2094:"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,
2966:, 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%.
8432:
8418:
33:
8456:
8444:
3995:
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
1910:
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,
1196:
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
2624:
4107:
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:
2841:
4541:
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
3165:
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.
769:
1271:
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
3607:. 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.
3182:
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.
1718:
4309:
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).
2869:
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.
3461:
2405:
2420:
4031:-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.
4313:
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
3519:
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
3233:
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
3186:
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
2671:
1011:
917:
1533:
1400:
5854:. 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.
4355:). 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.
2192:
4286:
2301:
570:
3065:
4695:
Mason, S. J., Graham, N. E. (2002). "Areas beneath the relative operating characteristics (ROC) and relative operating levels (ROL) curves: Statistical significance and interpretation".
3880:
461:
3989:
3815:
535:
458:
320:
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.,
1875:
3920:
815:
3686:
5998:
5046:
4384:
376:
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.
3720:
4922:
4893:
3299:
3272:
3228:
3002:
1564:
1257:
1226:
7553:
2051:
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,
926:, 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.
8058:
5726:
5092:
1287:
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
555:
482:
97:
4012:
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
6013:. 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.
5654:
Bergmann, Reinhard; Ludbrook, John; Spooren, Will P.J.M. (2000). "Different
Outcomes of the Wilcoxon–Mann–Whitney Test from Different Statistics Packages".
3307:
8208:
7832:
6473:
1051:(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
817:
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.
7606:
4092:
8045:
2619:{\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)}}},\,}
2310:
5241:
Zimmerman, Donald W. (1998-01-01). "Invalidation of
Parametric and Nonparametric Statistical Tests by Concurrent Violation of Two Assumptions".
4963:
Nakagawa, Shinichi; Cuthill, Innes C (2007). "Effect size, confidence interval and statistical significance: a practical guide for biologists".
3528:
tests a null hypothesis of equal means in two groups against an alternative of unequal means. Hence, except in special cases, the Mann–Whitney
3470:
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,
6057:
6018:
5859:
5452:
5417:
4787:
3161:
statistic can be seen in the case of the odd example used above, where two distributions that were significantly different on a Mann–Whitney
3153:
are randomly chosen observations from the two distributions. Both extreme values represent complete separation of the distributions, while a
6468:
6168:
1927:
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
5197:
Wendt, H.W. (1972). "Dealing with a common problem in social science: A simplified rank-biserial coefficient of correlation based on the
7072:
6220:
3125:
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
2836:{\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)}},}
1937:. 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
184:
8460:
4023:
Similarly, some authors (e.g., Conover) suggest transforming the data to ranks (if they are not already ranks) and then performing the
1966:
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.
5886:
4859:
939:
8487:
7855:
7747:
5906:
4425:
839:
833:
2055:"Median latencies in groups E and C were 153 and 247 ms; the distributions in the two groups differed significantly (Mann–Whitney
8033:
7907:
4323:
848:
8091:
7752:
7497:
6868:
6458:
4683:
3195:)). This simple difference formula is just the difference of the common language effect size of each group, and is as follows:
7082:
1452:
1319:
8142:
7354:
7161:
7050:
7008:
6247:
8385:
7344:
4167:
7394:
3551:
but not interval scaled, in which case the spacing between adjacent values of the scale cannot be assumed to be constant.
2138:
2113:
However it would be rare to find such an extensive report in a document whose major topic was not statistical inference.
51:
7936:
7885:
7870:
7860:
7729:
7601:
7568:
7349:
7179:
6130:
4198:
1178:
8005:
7306:
4374:
4071:
764:{\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}}
362:
358:
8280:
8081:
7060:
6729:
6193:
5714:
scipy.stats.mannwhitneyu(x, y, use_continuity=True): Computes the Mann–Whitney rank test on samples x and y.
4189:
2410:
The formula for the standard deviation is more complicated in the presence of tied ranks. If there are ties in ranks,
2243:
923:
353:
test as showing a difference in medians. Under this location shift assumption, we can also interpret the Mann–Whitney
8165:
8132:
4246:
2219:, is approximately a standard normal deviate whose significance can be checked in tables of the normal distribution.
5697:
8482:
8137:
7880:
7639:
7545:
7525:
7433:
7144:
6962:
6445:
6317:
6078:(December 1963). "On the estimation of relative potency in dilution(-direct) assays by distribution-free methods".
4212:
46:
7311:
7077:
6935:
3010:
1192:
For comparing two small sets of observations, a direct method is quick, and gives insight into the meaning of the
399:
test is applied to independent samples. The
Wilcoxon signed-rank test is applied to matched or dependent samples.
7897:
7665:
7386:
7240:
7169:
7089:
6947:
6928:
6636:
6357:
5363:
4379:
3820:
3574:
1899:
384:
164:
72:
8010:
3599:-test. For distributions sufficiently far from normal and for sufficiently large sample sizes, the Mann–Whitney
369:
of all possible differences between an observation in the first sample and an observation in the second sample.
8380:
8147:
7695:
7660:
7624:
7409:
6851:
6760:
6719:
6631:
6322:
6161:
4124:
3516:
1435:
Note that it doesn't matter which of the two samples is considered sample 1. An equally valid formula for
246:
211:
118:
7417:
4298:
The statistic appeared in a 1914 article by the German Gustav
Deuchler (with a missing term in the variance).
4035:
5127:
Herrnstein, Richard J.; Loveland, Donald H.; Cable, Cynthia (1976). "Natural
Concepts in Pigeons".
4128:
4091:
test is related to a number of other non-parametric statistical procedures. For example, it is equivalent to
3929:
3755:
487:
410:
8289:
7902:
7842:
7779:
7139:
7001:
6991:
6841:
6755:
4394:
4064:
3588:
3094:
8050:
7987:
4137:
372:
Otherwise, if both the dispersions and shapes of the distribution of both samples differ, the Mann–Whitney
8327:
8257:
7742:
7629:
6626:
6523:
6430:
6309:
6208:
5289:
4318:
and his student Donald Ransom
Whitney in 1947. This article discussed alternative hypotheses, including a
188:
8448:
7326:
41:
8492:
8352:
8294:
8237:
8063:
7956:
7865:
7591:
7475:
7334:
7216:
7208:
7023:
6919:
6897:
6856:
6821:
6788:
6734:
6709:
6664:
6603:
6563:
6365:
6188:
6140:
5992:
5961:
Kerby, D.S. (2014). "The simple difference formula: An approach to teaching nonparametric correlation".
5762:
Kruskal, William H. (September 1957). "Historical Notes on the
Wilcoxon Unpaired Two-Sample Test".
5627:
5040:
5009:
Kerby, D.S. (2014). "The simple difference formula: An approach to teaching nonparametric correlation".
933:
statistic can be generalized to a measure of a classifier's separation power for more than two classes:
8431:
7321:
1911:
from first to last crossing the finish line) is as follows, writing T for a tortoise and H for a hare:
3466:
This formula is useful when the data are not available, but when there is a published report, because
1830:
1184:
It is also easily calculated by hand, especially for small samples. There are two ways of doing this.
8275:
7850:
7799:
7775:
7737:
7655:
7634:
7586:
7465:
7443:
7412:
7198:
7149:
7067:
7040:
6996:
6952:
6714:
6490:
6370:
4936:
Wilkinson, Leland (1999). "Statistical methods in psychology journals: Guidelines and explanations".
4537:, See Table 2.1 of Pratt (1964) "Robustness of Some Procedures for the Two-Sample Location Problem."
4306:
3548:
392:
4747:"A Simple Generalisation of the Area Under the ROC Curve for Multiple Class Classification Problems"
8422:
8347:
8270:
7951:
7715:
7708:
7670:
7578:
7558:
7530:
7263:
7129:
7124:
7114:
7106:
6924:
6885:
6775:
6765:
6674:
6453:
6409:
6327:
6252:
6154:
5802:
4389:
4319:
4274:
4183:
4049:
3923:
2126:
1148:
218:
192:
7997:
5436:
Rank and pseudo-rank procedures for independent observations in factorial designs: Using R and SAS
4095:
correlation coefficient if one of the variables is binary (that is, it can only take two values).
3882:), without assuming that the distributions are the same under the null hypothesis (i.e., assuming
8436:
8247:
8101:
7946:
7822:
7719:
7703:
7680:
7457:
7191:
7174:
7134:
7045:
6940:
6902:
6873:
6833:
6793:
6739:
6656:
6342:
6337:
6095:
5980:
5925:
5829:
5779:
5679:
5671:
5600:
5566:
5547:
5380:
5179:
5028:
4988:
4916:
4887:
4608:
4483:
3885:
780:
5326:
5309:
3656:
3532:
test and the t-test do not test the same hypotheses and should be compared with this in mind.
2022:
A measure of the central tendencies of the two groups (means or medians; since the Mann–Whitney
4731:
3243:
An alternative formula for the rank-biserial can be used to calculate it from the Mann–Whitney
8342:
8312:
8304:
8124:
8115:
8040:
7971:
7827:
7812:
7787:
7675:
7616:
7482:
7470:
7096:
7013:
6957:
6880:
6724:
6646:
6425:
6299:
6053:
6014:
6006:
5897:
5882:
5855:
5508:
5448:
5413:
5339:
5331:
5258:
5144:
5109:
4980:
4855:
4832:
4824:
4783:
4712:
4666:
4518:
4013:
1713:{\displaystyle U_{1}+U_{2}=R_{1}-{n_{1}(n_{1}+1) \over 2}+R_{2}-{n_{2}(n_{2}+1) \over 2}.\,\!}
5953:
4230:
8367:
8322:
8086:
8073:
7966:
7941:
7875:
7807:
7685:
7293:
7186:
7119:
7032:
6979:
6798:
6669:
6463:
6262:
6229:
6103:
6087:
6045:
5970:
5941:
5915:
5819:
5811:
5771:
5663:
5590:
5582:
5539:
5498:
5488:
5440:
5405:
5372:
5361:(1981). "Rank Transformations as a Bridge Between Parametric and Nonparametric Statistics".
5321:
5250:
5210:
5171:
5136:
5101:
5072:
5018:
4972:
4945:
4816:
4758:
4704:
4656:
4646:
4598:
4565:
4508:
4492:
4452:
4434:
3697:
3179:
1156:
121:
6067:
6028:
5937:
5869:
4504:
4448:
3277:
3250:
3201:
2980:
2629:
where the left side is simply the variance and the right side is the adjustment for ties,
1559:
is the one used when consulting significance tables. The sum of the two values is given by
1235:
1204:
8284:
8028:
7890:
7817:
7492:
7366:
7339:
7316:
7285:
6912:
6907:
6861:
6591:
6242:
6107:
6063:
6024:
5945:
5933:
5865:
5640:
4500:
4479:-test? On assumptions for hypothesis tests and multiple interpretations of decision rules"
4456:
4444:
4421:"On a Test of Whether one of Two Random Variables is Stochastically Larger than the Other"
3996:
test) if it cannot be assumed that the distributions are equal under the null hypothesis.
1137:
196:
125:
5090:
Grissom RJ (1994). "Statistical analysis of ordinal categorical status after therapies".
4635:"Mann–Whitney test is not just a test of medians: differences in spread can be important"
361:
of the difference in central tendency between the two populations differs from zero. The
8233:
8228:
6691:
6621:
6267:
5949:
5797:
5358:
4661:
4634:
4552:
Divine, George W.; Norton, H. James; Barón, Anna E.; Juarez-Colunga, Elizabeth (2018).
4513:
4474:
4302:
4193:
4157:
2130:
540:
467:
82:
3456:{\displaystyle r=f-(1-f)=2f-1={2U_{1} \over n_{1}n_{2}}-1=1-{2U_{2} \over n_{1}n_{2}}}
8476:
8390:
8357:
8220:
8181:
7992:
7961:
7425:
7379:
6984:
6686:
6513:
6277:
6272:
6075:
5984:
5683:
5604:
5551:
5434:
5183:
5032:
4976:
4804:
4612:
4586:
4116:
3926:
and the Fligner–Policello test. Specifically, under the more general null hypothesis
3922:). To test between those hypotheses, better tests are available. Among those are the
6543:
6125:
6044:. Lecture Notes in Statistics. Vol. 199. New York: Springer. pp. xiv+232.
3615:
test will give very similar results to performing an ordinary parametric two-sample
2090:
A statement that does full justice to the statistical status of the test might run,
8332:
8265:
8242:
8157:
7487:
6783:
6681:
6616:
6558:
6480:
6435:
5744:
5667:
4369:
4992:
4570:
4553:
4305:
proposed both the one-sample signed rank and the two-sample rank sum test, in a
8375:
8337:
8020:
7921:
7783:
7596:
7563:
7055:
6972:
6967:
6611:
6568:
6548:
6528:
6518:
6287:
5305:
5076:
4364:
4242:
4143:
2939:
560:
6143:– Nonparametric effect size estimators (Copyright 2015 by Karl L. Weunsch)
5105:
4949:
4708:
4603:
7221:
6701:
6401:
6332:
6282:
6257:
6177:
6049:
5920:
5901:
5824:
5444:
5399:
5254:
5140:
4763:
4746:
4439:
4420:
4416:
4315:
4224:
268:
is different (larger, or smaller) than the probability of an observation from
176:
17:
5586:
5512:
5493:
5472:
5409:
5335:
5310:"The Importance of the Normality Assumption in Large Public Health Data Sets"
5262:
4828:
4716:
4651:
7374:
7226:
6846:
6641:
6553:
6538:
6533:
6498:
5214:
5063:
McGraw, K.O.; Wong, J.J. (1992). "A common language effect size statistic".
4177:
1129:
221:(i.e., one can at least say, of any two observations, which is the greater),
160:
71:"Wilcoxon rank-sum test" redirects here. For Wilcoxon signed-rank test, see
5543:
5343:
4984:
4836:
4670:
4522:
4201:
has implementations of this test through several packages. In the package
2400:{\displaystyle \sigma _{U}={\sqrt {n_{1}n_{2}(n_{1}+n_{2}+1) \over 12}}.\,}
5439:. Springer Series in Statistics. Cham: Springer International Publishing.
5113:
6890:
6508:
6385:
6380:
6375:
6347:
5148:
4264:
4256:
1903:
5595:
5503:
3494:, which is the same result as with the simple difference formula above.
8395:
8096:
6099:
5929:
5833:
5783:
5675:
5384:
5175:
4535:
3566:
3098:
1776:
5975:
5023:
4805:"Effect size estimates: Current use, calculations, and interpretation"
4038:
has been suggested as an appropriate non-parametric equivalent to the
8317:
7298:
7272:
7252:
6503:
6294:
5567:"Psychologists Should Use Brunner–Munzel's Instead of Mann–Whitney's
5473:"Psychologists Should Use Brunner–Munzel's Instead of Mann–Whitney's
4820:
4496:
4218:
4112:
4039:
3626:
Relative efficiencies of the Mann–Whitney test versus the two-sample
3616:
3565:-test to spuriously indicate significance because of the presence of
3525:
366:
6091:
5815:
5775:
5619:
5376:
4803:
Fritz, Catherine O.; Morris, Peter E.; Richler, Jennifer J. (2012).
4255:(Stata Corporation, College Station, TX) implements the test in its
3071:
4587:"What Hypotheses do "Nonparametric" Two-Group Tests Actually Test?"
3521:
4282:
4252:
4171:
3522:
Mann–Whitney U test#Assumptions and formal statement of hypotheses
3191:) minus its complement (i.e.: the proportion that is unfavorable (
2918:
used in the normal approximation is the mean of the two values of
1895:
1197:
larger). Count 0.5 for any ties. The sum of wins and ties is
3105:
by its maximum value for the given sample sizes, which is simply
1065:
will always be zero but, unlike in the two-class case, generally
6237:
4554:"The Wilcoxon–Mann–Whitney Procedure Fails as a Test of Medians"
4270:
4236:
4154:
statistic for a Wilcoxon two-sample, paired, or one-sample test.
2938:
It is a widely recommended practice for scientists to report an
1907:
557:, and both samples independent of each other. The corresponding
8206:
7773:
7520:
6819:
6589:
6206:
6150:
4004:
If one desires a simple shift interpretation, the Mann–Whitney
1155:
Alternatively, the null distribution can be approximated using
1006:{\displaystyle M={1 \over c(c-1)}\sum \mathrm {AUC} _{k,\ell }}
1827:
is the product of the sample sizes for the two samples (i.e.:
1166:, such as the sum of ranks in one of the samples, rather than
26:
6146:
6141:
Brief guide by experimental psychologist Karl L. Weunsch
5129:
Journal of Experimental Psychology: Animal Behavior Processes
4782:. New Jersey: Prentice Hall International, INC. p. 147.
2958:
One method of reporting the effect size for the Mann–Whitney
1043:
considers only the ranking of the items belonging to classes
5433:
Brunner, Edgar; Bathke, Arne C.; Konietschke, Frank (2018).
3174:
A method of reporting the effect size for the Mann–Whitney
1144:
In the case of small samples, the distribution is tabulated
912:{\displaystyle \mathrm {AUC} _{1}={U_{1} \over n_{1}n_{2}}}
391:, although both are nonparametric and involve summation of
195:
than the other, there are many other ways to formulate the
5575:
Advances in Methods and Practices in Psychological Science
5481:
Advances in Methods and Practices in Psychological Science
3577:
control when data are both heteroscedastic and non-normal.
3157:
of 0.5 represents complete overlap. The usefulness of the
4965:
Biological Reviews of the Cambridge Philosophical Society
2026:
test is an ordinal test, medians are usually recommended)
1147:
For sample sizes above ~20, approximation using the
5294:, John Wiley & Sons, 1980 (2nd Edition), pp. 225–226
4473:
Fay, Michael P.; Proschan, Michael A. (2010).
4056:
test in case of violated assumption of exchangeability.
1528:{\displaystyle U_{2}=R_{2}-{n_{2}(n_{2}+1) \over 2}\,\!}
1395:{\displaystyle U_{1}=R_{1}-{n_{1}(n_{1}+1) \over 2}\,\!}
40:
It has been suggested that portions of this article be
4267:(Cytel Software Corporation, Cambridge, Massachusetts)
2926:-statistic calculated will be same whichever value of
706:
611:
231:, the distributions of both populations are identical.
199:
and alternative hypotheses such that the Mann–Whitney
5800:(1945). "Individual comparisons by ranking methods".
4697:
Quarterly Journal of the Royal Meteorological Society
3932:
3888:
3823:
3758:
3700:
3659:
3310:
3280:
3253:
3204:
3013:
2983:
2674:
2423:
2313:
2246:
2141:
1833:
1567:
1455:
1322:
1238:
1207:
942:
851:
783:
573:
543:
490:
470:
413:
383:
test / Wilcoxon rank-sum test is not the same as the
179:
and Donald Ransom Whitney developed the Mann–Whitney
85:
8059:
Autoregressive conditional heteroskedasticity (ARCH)
4880:
Nonparametric statistics for the behavioral sciences
8366:
8303:
8256:
8219:
8174:
8156:
8123:
8114:
8072:
8019:
7980:
7929:
7920:
7841:
7798:
7728:
7694:
7648:
7615:
7577:
7544:
7456:
7365:
7284:
7239:
7207:
7160:
7105:
7031:
7022:
6832:
6774:
6748:
6700:
6655:
6602:
6489:
6444:
6418:
6400:
6356:
6308:
6228:
6219:
5162:Cureton, E.E. (1956). "Rank-biserial correlation".
3557:As it compares the sums of ranks, the Mann–Whitney
2187:{\displaystyle z={\frac {U-m_{U}}{\sigma _{U}}},\,}
2033:(perhaps with some measure of effect size, such as
6011:Nonparametrics: Statistical methods based on ranks
4909:Nonparametrics: Statistical Methods Based on Ranks
4745:Hand, David J.; Till, Robert J. (2001).
4027:-test on the transformed data, the version of the
3983:
3914:
3874:
3809:
3752:test is not valid for testing the null hypothesis
3714:
3680:
3455:
3293:
3266:
3222:
3059:
2996:
2835:
2618:
2399:
2295:
2186:
1969:the sum of the ranks achieved by the tortoises is
1869:
1712:
1527:
1394:
1251:
1220:
1005:
911:
809:
763:
549:
529:
476:
452:
260:The probability of an observation from population
91:
5879:Nonparametric Statistics: A Step-by-Step Approach
4468:
4466:
4411:
4409:
1709:
1524:
1391:
5727:"MannWhitneyUTest (Apache Commons Math 3.3 API)"
5231:, San Diego, CA: GraphPad Software, 2007, p. 123
4539:Journal of the American Statistical Association.
2296:{\displaystyle m_{U}={\frac {n_{1}n_{2}}{2}},\,}
2034:
245:Under the general formulation, the test is only
7607:Multivariate adaptive regression splines (MARS)
5764:Journal of the American Statistical Association
4192:has an implementation of this test provided by
4170:has an implementation of this test provided by
4127:'s statistics base-package implements the test
2646:is the total number of unique ranks with ties.
822:Area-under-curve (AUC) statistic for ROC curves
6042:: An approach based on spatial signs and ranks
3093:and widely used in studies of categorization (
2038:
1989:The sum of the ranks achieved by the hares is
349:, we can interpret a significant Mann–Whitney
206:A very general formulation is to assume that:
171:Assumptions and formal statement of hypotheses
6162:
6038:Multivariate nonparametric methods with
5620:"Violation of Proportional Odds is Not Fatal"
5093:Journal of Consulting and Clinical Psychology
5058:
5056:
4245:(StatsDirect Ltd, Manchester, UK) implements
3603:test is considerably more efficient than the
1162:Some books tabulate statistics equivalent to
922:Note that this is the same definition as the
8:
5997:: CS1 maint: DOI inactive as of June 2024 (
5571:Test as the Default Nonparametric Procedure"
5477:Test as the Default Nonparametric Procedure"
5045:: CS1 maint: DOI inactive as of June 2024 (
5004:
5002:
4103:In many software packages, the Mann–Whitney
3101:), and elsewhere, is calculated by dividing
3072:area under the curve (AUC) for the ROC curve
3060:{\displaystyle f={U_{1} \over n_{1}n_{2}}\,}
1947:. Note that the sum of these two values for
241:is that the distributions are not identical.
5850:Hettmansperger, T.P.; McKean, J.W. (1998).
5327:10.1146/annurev.publhealth.23.100901.140546
4921:: CS1 maint: numeric names: authors list (
4892:: CS1 maint: numeric names: authors list (
4809:Journal of Experimental Psychology: General
4735:, Pattern Recognition Letters, 27, 861–874.
4385:Kruskal–Wallis one-way analysis of variance
3875:{\displaystyle P(Y>X)+0.5P(Y=X)\neq 0.5}
2649:A more computationally-efficient form with
2014:In reporting the results of a Mann–Whitney
1100:) pairs, in effect using the average of AUC
8216:
8203:
8120:
7926:
7795:
7770:
7541:
7517:
7245:
7028:
6829:
6816:
6599:
6586:
6225:
6216:
6203:
6169:
6155:
6147:
4850:Myles Hollander; Douglas A. Wolfe (1999).
3624:
2946:Proportion of concordance out of all pairs
210:All the observations from both groups are
5974:
5919:
5823:
5594:
5502:
5492:
5325:
5022:
4907:Lehmann, Erich; D'Abrera, Howard (1975).
4762:
4660:
4650:
4602:
4569:
4512:
4438:
3931:
3906:
3893:
3887:
3822:
3757:
3704:
3699:
3670:
3664:
3658:
3515:test tests a null hypothesis of that the
3444:
3434:
3422:
3412:
3388:
3378:
3366:
3356:
3309:
3285:
3279:
3258:
3252:
3203:
3056:
3047:
3037:
3026:
3020:
3012:
2988:
2982:
2791:
2778:
2773:
2760:
2749:
2742:
2707:
2697:
2690:
2688:
2679:
2673:
2615:
2575:
2562:
2557:
2544:
2533:
2523:
2513:
2506:
2482:
2469:
2456:
2446:
2439:
2437:
2428:
2422:
2396:
2371:
2358:
2345:
2335:
2327:
2318:
2312:
2292:
2277:
2267:
2260:
2251:
2245:
2183:
2172:
2161:
2148:
2140:
1861:
1851:
1838:
1832:
1708:
1684:
1671:
1664:
1655:
1627:
1614:
1607:
1598:
1585:
1572:
1566:
1523:
1502:
1489:
1482:
1473:
1460:
1454:
1426:is the sum of the ranks in sample 1.
1390:
1369:
1356:
1349:
1340:
1327:
1321:
1243:
1237:
1232:for the other set is the converse (i.e.:
1212:
1206:
991:
980:
949:
941:
900:
890:
879:
873:
864:
853:
850:
801:
788:
782:
755:
726:
713:
705:
696:
686:
673:
660:
631:
618:
610:
601:
591:
578:
572:
542:
519:
514:
495:
489:
469:
442:
437:
418:
412:
264:exceeding an observation from population
136:from two populations, the probability of
84:
68:Nonparametric test of the null hypothesis
5852:Robust nonparametric statistical methods
5466:
5464:
3595:or about 0.95 when compared to the
5902:"Estimation of location based on ranks"
5308:; Emerson, Scott; Chen, Lu (May 2002).
4873:
4871:
4405:
3984:{\displaystyle P(Y>X)+0.5P(Y=X)=0.5}
3810:{\displaystyle P(Y>X)+0.5P(Y=X)=0.5}
3583:When normality holds, the Mann–Whitney
2950:The following measures are equivalent.
2922:. Therefore, the absolute value of the
2215:are the mean and standard deviation of
2117:Normal approximation and tie correction
1128:The test involves the calculation of a
929:Because of its probabilistic form, the
530:{\displaystyle Y_{1},\ldots ,Y_{n_{2}}}
453:{\displaystyle X_{1},\ldots ,X_{n_{1}}}
8133:Kaplan–Meier estimator (product limit)
5990:
5636:
5625:
5038:
4914:
4885:
4020:-test may give more reliable results.
3301:) and the sample sizes of each group:
1280:, where the unadjusted ranks would be
5243:The Journal of Experimental Education
5203:European Journal of Social Psychology
4067:, allowing for covariate-adjustment.
1419:is the sample size for sample 1, and
44:out into another article titled
7:
8443:
8143:Accelerated failure time (AFT) model
5877:Corder, G.W.; Foreman, D.I. (2014).
5618:Harrell, Frank (20 September 2020).
5526:Kasuya, Eiiti (2001). "Mann–Whitney
3817:against the alternative hypothesis
1302:is the total number of observations.
8455:
7738:Analysis of variance (ANOVA, anova)
5704:. The Scipy community. 24 July 2015
5398:Vaart, A. W. van der (1998-10-13).
4326:satisfied the pointwise inequality
1890:Illustration of calculation methods
365:for this two-sample problem is the
128:that, for randomly selected values
7833:Cochran–Mantel–Haenszel statistics
6459:Pearson product-moment correlation
5530:test when variances are unequal".
5291:Practical Nonparametric Statistics
4854:(2 ed.). Wiley-Interscience.
4419:; Whitney, Donald R. (1947).
1020:is the number of classes, and the
987:
984:
981:
860:
857:
854:
25:
5907:Annals of Mathematical Statistics
4852:Nonparametric Statistical Methods
4426:Annals of Mathematical Statistics
4324:cumulative distribution functions
4052:, outperforming the Mann–Whitney
3638:equals a number of distributions
834:receiver operating characteristic
8454:
8442:
8430:
8417:
8416:
4977:10.1111/j.1469-185X.2007.00027.x
2018:test, it is important to state:
1870:{\displaystyle U_{i}=n_{1}n_{2}}
249:when the following occurs under
155:Nonparametric tests used on two
31:
8092:Least-squares spectral analysis
5745:"JuliaStats/HypothesisTests.jl"
5277:Elements of Large Sample Theory
4732:An introduction to ROC analysis
2414:should be adjusted as follows:
1877:). In such a case, the "other"
1136:, whose distribution under the
191:being that one distribution is
144:is equal to the probability of
7073:Mean-unbiased minimum-variance
5668:10.1080/00031305.2000.10474513
5404:. Cambridge University Press.
5314:Annual Review of Public Health
4215:(SAS Institute Inc., Cary, NC)
4207:pvalue(MannWhitneyUTest(X, Y))
4063:test is a special case of the
3972:
3960:
3948:
3936:
3863:
3851:
3839:
3827:
3798:
3786:
3774:
3762:
3477:is the smaller of the two, so
3335:
3323:
2817:
2805:
2797:
2766:
2736:
2724:
2638:is the number of ties for the
2604:
2592:
2581:
2550:
2494:
2462:
2383:
2351:
1696:
1677:
1639:
1620:
1514:
1495:
1381:
1362:
970:
958:
738:
719:
643:
624:
564:is defined as the smaller of:
357:test as assessing whether the
272:exceeding an observation from
1:
8386:Geographic information system
7602:Simultaneous equations models
5702:SciPy v0.16.0 Reference Guide
4778:Zar, Jerrold H. (1998).
4684:Boston University (SPH), 2017
4571:10.1080/00031305.2017.1305291
4221:(MathSoft, Inc., Seattle, WA)
3622:on the rankings of the data.
3561:test is less likely than the
3492:= 1 – (2×10) / (10×10) = 0.80
1984:= 32 − (6×7)/2 = 32 − 21 = 11
1159:and Monte Carlo simulations.
203:test will give a valid test.
183:test under the assumption of
7569:Coefficient of determination
7180:Uniformly most powerful test
6121:Table of critical values of
4048:A more powerful test is the
3569:. However, the Mann–Whitney
3089:that is linearly related to
2010:Example statement of results
1991:11 + 10 + 9 + 8 + 7 + 1 = 46
830:statistic is related to the
8138:Proportional hazards models
8082:Spectral density estimation
8064:Vector autoregression (VAR)
7498:Maximum posterior estimator
6730:Randomized controlled trial
5077:10.1037/0033-2909.111.2.361
4882:. McGraw-Hill. p. 121.
4301:In a single paper in 1945,
4285:implements the test in its
4273:implements the test in its
4227:(StatSoft, Inc., Tulsa, OK)
4160:implements the test in its
4148:package will calculate the
3915:{\displaystyle F_{1}=F_{2}}
2954:Common language effect size
2035:common language effect size
1971:12 + 6 + 5 + 4 + 3 + 2 = 32
924:common language effect size
810:{\displaystyle R_{1},R_{2}}
234:The alternative hypothesis
217:The responses are at least
8509:
7898:Multivariate distributions
6318:Average absolute deviation
5698:"scipy.stats.mannwhitneyu"
5357:Conover, William J.;
5288:Conover, William J.;
5227:Motulsky, Harvey J.;
5106:10.1037/0022-006X.62.2.281
4950:10.1037/0003-066X.54.8.594
4709:10.1256/003590002320603584
4604:10.1177/1536867X1201200202
4475:"Wilcoxon–Mann–Whitney or
4287:Wilcoxon–Mann–Whitney Test
4121:in its Statistics Toolbox.
3681:{\displaystyle \pi ^{2}/9}
3543:test is preferable to the
3487:. This formula then gives
3178:test is with a measure of
1961:Using the indirect method:
1779:, we find that the sum is
1278:(1, 3.5, 3.5, 3.5, 3.5, 6)
224:Under the null hypothesis
115:Wilcoxon–Mann–Whitney test
70:
47:Probability of superiority
8412:
8215:
8202:
7886:Structural equation model
7794:
7769:
7540:
7516:
7248:
7222:Score/Lagrange multiplier
6828:
6815:
6637:Sample size determination
6598:
6585:
6215:
6202:
6184:
6050:10.1007/978-1-4419-0468-3
5656:The American Statistician
5471:Karch, Julian D. (2021).
5445:10.1007/978-3-030-02914-2
5364:The American Statistician
5255:10.1080/00220979809598344
5141:10.1037/0097-7403.2.4.285
4911:. Holden-Day. p. 20.
4558:The American Statistician
4380:Wilcoxon signed-rank test
3587:test has an (asymptotic)
3170:Rank-biserial correlation
2969:The relationship between
2942:for an inferential test.
2039:rank-biserial correlation
1898:is dissatisfied with his
1177:test is included in most
165:Wilcoxon signed-rank test
73:Wilcoxon signed-rank test
8488:Nonparametric statistics
8381:Environmental statistics
7903:Elliptical distributions
7696:Generalized linear model
7625:Simple linear regression
7395:Hodges–Lehmann estimator
6852:Probability distribution
6761:Stochastic approximation
6323:Coefficient of variation
5963:Comprehensive Psychology
5587:10.1177/2515245921999602
5494:10.1177/2515245921999602
5410:10.1017/cbo9780511802256
5279:, Springer, 1999, p. 176
5275:Lehamnn, Erich L.;
5011:Comprehensive Psychology
4652:10.1136/bmj.323.7309.391
4186:(SPSS Inc., Chicago, IL)
4180:(SPSS Inc., Chicago, IL)
4099:Software implementations
3547:-test when the data are
3517:probability distribution
3503:Comparison to Student's
3070:This is the same as the
8041:Cross-correlation (XCF)
7649:Non-standard predictors
7083:Lehmann–Scheffé theorem
6756:Adaptive clinical trial
5979:(inactive 2024-06-02).
5921:10.1214/aoms/1177704172
5215:10.1002/ejsp.2420020412
5027:(inactive 2024-06-02).
4878:Siegal, Sidney (1956).
4780:Biostatistical Analysis
4764:10.1023/A:1010920819831
4440:10.1214/aoms/1177730491
4395:Proportional odds model
4375:Kolmogorov–Smirnov test
4133:in its "stats" package.
4078:Related test statistics
4072:Kolmogorov–Smirnov test
4065:proportional odds model
3744:Different distributions
3498:Relation to other tests
3095:discrimination learning
2047:The significance level.
1915:T H H H H H T T T T T H
1092:measure sums over all (
363:Hodges–Lehmann estimate
359:Hodges–Lehmann estimate
8437:Mathematics portal
8258:Engineering statistics
8166:Nelson–Aalen estimator
7743:Analysis of covariance
7630:Ordinary least squares
7554:Pearson product-moment
6958:Statistical functional
6869:Empirical distribution
6702:Controlled experiments
6431:Frequency distribution
6209:Descriptive statistics
6131:Interactive calculator
6007:Lehmann, Erich L.
5954:euclid.aoms/1177704172
5635:Cite journal requires
5565:Karch, Julian (2021).
5544:10.1006/anbe.2001.1691
5065:Psychological Bulletin
4585:Conroy, Ronán (2012).
3985:
3916:
3876:
3811:
3716:
3715:{\displaystyle 3/\pi }
3682:
3502:
3457:
3295:
3268:
3224:
3061:
2998:
2837:
2765:
2620:
2549:
2401:
2297:
2188:
1906:was found to beat one
1871:
1714:
1529:
1396:
1253:
1222:
1007:
913:
811:
765:
551:
537:an i.i.d. sample from
531:
478:
454:
193:stochastically greater
189:alternative hypothesis
111:Wilcoxon rank-sum test
93:
8353:Population statistics
8295:System identification
8029:Autocorrelation (ACF)
7957:Exponential smoothing
7871:Discriminant analysis
7866:Canonical correlation
7730:Partition of variance
7592:Regression validation
7436:(Jonckheere–Terpstra)
7335:Likelihood-ratio test
7024:Frequentist inference
6936:Location–scale family
6857:Sampling distribution
6822:Statistical inference
6789:Cross-sectional study
6776:Observational studies
6735:Randomized experiment
6564:Stem-and-leaf display
6366:Central limit theorem
5401:Asymptotic Statistics
4938:American Psychologist
4729:Fawcett, Tom (2006);
4233:(Unistat Ltd, London)
4045:for equal variances.
3986:
3917:
3877:
3812:
3717:
3683:
3458:
3296:
3294:{\displaystyle U_{2}}
3269:
3267:{\displaystyle U_{1}}
3225:
3223:{\displaystyle r=f-u}
3062:
2999:
2997:{\displaystyle U_{1}}
2973:and the Mann–Whitney
2838:
2745:
2621:
2529:
2402:
2298:
2189:
1986:(same as method one).
1919:What is the value of
1872:
1823:The maximum value of
1715:
1545:The smaller value of
1530:
1397:
1254:
1252:{\displaystyle U_{2}}
1228:) for the first set.
1223:
1221:{\displaystyle U_{1}}
1008:
914:
812:
766:
552:
532:
479:
455:
103:Mann–Whitney–Wilcoxon
94:
8276:Probabilistic design
7861:Principal components
7704:Exponential families
7656:Nonlinear regression
7635:General linear model
7597:Mixed effects models
7587:Errors and residuals
7564:Confounding variable
7466:Bayesian probability
7444:Van der Waerden test
7434:Ordered alternative
7199:Multiple comparisons
7078:Rao–Blackwellization
7041:Estimating equations
6997:Statistical distance
6715:Factorial experiment
6248:Arithmetic-Geometric
6137:and its significance
5359:Iman, Ronald L.
4307:test of significance
3930:
3886:
3821:
3756:
3698:
3657:
3573:test may have worse
3308:
3278:
3251:
3202:
3011:
2981:
2672:
2421:
2311:
2244:
2139:
2129:. In that case, the
2127:normally distributed
1831:
1565:
1453:
1320:
1267:For larger samples:
1236:
1205:
1179:statistical packages
940:
849:
781:
571:
541:
488:
468:
411:
83:
8348:Official statistics
8271:Methods engineering
7952:Seasonal adjustment
7720:Poisson regressions
7640:Bayesian regression
7579:Regression analysis
7559:Partial correlation
7531:Regression analysis
7130:Prediction interval
7125:Likelihood interval
7115:Confidence interval
7107:Interval estimation
7068:Unbiased estimators
6886:Model specification
6766:Up-and-down designs
6454:Partial correlation
6410:Index of dispersion
6328:Interquartile range
6036:Oja, Hannu (2010).
5803:Biometrics Bulletin
4633:Hart, Anna (2001).
4417:Mann, Henry B.
4390:Brunner Munzel test
4320:stochastic ordering
4247:all common variants
4239:(SPSS Inc, Chicago)
4205:, this is found as
4082:
4050:Brunner-Munzel test
4036:Brown–Forsythe test
3991:, the Mann–Whitney
3639:
3085:A statistic called
2783:
2567:
2121:For large samples,
1149:normal distribution
1088:, which is why the
395:. The Mann–Whitney
187:responses with the
148:being greater than
140:being greater than
8368:Spatial statistics
8248:Medical statistics
8148:First hitting time
8102:Whittle likelihood
7753:Degrees of freedom
7748:Multivariate ANOVA
7681:Heteroscedasticity
7493:Bayesian estimator
7458:Bayesian inference
7307:Kolmogorov–Smirnov
7192:Randomization test
7162:Testing hypotheses
7135:Tolerance interval
7046:Maximum likelihood
6941:Exponential family
6874:Density estimation
6834:Statistical theory
6794:Natural experiment
6740:Scientific control
6657:Survey methodology
6343:Standard deviation
5825:10338.dmlcz/135688
5731:commons.apache.org
5176:10.1007/BF02289138
4703:(584): 2145–2166.
4484:Statistics Surveys
4203:HypothesisTests.jl
3981:
3912:
3872:
3807:
3712:
3678:
3625:
3524:). In contrast, a
3453:
3291:
3264:
3220:
3057:
2994:
2833:
2769:
2616:
2553:
2397:
2293:
2184:
2131:standardized value
1900:classic experiment
1867:
1710:
1525:
1392:
1282:(1, 2, 3, 4, 5, 6)
1274:(3, 5, 5, 5, 5, 8)
1249:
1218:
1003:
909:
807:
761:
746:
651:
547:
527:
474:
450:
89:
8483:Statistical tests
8470:
8469:
8408:
8407:
8404:
8403:
8343:National accounts
8313:Actuarial science
8305:Social statistics
8198:
8197:
8194:
8193:
8190:
8189:
8125:Survival function
8110:
8109:
7972:Granger causality
7813:Contingency table
7788:Survival analysis
7765:
7764:
7761:
7760:
7617:Linear regression
7512:
7511:
7508:
7507:
7483:Credible interval
7452:
7451:
7235:
7234:
7051:Method of moments
6920:Parametric family
6881:Statistical model
6811:
6810:
6807:
6806:
6725:Random assignment
6647:Statistical power
6581:
6580:
6577:
6576:
6426:Contingency table
6396:
6395:
6263:Generalized/power
6076:Sen, Pranab Kumar
6059:978-1-4419-0467-6
6020:978-0-387-35212-1
5976:10.2466/11.IT.3.1
5861:978-0-340-54937-7
5454:978-3-030-02912-8
5419:978-0-511-80225-6
5024:10.2466/11.IT.3.1
4789:978-0-13-082390-8
4645:(7309): 391–393.
4087:The Mann–Whitney
4059:The Mann–Whitney
4014:unequal variances
3748:The Mann–Whitney
3741:
3740:
3611:The Mann–Whitney
3539:The Mann–Whitney
3511:The Mann–Whitney
3451:
3395:
3054:
3004:) is as follows:
2828:
2821:
2717:
2682:
2610:
2608:
2501:
2431:
2391:
2390:
2287:
2178:
2125:is approximately
1881:would be 0.
1775:, and doing some
1703:
1646:
1521:
1388:
1308:is then given by:
1173:The Mann–Whitney
1157:permutation tests
1132:, usually called
974:
907:
745:
650:
550:{\displaystyle Y}
477:{\displaystyle X}
379:The Mann–Whitney
101:(also called the
92:{\displaystyle U}
64:
63:
59:
16:(Redirected from
8500:
8458:
8457:
8446:
8445:
8435:
8434:
8420:
8419:
8323:Crime statistics
8217:
8204:
8121:
8087:Fourier analysis
8074:Frequency domain
8054:
8001:
7967:Structural break
7927:
7876:Cluster analysis
7823:Log-linear model
7796:
7771:
7712:
7686:Homoscedasticity
7542:
7518:
7437:
7429:
7421:
7420:(Kruskal–Wallis)
7405:
7390:
7345:Cross validation
7330:
7312:Anderson–Darling
7259:
7246:
7217:Likelihood-ratio
7209:Parametric tests
7187:Permutation test
7170:1- & 2-tails
7061:Minimum distance
7033:Point estimation
7029:
6980:Optimal decision
6931:
6830:
6817:
6799:Quasi-experiment
6749:Adaptive designs
6600:
6587:
6464:Rank correlation
6226:
6217:
6204:
6171:
6164:
6157:
6148:
6111:
6071:
6032:
6002:
5996:
5988:
5978:
5957:
5923:
5892:
5873:
5838:
5837:
5827:
5794:
5788:
5787:
5770:(279): 356–360.
5759:
5753:
5752:
5741:
5735:
5734:
5723:
5717:
5716:
5711:
5709:
5694:
5688:
5687:
5651:
5645:
5644:
5638:
5633:
5631:
5623:
5615:
5609:
5608:
5598:
5562:
5556:
5555:
5538:(6): 1247–1249.
5532:Animal Behaviour
5523:
5517:
5516:
5506:
5496:
5468:
5459:
5458:
5430:
5424:
5423:
5395:
5389:
5388:
5354:
5348:
5347:
5329:
5304:Lumley, Thomas;
5301:
5295:
5286:
5280:
5273:
5267:
5266:
5238:
5232:
5229:Statistics Guide
5225:
5219:
5218:
5194:
5188:
5187:
5159:
5153:
5152:
5124:
5118:
5117:
5087:
5081:
5080:
5060:
5051:
5050:
5044:
5036:
5026:
5006:
4997:
4996:
4960:
4954:
4953:
4933:
4927:
4926:
4920:
4912:
4904:
4898:
4897:
4891:
4883:
4875:
4866:
4865:
4847:
4841:
4840:
4821:10.1037/a0024338
4800:
4794:
4793:
4775:
4769:
4768:
4766:
4751:Machine Learning
4742:
4736:
4727:
4721:
4720:
4692:
4686:
4681:
4675:
4674:
4664:
4654:
4630:
4624:
4623:
4621:
4619:
4606:
4582:
4576:
4575:
4573:
4549:
4543:
4533:
4527:
4526:
4516:
4497:10.1214/09-SS051
4470:
4461:
4460:
4442:
4413:
4354:
4277:
4259:
4208:
4204:
4163:
4153:
4146:
4140:
4131:
4119:
3990:
3988:
3987:
3982:
3921:
3919:
3918:
3913:
3911:
3910:
3898:
3897:
3881:
3879:
3878:
3873:
3816:
3814:
3813:
3808:
3721:
3719:
3718:
3713:
3708:
3687:
3685:
3684:
3679:
3674:
3669:
3668:
3640:
3594:
3493:
3486:
3462:
3460:
3459:
3454:
3452:
3450:
3449:
3448:
3439:
3438:
3428:
3427:
3426:
3413:
3396:
3394:
3393:
3392:
3383:
3382:
3372:
3371:
3370:
3357:
3300:
3298:
3297:
3292:
3290:
3289:
3273:
3271:
3270:
3265:
3263:
3262:
3239:
3229:
3227:
3226:
3221:
3180:rank correlation
3144:
3120:
3066:
3064:
3063:
3058:
3055:
3053:
3052:
3051:
3042:
3041:
3031:
3030:
3021:
3003:
3001:
3000:
2995:
2993:
2992:
2917:
2901:
2873:Note that since
2865:
2842:
2840:
2839:
2834:
2829:
2827:
2823:
2822:
2820:
2800:
2796:
2795:
2782:
2777:
2764:
2759:
2743:
2718:
2713:
2712:
2711:
2702:
2701:
2691:
2689:
2684:
2683:
2680:
2665:factored out is
2664:
2625:
2623:
2622:
2617:
2611:
2609:
2607:
2584:
2580:
2579:
2566:
2561:
2548:
2543:
2528:
2527:
2518:
2517:
2507:
2502:
2497:
2487:
2486:
2474:
2473:
2461:
2460:
2451:
2450:
2440:
2438:
2433:
2432:
2429:
2406:
2404:
2403:
2398:
2392:
2386:
2376:
2375:
2363:
2362:
2350:
2349:
2340:
2339:
2329:
2328:
2323:
2322:
2302:
2300:
2299:
2294:
2288:
2283:
2282:
2281:
2272:
2271:
2261:
2256:
2255:
2193:
2191:
2190:
2185:
2179:
2177:
2176:
2167:
2166:
2165:
2149:
2108:
2101:
2085:
2078:
2061:
2044:The sample sizes
2002:
1992:
1985:
1972:
1957:
1953:
1946:
1936:
1876:
1874:
1873:
1868:
1866:
1865:
1856:
1855:
1843:
1842:
1809:
1774:
1754:
1719:
1717:
1716:
1711:
1704:
1699:
1689:
1688:
1676:
1675:
1665:
1660:
1659:
1647:
1642:
1632:
1631:
1619:
1618:
1608:
1603:
1602:
1590:
1589:
1577:
1576:
1534:
1532:
1531:
1526:
1522:
1517:
1507:
1506:
1494:
1493:
1483:
1478:
1477:
1465:
1464:
1401:
1399:
1398:
1393:
1389:
1384:
1374:
1373:
1361:
1360:
1350:
1345:
1344:
1332:
1331:
1297:
1283:
1279:
1275:
1258:
1256:
1255:
1250:
1248:
1247:
1227:
1225:
1224:
1219:
1217:
1216:
1087:
1012:
1010:
1009:
1004:
1002:
1001:
990:
975:
973:
950:
918:
916:
915:
910:
908:
906:
905:
904:
895:
894:
884:
883:
874:
869:
868:
863:
816:
814:
813:
808:
806:
805:
793:
792:
770:
768:
767:
762:
760:
759:
747:
741:
731:
730:
718:
717:
707:
701:
700:
691:
690:
678:
677:
665:
664:
652:
646:
636:
635:
623:
622:
612:
606:
605:
596:
595:
583:
582:
556:
554:
553:
548:
536:
534:
533:
528:
526:
525:
524:
523:
500:
499:
483:
481:
480:
475:
459:
457:
456:
451:
449:
448:
447:
446:
423:
422:
348:
315:
295:
159:samples are the
122:statistical test
98:
96:
95:
90:
55:
35:
34:
27:
21:
8508:
8507:
8503:
8502:
8501:
8499:
8498:
8497:
8473:
8472:
8471:
8466:
8429:
8400:
8362:
8299:
8285:quality control
8252:
8234:Clinical trials
8211:
8186:
8170:
8158:Hazard function
8152:
8106:
8068:
8052:
8015:
8011:Breusch–Godfrey
7999:
7976:
7916:
7891:Factor analysis
7837:
7818:Graphical model
7790:
7757:
7724:
7710:
7690:
7644:
7611:
7573:
7536:
7535:
7504:
7448:
7435:
7427:
7419:
7403:
7388:
7367:Rank statistics
7361:
7340:Model selection
7328:
7286:Goodness of fit
7280:
7257:
7231:
7203:
7156:
7101:
7090:Median unbiased
7018:
6929:
6862:Order statistic
6824:
6803:
6770:
6744:
6696:
6651:
6594:
6592:Data collection
6573:
6485:
6440:
6414:
6392:
6352:
6304:
6221:Continuous data
6211:
6198:
6180:
6175:
6118:
6092:10.2307/2527532
6074:
6060:
6035:
6021:
6005:
5989:
5960:
5895:
5889:
5876:
5862:
5849:
5846:
5841:
5816:10.2307/3001968
5798:Wilcoxon, Frank
5796:
5795:
5791:
5776:10.2307/2280906
5761:
5760:
5756:
5743:
5742:
5738:
5725:
5724:
5720:
5707:
5705:
5696:
5695:
5691:
5653:
5652:
5648:
5634:
5624:
5617:
5616:
5612:
5564:
5563:
5559:
5525:
5524:
5520:
5470:
5469:
5462:
5455:
5432:
5431:
5427:
5420:
5397:
5396:
5392:
5377:10.2307/2683975
5356:
5355:
5351:
5303:
5302:
5298:
5287:
5283:
5274:
5270:
5240:
5239:
5235:
5226:
5222:
5196:
5195:
5191:
5161:
5160:
5156:
5126:
5125:
5121:
5089:
5088:
5084:
5062:
5061:
5054:
5037:
5008:
5007:
5000:
4962:
4961:
4957:
4935:
4934:
4930:
4913:
4906:
4905:
4901:
4884:
4877:
4876:
4869:
4862:
4849:
4848:
4844:
4802:
4801:
4797:
4790:
4777:
4776:
4772:
4744:
4743:
4739:
4728:
4724:
4694:
4693:
4689:
4682:
4678:
4632:
4631:
4627:
4617:
4615:
4584:
4583:
4579:
4551:
4550:
4546:
4534:
4530:
4472:
4471:
4464:
4415:
4414:
4407:
4403:
4361:
4348:
4335:
4327:
4296:
4275:
4257:
4206:
4202:
4161:
4149:
4144:
4138:
4136:The R function
4129:
4117:
4101:
4085:
4080:
4016:version of the
4002:
3928:
3927:
3902:
3889:
3884:
3883:
3819:
3818:
3754:
3753:
3746:
3696:
3695:
3660:
3655:
3654:
3592:
3509:
3500:
3488:
3483:
3478:
3476:
3440:
3430:
3429:
3418:
3414:
3384:
3374:
3373:
3362:
3358:
3306:
3305:
3281:
3276:
3275:
3254:
3249:
3248:
3234:
3200:
3199:
3172:
3126:
3119:
3112:
3106:
3083:
3043:
3033:
3032:
3022:
3009:
3008:
2984:
2979:
2978:
2956:
2948:
2936:
2915:
2909:
2903:
2900:
2894:
2887:
2880:
2874:
2864:
2857:
2847:
2801:
2787:
2744:
2723:
2719:
2703:
2693:
2692:
2675:
2670:
2669:
2662:
2656:
2650:
2637:
2585:
2571:
2519:
2509:
2508:
2478:
2465:
2452:
2442:
2441:
2424:
2419:
2418:
2367:
2354:
2341:
2331:
2330:
2314:
2309:
2308:
2273:
2263:
2262:
2247:
2242:
2241:
2236:
2227:
2214:
2205:
2168:
2157:
2150:
2137:
2136:
2119:
2103:
2095:
2080:
2076:
2069:
2063:
2056:
2012:
1999:
1994:
1990:
1982:
1977:
1970:
1955:
1948:
1943:
1938:
1933:
1928:
1892:
1887:
1857:
1847:
1834:
1829:
1828:
1821:
1808:
1802:
1795:
1788:
1782:
1773:
1766:
1756:
1744:
1737:
1731:
1680:
1667:
1666:
1651:
1623:
1610:
1609:
1594:
1581:
1568:
1563:
1562:
1558:
1551:
1498:
1485:
1484:
1469:
1456:
1451:
1450:
1425:
1418:
1365:
1352:
1351:
1336:
1323:
1318:
1317:
1288:
1281:
1277:
1273:
1239:
1234:
1233:
1208:
1203:
1202:
1151:is fairly good.
1138:null hypothesis
1126:
1119:
1109:
1086:
1076:
1066:
1064:
1042:
1032:
979:
954:
938:
937:
896:
886:
885:
875:
852:
847:
846:
832:area under the
824:
797:
784:
779:
778:
751:
722:
709:
708:
692:
682:
669:
656:
627:
614:
613:
597:
587:
574:
569:
568:
539:
538:
515:
510:
491:
486:
485:
466:
465:
438:
433:
414:
409:
408:
405:
338:
327:
321:
297:
277:
255:
240:
230:
173:
126:null hypothesis
81:
80:
76:
69:
60:
36:
32:
23:
22:
15:
12:
11:
5:
8506:
8504:
8496:
8495:
8490:
8485:
8475:
8474:
8468:
8467:
8465:
8464:
8452:
8440:
8426:
8413:
8410:
8409:
8406:
8405:
8402:
8401:
8399:
8398:
8393:
8388:
8383:
8378:
8372:
8370:
8364:
8363:
8361:
8360:
8355:
8350:
8345:
8340:
8335:
8330:
8325:
8320:
8315:
8309:
8307:
8301:
8300:
8298:
8297:
8292:
8287:
8278:
8273:
8268:
8262:
8260:
8254:
8253:
8251:
8250:
8245:
8240:
8231:
8229:Bioinformatics
8225:
8223:
8213:
8212:
8207:
8200:
8199:
8196:
8195:
8192:
8191:
8188:
8187:
8185:
8184:
8178:
8176:
8172:
8171:
8169:
8168:
8162:
8160:
8154:
8153:
8151:
8150:
8145:
8140:
8135:
8129:
8127:
8118:
8112:
8111:
8108:
8107:
8105:
8104:
8099:
8094:
8089:
8084:
8078:
8076:
8070:
8069:
8067:
8066:
8061:
8056:
8048:
8043:
8038:
8037:
8036:
8034:partial (PACF)
8025:
8023:
8017:
8016:
8014:
8013:
8008:
8003:
7995:
7990:
7984:
7982:
7981:Specific tests
7978:
7977:
7975:
7974:
7969:
7964:
7959:
7954:
7949:
7944:
7939:
7933:
7931:
7924:
7918:
7917:
7915:
7914:
7913:
7912:
7911:
7910:
7895:
7894:
7893:
7883:
7881:Classification
7878:
7873:
7868:
7863:
7858:
7853:
7847:
7845:
7839:
7838:
7836:
7835:
7830:
7828:McNemar's test
7825:
7820:
7815:
7810:
7804:
7802:
7792:
7791:
7774:
7767:
7766:
7763:
7762:
7759:
7758:
7756:
7755:
7750:
7745:
7740:
7734:
7732:
7726:
7725:
7723:
7722:
7706:
7700:
7698:
7692:
7691:
7689:
7688:
7683:
7678:
7673:
7668:
7666:Semiparametric
7663:
7658:
7652:
7650:
7646:
7645:
7643:
7642:
7637:
7632:
7627:
7621:
7619:
7613:
7612:
7610:
7609:
7604:
7599:
7594:
7589:
7583:
7581:
7575:
7574:
7572:
7571:
7566:
7561:
7556:
7550:
7548:
7538:
7537:
7534:
7533:
7528:
7522:
7521:
7514:
7513:
7510:
7509:
7506:
7505:
7503:
7502:
7501:
7500:
7490:
7485:
7480:
7479:
7478:
7473:
7462:
7460:
7454:
7453:
7450:
7449:
7447:
7446:
7441:
7440:
7439:
7431:
7423:
7407:
7404:(Mann–Whitney)
7399:
7398:
7397:
7384:
7383:
7382:
7371:
7369:
7363:
7362:
7360:
7359:
7358:
7357:
7352:
7347:
7337:
7332:
7329:(Shapiro–Wilk)
7324:
7319:
7314:
7309:
7304:
7296:
7290:
7288:
7282:
7281:
7279:
7278:
7270:
7261:
7249:
7243:
7241:Specific tests
7237:
7236:
7233:
7232:
7230:
7229:
7224:
7219:
7213:
7211:
7205:
7204:
7202:
7201:
7196:
7195:
7194:
7184:
7183:
7182:
7172:
7166:
7164:
7158:
7157:
7155:
7154:
7153:
7152:
7147:
7137:
7132:
7127:
7122:
7117:
7111:
7109:
7103:
7102:
7100:
7099:
7094:
7093:
7092:
7087:
7086:
7085:
7080:
7065:
7064:
7063:
7058:
7053:
7048:
7037:
7035:
7026:
7020:
7019:
7017:
7016:
7011:
7006:
7005:
7004:
6994:
6989:
6988:
6987:
6977:
6976:
6975:
6970:
6965:
6955:
6950:
6945:
6944:
6943:
6938:
6933:
6917:
6916:
6915:
6910:
6905:
6895:
6894:
6893:
6888:
6878:
6877:
6876:
6866:
6865:
6864:
6854:
6849:
6844:
6838:
6836:
6826:
6825:
6820:
6813:
6812:
6809:
6808:
6805:
6804:
6802:
6801:
6796:
6791:
6786:
6780:
6778:
6772:
6771:
6769:
6768:
6763:
6758:
6752:
6750:
6746:
6745:
6743:
6742:
6737:
6732:
6727:
6722:
6717:
6712:
6706:
6704:
6698:
6697:
6695:
6694:
6692:Standard error
6689:
6684:
6679:
6678:
6677:
6672:
6661:
6659:
6653:
6652:
6650:
6649:
6644:
6639:
6634:
6629:
6624:
6622:Optimal design
6619:
6614:
6608:
6606:
6596:
6595:
6590:
6583:
6582:
6579:
6578:
6575:
6574:
6572:
6571:
6566:
6561:
6556:
6551:
6546:
6541:
6536:
6531:
6526:
6521:
6516:
6511:
6506:
6501:
6495:
6493:
6487:
6486:
6484:
6483:
6478:
6477:
6476:
6471:
6461:
6456:
6450:
6448:
6442:
6441:
6439:
6438:
6433:
6428:
6422:
6420:
6419:Summary tables
6416:
6415:
6413:
6412:
6406:
6404:
6398:
6397:
6394:
6393:
6391:
6390:
6389:
6388:
6383:
6378:
6368:
6362:
6360:
6354:
6353:
6351:
6350:
6345:
6340:
6335:
6330:
6325:
6320:
6314:
6312:
6306:
6305:
6303:
6302:
6297:
6292:
6291:
6290:
6285:
6280:
6275:
6270:
6265:
6260:
6255:
6253:Contraharmonic
6250:
6245:
6234:
6232:
6223:
6213:
6212:
6207:
6200:
6199:
6197:
6196:
6191:
6185:
6182:
6181:
6176:
6174:
6173:
6166:
6159:
6151:
6145:
6144:
6138:
6128:
6117:
6116:External links
6114:
6113:
6112:
6086:(4): 532–552.
6072:
6058:
6033:
6019:
6003:
5958:
5914:(2): 598–611.
5896:Hodges, J.L.;
5893:
5888:978-1118840313
5887:
5874:
5860:
5845:
5842:
5840:
5839:
5789:
5754:
5751:. 30 May 2021.
5736:
5718:
5689:
5646:
5637:|journal=
5610:
5557:
5518:
5460:
5453:
5425:
5418:
5390:
5371:(3): 124–129.
5349:
5320:(1): 151–169.
5296:
5281:
5268:
5233:
5220:
5209:(4): 463–465.
5189:
5170:(3): 287–290.
5154:
5135:(4): 285–302.
5119:
5100:(2): 281–284.
5082:
5071:(2): 361–365.
5052:
4998:
4971:(4): 591–605.
4955:
4944:(8): 594–604.
4928:
4899:
4867:
4861:978-0471190455
4860:
4842:
4795:
4788:
4770:
4757:(2): 171–186.
4737:
4722:
4687:
4676:
4625:
4597:(2): 182–190.
4577:
4564:(3): 278–286.
4544:
4528:
4462:
4404:
4402:
4399:
4398:
4397:
4392:
4387:
4382:
4377:
4372:
4367:
4360:
4357:
4344:
4331:
4303:Frank Wilcoxon
4295:
4292:
4291:
4290:
4280:
4268:
4262:
4250:
4240:
4234:
4228:
4222:
4216:
4210:
4196:
4194:Apache Commons
4187:
4181:
4175:
4165:
4155:
4134:
4122:
4100:
4097:
4084:
4081:
4079:
4076:
4001:
3998:
3980:
3977:
3974:
3971:
3968:
3965:
3962:
3959:
3956:
3953:
3950:
3947:
3944:
3941:
3938:
3935:
3924:Brunner-Munzel
3909:
3905:
3901:
3896:
3892:
3871:
3868:
3865:
3862:
3859:
3856:
3853:
3850:
3847:
3844:
3841:
3838:
3835:
3832:
3829:
3826:
3806:
3803:
3800:
3797:
3794:
3791:
3788:
3785:
3782:
3779:
3776:
3773:
3770:
3767:
3764:
3761:
3745:
3742:
3739:
3738:
3735:
3731:
3730:
3727:
3723:
3722:
3711:
3707:
3703:
3693:
3689:
3688:
3677:
3673:
3667:
3663:
3652:
3648:
3647:
3644:
3609:
3608:
3581:
3578:
3555:
3552:
3537:
3508:
3501:
3499:
3496:
3481:
3474:
3464:
3463:
3447:
3443:
3437:
3433:
3425:
3421:
3417:
3411:
3408:
3405:
3402:
3399:
3391:
3387:
3381:
3377:
3369:
3365:
3361:
3355:
3352:
3349:
3346:
3343:
3340:
3337:
3334:
3331:
3328:
3325:
3322:
3319:
3316:
3313:
3288:
3284:
3261:
3257:
3231:
3230:
3219:
3216:
3213:
3210:
3207:
3171:
3168:
3117:
3110:
3082:
3076:
3068:
3067:
3050:
3046:
3040:
3036:
3029:
3025:
3019:
3016:
2991:
2987:
2977:(specifically
2955:
2952:
2947:
2944:
2935:
2932:
2913:
2907:
2898:
2892:
2885:
2878:
2862:
2855:
2844:
2843:
2832:
2826:
2819:
2816:
2813:
2810:
2807:
2804:
2799:
2794:
2790:
2786:
2781:
2776:
2772:
2768:
2763:
2758:
2755:
2752:
2748:
2741:
2738:
2735:
2732:
2729:
2726:
2722:
2716:
2710:
2706:
2700:
2696:
2687:
2678:
2660:
2654:
2633:
2627:
2626:
2614:
2606:
2603:
2600:
2597:
2594:
2591:
2588:
2583:
2578:
2574:
2570:
2565:
2560:
2556:
2552:
2547:
2542:
2539:
2536:
2532:
2526:
2522:
2516:
2512:
2505:
2500:
2496:
2493:
2490:
2485:
2481:
2477:
2472:
2468:
2464:
2459:
2455:
2449:
2445:
2436:
2427:
2408:
2407:
2395:
2389:
2385:
2382:
2379:
2374:
2370:
2366:
2361:
2357:
2353:
2348:
2344:
2338:
2334:
2326:
2321:
2317:
2305:
2304:
2291:
2286:
2280:
2276:
2270:
2266:
2259:
2254:
2250:
2232:
2223:
2210:
2201:
2195:
2194:
2182:
2175:
2171:
2164:
2160:
2156:
2153:
2147:
2144:
2118:
2115:
2111:
2110:
2088:
2087:
2074:
2067:
2049:
2048:
2045:
2042:
2027:
2011:
2008:
2007:
2006:
2005:
2004:
2001:= 46 − 21 = 25
1997:
1987:
1980:
1967:
1963:
1962:
1959:
1941:
1931:
1917:
1916:
1891:
1888:
1886:
1883:
1864:
1860:
1854:
1850:
1846:
1841:
1837:
1820:
1817:
1816:
1815:
1814:
1813:
1812:
1811:
1806:
1800:
1793:
1786:
1771:
1764:
1742:
1735:
1725:
1724:
1723:
1722:
1721:
1720:
1707:
1702:
1698:
1695:
1692:
1687:
1683:
1679:
1674:
1670:
1663:
1658:
1654:
1650:
1645:
1641:
1638:
1635:
1630:
1626:
1622:
1617:
1613:
1606:
1601:
1597:
1593:
1588:
1584:
1580:
1575:
1571:
1556:
1549:
1540:
1539:
1538:
1537:
1536:
1535:
1520:
1516:
1513:
1510:
1505:
1501:
1497:
1492:
1488:
1481:
1476:
1472:
1468:
1463:
1459:
1443:
1442:
1441:
1440:
1430:
1429:
1428:
1427:
1423:
1416:
1407:
1406:
1405:
1404:
1403:
1402:
1387:
1383:
1380:
1377:
1372:
1368:
1364:
1359:
1355:
1348:
1343:
1339:
1335:
1330:
1326:
1310:
1309:
1303:
1285:
1246:
1242:
1215:
1211:
1153:
1152:
1145:
1125:
1122:
1111:
1101:
1078:
1068:
1056:
1034:
1024:
1014:
1013:
1000:
997:
994:
989:
986:
983:
978:
972:
969:
966:
963:
960:
957:
953:
948:
945:
920:
919:
903:
899:
893:
889:
882:
878:
872:
867:
862:
859:
856:
823:
820:
819:
818:
804:
800:
796:
791:
787:
772:
771:
758:
754:
750:
744:
740:
737:
734:
729:
725:
721:
716:
712:
704:
699:
695:
689:
685:
681:
676:
672:
668:
663:
659:
655:
649:
645:
642:
639:
634:
630:
626:
621:
617:
609:
604:
600:
594:
590:
586:
581:
577:
546:
522:
518:
513:
509:
506:
503:
498:
494:
473:
445:
441:
436:
432:
429:
426:
421:
417:
404:
401:
336:
325:
318:
317:
253:
243:
242:
238:
232:
228:
222:
215:
214:of each other,
172:
169:
88:
67:
62:
61:
39:
37:
30:
24:
18:Mann–Whitney U
14:
13:
10:
9:
6:
4:
3:
2:
8505:
8494:
8491:
8489:
8486:
8484:
8481:
8480:
8478:
8463:
8462:
8453:
8451:
8450:
8441:
8439:
8438:
8433:
8427:
8425:
8424:
8415:
8414:
8411:
8397:
8394:
8392:
8391:Geostatistics
8389:
8387:
8384:
8382:
8379:
8377:
8374:
8373:
8371:
8369:
8365:
8359:
8358:Psychometrics
8356:
8354:
8351:
8349:
8346:
8344:
8341:
8339:
8336:
8334:
8331:
8329:
8326:
8324:
8321:
8319:
8316:
8314:
8311:
8310:
8308:
8306:
8302:
8296:
8293:
8291:
8288:
8286:
8282:
8279:
8277:
8274:
8272:
8269:
8267:
8264:
8263:
8261:
8259:
8255:
8249:
8246:
8244:
8241:
8239:
8235:
8232:
8230:
8227:
8226:
8224:
8222:
8221:Biostatistics
8218:
8214:
8210:
8205:
8201:
8183:
8182:Log-rank test
8180:
8179:
8177:
8173:
8167:
8164:
8163:
8161:
8159:
8155:
8149:
8146:
8144:
8141:
8139:
8136:
8134:
8131:
8130:
8128:
8126:
8122:
8119:
8117:
8113:
8103:
8100:
8098:
8095:
8093:
8090:
8088:
8085:
8083:
8080:
8079:
8077:
8075:
8071:
8065:
8062:
8060:
8057:
8055:
8053:(Box–Jenkins)
8049:
8047:
8044:
8042:
8039:
8035:
8032:
8031:
8030:
8027:
8026:
8024:
8022:
8018:
8012:
8009:
8007:
8006:Durbin–Watson
8004:
8002:
7996:
7994:
7991:
7989:
7988:Dickey–Fuller
7986:
7985:
7983:
7979:
7973:
7970:
7968:
7965:
7963:
7962:Cointegration
7960:
7958:
7955:
7953:
7950:
7948:
7945:
7943:
7940:
7938:
7937:Decomposition
7935:
7934:
7932:
7928:
7925:
7923:
7919:
7909:
7906:
7905:
7904:
7901:
7900:
7899:
7896:
7892:
7889:
7888:
7887:
7884:
7882:
7879:
7877:
7874:
7872:
7869:
7867:
7864:
7862:
7859:
7857:
7854:
7852:
7849:
7848:
7846:
7844:
7840:
7834:
7831:
7829:
7826:
7824:
7821:
7819:
7816:
7814:
7811:
7809:
7808:Cohen's kappa
7806:
7805:
7803:
7801:
7797:
7793:
7789:
7785:
7781:
7777:
7772:
7768:
7754:
7751:
7749:
7746:
7744:
7741:
7739:
7736:
7735:
7733:
7731:
7727:
7721:
7717:
7713:
7707:
7705:
7702:
7701:
7699:
7697:
7693:
7687:
7684:
7682:
7679:
7677:
7674:
7672:
7669:
7667:
7664:
7662:
7661:Nonparametric
7659:
7657:
7654:
7653:
7651:
7647:
7641:
7638:
7636:
7633:
7631:
7628:
7626:
7623:
7622:
7620:
7618:
7614:
7608:
7605:
7603:
7600:
7598:
7595:
7593:
7590:
7588:
7585:
7584:
7582:
7580:
7576:
7570:
7567:
7565:
7562:
7560:
7557:
7555:
7552:
7551:
7549:
7547:
7543:
7539:
7532:
7529:
7527:
7524:
7523:
7519:
7515:
7499:
7496:
7495:
7494:
7491:
7489:
7486:
7484:
7481:
7477:
7474:
7472:
7469:
7468:
7467:
7464:
7463:
7461:
7459:
7455:
7445:
7442:
7438:
7432:
7430:
7424:
7422:
7416:
7415:
7414:
7411:
7410:Nonparametric
7408:
7406:
7400:
7396:
7393:
7392:
7391:
7385:
7381:
7380:Sample median
7378:
7377:
7376:
7373:
7372:
7370:
7368:
7364:
7356:
7353:
7351:
7348:
7346:
7343:
7342:
7341:
7338:
7336:
7333:
7331:
7325:
7323:
7320:
7318:
7315:
7313:
7310:
7308:
7305:
7303:
7301:
7297:
7295:
7292:
7291:
7289:
7287:
7283:
7277:
7275:
7271:
7269:
7267:
7262:
7260:
7255:
7251:
7250:
7247:
7244:
7242:
7238:
7228:
7225:
7223:
7220:
7218:
7215:
7214:
7212:
7210:
7206:
7200:
7197:
7193:
7190:
7189:
7188:
7185:
7181:
7178:
7177:
7176:
7173:
7171:
7168:
7167:
7165:
7163:
7159:
7151:
7148:
7146:
7143:
7142:
7141:
7138:
7136:
7133:
7131:
7128:
7126:
7123:
7121:
7118:
7116:
7113:
7112:
7110:
7108:
7104:
7098:
7095:
7091:
7088:
7084:
7081:
7079:
7076:
7075:
7074:
7071:
7070:
7069:
7066:
7062:
7059:
7057:
7054:
7052:
7049:
7047:
7044:
7043:
7042:
7039:
7038:
7036:
7034:
7030:
7027:
7025:
7021:
7015:
7012:
7010:
7007:
7003:
7000:
6999:
6998:
6995:
6993:
6990:
6986:
6985:loss function
6983:
6982:
6981:
6978:
6974:
6971:
6969:
6966:
6964:
6961:
6960:
6959:
6956:
6954:
6951:
6949:
6946:
6942:
6939:
6937:
6934:
6932:
6926:
6923:
6922:
6921:
6918:
6914:
6911:
6909:
6906:
6904:
6901:
6900:
6899:
6896:
6892:
6889:
6887:
6884:
6883:
6882:
6879:
6875:
6872:
6871:
6870:
6867:
6863:
6860:
6859:
6858:
6855:
6853:
6850:
6848:
6845:
6843:
6840:
6839:
6837:
6835:
6831:
6827:
6823:
6818:
6814:
6800:
6797:
6795:
6792:
6790:
6787:
6785:
6782:
6781:
6779:
6777:
6773:
6767:
6764:
6762:
6759:
6757:
6754:
6753:
6751:
6747:
6741:
6738:
6736:
6733:
6731:
6728:
6726:
6723:
6721:
6718:
6716:
6713:
6711:
6708:
6707:
6705:
6703:
6699:
6693:
6690:
6688:
6687:Questionnaire
6685:
6683:
6680:
6676:
6673:
6671:
6668:
6667:
6666:
6663:
6662:
6660:
6658:
6654:
6648:
6645:
6643:
6640:
6638:
6635:
6633:
6630:
6628:
6625:
6623:
6620:
6618:
6615:
6613:
6610:
6609:
6607:
6605:
6601:
6597:
6593:
6588:
6584:
6570:
6567:
6565:
6562:
6560:
6557:
6555:
6552:
6550:
6547:
6545:
6542:
6540:
6537:
6535:
6532:
6530:
6527:
6525:
6522:
6520:
6517:
6515:
6514:Control chart
6512:
6510:
6507:
6505:
6502:
6500:
6497:
6496:
6494:
6492:
6488:
6482:
6479:
6475:
6472:
6470:
6467:
6466:
6465:
6462:
6460:
6457:
6455:
6452:
6451:
6449:
6447:
6443:
6437:
6434:
6432:
6429:
6427:
6424:
6423:
6421:
6417:
6411:
6408:
6407:
6405:
6403:
6399:
6387:
6384:
6382:
6379:
6377:
6374:
6373:
6372:
6369:
6367:
6364:
6363:
6361:
6359:
6355:
6349:
6346:
6344:
6341:
6339:
6336:
6334:
6331:
6329:
6326:
6324:
6321:
6319:
6316:
6315:
6313:
6311:
6307:
6301:
6298:
6296:
6293:
6289:
6286:
6284:
6281:
6279:
6276:
6274:
6271:
6269:
6266:
6264:
6261:
6259:
6256:
6254:
6251:
6249:
6246:
6244:
6241:
6240:
6239:
6236:
6235:
6233:
6231:
6227:
6224:
6222:
6218:
6214:
6210:
6205:
6201:
6195:
6192:
6190:
6187:
6186:
6183:
6179:
6172:
6167:
6165:
6160:
6158:
6153:
6152:
6149:
6142:
6139:
6136:
6132:
6129:
6127:
6124:
6120:
6119:
6115:
6109:
6105:
6101:
6097:
6093:
6089:
6085:
6081:
6077:
6073:
6069:
6065:
6061:
6055:
6051:
6047:
6043:
6039:
6034:
6030:
6026:
6022:
6016:
6012:
6008:
6004:
6000:
5994:
5986:
5982:
5977:
5972:
5969:: 11.IT.3.1.
5968:
5964:
5959:
5955:
5951:
5947:
5943:
5939:
5935:
5931:
5927:
5922:
5917:
5913:
5909:
5908:
5903:
5899:
5898:Lehmann, E.L.
5894:
5890:
5884:
5880:
5875:
5871:
5867:
5863:
5857:
5853:
5848:
5847:
5843:
5835:
5831:
5826:
5821:
5817:
5813:
5809:
5805:
5804:
5799:
5793:
5790:
5785:
5781:
5777:
5773:
5769:
5765:
5758:
5755:
5750:
5746:
5740:
5737:
5732:
5728:
5722:
5719:
5715:
5703:
5699:
5693:
5690:
5685:
5681:
5677:
5673:
5669:
5665:
5661:
5657:
5650:
5647:
5642:
5629:
5621:
5614:
5611:
5606:
5602:
5597:
5592:
5588:
5584:
5580:
5576:
5572:
5570:
5561:
5558:
5553:
5549:
5545:
5541:
5537:
5533:
5529:
5522:
5519:
5514:
5510:
5505:
5500:
5495:
5490:
5486:
5482:
5478:
5476:
5467:
5465:
5461:
5456:
5450:
5446:
5442:
5438:
5437:
5429:
5426:
5421:
5415:
5411:
5407:
5403:
5402:
5394:
5391:
5386:
5382:
5378:
5374:
5370:
5366:
5365:
5360:
5353:
5350:
5345:
5341:
5337:
5333:
5328:
5323:
5319:
5315:
5311:
5307:
5300:
5297:
5293:
5292:
5285:
5282:
5278:
5272:
5269:
5264:
5260:
5256:
5252:
5248:
5244:
5237:
5234:
5230:
5224:
5221:
5216:
5212:
5208:
5204:
5200:
5193:
5190:
5185:
5181:
5177:
5173:
5169:
5165:
5164:Psychometrika
5158:
5155:
5150:
5146:
5142:
5138:
5134:
5130:
5123:
5120:
5115:
5111:
5107:
5103:
5099:
5095:
5094:
5086:
5083:
5078:
5074:
5070:
5066:
5059:
5057:
5053:
5048:
5042:
5034:
5030:
5025:
5020:
5017:: 11.IT.3.1.
5016:
5012:
5005:
5003:
4999:
4994:
4990:
4986:
4982:
4978:
4974:
4970:
4966:
4959:
4956:
4951:
4947:
4943:
4939:
4932:
4929:
4924:
4918:
4910:
4903:
4900:
4895:
4889:
4881:
4874:
4872:
4868:
4863:
4857:
4853:
4846:
4843:
4838:
4834:
4830:
4826:
4822:
4818:
4814:
4810:
4806:
4799:
4796:
4791:
4785:
4781:
4774:
4771:
4765:
4760:
4756:
4752:
4748:
4741:
4738:
4734:
4733:
4726:
4723:
4718:
4714:
4710:
4706:
4702:
4698:
4691:
4688:
4685:
4680:
4677:
4672:
4668:
4663:
4658:
4653:
4648:
4644:
4640:
4636:
4629:
4626:
4614:
4610:
4605:
4600:
4596:
4592:
4591:Stata Journal
4588:
4581:
4578:
4572:
4567:
4563:
4559:
4555:
4548:
4545:
4540:
4536:
4532:
4529:
4524:
4520:
4515:
4510:
4506:
4502:
4498:
4494:
4490:
4486:
4485:
4480:
4478:
4469:
4467:
4463:
4458:
4454:
4450:
4446:
4441:
4436:
4432:
4428:
4427:
4422:
4418:
4412:
4410:
4406:
4400:
4396:
4393:
4391:
4388:
4386:
4383:
4381:
4378:
4376:
4373:
4371:
4368:
4366:
4363:
4362:
4358:
4356:
4352:
4347:
4343:
4339:
4334:
4330:
4325:
4321:
4317:
4311:
4308:
4304:
4299:
4293:
4288:
4284:
4281:
4278:
4272:
4269:
4266:
4263:
4260:
4254:
4251:
4248:
4244:
4241:
4238:
4235:
4232:
4229:
4226:
4223:
4220:
4217:
4214:
4211:
4200:
4197:
4195:
4191:
4188:
4185:
4182:
4179:
4176:
4173:
4169:
4166:
4162:PROC NPAR1WAY
4159:
4156:
4152:
4147:
4141:
4135:
4132:
4126:
4123:
4120:
4114:
4111:
4110:
4109:
4106:
4098:
4096:
4094:
4093:Kendall's tau
4090:
4083:Kendall's tau
4077:
4075:
4073:
4068:
4066:
4062:
4057:
4055:
4051:
4046:
4044:
4042:
4037:
4032:
4030:
4026:
4021:
4019:
4015:
4011:
4007:
3999:
3997:
3994:
3978:
3975:
3969:
3966:
3963:
3957:
3954:
3951:
3945:
3942:
3939:
3933:
3925:
3907:
3903:
3899:
3894:
3890:
3869:
3866:
3860:
3857:
3854:
3848:
3845:
3842:
3836:
3833:
3830:
3824:
3804:
3801:
3795:
3792:
3789:
3783:
3780:
3777:
3771:
3768:
3765:
3759:
3751:
3743:
3736:
3733:
3732:
3728:
3725:
3724:
3709:
3705:
3701:
3694:
3691:
3690:
3675:
3671:
3665:
3661:
3653:
3650:
3649:
3645:
3643:Distribution
3642:
3641:
3637:
3633:
3629:
3623:
3621:
3619:
3614:
3606:
3602:
3598:
3590:
3586:
3582:
3579:
3576:
3572:
3568:
3564:
3560:
3556:
3553:
3550:
3546:
3542:
3538:
3535:
3534:
3533:
3531:
3527:
3523:
3518:
3514:
3506:
3497:
3495:
3491:
3484:
3473:
3469:
3445:
3441:
3435:
3431:
3423:
3419:
3415:
3409:
3406:
3403:
3400:
3397:
3389:
3385:
3379:
3375:
3367:
3363:
3359:
3353:
3350:
3347:
3344:
3341:
3338:
3332:
3329:
3326:
3320:
3317:
3314:
3311:
3304:
3303:
3302:
3286:
3282:
3259:
3255:
3246:
3241:
3237:
3217:
3214:
3211:
3208:
3205:
3198:
3197:
3196:
3194:
3190:
3184:
3181:
3177:
3169:
3167:
3164:
3160:
3156:
3152:
3148:
3142:
3138:
3134:
3130:
3124:
3116:
3109:
3104:
3100:
3096:
3092:
3088:
3080:
3077:
3075:
3073:
3048:
3044:
3038:
3034:
3027:
3023:
3017:
3014:
3007:
3006:
3005:
2989:
2985:
2976:
2972:
2967:
2965:
2962:test is with
2961:
2953:
2951:
2945:
2943:
2941:
2933:
2931:
2929:
2925:
2921:
2912:
2906:
2897:
2891:
2884:
2877:
2871:
2867:
2861:
2854:
2850:
2830:
2824:
2814:
2811:
2808:
2802:
2792:
2788:
2784:
2779:
2774:
2770:
2761:
2756:
2753:
2750:
2746:
2739:
2733:
2730:
2727:
2720:
2714:
2708:
2704:
2698:
2694:
2685:
2676:
2668:
2667:
2666:
2659:
2653:
2647:
2645:
2642:th rank, and
2641:
2636:
2632:
2612:
2601:
2598:
2595:
2589:
2586:
2576:
2572:
2568:
2563:
2558:
2554:
2545:
2540:
2537:
2534:
2530:
2524:
2520:
2514:
2510:
2503:
2498:
2491:
2488:
2483:
2479:
2475:
2470:
2466:
2457:
2453:
2447:
2443:
2434:
2425:
2417:
2416:
2415:
2413:
2393:
2387:
2380:
2377:
2372:
2368:
2364:
2359:
2355:
2346:
2342:
2336:
2332:
2324:
2319:
2315:
2307:
2306:
2289:
2284:
2278:
2274:
2268:
2264:
2257:
2252:
2248:
2240:
2239:
2238:
2237:are given by
2235:
2231:
2226:
2222:
2218:
2213:
2209:
2204:
2200:
2180:
2173:
2169:
2162:
2158:
2154:
2151:
2145:
2142:
2135:
2134:
2133:
2132:
2128:
2124:
2116:
2114:
2106:
2099:
2093:
2092:
2091:
2086:two-tailed)."
2083:
2073:
2066:
2059:
2054:
2053:
2052:
2046:
2043:
2040:
2036:
2032:
2029:The value of
2028:
2025:
2021:
2020:
2019:
2017:
2009:
2000:
1993:, leading to
1988:
1983:
1975:
1974:
1968:
1965:
1964:
1960:
1951:
1944:
1934:
1926:
1925:
1924:
1922:
1914:
1913:
1912:
1909:
1905:
1902:in which one
1901:
1897:
1894:Suppose that
1889:
1884:
1882:
1880:
1862:
1858:
1852:
1848:
1844:
1839:
1835:
1826:
1818:
1805:
1799:
1792:
1785:
1781:
1780:
1778:
1770:
1763:
1759:
1752:
1748:
1741:
1734:
1730:Knowing that
1729:
1728:
1727:
1726:
1705:
1700:
1693:
1690:
1685:
1681:
1672:
1668:
1661:
1656:
1652:
1648:
1643:
1636:
1633:
1628:
1624:
1615:
1611:
1604:
1599:
1595:
1591:
1586:
1582:
1578:
1573:
1569:
1561:
1560:
1555:
1548:
1544:
1543:
1542:
1541:
1518:
1511:
1508:
1503:
1499:
1490:
1486:
1479:
1474:
1470:
1466:
1461:
1457:
1449:
1448:
1447:
1446:
1445:
1444:
1438:
1434:
1433:
1432:
1431:
1422:
1415:
1411:
1410:
1409:
1408:
1385:
1378:
1375:
1370:
1366:
1357:
1353:
1346:
1341:
1337:
1333:
1328:
1324:
1316:
1315:
1314:
1313:
1312:
1311:
1307:
1304:
1301:
1295:
1291:
1286:
1270:
1269:
1268:
1265:
1264:
1260:
1244:
1240:
1231:
1213:
1209:
1200:
1195:
1190:
1189:
1185:
1182:
1180:
1176:
1171:
1169:
1165:
1160:
1158:
1150:
1146:
1143:
1142:
1141:
1139:
1135:
1131:
1123:
1121:
1118:
1114:
1108:
1104:
1099:
1095:
1091:
1085:
1081:
1075:
1071:
1063:
1059:
1054:
1050:
1046:
1041:
1037:
1031:
1027:
1023:
1019:
998:
995:
992:
976:
967:
964:
961:
955:
951:
946:
943:
936:
935:
934:
932:
927:
925:
901:
897:
891:
887:
880:
876:
870:
865:
845:
844:
843:
841:
837:
835:
829:
821:
802:
798:
794:
789:
785:
777:
776:
775:
756:
752:
748:
742:
735:
732:
727:
723:
714:
710:
702:
697:
693:
687:
683:
679:
674:
670:
666:
661:
657:
653:
647:
640:
637:
632:
628:
619:
615:
607:
602:
598:
592:
588:
584:
579:
575:
567:
566:
565:
563:
562:
559:Mann–Whitney
544:
520:
516:
511:
507:
504:
501:
496:
492:
471:
463:
462:i.i.d. sample
443:
439:
434:
430:
427:
424:
419:
415:
402:
400:
398:
394:
390:
388:
382:
377:
375:
370:
368:
364:
360:
356:
352:
346:
342:
335:
331:
324:
313:
309:
305:
301:
293:
289:
285:
281:
275:
271:
267:
263:
259:
258:
257:
252:
248:
237:
233:
227:
223:
220:
216:
213:
209:
208:
207:
204:
202:
198:
194:
190:
186:
182:
178:
170:
168:
166:
162:
158:
153:
151:
147:
143:
139:
135:
131:
127:
123:
120:
119:nonparametric
116:
112:
108:
104:
100:
86:
79:Mann–Whitney
74:
66:
58:
53:
49:
48:
43:
38:
29:
28:
19:
8493:U-statistics
8459:
8447:
8428:
8421:
8333:Econometrics
8283: /
8266:Chemometrics
8243:Epidemiology
8236: /
8209:Applications
8051:ARIMA model
7998:Q-statistic
7947:Stationarity
7843:Multivariate
7786: /
7782: /
7780:Multivariate
7778: /
7718: /
7714: /
7488:Bayes factor
7401:
7387:Signed rank
7299:
7273:
7265:
7253:
6948:Completeness
6784:Cohort study
6682:Opinion poll
6617:Missing data
6604:Study design
6559:Scatter plot
6481:Scatter plot
6474:Spearman's ρ
6436:Grouped data
6134:
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5993:cite journal
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5706:. Retrieved
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4000:Alternatives
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3575:type I error
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3536:Ordinal data
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2934:Effect sizes
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1124:Calculations
1116:
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1106:
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8461:WikiProject
8376:Cartography
8338:Jurimetrics
8290:Reliability
8021:Time domain
8000:(Ljung–Box)
7922:Time-series
7800:Categorical
7784:Time-series
7776:Categorical
7711:(Bernoulli)
7546:Correlation
7526:Correlation
7322:Jarque–Bera
7294:Chi-squared
7056:M-estimator
7009:Asymptotics
6953:Sufficiency
6720:Interaction
6632:Replication
6612:Effect size
6569:Violin plot
6549:Radar chart
6529:Forest plot
6519:Correlogram
6469:Kendall's τ
4815:(1): 2–18.
4365:Lepage test
4322:(where the
4243:StatsDirect
4130:wilcox.test
3646:Efficiency
2940:effect size
2902:, the mean
1954:, which is
1263:Method two:
1188:Method one:
1033:term of AUC
561:U statistic
403:U statistic
212:independent
8477:Categories
8328:Demography
8046:ARMA model
7851:Regression
7428:(Friedman)
7389:(Wilcoxon)
7327:Normality
7317:Lilliefors
7264:Student's
7140:Resampling
7014:Robustness
7002:divergence
6992:Efficiency
6930:(monotone)
6925:Likelihood
6842:Population
6675:Stratified
6627:Population
6446:Dependence
6402:Count data
6333:Percentile
6310:Dispersion
6243:Arithmetic
6178:Statistics
6108:0119.15604
6080:Biometrics
5946:0203.21105
5844:References
4457:0041.26103
4316:Henry Mann
4225:STATISTICA
4164:procedure.
4145:rcompanion
3589:efficiency
3580:Efficiency
3554:Robustness
3135:) + 0.5 P(
3097:involving
1976:Therefore
1819:Properties
1140:is known:
389:-rank test
247:consistent
185:continuous
177:Henry Mann
57:(May 2024)
7709:Logistic
7476:posterior
7402:Rank sum
7150:Jackknife
7145:Bootstrap
6963:Bootstrap
6898:Parameter
6847:Statistic
6642:Statistic
6554:Run chart
6539:Pie chart
6534:Histogram
6524:Fan chart
6499:Bar chart
6381:L-moments
6268:Geometric
5985:120622013
5881:. Wiley.
5684:120473946
5605:235521799
5552:140209347
5513:2515-2459
5336:0163-7525
5263:0022-0973
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5033:120622013
4917:cite book
4888:cite book
4829:1939-2222
4717:1477-870X
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4142:from the
4139:wilcoxonZ
4070:See also
3867:≠
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3662:π
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428:…
385:Wilcoxon
276:; i.e.,
175:Although
161:sign test
157:dependent
8423:Category
8116:Survival
7993:Johansen
7716:Binomial
7671:Isotonic
7258:(normal)
6903:location
6710:Blocking
6665:Sampling
6544:Q–Q plot
6509:Box plot
6491:Graphics
6386:Skewness
6376:Kurtosis
6348:Variance
6278:Heronian
6273:Harmonic
6009:(2006).
5900:(1963).
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1904:tortoise
1885:Examples
1170:itself.
163:and the
8449:Commons
8396:Kriging
8281:Process
8238:studies
8097:Wavelet
7930:General
7097:Plug-in
6891:L space
6670:Cluster
6371:Moments
6189:Outline
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7418:1-way
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6194:Index
6126:(pdf)
6096:JSTOR
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4401:Notes
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3015:f
2990:1
2986:U
2975:U
2971:f
2964:f
2960:U
2928:U
2924:z
2920:U
2914:2
2911:n
2908:1
2905:n
2899:2
2896:n
2893:1
2890:n
2886:2
2883:U
2879:1
2876:U
2863:2
2860:n
2856:1
2853:n
2849:n
2831:,
2825:)
2818:)
2815:1
2809:n
2806:(
2803:n
2798:)
2793:k
2789:t
2780:3
2775:k
2771:t
2767:(
2762:K
2757:1
2754:=
2751:k
2737:)
2734:1
2731:+
2728:n
2725:(
2721:(
2709:2
2705:n
2699:1
2695:n
2686:=
2661:2
2658:n
2655:1
2652:n
2644:K
2640:k
2635:k
2631:t
2613:,
2605:)
2602:1
2596:n
2593:(
2590:n
2582:)
2577:k
2573:t
2564:3
2559:k
2555:t
2551:(
2546:K
2541:1
2538:=
2535:k
2525:2
2521:n
2515:1
2511:n
2495:)
2492:1
2489:+
2484:2
2480:n
2476:+
2471:1
2467:n
2463:(
2458:2
2454:n
2448:1
2444:n
2435:=
2412:σ
2394:.
2384:)
2381:1
2378:+
2373:2
2369:n
2365:+
2360:1
2356:n
2352:(
2347:2
2343:n
2337:1
2333:n
2325:=
2320:U
2290:,
2285:2
2279:2
2275:n
2269:1
2265:n
2258:=
2253:U
2249:m
2234:U
2230:σ
2225:U
2221:m
2217:U
2212:U
2208:σ
2203:U
2199:m
2181:,
2174:U
2163:U
2159:m
2152:U
2146:=
2143:z
2123:U
2105:ρ
2098:P
2096:2
2082:P
2075:2
2072:n
2068:1
2065:n
2058:U
2031:U
2024:U
2016:U
2003:.
1998:H
1996:U
1981:T
1979:U
1958:.
1950:U
1942:H
1940:U
1932:T
1930:U
1921:U
1879:U
1863:2
1859:n
1853:1
1849:n
1845:=
1840:i
1836:U
1825:U
1810:.
1807:2
1804:n
1801:1
1798:n
1794:2
1791:U
1787:1
1784:U
1772:2
1769:n
1765:1
1762:n
1758:N
1751:N
1749:(
1747:N
1743:2
1740:R
1736:1
1733:R
1706:.
1701:2
1697:)
1694:1
1691:+
1686:2
1682:n
1678:(
1673:2
1669:n
1657:2
1653:R
1649:+
1644:2
1640:)
1637:1
1634:+
1629:1
1625:n
1621:(
1616:1
1612:n
1600:1
1596:R
1592:=
1587:2
1583:U
1579:+
1574:1
1570:U
1557:2
1554:U
1550:1
1547:U
1519:2
1515:)
1512:1
1509:+
1504:2
1500:n
1496:(
1491:2
1487:n
1475:2
1471:R
1467:=
1462:2
1458:U
1437:U
1424:1
1421:R
1417:1
1414:n
1386:2
1382:)
1379:1
1376:+
1371:1
1367:n
1363:(
1358:1
1354:n
1342:1
1338:R
1334:=
1329:1
1325:U
1306:U
1300:N
1294:N
1292:(
1290:N
1245:2
1241:U
1230:U
1214:1
1210:U
1199:U
1194:U
1175:U
1168:U
1164:U
1134:U
1117:k
1115:,
1113:ℓ
1107:ℓ
1105:,
1103:k
1098:ℓ
1096:,
1094:k
1090:M
1084:k
1082:,
1080:ℓ
1074:ℓ
1072:,
1070:k
1062:k
1060:,
1058:k
1053:k
1049:ℓ
1045:k
1040:ℓ
1038:,
1036:k
1030:ℓ
1028:,
1026:k
1022:R
1018:c
996:,
993:k
988:C
985:U
982:A
971:)
968:1
962:c
959:(
956:c
952:1
947:=
944:M
931:U
902:2
898:n
892:1
888:n
881:1
877:U
871:=
866:1
861:C
858:U
855:A
838:(
828:U
803:2
799:R
795:,
790:1
786:R
757:2
753:R
743:2
739:)
736:1
733:+
728:2
724:n
720:(
715:2
711:n
703:+
698:2
694:n
688:1
684:n
680:=
675:2
671:U
667:,
662:1
658:R
648:2
644:)
641:1
638:+
633:1
629:n
625:(
620:1
616:n
608:+
603:2
599:n
593:1
589:n
585:=
580:1
576:U
545:Y
521:2
517:n
512:Y
508:,
502:,
497:1
493:Y
472:X
444:1
440:n
435:X
431:,
425:,
420:1
416:X
397:U
381:U
374:U
355:U
351:U
347:)
345:δ
341:x
339:(
337:2
334:F
330:x
328:(
326:1
323:F
316:.
312:Y
308:X
304:Y
300:X
294:)
292:X
288:Y
284:Y
280:X
274:X
270:Y
266:Y
262:X
254:1
251:H
239:1
236:H
229:0
226:H
201:U
181:U
150:X
146:Y
142:Y
138:X
134:Y
130:X
105:(
87:U
75:.
20:)
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