14709:
14695:
6400:
rank procedure; but increasing some of the values so as to break the ties, or breaking the ties in any way whatsoever, results in a sample that the test judges to be not significant. However, increasing all the observed values by the same amount cannot turn a significantly positive result into an insignificant one, nor an insignificant one into a significantly negative one. Furthermore, if the observations are distributed symmetrically, then the values of
14733:
14721:
1907:
8642:
6461:. The rank assigned to an observation depends on its absolute value and the tiebreaking rule. Observations with smaller absolute values are always given smaller ranks, just as in the standard rank-sum test. The tiebreaking rule is used to assign ranks to observations with the same absolute value. One advantage of tiebreaking rules is that they allow the use of standard tables for computing
1545:
8329:
122:
There are two variants of the signed-rank test. From a theoretical point of view, the one-sample test is more fundamental because the paired sample test is performed by converting the data to the situation of the one-sample test. However, most practical applications of the signed-rank test arise from
77:
of the differences between paired individuals cannot be assumed. Instead, it assumes a weaker hypothesis that the distribution of this difference is symmetric around a central value and it aims to test whether this center value differs significantly from zero. The
Wilcoxon test is a more powerful
6471:
breaks the ties at random. Under random tiebreaking, the null distribution is the same as when there are no ties, but the result of the test depends not only on the data but on additional random choices. Averaging the ranks over the possible random choices results in the average rank procedure. One
6399:
Under the average rank procedure, the null distribution is different in the presence of ties. The average rank procedure also has some disadvantages that are similar to those of the reduced sample procedure for zeros. It is possible that a sample can be judged significantly positive by the average
4984:
Wilcoxon's original paper did not address the question of observations (or, in the paired sample case, differences) that equal zero. However, in later surveys, he recommended removing zeros from the sample. Then the standard signed-rank test could be applied to the resulting data, as long as there
1433:
5603:
From the viewpoint of statistical efficiency, there is no perfect rule for handling zeros. Conover found examples of null and alternative hypotheses that show that neither
Wilcoxon's and Pratt's methods are uniformly better than the other. When comparing a discrete uniform distribution to a
5094:. Pratt argues that one would expect that decreasing the observations should certainly not make the data appear more positive. However, if the zero observation is decreased by an amount less than 2, or if all observations are decreased by an amount less than 1, then the signed ranks become:
2240:
5361:
A sample is significantly positive, not significant, or significantly negative, if and only if it is so when the zeros are assigned arbitrary non-zero signs, if and only if it is so when the zeros are replaced with non-zero values which are smaller in absolute value than any non-zero
9670:
11304:
is reported, an equivalent way to compute the rank correlation is with the difference in proportion between the two rank sums, which is the Kerby (2014) simple difference formula. To continue with the current example, the sample size is 9, so the total rank sum is 45.
1902:{\displaystyle {\begin{aligned}T^{+}&={\frac {n(n+1)}{2}}-T^{-}={\frac {n(n+1)}{4}}+{\frac {T}{2}},\\T^{-}&={\frac {n(n+1)}{2}}-T^{+}={\frac {n(n+1)}{4}}-{\frac {T}{2}},\\T&=T^{+}-T^{-}=2T^{+}-{\frac {n(n+1)}{2}}={\frac {n(n+1)}{2}}-2T^{-}.\end{aligned}}}
8637:{\displaystyle {\begin{aligned}\mathbf {E} &=\mathbf {E} ={\frac {n(n+1)}{4}},\\\mathbf {E} &=0,\\\operatorname {Var} (T^{+})&=\operatorname {Var} (T^{-})={\frac {n(n+1)(2n+1)}{24}},\\\operatorname {Var} (T)&={\frac {n(n+1)(2n+1)}{6}}.\end{aligned}}}
8833:
4949:(in the paired sample case). This is particularly common for discrete data. When this happens, the test procedure defined above is usually undefined because there is no way to uniquely rank the data. (The sole exception is if there is a single observation
5722:(except when there are no ties). The rank assigned to an observation is the average of the possible ranks it would have if the ties were broken in all possible ways. Once the ranks are assigned, the test statistic is computed in the same way as usual.
4198:
The null hypothesis of exchangeability can arise from a matched pair experiment with a treatment group and a control group. Randomizing the treatment and control within each pair makes the observations exchangeable. For an exchangeable distribution,
11425:
PROC UNIVARIATE includes the
Wilcoxon-Signed Rank Test in the frame titles "Tests for Location" as "Signed Rank". Even though this procedure calculates an S-Statistic rather than a W-Statistic, the resulting p-value can still be used for this
8958:
1261:
2080:
6472:
could also report the probability of rejection over all random choices. Random tiebreaking has the advantage that the probability that a sample is judged significantly positive does not decrease when some observations are increased.
11155:
3115:
Because the paired data test arises from taking paired differences, its null and alternative hypotheses can be derived from those of the one-sample test. In each case, they become assertions about the behavior of the differences
5901:
357:. Assume for simplicity that the observations in the sample have distinct absolute values and that no observation equals zero. (Zeros and ties introduce several complications; see below.) The test is performed as follows:
9564:
5604:
distribution where probabilities linearly increase from left to right, Pratt's method outperforms
Wilcoxon's. When testing a binomial distribution centered at zero to see whether the parameter of each Bernoulli trial is
5683:
to the observations, with two observations getting the same number if and only if they have the same absolute value. These numbers are conventionally called ranks even though the set of these numbers is not equal to
9559:
9428:
3052:
The hypothesis that the data are IID can be weakened. Each data point may be taken from a different distribution, as long as all the distributions are assumed to be continuous and symmetric about a common point
9153:
5409:
Pratt remarks that, when the signed-rank zero procedure is combined with the average rank procedure for resolving ties, the resulting test is a consistent test against the alternative hypothesis that, for all
4991:
Pratt observed that the reduced sample procedure can lead to paradoxical behavior. He gives the following example. Suppose that we are in the one-sample situation and have the following thirteen observations:
806:
8649:
11408:
also returns a logical value indicating the test decision. The result h = 1 indicates a rejection of the null hypothesis, and h = 0 indicates a failure to reject the null hypothesis at the 5% significance
4693:
4516:
1111:
8178:
7510:
6566:
as small as possible. Pratt observes that when ties are likely, the conservative tiebreaking procedure "presumably has low power, since it amounts to breaking all ties in favor of the null hypothesis."
11191:
11321:, or in this case 45 − 18 = 27. Next, the two rank-sum proportions are 27/45 = 60% and 18/45 = 40%. Finally, the rank correlation is the difference between the two proportions (.60 minus .40), hence
1027:
9569:
8334:
7876:
5186:
2085:
1550:
1266:
5405:, the probability of calling a set of observations significantly positive (respectively, significantly negative) is a non-decreasing (respectively, non-increasing) function of the observations.
5092:
208:
45:
of a population based on a sample of data, or to compare the locations of two populations using two matched samples. The one-sample version serves a purpose similar to that of the one-sample
10671:
2900:
2791:
2075:
8838:
11073:
674:
5329:
607:
4947:
4192:
4050:
3908:
428:
496:
5544:
5489:
1540:
10929:
10586:
10217:
8227:
565:
10012:
959:
542:
355:
9726:
7723:
7639:
6674:
6625:
1153:
10563:
10194:
9260:
5631:
7548:
7313:
7231:
7113:
6459:
5720:
7388:
4775:
4339:
4134:
3992:
3850:
3766:
3710:
3664:
3614:
3238:
2585:
262:
10955:
10609:
10240:
8056:
7953:
7035:
4282:
Because the one-sample test can be used as a one-sided test for stochastic dominance, the paired difference
Wilcoxon test can be used to compare the following hypotheses:
9032:
7993:
7264:
7075:
6394:
6188:
6028:
4881:
4815:
4568:
4391:
4277:
4237:
3525:
3447:
3369:
3291:
3154:
1473:
302:
7182:
6496:
tend to be more significant, ties are broken by assigning lower ranks to negative observations and higher ranks to positive ones. When the test makes positive values of
5241:
5139:
4088:
3551:
2540:
9854:
9822:
5276:
5045:
3946:
3804:
3473:
3395:
2483:
2426:
8960:
The technical underpinnings of these expansions are rather involved, because conventional
Edgeworth expansions apply to sums of IID continuous random variables, while
5728:
3105:
3078:
10510:
10476:
10141:
10107:
7756:
5434:
5403:
5383:
5206:
4841:
4594:
4417:
2926:
2817:
11234:
7339:
3317:
3192:
2984:
2369:
2017:
700:
633:
10986:
10637:
10039:
9928:
9901:
9790:
9327:
8985:
8324:
8297:
7903:
7783:
7575:
7140:
6992:
6945:
6875:
6796:
6354:
6274:
6136:
6109:
6082:
6055:
5982:
5955:
5928:
5666:
4974:
4729:
1981:
1954:
1256:
1226:
879:
6418:
5356:
2665:
2605:
2270:
9759:
6848:
6822:
6564:
6234:
11214:
10442:
10073:
9948:
9874:
9300:
9280:
9215:
9195:
9175:
9055:
8270:
8250:
8013:
7679:
7659:
7595:
6965:
6918:
6895:
6769:
6749:
6701:
6534:
6514:
6494:
6334:
6314:
6294:
6254:
6208:
5584:
5564:
5335:
Increasing the observed values does not make a significantly positive sample insignificant, and it does not make an insignificant sample significantly negative.
3046:
3015:
2958:
2837:
2732:
2712:
2688:
2645:
2625:
2514:
2457:
2400:
2343:
2311:
2291:
1927:
1196:
1173:
899:
848:
828:
727:
11079:
9433:
9157:
When zeros are present and the signed-rank zero procedure is used, or when ties are present and the average rank procedure is used, the null distribution of
5293:
This procedure includes the zeros when ranking the observations in the sample. However, it excludes them from the test statistic, or equivalently it defines
9332:
6423:
The other common option for handling ties is a tiebreaking procedure. In a tiebreaking procedure, the observations are assigned distinct ranks in the set
2254:
The one-sample
Wilcoxon signed-rank test can be used to test whether data comes from a symmetric population with a specified center (which corresponds to
9060:
732:
1428:{\displaystyle {\begin{aligned}T^{+}&=\sum _{1\leq i\leq n,\ X_{i}>0}R_{i},\\T^{-}&=\sum _{1\leq i\leq n,\ X_{i}<0}R_{i}.\end{aligned}}}
3080:. The data points are not required to be independent as long as the conditional distribution of each observation given the others is symmetric about
2235:{\displaystyle {\begin{aligned}T^{+}=\#\{W_{ij}>0\colon 1\leq i\leq j\leq n\},\\T^{-}=\#\{W_{ij}<0\colon 1\leq i\leq j\leq n\}.\end{aligned}}}
12319:
8061:
7393:
5600:
with equally spaced categories, the signed-rank zero procedure is more likely to maintain the Type I error rate than the reduced sample procedure.
9728:. Early authors such as Siegel followed Wilcoxon. This is appropriate for two-sided hypothesis tests, but it cannot be used for one-sided tests.
13830:
5668:
are used to calculate the test statistic. In the presence of ties, the ranks are not defined. There are two main approaches to resolving this.
5589:
The signed-rank zero procedure has the disadvantage that, when zeros occur, the null distribution of the test statistic changes, so tables of
14335:
6570:
The average rank procedure can disagree with tiebreaking procedures. Pratt gives the following example. Suppose that the observations are:
2670:
The restriction that the alternative distribution is symmetric is highly restrictive, but for one-sided tests it can be weakened. Say that
8987:
is a sum of non-identically distributed discrete random variables. The final result, however, is that the above expansion has an error of
14485:
9665:{\displaystyle {\begin{aligned}\operatorname {Var} (T)&=\sigma ^{2},\\\operatorname {Var} (T^{+})&=\sigma ^{2}/4.\end{aligned}}}
1986:
The positive-rank sum and negative-rank sum have alternative interpretations that are useful for the theory behind the test. Define the
14109:
12750:
10992:. Notice that pairs 3 and 9 are tied in absolute value. They would be ranked 1 and 2, so each gets the average of those ranks, 1.5.
4599:
4422:
1032:
13883:
2627:. If this median is unique, then the Wilcoxon signed-rank sum test becomes a test for the location of the median. When the mean of
12206:
Cureton, Edward E. (1967). "The normal approximation to the signed-rank sampling distribution when zero differences are present".
14322:
12387:
11161:
11395:
6703:
is computed for every possible way of breaking ties, and the final statistic is the mean of the tie-broken statistics. In the
5331:. Pratt proved that the signed-rank zero procedure has several desirable behaviors not shared by the reduced sample procedure:
964:
12344:
12063:
11778:
11624:
12745:
12445:
11255:
11245:
8252:, even the above recursion is too slow. In this case, the null distribution can be approximated. The null distributions of
13349:
12497:
12360:
9177:
changes. Cureton derived a normal approximation for this situation. Suppose that the original number of observations was
8828:{\displaystyle \Pr(T^{+}\leq k)\approx \Phi (t)+\phi (t){\Big (}{\frac {3n^{2}+3n-1}{10n(n+1)(2n+1)}}{\Big )}(t^{3}-3t),}
7788:
5144:
11313:
is 3 + 4 + 5 + 6 = 18. From this information alone, the remaining rank sum can be computed, because it is the total sum
5050:
129:
8646:
Better approximations can be produced using
Edgeworth expansions. Using a fourth-order Edgeworth expansion shows that:
7906:
82:
because it considers the magnitude of the differences, but it requires this moderately strong assumption of symmetry.
14764:
14132:
14024:
11686:
11572:
11470:
10643:
9220:
308:
scale, a type of scale that carries more information than an ordinal scale but may have less than an interval scale.
14737:
14310:
14184:
6771:, the distribution may be computed exactly. Under the null hypothesis that the data is symmetric about zero, each
2846:
2737:
2022:
304:. In general, it must be possible to rank the differences between the pairs. This requires that the data be on an
14368:
14029:
13774:
13145:
12735:
11389:
10014:. When consideration is restricted to continuous distributions, this is a minimum variance unbiased estimator of
14419:
13631:
13438:
13327:
13285:
10998:
638:
38:
13359:
6476:
breaks the ties in favor of the null hypothesis. When performing a one-sided test in which negative values of
5296:
2266:). If the population center is known, then it can be used to test whether data is symmetric about its center.
574:
14662:
13621:
12524:
8180:
with similar boundary conditions. There is also a recursive formula for the cumulative distribution function
4886:
4139:
3997:
3855:
2928:. Then the Wilcoxon signed-rank sum test can also be used for the following null and alternative hypotheses:
364:
12290:
Kerby, Dave S. (2014), "The simple difference formula: An approach to teaching nonparametric correlation.",
11373:
9930:
but with the modified ranks in place of the ordinary ranks. The probability that the sum of two independent
435:
210:. Each data point in the sample is a pair of measurements. In the simplest case, the measurements are on an
14213:
14162:
14147:
14137:
14006:
13878:
13845:
13671:
13626:
13456:
5671:
The most common procedure for handling ties, and the one originally recommended by
Wilcoxon, is called the
5494:
5439:
1478:
10914:
10571:
10202:
8183:
6751:
under the null hypothesis. There is no closed formula for this distribution. However, for small values of
550:
14725:
14557:
14358:
14282:
13583:
13337:
13006:
12470:
9953:
904:
12407:
501:
314:
14759:
14442:
14414:
14409:
14157:
13916:
13822:
13802:
13710:
13421:
13239:
12722:
12594:
12413:
Kerby, D. S. (2014). The simple difference formula: An approach to teaching nonparametric correlation.
9682:
95:
7684:
7600:
6638:
6589:
1116:
14174:
13942:
13588:
13517:
13446:
13366:
13354:
13224:
13212:
13205:
12913:
12634:
12402:
10516:
10147:
5607:
4976:
which is zero and no other zeros or ties.) Because of this, the test statistic needs to be modified.
31:
8953:{\displaystyle t={\frac {k+{\tfrac {1}{2}}-{\frac {n(n+1)}{4}}}{\sqrt {\frac {n(n+1)(2n+1)}{24}}}}.}
7515:
7269:
7187:
7080:
6426:
5687:
14657:
14424:
14287:
13972:
13937:
13901:
13686:
13128:
13037:
12996:
12908:
12599:
12438:
11512:
11334:
7344:
5000:
The reduced sample procedure removes the zero. To the remaining data, it assigns the signed ranks:
4734:
4298:
4093:
3951:
3809:
3715:
3669:
3623:
3573:
3197:
2549:
221:
10938:
10592:
10223:
8018:
7915:
6997:
6679:
Two other options for handling ties are based around averaging the results of tiebreaking. In the
14566:
14179:
14119:
14056:
13694:
13678:
13416:
13278:
13268:
13118:
13032:
11436:
8990:
7958:
7236:
7040:
6359:
6141:
5987:
4846:
4780:
4533:
4356:
4242:
4202:
3490:
3412:
3334:
3256:
3119:
1438:
267:
218:, and the paired sample test is converted to a one-sample test by replacing each pair of numbers
11562:
7145:
5211:
5109:
4067:
3530:
2519:
14604:
14534:
14327:
14264:
14019:
13906:
12903:
12800:
12707:
12586:
12485:
11376:
includes implementation of the
Wilcoxon signed-rank test in C++, C#, Delphi, Visual Basic, etc.
9827:
9795:
5246:
5015:
3925:
3783:
3452:
3374:
2462:
2405:
544:: The rank of the smallest observation is one, the rank of the next smallest is two, and so on.
54:
20:
3083:
3056:
14769:
14629:
14571:
14514:
14340:
14233:
14142:
13868:
13752:
13611:
13603:
13493:
13485:
13300:
13196:
13174:
13133:
13098:
13065:
13011:
12986:
12941:
12880:
12840:
12642:
12465:
12334:
11901:
Conover, William Jay (1973). "On Methods of Handling Ties in the Wilcoxon Signed-Rank Test".
10482:
10448:
10113:
10079:
7728:
5413:
5388:
5368:
5191:
4820:
4573:
4396:
2905:
2796:
11219:
7318:
3296:
3162:
2963:
2348:
1992:
679:
612:
14552:
14127:
14076:
14052:
14014:
13932:
13911:
13863:
13742:
13720:
13689:
13598:
13475:
13426:
13344:
13317:
13273:
13229:
12991:
12767:
12647:
10964:
10615:
10017:
9906:
9879:
9768:
9305:
8963:
8302:
8275:
7881:
7761:
7553:
7118:
6970:
6923:
6853:
6774:
6339:
6259:
6190:. Formally, suppose that there is a set of observations all having the same absolute value
6114:
6087:
6060:
6033:
5960:
5933:
5906:
5644:
4952:
4707:
1959:
1932:
1234:
1204:
857:
6403:
5341:
2650:
2590:
2269:
To explain the null and alternative hypotheses formally, assume that the data consists of
8:
14699:
14624:
14547:
14228:
13992:
13985:
13947:
13855:
13835:
13807:
13540:
13406:
13401:
13391:
13383:
13201:
13162:
13052:
13042:
12951:
12730:
12686:
12604:
12529:
12431:
12418:
11882:
Derrick, B; White, P (2017). "Comparing Two Samples from an Individual Likert Question".
11487:
11150:{\displaystyle |W|<W_{\operatorname {crit} (\alpha =0.05,\ 9{\text{, two-sided}})}=15}
9738:
6827:
6801:
6539:
6213:
74:
46:
11398:
includes an implementation of the Wilcoxon signed-rank test in C# for .NET applications.
5141:. Therefore the sample would be judged significantly positive at any significance level
4393:
are stochastically smaller than a distribution symmetric about zero, that is, for every
102:(1956) in his influential textbook on non-parametric statistics. Siegel used the symbol
14713:
14524:
14378:
14274:
14223:
14099:
13996:
13980:
13957:
13734:
13468:
13451:
13411:
13322:
13217:
13179:
13150:
13110:
13070:
13016:
12933:
12619:
12614:
11543:
11199:
10427:
10058:
9933:
9859:
9285:
9265:
9200:
9180:
9160:
9040:
8255:
8235:
7998:
7664:
7644:
7580:
6950:
6903:
6880:
6754:
6734:
6686:
6519:
6499:
6479:
6319:
6299:
6279:
6239:
6193:
5569:
5549:
4570:
are stochastically larger than a distribution symmetric about zero, that is, for every
3031:
3000:
2943:
2822:
2717:
2697:
2673:
2630:
2610:
2499:
2442:
2385:
2328:
2296:
2276:
1912:
1181:
1158:
884:
833:
813:
712:
311:
The data for a one-sample test is a sample in which each observation is a real number:
42:
11837:
Pratt, J. (1959). "Remarks on zeros and ties in the Wilcoxon signed rank procedures".
6798:
is exactly as likely to be positive as it is negative. Therefore the probability that
4279:, and therefore, under the null hypothesis, the distribution is symmetric about zero.
2313:
can be assumed symmetric, then the null and alternative hypotheses are the following:
14708:
14619:
14589:
14581:
14401:
14392:
14317:
14248:
14104:
14089:
14064:
13952:
13893:
13759:
13747:
13373:
13290:
13234:
13157:
13001:
12923:
12702:
12576:
12340:
12313:
12173:
12156:
12059:
11774:
11682:
11620:
11568:
11466:
11401:
11293:= (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9) = 45. Hence, the rank correlation is 9/45, so
14644:
14599:
14363:
14350:
14243:
14218:
14152:
14084:
13962:
13570:
13463:
13396:
13309:
13256:
13075:
12946:
12740:
12624:
12539:
12506:
12299:
12219:
12215:
12168:
11914:
11910:
11850:
11846:
11535:
11527:
11412:
5047:, and therefore the sample is not significantly positive at any significance level
106:
for the test statistic, and consequently, the test is sometimes referred to as the
14561:
14305:
14167:
14094:
13769:
13643:
13616:
13593:
13562:
13189:
13184:
13138:
12868:
12519:
11766:
6824:
under the null hypothesis is equal to the number of sign combinations that yield
5896:{\displaystyle |X_{3}|<|X_{2}|=|X_{5}|<|X_{6}|<|X_{1}|=|X_{4}|=|X_{7}|.}
14051:
4817:. It can also happen that there are tied observations. This means that for some
14510:
14505:
12968:
12898:
12544:
11422:
10958:
703:
211:
91:
69:-test for dependent samples"). The Wilcoxon test is a good alternative to the
11539:
6516:
significant, ties are broken the other way, and when large absolute values of
14753:
14667:
14634:
14497:
14458:
14269:
14238:
13702:
13656:
13261:
12963:
12790:
12554:
12549:
12392:
10932:
568:
99:
14609:
14542:
14519:
14434:
13764:
13060:
12958:
12893:
12835:
12820:
12757:
12712:
5597:
2263:
1983:
carry the same information, any of them may be used as the test statistic.
12410:- Nonparametric effect size estimators (Copyright 2015 by Karl L. Weunsch)
6711:-value is computed for every possible way of breaking ties, and the final
3557:
These can also be expressed more directly in terms of the original pairs:
14652:
14614:
14297:
14198:
14060:
13873:
13840:
13332:
13249:
13244:
12888:
12845:
12825:
12805:
12795:
12564:
11382:
implements various one-tailed and two-tailed versions of the test in the
11251:
5338:
If the distribution of the observations is symmetric, then the values of
215:
13498:
12978:
12678:
12609:
12559:
12534:
12454:
12397:
11547:
11379:
9554:{\displaystyle \sigma ^{2}={\frac {n(n+1)(2n+1)-z(z+1)(2z+1)-c/2}{6}},}
12304:
11392:
includes an implementation of the Wilcoxon signed-rank test in Python.
13651:
13503:
13123:
12918:
12830:
12815:
12810:
12775:
11441:
9423:{\displaystyle \mathbf {E} ={\frac {n(n+1)}{4}}-{\frac {z(z+1)}{4}}.}
79:
34:
11531:
13167:
12785:
12662:
12657:
12652:
11265:
is reported, the rank correlation r is equal to the test statistic
6967:. However, there is an efficient recursion for the distribution of
3017:
is stochastically smaller than a distribution symmetric about zero.
11415:
HypothesisTests package includes the Wilcoxon signed-rank test as
9679:
Wilcoxon originally defined the Wilcoxon rank-sum statistic to be
9148:{\displaystyle M(t)={\frac {1}{2^{n}}}\prod _{j=1}^{n}(1+e^{jt}).}
3048:
is stochastically larger than a distribution symmetric about zero.
14672:
14373:
10989:
801:{\displaystyle T=\sum _{i=1}^{N}\operatorname {sgn}(X_{i})R_{i}.}
6627:. On the other hand, any tiebreaking rule will assign the ranks
4704:
In real data, it sometimes happens that there is an observation
14594:
13575:
13549:
13529:
12780:
12571:
11488:"Wilcoxon signed-rank test - Handbook of Biological Statistics"
11193:
that the median of pairwise differences is different from zero.
2692:
stochastically smaller than a distribution symmetric about zero
2255:
70:
9950:-distributed random variables is positive can be estimated as
7758:. Under the null hypothesis, the probability mass function of
4688:{\displaystyle Pr(X_{i}<Y_{i}-x)\leq \Pr(X_{i}>Y_{i}+x)}
4511:{\displaystyle Pr(X_{i}<Y_{i}-x)\geq \Pr(X_{i}>Y_{i}+x)}
2841:
stochastically larger than a distribution symmetric about zero
1106:{\displaystyle |X_{\sigma (1)}|<\dots <|X_{\sigma (n)}|}
12423:
9034:, just like a conventional fourth-order Edgeworth expansion.
8173:{\displaystyle 2p_{n}(t^{+})=p_{n-1}(t^{+})+p_{n-1}(t^{+}-n)}
7505:{\displaystyle u_{n}(t^{+})=u_{n-1}(t^{+})+u_{n-1}(t^{+}-n).}
6586:
This sample is significantly positive at the one-sided level
6356:. The average rank procedure therefore assigns them the rank
94:(1892–1965) who, in a single paper, proposed both it and the
12403:
A table of critical values for the Wilcoxon signed-rank test
2667:, and the test is also a test for the location of the mean.
12514:
5546:
differ by at least a fixed constant that is independent of
5385:, and for a test which is randomized to have level exactly
2259:
12339:. Springer Science & Business Media. pp. 99–100.
11186:{\displaystyle \therefore {\text{failed to reject }}H_{0}}
6578:
The average rank procedure assigns these the signed ranks
5282:
an insignificant sample causes it to appear significantly
6296:. If the ties among the observations with absolute value
3240:. In this case, the null and alternative hypotheses are:
98:
for two independent samples. The test was popularized by
12408:
Brief guide by experimental psychologist Karl L. Weunsch
6877:. This can be used to compute the exact distribution of
6316:
were broken, then these observations would occupy ranks
5098:−1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, −13.
1022:{\displaystyle \sigma :\{1,\dots ,n\}\to \{1,\dots ,n\}}
126:
For a paired sample test, the data consists of a sample
12157:"Edgeworth approximations for rank sum test statistics"
11759:
11404:
implements this test using "Wilcoxon rank sum test" as
9282:
of each group of tied observations. The expectation of
6276:
observations have absolute value less than or equal to
8858:
5612:
2027:
11765:
11564:
Non-parametric statistics for the behavioral sciences
11222:
11202:
11164:
11082:
11001:
10967:
10941:
10917:
10646:
10618:
10595:
10574:
10519:
10485:
10451:
10430:
10226:
10205:
10150:
10116:
10082:
10061:
10020:
9956:
9936:
9909:
9882:
9862:
9830:
9798:
9771:
9741:
9685:
9567:
9436:
9335:
9308:
9288:
9268:
9223:
9203:
9183:
9163:
9063:
9043:
8993:
8966:
8841:
8652:
8332:
8305:
8278:
8258:
8238:
8186:
8064:
8021:
8001:
7961:
7918:
7884:
7791:
7764:
7731:
7687:
7667:
7647:
7603:
7583:
7556:
7518:
7396:
7347:
7321:
7272:
7239:
7190:
7148:
7121:
7083:
7043:
7000:
6973:
6953:
6926:
6906:
6883:
6856:
6830:
6804:
6777:
6757:
6737:
6689:
6641:
6592:
6542:
6522:
6502:
6482:
6429:
6406:
6362:
6342:
6322:
6302:
6282:
6262:
6242:
6216:
6196:
6144:
6117:
6090:
6063:
6036:
5990:
5963:
5936:
5909:
5731:
5690:
5647:
5610:
5572:
5552:
5497:
5442:
5416:
5391:
5371:
5344:
5299:
5249:
5214:
5194:
5147:
5112:
5053:
5018:
4955:
4889:
4849:
4823:
4783:
4737:
4710:
4602:
4576:
4536:
4425:
4399:
4359:
4301:
4245:
4205:
4142:
4096:
4070:
4000:
3954:
3928:
3858:
3812:
3786:
3718:
3672:
3626:
3576:
3533:
3493:
3455:
3415:
3377:
3337:
3299:
3259:
3200:
3165:
3122:
3086:
3059:
3034:
3003:
2966:
2946:
2908:
2849:
2825:
2799:
2740:
2720:
2700:
2676:
2653:
2633:
2613:
2593:
2552:
2522:
2502:
2465:
2445:
2408:
2388:
2351:
2331:
2299:
2279:
2083:
2025:
1995:
1962:
1935:
1915:
1548:
1481:
1441:
1264:
1237:
1207:
1198:
is closely related to two other test statistics. The
1184:
1161:
1119:
1035:
967:
907:
887:
860:
836:
816:
735:
715:
682:
641:
615:
577:
553:
504:
438:
367:
317:
270:
224:
132:
14336:
Autoregressive conditional heteroskedasticity (ARCH)
11246:
Mann–Whitney_U_test § Rank-biserial_correlation
9735:, it is also possible to assign ranks between 0 and
8326:
are asymptotically normal with means and variances:
6947:
sums, which is intractable for all but the smallest
6920:
by considering all possibilities requires computing
6850:
divided by the number of possible sign combinations
11884:
International Journal of Mathematics and Statistics
7871:{\displaystyle \Pr(T^{+}=t^{+})=u_{n}(t^{+})/2^{n}}
5725:For example, suppose that the observations satisfy
5181:{\displaystyle \alpha >109/2^{13}\approx 0.0133}
13798:
12054:Gibbons, Jean D.; Chakraborti, Subhabrata (2011).
12053:
11289:= 9. The sample size of 9 has a total rank sum of
11285:. Using the above example, the test statistic is
11228:
11208:
11185:
11149:
11067:
10980:
10949:
10923:
10665:
10631:
10603:
10580:
10557:
10504:
10470:
10436:
10234:
10211:
10188:
10135:
10101:
10067:
10033:
10006:
9942:
9922:
9895:
9868:
9848:
9816:
9784:
9753:
9720:
9664:
9553:
9422:
9321:
9294:
9274:
9254:
9209:
9189:
9169:
9147:
9049:
9026:
8979:
8952:
8827:
8636:
8318:
8291:
8264:
8244:
8221:
8172:
8050:
8007:
7987:
7947:
7897:
7870:
7777:
7750:
7717:
7673:
7653:
7633:
7589:
7569:
7542:
7504:
7382:
7333:
7307:
7258:
7225:
7176:
7134:
7107:
7069:
7029:
6986:
6959:
6939:
6912:
6889:
6869:
6842:
6816:
6790:
6763:
6743:
6722:
6695:
6668:
6619:
6558:
6528:
6508:
6488:
6453:
6412:
6388:
6348:
6328:
6308:
6288:
6268:
6248:
6228:
6202:
6182:
6130:
6103:
6076:
6049:
6022:
5976:
5949:
5922:
5895:
5714:
5660:
5625:
5578:
5558:
5538:
5483:
5428:
5397:
5377:
5350:
5323:
5270:
5235:
5200:
5180:
5133:
5087:{\displaystyle \alpha <55/2^{12}\approx 0.0134}
5086:
5039:
4996:0, 2, 3, 4, 6, 7, 8, 9, 11, 14, 15, 17, −18.
4968:
4941:
4875:
4835:
4809:
4769:
4723:
4687:
4588:
4562:
4510:
4411:
4385:
4333:
4271:
4231:
4186:
4128:
4082:
4044:
3986:
3940:
3902:
3844:
3798:
3760:
3704:
3658:
3608:
3545:
3519:
3467:
3441:
3389:
3363:
3311:
3285:
3232:
3194:be the joint cumulative distribution of the pairs
3186:
3148:
3099:
3072:
3040:
3009:
2978:
2952:
2920:
2894:
2831:
2811:
2785:
2726:
2706:
2682:
2659:
2639:
2619:
2599:
2579:
2534:
2508:
2477:
2451:
2420:
2394:
2363:
2337:
2305:
2285:
2244:
2234:
2069:
2011:
1975:
1948:
1921:
1901:
1534:
1467:
1427:
1250:
1220:
1190:
1167:
1147:
1105:
1021:
953:
893:
873:
842:
822:
800:
721:
694:
668:
627:
601:
559:
536:
490:
422:
349:
296:
256:
203:{\displaystyle (X_{1},Y_{1}),\dots ,(X_{n},Y_{n})}
202:
8792:
8710:
6420:which the test does not reject form an interval.
14751:
9686:
8653:
8187:
7792:
7037:to be the number of sign combinations for which
5498:
5443:
5358:which the test does not reject form an interval.
4647:
4470:
2874:
2850:
2765:
2741:
2553:
13884:Multivariate adaptive regression splines (MARS)
12208:Journal of the American Statistical Association
11903:Journal of the American Statistical Association
11839:Journal of the American Statistical Association
10666:{\displaystyle \operatorname {sgn} \cdot R_{i}}
6536:are significant, ties are broken so as to make
12417:, volume 3, article 1. doi:10.2466/11.IT.3.1.
12393:Example of using the Wilcoxon signed-rank test
11773:(Third ed.). John Wiley & Sons, Inc.
11676:
11670:
850:to its distribution under the null hypothesis.
12439:
11824:Some Rapid Approximate Statistical Procedures
6731:-values requires knowing the distribution of
5679:This procedure assigns numbers between 1 and
5004:1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, −12.
12318:: CS1 maint: DOI inactive as of June 2024 (
12285:
12283:
11881:
11614:
11465:(3rd ed.). John Wiley & Sons, Inc.
11370:, where x and y are vectors of equal length.
7712:
7688:
7628:
7604:
7537:
7519:
7512:The formula is true because every subset of
7102:
7084:
7077:. This is equal to the number of subsets of
6448:
6430:
5709:
5691:
5641:When the data does not have ties, the ranks
2895:{\displaystyle \Pr(X<-x)\leq \Pr(X>x)}
2786:{\displaystyle \Pr(X<-x)\geq \Pr(X>x)}
2222:
2176:
2150:
2104:
2070:{\displaystyle {\tfrac {1}{2}}(X_{i}+X_{j})}
1016:
998:
992:
974:
12097:Pratt and Gibbons, pp. 148–149, pp. 186–187
11875:
11769:; Wolfe, Douglas A.; Chicken, Eric (2014).
11513:"Individual comparisons by ranking methods"
11328:
6236:observations have absolute value less than
498:, and use this sorted list to assign ranks
12484:
12446:
12432:
12361:"Wilcox signed-rank test: SAS instruction"
12150:
12148:
12058:(Fifth ed.). Chapman & Hall/CRC.
11337:includes an implementation of the test as
11254:for the signed-rank test, one can use the
4731:in the sample which equals zero or a pair
3712:have the same distribution. Equivalently,
1475:equals the sum of all the ranks, which is
13097:
12303:
12280:
12172:
11615:Pratt, John W.; Gibbons, Jean D. (1981).
11567:. New York: McGraw-Hill. pp. 75–83.
11068:{\displaystyle W=1.5+1.5-3-4-5-6+7+8+9=9}
9731:Instead of assigning ranks between 1 and
9674:
7995:under the null hypothesis when there are
5633:, Wilcoxon's method outperforms Pratt's.
1542:, these three statistics are related by:
669:{\displaystyle \operatorname {sgn}(x)=-1}
12332:
11821:
11510:
11309:is the smaller of the two rank sums, so
9302:is still zero, while the expectation of
5324:{\displaystyle \operatorname {sgn}(0)=0}
602:{\displaystyle \operatorname {sgn}(x)=1}
12205:
12199:
12154:
12145:
11900:
11504:
11460:
7597:, in which case it is also a subset of
4942:{\displaystyle X_{i}-Y_{i}=X_{j}-Y_{j}}
4187:{\displaystyle (Y_{i}+\mu ,X_{i}-\mu )}
4045:{\displaystyle (Y_{i}+\mu ,X_{i}-\mu )}
3903:{\displaystyle (Y_{i}+\mu ,X_{i}-\mu )}
2271:independent and identically distributed
423:{\displaystyle |X_{1}|,\dots ,|X_{n}|.}
14752:
14410:Kaplan–Meier estimator (product limit)
11560:
11454:
7142:. The base cases of the recursion are
491:{\displaystyle |X_{1}|,\dots ,|X_{n}|}
14483:
14050:
13797:
13096:
12866:
12483:
12427:
12289:
11836:
11830:
11554:
9824:, and the modified negative-rank sum
7681:from the subset produces a subset of
6715:-value is the mean of the tie-broken
5539:{\displaystyle \Pr(X_{i}+X_{j}<0)}
5484:{\displaystyle \Pr(X_{i}+X_{j}>0)}
4985:were no ties. This is now called the
1535:{\displaystyle 1+2+\dots +n=n(n+1)/2}
14720:
14420:Accelerated failure time (AFT) model
11679:Statistical Inference Based on Ranks
10924:{\displaystyle \operatorname {sgn} }
10581:{\displaystyle \operatorname {sgn} }
10212:{\displaystyle \operatorname {sgn} }
9262:where the sum is over all the sizes
8222:{\displaystyle \Pr(T^{+}\leq t^{+})}
560:{\displaystyle \operatorname {sgn} }
14732:
14015:Analysis of variance (ANOVA, anova)
12867:
12056:Nonparametric Statistical Inference
10007:{\displaystyle 2T_{0}^{+}/(n(n-1))}
6582:2.5, 2.5, 2.5, 2.5, 5, 6, −7.
5365:For a fixed significance threshold
3110:
954:{\displaystyle |X_{j}|\leq |X_{i}|}
53:. For two matched samples, it is a
13:
14110:Cochran–Mantel–Haenszel statistics
12736:Pearson product-moment correlation
12161:Statistics and Probability Letters
11677:Hettmansperger, Thomas P. (1984).
11463:Practical nonparametric statistics
9037:The moment generating function of
8681:
7905:is closely related to the integer
2249:
2173:
2101:
537:{\displaystyle R_{1},\dots ,R_{n}}
350:{\displaystyle X_{1},\dots ,X_{n}}
14:
14781:
12381:
11771:Nonparametric Statistical Methods
9792:, the modified positive-rank sum
9721:{\displaystyle \min(T^{+},T^{-})}
8015:observations in the sample, then
4699:
4521:One-sided alternative hypothesis
4344:One-sided alternative hypothesis
4055:Two-sided alternative hypothesis
3913:One-sided alternative hypothesis
3771:One-sided alternative hypothesis
3478:Two-sided alternative hypothesis
3400:One-sided alternative hypothesis
3322:One-sided alternative hypothesis
3020:One-sided alternative hypothesis
2989:One-sided alternative hypothesis
2488:Two-sided alternative hypothesis
2431:One-sided alternative hypothesis
2374:One-sided alternative hypothesis
117:
14731:
14719:
14707:
14694:
14693:
14484:
11617:Concepts of Nonparametric Theory
9337:
8424:
8366:
8338:
7718:{\displaystyle \{1,\dots ,n-1\}}
7634:{\displaystyle \{1,\dots ,n-1\}}
6669:{\displaystyle \alpha =14/2^{7}}
6620:{\displaystyle \alpha =14/2^{7}}
1148:{\displaystyle R_{\sigma (i)}=i}
214:. Then they may be converted to
14369:Least-squares spectral analysis
12353:
12326:
12271:
12262:
12253:
12244:
12235:
12226:
12190:
12181:
12136:
12127:
12118:
12109:
12100:
12091:
12082:
12073:
12047:
12038:
12029:
12020:
12011:
12002:
11999:Pratt and Gibbons, pp. 163, 166
11993:
11984:
11975:
11966:
11957:
11948:
11939:
11930:
11921:
11894:
11866:
11857:
11815:
11806:
11797:
11788:
11750:
11741:
11732:
11723:
11714:
11705:
11696:
11661:
11652:
11643:
11416:
11405:
11338:
10558:{\displaystyle x_{2,i}-x_{1,i}}
10189:{\displaystyle x_{2,i}-x_{1,i}}
9765:. The modified signed-rank sum
9255:{\displaystyle c=\sum t^{3}-t,}
8058:satisfies a similar recursion:
6723:Computing the null distribution
5626:{\displaystyle {\tfrac {1}{2}}}
5593:-values can no longer be used.
2245:Null and alternative hypotheses
13350:Mean-unbiased minimum-variance
12453:
12388:Wilcoxon Signed-Rank Test in R
12336:Introductory Statistics with R
12220:10.1080/01621459.1967.10500917
12088:Pratt and Gibbons, pp. 148–149
11972:Pratt and Gibbons, pp. 168–169
11915:10.1080/01621459.1973.10481460
11851:10.1080/01621459.1959.10501526
11729:Pratt and Gibbons, pp. 155–156
11702:Pratt and Gibbons, pp. 146–147
11634:
11608:
11599:
11590:
11581:
11480:
11269:divided by the total rank sum
11239:
11136:
11110:
11092:
11084:
10001:
9998:
9986:
9980:
9715:
9689:
9630:
9617:
9584:
9578:
9525:
9510:
9507:
9495:
9486:
9471:
9468:
9456:
9408:
9396:
9378:
9366:
9354:
9341:
9139:
9117:
9073:
9067:
9021:
8997:
8936:
8921:
8918:
8906:
8890:
8878:
8819:
8797:
8784:
8769:
8766:
8754:
8705:
8699:
8690:
8684:
8675:
8656:
8618:
8603:
8600:
8588:
8572:
8566:
8544:
8529:
8526:
8514:
8502:
8489:
8473:
8460:
8434:
8428:
8407:
8395:
8383:
8370:
8355:
8342:
8216:
8190:
8167:
8148:
8126:
8113:
8091:
8078:
8045:
8032:
7942:
7929:
7850:
7837:
7821:
7795:
7543:{\displaystyle \{1,\dots ,n\}}
7496:
7477:
7455:
7442:
7420:
7407:
7369:
7357:
7308:{\displaystyle u_{n}(t^{+})=0}
7296:
7283:
7226:{\displaystyle u_{0}(t^{+})=0}
7214:
7201:
7165:
7159:
7108:{\displaystyle \{1,\dots ,n\}}
7024:
7011:
6900:Computing the distribution of
6552:
6544:
6454:{\displaystyle \{1,\dots ,n\}}
6375:
6363:
6163:
6145:
6003:
5991:
5886:
5871:
5863:
5848:
5840:
5825:
5817:
5802:
5794:
5779:
5771:
5756:
5748:
5733:
5715:{\displaystyle \{1,\dots ,n\}}
5533:
5501:
5478:
5446:
5312:
5306:
4764:
4738:
4682:
4650:
4641:
4609:
4505:
4473:
4464:
4432:
4328:
4302:
4181:
4143:
4123:
4097:
4039:
4001:
3981:
3955:
3897:
3859:
3839:
3813:
3755:
3743:
3734:
3722:
3699:
3673:
3653:
3627:
3603:
3577:
3227:
3201:
3181:
3169:
2889:
2877:
2868:
2853:
2780:
2768:
2759:
2744:
2568:
2556:
2064:
2038:
1867:
1855:
1837:
1825:
1738:
1726:
1695:
1683:
1631:
1619:
1588:
1576:
1521:
1509:
1134:
1128:
1099:
1093:
1087:
1075:
1061:
1055:
1049:
1037:
995:
947:
932:
924:
909:
854:The ranks are defined so that
782:
769:
654:
648:
590:
584:
484:
469:
455:
440:
413:
398:
384:
369:
251:
225:
197:
171:
159:
133:
39:statistical hypothesis testing
1:
14663:Geographic information system
13879:Simultaneous equations models
12398:An online version of the test
11447:
10418:order by absolute difference
7383:{\displaystyle t>n(n+1)/2}
5289:Pratt therefore proposed the
4770:{\displaystyle (X_{i},Y_{i})}
4334:{\displaystyle (X_{i},Y_{i})}
4239:has the same distribution as
4129:{\displaystyle (X_{i},Y_{i})}
3987:{\displaystyle (X_{i},Y_{i})}
3845:{\displaystyle (X_{i},Y_{i})}
3761:{\displaystyle F(x,y)=F(y,x)}
3705:{\displaystyle (Y_{i},X_{i})}
3659:{\displaystyle (X_{i},Y_{i})}
3609:{\displaystyle (X_{i},Y_{i})}
3233:{\displaystyle (X_{i},Y_{i})}
2714:-distributed random variable
2647:is defined, then the mean is
2580:{\displaystyle \Pr(X=\mu )=0}
257:{\displaystyle (X_{i},Y_{i})}
65:-test for matched pairs" or "
13846:Coefficient of determination
13457:Uniformly most powerful test
12174:10.1016/0167-7152(95)00164-H
11511:Wilcoxon, Frank (Dec 1945).
10950:{\displaystyle {\text{abs}}}
10604:{\displaystyle {\text{abs}}}
10235:{\displaystyle {\text{abs}}}
9197:and the number of zeros was
8051:{\displaystyle p_{n}(t^{+})}
7948:{\displaystyle p_{n}(t^{+})}
7030:{\displaystyle u_{n}(t^{+})}
6635:At the same one-sided level
4883:(in the one-sample case) or
2273:samples from a distribution
7:
14415:Proportional hazards models
14359:Spectral density estimation
14341:Vector autoregression (VAR)
13775:Maximum posterior estimator
13007:Randomized controlled trial
11430:
11417:value(SignedRankTest(x, y))
10048:
9856:are defined analogously to
9027:{\displaystyle O(n^{-3/2})}
7988:{\displaystyle T^{+}=t^{+}}
7390:. The recursive formula is
7259:{\displaystyle t^{+}\neq 0}
7070:{\displaystyle T^{+}=t^{+}}
6897:under the null hypothesis.
6683:method, the test statistic
6676:, this is not significant.
6631:1, 2, 3, 4, 5, 6, −7.
6574:1, 1, 1, 1, 2, 3, −4.
6389:{\displaystyle (k+\ell )/2}
6183:{\displaystyle (5+6+7)/3=6}
6023:{\displaystyle (2+3)/2=2.5}
5291:signed-rank zero procedure.
4876:{\displaystyle X_{i}=X_{j}}
4810:{\displaystyle X_{i}=Y_{i}}
4563:{\displaystyle X_{i}-Y_{i}}
4386:{\displaystyle X_{i}-Y_{i}}
4272:{\displaystyle Y_{i}-X_{i}}
4232:{\displaystyle X_{i}-Y_{i}}
4194:have the same distribution.
4052:have the same distribution.
3910:have the same distribution.
3520:{\displaystyle X_{i}-Y_{i}}
3442:{\displaystyle X_{i}-Y_{i}}
3364:{\displaystyle X_{i}-Y_{i}}
3286:{\displaystyle X_{i}-Y_{i}}
3149:{\displaystyle X_{i}-Y_{i}}
1468:{\displaystyle T^{+}+T^{-}}
297:{\displaystyle X_{i}-Y_{i}}
16:Statistical hypothesis test
10:
14786:
14175:Multivariate distributions
12595:Average absolute deviation
11437:Mann–Whitney–Wilcoxon test
11243:
11216:-value for this result is
10044:
7177:{\displaystyle u_{0}(0)=1}
5236:{\displaystyle 109/2^{13}}
5188:. The paradox is that, if
5134:{\displaystyle 109/2^{13}}
4083:{\displaystyle \mu \neq 0}
3546:{\displaystyle \mu \neq 0}
2535:{\displaystyle \mu \neq 0}
85:
61:-test (also known as the "
57:like the paired Student's
18:
14689:
14643:
14580:
14533:
14496:
14492:
14479:
14451:
14433:
14400:
14391:
14349:
14296:
14257:
14206:
14197:
14163:Structural equation model
14118:
14075:
14071:
14046:
14005:
13971:
13925:
13892:
13854:
13821:
13817:
13793:
13733:
13642:
13561:
13525:
13516:
13499:Score/Lagrange multiplier
13484:
13437:
13382:
13308:
13299:
13109:
13105:
13092:
13051:
13025:
12977:
12932:
12914:Sample size determination
12879:
12875:
12862:
12766:
12721:
12695:
12677:
12633:
12585:
12505:
12496:
12492:
12479:
12461:
12277:Pratt and Gibbons, p. 191
12268:Pratt and Gibbons, p. 159
12259:Pratt and Gibbons, p. 158
12232:Pratt and Gibbons, p. 193
12155:Kolassa, John E. (1995).
12142:Pratt and Gibbons, p. 149
12133:Pratt and Gibbons, p. 187
12124:Pratt and Gibbons, p. 187
12115:Pratt and Gibbons, p. 187
12026:Pratt and Gibbons, p. 171
12017:Pratt and Gibbons, p. 166
11990:Pratt and Gibbons, p. 170
11945:Pratt and Gibbons, p. 164
11927:Pratt and Gibbons, p. 162
11812:Hettmansperger, pp. 49–50
11803:Pratt and Gibbons, p. 147
11794:Pratt and Gibbons, p. 147
11747:Pratt and Gibbons, p. 155
11738:Hettmansperger, pp. 49–50
11711:Hettmansperger, pp. 30–31
11681:. John Wiley & Sons.
11658:Pratt and Gibbons, p. 150
11649:Pratt and Gibbons, p. 148
11640:Pratt and Gibbons, p. 148
11256:rank-biserial correlation
10513:
10479:
10445:
10424:
10144:
10110:
10076:
10055:
9849:{\displaystyle T_{0}^{-}}
9817:{\displaystyle T_{0}^{+}}
7661:, in which case removing
5271:{\displaystyle 55/2^{12}}
5040:{\displaystyle 55/2^{12}}
4987:reduced sample procedure.
3941:{\displaystyle \mu >0}
3799:{\displaystyle \mu <0}
3468:{\displaystyle \mu >0}
3390:{\displaystyle \mu <0}
2478:{\displaystyle \mu >0}
2421:{\displaystyle \mu <0}
28:Wilcoxon signed-rank test
14765:Nonparametric statistics
14658:Environmental statistics
14180:Elliptical distributions
13973:Generalized linear model
13902:Simple linear regression
13672:Hodges–Lehmann estimator
13129:Probability distribution
13038:Stochastic approximation
12600:Coefficient of variation
12415:Comprehensive Psychology
12333:Dalgaard, Peter (2008).
12292:Comprehensive Psychology
11822:Wilcoxon, Frank (1949).
11329:Software implementations
9217:. The tie correction is
7955:is the probability that
7577:either does not contain
6474:Conservative tiebreaking
6057:is assigned rank 4, and
4979:
3100:{\displaystyle \mu _{0}}
3073:{\displaystyle \mu _{0}}
90:The test is named after
41:used either to test the
19:Not to be confused with
14318:Cross-correlation (XCF)
13926:Non-standard predictors
13360:Lehmann–Scheffé theorem
13033:Adaptive clinical trial
11561:Siegel, Sidney (1956).
11492:www.biostathandbook.com
11461:Conover, W. J. (1999).
10505:{\displaystyle x_{1,i}}
10471:{\displaystyle x_{2,i}}
10136:{\displaystyle x_{1,i}}
10102:{\displaystyle x_{2,i}}
9057:has the exact formula:
7751:{\displaystyle t^{+}-n}
5636:
5429:{\displaystyle i\neq j}
5398:{\displaystyle \alpha }
5378:{\displaystyle \alpha }
5201:{\displaystyle \alpha }
4836:{\displaystyle i\neq j}
4589:{\displaystyle x\geq 0}
4412:{\displaystyle x\geq 0}
2921:{\displaystyle x\geq 0}
2812:{\displaystyle x\geq 0}
14714:Mathematics portal
14535:Engineering statistics
14443:Nelson–Aalen estimator
14020:Analysis of covariance
13907:Ordinary least squares
13831:Pearson product-moment
13235:Statistical functional
13146:Empirical distribution
12979:Controlled experiments
12708:Frequency distribution
12486:Descriptive statistics
12106:Hettmansperger, p. 171
11826:. American Cynamic Co.
11300:If the test statistic
11261:If the test statistic
11230:
11229:{\displaystyle 0.6113}
11210:
11187:
11170:failed to reject
11151:
11069:
10982:
10951:
10925:
10667:
10633:
10605:
10582:
10559:
10506:
10472:
10438:
10236:
10213:
10190:
10137:
10103:
10069:
10035:
10008:
9944:
9924:
9897:
9870:
9850:
9818:
9786:
9755:
9722:
9675:Alternative statistics
9666:
9555:
9424:
9323:
9296:
9276:
9256:
9211:
9191:
9171:
9149:
9116:
9051:
9028:
8981:
8954:
8829:
8638:
8320:
8293:
8266:
8246:
8223:
8174:
8052:
8009:
7989:
7949:
7899:
7872:
7779:
7752:
7719:
7675:
7655:
7635:
7591:
7571:
7544:
7506:
7384:
7335:
7334:{\displaystyle t<0}
7309:
7260:
7227:
7178:
7136:
7109:
7071:
7031:
6988:
6961:
6941:
6914:
6891:
6871:
6844:
6818:
6792:
6765:
6745:
6697:
6670:
6621:
6560:
6530:
6510:
6490:
6455:
6414:
6390:
6350:
6330:
6310:
6290:
6270:
6250:
6230:
6204:
6184:
6132:
6105:
6078:
6051:
6024:
5978:
5951:
5924:
5897:
5716:
5662:
5627:
5596:When the data is on a
5580:
5560:
5540:
5485:
5430:
5399:
5379:
5352:
5325:
5272:
5237:
5202:
5182:
5135:
5088:
5041:
4970:
4943:
4877:
4837:
4811:
4771:
4725:
4689:
4590:
4564:
4512:
4413:
4387:
4335:
4273:
4233:
4188:
4130:
4084:
4046:
3988:
3942:
3904:
3846:
3800:
3762:
3706:
3660:
3610:
3547:
3521:
3469:
3443:
3391:
3365:
3313:
3312:{\displaystyle \mu =0}
3287:
3234:
3188:
3187:{\displaystyle F(x,y)}
3150:
3101:
3074:
3042:
3011:
2980:
2979:{\displaystyle \mu =0}
2954:
2922:
2896:
2833:
2813:
2787:
2728:
2708:
2684:
2661:
2641:
2621:
2601:
2581:
2536:
2510:
2479:
2453:
2422:
2396:
2365:
2364:{\displaystyle \mu =0}
2339:
2307:
2287:
2236:
2071:
2013:
2012:{\displaystyle W_{ij}}
1977:
1950:
1923:
1903:
1536:
1469:
1429:
1252:
1222:
1192:
1169:
1149:
1107:
1023:
955:
895:
875:
844:
824:
802:
762:
723:
696:
695:{\displaystyle x<0}
670:
629:
628:{\displaystyle x>0}
603:
561:
538:
492:
424:
351:
298:
258:
204:
55:paired difference test
21:Wilcoxon rank-sum test
14630:Population statistics
14572:System identification
14306:Autocorrelation (ACF)
14234:Exponential smoothing
14148:Discriminant analysis
14143:Canonical correlation
14007:Partition of variance
13869:Regression validation
13713:(Jonckheere–Terpstra)
13612:Likelihood-ratio test
13301:Frequentist inference
13213:Location–scale family
13134:Sampling distribution
13099:Statistical inference
13066:Cross-sectional study
13053:Observational studies
13012:Randomized experiment
12841:Stem-and-leaf display
12643:Central limit theorem
12308:(inactive 2024-06-26)
12196:Hettmansperger, p. 35
12187:Hettmansperger, p. 37
12079:Hettmansperger, p. 34
11231:
11211:
11188:
11152:
11070:
10983:
10981:{\displaystyle R_{i}}
10952:
10926:
10668:
10634:
10632:{\displaystyle R_{i}}
10606:
10583:
10560:
10507:
10473:
10439:
10237:
10214:
10191:
10138:
10104:
10070:
10036:
10034:{\displaystyle p_{2}}
10009:
9945:
9925:
9923:{\displaystyle T^{-}}
9898:
9896:{\displaystyle T^{+}}
9871:
9851:
9819:
9787:
9785:{\displaystyle T_{0}}
9756:
9723:
9667:
9556:
9425:
9324:
9322:{\displaystyle T^{+}}
9297:
9277:
9257:
9212:
9192:
9172:
9150:
9096:
9052:
9029:
8982:
8980:{\displaystyle T^{+}}
8955:
8830:
8639:
8321:
8319:{\displaystyle T^{-}}
8294:
8292:{\displaystyle T^{+}}
8267:
8247:
8224:
8175:
8053:
8010:
7990:
7950:
7900:
7898:{\displaystyle u_{n}}
7873:
7780:
7778:{\displaystyle T^{+}}
7753:
7720:
7676:
7656:
7641:, or it does contain
7636:
7592:
7572:
7570:{\displaystyle t^{+}}
7545:
7507:
7385:
7336:
7310:
7261:
7228:
7179:
7137:
7135:{\displaystyle t^{+}}
7110:
7072:
7032:
6989:
6987:{\displaystyle T^{+}}
6962:
6942:
6940:{\displaystyle 2^{n}}
6915:
6892:
6872:
6870:{\displaystyle 2^{n}}
6845:
6819:
6793:
6791:{\displaystyle X_{i}}
6766:
6746:
6698:
6671:
6622:
6561:
6531:
6511:
6491:
6456:
6415:
6391:
6351:
6349:{\displaystyle \ell }
6331:
6311:
6291:
6271:
6269:{\displaystyle \ell }
6251:
6231:
6205:
6185:
6133:
6131:{\displaystyle X_{7}}
6106:
6104:{\displaystyle X_{4}}
6079:
6077:{\displaystyle X_{1}}
6052:
6050:{\displaystyle X_{6}}
6025:
5979:
5977:{\displaystyle X_{5}}
5952:
5950:{\displaystyle X_{2}}
5925:
5923:{\displaystyle X_{3}}
5898:
5717:
5663:
5661:{\displaystyle R_{i}}
5628:
5581:
5561:
5541:
5486:
5431:
5400:
5380:
5353:
5326:
5273:
5238:
5203:
5183:
5136:
5102:This has a one-sided
5089:
5042:
5008:This has a one-sided
4971:
4969:{\displaystyle X_{i}}
4944:
4878:
4838:
4812:
4772:
4726:
4724:{\displaystyle X_{i}}
4690:
4591:
4565:
4513:
4414:
4388:
4336:
4274:
4234:
4189:
4131:
4085:
4047:
3989:
3943:
3905:
3847:
3801:
3763:
3707:
3661:
3611:
3548:
3522:
3470:
3444:
3392:
3366:
3314:
3288:
3235:
3189:
3151:
3102:
3075:
3043:
3012:
2981:
2955:
2923:
2897:
2834:
2814:
2788:
2729:
2709:
2685:
2662:
2642:
2622:
2602:
2582:
2537:
2511:
2480:
2454:
2423:
2397:
2366:
2340:
2308:
2288:
2237:
2072:
2014:
1978:
1976:{\displaystyle T^{-}}
1951:
1949:{\displaystyle T^{+}}
1924:
1904:
1537:
1470:
1430:
1253:
1251:{\displaystyle T^{-}}
1223:
1221:{\displaystyle T^{+}}
1193:
1170:
1150:
1108:
1024:
956:
896:
876:
874:{\displaystyle R_{i}}
845:
825:
803:
742:
724:
697:
671:
630:
604:
562:
539:
493:
425:
352:
299:
259:
205:
14553:Probabilistic design
14138:Principal components
13981:Exponential families
13933:Nonlinear regression
13912:General linear model
13874:Mixed effects models
13864:Errors and residuals
13841:Confounding variable
13743:Bayesian probability
13721:Van der Waerden test
13711:Ordered alternative
13476:Multiple comparisons
13355:Rao–Blackwellization
13318:Estimating equations
13274:Statistical distance
12992:Factorial experiment
12525:Arithmetic-Geometric
11954:Conover, pp. 358–359
11936:Conover, pp. 352–353
11667:Conover, pp. 352–357
11220:
11200:
11162:
11080:
10999:
10965:
10939:
10915:
10644:
10616:
10593:
10572:
10517:
10483:
10449:
10428:
10224:
10203:
10148:
10114:
10080:
10059:
10018:
9954:
9934:
9907:
9880:
9860:
9828:
9796:
9769:
9739:
9683:
9565:
9434:
9333:
9306:
9286:
9266:
9221:
9201:
9181:
9161:
9061:
9041:
8991:
8964:
8839:
8650:
8330:
8303:
8276:
8256:
8236:
8184:
8062:
8019:
7999:
7959:
7916:
7882:
7789:
7762:
7729:
7685:
7665:
7645:
7601:
7581:
7554:
7516:
7394:
7345:
7319:
7270:
7237:
7188:
7146:
7119:
7081:
7041:
6998:
6971:
6951:
6924:
6904:
6881:
6854:
6828:
6802:
6775:
6755:
6735:
6687:
6639:
6590:
6540:
6520:
6500:
6480:
6427:
6413:{\displaystyle \mu }
6404:
6360:
6340:
6320:
6300:
6280:
6260:
6240:
6214:
6194:
6142:
6115:
6088:
6061:
6034:
5988:
5961:
5934:
5930:is assigned rank 1,
5907:
5729:
5688:
5645:
5608:
5570:
5550:
5495:
5440:
5414:
5389:
5369:
5351:{\displaystyle \mu }
5342:
5297:
5247:
5212:
5192:
5145:
5110:
5051:
5016:
4953:
4887:
4847:
4821:
4781:
4735:
4708:
4600:
4574:
4534:
4423:
4397:
4357:
4299:
4243:
4203:
4140:
4094:
4068:
3998:
3952:
3926:
3856:
3810:
3784:
3716:
3670:
3624:
3574:
3531:
3527:are symmetric about
3491:
3453:
3449:are symmetric about
3413:
3375:
3371:are symmetric about
3335:
3297:
3293:are symmetric about
3257:
3198:
3163:
3120:
3084:
3057:
3032:
3001:
2964:
2944:
2906:
2847:
2823:
2797:
2738:
2718:
2698:
2674:
2660:{\displaystyle \mu }
2651:
2631:
2611:
2600:{\displaystyle \mu }
2591:
2550:
2520:
2500:
2463:
2443:
2406:
2386:
2349:
2329:
2297:
2277:
2081:
2023:
1993:
1960:
1933:
1913:
1546:
1479:
1439:
1262:
1235:
1205:
1182:
1178:The signed-rank sum
1159:
1117:
1033:
965:
961:. Additionally, if
905:
885:
858:
834:
830:-value by comparing
814:
733:
713:
680:
639:
613:
575:
551:
502:
436:
365:
315:
268:
222:
130:
14625:Official statistics
14548:Methods engineering
14229:Seasonal adjustment
13997:Poisson regressions
13917:Bayesian regression
13856:Regression analysis
13836:Partial correlation
13808:Regression analysis
13407:Prediction interval
13402:Likelihood interval
13392:Confidence interval
13384:Interval estimation
13345:Unbiased estimators
13163:Model specification
13043:Up-and-down designs
12731:Partial correlation
12687:Index of dispersion
12605:Interquartile range
12365:www.stat.purdue.edu
11619:. Springer-Verlag.
11520:Biometrics Bulletin
9974:
9845:
9813:
9761:. These are called
9754:{\displaystyle n-1}
6843:{\displaystyle T=t}
6817:{\displaystyle T=t}
6705:average probability
6559:{\displaystyle |T|}
6229:{\displaystyle k-1}
2960:is symmetric about
2516:is symmetric about
2459:is symmetric about
2402:is symmetric about
2345:is symmetric about
78:alternative to the
75:normal distribution
14645:Spatial statistics
14525:Medical statistics
14425:First hitting time
14379:Whittle likelihood
14030:Degrees of freedom
14025:Multivariate ANOVA
13958:Heteroscedasticity
13770:Bayesian estimator
13735:Bayesian inference
13584:Kolmogorov–Smirnov
13469:Randomization test
13439:Testing hypotheses
13412:Tolerance interval
13323:Maximum likelihood
13218:Exponential family
13151:Density estimation
13111:Statistical theory
13071:Natural experiment
13017:Scientific control
12934:Survey methodology
12620:Standard deviation
12214:(319): 1068–1069.
11981:Pratt, pp. 661–662
11540:10338.dmlcz/135688
11226:
11206:
11183:
11147:
11065:
10978:
10947:
10921:
10663:
10629:
10601:
10578:
10555:
10502:
10468:
10434:
10232:
10209:
10186:
10133:
10099:
10065:
10031:
10004:
9960:
9940:
9920:
9893:
9866:
9846:
9831:
9814:
9799:
9782:
9751:
9718:
9662:
9660:
9551:
9420:
9319:
9292:
9272:
9252:
9207:
9187:
9167:
9145:
9047:
9024:
8977:
8950:
8867:
8825:
8634:
8632:
8316:
8289:
8262:
8242:
8219:
8170:
8048:
8005:
7985:
7945:
7907:partition function
7895:
7868:
7775:
7748:
7715:
7671:
7651:
7631:
7587:
7567:
7540:
7502:
7380:
7331:
7305:
7256:
7223:
7174:
7132:
7105:
7067:
7027:
6984:
6957:
6937:
6910:
6887:
6867:
6840:
6814:
6788:
6761:
6741:
6693:
6666:
6617:
6556:
6526:
6506:
6486:
6469:Random tiebreaking
6451:
6410:
6386:
6346:
6326:
6306:
6286:
6266:
6246:
6226:
6200:
6180:
6138:are assigned rank
6128:
6101:
6074:
6047:
6020:
5984:are assigned rank
5974:
5947:
5920:
5893:
5712:
5677:midrank procedure.
5658:
5623:
5621:
5576:
5556:
5536:
5481:
5426:
5395:
5375:
5348:
5321:
5268:
5233:
5198:
5178:
5131:
5084:
5037:
4966:
4939:
4873:
4833:
4807:
4767:
4721:
4685:
4586:
4560:
4508:
4409:
4383:
4331:
4269:
4229:
4184:
4126:
4080:
4042:
3984:
3938:
3900:
3842:
3796:
3758:
3702:
3656:
3606:
3543:
3517:
3465:
3439:
3387:
3361:
3309:
3283:
3230:
3184:
3146:
3097:
3070:
3038:
3007:
2976:
2950:
2918:
2892:
2829:
2809:
2783:
2724:
2704:
2680:
2657:
2637:
2617:
2597:
2577:
2532:
2506:
2475:
2449:
2418:
2392:
2361:
2335:
2303:
2283:
2232:
2230:
2067:
2036:
2009:
1973:
1946:
1919:
1899:
1897:
1532:
1465:
1425:
1423:
1407:
1329:
1248:
1218:
1188:
1165:
1145:
1103:
1019:
951:
891:
871:
840:
820:
798:
719:
692:
666:
625:
599:
557:
534:
488:
420:
347:
294:
264:by its difference
254:
200:
14760:Statistical tests
14747:
14746:
14685:
14684:
14681:
14680:
14620:National accounts
14590:Actuarial science
14582:Social statistics
14475:
14474:
14471:
14470:
14467:
14466:
14402:Survival function
14387:
14386:
14249:Granger causality
14090:Contingency table
14065:Survival analysis
14042:
14041:
14038:
14037:
13894:Linear regression
13789:
13788:
13785:
13784:
13760:Credible interval
13729:
13728:
13512:
13511:
13328:Method of moments
13197:Parametric family
13158:Statistical model
13088:
13087:
13084:
13083:
13002:Random assignment
12924:Statistical power
12858:
12857:
12854:
12853:
12703:Contingency table
12673:
12672:
12540:Generalized/power
12346:978-0-387-79053-4
12305:10.2466/11.IT.3.1
12065:978-1-4200-7762-9
11780:978-0-470-38737-5
11626:978-1-4612-5933-6
11209:{\displaystyle p}
11171:
11134:
11127:
10945:
10910:
10909:
10906:
10905:
10599:
10437:{\displaystyle i}
10415:
10414:
10230:
10068:{\displaystyle i}
9943:{\displaystyle F}
9869:{\displaystyle T}
9546:
9415:
9385:
9295:{\displaystyle T}
9275:{\displaystyle t}
9210:{\displaystyle z}
9190:{\displaystyle n}
9170:{\displaystyle T}
9094:
9050:{\displaystyle T}
8945:
8944:
8943:
8897:
8866:
8788:
8625:
8551:
8414:
8265:{\displaystyle T}
8245:{\displaystyle n}
8008:{\displaystyle n}
7674:{\displaystyle n}
7654:{\displaystyle n}
7590:{\displaystyle n}
6960:{\displaystyle n}
6913:{\displaystyle T}
6890:{\displaystyle T}
6764:{\displaystyle n}
6744:{\displaystyle T}
6696:{\displaystyle T}
6681:average statistic
6529:{\displaystyle T}
6509:{\displaystyle T}
6489:{\displaystyle T}
6329:{\displaystyle k}
6309:{\displaystyle v}
6289:{\displaystyle v}
6249:{\displaystyle v}
6203:{\displaystyle v}
5620:
5579:{\displaystyle j}
5559:{\displaystyle i}
4341:are exchangeable.
4295:The observations
3570:The observations
3487:The observations
3409:The observations
3331:The observations
3253:The observations
3041:{\displaystyle F}
3010:{\displaystyle F}
2953:{\displaystyle F}
2832:{\displaystyle F}
2727:{\displaystyle X}
2707:{\displaystyle F}
2683:{\displaystyle F}
2640:{\displaystyle F}
2620:{\displaystyle F}
2509:{\displaystyle F}
2452:{\displaystyle F}
2395:{\displaystyle F}
2338:{\displaystyle F}
2306:{\displaystyle F}
2286:{\displaystyle F}
2035:
1922:{\displaystyle T}
1874:
1844:
1758:
1745:
1702:
1651:
1638:
1595:
1389:
1364:
1311:
1286:
1230:negative-rank sum
1200:positive-rank sum
1191:{\displaystyle T}
1168:{\displaystyle i}
894:{\displaystyle j}
881:is the number of
843:{\displaystyle T}
823:{\displaystyle p}
722:{\displaystyle T}
14777:
14735:
14734:
14723:
14722:
14712:
14711:
14697:
14696:
14600:Crime statistics
14494:
14493:
14481:
14480:
14398:
14397:
14364:Fourier analysis
14351:Frequency domain
14331:
14278:
14244:Structural break
14204:
14203:
14153:Cluster analysis
14100:Log-linear model
14073:
14072:
14048:
14047:
13989:
13963:Homoscedasticity
13819:
13818:
13795:
13794:
13714:
13706:
13698:
13697:(Kruskal–Wallis)
13682:
13667:
13622:Cross validation
13607:
13589:Anderson–Darling
13536:
13523:
13522:
13494:Likelihood-ratio
13486:Parametric tests
13464:Permutation test
13447:1- & 2-tails
13338:Minimum distance
13310:Point estimation
13306:
13305:
13257:Optimal decision
13208:
13107:
13106:
13094:
13093:
13076:Quasi-experiment
13026:Adaptive designs
12877:
12876:
12864:
12863:
12741:Rank correlation
12503:
12502:
12494:
12493:
12481:
12480:
12448:
12441:
12434:
12425:
12424:
12375:
12374:
12372:
12371:
12357:
12351:
12350:
12330:
12324:
12323:
12317:
12309:
12307:
12287:
12278:
12275:
12269:
12266:
12260:
12257:
12251:
12248:
12242:
12239:
12233:
12230:
12224:
12223:
12203:
12197:
12194:
12188:
12185:
12179:
12178:
12176:
12152:
12143:
12140:
12134:
12131:
12125:
12122:
12116:
12113:
12107:
12104:
12098:
12095:
12089:
12086:
12080:
12077:
12071:
12069:
12051:
12045:
12042:
12036:
12033:
12027:
12024:
12018:
12015:
12009:
12006:
12000:
11997:
11991:
11988:
11982:
11979:
11973:
11970:
11964:
11961:
11955:
11952:
11946:
11943:
11937:
11934:
11928:
11925:
11919:
11918:
11909:(344): 985–988.
11898:
11892:
11891:
11879:
11873:
11870:
11864:
11861:
11855:
11854:
11845:(287): 655–667.
11834:
11828:
11827:
11819:
11813:
11810:
11804:
11801:
11795:
11792:
11786:
11784:
11767:Hollander, Myles
11763:
11757:
11754:
11748:
11745:
11739:
11736:
11730:
11727:
11721:
11718:
11712:
11709:
11703:
11700:
11694:
11692:
11674:
11668:
11665:
11659:
11656:
11650:
11647:
11641:
11638:
11632:
11630:
11612:
11606:
11603:
11597:
11594:
11588:
11585:
11579:
11578:
11558:
11552:
11551:
11517:
11508:
11502:
11501:
11499:
11498:
11484:
11478:
11476:
11458:
11418:
11407:
11385:
11369:
11368:
11365:
11362:
11359:
11356:
11353:
11350:
11347:
11344:
11341:
11235:
11233:
11232:
11227:
11215:
11213:
11212:
11207:
11192:
11190:
11189:
11184:
11182:
11181:
11172:
11169:
11156:
11154:
11153:
11148:
11140:
11139:
11135:
11132:
11125:
11095:
11087:
11074:
11072:
11071:
11066:
10987:
10985:
10984:
10979:
10977:
10976:
10956:
10954:
10953:
10948:
10946:
10943:
10930:
10928:
10927:
10922:
10672:
10670:
10669:
10664:
10662:
10661:
10638:
10636:
10635:
10630:
10628:
10627:
10610:
10608:
10607:
10602:
10600:
10597:
10587:
10585:
10584:
10579:
10564:
10562:
10561:
10556:
10554:
10553:
10535:
10534:
10511:
10509:
10508:
10503:
10501:
10500:
10477:
10475:
10474:
10469:
10467:
10466:
10443:
10441:
10440:
10435:
10422:
10421:
10241:
10239:
10238:
10233:
10231:
10228:
10218:
10216:
10215:
10210:
10195:
10193:
10192:
10187:
10185:
10184:
10166:
10165:
10142:
10140:
10139:
10134:
10132:
10131:
10108:
10106:
10105:
10100:
10098:
10097:
10074:
10072:
10071:
10066:
10053:
10052:
10049:
10040:
10038:
10037:
10032:
10030:
10029:
10013:
10011:
10010:
10005:
9979:
9973:
9968:
9949:
9947:
9946:
9941:
9929:
9927:
9926:
9921:
9919:
9918:
9902:
9900:
9899:
9894:
9892:
9891:
9875:
9873:
9872:
9867:
9855:
9853:
9852:
9847:
9844:
9839:
9823:
9821:
9820:
9815:
9812:
9807:
9791:
9789:
9788:
9783:
9781:
9780:
9760:
9758:
9757:
9752:
9727:
9725:
9724:
9719:
9714:
9713:
9701:
9700:
9671:
9669:
9668:
9663:
9661:
9654:
9649:
9648:
9629:
9628:
9603:
9602:
9560:
9558:
9557:
9552:
9547:
9542:
9538:
9451:
9446:
9445:
9429:
9427:
9426:
9421:
9416:
9411:
9391:
9386:
9381:
9361:
9353:
9352:
9340:
9328:
9326:
9325:
9320:
9318:
9317:
9301:
9299:
9298:
9293:
9281:
9279:
9278:
9273:
9261:
9259:
9258:
9253:
9242:
9241:
9216:
9214:
9213:
9208:
9196:
9194:
9193:
9188:
9176:
9174:
9173:
9168:
9154:
9152:
9151:
9146:
9138:
9137:
9115:
9110:
9095:
9093:
9092:
9080:
9056:
9054:
9053:
9048:
9033:
9031:
9030:
9025:
9020:
9019:
9015:
8986:
8984:
8983:
8978:
8976:
8975:
8959:
8957:
8956:
8951:
8946:
8939:
8901:
8900:
8899:
8898:
8893:
8873:
8868:
8859:
8849:
8834:
8832:
8831:
8826:
8809:
8808:
8796:
8795:
8789:
8787:
8746:
8730:
8729:
8716:
8714:
8713:
8668:
8667:
8643:
8641:
8640:
8635:
8633:
8626:
8621:
8583:
8552:
8547:
8509:
8501:
8500:
8472:
8471:
8427:
8415:
8410:
8390:
8382:
8381:
8369:
8354:
8353:
8341:
8325:
8323:
8322:
8317:
8315:
8314:
8298:
8296:
8295:
8290:
8288:
8287:
8271:
8269:
8268:
8263:
8251:
8249:
8248:
8243:
8228:
8226:
8225:
8220:
8215:
8214:
8202:
8201:
8179:
8177:
8176:
8171:
8160:
8159:
8147:
8146:
8125:
8124:
8112:
8111:
8090:
8089:
8077:
8076:
8057:
8055:
8054:
8049:
8044:
8043:
8031:
8030:
8014:
8012:
8011:
8006:
7994:
7992:
7991:
7986:
7984:
7983:
7971:
7970:
7954:
7952:
7951:
7946:
7941:
7940:
7928:
7927:
7904:
7902:
7901:
7896:
7894:
7893:
7877:
7875:
7874:
7869:
7867:
7866:
7857:
7849:
7848:
7836:
7835:
7820:
7819:
7807:
7806:
7784:
7782:
7781:
7776:
7774:
7773:
7757:
7755:
7754:
7749:
7741:
7740:
7724:
7722:
7721:
7716:
7680:
7678:
7677:
7672:
7660:
7658:
7657:
7652:
7640:
7638:
7637:
7632:
7596:
7594:
7593:
7588:
7576:
7574:
7573:
7568:
7566:
7565:
7549:
7547:
7546:
7541:
7511:
7509:
7508:
7503:
7489:
7488:
7476:
7475:
7454:
7453:
7441:
7440:
7419:
7418:
7406:
7405:
7389:
7387:
7386:
7381:
7376:
7340:
7338:
7337:
7332:
7314:
7312:
7311:
7306:
7295:
7294:
7282:
7281:
7265:
7263:
7262:
7257:
7249:
7248:
7232:
7230:
7229:
7224:
7213:
7212:
7200:
7199:
7183:
7181:
7180:
7175:
7158:
7157:
7141:
7139:
7138:
7133:
7131:
7130:
7114:
7112:
7111:
7106:
7076:
7074:
7073:
7068:
7066:
7065:
7053:
7052:
7036:
7034:
7033:
7028:
7023:
7022:
7010:
7009:
6993:
6991:
6990:
6985:
6983:
6982:
6966:
6964:
6963:
6958:
6946:
6944:
6943:
6938:
6936:
6935:
6919:
6917:
6916:
6911:
6896:
6894:
6893:
6888:
6876:
6874:
6873:
6868:
6866:
6865:
6849:
6847:
6846:
6841:
6823:
6821:
6820:
6815:
6797:
6795:
6794:
6789:
6787:
6786:
6770:
6768:
6767:
6762:
6750:
6748:
6747:
6742:
6702:
6700:
6699:
6694:
6675:
6673:
6672:
6667:
6665:
6664:
6655:
6626:
6624:
6623:
6618:
6616:
6615:
6606:
6565:
6563:
6562:
6557:
6555:
6547:
6535:
6533:
6532:
6527:
6515:
6513:
6512:
6507:
6495:
6493:
6492:
6487:
6460:
6458:
6457:
6452:
6419:
6417:
6416:
6411:
6395:
6393:
6392:
6387:
6382:
6355:
6353:
6352:
6347:
6335:
6333:
6332:
6327:
6315:
6313:
6312:
6307:
6295:
6293:
6292:
6287:
6275:
6273:
6272:
6267:
6255:
6253:
6252:
6247:
6235:
6233:
6232:
6227:
6209:
6207:
6206:
6201:
6189:
6187:
6186:
6181:
6170:
6137:
6135:
6134:
6129:
6127:
6126:
6110:
6108:
6107:
6102:
6100:
6099:
6083:
6081:
6080:
6075:
6073:
6072:
6056:
6054:
6053:
6048:
6046:
6045:
6029:
6027:
6026:
6021:
6010:
5983:
5981:
5980:
5975:
5973:
5972:
5956:
5954:
5953:
5948:
5946:
5945:
5929:
5927:
5926:
5921:
5919:
5918:
5902:
5900:
5899:
5894:
5889:
5884:
5883:
5874:
5866:
5861:
5860:
5851:
5843:
5838:
5837:
5828:
5820:
5815:
5814:
5805:
5797:
5792:
5791:
5782:
5774:
5769:
5768:
5759:
5751:
5746:
5745:
5736:
5721:
5719:
5718:
5713:
5667:
5665:
5664:
5659:
5657:
5656:
5632:
5630:
5629:
5624:
5622:
5613:
5585:
5583:
5582:
5577:
5565:
5563:
5562:
5557:
5545:
5543:
5542:
5537:
5526:
5525:
5513:
5512:
5490:
5488:
5487:
5482:
5471:
5470:
5458:
5457:
5435:
5433:
5432:
5427:
5404:
5402:
5401:
5396:
5384:
5382:
5381:
5376:
5357:
5355:
5354:
5349:
5330:
5328:
5327:
5322:
5277:
5275:
5274:
5269:
5267:
5266:
5257:
5242:
5240:
5239:
5234:
5232:
5231:
5222:
5207:
5205:
5204:
5199:
5187:
5185:
5184:
5179:
5171:
5170:
5161:
5140:
5138:
5137:
5132:
5130:
5129:
5120:
5093:
5091:
5090:
5085:
5077:
5076:
5067:
5046:
5044:
5043:
5038:
5036:
5035:
5026:
4975:
4973:
4972:
4967:
4965:
4964:
4948:
4946:
4945:
4940:
4938:
4937:
4925:
4924:
4912:
4911:
4899:
4898:
4882:
4880:
4879:
4874:
4872:
4871:
4859:
4858:
4842:
4840:
4839:
4834:
4816:
4814:
4813:
4808:
4806:
4805:
4793:
4792:
4776:
4774:
4773:
4768:
4763:
4762:
4750:
4749:
4730:
4728:
4727:
4722:
4720:
4719:
4694:
4692:
4691:
4686:
4675:
4674:
4662:
4661:
4634:
4633:
4621:
4620:
4595:
4593:
4592:
4587:
4569:
4567:
4566:
4561:
4559:
4558:
4546:
4545:
4530:The differences
4517:
4515:
4514:
4509:
4498:
4497:
4485:
4484:
4457:
4456:
4444:
4443:
4418:
4416:
4415:
4410:
4392:
4390:
4389:
4384:
4382:
4381:
4369:
4368:
4353:The differences
4340:
4338:
4337:
4332:
4327:
4326:
4314:
4313:
4286:Null hypothesis
4278:
4276:
4275:
4270:
4268:
4267:
4255:
4254:
4238:
4236:
4235:
4230:
4228:
4227:
4215:
4214:
4193:
4191:
4190:
4185:
4174:
4173:
4155:
4154:
4135:
4133:
4132:
4127:
4122:
4121:
4109:
4108:
4089:
4087:
4086:
4081:
4051:
4049:
4048:
4043:
4032:
4031:
4013:
4012:
3993:
3991:
3990:
3985:
3980:
3979:
3967:
3966:
3947:
3945:
3944:
3939:
3909:
3907:
3906:
3901:
3890:
3889:
3871:
3870:
3851:
3849:
3848:
3843:
3838:
3837:
3825:
3824:
3805:
3803:
3802:
3797:
3767:
3765:
3764:
3759:
3711:
3709:
3708:
3703:
3698:
3697:
3685:
3684:
3665:
3663:
3662:
3657:
3652:
3651:
3639:
3638:
3615:
3613:
3612:
3607:
3602:
3601:
3589:
3588:
3561:Null hypothesis
3552:
3550:
3549:
3544:
3526:
3524:
3523:
3518:
3516:
3515:
3503:
3502:
3474:
3472:
3471:
3466:
3448:
3446:
3445:
3440:
3438:
3437:
3425:
3424:
3396:
3394:
3393:
3388:
3370:
3368:
3367:
3362:
3360:
3359:
3347:
3346:
3318:
3316:
3315:
3310:
3292:
3290:
3289:
3284:
3282:
3281:
3269:
3268:
3244:Null hypothesis
3239:
3237:
3236:
3231:
3226:
3225:
3213:
3212:
3193:
3191:
3190:
3185:
3155:
3153:
3152:
3147:
3145:
3144:
3132:
3131:
3111:Paired data test
3106:
3104:
3103:
3098:
3096:
3095:
3079:
3077:
3076:
3071:
3069:
3068:
3047:
3045:
3044:
3039:
3016:
3014:
3013:
3008:
2985:
2983:
2982:
2977:
2959:
2957:
2956:
2951:
2932:Null hypothesis
2927:
2925:
2924:
2919:
2901:
2899:
2898:
2893:
2838:
2836:
2835:
2830:
2818:
2816:
2815:
2810:
2792:
2790:
2789:
2784:
2733:
2731:
2730:
2725:
2713:
2711:
2710:
2705:
2689:
2687:
2686:
2681:
2666:
2664:
2663:
2658:
2646:
2644:
2643:
2638:
2626:
2624:
2623:
2618:
2606:
2604:
2603:
2598:
2586:
2584:
2583:
2578:
2541:
2539:
2538:
2533:
2515:
2513:
2512:
2507:
2484:
2482:
2481:
2476:
2458:
2456:
2455:
2450:
2427:
2425:
2424:
2419:
2401:
2399:
2398:
2393:
2370:
2368:
2367:
2362:
2344:
2342:
2341:
2336:
2317:Null hypothesis
2312:
2310:
2309:
2304:
2292:
2290:
2289:
2284:
2241:
2239:
2238:
2233:
2231:
2191:
2190:
2169:
2168:
2119:
2118:
2097:
2096:
2076:
2074:
2073:
2068:
2063:
2062:
2050:
2049:
2037:
2028:
2018:
2016:
2015:
2010:
2008:
2007:
1982:
1980:
1979:
1974:
1972:
1971:
1955:
1953:
1952:
1947:
1945:
1944:
1928:
1926:
1925:
1920:
1908:
1906:
1905:
1900:
1898:
1891:
1890:
1875:
1870:
1850:
1845:
1840:
1820:
1815:
1814:
1799:
1798:
1786:
1785:
1759:
1751:
1746:
1741:
1721:
1716:
1715:
1703:
1698:
1678:
1669:
1668:
1652:
1644:
1639:
1634:
1614:
1609:
1608:
1596:
1591:
1571:
1562:
1561:
1541:
1539:
1538:
1533:
1528:
1474:
1472:
1471:
1466:
1464:
1463:
1451:
1450:
1434:
1432:
1431:
1426:
1424:
1417:
1416:
1406:
1399:
1398:
1387:
1356:
1355:
1339:
1338:
1328:
1321:
1320:
1309:
1278:
1277:
1257:
1255:
1254:
1249:
1247:
1246:
1227:
1225:
1224:
1219:
1217:
1216:
1197:
1195:
1194:
1189:
1174:
1172:
1171:
1166:
1154:
1152:
1151:
1146:
1138:
1137:
1112:
1110:
1109:
1104:
1102:
1097:
1096:
1078:
1064:
1059:
1058:
1040:
1028:
1026:
1025:
1020:
960:
958:
957:
952:
950:
945:
944:
935:
927:
922:
921:
912:
900:
898:
897:
892:
880:
878:
877:
872:
870:
869:
849:
847:
846:
841:
829:
827:
826:
821:
807:
805:
804:
799:
794:
793:
781:
780:
761:
756:
728:
726:
725:
720:
701:
699:
698:
693:
675:
673:
672:
667:
634:
632:
631:
626:
608:
606:
605:
600:
566:
564:
563:
558:
543:
541:
540:
535:
533:
532:
514:
513:
497:
495:
494:
489:
487:
482:
481:
472:
458:
453:
452:
443:
429:
427:
426:
421:
416:
411:
410:
401:
387:
382:
381:
372:
356:
354:
353:
348:
346:
345:
327:
326:
303:
301:
300:
295:
293:
292:
280:
279:
263:
261:
260:
255:
250:
249:
237:
236:
209:
207:
206:
201:
196:
195:
183:
182:
158:
157:
145:
144:
14785:
14784:
14780:
14779:
14778:
14776:
14775:
14774:
14750:
14749:
14748:
14743:
14706:
14677:
14639:
14576:
14562:quality control
14529:
14511:Clinical trials
14488:
14463:
14447:
14435:Hazard function
14429:
14383:
14345:
14329:
14292:
14288:Breusch–Godfrey
14276:
14253:
14193:
14168:Factor analysis
14114:
14095:Graphical model
14067:
14034:
14001:
13987:
13967:
13921:
13888:
13850:
13813:
13812:
13781:
13725:
13712:
13704:
13696:
13680:
13665:
13644:Rank statistics
13638:
13617:Model selection
13605:
13563:Goodness of fit
13557:
13534:
13508:
13480:
13433:
13378:
13367:Median unbiased
13295:
13206:
13139:Order statistic
13101:
13080:
13047:
13021:
12973:
12928:
12871:
12869:Data collection
12850:
12762:
12717:
12691:
12669:
12629:
12581:
12498:Continuous data
12488:
12475:
12457:
12452:
12419:link to article
12384:
12379:
12378:
12369:
12367:
12359:
12358:
12354:
12347:
12331:
12327:
12311:
12310:
12288:
12281:
12276:
12272:
12267:
12263:
12258:
12254:
12249:
12245:
12241:Wilcoxon, p. 82
12240:
12236:
12231:
12227:
12204:
12200:
12195:
12191:
12186:
12182:
12153:
12146:
12141:
12137:
12132:
12128:
12123:
12119:
12114:
12110:
12105:
12101:
12096:
12092:
12087:
12083:
12078:
12074:
12066:
12052:
12048:
12043:
12039:
12034:
12030:
12025:
12021:
12016:
12012:
12007:
12003:
11998:
11994:
11989:
11985:
11980:
11976:
11971:
11967:
11962:
11958:
11953:
11949:
11944:
11940:
11935:
11931:
11926:
11922:
11899:
11895:
11880:
11876:
11871:
11867:
11862:
11858:
11835:
11831:
11820:
11816:
11811:
11807:
11802:
11798:
11793:
11789:
11781:
11764:
11760:
11756:Conover, p. 354
11755:
11751:
11746:
11742:
11737:
11733:
11728:
11724:
11720:Conover, p. 353
11719:
11715:
11710:
11706:
11701:
11697:
11689:
11675:
11671:
11666:
11662:
11657:
11653:
11648:
11644:
11639:
11635:
11627:
11613:
11609:
11605:Conover, p. 353
11604:
11600:
11595:
11591:
11587:Conover, p. 352
11586:
11582:
11575:
11559:
11555:
11532:10.2307/3001968
11515:
11509:
11505:
11496:
11494:
11486:
11485:
11481:
11473:
11459:
11455:
11450:
11433:
11406:= signrank(x,y)
11383:
11366:
11363:
11360:
11357:
11354:
11351:
11348:
11345:
11342:
11339:
11331:
11248:
11242:
11221:
11218:
11217:
11201:
11198:
11197:
11177:
11173:
11168:
11163:
11160:
11159:
11131:
11103:
11099:
11091:
11083:
11081:
11078:
11077:
11000:
10997:
10996:
10972:
10968:
10966:
10963:
10962:
10942:
10940:
10937:
10936:
10916:
10913:
10912:
10657:
10653:
10645:
10642:
10641:
10623:
10619:
10617:
10614:
10613:
10596:
10594:
10591:
10590:
10573:
10570:
10569:
10543:
10539:
10524:
10520:
10518:
10515:
10514:
10490:
10486:
10484:
10481:
10480:
10456:
10452:
10450:
10447:
10446:
10429:
10426:
10425:
10227:
10225:
10222:
10221:
10204:
10201:
10200:
10174:
10170:
10155:
10151:
10149:
10146:
10145:
10121:
10117:
10115:
10112:
10111:
10087:
10083:
10081:
10078:
10077:
10060:
10057:
10056:
10047:
10025:
10021:
10019:
10016:
10015:
9975:
9969:
9964:
9955:
9952:
9951:
9935:
9932:
9931:
9914:
9910:
9908:
9905:
9904:
9887:
9883:
9881:
9878:
9877:
9861:
9858:
9857:
9840:
9835:
9829:
9826:
9825:
9808:
9803:
9797:
9794:
9793:
9776:
9772:
9770:
9767:
9766:
9740:
9737:
9736:
9709:
9705:
9696:
9692:
9684:
9681:
9680:
9677:
9659:
9658:
9650:
9644:
9640:
9633:
9624:
9620:
9608:
9607:
9598:
9594:
9587:
9568:
9566:
9563:
9562:
9534:
9452:
9450:
9441:
9437:
9435:
9432:
9431:
9392:
9390:
9362:
9360:
9348:
9344:
9336:
9334:
9331:
9330:
9313:
9309:
9307:
9304:
9303:
9287:
9284:
9283:
9267:
9264:
9263:
9237:
9233:
9222:
9219:
9218:
9202:
9199:
9198:
9182:
9179:
9178:
9162:
9159:
9158:
9130:
9126:
9111:
9100:
9088:
9084:
9079:
9062:
9059:
9058:
9042:
9039:
9038:
9011:
9004:
9000:
8992:
8989:
8988:
8971:
8967:
8965:
8962:
8961:
8902:
8874:
8872:
8857:
8850:
8848:
8840:
8837:
8836:
8804:
8800:
8791:
8790:
8747:
8725:
8721:
8717:
8715:
8709:
8708:
8663:
8659:
8651:
8648:
8647:
8631:
8630:
8584:
8582:
8575:
8557:
8556:
8510:
8508:
8496:
8492:
8476:
8467:
8463:
8451:
8450:
8437:
8423:
8420:
8419:
8391:
8389:
8377:
8373:
8365:
8358:
8349:
8345:
8337:
8333:
8331:
8328:
8327:
8310:
8306:
8304:
8301:
8300:
8283:
8279:
8277:
8274:
8273:
8257:
8254:
8253:
8237:
8234:
8233:
8232:For very large
8210:
8206:
8197:
8193:
8185:
8182:
8181:
8155:
8151:
8136:
8132:
8120:
8116:
8101:
8097:
8085:
8081:
8072:
8068:
8063:
8060:
8059:
8039:
8035:
8026:
8022:
8020:
8017:
8016:
8000:
7997:
7996:
7979:
7975:
7966:
7962:
7960:
7957:
7956:
7936:
7932:
7923:
7919:
7917:
7914:
7913:
7889:
7885:
7883:
7880:
7879:
7878:. The function
7862:
7858:
7853:
7844:
7840:
7831:
7827:
7815:
7811:
7802:
7798:
7790:
7787:
7786:
7769:
7765:
7763:
7760:
7759:
7736:
7732:
7730:
7727:
7726:
7686:
7683:
7682:
7666:
7663:
7662:
7646:
7643:
7642:
7602:
7599:
7598:
7582:
7579:
7578:
7561:
7557:
7555:
7552:
7551:
7517:
7514:
7513:
7484:
7480:
7465:
7461:
7449:
7445:
7430:
7426:
7414:
7410:
7401:
7397:
7395:
7392:
7391:
7372:
7346:
7343:
7342:
7320:
7317:
7316:
7290:
7286:
7277:
7273:
7271:
7268:
7267:
7244:
7240:
7238:
7235:
7234:
7208:
7204:
7195:
7191:
7189:
7186:
7185:
7153:
7149:
7147:
7144:
7143:
7126:
7122:
7120:
7117:
7116:
7082:
7079:
7078:
7061:
7057:
7048:
7044:
7042:
7039:
7038:
7018:
7014:
7005:
7001:
6999:
6996:
6995:
6978:
6974:
6972:
6969:
6968:
6952:
6949:
6948:
6931:
6927:
6925:
6922:
6921:
6905:
6902:
6901:
6882:
6879:
6878:
6861:
6857:
6855:
6852:
6851:
6829:
6826:
6825:
6803:
6800:
6799:
6782:
6778:
6776:
6773:
6772:
6756:
6753:
6752:
6736:
6733:
6732:
6725:
6688:
6685:
6684:
6660:
6656:
6651:
6640:
6637:
6636:
6611:
6607:
6602:
6591:
6588:
6587:
6551:
6543:
6541:
6538:
6537:
6521:
6518:
6517:
6501:
6498:
6497:
6481:
6478:
6477:
6428:
6425:
6424:
6405:
6402:
6401:
6378:
6361:
6358:
6357:
6341:
6338:
6337:
6321:
6318:
6317:
6301:
6298:
6297:
6281:
6278:
6277:
6261:
6258:
6257:
6241:
6238:
6237:
6215:
6212:
6211:
6195:
6192:
6191:
6166:
6143:
6140:
6139:
6122:
6118:
6116:
6113:
6112:
6095:
6091:
6089:
6086:
6085:
6068:
6064:
6062:
6059:
6058:
6041:
6037:
6035:
6032:
6031:
6006:
5989:
5986:
5985:
5968:
5964:
5962:
5959:
5958:
5941:
5937:
5935:
5932:
5931:
5914:
5910:
5908:
5905:
5904:
5885:
5879:
5875:
5870:
5862:
5856:
5852:
5847:
5839:
5833:
5829:
5824:
5816:
5810:
5806:
5801:
5793:
5787:
5783:
5778:
5770:
5764:
5760:
5755:
5747:
5741:
5737:
5732:
5730:
5727:
5726:
5689:
5686:
5685:
5652:
5648:
5646:
5643:
5642:
5639:
5611:
5609:
5606:
5605:
5571:
5568:
5567:
5551:
5548:
5547:
5521:
5517:
5508:
5504:
5496:
5493:
5492:
5466:
5462:
5453:
5449:
5441:
5438:
5437:
5415:
5412:
5411:
5390:
5387:
5386:
5370:
5367:
5366:
5343:
5340:
5339:
5298:
5295:
5294:
5262:
5258:
5253:
5248:
5245:
5244:
5227:
5223:
5218:
5213:
5210:
5209:
5193:
5190:
5189:
5166:
5162:
5157:
5146:
5143:
5142:
5125:
5121:
5116:
5111:
5108:
5107:
5072:
5068:
5063:
5052:
5049:
5048:
5031:
5027:
5022:
5017:
5014:
5013:
4982:
4960:
4956:
4954:
4951:
4950:
4933:
4929:
4920:
4916:
4907:
4903:
4894:
4890:
4888:
4885:
4884:
4867:
4863:
4854:
4850:
4848:
4845:
4844:
4822:
4819:
4818:
4801:
4797:
4788:
4784:
4782:
4779:
4778:
4758:
4754:
4745:
4741:
4736:
4733:
4732:
4715:
4711:
4709:
4706:
4705:
4702:
4670:
4666:
4657:
4653:
4629:
4625:
4616:
4612:
4601:
4598:
4597:
4575:
4572:
4571:
4554:
4550:
4541:
4537:
4535:
4532:
4531:
4527:
4493:
4489:
4480:
4476:
4452:
4448:
4439:
4435:
4424:
4421:
4420:
4398:
4395:
4394:
4377:
4373:
4364:
4360:
4358:
4355:
4354:
4350:
4322:
4318:
4309:
4305:
4300:
4297:
4296:
4292:
4263:
4259:
4250:
4246:
4244:
4241:
4240:
4223:
4219:
4210:
4206:
4204:
4201:
4200:
4169:
4165:
4150:
4146:
4141:
4138:
4137:
4117:
4113:
4104:
4100:
4095:
4092:
4091:
4069:
4066:
4065:
4061:
4027:
4023:
4008:
4004:
3999:
3996:
3995:
3975:
3971:
3962:
3958:
3953:
3950:
3949:
3927:
3924:
3923:
3919:
3885:
3881:
3866:
3862:
3857:
3854:
3853:
3833:
3829:
3820:
3816:
3811:
3808:
3807:
3785:
3782:
3781:
3777:
3717:
3714:
3713:
3693:
3689:
3680:
3676:
3671:
3668:
3667:
3647:
3643:
3634:
3630:
3625:
3622:
3621:
3620:, meaning that
3597:
3593:
3584:
3580:
3575:
3572:
3571:
3567:
3532:
3529:
3528:
3511:
3507:
3498:
3494:
3492:
3489:
3488:
3484:
3454:
3451:
3450:
3433:
3429:
3420:
3416:
3414:
3411:
3410:
3406:
3376:
3373:
3372:
3355:
3351:
3342:
3338:
3336:
3333:
3332:
3328:
3298:
3295:
3294:
3277:
3273:
3264:
3260:
3258:
3255:
3254:
3250:
3221:
3217:
3208:
3204:
3199:
3196:
3195:
3164:
3161:
3160:
3140:
3136:
3127:
3123:
3121:
3118:
3117:
3113:
3091:
3087:
3085:
3082:
3081:
3064:
3060:
3058:
3055:
3054:
3033:
3030:
3029:
3026:
3002:
2999:
2998:
2995:
2965:
2962:
2961:
2945:
2942:
2941:
2938:
2907:
2904:
2903:
2848:
2845:
2844:
2824:
2821:
2820:
2798:
2795:
2794:
2739:
2736:
2735:
2719:
2716:
2715:
2699:
2696:
2695:
2675:
2672:
2671:
2652:
2649:
2648:
2632:
2629:
2628:
2612:
2609:
2608:
2607:is a median of
2592:
2589:
2588:
2551:
2548:
2547:
2546:If in addition
2521:
2518:
2517:
2501:
2498:
2497:
2494:
2464:
2461:
2460:
2444:
2441:
2440:
2437:
2407:
2404:
2403:
2387:
2384:
2383:
2380:
2350:
2347:
2346:
2330:
2327:
2326:
2323:
2298:
2295:
2294:
2278:
2275:
2274:
2252:
2250:One-sample test
2247:
2229:
2228:
2183:
2179:
2164:
2160:
2157:
2156:
2111:
2107:
2092:
2088:
2084:
2082:
2079:
2078:
2058:
2054:
2045:
2041:
2026:
2024:
2021:
2020:
2000:
1996:
1994:
1991:
1990:
1967:
1963:
1961:
1958:
1957:
1940:
1936:
1934:
1931:
1930:
1914:
1911:
1910:
1896:
1895:
1886:
1882:
1851:
1849:
1821:
1819:
1810:
1806:
1794:
1790:
1781:
1777:
1770:
1764:
1763:
1750:
1722:
1720:
1711:
1707:
1679:
1677:
1670:
1664:
1660:
1657:
1656:
1643:
1615:
1613:
1604:
1600:
1572:
1570:
1563:
1557:
1553:
1549:
1547:
1544:
1543:
1524:
1480:
1477:
1476:
1459:
1455:
1446:
1442:
1440:
1437:
1436:
1422:
1421:
1412:
1408:
1394:
1390:
1368:
1357:
1351:
1347:
1344:
1343:
1334:
1330:
1316:
1312:
1290:
1279:
1273:
1269:
1265:
1263:
1260:
1259:
1258:are defined by
1242:
1238:
1236:
1233:
1232:
1212:
1208:
1206:
1203:
1202:
1183:
1180:
1179:
1160:
1157:
1156:
1124:
1120:
1118:
1115:
1114:
1098:
1083:
1079:
1074:
1060:
1045:
1041:
1036:
1034:
1031:
1030:
966:
963:
962:
946:
940:
936:
931:
923:
917:
913:
908:
906:
903:
902:
886:
883:
882:
865:
861:
859:
856:
855:
835:
832:
831:
815:
812:
811:
789:
785:
776:
772:
757:
746:
734:
731:
730:
714:
711:
710:
708:signed-rank sum
681:
678:
677:
640:
637:
636:
614:
611:
610:
576:
573:
572:
552:
549:
548:
528:
524:
509:
505:
503:
500:
499:
483:
477:
473:
468:
454:
448:
444:
439:
437:
434:
433:
412:
406:
402:
397:
383:
377:
373:
368:
366:
363:
362:
341:
337:
322:
318:
316:
313:
312:
288:
284:
275:
271:
269:
266:
265:
245:
241:
232:
228:
223:
220:
219:
191:
187:
178:
174:
153:
149:
140:
136:
131:
128:
127:
120:
88:
24:
17:
12:
11:
5:
14783:
14773:
14772:
14767:
14762:
14745:
14744:
14742:
14741:
14729:
14717:
14703:
14690:
14687:
14686:
14683:
14682:
14679:
14678:
14676:
14675:
14670:
14665:
14660:
14655:
14649:
14647:
14641:
14640:
14638:
14637:
14632:
14627:
14622:
14617:
14612:
14607:
14602:
14597:
14592:
14586:
14584:
14578:
14577:
14575:
14574:
14569:
14564:
14555:
14550:
14545:
14539:
14537:
14531:
14530:
14528:
14527:
14522:
14517:
14508:
14506:Bioinformatics
14502:
14500:
14490:
14489:
14477:
14476:
14473:
14472:
14469:
14468:
14465:
14464:
14462:
14461:
14455:
14453:
14449:
14448:
14446:
14445:
14439:
14437:
14431:
14430:
14428:
14427:
14422:
14417:
14412:
14406:
14404:
14395:
14389:
14388:
14385:
14384:
14382:
14381:
14376:
14371:
14366:
14361:
14355:
14353:
14347:
14346:
14344:
14343:
14338:
14333:
14325:
14320:
14315:
14314:
14313:
14311:partial (PACF)
14302:
14300:
14294:
14293:
14291:
14290:
14285:
14280:
14272:
14267:
14261:
14259:
14258:Specific tests
14255:
14254:
14252:
14251:
14246:
14241:
14236:
14231:
14226:
14221:
14216:
14210:
14208:
14201:
14195:
14194:
14192:
14191:
14190:
14189:
14188:
14187:
14172:
14171:
14170:
14160:
14158:Classification
14155:
14150:
14145:
14140:
14135:
14130:
14124:
14122:
14116:
14115:
14113:
14112:
14107:
14105:McNemar's test
14102:
14097:
14092:
14087:
14081:
14079:
14069:
14068:
14044:
14043:
14040:
14039:
14036:
14035:
14033:
14032:
14027:
14022:
14017:
14011:
14009:
14003:
14002:
14000:
13999:
13983:
13977:
13975:
13969:
13968:
13966:
13965:
13960:
13955:
13950:
13945:
13943:Semiparametric
13940:
13935:
13929:
13927:
13923:
13922:
13920:
13919:
13914:
13909:
13904:
13898:
13896:
13890:
13889:
13887:
13886:
13881:
13876:
13871:
13866:
13860:
13858:
13852:
13851:
13849:
13848:
13843:
13838:
13833:
13827:
13825:
13815:
13814:
13811:
13810:
13805:
13799:
13791:
13790:
13787:
13786:
13783:
13782:
13780:
13779:
13778:
13777:
13767:
13762:
13757:
13756:
13755:
13750:
13739:
13737:
13731:
13730:
13727:
13726:
13724:
13723:
13718:
13717:
13716:
13708:
13700:
13684:
13681:(Mann–Whitney)
13676:
13675:
13674:
13661:
13660:
13659:
13648:
13646:
13640:
13639:
13637:
13636:
13635:
13634:
13629:
13624:
13614:
13609:
13606:(Shapiro–Wilk)
13601:
13596:
13591:
13586:
13581:
13573:
13567:
13565:
13559:
13558:
13556:
13555:
13547:
13538:
13526:
13520:
13518:Specific tests
13514:
13513:
13510:
13509:
13507:
13506:
13501:
13496:
13490:
13488:
13482:
13481:
13479:
13478:
13473:
13472:
13471:
13461:
13460:
13459:
13449:
13443:
13441:
13435:
13434:
13432:
13431:
13430:
13429:
13424:
13414:
13409:
13404:
13399:
13394:
13388:
13386:
13380:
13379:
13377:
13376:
13371:
13370:
13369:
13364:
13363:
13362:
13357:
13342:
13341:
13340:
13335:
13330:
13325:
13314:
13312:
13303:
13297:
13296:
13294:
13293:
13288:
13283:
13282:
13281:
13271:
13266:
13265:
13264:
13254:
13253:
13252:
13247:
13242:
13232:
13227:
13222:
13221:
13220:
13215:
13210:
13194:
13193:
13192:
13187:
13182:
13172:
13171:
13170:
13165:
13155:
13154:
13153:
13143:
13142:
13141:
13131:
13126:
13121:
13115:
13113:
13103:
13102:
13090:
13089:
13086:
13085:
13082:
13081:
13079:
13078:
13073:
13068:
13063:
13057:
13055:
13049:
13048:
13046:
13045:
13040:
13035:
13029:
13027:
13023:
13022:
13020:
13019:
13014:
13009:
13004:
12999:
12994:
12989:
12983:
12981:
12975:
12974:
12972:
12971:
12969:Standard error
12966:
12961:
12956:
12955:
12954:
12949:
12938:
12936:
12930:
12929:
12927:
12926:
12921:
12916:
12911:
12906:
12901:
12899:Optimal design
12896:
12891:
12885:
12883:
12873:
12872:
12860:
12859:
12856:
12855:
12852:
12851:
12849:
12848:
12843:
12838:
12833:
12828:
12823:
12818:
12813:
12808:
12803:
12798:
12793:
12788:
12783:
12778:
12772:
12770:
12764:
12763:
12761:
12760:
12755:
12754:
12753:
12748:
12738:
12733:
12727:
12725:
12719:
12718:
12716:
12715:
12710:
12705:
12699:
12697:
12696:Summary tables
12693:
12692:
12690:
12689:
12683:
12681:
12675:
12674:
12671:
12670:
12668:
12667:
12666:
12665:
12660:
12655:
12645:
12639:
12637:
12631:
12630:
12628:
12627:
12622:
12617:
12612:
12607:
12602:
12597:
12591:
12589:
12583:
12582:
12580:
12579:
12574:
12569:
12568:
12567:
12562:
12557:
12552:
12547:
12542:
12537:
12532:
12530:Contraharmonic
12527:
12522:
12511:
12509:
12500:
12490:
12489:
12477:
12476:
12474:
12473:
12468:
12462:
12459:
12458:
12451:
12450:
12443:
12436:
12428:
12422:
12421:
12411:
12405:
12400:
12395:
12390:
12383:
12382:External links
12380:
12377:
12376:
12352:
12345:
12325:
12279:
12270:
12261:
12252:
12243:
12234:
12225:
12198:
12189:
12180:
12167:(2): 169–171.
12144:
12135:
12126:
12117:
12108:
12099:
12090:
12081:
12072:
12064:
12046:
12037:
12028:
12019:
12010:
12001:
11992:
11983:
11974:
11965:
11956:
11947:
11938:
11929:
11920:
11893:
11874:
11865:
11856:
11829:
11814:
11805:
11796:
11787:
11779:
11758:
11749:
11740:
11731:
11722:
11713:
11704:
11695:
11687:
11669:
11660:
11651:
11642:
11633:
11625:
11607:
11598:
11589:
11580:
11573:
11553:
11503:
11479:
11471:
11452:
11451:
11449:
11446:
11445:
11444:
11439:
11432:
11429:
11428:
11427:
11420:
11410:
11399:
11393:
11387:
11377:
11371:
11330:
11327:
11250:To compute an
11244:Main article:
11241:
11238:
11237:
11236:
11225:
11205:
11194:
11180:
11176:
11167:
11157:
11146:
11143:
11138:
11130:
11124:
11121:
11118:
11115:
11112:
11109:
11106:
11102:
11098:
11094:
11090:
11086:
11075:
11064:
11061:
11058:
11055:
11052:
11049:
11046:
11043:
11040:
11037:
11034:
11031:
11028:
11025:
11022:
11019:
11016:
11013:
11010:
11007:
11004:
10975:
10971:
10959:absolute value
10920:
10908:
10907:
10904:
10903:
10900:
10897:
10894:
10891:
10888:
10885:
10881:
10880:
10877:
10874:
10871:
10868:
10865:
10862:
10858:
10857:
10854:
10851:
10848:
10845:
10842:
10839:
10835:
10834:
10831:
10828:
10825:
10822:
10819:
10816:
10812:
10811:
10808:
10805:
10802:
10799:
10796:
10793:
10789:
10788:
10785:
10782:
10779:
10776:
10773:
10770:
10766:
10765:
10762:
10759:
10756:
10753:
10750:
10747:
10743:
10742:
10739:
10736:
10733:
10730:
10727:
10724:
10720:
10719:
10716:
10713:
10710:
10707:
10704:
10701:
10697:
10696:
10693:
10690:
10687:
10684:
10681:
10678:
10674:
10673:
10660:
10656:
10652:
10649:
10639:
10626:
10622:
10611:
10588:
10577:
10566:
10565:
10552:
10549:
10546:
10542:
10538:
10533:
10530:
10527:
10523:
10512:
10499:
10496:
10493:
10489:
10478:
10465:
10462:
10459:
10455:
10444:
10433:
10419:
10416:
10413:
10412:
10409:
10406:
10403:
10400:
10396:
10395:
10392:
10389:
10386:
10383:
10379:
10378:
10375:
10372:
10369:
10366:
10362:
10361:
10358:
10355:
10352:
10349:
10345:
10344:
10341:
10338:
10335:
10332:
10328:
10327:
10324:
10321:
10318:
10315:
10311:
10310:
10307:
10304:
10301:
10298:
10294:
10293:
10290:
10287:
10284:
10281:
10277:
10276:
10273:
10270:
10267:
10264:
10260:
10259:
10256:
10253:
10250:
10247:
10243:
10242:
10219:
10208:
10197:
10196:
10183:
10180:
10177:
10173:
10169:
10164:
10161:
10158:
10154:
10143:
10130:
10127:
10124:
10120:
10109:
10096:
10093:
10090:
10086:
10075:
10064:
10046:
10043:
10028:
10024:
10003:
10000:
9997:
9994:
9991:
9988:
9985:
9982:
9978:
9972:
9967:
9963:
9959:
9939:
9917:
9913:
9890:
9886:
9865:
9843:
9838:
9834:
9811:
9806:
9802:
9779:
9775:
9763:modified ranks
9750:
9747:
9744:
9717:
9712:
9708:
9704:
9699:
9695:
9691:
9688:
9676:
9673:
9657:
9653:
9647:
9643:
9639:
9636:
9634:
9632:
9627:
9623:
9619:
9616:
9613:
9610:
9609:
9606:
9601:
9597:
9593:
9590:
9588:
9586:
9583:
9580:
9577:
9574:
9571:
9570:
9550:
9545:
9541:
9537:
9533:
9530:
9527:
9524:
9521:
9518:
9515:
9512:
9509:
9506:
9503:
9500:
9497:
9494:
9491:
9488:
9485:
9482:
9479:
9476:
9473:
9470:
9467:
9464:
9461:
9458:
9455:
9449:
9444:
9440:
9419:
9414:
9410:
9407:
9404:
9401:
9398:
9395:
9389:
9384:
9380:
9377:
9374:
9371:
9368:
9365:
9359:
9356:
9351:
9347:
9343:
9339:
9316:
9312:
9291:
9271:
9251:
9248:
9245:
9240:
9236:
9232:
9229:
9226:
9206:
9186:
9166:
9144:
9141:
9136:
9133:
9129:
9125:
9122:
9119:
9114:
9109:
9106:
9103:
9099:
9091:
9087:
9083:
9078:
9075:
9072:
9069:
9066:
9046:
9023:
9018:
9014:
9010:
9007:
9003:
8999:
8996:
8974:
8970:
8949:
8942:
8938:
8935:
8932:
8929:
8926:
8923:
8920:
8917:
8914:
8911:
8908:
8905:
8896:
8892:
8889:
8886:
8883:
8880:
8877:
8871:
8865:
8862:
8856:
8853:
8847:
8844:
8824:
8821:
8818:
8815:
8812:
8807:
8803:
8799:
8794:
8786:
8783:
8780:
8777:
8774:
8771:
8768:
8765:
8762:
8759:
8756:
8753:
8750:
8745:
8742:
8739:
8736:
8733:
8728:
8724:
8720:
8712:
8707:
8704:
8701:
8698:
8695:
8692:
8689:
8686:
8683:
8680:
8677:
8674:
8671:
8666:
8662:
8658:
8655:
8629:
8624:
8620:
8617:
8614:
8611:
8608:
8605:
8602:
8599:
8596:
8593:
8590:
8587:
8581:
8578:
8576:
8574:
8571:
8568:
8565:
8562:
8559:
8558:
8555:
8550:
8546:
8543:
8540:
8537:
8534:
8531:
8528:
8525:
8522:
8519:
8516:
8513:
8507:
8504:
8499:
8495:
8491:
8488:
8485:
8482:
8479:
8477:
8475:
8470:
8466:
8462:
8459:
8456:
8453:
8452:
8449:
8446:
8443:
8440:
8438:
8436:
8433:
8430:
8426:
8422:
8421:
8418:
8413:
8409:
8406:
8403:
8400:
8397:
8394:
8388:
8385:
8380:
8376:
8372:
8368:
8364:
8361:
8359:
8357:
8352:
8348:
8344:
8340:
8336:
8335:
8313:
8309:
8286:
8282:
8261:
8241:
8218:
8213:
8209:
8205:
8200:
8196:
8192:
8189:
8169:
8166:
8163:
8158:
8154:
8150:
8145:
8142:
8139:
8135:
8131:
8128:
8123:
8119:
8115:
8110:
8107:
8104:
8100:
8096:
8093:
8088:
8084:
8080:
8075:
8071:
8067:
8047:
8042:
8038:
8034:
8029:
8025:
8004:
7982:
7978:
7974:
7969:
7965:
7944:
7939:
7935:
7931:
7926:
7922:
7892:
7888:
7865:
7861:
7856:
7852:
7847:
7843:
7839:
7834:
7830:
7826:
7823:
7818:
7814:
7810:
7805:
7801:
7797:
7794:
7772:
7768:
7747:
7744:
7739:
7735:
7725:which sums to
7714:
7711:
7708:
7705:
7702:
7699:
7696:
7693:
7690:
7670:
7650:
7630:
7627:
7624:
7621:
7618:
7615:
7612:
7609:
7606:
7586:
7564:
7560:
7550:which sums to
7539:
7536:
7533:
7530:
7527:
7524:
7521:
7501:
7498:
7495:
7492:
7487:
7483:
7479:
7474:
7471:
7468:
7464:
7460:
7457:
7452:
7448:
7444:
7439:
7436:
7433:
7429:
7425:
7422:
7417:
7413:
7409:
7404:
7400:
7379:
7375:
7371:
7368:
7365:
7362:
7359:
7356:
7353:
7350:
7330:
7327:
7324:
7304:
7301:
7298:
7293:
7289:
7285:
7280:
7276:
7255:
7252:
7247:
7243:
7222:
7219:
7216:
7211:
7207:
7203:
7198:
7194:
7173:
7170:
7167:
7164:
7161:
7156:
7152:
7129:
7125:
7104:
7101:
7098:
7095:
7092:
7089:
7086:
7064:
7060:
7056:
7051:
7047:
7026:
7021:
7017:
7013:
7008:
7004:
6981:
6977:
6956:
6934:
6930:
6909:
6886:
6864:
6860:
6839:
6836:
6833:
6813:
6810:
6807:
6785:
6781:
6760:
6740:
6724:
6721:
6692:
6663:
6659:
6654:
6650:
6647:
6644:
6633:
6632:
6614:
6610:
6605:
6601:
6598:
6595:
6584:
6583:
6576:
6575:
6554:
6550:
6546:
6525:
6505:
6485:
6450:
6447:
6444:
6441:
6438:
6435:
6432:
6409:
6385:
6381:
6377:
6374:
6371:
6368:
6365:
6345:
6325:
6305:
6285:
6265:
6245:
6225:
6222:
6219:
6199:
6179:
6176:
6173:
6169:
6165:
6162:
6159:
6156:
6153:
6150:
6147:
6125:
6121:
6098:
6094:
6071:
6067:
6044:
6040:
6019:
6016:
6013:
6009:
6005:
6002:
5999:
5996:
5993:
5971:
5967:
5944:
5940:
5917:
5913:
5903:In this case,
5892:
5888:
5882:
5878:
5873:
5869:
5865:
5859:
5855:
5850:
5846:
5842:
5836:
5832:
5827:
5823:
5819:
5813:
5809:
5804:
5800:
5796:
5790:
5786:
5781:
5777:
5773:
5767:
5763:
5758:
5754:
5750:
5744:
5740:
5735:
5711:
5708:
5705:
5702:
5699:
5696:
5693:
5655:
5651:
5638:
5635:
5619:
5616:
5575:
5555:
5535:
5532:
5529:
5524:
5520:
5516:
5511:
5507:
5503:
5500:
5480:
5477:
5474:
5469:
5465:
5461:
5456:
5452:
5448:
5445:
5425:
5422:
5419:
5407:
5406:
5394:
5374:
5363:
5359:
5347:
5336:
5320:
5317:
5314:
5311:
5308:
5305:
5302:
5265:
5261:
5256:
5252:
5230:
5226:
5221:
5217:
5197:
5177:
5174:
5169:
5165:
5160:
5156:
5153:
5150:
5128:
5124:
5119:
5115:
5100:
5099:
5083:
5080:
5075:
5071:
5066:
5062:
5059:
5056:
5034:
5030:
5025:
5021:
5006:
5005:
4998:
4997:
4981:
4978:
4963:
4959:
4936:
4932:
4928:
4923:
4919:
4915:
4910:
4906:
4902:
4897:
4893:
4870:
4866:
4862:
4857:
4853:
4832:
4829:
4826:
4804:
4800:
4796:
4791:
4787:
4766:
4761:
4757:
4753:
4748:
4744:
4740:
4718:
4714:
4701:
4700:Zeros and ties
4698:
4697:
4696:
4684:
4681:
4678:
4673:
4669:
4665:
4660:
4656:
4652:
4649:
4646:
4643:
4640:
4637:
4632:
4628:
4624:
4619:
4615:
4611:
4608:
4605:
4585:
4582:
4579:
4557:
4553:
4549:
4544:
4540:
4528:
4525:
4519:
4507:
4504:
4501:
4496:
4492:
4488:
4483:
4479:
4475:
4472:
4469:
4466:
4463:
4460:
4455:
4451:
4447:
4442:
4438:
4434:
4431:
4428:
4408:
4405:
4402:
4380:
4376:
4372:
4367:
4363:
4351:
4348:
4342:
4330:
4325:
4321:
4317:
4312:
4308:
4304:
4293:
4290:
4266:
4262:
4258:
4253:
4249:
4226:
4222:
4218:
4213:
4209:
4196:
4195:
4183:
4180:
4177:
4172:
4168:
4164:
4161:
4158:
4153:
4149:
4145:
4125:
4120:
4116:
4112:
4107:
4103:
4099:
4079:
4076:
4073:
4062:
4059:
4053:
4041:
4038:
4035:
4030:
4026:
4022:
4019:
4016:
4011:
4007:
4003:
3983:
3978:
3974:
3970:
3965:
3961:
3957:
3937:
3934:
3931:
3920:
3917:
3911:
3899:
3896:
3893:
3888:
3884:
3880:
3877:
3874:
3869:
3865:
3861:
3841:
3836:
3832:
3828:
3823:
3819:
3815:
3795:
3792:
3789:
3778:
3775:
3769:
3757:
3754:
3751:
3748:
3745:
3742:
3739:
3736:
3733:
3730:
3727:
3724:
3721:
3701:
3696:
3692:
3688:
3683:
3679:
3675:
3655:
3650:
3646:
3642:
3637:
3633:
3629:
3605:
3600:
3596:
3592:
3587:
3583:
3579:
3568:
3565:
3555:
3554:
3542:
3539:
3536:
3514:
3510:
3506:
3501:
3497:
3485:
3482:
3476:
3464:
3461:
3458:
3436:
3432:
3428:
3423:
3419:
3407:
3404:
3398:
3386:
3383:
3380:
3358:
3354:
3350:
3345:
3341:
3329:
3326:
3320:
3308:
3305:
3302:
3280:
3276:
3272:
3267:
3263:
3251:
3248:
3229:
3224:
3220:
3216:
3211:
3207:
3203:
3183:
3180:
3177:
3174:
3171:
3168:
3143:
3139:
3135:
3130:
3126:
3112:
3109:
3094:
3090:
3067:
3063:
3050:
3049:
3037:
3027:
3024:
3018:
3006:
2996:
2993:
2987:
2975:
2972:
2969:
2949:
2939:
2936:
2917:
2914:
2911:
2891:
2888:
2885:
2882:
2879:
2876:
2873:
2870:
2867:
2864:
2861:
2858:
2855:
2852:
2828:
2808:
2805:
2802:
2782:
2779:
2776:
2773:
2770:
2767:
2764:
2761:
2758:
2755:
2752:
2749:
2746:
2743:
2723:
2703:
2679:
2656:
2636:
2616:
2596:
2576:
2573:
2570:
2567:
2564:
2561:
2558:
2555:
2544:
2543:
2531:
2528:
2525:
2505:
2495:
2492:
2486:
2474:
2471:
2468:
2448:
2438:
2435:
2429:
2417:
2414:
2411:
2391:
2381:
2378:
2372:
2360:
2357:
2354:
2334:
2324:
2321:
2302:
2282:
2251:
2248:
2246:
2243:
2227:
2224:
2221:
2218:
2215:
2212:
2209:
2206:
2203:
2200:
2197:
2194:
2189:
2186:
2182:
2178:
2175:
2172:
2167:
2163:
2159:
2158:
2155:
2152:
2149:
2146:
2143:
2140:
2137:
2134:
2131:
2128:
2125:
2122:
2117:
2114:
2110:
2106:
2103:
2100:
2095:
2091:
2087:
2086:
2066:
2061:
2057:
2053:
2048:
2044:
2040:
2034:
2031:
2006:
2003:
1999:
1970:
1966:
1943:
1939:
1918:
1894:
1889:
1885:
1881:
1878:
1873:
1869:
1866:
1863:
1860:
1857:
1854:
1848:
1843:
1839:
1836:
1833:
1830:
1827:
1824:
1818:
1813:
1809:
1805:
1802:
1797:
1793:
1789:
1784:
1780:
1776:
1773:
1771:
1769:
1766:
1765:
1762:
1757:
1754:
1749:
1744:
1740:
1737:
1734:
1731:
1728:
1725:
1719:
1714:
1710:
1706:
1701:
1697:
1694:
1691:
1688:
1685:
1682:
1676:
1673:
1671:
1667:
1663:
1659:
1658:
1655:
1650:
1647:
1642:
1637:
1633:
1630:
1627:
1624:
1621:
1618:
1612:
1607:
1603:
1599:
1594:
1590:
1587:
1584:
1581:
1578:
1575:
1569:
1566:
1564:
1560:
1556:
1552:
1551:
1531:
1527:
1523:
1520:
1517:
1514:
1511:
1508:
1505:
1502:
1499:
1496:
1493:
1490:
1487:
1484:
1462:
1458:
1454:
1449:
1445:
1420:
1415:
1411:
1405:
1402:
1397:
1393:
1386:
1383:
1380:
1377:
1374:
1371:
1367:
1363:
1360:
1358:
1354:
1350:
1346:
1345:
1342:
1337:
1333:
1327:
1324:
1319:
1315:
1308:
1305:
1302:
1299:
1296:
1293:
1289:
1285:
1282:
1280:
1276:
1272:
1268:
1267:
1245:
1241:
1215:
1211:
1187:
1164:
1144:
1141:
1136:
1133:
1130:
1127:
1123:
1101:
1095:
1092:
1089:
1086:
1082:
1077:
1073:
1070:
1067:
1063:
1057:
1054:
1051:
1048:
1044:
1039:
1018:
1015:
1012:
1009:
1006:
1003:
1000:
997:
994:
991:
988:
985:
982:
979:
976:
973:
970:
949:
943:
939:
934:
930:
926:
920:
916:
911:
890:
868:
864:
852:
851:
839:
819:
808:
797:
792:
788:
784:
779:
775:
771:
768:
765:
760:
755:
752:
749:
745:
741:
738:
718:
704:test statistic
691:
688:
685:
665:
662:
659:
656:
653:
650:
647:
644:
624:
621:
618:
598:
595:
592:
589:
586:
583:
580:
556:
545:
531:
527:
523:
520:
517:
512:
508:
486:
480:
476:
471:
467:
464:
461:
457:
451:
447:
442:
430:
419:
415:
409:
405:
400:
396:
393:
390:
386:
380:
376:
371:
344:
340:
336:
333:
330:
325:
321:
306:ordered metric
291:
287:
283:
278:
274:
253:
248:
244:
240:
235:
231:
227:
212:interval scale
199:
194:
190:
186:
181:
177:
173:
170:
167:
164:
161:
156:
152:
148:
143:
139:
135:
119:
118:Test procedure
116:
92:Frank Wilcoxon
87:
84:
32:non-parametric
15:
9:
6:
4:
3:
2:
14782:
14771:
14768:
14766:
14763:
14761:
14758:
14757:
14755:
14740:
14739:
14730:
14728:
14727:
14718:
14716:
14715:
14710:
14704:
14702:
14701:
14692:
14691:
14688:
14674:
14671:
14669:
14668:Geostatistics
14666:
14664:
14661:
14659:
14656:
14654:
14651:
14650:
14648:
14646:
14642:
14636:
14635:Psychometrics
14633:
14631:
14628:
14626:
14623:
14621:
14618:
14616:
14613:
14611:
14608:
14606:
14603:
14601:
14598:
14596:
14593:
14591:
14588:
14587:
14585:
14583:
14579:
14573:
14570:
14568:
14565:
14563:
14559:
14556:
14554:
14551:
14549:
14546:
14544:
14541:
14540:
14538:
14536:
14532:
14526:
14523:
14521:
14518:
14516:
14512:
14509:
14507:
14504:
14503:
14501:
14499:
14498:Biostatistics
14495:
14491:
14487:
14482:
14478:
14460:
14459:Log-rank test
14457:
14456:
14454:
14450:
14444:
14441:
14440:
14438:
14436:
14432:
14426:
14423:
14421:
14418:
14416:
14413:
14411:
14408:
14407:
14405:
14403:
14399:
14396:
14394:
14390:
14380:
14377:
14375:
14372:
14370:
14367:
14365:
14362:
14360:
14357:
14356:
14354:
14352:
14348:
14342:
14339:
14337:
14334:
14332:
14330:(Box–Jenkins)
14326:
14324:
14321:
14319:
14316:
14312:
14309:
14308:
14307:
14304:
14303:
14301:
14299:
14295:
14289:
14286:
14284:
14283:Durbin–Watson
14281:
14279:
14273:
14271:
14268:
14266:
14265:Dickey–Fuller
14263:
14262:
14260:
14256:
14250:
14247:
14245:
14242:
14240:
14239:Cointegration
14237:
14235:
14232:
14230:
14227:
14225:
14222:
14220:
14217:
14215:
14214:Decomposition
14212:
14211:
14209:
14205:
14202:
14200:
14196:
14186:
14183:
14182:
14181:
14178:
14177:
14176:
14173:
14169:
14166:
14165:
14164:
14161:
14159:
14156:
14154:
14151:
14149:
14146:
14144:
14141:
14139:
14136:
14134:
14131:
14129:
14126:
14125:
14123:
14121:
14117:
14111:
14108:
14106:
14103:
14101:
14098:
14096:
14093:
14091:
14088:
14086:
14085:Cohen's kappa
14083:
14082:
14080:
14078:
14074:
14070:
14066:
14062:
14058:
14054:
14049:
14045:
14031:
14028:
14026:
14023:
14021:
14018:
14016:
14013:
14012:
14010:
14008:
14004:
13998:
13994:
13990:
13984:
13982:
13979:
13978:
13976:
13974:
13970:
13964:
13961:
13959:
13956:
13954:
13951:
13949:
13946:
13944:
13941:
13939:
13938:Nonparametric
13936:
13934:
13931:
13930:
13928:
13924:
13918:
13915:
13913:
13910:
13908:
13905:
13903:
13900:
13899:
13897:
13895:
13891:
13885:
13882:
13880:
13877:
13875:
13872:
13870:
13867:
13865:
13862:
13861:
13859:
13857:
13853:
13847:
13844:
13842:
13839:
13837:
13834:
13832:
13829:
13828:
13826:
13824:
13820:
13816:
13809:
13806:
13804:
13801:
13800:
13796:
13792:
13776:
13773:
13772:
13771:
13768:
13766:
13763:
13761:
13758:
13754:
13751:
13749:
13746:
13745:
13744:
13741:
13740:
13738:
13736:
13732:
13722:
13719:
13715:
13709:
13707:
13701:
13699:
13693:
13692:
13691:
13688:
13687:Nonparametric
13685:
13683:
13677:
13673:
13670:
13669:
13668:
13662:
13658:
13657:Sample median
13655:
13654:
13653:
13650:
13649:
13647:
13645:
13641:
13633:
13630:
13628:
13625:
13623:
13620:
13619:
13618:
13615:
13613:
13610:
13608:
13602:
13600:
13597:
13595:
13592:
13590:
13587:
13585:
13582:
13580:
13578:
13574:
13572:
13569:
13568:
13566:
13564:
13560:
13554:
13552:
13548:
13546:
13544:
13539:
13537:
13532:
13528:
13527:
13524:
13521:
13519:
13515:
13505:
13502:
13500:
13497:
13495:
13492:
13491:
13489:
13487:
13483:
13477:
13474:
13470:
13467:
13466:
13465:
13462:
13458:
13455:
13454:
13453:
13450:
13448:
13445:
13444:
13442:
13440:
13436:
13428:
13425:
13423:
13420:
13419:
13418:
13415:
13413:
13410:
13408:
13405:
13403:
13400:
13398:
13395:
13393:
13390:
13389:
13387:
13385:
13381:
13375:
13372:
13368:
13365:
13361:
13358:
13356:
13353:
13352:
13351:
13348:
13347:
13346:
13343:
13339:
13336:
13334:
13331:
13329:
13326:
13324:
13321:
13320:
13319:
13316:
13315:
13313:
13311:
13307:
13304:
13302:
13298:
13292:
13289:
13287:
13284:
13280:
13277:
13276:
13275:
13272:
13270:
13267:
13263:
13262:loss function
13260:
13259:
13258:
13255:
13251:
13248:
13246:
13243:
13241:
13238:
13237:
13236:
13233:
13231:
13228:
13226:
13223:
13219:
13216:
13214:
13211:
13209:
13203:
13200:
13199:
13198:
13195:
13191:
13188:
13186:
13183:
13181:
13178:
13177:
13176:
13173:
13169:
13166:
13164:
13161:
13160:
13159:
13156:
13152:
13149:
13148:
13147:
13144:
13140:
13137:
13136:
13135:
13132:
13130:
13127:
13125:
13122:
13120:
13117:
13116:
13114:
13112:
13108:
13104:
13100:
13095:
13091:
13077:
13074:
13072:
13069:
13067:
13064:
13062:
13059:
13058:
13056:
13054:
13050:
13044:
13041:
13039:
13036:
13034:
13031:
13030:
13028:
13024:
13018:
13015:
13013:
13010:
13008:
13005:
13003:
13000:
12998:
12995:
12993:
12990:
12988:
12985:
12984:
12982:
12980:
12976:
12970:
12967:
12965:
12964:Questionnaire
12962:
12960:
12957:
12953:
12950:
12948:
12945:
12944:
12943:
12940:
12939:
12937:
12935:
12931:
12925:
12922:
12920:
12917:
12915:
12912:
12910:
12907:
12905:
12902:
12900:
12897:
12895:
12892:
12890:
12887:
12886:
12884:
12882:
12878:
12874:
12870:
12865:
12861:
12847:
12844:
12842:
12839:
12837:
12834:
12832:
12829:
12827:
12824:
12822:
12819:
12817:
12814:
12812:
12809:
12807:
12804:
12802:
12799:
12797:
12794:
12792:
12791:Control chart
12789:
12787:
12784:
12782:
12779:
12777:
12774:
12773:
12771:
12769:
12765:
12759:
12756:
12752:
12749:
12747:
12744:
12743:
12742:
12739:
12737:
12734:
12732:
12729:
12728:
12726:
12724:
12720:
12714:
12711:
12709:
12706:
12704:
12701:
12700:
12698:
12694:
12688:
12685:
12684:
12682:
12680:
12676:
12664:
12661:
12659:
12656:
12654:
12651:
12650:
12649:
12646:
12644:
12641:
12640:
12638:
12636:
12632:
12626:
12623:
12621:
12618:
12616:
12613:
12611:
12608:
12606:
12603:
12601:
12598:
12596:
12593:
12592:
12590:
12588:
12584:
12578:
12575:
12573:
12570:
12566:
12563:
12561:
12558:
12556:
12553:
12551:
12548:
12546:
12543:
12541:
12538:
12536:
12533:
12531:
12528:
12526:
12523:
12521:
12518:
12517:
12516:
12513:
12512:
12510:
12508:
12504:
12501:
12499:
12495:
12491:
12487:
12482:
12478:
12472:
12469:
12467:
12464:
12463:
12460:
12456:
12449:
12444:
12442:
12437:
12435:
12430:
12429:
12426:
12420:
12416:
12412:
12409:
12406:
12404:
12401:
12399:
12396:
12394:
12391:
12389:
12386:
12385:
12366:
12362:
12356:
12348:
12342:
12338:
12337:
12329:
12321:
12315:
12306:
12301:
12298:: 11.IT.3.1,
12297:
12293:
12286:
12284:
12274:
12265:
12256:
12250:Siegel, p. 76
12247:
12238:
12229:
12221:
12217:
12213:
12209:
12202:
12193:
12184:
12175:
12170:
12166:
12162:
12158:
12151:
12149:
12139:
12130:
12121:
12112:
12103:
12094:
12085:
12076:
12067:
12061:
12057:
12050:
12044:Pratt, p. 660
12041:
12035:Pratt, p. 661
12032:
12023:
12014:
12008:Pratt, p. 660
12005:
11996:
11987:
11978:
11969:
11963:Pratt, p. 660
11960:
11951:
11942:
11933:
11924:
11916:
11912:
11908:
11904:
11897:
11889:
11885:
11878:
11872:Pratt, p. 663
11869:
11863:Pratt, p. 659
11860:
11852:
11848:
11844:
11840:
11833:
11825:
11818:
11809:
11800:
11791:
11782:
11776:
11772:
11768:
11762:
11753:
11744:
11735:
11726:
11717:
11708:
11699:
11690:
11688:0-471-88474-X
11684:
11680:
11673:
11664:
11655:
11646:
11637:
11628:
11622:
11618:
11611:
11602:
11596:Siegel, p. 76
11593:
11584:
11576:
11574:9780070573482
11570:
11566:
11565:
11557:
11549:
11545:
11541:
11537:
11533:
11529:
11525:
11521:
11514:
11507:
11493:
11489:
11483:
11474:
11472:0-471-16068-7
11468:
11464:
11457:
11453:
11443:
11440:
11438:
11435:
11434:
11424:
11421:
11414:
11411:
11403:
11400:
11397:
11394:
11391:
11388:
11384:wilcoxon_test
11381:
11378:
11375:
11372:
11336:
11333:
11332:
11326:
11324:
11320:
11316:
11312:
11308:
11303:
11298:
11296:
11292:
11288:
11284:
11280:
11277: =
11276:
11272:
11268:
11264:
11259:
11257:
11253:
11247:
11223:
11203:
11195:
11178:
11174:
11165:
11158:
11144:
11141:
11128:
11122:
11119:
11116:
11113:
11107:
11104:
11100:
11096:
11088:
11076:
11062:
11059:
11056:
11053:
11050:
11047:
11044:
11041:
11038:
11035:
11032:
11029:
11026:
11023:
11020:
11017:
11014:
11011:
11008:
11005:
11002:
10995:
10994:
10993:
10991:
10973:
10969:
10960:
10934:
10933:sign function
10918:
10901:
10898:
10895:
10892:
10889:
10886:
10883:
10882:
10878:
10875:
10872:
10869:
10866:
10863:
10860:
10859:
10855:
10852:
10849:
10846:
10843:
10840:
10837:
10836:
10832:
10829:
10826:
10823:
10820:
10817:
10814:
10813:
10809:
10806:
10803:
10800:
10797:
10794:
10791:
10790:
10786:
10783:
10780:
10777:
10774:
10771:
10768:
10767:
10763:
10760:
10757:
10754:
10751:
10748:
10745:
10744:
10740:
10737:
10734:
10731:
10728:
10725:
10722:
10721:
10717:
10714:
10711:
10708:
10705:
10702:
10699:
10698:
10694:
10691:
10688:
10685:
10682:
10679:
10676:
10675:
10658:
10654:
10650:
10647:
10640:
10624:
10620:
10612:
10589:
10575:
10568:
10567:
10550:
10547:
10544:
10540:
10536:
10531:
10528:
10525:
10521:
10497:
10494:
10491:
10487:
10463:
10460:
10457:
10453:
10431:
10423:
10420:
10417:
10410:
10407:
10404:
10401:
10398:
10397:
10393:
10390:
10387:
10384:
10381:
10380:
10376:
10373:
10370:
10367:
10364:
10363:
10359:
10356:
10353:
10350:
10347:
10346:
10342:
10339:
10336:
10333:
10330:
10329:
10325:
10322:
10319:
10316:
10313:
10312:
10308:
10305:
10302:
10299:
10296:
10295:
10291:
10288:
10285:
10282:
10279:
10278:
10274:
10271:
10268:
10265:
10262:
10261:
10257:
10254:
10251:
10248:
10245:
10244:
10220:
10206:
10199:
10198:
10181:
10178:
10175:
10171:
10167:
10162:
10159:
10156:
10152:
10128:
10125:
10122:
10118:
10094:
10091:
10088:
10084:
10062:
10054:
10051:
10050:
10042:
10026:
10022:
9995:
9992:
9989:
9983:
9976:
9970:
9965:
9961:
9957:
9937:
9915:
9911:
9888:
9884:
9863:
9841:
9836:
9832:
9809:
9804:
9800:
9777:
9773:
9764:
9748:
9745:
9742:
9734:
9729:
9710:
9706:
9702:
9697:
9693:
9672:
9655:
9651:
9645:
9641:
9637:
9635:
9625:
9621:
9614:
9611:
9604:
9599:
9595:
9591:
9589:
9581:
9575:
9572:
9548:
9543:
9539:
9535:
9531:
9528:
9522:
9519:
9516:
9513:
9504:
9501:
9498:
9492:
9489:
9483:
9480:
9477:
9474:
9465:
9462:
9459:
9453:
9447:
9442:
9438:
9417:
9412:
9405:
9402:
9399:
9393:
9387:
9382:
9375:
9372:
9369:
9363:
9357:
9349:
9345:
9314:
9310:
9289:
9269:
9249:
9246:
9243:
9238:
9234:
9230:
9227:
9224:
9204:
9184:
9164:
9155:
9142:
9134:
9131:
9127:
9123:
9120:
9112:
9107:
9104:
9101:
9097:
9089:
9085:
9081:
9076:
9070:
9064:
9044:
9035:
9016:
9012:
9008:
9005:
9001:
8994:
8972:
8968:
8947:
8940:
8933:
8930:
8927:
8924:
8915:
8912:
8909:
8903:
8894:
8887:
8884:
8881:
8875:
8869:
8863:
8860:
8854:
8851:
8845:
8842:
8822:
8816:
8813:
8810:
8805:
8801:
8781:
8778:
8775:
8772:
8763:
8760:
8757:
8751:
8748:
8743:
8740:
8737:
8734:
8731:
8726:
8722:
8718:
8702:
8696:
8693:
8687:
8678:
8672:
8669:
8664:
8660:
8644:
8627:
8622:
8615:
8612:
8609:
8606:
8597:
8594:
8591:
8585:
8579:
8577:
8569:
8563:
8560:
8553:
8548:
8541:
8538:
8535:
8532:
8523:
8520:
8517:
8511:
8505:
8497:
8493:
8486:
8483:
8480:
8478:
8468:
8464:
8457:
8454:
8447:
8444:
8441:
8439:
8431:
8416:
8411:
8404:
8401:
8398:
8392:
8386:
8378:
8374:
8362:
8360:
8350:
8346:
8311:
8307:
8284:
8280:
8259:
8239:
8230:
8211:
8207:
8203:
8198:
8194:
8164:
8161:
8156:
8152:
8143:
8140:
8137:
8133:
8129:
8121:
8117:
8108:
8105:
8102:
8098:
8094:
8086:
8082:
8073:
8069:
8065:
8040:
8036:
8027:
8023:
8002:
7980:
7976:
7972:
7967:
7963:
7937:
7933:
7924:
7920:
7910:
7908:
7890:
7886:
7863:
7859:
7854:
7845:
7841:
7832:
7828:
7824:
7816:
7812:
7808:
7803:
7799:
7770:
7766:
7745:
7742:
7737:
7733:
7709:
7706:
7703:
7700:
7697:
7694:
7691:
7668:
7648:
7625:
7622:
7619:
7616:
7613:
7610:
7607:
7584:
7562:
7558:
7534:
7531:
7528:
7525:
7522:
7499:
7493:
7490:
7485:
7481:
7472:
7469:
7466:
7462:
7458:
7450:
7446:
7437:
7434:
7431:
7427:
7423:
7415:
7411:
7402:
7398:
7377:
7373:
7366:
7363:
7360:
7354:
7351:
7348:
7328:
7325:
7322:
7302:
7299:
7291:
7287:
7278:
7274:
7253:
7250:
7245:
7241:
7220:
7217:
7209:
7205:
7196:
7192:
7171:
7168:
7162:
7154:
7150:
7127:
7123:
7115:which sum to
7099:
7096:
7093:
7090:
7087:
7062:
7058:
7054:
7049:
7045:
7019:
7015:
7006:
7002:
6979:
6975:
6954:
6932:
6928:
6907:
6898:
6884:
6862:
6858:
6837:
6834:
6831:
6811:
6808:
6805:
6783:
6779:
6758:
6738:
6730:
6720:
6718:
6714:
6710:
6706:
6690:
6682:
6677:
6661:
6657:
6652:
6648:
6645:
6642:
6630:
6629:
6628:
6612:
6608:
6603:
6599:
6596:
6593:
6581:
6580:
6579:
6573:
6572:
6571:
6568:
6548:
6523:
6503:
6483:
6475:
6470:
6466:
6464:
6445:
6442:
6439:
6436:
6433:
6421:
6407:
6397:
6383:
6379:
6372:
6369:
6366:
6343:
6323:
6303:
6283:
6263:
6243:
6223:
6220:
6217:
6197:
6177:
6174:
6171:
6167:
6160:
6157:
6154:
6151:
6148:
6123:
6119:
6096:
6092:
6069:
6065:
6042:
6038:
6017:
6014:
6011:
6007:
6000:
5997:
5994:
5969:
5965:
5942:
5938:
5915:
5911:
5890:
5880:
5876:
5867:
5857:
5853:
5844:
5834:
5830:
5821:
5811:
5807:
5798:
5788:
5784:
5775:
5765:
5761:
5752:
5742:
5738:
5723:
5706:
5703:
5700:
5697:
5694:
5682:
5678:
5674:
5669:
5653:
5649:
5634:
5617:
5614:
5601:
5599:
5594:
5592:
5587:
5573:
5553:
5530:
5527:
5522:
5518:
5514:
5509:
5505:
5475:
5472:
5467:
5463:
5459:
5454:
5450:
5423:
5420:
5417:
5392:
5372:
5364:
5360:
5345:
5337:
5334:
5333:
5332:
5318:
5315:
5309:
5303:
5300:
5292:
5287:
5285:
5281:
5263:
5259:
5254:
5250:
5228:
5224:
5219:
5215:
5195:
5175:
5172:
5167:
5163:
5158:
5154:
5151:
5148:
5126:
5122:
5117:
5113:
5105:
5097:
5096:
5095:
5081:
5078:
5073:
5069:
5064:
5060:
5057:
5054:
5032:
5028:
5023:
5019:
5011:
5003:
5002:
5001:
4995:
4994:
4993:
4989:
4988:
4977:
4961:
4957:
4934:
4930:
4926:
4921:
4917:
4913:
4908:
4904:
4900:
4895:
4891:
4868:
4864:
4860:
4855:
4851:
4830:
4827:
4824:
4802:
4798:
4794:
4789:
4785:
4759:
4755:
4751:
4746:
4742:
4716:
4712:
4679:
4676:
4671:
4667:
4663:
4658:
4654:
4644:
4638:
4635:
4630:
4626:
4622:
4617:
4613:
4606:
4603:
4583:
4580:
4577:
4555:
4551:
4547:
4542:
4538:
4529:
4524:
4520:
4502:
4499:
4494:
4490:
4486:
4481:
4477:
4467:
4461:
4458:
4453:
4449:
4445:
4440:
4436:
4429:
4426:
4406:
4403:
4400:
4378:
4374:
4370:
4365:
4361:
4352:
4347:
4343:
4323:
4319:
4315:
4310:
4306:
4294:
4289:
4285:
4284:
4283:
4280:
4264:
4260:
4256:
4251:
4247:
4224:
4220:
4216:
4211:
4207:
4178:
4175:
4170:
4166:
4162:
4159:
4156:
4151:
4147:
4118:
4114:
4110:
4105:
4101:
4077:
4074:
4071:
4063:
4058:
4054:
4036:
4033:
4028:
4024:
4020:
4017:
4014:
4009:
4005:
3976:
3972:
3968:
3963:
3959:
3935:
3932:
3929:
3921:
3916:
3912:
3894:
3891:
3886:
3882:
3878:
3875:
3872:
3867:
3863:
3834:
3830:
3826:
3821:
3817:
3793:
3790:
3787:
3779:
3774:
3770:
3752:
3749:
3746:
3740:
3737:
3731:
3728:
3725:
3719:
3694:
3690:
3686:
3681:
3677:
3648:
3644:
3640:
3635:
3631:
3619:
3598:
3594:
3590:
3585:
3581:
3569:
3564:
3560:
3559:
3558:
3540:
3537:
3534:
3512:
3508:
3504:
3499:
3495:
3486:
3481:
3477:
3462:
3459:
3456:
3434:
3430:
3426:
3421:
3417:
3408:
3403:
3399:
3384:
3381:
3378:
3356:
3352:
3348:
3343:
3339:
3330:
3325:
3321:
3306:
3303:
3300:
3278:
3274:
3270:
3265:
3261:
3252:
3247:
3243:
3242:
3241:
3222:
3218:
3214:
3209:
3205:
3178:
3175:
3172:
3166:
3157:
3141:
3137:
3133:
3128:
3124:
3108:
3092:
3088:
3065:
3061:
3035:
3028:
3023:
3019:
3004:
2997:
2992:
2988:
2973:
2970:
2967:
2947:
2940:
2935:
2931:
2930:
2929:
2915:
2912:
2909:
2886:
2883:
2880:
2871:
2865:
2862:
2859:
2856:
2842:
2826:
2819:. Similarly,
2806:
2803:
2800:
2777:
2774:
2771:
2762:
2756:
2753:
2750:
2747:
2721:
2701:
2693:
2677:
2668:
2654:
2634:
2614:
2594:
2574:
2571:
2565:
2562:
2559:
2529:
2526:
2523:
2503:
2496:
2491:
2487:
2472:
2469:
2466:
2446:
2439:
2434:
2430:
2415:
2412:
2409:
2389:
2382:
2377:
2373:
2358:
2355:
2352:
2332:
2325:
2320:
2316:
2315:
2314:
2300:
2280:
2272:
2267:
2265:
2261:
2257:
2242:
2225:
2219:
2216:
2213:
2210:
2207:
2204:
2201:
2198:
2195:
2192:
2187:
2184:
2180:
2170:
2165:
2161:
2153:
2147:
2144:
2141:
2138:
2135:
2132:
2129:
2126:
2123:
2120:
2115:
2112:
2108:
2098:
2093:
2089:
2059:
2055:
2051:
2046:
2042:
2032:
2029:
2004:
2001:
1997:
1989:
1988:Walsh average
1984:
1968:
1964:
1941:
1937:
1916:
1892:
1887:
1883:
1879:
1876:
1871:
1864:
1861:
1858:
1852:
1846:
1841:
1834:
1831:
1828:
1822:
1816:
1811:
1807:
1803:
1800:
1795:
1791:
1787:
1782:
1778:
1774:
1772:
1767:
1760:
1755:
1752:
1747:
1742:
1735:
1732:
1729:
1723:
1717:
1712:
1708:
1704:
1699:
1692:
1689:
1686:
1680:
1674:
1672:
1665:
1661:
1653:
1648:
1645:
1640:
1635:
1628:
1625:
1622:
1616:
1610:
1605:
1601:
1597:
1592:
1585:
1582:
1579:
1573:
1567:
1565:
1558:
1554:
1529:
1525:
1518:
1515:
1512:
1506:
1503:
1500:
1497:
1494:
1491:
1488:
1485:
1482:
1460:
1456:
1452:
1447:
1443:
1418:
1413:
1409:
1403:
1400:
1395:
1391:
1384:
1381:
1378:
1375:
1372:
1369:
1365:
1361:
1359:
1352:
1348:
1340:
1335:
1331:
1325:
1322:
1317:
1313:
1306:
1303:
1300:
1297:
1294:
1291:
1287:
1283:
1281:
1274:
1270:
1243:
1239:
1231:
1213:
1209:
1201:
1185:
1176:
1162:
1142:
1139:
1131:
1125:
1121:
1090:
1084:
1080:
1071:
1068:
1065:
1052:
1046:
1042:
1029:is such that
1013:
1010:
1007:
1004:
1001:
989:
986:
983:
980:
977:
971:
968:
941:
937:
928:
918:
914:
888:
866:
862:
837:
817:
809:
795:
790:
786:
777:
773:
766:
763:
758:
753:
750:
747:
743:
739:
736:
716:
709:
705:
689:
686:
683:
663:
660:
657:
651:
645:
642:
622:
619:
616:
596:
593:
587:
581:
578:
570:
569:sign function
554:
546:
529:
525:
521:
518:
515:
510:
506:
478:
474:
465:
462:
459:
449:
445:
431:
417:
407:
403:
394:
391:
388:
378:
374:
360:
359:
358:
342:
338:
334:
331:
328:
323:
319:
309:
307:
289:
285:
281:
276:
272:
246:
242:
238:
233:
229:
217:
213:
192:
188:
184:
179:
175:
168:
165:
162:
154:
150:
146:
141:
137:
124:
123:paired data.
115:
113:
111:
105:
101:
100:Sidney Siegel
97:
96:rank-sum test
93:
83:
81:
76:
72:
68:
64:
60:
56:
52:
50:
44:
40:
36:
33:
29:
22:
14770:U-statistics
14736:
14724:
14705:
14698:
14610:Econometrics
14560: /
14543:Chemometrics
14520:Epidemiology
14513: /
14486:Applications
14328:ARIMA model
14275:Q-statistic
14224:Stationarity
14120:Multivariate
14063: /
14059: /
14057:Multivariate
14055: /
13995: /
13991: /
13765:Bayes factor
13664:Signed rank
13663:
13576:
13550:
13542:
13530:
13225:Completeness
13061:Cohort study
12959:Opinion poll
12894:Missing data
12881:Study design
12836:Scatter plot
12758:Scatter plot
12751:Spearman's ρ
12713:Grouped data
12414:
12368:. Retrieved
12364:
12355:
12335:
12328:
12295:
12291:
12273:
12264:
12255:
12246:
12237:
12228:
12211:
12207:
12201:
12192:
12183:
12164:
12160:
12138:
12129:
12120:
12111:
12102:
12093:
12084:
12075:
12055:
12049:
12040:
12031:
12022:
12013:
12004:
11995:
11986:
11977:
11968:
11959:
11950:
11941:
11932:
11923:
11906:
11902:
11896:
11887:
11883:
11877:
11868:
11859:
11842:
11838:
11832:
11823:
11817:
11808:
11799:
11790:
11770:
11761:
11752:
11743:
11734:
11725:
11716:
11707:
11698:
11693:, pp. 32, 50
11678:
11672:
11663:
11654:
11645:
11636:
11616:
11610:
11601:
11592:
11583:
11563:
11556:
11526:(6): 80–83.
11523:
11519:
11506:
11495:. Retrieved
11491:
11482:
11462:
11456:
11322:
11318:
11314:
11310:
11306:
11301:
11299:
11294:
11290:
11286:
11282:
11278:
11274:
11270:
11266:
11262:
11260:
11249:
10911:
9762:
9732:
9730:
9678:
9156:
9036:
8645:
8231:
7911:
6899:
6728:
6726:
6716:
6712:
6708:
6707:method, the
6704:
6680:
6678:
6634:
6585:
6577:
6569:
6473:
6468:
6467:
6462:
6422:
6398:
5724:
5680:
5676:
5673:average rank
5672:
5670:
5640:
5602:
5598:Likert scale
5595:
5590:
5588:
5408:
5362:observation.
5290:
5288:
5283:
5279:
5103:
5101:
5009:
5007:
4999:
4990:
4986:
4983:
4703:
4522:
4345:
4287:
4281:
4197:
4090:, the pairs
4056:
3948:, the pairs
3914:
3806:, the pairs
3772:
3618:exchangeable
3617:
3562:
3556:
3479:
3401:
3323:
3245:
3158:
3114:
3051:
3021:
2990:
2933:
2840:
2691:
2669:
2545:
2489:
2432:
2375:
2318:
2268:
2264:pseudomedian
2253:
1987:
1985:
1229:
1199:
1177:
853:
707:
310:
305:
216:real numbers
125:
121:
109:
107:
103:
89:
66:
62:
58:
48:
27:
25:
14738:WikiProject
14653:Cartography
14615:Jurimetrics
14567:Reliability
14298:Time domain
14277:(Ljung–Box)
14199:Time-series
14077:Categorical
14061:Time-series
14053:Categorical
13988:(Bernoulli)
13823:Correlation
13803:Correlation
13599:Jarque–Bera
13571:Chi-squared
13333:M-estimator
13286:Asymptotics
13230:Sufficiency
12997:Interaction
12909:Replication
12889:Effect size
12846:Violin plot
12826:Radar chart
12806:Forest plot
12796:Correlogram
12746:Kendall's τ
11785:, pp. 39–41
11340:wilcox.test
11252:effect size
11240:Effect size
11133:, two-sided
6256:, and that
5208:is between
567:denote the
14754:Categories
14605:Demography
14323:ARMA model
14128:Regression
13705:(Friedman)
13666:(Wilcoxon)
13604:Normality
13594:Lilliefors
13541:Student's
13417:Resampling
13291:Robustness
13279:divergence
13269:Efficiency
13207:(monotone)
13202:Likelihood
13119:Population
12952:Stratified
12904:Population
12723:Dependence
12679:Count data
12610:Percentile
12587:Dispersion
12520:Arithmetic
12455:Statistics
12370:2023-08-24
11890:(3): 1–13.
11497:2021-09-02
11448:References
11396:Accord.NET
11380:GNU Octave
11273:, or
7785:satisfies
6727:Computing
5280:decreasing
5106:-value of
5012:-value of
4843:, we have
2734:satisfies
901:for which
810:Produce a
47:Student's
13986:Logistic
13753:posterior
13679:Rank sum
13427:Jackknife
13422:Bootstrap
13240:Bootstrap
13175:Parameter
13124:Statistic
12919:Statistic
12831:Run chart
12816:Pie chart
12811:Histogram
12801:Fan chart
12776:Bar chart
12658:L-moments
12545:Geometric
11442:Sign test
11386:function.
11166:∴
11114:α
11108:
11036:−
11030:−
11024:−
11018:−
10833: –6
10824: –1
10810: –5
10801: –1
10787: –4
10778: –1
10764: –3
10755: –1
10651:⋅
10537:−
10408: –1
10374: –1
10340: –1
10272: –1
10168:−
9993:−
9916:−
9842:−
9746:−
9711:−
9642:σ
9615:
9596:σ
9576:
9529:−
9490:−
9439:σ
9388:−
9244:−
9231:∑
9098:∏
9006:−
8870:−
8811:−
8741:−
8697:ϕ
8682:Φ
8679:≈
8670:≤
8564:
8498:−
8487:
8458:
8379:−
8312:−
8204:≤
8162:−
8141:−
8106:−
7743:−
7707:−
7698:…
7623:−
7614:…
7529:…
7491:−
7470:−
7435:−
7251:≠
7094:…
6994:. Define
6719:-values.
6643:α
6594:α
6465:-values.
6440:…
6408:μ
6373:ℓ
6344:ℓ
6264:ℓ
6221:−
5701:…
5421:≠
5393:α
5373:α
5346:μ
5304:
5196:α
5173:≈
5149:α
5079:≈
5055:α
4927:−
4901:−
4828:≠
4645:≤
4636:−
4581:≥
4548:−
4468:≥
4459:−
4404:≥
4371:−
4257:−
4217:−
4179:μ
4176:−
4160:μ
4075:≠
4072:μ
4064:For some
4037:μ
4034:−
4018:μ
3930:μ
3922:For some
3895:μ
3892:−
3876:μ
3788:μ
3780:For some
3538:≠
3535:μ
3505:−
3457:μ
3427:−
3379:μ
3349:−
3301:μ
3271:−
3134:−
3089:μ
3062:μ
2968:μ
2913:≥
2872:≤
2863:−
2804:≥
2763:≥
2754:−
2655:μ
2595:μ
2566:μ
2527:≠
2524:μ
2467:μ
2410:μ
2353:μ
2217:≤
2211:≤
2205:≤
2199::
2174:#
2166:−
2145:≤
2139:≤
2133:≤
2127::
2102:#
1969:−
1888:−
1877:−
1817:−
1796:−
1788:−
1748:−
1705:−
1666:−
1606:−
1598:−
1495:⋯
1461:−
1379:≤
1373:≤
1366:∑
1353:−
1301:≤
1295:≤
1288:∑
1244:−
1126:σ
1085:σ
1069:⋯
1047:σ
1008:…
996:→
984:…
969:σ
929:≤
767:
744:∑
661:−
646:
582:
519:…
463:…
392:…
332:…
282:−
166:…
108:Wilcoxon
80:sign test
73:when the
35:rank test
14700:Category
14393:Survival
14270:Johansen
13993:Binomial
13948:Isotonic
13535:(normal)
13180:location
12987:Blocking
12942:Sampling
12821:Q–Q plot
12786:Box plot
12768:Graphics
12663:Skewness
12653:Kurtosis
12625:Variance
12555:Heronian
12550:Harmonic
12314:citation
12070:, p. 194
11631:, p. 148
11477:, p. 350
11431:See also
11297:= 0.20.
7315:for all
7233:for all
6336:through
5284:positive
2902:for all
2793:for all
2077:. Then:
1909:Because
1435:Because
1228:and the
1155:for all
361:Compute
43:location
14726:Commons
14673:Kriging
14558:Process
14515:studies
14374:Wavelet
14207:General
13374:Plug-in
13168:L space
12947:Cluster
12648:Moments
12466:Outline
11548:3001968
11325:= .20.
10988:is the
10957:is the
10931:is the
10695:
10692:
10686:
10323:
10045:Example
6210:, that
5278:, then
2587:, then
1113:, then
706:is the
86:History
14595:Census
14185:Normal
14133:Manova
13953:Robust
13703:2-way
13695:1-way
13533:-test
13204:
12781:Biplot
12572:Median
12565:Lehmer
12507:Center
12343:
12062:
11777:
11685:
11623:
11571:
11546:
11469:
11409:level.
11402:MATLAB
11374:ALGLIB
11358:paired
11317:minus
11224:0.6113
11126:
10961:, and
9903:, and
8835:where
8299:, and
7266:, and
6111:, and
5176:0.0133
5082:0.0134
2694:if an
2256:median
2019:to be
1956:, and
1388:
1310:
702:. The
71:t-test
14219:Trend
13748:prior
13690:anova
13579:-test
13553:-test
13545:-test
13452:Power
13397:Pivot
13190:shape
13185:scale
12635:Shape
12615:Range
12560:Heinz
12535:Cubic
12471:Index
11544:JSTOR
11516:(PDF)
11426:test.
11413:Julia
11390:SciPy
9561:then
4980:Zeros
4777:with
2293:. If
432:Sort
112:-test
51:-test
30:is a
14452:Test
13652:Sign
13504:Wald
12577:Mode
12515:Mean
12341:ISBN
12320:link
12060:ISBN
11775:ISBN
11683:ISBN
11621:ISBN
11569:ISBN
11467:ISBN
11364:TRUE
11196:The
11120:0.05
11105:crit
11097:<
10990:rank
10890:120
10887:140
10867:123
10864:140
10844:110
10841:125
10821:137
10818:125
10798:145
10795:135
10775:124
10772:115
10752:122
10749:115
10741:1.5
10738:1.5
10729:135
10726:140
10718:1.5
10715:1.5
10706:125
10703:130
10683:140
10680:140
10405:145
10402:135
10388:135
10385:140
10371:137
10368:125
10354:123
10351:140
10337:124
10334:115
10320:140
10317:140
10303:120
10300:140
10286:125
10283:130
10269:122
10266:115
10252:110
10249:125
7352:>
7326:<
5957:and
5822:<
5799:<
5753:<
5637:Ties
5566:and
5528:<
5491:and
5473:>
5243:and
5152:>
5058:<
4664:>
4623:<
4487:>
4446:<
4136:and
3994:and
3933:>
3852:and
3791:<
3666:and
3616:are
3460:>
3382:<
3159:Let
2884:>
2860:<
2775:>
2751:<
2470:>
2413:<
2262:and
2260:mean
2193:<
2121:>
1401:<
1323:>
1072:<
1066:<
687:<
635:and
620:>
547:Let
37:for
26:The
13632:BIC
13627:AIC
12300:doi
12216:doi
12169:doi
11911:doi
11847:doi
11536:hdl
11528:doi
11423:SAS
11015:1.5
11009:1.5
10944:abs
10919:sgn
10896:20
10873:17
10850:15
10827:12
10804:10
10792:10
10648:sgn
10598:abs
10576:sgn
10411:10
10399:10
10377:12
10360:17
10309:20
10258:15
10229:abs
10207:sgn
9687:min
9612:Var
9573:Var
9430:If
9329:is
8561:Var
8484:Var
8455:Var
7912:If
7341:or
6018:2.5
5675:or
5301:sgn
5216:109
5155:109
5114:109
2843:if
2839:is
2690:is
764:sgn
676:if
643:sgn
609:if
579:sgn
555:sgn
14756::
12363:.
12316:}}
12312:{{
12294:,
12282:^
12212:62
12210:.
12165:24
12163:.
12159:.
12147:^
11907:68
11905:.
11888:18
11886:.
11843:54
11841:.
11542:.
11534:.
11522:.
11518:.
11490:.
11258:.
11145:15
10935:,
10902:9
10899:9
10893:1
10884:4
10879:8
10876:8
10870:1
10861:7
10856:7
10853:7
10847:1
10838:1
10830:6
10815:8
10807:5
10784:4
10781:9
10769:6
10761:3
10758:7
10746:2
10735:5
10732:1
10723:9
10712:5
10709:1
10700:3
10689:0
10677:5
10394:5
10391:1
10382:9
10365:8
10357:1
10348:7
10343:9
10331:6
10326:0
10314:5
10306:1
10297:4
10292:5
10289:1
10280:3
10275:7
10263:2
10255:1
10246:1
10041:.
9876:,
9656:4.
8941:24
8749:10
8654:Pr
8549:24
8272:,
8229:.
8188:Pr
7909:.
7793:Pr
7184:,
6649:14
6600:14
6396:.
6084:,
6030:,
5586:.
5499:Pr
5444:Pr
5436:,
5286:.
5264:12
5251:55
5229:13
5168:13
5127:13
5074:12
5061:55
5033:12
5020:55
4648:Pr
4596:,
4471:Pr
4419:,
3156:.
3107:.
2875:Pr
2851:Pr
2766:Pr
2742:Pr
2554:Pr
2258:,
1929:,
1175:.
729::
571::
114:.
13577:G
13551:F
13543:t
13531:Z
13250:V
13245:U
12447:e
12440:t
12433:v
12373:.
12349:.
12322:)
12302::
12296:3
12222:.
12218::
12177:.
12171::
12068:.
11917:.
11913::
11853:.
11849::
11783:.
11691:.
11629:.
11577:.
11550:.
11538::
11530::
11524:1
11500:.
11475:.
11419:.
11367:)
11361:=
11355:,
11352:y
11349:,
11346:x
11343:(
11335:R
11323:r
11319:T
11315:S
11311:T
11307:T
11302:T
11295:r
11291:S
11287:T
11283:S
11281:/
11279:T
11275:r
11271:S
11267:T
11263:T
11204:p
11179:0
11175:H
11142:=
11137:)
11129:9
11123:,
11117:=
11111:(
11101:W
11093:|
11089:W
11085:|
11063:9
11060:=
11057:9
11054:+
11051:8
11048:+
11045:7
11042:+
11039:6
11033:5
11027:4
11021:3
11012:+
11006:=
11003:W
10974:i
10970:R
10659:i
10655:R
10625:i
10621:R
10551:i
10548:,
10545:1
10541:x
10532:i
10529:,
10526:2
10522:x
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10027:2
10023:p
10002:)
9999:)
9996:1
9990:n
9987:(
9984:n
9981:(
9977:/
9971:+
9966:0
9962:T
9958:2
9938:F
9912:T
9889:+
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9864:T
9837:0
9833:T
9810:+
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9801:T
9778:0
9774:T
9749:1
9743:n
9733:n
9716:)
9707:T
9703:,
9698:+
9694:T
9690:(
9652:/
9646:2
9638:=
9631:)
9626:+
9622:T
9618:(
9605:,
9600:2
9592:=
9585:)
9582:T
9579:(
9549:,
9544:6
9540:2
9536:/
9532:c
9526:)
9523:1
9520:+
9517:z
9514:2
9511:(
9508:)
9505:1
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9499:z
9496:(
9493:z
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9484:1
9481:+
9478:n
9475:2
9472:(
9469:)
9466:1
9463:+
9460:n
9457:(
9454:n
9448:=
9443:2
9418:.
9413:4
9409:)
9406:1
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9400:z
9397:(
9394:z
9383:4
9379:)
9376:1
9373:+
9370:n
9367:(
9364:n
9358:=
9355:]
9350:+
9346:T
9342:[
9338:E
9315:+
9311:T
9290:T
9270:t
9250:,
9247:t
9239:3
9235:t
9228:=
9225:c
9205:z
9185:n
9165:T
9143:.
9140:)
9135:t
9132:j
9128:e
9124:+
9121:1
9118:(
9113:n
9108:1
9105:=
9102:j
9090:n
9086:2
9082:1
9077:=
9074:)
9071:t
9068:(
9065:M
9045:T
9022:)
9017:2
9013:/
9009:3
9002:n
8998:(
8995:O
8973:+
8969:T
8948:.
8937:)
8934:1
8931:+
8928:n
8925:2
8922:(
8919:)
8916:1
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8910:n
8907:(
8904:n
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8891:)
8888:1
8885:+
8882:n
8879:(
8876:n
8864:2
8861:1
8855:+
8852:k
8846:=
8843:t
8823:,
8820:)
8817:t
8814:3
8806:3
8802:t
8798:(
8793:)
8785:)
8782:1
8779:+
8776:n
8773:2
8770:(
8767:)
8764:1
8761:+
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8755:(
8752:n
8744:1
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8735:3
8732:+
8727:2
8723:n
8719:3
8711:(
8706:)
8703:t
8700:(
8694:+
8691:)
8688:t
8685:(
8676:)
8673:k
8665:+
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8657:(
8628:.
8623:6
8619:)
8616:1
8613:+
8610:n
8607:2
8604:(
8601:)
8598:1
8595:+
8592:n
8589:(
8586:n
8580:=
8573:)
8570:T
8567:(
8554:,
8545:)
8542:1
8539:+
8536:n
8533:2
8530:(
8527:)
8524:1
8521:+
8518:n
8515:(
8512:n
8506:=
8503:)
8494:T
8490:(
8481:=
8474:)
8469:+
8465:T
8461:(
8448:,
8445:0
8442:=
8435:]
8432:T
8429:[
8425:E
8417:,
8412:4
8408:)
8405:1
8402:+
8399:n
8396:(
8393:n
8387:=
8384:]
8375:T
8371:[
8367:E
8363:=
8356:]
8351:+
8347:T
8343:[
8339:E
8308:T
8285:+
8281:T
8260:T
8240:n
8217:)
8212:+
8208:t
8199:+
8195:T
8191:(
8168:)
8165:n
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8144:1
8138:n
8134:p
8130:+
8127:)
8122:+
8118:t
8114:(
8109:1
8103:n
8099:p
8095:=
8092:)
8087:+
8083:t
8079:(
8074:n
8070:p
8066:2
8046:)
8041:+
8037:t
8033:(
8028:n
8024:p
8003:n
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7977:t
7973:=
7968:+
7964:T
7943:)
7938:+
7934:t
7930:(
7925:n
7921:p
7891:n
7887:u
7864:n
7860:2
7855:/
7851:)
7846:+
7842:t
7838:(
7833:n
7829:u
7825:=
7822:)
7817:+
7813:t
7809:=
7804:+
7800:T
7796:(
7771:+
7767:T
7746:n
7738:+
7734:t
7713:}
7710:1
7704:n
7701:,
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7689:{
7669:n
7649:n
7629:}
7626:1
7620:n
7617:,
7611:,
7608:1
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7585:n
7563:+
7559:t
7538:}
7535:n
7532:,
7526:,
7523:1
7520:{
7500:.
7497:)
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7473:1
7467:n
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7459:+
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7438:1
7432:n
7428:u
7424:=
7421:)
7416:+
7412:t
7408:(
7403:n
7399:u
7378:2
7374:/
7370:)
7367:1
7364:+
7361:n
7358:(
7355:n
7349:t
7329:0
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7303:0
7300:=
7297:)
7292:+
7288:t
7284:(
7279:n
7275:u
7254:0
7246:+
7242:t
7221:0
7218:=
7215:)
7210:+
7206:t
7202:(
7197:0
7193:u
7172:1
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7166:)
7163:0
7160:(
7155:0
7151:u
7128:+
7124:t
7103:}
7100:n
7097:,
7091:,
7088:1
7085:{
7063:+
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7050:+
7046:T
7025:)
7020:+
7016:t
7012:(
7007:n
7003:u
6980:+
6976:T
6955:n
6933:n
6929:2
6908:T
6885:T
6863:n
6859:2
6838:t
6835:=
6832:T
6812:t
6809:=
6806:T
6784:i
6780:X
6759:n
6739:T
6729:p
6717:p
6713:p
6709:p
6691:T
6662:7
6658:2
6653:/
6646:=
6613:7
6609:2
6604:/
6597:=
6553:|
6549:T
6545:|
6524:T
6504:T
6484:T
6463:p
6449:}
6446:n
6443:,
6437:,
6434:1
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6380:/
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6367:k
6364:(
6324:k
6304:v
6284:v
6244:v
6224:1
6218:k
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6175:=
6172:3
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6164:)
6161:7
6158:+
6155:6
6152:+
6149:5
6146:(
6124:7
6120:X
6097:4
6093:X
6070:1
6066:X
6043:6
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6015:=
6012:2
6008:/
6004:)
6001:3
5998:+
5995:2
5992:(
5970:5
5966:X
5943:2
5939:X
5916:3
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5891:.
5887:|
5881:7
5877:X
5872:|
5868:=
5864:|
5858:4
5854:X
5849:|
5845:=
5841:|
5835:1
5831:X
5826:|
5818:|
5812:6
5808:X
5803:|
5795:|
5789:5
5785:X
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5776:=
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5766:2
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5749:|
5743:3
5739:X
5734:|
5710:}
5707:n
5704:,
5698:,
5695:1
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5681:n
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5618:2
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5476:0
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5460:+
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5424:j
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5319:0
5316:=
5313:)
5310:0
5307:(
5260:2
5255:/
5225:2
5220:/
5164:2
5159:/
5123:2
5118:/
5104:p
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5065:/
5029:2
5024:/
5010:p
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4914:=
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4861:=
4856:i
4852:X
4831:j
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4795:=
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4717:i
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3279:i
3275:Y
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2994:1
2991:H
2986:.
2974:0
2971:=
2948:F
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2910:x
2890:)
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2869:)
2866:x
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2854:(
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2807:0
2801:x
2781:)
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2769:(
2760:)
2757:x
2748:X
2745:(
2722:X
2702:F
2678:F
2635:F
2615:F
2575:0
2572:=
2569:)
2563:=
2560:X
2557:(
2542:.
2530:0
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2493:3
2490:H
2485:.
2473:0
2447:F
2436:2
2433:H
2428:.
2416:0
2390:F
2379:1
2376:H
2371:.
2359:0
2356:=
2333:F
2322:0
2319:H
2301:F
2281:F
2226:.
2223:}
2220:n
2214:j
2208:i
2202:1
2196:0
2188:j
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2181:W
2177:{
2171:=
2162:T
2154:,
2151:}
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2052:+
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1942:+
1938:T
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1893:.
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1868:)
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1862:+
1859:n
1856:(
1853:n
1847:=
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1838:)
1835:1
1832:+
1829:n
1826:(
1823:n
1812:+
1808:T
1804:2
1801:=
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1783:+
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1775:=
1768:T
1761:,
1756:2
1753:T
1743:4
1739:)
1736:1
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1730:n
1727:(
1724:n
1718:=
1713:+
1709:T
1700:2
1696:)
1693:1
1690:+
1687:n
1684:(
1681:n
1675:=
1662:T
1654:,
1649:2
1646:T
1641:+
1636:4
1632:)
1629:1
1626:+
1623:n
1620:(
1617:n
1611:=
1602:T
1593:2
1589:)
1586:1
1583:+
1580:n
1577:(
1574:n
1568:=
1559:+
1555:T
1530:2
1526:/
1522:)
1519:1
1516:+
1513:n
1510:(
1507:n
1504:=
1501:n
1498:+
1492:+
1489:2
1486:+
1483:1
1457:T
1453:+
1448:+
1444:T
1419:.
1414:i
1410:R
1404:0
1396:i
1392:X
1385:,
1382:n
1376:i
1370:1
1362:=
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1341:,
1336:i
1332:R
1326:0
1318:i
1314:X
1307:,
1304:n
1298:i
1292:1
1284:=
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1271:T
1240:T
1214:+
1210:T
1186:T
1163:i
1143:i
1140:=
1135:)
1132:i
1129:(
1122:R
1100:|
1094:)
1091:n
1088:(
1081:X
1076:|
1062:|
1056:)
1053:1
1050:(
1043:X
1038:|
1017:}
1014:n
1011:,
1005:,
1002:1
999:{
993:}
990:n
987:,
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978:1
975:{
972::
948:|
942:i
938:X
933:|
925:|
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910:|
889:j
867:i
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838:T
818:p
796:.
791:i
787:R
783:)
778:i
774:X
770:(
759:N
754:1
751:=
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740:=
737:T
717:T
690:0
684:x
664:1
658:=
655:)
652:x
649:(
623:0
617:x
597:1
594:=
591:)
588:x
585:(
530:n
526:R
522:,
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485:|
479:n
475:X
470:|
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418:.
414:|
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399:|
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385:|
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370:|
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335:,
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290:i
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230:X
226:(
198:)
193:n
189:Y
185:,
180:n
176:X
172:(
169:,
163:,
160:)
155:1
151:Y
147:,
142:1
138:X
134:(
110:T
104:T
67:t
63:t
59:t
49:t
23:.
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