test is applied to matched or dependent samples.
8080:
7741:
7486:
6857:
6447:
3809:
3588:-test. For distributions sufficiently far from normal and for sufficiently large sample sizes, the Mann–Whitney
358:
of all possible differences between an observation in the first sample and an observation in the second sample.
8131:
7343:
7150:
7039:
6997:
1424:
Note that it doesn't matter which of the two samples is considered sample 1. An equally valid formula for
7071:
4287:
The statistic appeared in a 1914 article by the German Gustav
Deuchler (with a missing term in the variance).
8374:
7333:
6236:
5116:
Herrnstein, Richard J.; Loveland, Donald H.; Cable, Cynthia (1976). "Natural
Concepts in Pigeons".
4156:
4117:
4080:
test is related to a number of other non-parametric statistical procedures. For example, it is equivalent to
3918:
3744:
476:
399:
7925:
7874:
7859:
7849:
7718:
7590:
7557:
7383:
7338:
7168:
4187:
4126:
1167:
361:
Otherwise, if both the dispersions and shapes of the distribution of both samples differ, the Mann–Whitney
40:
5278:
4307:
and his student Donald Ransom
Whitney in 1947. This article discussed alternative hypotheses, including a
8437:
8269:
8070:
7994:
7295:
7049:
6718:
6182:
4363:
4178:
4060:
3537:
912:
351:
347:
235:
6129:
5950:
Kerby, D.S. (2014). "The simple difference formula: An approach to teaching nonparametric correlation".
5751:
Kruskal, William H. (September 1957). "Historical Notes on the
Wilcoxon Unpaired Two-Sample Test".
4998:
Kerby, D.S. (2014). "The simple difference formula: An approach to teaching nonparametric correlation".
922:
statistic can be generalized to a measure of a classifier's separation power for more than two classes:
8471:
8154:
8126:
8121:
7869:
7628:
7534:
7514:
7422:
7133:
6951:
6434:
6306:
4201:
1900:
from first to last crossing the finish line) is as follows, writing T for a tortoise and H for a hare:
35:
3455:
This formula is useful when the data are not available, but when there is a published report, because
1819:
1173:
It is also easily calculated by hand, especially for small samples. There are two ways of doing this.
7886:
7654:
7375:
7300:
7229:
7158:
7078:
7066:
6936:
6924:
6917:
6625:
6346:
5352:
4925:
Wilkinson, Leland (1999). "Statistical methods in psychology journals: Guidelines and explanations".
4526:, See Table 2.1 of Pratt (1964) "Robustness of Some Procedures for the Two-Sample Location Problem."
4368:
3563:
1888:
373:
153:
61:
5629:
4736:"A Simple Generalisation of the Area Under the ROC Curve for Multiple Class Classification Problems"
8369:
8136:
7999:
7684:
7649:
7613:
7398:
6840:
6749:
6708:
6620:
6311:
6150:
4263:
4113:
3505:
200:
107:
5425:
Rank and pseudo-rank procedures for independent observations in factorial designs: Using R and SAS
4084:
correlation coefficient if one of the variables is binary (that is, it can only take two values).
3871:), without assuming that the distributions are the same under the null hypothesis (i.e., assuming
8278:
7891:
7831:
7768:
7406:
7128:
6990:
6980:
6830:
6744:
5555:
4383:
4053:
4024:
3874:
3577:
3083:
769:
5315:
5298:
3645:
3521:
test and the t-test do not test the same hypotheses and should be compared with this in mind.
2011:
A measure of the central tendencies of the two groups (means or medians; since the Mann–Whitney
8316:
8246:
8039:
7976:
7731:
7618:
6615:
6512:
6419:
6298:
6197:
4720:
3232:
An alternative formula for the rank-biserial can be used to calculate it from the Mann–Whitney
177:
1702:{\displaystyle U_{1}+U_{2}=R_{1}-{n_{1}(n_{1}+1) \over 2}+R_{2}-{n_{2}(n_{2}+1) \over 2}.\,\!}
8481:
8341:
8283:
8226:
8052:
7945:
7854:
7580:
7464:
7323:
7315:
7205:
7197:
7012:
6908:
6886:
6845:
6810:
6777:
6723:
6698:
6653:
6592:
6552:
6354:
6177:
5981:
5942:
5616:
5029:
5350:(1981). "Rank Transformations as a Bridge Between Parametric and Nonparametric Statistics".
3686:
8264:
7839:
7788:
7764:
7726:
7644:
7623:
7575:
7454:
7432:
7401:
7310:
7187:
7138:
7056:
7029:
6985:
6941:
6703:
6479:
6359:
6056:
6017:
5926:
5858:
4493:
4437:
4295:
3266:
3239:
3190:
2969:
2618:
where the left side is simply the variance and the right side is the adjustment for ties,
1548:
is the one used when consulting significance tables. The sum of the two values is given by
1224:
1193:
381:
6096:
5934:
4468:-test? On assumptions for hypothesis tests and multiple interpretations of decision rules"
4445:
4410:"On a Test of Whether one of Two Random Variables is Stochastically Larger than the Other"
3985:
test) if it cannot be assumed that the distributions are equal under the null hypothesis.
8:
8411:
8336:
8259:
7940:
7704:
7697:
7659:
7567:
7547:
7519:
7252:
7118:
7113:
7103:
7095:
6913:
6874:
6764:
6754:
6663:
6442:
6398:
6316:
6241:
6143:
5791:
5079:
Grissom RJ (1994). "Statistical analysis of ordinal categorical status after therapies".
4624:"Mann–Whitney test is not just a test of medians: differences in spread can be important"
4378:
4308:
4172:
4038:
3912:
2115:
1137:
350:
of the difference in central tendency between the two populations differs from zero. The
207:
181:
30:
8425:
8236:
8090:
7986:
7935:
7811:
7708:
7692:
7669:
7446:
7180:
7163:
7123:
7034:
6929:
6891:
6862:
6822:
6782:
6728:
6645:
6331:
6326:
6084:
5969:
5914:
5818:
5768:
5668:
5660:
5589:
5536:
5369:
5168:
5017:
4977:
4905:
4876:
4650:
4623:
4597:
4541:
Divine, George W.; Norton, H. James; Barón, Anna E.; Juarez-Colunga, Elizabeth (2018).
4502:
4472:
4463:
529:
456:
71:
3445:{\displaystyle r=f-(1-f)=2f-1={2U_{1} \over n_{1}n_{2}}-1=1-{2U_{2} \over n_{1}n_{2}}}
8420:
8331:
8301:
8293:
8113:
8104:
8029:
7960:
7816:
7801:
7776:
7664:
7605:
7471:
7459:
7085:
7002:
6946:
6869:
6713:
6635:
6414:
6288:
6042:
6003:
5995:
5973:
5886:
5871:
5844:
5672:
5593:
5540:
5497:
5437:
5423:
5402:
5328:
5320:
5247:
5172:
5133:
5098:
5021:
4969:
4965:
4844:
4821:
4813:
4793:
4772:
4701:
4655:
4601:
4575:
4507:
4105:
4002:
3915:
and the Fligner–Policello test. Specifically, under the more general null hypothesis
3911:). To test between those hypotheses, better tests are available. Among those are the
6114:
6033:. Lecture Notes in Statistics. Vol. 199. New York: Springer. pp. xiv+232.
3604:
test will give very similar results to performing an ordinary parametric two-sample
2079:
A statement that does full justice to the statistical status of the test might run,
8356:
8311:
8075:
8062:
7955:
7930:
7864:
7796:
7674:
7282:
7175:
7108:
7021:
6968:
6787:
6658:
6452:
6336:
6251:
6218:
6092:
6076:
6034:
5959:
5930:
5904:
5808:
5800:
5760:
5733:
5656:
5652:
5579:
5571:
5528:
5487:
5477:
5429:
5394:
5361:
5310:
5239:
5199:
5160:
5125:
5090:
5061:
5007:
4961:
4934:
4805:
4747:
4693:
4645:
4635:
4587:
4554:
4497:
4481:
4441:
4423:
3168:
1145:
110:
4981:
4559:
4542:
8273:
8017:
7879:
7806:
7481:
7355:
7328:
7305:
7274:
6901:
6896:
6850:
6580:
6231:
6052:
6013:
5922:
5854:
4489:
4433:
4294:
proposed both the one-sample signed rank and the two-sample rank sum test, in a
1126:
185:
114:
7763:
5065:
4132:
8222:
8217:
6680:
6610:
6256:
6132:– Nonparametric effect size estimators (Copyright 2015 by Karl L. Weunsch)
5938:
5786:
5347:
5094:
4938:
4697:
4592:
4291:
4182:
4146:
2119:
6038:
5909:
5890:
5813:
5433:
5388:
5243:
5129:
4752:
4735:
4428:
4409:
257:
is different (larger, or smaller) than the probability of an observation from
8465:
8379:
8346:
8209:
8170:
7981:
7950:
7414:
7368:
6973:
6675:
6502:
6266:
6261:
6064:
5575:
5501:
5482:
5461:
5398:
5324:
5299:"The Importance of the Normality Assumption in Large Public Health Data Sets"
5251:
4817:
4705:
4640:
5203:
5052:
McGraw, K.O.; Wong, J.J. (1992). "A common language effect size statistic".
210:(i.e., one can at least say, of any two observations, which is the greater),
60:"Wilcoxon rank-sum test" redirects here. For Wilcoxon signed-rank test, see
8321:
8254:
8231:
8146:
7476:
6772:
6670:
6605:
6547:
6532:
6469:
6424:
5532:
5332:
4973:
4825:
4659:
4511:
4358:
4190:
has implementations of this test through several packages. In the package
2389:{\displaystyle \sigma _{U}={\sqrt {n_{1}n_{2}(n_{1}+n_{2}+1) \over 12}}.\,}
5428:. Springer Series in Statistics. Cham: Springer International Publishing.
5102:
8364:
8326:
8009:
7910:
7772:
7585:
7552:
7044:
6961:
6956:
6600:
6557:
6537:
6517:
6507:
6276:
5294:
5137:
4353:
4245:
4231:
2928:
549:
5584:
5492:
3483:, which is the same result as with the simple difference formula above.
7210:
6690:
6390:
6321:
6271:
6246:
6166:
6088:
5918:
5822:
5772:
5664:
5373:
5164:
4524:
4405:
4304:
4213:
165:
5964:
5012:
4794:"Effect size estimates: Current use, calculations, and interpretation"
4027:
has been suggested as an appropriate non-parametric equivalent to the
7363:
7215:
6835:
6630:
6542:
6527:
6522:
6487:
5556:"Psychologists Should Use Brunner–Munzel's Instead of Mann–Whitney's
5462:"Psychologists Should Use Brunner–Munzel's Instead of Mann–Whitney's
4809:
4485:
4166:
3615:
Relative efficiencies of the Mann–Whitney test versus the two-sample
3554:-test to spuriously indicate significance because of the presence of
1118:
149:
6080:
5804:
5764:
5608:
5365:
4792:
Fritz, Catherine O.; Morris, Peter E.; Richler, Jennifer J. (2012).
4244:(Stata Corporation, College Station, TX) implements the test in its
3060:
6879:
6497:
6374:
6369:
6364:
4576:"What Hypotheses do "Nonparametric" Two-Group Tests Actually Test?"
4253:
3510:
1892:
3511:
Mann–Whitney U test#Assumptions and formal statement of hypotheses
3180:) minus its complement (i.e.: the proportion that is unfavorable (
2907:
used in the normal approximation is the mean of the two values of
1186:
larger). Count 0.5 for any ties. The sum of wins and ties is
810:
8384:
8085:
3555:
3094:
by its maximum value for the given sample sizes, which is simply
3087:
1765:
1054:
will always be zero but, unlike in the two-class case, generally
159:
4543:"The Wilcoxon–Mann–Whitney Procedure Fails as a Test of Medians"
4143:
statistic for a Wilcoxon two-sample, paired, or one-sample test.
2927:
It is a widely recommended practice for scientists to report an
546:, and both samples independent of each other. The corresponding
8306:
7287:
7261:
7241:
6492:
6283:
4838:
4540:
4207:
4101:
4028:
3993:
If one desires a simple shift interpretation, the Mann–Whitney
3605:
3514:
1144:
Alternatively, the null distribution can be approximated using
995:{\displaystyle M={1 \over c(c-1)}\sum \mathrm {AUC} _{k,\ell }}
355:
1816:
is the product of the sample sizes for the two samples (i.e.:
1155:, such as the sum of ranks in one of the samples, rather than
6135:
6130:
Brief guide by experimental psychologist Karl L. Weunsch
5118:
Journal of Experimental Psychology: Animal Behavior Processes
4771:. New Jersey: Prentice Hall International, INC. p. 147.
4271:
4241:
4160:
2947:
One method of reporting the effect size for the Mann–Whitney
2934:
1884:
1032:
considers only the ranking of the items belonging to classes
5422:
Brunner, Edgar; Bathke, Arne C.; Konietschke, Frank (2018).
6226:
4259:
4225:
3163:
A method of reporting the effect size for the Mann–Whitney
1896:
1133:
In the case of small samples, the distribution is tabulated
901:{\displaystyle \mathrm {AUC} _{1}={U_{1} \over n_{1}n_{2}}}
380:, although both are nonparametric and involve summation of
184:
than the other, there are many other ways to formulate the
5564:
Advances in Methods and Practices in Psychological Science
5470:
Advances in Methods and Practices in Psychological Science
3566:
control when data are both heteroscedastic and non-normal.
3146:
of 0.5 represents complete overlap. The usefulness of the
2105:
4954:
Biological Reviews of the Cambridge Philosophical Society
4683:
2015:
test is an ordinal test, medians are usually recommended)
1136:
For sample sizes above ~20, approximation using the
5421:
5283:, John Wiley & Sons, 1980 (2nd Edition), pp. 225–226
4951:
4462:
Fay, Michael P.; Proschan, Michael A. (2010).
4045:
test in case of violated assumption of exchangeability.
1517:{\displaystyle U_{2}=R_{2}-{n_{2}(n_{2}+1) \over 2}\,\!}
1384:{\displaystyle U_{1}=R_{1}-{n_{1}(n_{1}+1) \over 2}\,\!}
5642:
5115:
29:
It has been suggested that portions of this article be
5292:
4256:(Cytel Software Corporation, Cambridge, Massachusetts)
2915:-statistic calculated will be same whichever value of
1878:
695:
600:
220:, the distributions of both populations are identical.
188:
and alternative hypotheses such that the Mann–Whitney
5789:(1945). "Individual comparisons by ranking methods".
4686:
Quarterly Journal of the Royal Meteorological Society
3921:
3877:
3812:
3747:
3689:
3648:
3299:
3269:
3242:
3193:
3002:
2972:
2663:
2412:
2302:
2235:
2130:
1822:
1556:
1444:
1311:
1227:
1196:
931:
840:
772:
562:
532:
479:
459:
402:
372:
test / Wilcoxon rank-sum test is not the same as the
168:
and Donald Ransom Whitney developed the Mann–Whitney
74:
8048:
Autoregressive conditional heteroskedasticity (ARCH)
5838:
4869:
Nonparametric statistics for the behavioral sciences
5151:Cureton, E.E. (1956). "Rank-biserial correlation".
3546:As it compares the sums of ranks, the Mann–Whitney
2176:{\displaystyle z={\frac {U-m_{U}}{\sigma _{U}}},\,}
2022:(perhaps with some measure of effect size, such as
7510:
6000:Nonparametrics: Statistical methods based on ranks
4898:Nonparametrics: Statistical Methods Based on Ranks
4791:
4734:Hand, David J.; Till, Robert J. (2001).
4016:-test on the transformed data, the version of the
3972:
3903:
3863:
3798:
3741:test is not valid for testing the null hypothesis
3703:
3669:
3444:
3282:
3255:
3211:
3048:
2985:
2824:
2607:
2388:
2284:
2175:
1958:the sum of the ranks achieved by the tortoises is
1858:
1701:
1516:
1383:
1240:
1209:
994:
900:
798:
752:
538:
518:
465:
441:
249:The probability of an observation from population
80:
5868:Nonparametric Statistics: A Step-by-Step Approach
4457:
4455:
4400:
4398:
3491:
1698:
1513:
1380:
8463:
5716:"MannWhitneyUTest (Apache Commons Math 3.3 API)"
5220:, San Diego, CA: GraphPad Software, 2007, p. 123
4528:Journal of the American Statistical Association.
2285:{\displaystyle m_{U}={\frac {n_{1}n_{2}}{2}},\,}
2023:
234:Under the general formulation, the test is only
7596:Multivariate adaptive regression splines (MARS)
5753:Journal of the American Statistical Association
4895:
4181:has an implementation of this test provided by
4159:has an implementation of this test provided by
4116:'s statistics base-package implements the test
2635:is the total number of unique ranks with ties.
1998:
811:Area-under-curve (AUC) statistic for ROC curves
6031:: An approach based on spatial signs and ranks
4452:
4395:
3082:and widely used in studies of categorization (
2942:
2027:
1978:The sum of the ranks achieved by the hares is
338:, we can interpret a significant Mann–Whitney
195:A very general formulation is to assume that:
160:Assumptions and formal statement of hypotheses
6151:
6027:Multivariate nonparametric methods with
5884:
5865:
5609:"Violation of Proportional Odds is Not Fatal"
5210:
5082:Journal of Consulting and Clinical Psychology
5047:
5045:
4918:
4234:(StatsDirect Ltd, Manchester, UK) implements
3592:test is considerably more efficient than the
1151:Some books tabulate statistics equivalent to
911:Note that this is the same definition as the
5986:: CS1 maint: DOI inactive as of June 2024 (
5779:
5560:Test as the Default Nonparametric Procedure"
5466:Test as the Default Nonparametric Procedure"
5271:
5258:
5034:: CS1 maint: DOI inactive as of June 2024 (
4993:
4991:
4945:
4092:In many software packages, the Mann–Whitney
3158:
3090:), and elsewhere, is calculated by dividing
3061:area under the curve (AUC) for the ROC curve
3049:{\displaystyle f={U_{1} \over n_{1}n_{2}}\,}
1936:. Note that the sum of these two values for
230:is that the distributions are not identical.
5839:Hettmansperger, T.P.; McKean, J.W. (1998).
5345:
5316:10.1146/annurev.publhealth.23.100901.140546
4910:: CS1 maint: numeric names: authors list (
4881:: CS1 maint: numeric names: authors list (
4798:Journal of Experimental Psychology: General
4724:, Pattern Recognition Letters, 27, 861–874.
4461:
4404:
4374:Kruskal–Wallis one-way analysis of variance
4087:
3864:{\displaystyle P(Y>X)+0.5P(Y=X)\neq 0.5}
2638:A more computationally-efficient form with
2003:In reporting the results of a Mann–Whitney
1089:) pairs, in effect using the average of AUC
6196:
6158:
6144:
5078:
5051:
5042:
4839:Myles Hollander; Douglas A. Wolfe (1999).
4066:
3732:
3486:
2935:Proportion of concordance out of all pairs
199:All the observations from both groups are
6809:
5963:
5908:
5812:
5744:
5583:
5491:
5481:
5314:
5229:
5011:
4988:
4924:
4896:Lehmann, Erich; D'Abrera, Howard (1975).
4751:
4649:
4639:
4591:
4558:
4501:
4427:
3504:test tests a null hypothesis of that the
3045:
2604:
2385:
2281:
2172:
1697:
1512:
1415:is the sum of the ranks in sample 1.
1379:
1221:for the other set is the converse (i.e.:
253:exceeding an observation from population
125:from two populations, the probability of
57:Nonparametric test of the null hypothesis
5841:Robust nonparametric statistical methods
5785:
5455:
5453:
4733:
3584:or about 0.95 when compared to the
5994:
5891:"Estimation of location based on ranks"
5750:
5606:
5297:; Emerson, Scott; Chen, Lu (May 2002).
5150:
4862:
4860:
3973:{\displaystyle P(Y>X)+0.5P(Y=X)=0.5}
3799:{\displaystyle P(Y>X)+0.5P(Y=X)=0.5}
3572:When normality holds, the Mann–Whitney
2939:The following measures are equivalent.
2911:. Therefore, the absolute value of the
2204:are the mean and standard deviation of
2106:Normal approximation and tie correction
1117:The test involves the calculation of a
918:Because of its probabilistic form, the
519:{\displaystyle Y_{1},\ldots ,Y_{n_{2}}}
442:{\displaystyle X_{1},\ldots ,X_{n_{1}}}
8464:
8122:Kaplan–Meier estimator (product limit)
5514:
4712:
4573:
4009:-test may give more reliable results.
3290:) and the sample sizes of each group:
1269:, where the unadjusted ranks would be
8195:
7762:
7509:
6808:
6578:
6195:
6139:
5949:
5553:
5459:
5450:
5386:
5232:The Journal of Experimental Education
5192:European Journal of Social Psychology
5185:
5109:
4997:
4056:, allowing for covariate-adjustment.
1408:is the sample size for sample 1, and
33:out into another article titled
8432:
8132:Accelerated failure time (AFT) model
5866:Corder, G.W.; Foreman, D.I. (2014).
5607:Harrell, Frank (20 September 2020).
5515:Kasuya, Eiiti (2001). "Mann–Whitney
4857:
4621:
3806:against the alternative hypothesis
1291:is the total number of observations.
15:
8444:
7727:Analysis of variance (ANOVA, anova)
6579:
6063:
6024:
5693:. The Scipy community. 24 July 2015
5387:Vaart, A. W. van der (1998-10-13).
4766:
4315:satisfied the pointwise inequality
1879:Illustration of calculation methods
354:for this two-sample problem is the
117:that, for randomly selected values
13:
7822:Cochran–Mantel–Haenszel statistics
6448:Pearson product-moment correlation
5519:test when variances are unequal".
5280:Practical Nonparametric Statistics
4866:
4843:(2 ed.). Wiley-Interscience.
4408:; Whitney, Donald R. (1947).
1009:is the number of classes, and the
976:
973:
970:
849:
846:
843:
14:
8493:
6104:
5896:Annals of Mathematical Statistics
4841:Nonparametric Statistical Methods
4415:Annals of Mathematical Statistics
4313:cumulative distribution functions
4041:, outperforming the Mann–Whitney
3627:equals a number of distributions
823:receiver operating characteristic
8443:
8431:
8419:
8406:
8405:
8196:
4966:10.1111/j.1469-185X.2007.00027.x
4071:
2007:test, it is important to state:
1859:{\displaystyle U_{i}=n_{1}n_{2}}
238:when the following occurs under
144:Nonparametric tests used on two
20:
8081:Least-squares spectral analysis
5734:"JuliaStats/HypothesisTests.jl"
5726:
5708:
5679:
5636:
5600:
5547:
5508:
5415:
5380:
5339:
5286:
5266:Elements of Large Sample Theory
5223:
5179:
5144:
5072:
4889:
4832:
4785:
4760:
4727:
4721:An introduction to ROC analysis
3988:
2922:
2403:should be adjusted as follows:
1866:). In such a case, the "other"
1125:, whose distribution under the
1112:
180:being that one distribution is
133:is equal to the probability of
7062:Mean-unbiased minimum-variance
6165:
5657:10.1080/00031305.2000.10474513
5393:. Cambridge University Press.
5303:Annual Review of Public Health
4677:
4666:
4615:
4567:
4534:
4518:
4204:(SAS Institute Inc., Cary, NC)
4196:pvalue(MannWhitneyUTest(X, Y))
4052:test is a special case of the
3961:
3949:
3937:
3925:
3852:
3840:
3828:
3816:
3787:
3775:
3763:
3751:
3466:is the smaller of the two, so
3324:
3312:
3066:
2806:
2794:
2786:
2755:
2725:
2713:
2627:is the number of ties for the
2593:
2581:
2570:
2539:
2483:
2451:
2372:
2340:
1685:
1666:
1628:
1609:
1503:
1484:
1370:
1351:
959:
947:
727:
708:
632:
613:
553:is defined as the smaller of:
391:
346:test as assessing whether the
261:exceeding an observation from
1:
8375:Geographic information system
7591:Simultaneous equations models
5832:
5691:SciPy v0.16.0 Reference Guide
4767:Zar, Jerrold H. (1998).
4673:Boston University (SPH), 2017
4560:10.1080/00031305.2017.1305291
4210:(MathSoft, Inc., Seattle, WA)
3611:on the rankings of the data.
3550:test is less likely than the
3481:= 1 – (2×10) / (10×10) = 0.80
1973:= 32 − (6×7)/2 = 32 − 21 = 11
1807:
1148:and Monte Carlo simulations.
192:test will give a valid test.
172:test under the assumption of
7558:Coefficient of determination
7169:Uniformly most powerful test
6110:Table of critical values of
4037:A more powerful test is the
3558:. However, the Mann–Whitney
3078:that is linearly related to
1999:Example statement of results
1980:11 + 10 + 9 + 8 + 7 + 1 = 46
819:statistic is related to the
7:
8127:Proportional hazards models
8071:Spectral density estimation
8053:Vector autoregression (VAR)
7487:Maximum posterior estimator
6719:Randomized controlled trial
5066:10.1037/0033-2909.111.2.361
4871:. McGraw-Hill. p. 121.
4347:
4290:In a single paper in 1945,
4274:implements the test in its
4262:implements the test in its
4216:(StatSoft, Inc., Tulsa, OK)
4149:implements the test in its
4137:package will calculate the
3904:{\displaystyle F_{1}=F_{2}}
2943:Common language effect size
2024:common language effect size
1960:12 + 6 + 5 + 4 + 3 + 2 = 32
1873:
913:common language effect size
799:{\displaystyle R_{1},R_{2}}
223:The alternative hypothesis
206:The responses are at least
10:
8498:
7887:Multivariate distributions
6307:Average absolute deviation
5687:"scipy.stats.mannwhitneyu"
5346:Conover, William J.;
5277:Conover, William J.;
5216:Motulsky, Harvey J.;
5095:10.1037/0022-006X.62.2.281
4939:10.1037/0003-066X.54.8.594
4698:10.1256/003590002320603584
4593:10.1177/1536867X1201200202
4464:"Wilcoxon–Mann–Whitney or
4282:
4276:Wilcoxon–Mann–Whitney Test
4110:in its Statistics Toolbox.
3670:{\displaystyle \pi ^{2}/9}
3532:test is preferable to the
3476:. This formula then gives
3167:test is with a measure of
1950:Using the indirect method:
1768:, we find that the sum is
1267:(1, 3.5, 3.5, 3.5, 3.5, 6)
213:Under the null hypothesis
104:Wilcoxon–Mann–Whitney test
59:
36:Probability of superiority
8401:
8355:
8292:
8245:
8208:
8204:
8191:
8163:
8145:
8112:
8103:
8061:
8008:
7969:
7918:
7909:
7875:Structural equation model
7830:
7787:
7783:
7758:
7717:
7683:
7637:
7604:
7566:
7533:
7529:
7505:
7445:
7354:
7273:
7237:
7228:
7211:Score/Lagrange multiplier
7196:
7149:
7094:
7020:
7011:
6821:
6817:
6804:
6763:
6737:
6689:
6644:
6626:Sample size determination
6591:
6587:
6574:
6478:
6433:
6407:
6389:
6345:
6297:
6217:
6208:
6204:
6191:
6173:
6039:10.1007/978-1-4419-0468-3
5645:The American Statistician
5460:Karch, Julian D. (2021).
5434:10.1007/978-3-030-02914-2
5353:The American Statistician
5244:10.1080/00220979809598344
5130:10.1037/0097-7403.2.4.285
4900:. Holden-Day. p. 20.
4547:The American Statistician
4369:Wilcoxon signed-rank test
3576:test has an (asymptotic)
3159:Rank-biserial correlation
2958:The relationship between
2931:for an inferential test.
2028:rank-biserial correlation
1887:is dissatisfied with his
1166:test is included in most
154:Wilcoxon signed-rank test
62:Wilcoxon signed-rank test
8477:Nonparametric statistics
8370:Environmental statistics
7892:Elliptical distributions
7685:Generalized linear model
7614:Simple linear regression
7384:Hodges–Lehmann estimator
6841:Probability distribution
6750:Stochastic approximation
6312:Coefficient of variation
5952:Comprehensive Psychology
5576:10.1177/2515245921999602
5483:10.1177/2515245921999602
5399:10.1017/cbo9780511802256
5268:, Springer, 1999, p. 176
5264:Lehamnn, Erich L.;
5000:Comprehensive Psychology
4641:10.1136/bmj.323.7309.391
4389:
4175:(SPSS Inc., Chicago, IL)
4169:(SPSS Inc., Chicago, IL)
4088:Software implementations
3536:-test when the data are
3506:probability distribution
3492:Comparison to Student's
3059:This is the same as the
8030:Cross-correlation (XCF)
7638:Non-standard predictors
7072:Lehmann–Scheffé theorem
6745:Adaptive clinical trial
5968:(inactive 2024-06-02).
5910:10.1214/aoms/1177704172
5204:10.1002/ejsp.2420020412
5016:(inactive 2024-06-02).
4867:Siegal, Sidney (1956).
4769:Biostatistical Analysis
4753:10.1023/A:1010920819831
4429:10.1214/aoms/1177730491
4384:Proportional odds model
4364:Kolmogorov–Smirnov test
4122:in its "stats" package.
4067:Related test statistics
4061:Kolmogorov–Smirnov test
4054:proportional odds model
3733:Different distributions
3487:Relation to other tests
3084:discrimination learning
2036:The significance level.
1904:T H H H H H T T T T T H
1081:measure sums over all (
352:Hodges–Lehmann estimate
348:Hodges–Lehmann estimate
8426:Mathematics portal
8247:Engineering statistics
8155:Nelson–Aalen estimator
7732:Analysis of covariance
7619:Ordinary least squares
7543:Pearson product-moment
6947:Statistical functional
6858:Empirical distribution
6691:Controlled experiments
6420:Frequency distribution
6198:Descriptive statistics
6120:Interactive calculator
5996:Lehmann, Erich L.
5943:euclid.aoms/1177704172
5624:Cite journal requires
5554:Karch, Julian (2021).
5533:10.1006/anbe.2001.1691
5054:Psychological Bulletin
4574:Conroy, Ronán (2012).
3974:
3905:
3865:
3800:
3705:
3704:{\displaystyle 3/\pi }
3671:
3446:
3284:
3257:
3213:
3050:
2987:
2826:
2754:
2609:
2538:
2390:
2286:
2177:
1895:was found to beat one
1860:
1703:
1518:
1385:
1242:
1211:
996:
902:
800:
754:
540:
526:an i.i.d. sample from
520:
467:
443:
182:stochastically greater
178:alternative hypothesis
100:Wilcoxon rank-sum test
82:
8342:Population statistics
8284:System identification
8018:Autocorrelation (ACF)
7946:Exponential smoothing
7860:Discriminant analysis
7855:Canonical correlation
7719:Partition of variance
7581:Regression validation
7425:(Jonckheere–Terpstra)
7324:Likelihood-ratio test
7013:Frequentist inference
6925:Location–scale family
6846:Sampling distribution
6811:Statistical inference
6778:Cross-sectional study
6765:Observational studies
6724:Randomized experiment
6553:Stem-and-leaf display
6355:Central limit theorem
5390:Asymptotic Statistics
4927:American Psychologist
4718:Fawcett, Tom (2006);
4222:(Unistat Ltd, London)
4034:for equal variances.
3975:
3906:
3866:
3801:
3706:
3672:
3447:
3285:
3283:{\displaystyle U_{2}}
3258:
3256:{\displaystyle U_{1}}
3214:
3212:{\displaystyle r=f-u}
3051:
2988:
2986:{\displaystyle U_{1}}
2962:and the Mann–Whitney
2827:
2734:
2610:
2518:
2391:
2287:
2178:
1975:(same as method one).
1908:What is the value of
1861:
1812:The maximum value of
1704:
1534:The smaller value of
1519:
1386:
1243:
1241:{\displaystyle U_{2}}
1217:) for the first set.
1212:
1210:{\displaystyle U_{1}}
997:
903:
801:
755:
541:
521:
468:
444:
92:Mann–Whitney–Wilcoxon
83:
8265:Probabilistic design
7850:Principal components
7693:Exponential families
7645:Nonlinear regression
7624:General linear model
7586:Mixed effects models
7576:Errors and residuals
7553:Confounding variable
7455:Bayesian probability
7433:Van der Waerden test
7423:Ordered alternative
7188:Multiple comparisons
7067:Rao–Blackwellization
7030:Estimating equations
6986:Statistical distance
6704:Factorial experiment
6237:Arithmetic-Geometric
6126:and its significance
5348:Iman, Ronald L.
4296:test of significance
3919:
3875:
3810:
3745:
3687:
3646:
3562:test may have worse
3297:
3267:
3240:
3191:
3000:
2970:
2661:
2410:
2300:
2233:
2128:
2118:. In that case, the
2116:normally distributed
1820:
1554:
1442:
1309:
1256:For larger samples:
1225:
1194:
1168:statistical packages
929:
838:
770:
560:
530:
477:
457:
400:
72:
8337:Official statistics
8260:Methods engineering
7941:Seasonal adjustment
7709:Poisson regressions
7629:Bayesian regression
7568:Regression analysis
7548:Partial correlation
7520:Regression analysis
7119:Prediction interval
7114:Likelihood interval
7104:Confidence interval
7096:Interval estimation
7057:Unbiased estimators
6875:Model specification
6755:Up-and-down designs
6443:Partial correlation
6399:Index of dispersion
6317:Interquartile range
6025:Oja, Hannu (2010).
5792:Biometrics Bulletin
4622:Hart, Anna (2001).
4406:Mann, Henry B.
4379:Brunner Munzel test
4309:stochastic ordering
4236:all common variants
4228:(SPSS Inc, Chicago)
4194:, this is found as
4039:Brunner-Munzel test
4025:Brown–Forsythe test
3980:, the Mann–Whitney
3628:
3074:A statistic called
2772:
2556:
2110:For large samples,
1138:normal distribution
1077:, which is why the
384:. The Mann–Whitney
176:responses with the
137:being greater than
129:being greater than
8357:Spatial statistics
8237:Medical statistics
8137:First hitting time
8091:Whittle likelihood
7742:Degrees of freedom
7737:Multivariate ANOVA
7670:Heteroscedasticity
7482:Bayesian estimator
7447:Bayesian inference
7296:Kolmogorov–Smirnov
7181:Randomization test
7151:Testing hypotheses
7124:Tolerance interval
7035:Maximum likelihood
6930:Exponential family
6863:Density estimation
6823:Statistical theory
6783:Natural experiment
6729:Scientific control
6646:Survey methodology
6332:Standard deviation
5814:10338.dmlcz/135688
5720:commons.apache.org
5165:10.1007/BF02289138
4692:(584): 2145–2166.
4473:Statistics Surveys
4192:HypothesisTests.jl
3970:
3901:
3861:
3796:
3701:
3667:
3614:
3513:). In contrast, a
3442:
3280:
3253:
3209:
3046:
2983:
2822:
2758:
2605:
2542:
2386:
2282:
2173:
2120:standardized value
1889:classic experiment
1856:
1699:
1514:
1381:
1271:(1, 2, 3, 4, 5, 6)
1263:(3, 5, 5, 5, 5, 8)
1238:
1207:
992:
898:
796:
750:
735:
640:
536:
516:
463:
439:
78:
8472:Statistical tests
8459:
8458:
8397:
8396:
8393:
8392:
8332:National accounts
8302:Actuarial science
8294:Social statistics
8187:
8186:
8183:
8182:
8179:
8178:
8114:Survival function
8099:
8098:
7961:Granger causality
7802:Contingency table
7777:Survival analysis
7754:
7753:
7750:
7749:
7606:Linear regression
7501:
7500:
7497:
7496:
7472:Credible interval
7441:
7440:
7224:
7223:
7040:Method of moments
6909:Parametric family
6870:Statistical model
6800:
6799:
6796:
6795:
6714:Random assignment
6636:Statistical power
6570:
6569:
6566:
6565:
6415:Contingency table
6385:
6384:
6252:Generalized/power
6065:Sen, Pranab Kumar
6048:978-1-4419-0467-6
6009:978-0-387-35212-1
5965:10.2466/11.IT.3.1
5850:978-0-340-54937-7
5443:978-3-030-02912-8
5408:978-0-511-80225-6
5013:10.2466/11.IT.3.1
4778:978-0-13-082390-8
4634:(7309): 391–393.
4076:The Mann–Whitney
4048:The Mann–Whitney
4003:unequal variances
3737:The Mann–Whitney
3730:
3729:
3600:The Mann–Whitney
3528:The Mann–Whitney
3500:The Mann–Whitney
3440:
3384:
3043:
2993:) is as follows:
2817:
2810:
2706:
2671:
2599:
2597:
2490:
2420:
2380:
2379:
2276:
2167:
2114:is approximately
1870:would be 0.
1764:, and doing some
1692:
1635:
1510:
1377:
1297:is then given by:
1162:The Mann–Whitney
1146:permutation tests
1121:, usually called
963:
896:
734:
639:
539:{\displaystyle Y}
466:{\displaystyle X}
368:The Mann–Whitney
90:(also called the
81:{\displaystyle U}
53:
52:
48:
8489:
8447:
8446:
8435:
8434:
8424:
8423:
8409:
8408:
8312:Crime statistics
8206:
8205:
8193:
8192:
8110:
8109:
8076:Fourier analysis
8063:Frequency domain
8043:
7990:
7956:Structural break
7916:
7915:
7865:Cluster analysis
7812:Log-linear model
7785:
7784:
7760:
7759:
7701:
7675:Homoscedasticity
7531:
7530:
7507:
7506:
7426:
7418:
7410:
7409:(Kruskal–Wallis)
7394:
7379:
7334:Cross validation
7319:
7301:Anderson–Darling
7248:
7235:
7234:
7206:Likelihood-ratio
7198:Parametric tests
7176:Permutation test
7159:1- & 2-tails
7050:Minimum distance
7022:Point estimation
7018:
7017:
6969:Optimal decision
6920:
6819:
6818:
6806:
6805:
6788:Quasi-experiment
6738:Adaptive designs
6589:
6588:
6576:
6575:
6453:Rank correlation
6215:
6214:
6206:
6205:
6193:
6192:
6160:
6153:
6146:
6137:
6136:
6100:
6060:
6021:
5991:
5985:
5977:
5967:
5946:
5912:
5881:
5862:
5827:
5826:
5816:
5783:
5777:
5776:
5759:(279): 356–360.
5748:
5742:
5741:
5730:
5724:
5723:
5712:
5706:
5705:
5700:
5698:
5683:
5677:
5676:
5640:
5634:
5633:
5627:
5622:
5620:
5612:
5604:
5598:
5597:
5587:
5551:
5545:
5544:
5527:(6): 1247–1249.
5521:Animal Behaviour
5512:
5506:
5505:
5495:
5485:
5457:
5448:
5447:
5419:
5413:
5412:
5384:
5378:
5377:
5343:
5337:
5336:
5318:
5293:Lumley, Thomas;
5290:
5284:
5275:
5269:
5262:
5256:
5255:
5227:
5221:
5218:Statistics Guide
5214:
5208:
5207:
5183:
5177:
5176:
5148:
5142:
5141:
5113:
5107:
5106:
5076:
5070:
5069:
5049:
5040:
5039:
5033:
5025:
5015:
4995:
4986:
4985:
4949:
4943:
4942:
4922:
4916:
4915:
4909:
4901:
4893:
4887:
4886:
4880:
4872:
4864:
4855:
4854:
4836:
4830:
4829:
4810:10.1037/a0024338
4789:
4783:
4782:
4764:
4758:
4757:
4755:
4740:Machine Learning
4731:
4725:
4716:
4710:
4709:
4681:
4675:
4670:
4664:
4663:
4653:
4643:
4619:
4613:
4612:
4610:
4608:
4595:
4571:
4565:
4564:
4562:
4538:
4532:
4522:
4516:
4515:
4505:
4486:10.1214/09-SS051
4459:
4450:
4449:
4431:
4402:
4343:
4266:
4248:
4197:
4193:
4152:
4142:
4135:
4129:
4120:
4108:
3979:
3977:
3976:
3971:
3910:
3908:
3907:
3902:
3900:
3899:
3887:
3886:
3870:
3868:
3867:
3862:
3805:
3803:
3802:
3797:
3710:
3708:
3707:
3702:
3697:
3676:
3674:
3673:
3668:
3663:
3658:
3657:
3629:
3613:
3583:
3482:
3475:
3451:
3449:
3448:
3443:
3441:
3439:
3438:
3437:
3428:
3427:
3417:
3416:
3415:
3402:
3385:
3383:
3382:
3381:
3372:
3371:
3361:
3360:
3359:
3346:
3289:
3287:
3286:
3281:
3279:
3278:
3262:
3260:
3259:
3254:
3252:
3251:
3228:
3218:
3216:
3215:
3210:
3169:rank correlation
3133:
3109:
3055:
3053:
3052:
3047:
3044:
3042:
3041:
3040:
3031:
3030:
3020:
3019:
3010:
2992:
2990:
2989:
2984:
2982:
2981:
2906:
2890:
2862:Note that since
2854:
2831:
2829:
2828:
2823:
2818:
2816:
2812:
2811:
2809:
2789:
2785:
2784:
2771:
2766:
2753:
2748:
2732:
2707:
2702:
2701:
2700:
2691:
2690:
2680:
2678:
2673:
2672:
2669:
2654:factored out is
2653:
2614:
2612:
2611:
2606:
2600:
2598:
2596:
2573:
2569:
2568:
2555:
2550:
2537:
2532:
2517:
2516:
2507:
2506:
2496:
2491:
2486:
2476:
2475:
2463:
2462:
2450:
2449:
2440:
2439:
2429:
2427:
2422:
2421:
2418:
2395:
2393:
2392:
2387:
2381:
2375:
2365:
2364:
2352:
2351:
2339:
2338:
2329:
2328:
2318:
2317:
2312:
2311:
2291:
2289:
2288:
2283:
2277:
2272:
2271:
2270:
2261:
2260:
2250:
2245:
2244:
2182:
2180:
2179:
2174:
2168:
2166:
2165:
2156:
2155:
2154:
2138:
2097:
2090:
2074:
2067:
2050:
2033:The sample sizes
1991:
1981:
1974:
1961:
1946:
1942:
1935:
1925:
1865:
1863:
1862:
1857:
1855:
1854:
1845:
1844:
1832:
1831:
1798:
1763:
1743:
1708:
1706:
1705:
1700:
1693:
1688:
1678:
1677:
1665:
1664:
1654:
1649:
1648:
1636:
1631:
1621:
1620:
1608:
1607:
1597:
1592:
1591:
1579:
1578:
1566:
1565:
1523:
1521:
1520:
1515:
1511:
1506:
1496:
1495:
1483:
1482:
1472:
1467:
1466:
1454:
1453:
1390:
1388:
1387:
1382:
1378:
1373:
1363:
1362:
1350:
1349:
1339:
1334:
1333:
1321:
1320:
1286:
1272:
1268:
1264:
1247:
1245:
1244:
1239:
1237:
1236:
1216:
1214:
1213:
1208:
1206:
1205:
1076:
1001:
999:
998:
993:
991:
990:
979:
964:
962:
939:
907:
905:
904:
899:
897:
895:
894:
893:
884:
883:
873:
872:
863:
858:
857:
852:
805:
803:
802:
797:
795:
794:
782:
781:
759:
757:
756:
751:
749:
748:
736:
730:
720:
719:
707:
706:
696:
690:
689:
680:
679:
667:
666:
654:
653:
641:
635:
625:
624:
612:
611:
601:
595:
594:
585:
584:
572:
571:
545:
543:
542:
537:
525:
523:
522:
517:
515:
514:
513:
512:
489:
488:
472:
470:
469:
464:
448:
446:
445:
440:
438:
437:
436:
435:
412:
411:
337:
304:
284:
148:samples are the
111:statistical test
87:
85:
84:
79:
44:
24:
23:
16:
8497:
8496:
8492:
8491:
8490:
8488:
8487:
8486:
8462:
8461:
8460:
8455:
8418:
8389:
8351:
8288:
8274:quality control
8241:
8223:Clinical trials
8200:
8175:
8159:
8147:Hazard function
8141:
8095:
8057:
8041:
8004:
8000:Breusch–Godfrey
7988:
7965:
7905:
7880:Factor analysis
7826:
7807:Graphical model
7779:
7746:
7713:
7699:
7679:
7633:
7600:
7562:
7525:
7524:
7493:
7437:
7424:
7416:
7408:
7392:
7377:
7356:Rank statistics
7350:
7329:Model selection
7317:
7275:Goodness of fit
7269:
7246:
7220:
7192:
7145:
7090:
7079:Median unbiased
7007:
6918:
6851:Order statistic
6813:
6792:
6759:
6733:
6685:
6640:
6583:
6581:Data collection
6562:
6474:
6429:
6403:
6381:
6341:
6293:
6210:Continuous data
6200:
6187:
6169:
6164:
6107:
6081:10.2307/2527532
6049:
6010:
5979:
5978:
5878:
5851:
5835:
5830:
5805:10.2307/3001968
5787:Wilcoxon, Frank
5784:
5780:
5765:10.2307/2280906
5749:
5745:
5732:
5731:
5727:
5714:
5713:
5709:
5696:
5694:
5685:
5684:
5680:
5641:
5637:
5625:
5623:
5614:
5613:
5605:
5601:
5552:
5548:
5513:
5509:
5458:
5451:
5444:
5420:
5416:
5409:
5385:
5381:
5366:10.2307/2683975
5344:
5340:
5291:
5287:
5276:
5272:
5263:
5259:
5228:
5224:
5215:
5211:
5184:
5180:
5149:
5145:
5114:
5110:
5077:
5073:
5050:
5043:
5027:
5026:
4996:
4989:
4950:
4946:
4923:
4919:
4903:
4902:
4894:
4890:
4874:
4873:
4865:
4858:
4851:
4837:
4833:
4790:
4786:
4779:
4765:
4761:
4732:
4728:
4717:
4713:
4682:
4678:
4671:
4667:
4620:
4616:
4606:
4604:
4572:
4568:
4539:
4535:
4523:
4519:
4460:
4453:
4403:
4396:
4392:
4350:
4337:
4324:
4316:
4285:
4264:
4246:
4195:
4191:
4150:
4138:
4133:
4127:
4125:The R function
4118:
4106:
4090:
4074:
4069:
4005:version of the
3991:
3920:
3917:
3916:
3895:
3891:
3882:
3878:
3876:
3873:
3872:
3811:
3808:
3807:
3746:
3743:
3742:
3735:
3693:
3688:
3685:
3684:
3659:
3653:
3649:
3647:
3644:
3643:
3581:
3498:
3489:
3477:
3472:
3467:
3465:
3433:
3429:
3423:
3419:
3418:
3411:
3407:
3403:
3401:
3377:
3373:
3367:
3363:
3362:
3355:
3351:
3347:
3345:
3298:
3295:
3294:
3274:
3270:
3268:
3265:
3264:
3247:
3243:
3241:
3238:
3237:
3223:
3192:
3189:
3188:
3161:
3115:
3108:
3101:
3095:
3072:
3036:
3032:
3026:
3022:
3021:
3015:
3011:
3009:
3001:
2998:
2997:
2977:
2973:
2971:
2968:
2967:
2945:
2937:
2925:
2904:
2898:
2892:
2889:
2883:
2876:
2869:
2863:
2853:
2846:
2836:
2790:
2780:
2776:
2767:
2762:
2749:
2738:
2733:
2731:
2712:
2708:
2696:
2692:
2686:
2682:
2681:
2679:
2677:
2668:
2664:
2662:
2659:
2658:
2651:
2645:
2639:
2626:
2574:
2564:
2560:
2551:
2546:
2533:
2522:
2512:
2508:
2502:
2498:
2497:
2495:
2471:
2467:
2458:
2454:
2445:
2441:
2435:
2431:
2430:
2428:
2426:
2417:
2413:
2411:
2408:
2407:
2360:
2356:
2347:
2343:
2334:
2330:
2324:
2320:
2319:
2316:
2307:
2303:
2301:
2298:
2297:
2266:
2262:
2256:
2252:
2251:
2249:
2240:
2236:
2234:
2231:
2230:
2225:
2216:
2203:
2194:
2161:
2157:
2150:
2146:
2139:
2137:
2129:
2126:
2125:
2108:
2092:
2084:
2069:
2065:
2058:
2052:
2045:
2001:
1988:
1983:
1979:
1971:
1966:
1959:
1944:
1937:
1932:
1927:
1922:
1917:
1881:
1876:
1850:
1846:
1840:
1836:
1827:
1823:
1821:
1818:
1817:
1810:
1797:
1791:
1784:
1777:
1771:
1762:
1755:
1745:
1733:
1726:
1720:
1673:
1669:
1660:
1656:
1655:
1653:
1644:
1640:
1616:
1612:
1603:
1599:
1598:
1596:
1587:
1583:
1574:
1570:
1561:
1557:
1555:
1552:
1551:
1547:
1540:
1491:
1487:
1478:
1474:
1473:
1471:
1462:
1458:
1449:
1445:
1443:
1440:
1439:
1414:
1407:
1358:
1354:
1345:
1341:
1340:
1338:
1329:
1325:
1316:
1312:
1310:
1307:
1306:
1277:
1270:
1266:
1262:
1232:
1228:
1226:
1223:
1222:
1201:
1197:
1195:
1192:
1191:
1140:is fairly good.
1127:null hypothesis
1115:
1108:
1098:
1075:
1065:
1055:
1053:
1031:
1021:
980:
969:
968:
943:
938:
930:
927:
926:
889:
885:
879:
875:
874:
868:
864:
862:
853:
842:
841:
839:
836:
835:
821:area under the
813:
790:
786:
777:
773:
771:
768:
767:
744:
740:
715:
711:
702:
698:
697:
694:
685:
681:
675:
671:
662:
658:
649:
645:
620:
616:
607:
603:
602:
599:
590:
586:
580:
576:
567:
563:
561:
558:
557:
531:
528:
527:
508:
504:
503:
499:
484:
480:
478:
475:
474:
458:
455:
454:
431:
427:
426:
422:
407:
403:
401:
398:
397:
394:
327:
316:
310:
286:
266:
244:
229:
219:
162:
115:null hypothesis
73:
70:
69:
65:
58:
49:
25:
21:
12:
11:
5:
8495:
8485:
8484:
8479:
8474:
8457:
8456:
8454:
8453:
8441:
8429:
8415:
8402:
8399:
8398:
8395:
8394:
8391:
8390:
8388:
8387:
8382:
8377:
8372:
8367:
8361:
8359:
8353:
8352:
8350:
8349:
8344:
8339:
8334:
8329:
8324:
8319:
8314:
8309:
8304:
8298:
8296:
8290:
8289:
8287:
8286:
8281:
8276:
8267:
8262:
8257:
8251:
8249:
8243:
8242:
8240:
8239:
8234:
8229:
8220:
8218:Bioinformatics
8214:
8212:
8202:
8201:
8189:
8188:
8185:
8184:
8181:
8180:
8177:
8176:
8174:
8173:
8167:
8165:
8161:
8160:
8158:
8157:
8151:
8149:
8143:
8142:
8140:
8139:
8134:
8129:
8124:
8118:
8116:
8107:
8101:
8100:
8097:
8096:
8094:
8093:
8088:
8083:
8078:
8073:
8067:
8065:
8059:
8058:
8056:
8055:
8050:
8045:
8037:
8032:
8027:
8026:
8025:
8023:partial (PACF)
8014:
8012:
8006:
8005:
8003:
8002:
7997:
7992:
7984:
7979:
7973:
7971:
7970:Specific tests
7967:
7966:
7964:
7963:
7958:
7953:
7948:
7943:
7938:
7933:
7928:
7922:
7920:
7913:
7907:
7906:
7904:
7903:
7902:
7901:
7900:
7899:
7884:
7883:
7882:
7872:
7870:Classification
7867:
7862:
7857:
7852:
7847:
7842:
7836:
7834:
7828:
7827:
7825:
7824:
7819:
7817:McNemar's test
7814:
7809:
7804:
7799:
7793:
7791:
7781:
7780:
7756:
7755:
7752:
7751:
7748:
7747:
7745:
7744:
7739:
7734:
7729:
7723:
7721:
7715:
7714:
7712:
7711:
7695:
7689:
7687:
7681:
7680:
7678:
7677:
7672:
7667:
7662:
7657:
7655:Semiparametric
7652:
7647:
7641:
7639:
7635:
7634:
7632:
7631:
7626:
7621:
7616:
7610:
7608:
7602:
7601:
7599:
7598:
7593:
7588:
7583:
7578:
7572:
7570:
7564:
7563:
7561:
7560:
7555:
7550:
7545:
7539:
7537:
7527:
7526:
7523:
7522:
7517:
7511:
7503:
7502:
7499:
7498:
7495:
7494:
7492:
7491:
7490:
7489:
7479:
7474:
7469:
7468:
7467:
7462:
7451:
7449:
7443:
7442:
7439:
7438:
7436:
7435:
7430:
7429:
7428:
7420:
7412:
7396:
7393:(Mann–Whitney)
7388:
7387:
7386:
7373:
7372:
7371:
7360:
7358:
7352:
7351:
7349:
7348:
7347:
7346:
7341:
7336:
7326:
7321:
7318:(Shapiro–Wilk)
7313:
7308:
7303:
7298:
7293:
7285:
7279:
7277:
7271:
7270:
7268:
7267:
7259:
7250:
7238:
7232:
7230:Specific tests
7226:
7225:
7222:
7221:
7219:
7218:
7213:
7208:
7202:
7200:
7194:
7193:
7191:
7190:
7185:
7184:
7183:
7173:
7172:
7171:
7161:
7155:
7153:
7147:
7146:
7144:
7143:
7142:
7141:
7136:
7126:
7121:
7116:
7111:
7106:
7100:
7098:
7092:
7091:
7089:
7088:
7083:
7082:
7081:
7076:
7075:
7074:
7069:
7054:
7053:
7052:
7047:
7042:
7037:
7026:
7024:
7015:
7009:
7008:
7006:
7005:
7000:
6995:
6994:
6993:
6983:
6978:
6977:
6976:
6966:
6965:
6964:
6959:
6954:
6944:
6939:
6934:
6933:
6932:
6927:
6922:
6906:
6905:
6904:
6899:
6894:
6884:
6883:
6882:
6877:
6867:
6866:
6865:
6855:
6854:
6853:
6843:
6838:
6833:
6827:
6825:
6815:
6814:
6802:
6801:
6798:
6797:
6794:
6793:
6791:
6790:
6785:
6780:
6775:
6769:
6767:
6761:
6760:
6758:
6757:
6752:
6747:
6741:
6739:
6735:
6734:
6732:
6731:
6726:
6721:
6716:
6711:
6706:
6701:
6695:
6693:
6687:
6686:
6684:
6683:
6681:Standard error
6678:
6673:
6668:
6667:
6666:
6661:
6650:
6648:
6642:
6641:
6639:
6638:
6633:
6628:
6623:
6618:
6613:
6611:Optimal design
6608:
6603:
6597:
6595:
6585:
6584:
6572:
6571:
6568:
6567:
6564:
6563:
6561:
6560:
6555:
6550:
6545:
6540:
6535:
6530:
6525:
6520:
6515:
6510:
6505:
6500:
6495:
6490:
6484:
6482:
6476:
6475:
6473:
6472:
6467:
6466:
6465:
6460:
6450:
6445:
6439:
6437:
6431:
6430:
6428:
6427:
6422:
6417:
6411:
6409:
6408:Summary tables
6405:
6404:
6402:
6401:
6395:
6393:
6387:
6386:
6383:
6382:
6380:
6379:
6378:
6377:
6372:
6367:
6357:
6351:
6349:
6343:
6342:
6340:
6339:
6334:
6329:
6324:
6319:
6314:
6309:
6303:
6301:
6295:
6294:
6292:
6291:
6286:
6281:
6280:
6279:
6274:
6269:
6264:
6259:
6254:
6249:
6244:
6242:Contraharmonic
6239:
6234:
6223:
6221:
6212:
6202:
6201:
6189:
6188:
6186:
6185:
6180:
6174:
6171:
6170:
6163:
6162:
6155:
6148:
6140:
6134:
6133:
6127:
6117:
6106:
6105:External links
6103:
6102:
6101:
6075:(4): 532–552.
6061:
6047:
6022:
6008:
5992:
5947:
5903:(2): 598–611.
5885:Hodges, J.L.;
5882:
5877:978-1118840313
5876:
5863:
5849:
5834:
5831:
5829:
5828:
5778:
5743:
5740:. 30 May 2021.
5725:
5707:
5678:
5635:
5626:|journal=
5599:
5546:
5507:
5449:
5442:
5414:
5407:
5379:
5360:(3): 124–129.
5338:
5309:(1): 151–169.
5285:
5270:
5257:
5222:
5209:
5198:(4): 463–465.
5178:
5159:(3): 287–290.
5143:
5124:(4): 285–302.
5108:
5089:(2): 281–284.
5071:
5060:(2): 361–365.
5041:
4987:
4960:(4): 591–605.
4944:
4933:(8): 594–604.
4917:
4888:
4856:
4850:978-0471190455
4849:
4831:
4784:
4777:
4759:
4746:(2): 171–186.
4726:
4711:
4676:
4665:
4614:
4586:(2): 182–190.
4566:
4553:(3): 278–286.
4533:
4517:
4451:
4393:
4391:
4388:
4387:
4386:
4381:
4376:
4371:
4366:
4361:
4356:
4349:
4346:
4333:
4320:
4292:Frank Wilcoxon
4284:
4281:
4280:
4279:
4269:
4257:
4251:
4239:
4229:
4223:
4217:
4211:
4205:
4199:
4185:
4183:Apache Commons
4176:
4170:
4164:
4154:
4144:
4123:
4111:
4089:
4086:
4073:
4070:
4068:
4065:
3990:
3987:
3969:
3966:
3963:
3960:
3957:
3954:
3951:
3948:
3945:
3942:
3939:
3936:
3933:
3930:
3927:
3924:
3913:Brunner-Munzel
3898:
3894:
3890:
3885:
3881:
3860:
3857:
3854:
3851:
3848:
3845:
3842:
3839:
3836:
3833:
3830:
3827:
3824:
3821:
3818:
3815:
3795:
3792:
3789:
3786:
3783:
3780:
3777:
3774:
3771:
3768:
3765:
3762:
3759:
3756:
3753:
3750:
3734:
3731:
3728:
3727:
3724:
3720:
3719:
3716:
3712:
3711:
3700:
3696:
3692:
3682:
3678:
3677:
3666:
3662:
3656:
3652:
3641:
3637:
3636:
3633:
3598:
3597:
3570:
3567:
3544:
3541:
3526:
3497:
3490:
3488:
3485:
3470:
3463:
3453:
3452:
3436:
3432:
3426:
3422:
3414:
3410:
3406:
3400:
3397:
3394:
3391:
3388:
3380:
3376:
3370:
3366:
3358:
3354:
3350:
3344:
3341:
3338:
3335:
3332:
3329:
3326:
3323:
3320:
3317:
3314:
3311:
3308:
3305:
3302:
3277:
3273:
3250:
3246:
3220:
3219:
3208:
3205:
3202:
3199:
3196:
3160:
3157:
3106:
3099:
3071:
3065:
3057:
3056:
3039:
3035:
3029:
3025:
3018:
3014:
3008:
3005:
2980:
2976:
2966:(specifically
2944:
2941:
2936:
2933:
2924:
2921:
2902:
2896:
2887:
2881:
2874:
2867:
2851:
2844:
2833:
2832:
2821:
2815:
2808:
2805:
2802:
2799:
2796:
2793:
2788:
2783:
2779:
2775:
2770:
2765:
2761:
2757:
2752:
2747:
2744:
2741:
2737:
2730:
2727:
2724:
2721:
2718:
2715:
2711:
2705:
2699:
2695:
2689:
2685:
2676:
2667:
2649:
2643:
2622:
2616:
2615:
2603:
2595:
2592:
2589:
2586:
2583:
2580:
2577:
2572:
2567:
2563:
2559:
2554:
2549:
2545:
2541:
2536:
2531:
2528:
2525:
2521:
2515:
2511:
2505:
2501:
2494:
2489:
2485:
2482:
2479:
2474:
2470:
2466:
2461:
2457:
2453:
2448:
2444:
2438:
2434:
2425:
2416:
2397:
2396:
2384:
2378:
2374:
2371:
2368:
2363:
2359:
2355:
2350:
2346:
2342:
2337:
2333:
2327:
2323:
2315:
2310:
2306:
2294:
2293:
2280:
2275:
2269:
2265:
2259:
2255:
2248:
2243:
2239:
2221:
2212:
2199:
2190:
2184:
2183:
2171:
2164:
2160:
2153:
2149:
2145:
2142:
2136:
2133:
2107:
2104:
2100:
2099:
2077:
2076:
2063:
2056:
2038:
2037:
2034:
2031:
2016:
2000:
1997:
1996:
1995:
1994:
1993:
1990:= 46 − 21 = 25
1986:
1976:
1969:
1956:
1952:
1951:
1948:
1930:
1920:
1906:
1905:
1880:
1877:
1875:
1872:
1853:
1849:
1843:
1839:
1835:
1830:
1826:
1809:
1806:
1805:
1804:
1803:
1802:
1801:
1800:
1795:
1789:
1782:
1775:
1760:
1753:
1731:
1724:
1714:
1713:
1712:
1711:
1710:
1709:
1696:
1691:
1687:
1684:
1681:
1676:
1672:
1668:
1663:
1659:
1652:
1647:
1643:
1639:
1634:
1630:
1627:
1624:
1619:
1615:
1611:
1606:
1602:
1595:
1590:
1586:
1582:
1577:
1573:
1569:
1564:
1560:
1545:
1538:
1529:
1528:
1527:
1526:
1525:
1524:
1509:
1505:
1502:
1499:
1494:
1490:
1486:
1481:
1477:
1470:
1465:
1461:
1457:
1452:
1448:
1432:
1431:
1430:
1429:
1419:
1418:
1417:
1416:
1412:
1405:
1396:
1395:
1394:
1393:
1392:
1391:
1376:
1372:
1369:
1366:
1361:
1357:
1353:
1348:
1344:
1337:
1332:
1328:
1324:
1319:
1315:
1299:
1298:
1292:
1274:
1235:
1231:
1204:
1200:
1142:
1141:
1134:
1114:
1111:
1100:
1090:
1067:
1057:
1045:
1023:
1013:
1003:
1002:
989:
986:
983:
978:
975:
972:
967:
961:
958:
955:
952:
949:
946:
942:
937:
934:
909:
908:
892:
888:
882:
878:
871:
867:
861:
856:
851:
848:
845:
812:
809:
808:
807:
793:
789:
785:
780:
776:
761:
760:
747:
743:
739:
733:
729:
726:
723:
718:
714:
710:
705:
701:
693:
688:
684:
678:
674:
670:
665:
661:
657:
652:
648:
644:
638:
634:
631:
628:
623:
619:
615:
610:
606:
598:
593:
589:
583:
579:
575:
570:
566:
535:
511:
507:
502:
498:
495:
492:
487:
483:
462:
434:
430:
425:
421:
418:
415:
410:
406:
393:
390:
325:
314:
307:
306:
242:
232:
231:
227:
221:
217:
211:
204:
203:of each other,
161:
158:
77:
56:
51:
50:
28:
26:
19:
9:
6:
4:
3:
2:
8494:
8483:
8480:
8478:
8475:
8473:
8470:
8469:
8467:
8452:
8451:
8442:
8440:
8439:
8430:
8428:
8427:
8422:
8416:
8414:
8413:
8404:
8403:
8400:
8386:
8383:
8381:
8380:Geostatistics
8378:
8376:
8373:
8371:
8368:
8366:
8363:
8362:
8360:
8358:
8354:
8348:
8347:Psychometrics
8345:
8343:
8340:
8338:
8335:
8333:
8330:
8328:
8325:
8323:
8320:
8318:
8315:
8313:
8310:
8308:
8305:
8303:
8300:
8299:
8297:
8295:
8291:
8285:
8282:
8280:
8277:
8275:
8271:
8268:
8266:
8263:
8261:
8258:
8256:
8253:
8252:
8250:
8248:
8244:
8238:
8235:
8233:
8230:
8228:
8224:
8221:
8219:
8216:
8215:
8213:
8211:
8210:Biostatistics
8207:
8203:
8199:
8194:
8190:
8172:
8171:Log-rank test
8169:
8168:
8166:
8162:
8156:
8153:
8152:
8150:
8148:
8144:
8138:
8135:
8133:
8130:
8128:
8125:
8123:
8120:
8119:
8117:
8115:
8111:
8108:
8106:
8102:
8092:
8089:
8087:
8084:
8082:
8079:
8077:
8074:
8072:
8069:
8068:
8066:
8064:
8060:
8054:
8051:
8049:
8046:
8044:
8042:(Box–Jenkins)
8038:
8036:
8033:
8031:
8028:
8024:
8021:
8020:
8019:
8016:
8015:
8013:
8011:
8007:
8001:
7998:
7996:
7995:Durbin–Watson
7993:
7991:
7985:
7983:
7980:
7978:
7977:Dickey–Fuller
7975:
7974:
7972:
7968:
7962:
7959:
7957:
7954:
7952:
7951:Cointegration
7949:
7947:
7944:
7942:
7939:
7937:
7934:
7932:
7929:
7927:
7926:Decomposition
7924:
7923:
7921:
7917:
7914:
7912:
7908:
7898:
7895:
7894:
7893:
7890:
7889:
7888:
7885:
7881:
7878:
7877:
7876:
7873:
7871:
7868:
7866:
7863:
7861:
7858:
7856:
7853:
7851:
7848:
7846:
7843:
7841:
7838:
7837:
7835:
7833:
7829:
7823:
7820:
7818:
7815:
7813:
7810:
7808:
7805:
7803:
7800:
7798:
7797:Cohen's kappa
7795:
7794:
7792:
7790:
7786:
7782:
7778:
7774:
7770:
7766:
7761:
7757:
7743:
7740:
7738:
7735:
7733:
7730:
7728:
7725:
7724:
7722:
7720:
7716:
7710:
7706:
7702:
7696:
7694:
7691:
7690:
7688:
7686:
7682:
7676:
7673:
7671:
7668:
7666:
7663:
7661:
7658:
7656:
7653:
7651:
7650:Nonparametric
7648:
7646:
7643:
7642:
7640:
7636:
7630:
7627:
7625:
7622:
7620:
7617:
7615:
7612:
7611:
7609:
7607:
7603:
7597:
7594:
7592:
7589:
7587:
7584:
7582:
7579:
7577:
7574:
7573:
7571:
7569:
7565:
7559:
7556:
7554:
7551:
7549:
7546:
7544:
7541:
7540:
7538:
7536:
7532:
7528:
7521:
7518:
7516:
7513:
7512:
7508:
7504:
7488:
7485:
7484:
7483:
7480:
7478:
7475:
7473:
7470:
7466:
7463:
7461:
7458:
7457:
7456:
7453:
7452:
7450:
7448:
7444:
7434:
7431:
7427:
7421:
7419:
7413:
7411:
7405:
7404:
7403:
7400:
7399:Nonparametric
7397:
7395:
7389:
7385:
7382:
7381:
7380:
7374:
7370:
7369:Sample median
7367:
7366:
7365:
7362:
7361:
7359:
7357:
7353:
7345:
7342:
7340:
7337:
7335:
7332:
7331:
7330:
7327:
7325:
7322:
7320:
7314:
7312:
7309:
7307:
7304:
7302:
7299:
7297:
7294:
7292:
7290:
7286:
7284:
7281:
7280:
7278:
7276:
7272:
7266:
7264:
7260:
7258:
7256:
7251:
7249:
7244:
7240:
7239:
7236:
7233:
7231:
7227:
7217:
7214:
7212:
7209:
7207:
7204:
7203:
7201:
7199:
7195:
7189:
7186:
7182:
7179:
7178:
7177:
7174:
7170:
7167:
7166:
7165:
7162:
7160:
7157:
7156:
7154:
7152:
7148:
7140:
7137:
7135:
7132:
7131:
7130:
7127:
7125:
7122:
7120:
7117:
7115:
7112:
7110:
7107:
7105:
7102:
7101:
7099:
7097:
7093:
7087:
7084:
7080:
7077:
7073:
7070:
7068:
7065:
7064:
7063:
7060:
7059:
7058:
7055:
7051:
7048:
7046:
7043:
7041:
7038:
7036:
7033:
7032:
7031:
7028:
7027:
7025:
7023:
7019:
7016:
7014:
7010:
7004:
7001:
6999:
6996:
6992:
6989:
6988:
6987:
6984:
6982:
6979:
6975:
6974:loss function
6972:
6971:
6970:
6967:
6963:
6960:
6958:
6955:
6953:
6950:
6949:
6948:
6945:
6943:
6940:
6938:
6935:
6931:
6928:
6926:
6923:
6921:
6915:
6912:
6911:
6910:
6907:
6903:
6900:
6898:
6895:
6893:
6890:
6889:
6888:
6885:
6881:
6878:
6876:
6873:
6872:
6871:
6868:
6864:
6861:
6860:
6859:
6856:
6852:
6849:
6848:
6847:
6844:
6842:
6839:
6837:
6834:
6832:
6829:
6828:
6826:
6824:
6820:
6816:
6812:
6807:
6803:
6789:
6786:
6784:
6781:
6779:
6776:
6774:
6771:
6770:
6768:
6766:
6762:
6756:
6753:
6751:
6748:
6746:
6743:
6742:
6740:
6736:
6730:
6727:
6725:
6722:
6720:
6717:
6715:
6712:
6710:
6707:
6705:
6702:
6700:
6697:
6696:
6694:
6692:
6688:
6682:
6679:
6677:
6676:Questionnaire
6674:
6672:
6669:
6665:
6662:
6660:
6657:
6656:
6655:
6652:
6651:
6649:
6647:
6643:
6637:
6634:
6632:
6629:
6627:
6624:
6622:
6619:
6617:
6614:
6612:
6609:
6607:
6604:
6602:
6599:
6598:
6596:
6594:
6590:
6586:
6582:
6577:
6573:
6559:
6556:
6554:
6551:
6549:
6546:
6544:
6541:
6539:
6536:
6534:
6531:
6529:
6526:
6524:
6521:
6519:
6516:
6514:
6511:
6509:
6506:
6504:
6503:Control chart
6501:
6499:
6496:
6494:
6491:
6489:
6486:
6485:
6483:
6481:
6477:
6471:
6468:
6464:
6461:
6459:
6456:
6455:
6454:
6451:
6449:
6446:
6444:
6441:
6440:
6438:
6436:
6432:
6426:
6423:
6421:
6418:
6416:
6413:
6412:
6410:
6406:
6400:
6397:
6396:
6394:
6392:
6388:
6376:
6373:
6371:
6368:
6366:
6363:
6362:
6361:
6358:
6356:
6353:
6352:
6350:
6348:
6344:
6338:
6335:
6333:
6330:
6328:
6325:
6323:
6320:
6318:
6315:
6313:
6310:
6308:
6305:
6304:
6302:
6300:
6296:
6290:
6287:
6285:
6282:
6278:
6275:
6273:
6270:
6268:
6265:
6263:
6260:
6258:
6255:
6253:
6250:
6248:
6245:
6243:
6240:
6238:
6235:
6233:
6230:
6229:
6228:
6225:
6224:
6222:
6220:
6216:
6213:
6211:
6207:
6203:
6199:
6194:
6190:
6184:
6181:
6179:
6176:
6175:
6172:
6168:
6161:
6156:
6154:
6149:
6147:
6142:
6141:
6138:
6131:
6128:
6125:
6121:
6118:
6116:
6113:
6109:
6108:
6098:
6094:
6090:
6086:
6082:
6078:
6074:
6070:
6066:
6062:
6058:
6054:
6050:
6044:
6040:
6036:
6032:
6028:
6023:
6019:
6015:
6011:
6005:
6001:
5997:
5993:
5989:
5983:
5975:
5971:
5966:
5961:
5958:: 11.IT.3.1.
5957:
5953:
5948:
5944:
5940:
5936:
5932:
5928:
5924:
5920:
5916:
5911:
5906:
5902:
5898:
5897:
5892:
5888:
5887:Lehmann, E.L.
5883:
5879:
5873:
5869:
5864:
5860:
5856:
5852:
5846:
5842:
5837:
5836:
5824:
5820:
5815:
5810:
5806:
5802:
5798:
5794:
5793:
5788:
5782:
5774:
5770:
5766:
5762:
5758:
5754:
5747:
5739:
5735:
5729:
5721:
5717:
5711:
5704:
5692:
5688:
5682:
5674:
5670:
5666:
5662:
5658:
5654:
5650:
5646:
5639:
5631:
5618:
5610:
5603:
5595:
5591:
5586:
5581:
5577:
5573:
5569:
5565:
5561:
5559:
5550:
5542:
5538:
5534:
5530:
5526:
5522:
5518:
5511:
5503:
5499:
5494:
5489:
5484:
5479:
5475:
5471:
5467:
5465:
5456:
5454:
5445:
5439:
5435:
5431:
5427:
5426:
5418:
5410:
5404:
5400:
5396:
5392:
5391:
5383:
5375:
5371:
5367:
5363:
5359:
5355:
5354:
5349:
5342:
5334:
5330:
5326:
5322:
5317:
5312:
5308:
5304:
5300:
5296:
5289:
5282:
5281:
5274:
5267:
5261:
5253:
5249:
5245:
5241:
5237:
5233:
5226:
5219:
5213:
5205:
5201:
5197:
5193:
5189:
5182:
5174:
5170:
5166:
5162:
5158:
5154:
5153:Psychometrika
5147:
5139:
5135:
5131:
5127:
5123:
5119:
5112:
5104:
5100:
5096:
5092:
5088:
5084:
5083:
5075:
5067:
5063:
5059:
5055:
5048:
5046:
5037:
5031:
5023:
5019:
5014:
5009:
5006:: 11.IT.3.1.
5005:
5001:
4994:
4992:
4983:
4979:
4975:
4971:
4967:
4963:
4959:
4955:
4948:
4940:
4936:
4932:
4928:
4921:
4913:
4907:
4899:
4892:
4884:
4878:
4870:
4863:
4861:
4852:
4846:
4842:
4835:
4827:
4823:
4819:
4815:
4811:
4807:
4803:
4799:
4795:
4788:
4780:
4774:
4770:
4763:
4754:
4749:
4745:
4741:
4737:
4730:
4723:
4722:
4715:
4707:
4703:
4699:
4695:
4691:
4687:
4680:
4674:
4669:
4661:
4657:
4652:
4647:
4642:
4637:
4633:
4629:
4625:
4618:
4603:
4599:
4594:
4589:
4585:
4581:
4580:Stata Journal
4577:
4570:
4561:
4556:
4552:
4548:
4544:
4537:
4529:
4525:
4521:
4513:
4509:
4504:
4499:
4495:
4491:
4487:
4483:
4479:
4475:
4474:
4469:
4467:
4458:
4456:
4447:
4443:
4439:
4435:
4430:
4425:
4421:
4417:
4416:
4411:
4407:
4401:
4399:
4394:
4385:
4382:
4380:
4377:
4375:
4372:
4370:
4367:
4365:
4362:
4360:
4357:
4355:
4352:
4351:
4345:
4341:
4336:
4332:
4328:
4323:
4319:
4314:
4310:
4306:
4300:
4297:
4293:
4288:
4277:
4273:
4270:
4267:
4261:
4258:
4255:
4252:
4249:
4243:
4240:
4237:
4233:
4230:
4227:
4224:
4221:
4218:
4215:
4212:
4209:
4206:
4203:
4200:
4189:
4186:
4184:
4180:
4177:
4174:
4171:
4168:
4165:
4162:
4158:
4155:
4151:PROC NPAR1WAY
4148:
4145:
4141:
4136:
4130:
4124:
4121:
4115:
4112:
4109:
4103:
4100:
4099:
4098:
4095:
4085:
4083:
4082:Kendall's tau
4079:
4072:Kendall's tau
4064:
4062:
4057:
4055:
4051:
4046:
4044:
4040:
4035:
4033:
4031:
4026:
4021:
4019:
4015:
4010:
4008:
4004:
4000:
3996:
3986:
3983:
3967:
3964:
3958:
3955:
3952:
3946:
3943:
3940:
3934:
3931:
3928:
3922:
3914:
3896:
3892:
3888:
3883:
3879:
3858:
3855:
3849:
3846:
3843:
3837:
3834:
3831:
3825:
3822:
3819:
3813:
3793:
3790:
3784:
3781:
3778:
3772:
3769:
3766:
3760:
3757:
3754:
3748:
3740:
3725:
3722:
3721:
3717:
3714:
3713:
3698:
3694:
3690:
3683:
3680:
3679:
3664:
3660:
3654:
3650:
3642:
3639:
3638:
3634:
3632:Distribution
3631:
3630:
3626:
3622:
3618:
3612:
3610:
3608:
3603:
3595:
3591:
3587:
3579:
3575:
3571:
3568:
3565:
3561:
3557:
3553:
3549:
3545:
3542:
3539:
3535:
3531:
3527:
3524:
3523:
3522:
3520:
3516:
3512:
3507:
3503:
3495:
3484:
3480:
3473:
3462:
3458:
3434:
3430:
3424:
3420:
3412:
3408:
3404:
3398:
3395:
3392:
3389:
3386:
3378:
3374:
3368:
3364:
3356:
3352:
3348:
3342:
3339:
3336:
3333:
3330:
3327:
3321:
3318:
3315:
3309:
3306:
3303:
3300:
3293:
3292:
3291:
3275:
3271:
3248:
3244:
3235:
3230:
3226:
3206:
3203:
3200:
3197:
3194:
3187:
3186:
3185:
3183:
3179:
3173:
3170:
3166:
3156:
3153:
3149:
3145:
3141:
3137:
3131:
3127:
3123:
3119:
3113:
3105:
3098:
3093:
3089:
3085:
3081:
3077:
3069:
3064:
3062:
3037:
3033:
3027:
3023:
3016:
3012:
3006:
3003:
2996:
2995:
2994:
2978:
2974:
2965:
2961:
2956:
2954:
2951:test is with
2950:
2940:
2932:
2930:
2920:
2918:
2914:
2910:
2901:
2895:
2886:
2880:
2873:
2866:
2860:
2856:
2850:
2843:
2839:
2819:
2813:
2803:
2800:
2797:
2791:
2781:
2777:
2773:
2768:
2763:
2759:
2750:
2745:
2742:
2739:
2735:
2728:
2722:
2719:
2716:
2709:
2703:
2697:
2693:
2687:
2683:
2674:
2665:
2657:
2656:
2655:
2648:
2642:
2636:
2634:
2631:th rank, and
2630:
2625:
2621:
2601:
2590:
2587:
2584:
2578:
2575:
2565:
2561:
2557:
2552:
2547:
2543:
2534:
2529:
2526:
2523:
2519:
2513:
2509:
2503:
2499:
2492:
2487:
2480:
2477:
2472:
2468:
2464:
2459:
2455:
2446:
2442:
2436:
2432:
2423:
2414:
2406:
2405:
2404:
2402:
2382:
2376:
2369:
2366:
2361:
2357:
2353:
2348:
2344:
2335:
2331:
2325:
2321:
2313:
2308:
2304:
2296:
2295:
2278:
2273:
2267:
2263:
2257:
2253:
2246:
2241:
2237:
2229:
2228:
2227:
2226:are given by
2224:
2220:
2215:
2211:
2207:
2202:
2198:
2193:
2189:
2169:
2162:
2158:
2151:
2147:
2143:
2140:
2134:
2131:
2124:
2123:
2122:
2121:
2117:
2113:
2103:
2095:
2088:
2082:
2081:
2080:
2075:two-tailed)."
2072:
2062:
2055:
2048:
2043:
2042:
2041:
2035:
2032:
2029:
2025:
2021:
2018:The value of
2017:
2014:
2010:
2009:
2008:
2006:
1989:
1982:, leading to
1977:
1972:
1964:
1963:
1957:
1954:
1953:
1949:
1940:
1933:
1923:
1915:
1914:
1913:
1911:
1903:
1902:
1901:
1898:
1894:
1891:in which one
1890:
1886:
1883:Suppose that
1871:
1869:
1851:
1847:
1841:
1837:
1833:
1828:
1824:
1815:
1794:
1788:
1781:
1774:
1770:
1769:
1767:
1759:
1752:
1748:
1741:
1737:
1730:
1723:
1719:Knowing that
1718:
1717:
1716:
1715:
1694:
1689:
1682:
1679:
1674:
1670:
1661:
1657:
1650:
1645:
1641:
1637:
1632:
1625:
1622:
1617:
1613:
1604:
1600:
1593:
1588:
1584:
1580:
1575:
1571:
1567:
1562:
1558:
1550:
1549:
1544:
1537:
1533:
1532:
1531:
1530:
1507:
1500:
1497:
1492:
1488:
1479:
1475:
1468:
1463:
1459:
1455:
1450:
1446:
1438:
1437:
1436:
1435:
1434:
1433:
1427:
1423:
1422:
1421:
1420:
1411:
1404:
1400:
1399:
1398:
1397:
1374:
1367:
1364:
1359:
1355:
1346:
1342:
1335:
1330:
1326:
1322:
1317:
1313:
1305:
1304:
1303:
1302:
1301:
1300:
1296:
1293:
1290:
1284:
1280:
1275:
1259:
1258:
1257:
1254:
1253:
1249:
1233:
1229:
1220:
1202:
1198:
1189:
1184:
1179:
1178:
1174:
1171:
1169:
1165:
1160:
1158:
1154:
1149:
1147:
1139:
1135:
1132:
1131:
1130:
1128:
1124:
1120:
1110:
1107:
1103:
1097:
1093:
1088:
1084:
1080:
1074:
1070:
1064:
1060:
1052:
1048:
1043:
1039:
1035:
1030:
1026:
1020:
1016:
1012:
1008:
987:
984:
981:
965:
956:
953:
950:
944:
940:
935:
932:
925:
924:
923:
921:
916:
914:
890:
886:
880:
876:
869:
865:
859:
854:
834:
833:
832:
830:
826:
824:
818:
791:
787:
783:
778:
774:
766:
765:
764:
745:
741:
737:
731:
724:
721:
716:
712:
703:
699:
691:
686:
682:
676:
672:
668:
663:
659:
655:
650:
646:
642:
636:
629:
626:
621:
617:
608:
604:
596:
591:
587:
581:
577:
573:
568:
564:
556:
555:
554:
552:
551:
548:Mann–Whitney
533:
509:
505:
500:
496:
493:
490:
485:
481:
460:
452:
451:i.i.d. sample
432:
428:
423:
419:
416:
413:
408:
404:
389:
387:
383:
379:
377:
371:
366:
364:
359:
357:
353:
349:
345:
341:
335:
331:
324:
320:
313:
302:
298:
294:
290:
282:
278:
274:
270:
264:
260:
256:
252:
248:
247:
246:
241:
237:
226:
222:
216:
212:
209:
205:
202:
198:
197:
196:
193:
191:
187:
183:
179:
175:
171:
167:
157:
155:
151:
147:
142:
140:
136:
132:
128:
124:
120:
116:
112:
109:
108:nonparametric
105:
101:
97:
93:
89:
75:
68:Mann–Whitney
63:
55:
47:
42:
38:
37:
32:
27:
18:
17:
8482:U-statistics
8448:
8436:
8417:
8410:
8322:Econometrics
8272: /
8255:Chemometrics
8232:Epidemiology
8225: /
8198:Applications
8040:ARIMA model
7987:Q-statistic
7936:Stationarity
7832:Multivariate
7775: /
7771: /
7769:Multivariate
7767: /
7707: /
7703: /
7477:Bayes factor
7390:
7376:Signed rank
7288:
7262:
7254:
7242:
6937:Completeness
6773:Cohort study
6671:Opinion poll
6606:Missing data
6593:Study design
6548:Scatter plot
6470:Scatter plot
6463:Spearman's ρ
6425:Grouped data
6123:
6111:
6072:
6068:
6030:
6026:
5999:
5982:cite journal
5955:
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5695:. Retrieved
5690:
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5648:
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5617:cite journal
5602:
5585:1887/3209569
5567:
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5295:Diehr, Paula
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3989:Alternatives
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3564:type I error
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3525:Ordinal data
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2923:Effect sizes
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1113:Calculations
1105:
1101:
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8450:WikiProject
8365:Cartography
8327:Jurimetrics
8279:Reliability
8010:Time domain
7989:(Ljung–Box)
7911:Time-series
7789:Categorical
7773:Time-series
7765:Categorical
7700:(Bernoulli)
7535:Correlation
7515:Correlation
7311:Jarque–Bera
7283:Chi-squared
7045:M-estimator
6998:Asymptotics
6942:Sufficiency
6709:Interaction
6621:Replication
6601:Effect size
6558:Violin plot
6538:Radar chart
6518:Forest plot
6508:Correlogram
6458:Kendall's τ
4804:(1): 2–18.
4354:Lepage test
4311:(where the
4232:StatsDirect
4119:wilcox.test
3635:Efficiency
2929:effect size
2891:, the mean
1943:, which is
1252:Method two:
1177:Method one:
1022:term of AUC
550:U statistic
392:U statistic
201:independent
8466:Categories
8317:Demography
8035:ARMA model
7840:Regression
7417:(Friedman)
7378:(Wilcoxon)
7316:Normality
7306:Lilliefors
7253:Student's
7129:Resampling
7003:Robustness
6991:divergence
6981:Efficiency
6919:(monotone)
6914:Likelihood
6831:Population
6664:Stratified
6616:Population
6435:Dependence
6391:Count data
6322:Percentile
6299:Dispersion
6232:Arithmetic
6167:Statistics
6097:0119.15604
6069:Biometrics
5935:0203.21105
5833:References
4446:0041.26103
4305:Henry Mann
4214:STATISTICA
4153:procedure.
4134:rcompanion
3578:efficiency
3569:Efficiency
3543:Robustness
3124:) + 0.5 P(
3086:involving
1965:Therefore
1808:Properties
1129:is known:
378:-rank test
236:consistent
174:continuous
166:Henry Mann
46:(May 2024)
7698:Logistic
7465:posterior
7391:Rank sum
7139:Jackknife
7134:Bootstrap
6952:Bootstrap
6887:Parameter
6836:Statistic
6631:Statistic
6543:Run chart
6528:Pie chart
6523:Histogram
6513:Fan chart
6488:Bar chart
6370:L-moments
6257:Geometric
5974:120622013
5870:. Wiley.
5673:120473946
5594:235521799
5541:140209347
5502:2515-2459
5325:0163-7525
5252:0022-0973
5173:122500836
5022:120622013
4906:cite book
4877:cite book
4818:1939-2222
4706:1477-870X
4602:118445807
4268:function.
4167:SigmaStat
4131:from the
4128:wilcoxonZ
4059:See also
3856:≠
3699:π
3651:π
3640:Logistic
3619:-test if
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2801:−
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2736:∑
2729:−
2666:σ
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2415:σ
2305:σ
2159:σ
2144:−
2073:< 0.05
1651:−
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1469:−
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1119:statistic
988:ℓ
966:∑
954:−
738:−
643:−
494:…
417:…
374:Wilcoxon
265:; i.e.,
164:Although
150:sign test
146:dependent
8412:Category
8105:Survival
7982:Johansen
7705:Binomial
7660:Isotonic
7247:(normal)
6892:location
6699:Blocking
6654:Sampling
6533:Q–Q plot
6498:Box plot
6480:Graphics
6375:Skewness
6365:Kurtosis
6337:Variance
6267:Heronian
6262:Harmonic
5998:(2006).
5889:(1963).
5333:11910059
4974:17944619
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4348:See also
4265:WILCOXON
4254:StatXact
4250:command.
3723:Uniform
3715:Laplace
3556:outliers
3236:(either
3134:, where
3088:concepts
1893:tortoise
1874:Examples
1159:itself.
152:and the
8438:Commons
8385:Kriging
8270:Process
8227:studies
8086:Wavelet
7919:General
7086:Plug-in
6880:L space
6659:Cluster
6360:Moments
6178:Outline
6089:2527532
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4283:History
4247:ranksum
4220:UNISTAT
4107:ranksum
3681:Normal
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1766:algebra
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303:) ≠ 0.5
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113:of the
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96:MWW/MWU
41:Discuss
8307:Census
7897:Normal
7845:Manova
7665:Robust
7415:2-way
7407:1-way
7245:-test
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6284:Median
6277:Lehmer
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6272:Heinz
6247:Cubic
6183:Index
6115:(pdf)
6085:JSTOR
5970:S2CID
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5590:S2CID
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4390:Notes
4278:node.
4272:KNIME
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2888:2
2885:n
2882:1
2879:n
2875:2
2872:U
2868:1
2865:U
2852:2
2849:n
2845:1
2842:n
2838:n
2820:,
2814:)
2807:)
2804:1
2798:n
2795:(
2792:n
2787:)
2782:k
2778:t
2769:3
2764:k
2760:t
2756:(
2751:K
2746:1
2743:=
2740:k
2726:)
2723:1
2720:+
2717:n
2714:(
2710:(
2698:2
2694:n
2688:1
2684:n
2675:=
2650:2
2647:n
2644:1
2641:n
2633:K
2629:k
2624:k
2620:t
2602:,
2594:)
2591:1
2585:n
2582:(
2579:n
2571:)
2566:k
2562:t
2553:3
2548:k
2544:t
2540:(
2535:K
2530:1
2527:=
2524:k
2514:2
2510:n
2504:1
2500:n
2484:)
2481:1
2478:+
2473:2
2469:n
2465:+
2460:1
2456:n
2452:(
2447:2
2443:n
2437:1
2433:n
2424:=
2401:σ
2383:.
2373:)
2370:1
2367:+
2362:2
2358:n
2354:+
2349:1
2345:n
2341:(
2336:2
2332:n
2326:1
2322:n
2314:=
2309:U
2279:,
2274:2
2268:2
2264:n
2258:1
2254:n
2247:=
2242:U
2238:m
2223:U
2219:σ
2214:U
2210:m
2206:U
2201:U
2197:σ
2192:U
2188:m
2170:,
2163:U
2152:U
2148:m
2141:U
2135:=
2132:z
2112:U
2094:ρ
2087:P
2085:2
2071:P
2064:2
2061:n
2057:1
2054:n
2047:U
2020:U
2013:U
2005:U
1992:.
1987:H
1985:U
1970:T
1968:U
1947:.
1939:U
1931:H
1929:U
1921:T
1919:U
1910:U
1868:U
1852:2
1848:n
1842:1
1838:n
1834:=
1829:i
1825:U
1814:U
1799:.
1796:2
1793:n
1790:1
1787:n
1783:2
1780:U
1776:1
1773:U
1761:2
1758:n
1754:1
1751:n
1747:N
1740:N
1738:(
1736:N
1732:2
1729:R
1725:1
1722:R
1695:.
1690:2
1686:)
1683:1
1680:+
1675:2
1671:n
1667:(
1662:2
1658:n
1646:2
1642:R
1638:+
1633:2
1629:)
1626:1
1623:+
1618:1
1614:n
1610:(
1605:1
1601:n
1589:1
1585:R
1581:=
1576:2
1572:U
1568:+
1563:1
1559:U
1546:2
1543:U
1539:1
1536:U
1508:2
1504:)
1501:1
1498:+
1493:2
1489:n
1485:(
1480:2
1476:n
1464:2
1460:R
1456:=
1451:2
1447:U
1426:U
1413:1
1410:R
1406:1
1403:n
1375:2
1371:)
1368:1
1365:+
1360:1
1356:n
1352:(
1347:1
1343:n
1331:1
1327:R
1323:=
1318:1
1314:U
1295:U
1289:N
1283:N
1281:(
1279:N
1234:2
1230:U
1219:U
1203:1
1199:U
1188:U
1183:U
1164:U
1157:U
1153:U
1123:U
1106:k
1104:,
1102:ℓ
1096:ℓ
1094:,
1092:k
1087:ℓ
1085:,
1083:k
1079:M
1073:k
1071:,
1069:ℓ
1063:ℓ
1061:,
1059:k
1051:k
1049:,
1047:k
1042:k
1038:ℓ
1034:k
1029:ℓ
1027:,
1025:k
1019:ℓ
1017:,
1015:k
1011:R
1007:c
985:,
982:k
977:C
974:U
971:A
960:)
957:1
951:c
948:(
945:c
941:1
936:=
933:M
920:U
891:2
887:n
881:1
877:n
870:1
866:U
860:=
855:1
850:C
847:U
844:A
827:(
817:U
792:2
788:R
784:,
779:1
775:R
746:2
742:R
732:2
728:)
725:1
722:+
717:2
713:n
709:(
704:2
700:n
692:+
687:2
683:n
677:1
673:n
669:=
664:2
660:U
656:,
651:1
647:R
637:2
633:)
630:1
627:+
622:1
618:n
614:(
609:1
605:n
597:+
592:2
588:n
582:1
578:n
574:=
569:1
565:U
534:Y
510:2
506:n
501:Y
497:,
491:,
486:1
482:Y
461:X
433:1
429:n
424:X
420:,
414:,
409:1
405:X
386:U
370:U
363:U
344:U
340:U
336:)
334:δ
330:x
328:(
326:2
323:F
319:x
317:(
315:1
312:F
305:.
301:Y
297:X
293:Y
289:X
283:)
281:X
277:Y
273:Y
269:X
263:X
259:Y
255:Y
251:X
243:1
240:H
228:1
225:H
218:0
215:H
190:U
170:U
139:X
135:Y
131:Y
127:X
123:Y
119:X
94:(
76:U
64:.
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