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Rank correlation

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3804: 3064: 3799:{\displaystyle {\begin{aligned}{\frac {1}{n^{2}}}\sum _{i,j=1}^{n}(r_{j}-r_{i})(s_{j}-s_{i})&=2\left({\frac {1}{n^{2}}}\cdot n\sum _{i=1}^{n}r_{i}s_{i}-({\frac {1}{n}}\sum _{i=1}^{n}r_{i})({\frac {1}{n}}\sum _{j=1}^{n}s_{j})\right)\\&={\frac {1}{n}}\sum _{i=1}^{n}(r_{i}^{2}+s_{i}^{2}-d_{i}^{2})-2(\mathbb {E} )^{2}\\&={\frac {1}{n}}\sum _{i=1}^{n}r_{i}^{2}+{\frac {1}{n}}\sum _{i=1}^{n}s_{i}^{2}-{\frac {1}{n}}\sum _{i=1}^{n}d_{i}^{2}-2(\mathbb {E} )^{2}\\&=2(\mathbb {E} -(\mathbb {E} )^{2})-{\frac {1}{n}}\sum _{i=1}^{n}d_{i}^{2}\\\end{aligned}}} 7199: 4266: 7185: 4653:
Group A is faster than the runner from Group B. There are a total of 20 pairs, and 19 pairs support the hypothesis. The only pair that does not support the hypothesis are the two runners with ranks 5 and 6, because in this pair, the runner from Group B had the faster time. By the Kerby simple difference formula, 95% of the data support the hypothesis (19 of 20 pairs), and 5% do not support (1 of 20 pairs), so the rank correlation is
7223: 3815: 7211: 4261:{\displaystyle {\begin{aligned}{\frac {1}{n^{2}}}\sum _{i,j=1}^{n}(r_{j}-r_{i})^{2}&={\frac {1}{n^{2}}}\cdot n\sum _{i,j=1}^{n}(r_{i}^{2}+r_{j}^{2}-2r_{i}r_{j})\\&=2{\frac {1}{n}}\sum _{i=1}^{n}r_{i}^{2}-2({\frac {1}{n}}\sum _{i=1}^{n}r_{i})({\frac {1}{n}}\sum _{j=1}^{n}r_{j})\\&=2(\mathbb {E} -(\mathbb {E} )^{2})\\\end{aligned}}} 78:
If, for example, one variable is the identity of a college basketball program and another variable is the identity of a college football program, one could test for a relationship between the poll rankings of the two types of program: do colleges with a higher-ranked basketball program tend to have a
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The analysis is conducted on pairs, defined as a member of one group compared to a member of the other group. For example, the fastest runner in the study is a member of four pairs: (1,5), (1,7), (1,8), and (1,9). All four of these pairs support the hypothesis, because in each pair the runner from
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Kerby showed that this rank correlation can be expressed in terms of two concepts: the percent of data that support a stated hypothesis, and the percent of data that do not support it. The Kerby simple difference formula states that the rank correlation can be expressed as the difference between the
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To illustrate the computation, suppose a coach trains long-distance runners for one month using two methods. Group A has 5 runners, and Group B has 4 runners. The stated hypothesis is that method A produces faster runners. The race to assess the results finds that the runners from Group A do
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If there is only one variable, the identity of a college football program, but it is subject to two different poll rankings (say, one by coaches and one by sportswriters), then the similarity of the two different polls' rankings can be measured with a rank correlation coefficient.
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higher-ranked football program? A rank correlation coefficient can measure that relationship, and the measure of significance of the rank correlation coefficient can show whether the measured relationship is small enough to likely be a coincidence.
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Dave Kerby (2014) recommended the rank-biserial as the measure to introduce students to rank correlation, because the general logic can be explained at an introductory level. The rank-biserial is the correlation used with the
4668:= 0 indicates that half the pairs favor the hypothesis and half do not; in other words, the sample groups do not differ in ranks, so there is no evidence that they come from two different populations. An effect size of 1801: 1290: 2883: 2878: 4593:, a method commonly covered in introductory college courses on statistics. The data for this test consists of two groups; and for each member of the groups, the outcome is ranked for the study as a whole. 4579:
between two normal variables” (p. 91). The rank-biserial correlation had been introduced nine years before by Edward Cureton (1956) as a measure of rank correlation when the ranks are in two groups.
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variables or different rankings of the same variable, where a "ranking" is the assignment of the ordering labels "first", "second", "third", etc. to different observations of a particular variable. A
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indeed run faster, with the following ranks: 1, 2, 3, 4, and 6. The slower runners from Group B thus have ranks of 5, 7, 8, and 9.
6373: 4447:{\displaystyle \Gamma =1-{\frac {\sum _{i=1}^{n}d_{i}^{2}}{2n\mathrm {Var} (U)}}=1-{\frac {6\sum _{i=1}^{n}d_{i}^{2}}{n(n^{2}-1)}}} 1643: 6812: 914:{\displaystyle \Gamma ={\frac {\sum _{i,j=1}^{n}a_{ij}b_{ij}}{\sqrt {\sum _{i,j=1}^{n}a_{ij}^{2}\sum _{i,j=1}^{n}b_{ij}^{2}}}}} 2744: 5235: 4935: 4575:, the ranking variable, which estimates Spearman's rho between X and Y in the same way that biserial r estimates Pearson's 4542: 7249: 5839: 4987: 17: 4825:
Glass, Gene V. (1965). "A ranking variable analogue of biserial correlation: implications for short-cut item analysis".
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of objects. Thus we can look at observed rankings as data obtained when the sample space is (identified with) a
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implies increasing agreement between rankings. The coefficient is inside the interval and assumes the value:
7152: 6111: 5014: 114:—in the column variable), a rank correlation measures the relationship between income and educational level. 6703: 6652: 6637: 6627: 6496: 6368: 6335: 6161: 6116: 5946: 4864:
Kerby, Dave S. (2014). "The Simple Difference Formula: An Approach to Teaching Nonparametric Correlation".
177:−1 if the disagreement between the two rankings is perfect; one ranking is the reverse of the other. 7215: 7047: 6848: 6772: 6073: 5827: 5496: 4960: 2133: 2076: 365: 319: 4907: 2499: 1049: 1011: 6932: 6904: 6899: 6647: 6406: 6312: 6292: 6200: 5911: 5729: 5212: 5084: 4460: 2592: 1599: 1586:{\displaystyle a_{ij}=\operatorname {sgn}(r_{j}-r_{i}),\quad b_{ij}=\operatorname {sgn}(s_{j}-s_{i}).} 6664: 6432: 6153: 6078: 6007: 5936: 5856: 5844: 5714: 5702: 5695: 5403: 5124: 2230: 2192: 1760: 1649: 1296:. In particular, the general correlation coefficient is the cosine of the angle between the matrices 649: 67: 600: 551: 7147: 6914: 6777: 6462: 6427: 6391: 6176: 5618: 5527: 5486: 5398: 5089: 4928: 2662: 55: 7056: 6669: 6609: 6546: 6184: 6168: 5906: 5768: 5758: 5608: 5522: 4590: 2721:
from discrete mathematics, it is easy to see that for the uniformly distributed random variable,
1223: 63: 969: 927: 7094: 7024: 6817: 6754: 6509: 6396: 5393: 5290: 5197: 5076: 4975: 1171:{\displaystyle \Gamma ={\frac {\langle A,B\rangle _{\rm {F}}}{\|A\|_{\rm {F}}\|B\|_{\rm {F}}}}} 1687: 7119: 7061: 7004: 6830: 6723: 6632: 6358: 6242: 6101: 6093: 5983: 5975: 5790: 5686: 5664: 5623: 5588: 5555: 5501: 5476: 5431: 5370: 5330: 5132: 4955: 4887: 4672:= 0 can be said to describe no relationship between group membership and the members' ranks. 3041: 727: 1730: 521: 471: 7042: 6617: 6566: 6542: 6504: 6422: 6401: 6353: 6232: 6210: 6179: 6088: 5965: 5916: 5834: 5807: 5763: 5719: 5481: 5257: 5137: 4611: 4550: 4516: 2295: 2268: 1931: 1904: 1380: 1353: 236: 216: 198: 8: 7189: 7114: 7037: 6718: 6482: 6475: 6437: 6345: 6325: 6297: 6030: 5896: 5891: 5881: 5873: 5691: 5652: 5542: 5532: 5441: 5220: 5176: 5094: 5019: 4921: 2636: 2486:{\displaystyle \Gamma ={\frac {\sum (r_{j}-r_{i})(s_{j}-s_{i})}{\sum (r_{j}-r_{i})^{2}}}} 701: 7203: 7014: 6868: 6764: 6713: 6589: 6486: 6470: 6447: 6224: 5958: 5941: 5901: 5812: 5707: 5669: 5640: 5600: 5560: 5506: 5423: 5109: 5104: 4838: 4777: 4711: 2724: 2700: 2572: 2552: 2342: 2322: 1998: 1978: 1958: 1447: 1427: 1407: 1319: 1299: 501: 451: 431: 411: 299: 279: 259: 171:
1 if the agreement between the two rankings is perfect; the two rankings are the same.
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measures the degree of similarity between two rankings, and can be used to assess the
7259: 7198: 7109: 7079: 7071: 6891: 6882: 6807: 6738: 6594: 6579: 6554: 6442: 6383: 6249: 6237: 5863: 5780: 5724: 5647: 5491: 5413: 5192: 5066: 4851: 4812: 4811:, Lecture Notes-Monograph Series, Hayward, CA: Institute of Mathematical Statistics, 4794: 4781: 4734: 190: 87: 7134: 7089: 6853: 6840: 6733: 6708: 6642: 6574: 6452: 6060: 5953: 5886: 5799: 5746: 5565: 5436: 5114: 5029: 4996: 4873: 4834: 4769: 4703: 276:
objects, which are being considered in relation to two properties, represented by
7051: 6795: 6657: 6584: 6259: 6133: 6106: 6083: 6052: 5679: 5674: 5628: 5358: 5009: 4689: 194: 6541: 4664:= 1, which means that 100% of the pairs favor the hypothesis. A correlation of 7000: 6995: 5458: 5388: 5034: 1293: 4547:
Gene Glass (1965) noted that the rank-biserial can be derived from Spearman's
548:. The only requirement for these functions is that they be anti-symmetric, so 7243: 7157: 7124: 6987: 6948: 6759: 6728: 6192: 6146: 5751: 5453: 5280: 5044: 5039: 4760: 59: 1642:
is the number of concordant pairs minus the number of discordant pairs (see
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Equivalently, if all coefficients are collected into matrices
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methods of significance that use rank correlation are the
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Brief guide by experimental psychologist Karl L. Weunsch
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Cureton, Edward E. (1956). "Rank-biserial correlation".
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be a uniformly distributed discrete random variables on
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Inner product space § Norms on inner product spaces
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of the relation between them. For example, two common
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Autoregressive conditional heteroskedasticity (ARCH)
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Mann–Whitney_U_test § Rank-biserial_correlation
2015:-quality respectively, we may consider the matrices 1896: 4809:
Group Representations in Probability and Statistics
1345: 6288: 4632: 4583: 4559: 4525: 4510:is the difference between ranks, which is exactly 4502: 4446: 4260: 3798: 3050: 3030: 2872: 2787: 2733: 2709: 2689: 2651: 2625: 2581: 2561: 2541: 2485: 2351: 2331: 2311: 2284: 2257: 2219: 2178: 2121: 2062: 2007: 1987: 1967: 1947: 1920: 1885: 1787: 1749: 1719: 1676: 1634: 1585: 1456: 1436: 1416: 1396: 1369: 1328: 1308: 1284: 1214: 1170: 1076: 1038: 1000: 958: 913: 736: 716: 690: 638: 589: 540: 510: 490: 460: 440: 420: 400: 354: 308: 288: 268: 245: 225: 208: 7241: 4601:) minus the proportion of unfavorable evidence ( 2063:{\displaystyle a,b\in M(n\times n;\mathbb {R} )} 724:.) Then the generalized correlation coefficient 6374:Multivariate adaptive regression splines (MARS) 4695:Journal of the American Statistical Association 4929: 4643: 1215:{\displaystyle \langle A,B\rangle _{\rm {F}}} 174:0 if the rankings are completely independent. 38:is any of several statistics that measure an 4892:: CS1 maint: DOI inactive as of June 2024 ( 4567:. "One can derive a coefficient defined on 4536: 2620: 2596: 2549:denote the difference in the ranks for each 1269: 1256: 1240: 1233: 1201: 1188: 1154: 1147: 1136: 1129: 1116: 1103: 383: 369: 337: 323: 4692:(1958). "Ordinal Measures of Association". 117: 4974: 4936: 4922: 4722: 1464:-quality respectively, then we can define 102:in the row variable and educational level— 5587: 4877: 4793:, Cambridge: Cambridge University Press, 4660:The maximum value for the correlation is 4228: 4201: 3717: 3690: 3650: 3463: 2803: 2749: 2053: 1817: 1059: 1021: 4806: 1644:Kendall tau rank correlation coefficient 182: 4845: 4788: 4757: 4728: 4688: 4512:Spearman's rank correlation coefficient 14: 7242: 6900:Kaplan–Meier estimator (product limit) 4791:The Cambridge Dictionary of Statistics 408:. To any pair of individuals, say the 6973: 6540: 6287: 5586: 5356: 4973: 4917: 4863: 4824: 7210: 6910:Accelerated failure time (AFT) model 201:, making the symmetric group into a 27:Statistic comparing ordinal rankings 7222: 6505:Analysis of variance (ANOVA, anova) 5357: 2179:{\displaystyle b_{ij}:=s_{j}-s_{i}} 2122:{\displaystyle a_{ij}:=r_{j}-r_{i}} 401:{\displaystyle \{y_{i}\}_{i\leq n}} 355:{\displaystyle \{x_{i}\}_{i\leq n}} 24: 6600:Cochran–Mantel–Haenszel statistics 5226:Pearson product-moment correlation 4839:10.1111/j.1745-3984.1965.tb00396.x 4827:Journal of Educational Measurement 4751: 4597:proportion of favorable evidence ( 4347: 4344: 4341: 4281: 3045: 2894: 2891: 2888: 2542:{\displaystyle d_{i}:=r_{i}-s_{i}} 2369: 1805: 1274: 1245: 1206: 1159: 1141: 1121: 1094: 1077:{\displaystyle B^{\textsf {T}}=-B} 1039:{\displaystyle A^{\textsf {T}}=-A} 754: 731: 25: 7271: 4901: 4503:{\displaystyle d_{i}=r_{i}-s_{i}} 2626:{\displaystyle \{1,2,\ldots ,n\}} 2496:To simplify this expression, let 1897:Spearman’s ρ as a particular case 1635:{\displaystyle \sum a_{ij}b_{ij}} 7221: 7209: 7197: 7184: 7183: 6974: 4571:, the dichotomous variable, and 1346:Kendall's τ as a particular case 6859:Least-squares spectral analysis 4584:Kerby simple difference formula 2258:{\displaystyle \sum b_{ij}^{2}} 2220:{\displaystyle \sum a_{ij}^{2}} 1788:{\displaystyle \sum b_{ij}^{2}} 1677:{\displaystyle \sum a_{ij}^{2}} 1528: 691:{\displaystyle a_{ij}=b_{ij}=0} 209:General correlation coefficient 163:An increasing rank correlation 5840:Mean-unbiased minimum-variance 4943: 4682: 4438: 4419: 4357: 4351: 4251: 4242: 4238: 4232: 4224: 4218: 4205: 4197: 4181: 4137: 4134: 4090: 4022: 3960: 3894: 3867: 3740: 3731: 3727: 3721: 3713: 3707: 3694: 3686: 3664: 3660: 3654: 3646: 3477: 3473: 3467: 3459: 3450: 3396: 3347: 3303: 3300: 3256: 3171: 3145: 3142: 3116: 2987: 2975: 2972: 2960: 2944: 2929: 2926: 2914: 2904: 2898: 2860: 2845: 2842: 2830: 2820: 2807: 2759: 2753: 2471: 2444: 2436: 2410: 2407: 2381: 2057: 2037: 1866: 1854: 1846: 1843: 1835: 1829: 1821: 1818: 1706: 1694: 1577: 1551: 1522: 1496: 995: 979: 953: 937: 639:{\displaystyle b_{ij}=-b_{ji}} 590:{\displaystyle a_{ij}=-a_{ji}} 122:Some of the more popular rank 13: 1: 7153:Geographic information system 6369:Simultaneous equations models 4675: 2690:{\displaystyle 1,2,\ldots ,n} 316:, forming the sets of values 213:Kendall 1970 showed that his 185:, a ranking can be seen as a 6336:Coefficient of determination 5947:Uniformly most powerful test 52:rank correlation coefficient 7: 6905:Proportional hazards models 6849:Spectral density estimation 6831:Vector autoregression (VAR) 6265:Maximum posterior estimator 5497:Randomized controlled trial 4729:Kendall, Maurice G (1970). 646:. (Note that in particular 42:— the relationship between 10: 7276: 7250:Covariance and correlation 6665:Multivariate distributions 5085:Average absolute deviation 4644:Example and interpretation 4540: 1840:number of discordant pairs 1826:number of concordant pairs 1338: 1001:{\displaystyle B=(b_{ij})} 959:{\displaystyle A=(a_{ij})} 197:. We can then introduce a 73: 7179: 7133: 7070: 7023: 6986: 6982: 6969: 6941: 6923: 6890: 6881: 6839: 6786: 6747: 6696: 6687: 6653:Structural equation model 6608: 6565: 6561: 6536: 6495: 6461: 6415: 6382: 6344: 6311: 6307: 6283: 6223: 6132: 6051: 6015: 6006: 5989:Score/Lagrange multiplier 5974: 5927: 5872: 5798: 5789: 5599: 5595: 5582: 5541: 5515: 5467: 5422: 5404:Sample size determination 5369: 5365: 5352: 5256: 5211: 5185: 5167: 5123: 5075: 4995: 4986: 4982: 4969: 4951: 4657:= .95 − .05 = .90. 4537:Rank-biserial correlation 2659:are just permutations of 1975:-member according to the 1424:-member according to the 256:Suppose we have a set of 86:As another example, in a 68:Wilcoxon signed-rank test 7255:Nonparametric statistics 7148:Environmental statistics 6670:Elliptical distributions 6463:Generalized linear model 6392:Simple linear regression 6162:Hodges–Lehmann estimator 5619:Probability distribution 5528:Stochastic approximation 5090:Coefficient of variation 4866:Comprehensive Psychology 4848:Rank Correlation Methods 4731:Rank Correlation Methods 1720:{\displaystyle n(n-1)/2} 118:Correlation coefficients 6808:Cross-correlation (XCF) 6416:Non-standard predictors 5850:Lehmann–ScheffĂ© theorem 5523:Adaptive clinical trial 4846:Kendall, M. G. (1970), 4789:Everitt, B. S. (2002), 4733:(4 ed.). Griffin. 3051:{\displaystyle \Gamma } 1224:Frobenius inner product 737:{\displaystyle \Gamma } 7204:Mathematics portal 7025:Engineering statistics 6933:Nelson–Aalen estimator 6510:Analysis of covariance 6397:Ordinary least squares 6321:Pearson product-moment 5725:Statistical functional 5636:Empirical distribution 5469:Controlled experiments 5198:Frequency distribution 4976:Descriptive statistics 4882:(inactive 2024-06-26). 4634: 4561: 4527: 4504: 4448: 4398: 4316: 4262: 4170: 4123: 4068: 3959: 3866: 3800: 3776: 3624: 3575: 3526: 3395: 3336: 3289: 3232: 3115: 3052: 3032: 2874: 2789: 2735: 2711: 2691: 2653: 2627: 2583: 2563: 2543: 2487: 2353: 2333: 2313: 2286: 2265:are equal, since both 2259: 2221: 2180: 2123: 2064: 2009: 1989: 1969: 1949: 1922: 1887: 1789: 1751: 1750:{\displaystyle a_{ij}} 1727:, the number of terms 1721: 1678: 1636: 1587: 1458: 1438: 1418: 1398: 1371: 1330: 1310: 1286: 1216: 1172: 1078: 1040: 1002: 960: 915: 889: 844: 789: 738: 718: 692: 640: 591: 542: 541:{\displaystyle b_{ij}} 512: 492: 491:{\displaystyle a_{ij}} 462: 442: 422: 402: 356: 310: 290: 270: 247: 227: 147:Goodman and Kruskal's 7120:Population statistics 7062:System identification 6796:Autocorrelation (ACF) 6724:Exponential smoothing 6638:Discriminant analysis 6633:Canonical correlation 6497:Partition of variance 6359:Regression validation 6203:(Jonckheere–Terpstra) 6102:Likelihood-ratio test 5791:Frequentist inference 5703:Location–scale family 5624:Sampling distribution 5589:Statistical inference 5556:Cross-sectional study 5543:Observational studies 5502:Randomized experiment 5331:Stem-and-leaf display 5133:Central limit theorem 4807:Diaconis, P. (1988), 4635: 4633:{\displaystyle r=f-u} 4562: 4560:{\displaystyle \rho } 4528: 4526:{\displaystyle \rho } 4505: 4449: 4378: 4296: 4263: 4150: 4103: 4048: 3933: 3840: 3801: 3756: 3604: 3555: 3506: 3375: 3316: 3269: 3212: 3089: 3053: 3033: 2875: 2790: 2736: 2712: 2692: 2654: 2628: 2584: 2564: 2544: 2488: 2354: 2334: 2314: 2312:{\displaystyle s_{i}} 2287: 2285:{\displaystyle r_{i}} 2260: 2222: 2181: 2124: 2065: 2010: 1990: 1970: 1955:are the ranks of the 1950: 1948:{\displaystyle s_{i}} 1923: 1921:{\displaystyle r_{i}} 1888: 1795:. Thus in this case, 1790: 1752: 1722: 1679: 1637: 1588: 1459: 1439: 1419: 1404:are the ranks of the 1399: 1397:{\displaystyle s_{i}} 1372: 1370:{\displaystyle r_{i}} 1331: 1311: 1287: 1217: 1173: 1079: 1041: 1003: 961: 916: 863: 818: 763: 739: 719: 693: 641: 592: 543: 513: 493: 463: 443: 423: 403: 357: 311: 291: 271: 248: 246:{\displaystyle \rho } 233:(tau) and Spearman's 228: 226:{\displaystyle \tau } 7043:Probabilistic design 6628:Principal components 6471:Exponential families 6423:Nonlinear regression 6402:General linear model 6364:Mixed effects models 6354:Errors and residuals 6331:Confounding variable 6233:Bayesian probability 6211:Van der Waerden test 6201:Ordered alternative 5966:Multiple comparisons 5845:Rao–Blackwellization 5808:Estimating equations 5764:Statistical distance 5482:Factorial experiment 5015:Arithmetic-Geometric 4612: 4551: 4517: 4461: 4278: 3816: 3065: 3042: 2884: 2799: 2745: 2725: 2701: 2663: 2637: 2593: 2573: 2553: 2500: 2366: 2343: 2323: 2296: 2269: 2231: 2193: 2134: 2077: 2019: 1999: 1979: 1959: 1932: 1905: 1802: 1761: 1731: 1688: 1650: 1600: 1471: 1448: 1428: 1408: 1381: 1354: 1320: 1300: 1230: 1185: 1091: 1050: 1012: 970: 928: 751: 728: 702: 650: 601: 552: 522: 502: 472: 452: 432: 412: 366: 320: 300: 280: 260: 237: 217: 7115:Official statistics 7038:Methods engineering 6719:Seasonal adjustment 6487:Poisson regressions 6407:Bayesian regression 6346:Regression analysis 6326:Partial correlation 6298:Regression analysis 5897:Prediction interval 5892:Likelihood interval 5882:Confidence interval 5874:Interval estimation 5835:Unbiased estimators 5653:Model specification 5533:Up-and-down designs 5221:Partial correlation 5177:Index of dispersion 5095:Interquartile range 4850:, London: Griffin, 4690:Kruskal, William H. 4591:Mann–Whitney U test 4413: 4331: 4083: 3995: 3977: 3791: 3639: 3590: 3541: 3449: 3431: 3413: 2652:{\displaystyle r,s} 2254: 2216: 1784: 1673: 907: 862: 717:{\displaystyle i=j} 518:-score, denoted by 468:-score, denoted by 126:statistics include 64:Mann–Whitney U test 40:ordinal association 18:Ordinal association 7135:Spatial statistics 7015:Medical statistics 6915:First hitting time 6869:Whittle likelihood 6520:Degrees of freedom 6515:Multivariate ANOVA 6448:Heteroscedasticity 6260:Bayesian estimator 6225:Bayesian inference 6074:Kolmogorov–Smirnov 5959:Randomization test 5929:Testing hypotheses 5902:Tolerance interval 5813:Maximum likelihood 5708:Exponential family 5641:Density estimation 5601:Statistical theory 5561:Natural experiment 5507:Scientific control 5424:Survey methodology 5110:Standard deviation 4774:10.1007/BF02289138 4630: 4557: 4523: 4500: 4444: 4399: 4317: 4258: 4256: 4069: 3981: 3963: 3796: 3794: 3777: 3625: 3576: 3527: 3435: 3417: 3399: 3048: 3028: 3027: 3026: 3025: 2870: 2869: 2785: 2784: 2731: 2707: 2687: 2649: 2633:. Since the ranks 2623: 2579: 2559: 2539: 2483: 2349: 2329: 2309: 2282: 2255: 2237: 2217: 2199: 2176: 2119: 2060: 2005: 1985: 1965: 1945: 1918: 1883: 1785: 1767: 1747: 1717: 1674: 1656: 1632: 1583: 1454: 1434: 1414: 1394: 1367: 1326: 1306: 1282: 1212: 1168: 1074: 1036: 998: 956: 911: 890: 845: 734: 714: 688: 636: 587: 538: 508: 488: 458: 438: 418: 398: 352: 306: 286: 266: 243: 223: 7237: 7236: 7175: 7174: 7171: 7170: 7110:National accounts 7080:Actuarial science 7072:Social statistics 6965: 6964: 6961: 6960: 6957: 6956: 6892:Survival function 6877: 6876: 6739:Granger causality 6580:Contingency table 6555:Survival analysis 6532: 6531: 6528: 6527: 6384:Linear regression 6279: 6278: 6275: 6274: 6250:Credible interval 6219: 6218: 6002: 6001: 5818:Method of moments 5687:Parametric family 5648:Statistical model 5578: 5577: 5574: 5573: 5492:Random assignment 5414:Statistical power 5348: 5347: 5344: 5343: 5193:Contingency table 5163: 5162: 5030:Generalized/power 4879:10.2466/11.IT.3.1 4442: 4361: 4148: 4101: 4046: 3925: 3838: 3754: 3602: 3553: 3504: 3373: 3314: 3267: 3204: 3087: 3023: 2994: 2951: 2867: 2782: 2734:{\displaystyle U} 2719:summation results 2710:{\displaystyle U} 2582:{\displaystyle U} 2562:{\displaystyle i} 2481: 2352:{\displaystyle n} 2332:{\displaystyle 1} 2008:{\displaystyle y} 1988:{\displaystyle x} 1968:{\displaystyle i} 1878: 1870: 1841: 1827: 1457:{\displaystyle y} 1437:{\displaystyle x} 1417:{\displaystyle i} 1329:{\displaystyle B} 1309:{\displaystyle A} 1280: 1166: 1061: 1023: 909: 908: 511:{\displaystyle y} 461:{\displaystyle x} 441:{\displaystyle j} 421:{\displaystyle i} 309:{\displaystyle y} 289:{\displaystyle x} 269:{\displaystyle n} 88:contingency table 16:(Redirected from 7267: 7225: 7224: 7213: 7212: 7202: 7201: 7187: 7186: 7090:Crime statistics 6984: 6983: 6971: 6970: 6888: 6887: 6854:Fourier analysis 6841:Frequency domain 6821: 6768: 6734:Structural break 6694: 6693: 6643:Cluster analysis 6590:Log-linear model 6563: 6562: 6538: 6537: 6479: 6453:Homoscedasticity 6309: 6308: 6285: 6284: 6204: 6196: 6188: 6187:(Kruskal–Wallis) 6172: 6157: 6112:Cross validation 6097: 6079:Anderson–Darling 6026: 6013: 6012: 5984:Likelihood-ratio 5976:Parametric tests 5954:Permutation test 5937:1- & 2-tails 5828:Minimum distance 5800:Point estimation 5796: 5795: 5747:Optimal decision 5698: 5597: 5596: 5584: 5583: 5566:Quasi-experiment 5516:Adaptive designs 5367: 5366: 5354: 5353: 5231:Rank correlation 4993: 4992: 4984: 4983: 4971: 4970: 4938: 4931: 4924: 4915: 4914: 4897: 4891: 4883: 4881: 4872:(1): 11.IT.3.1. 4860: 4842: 4821: 4803: 4785: 4745: 4744: 4726: 4720: 4719: 4702:(284): 814–861. 4686: 4639: 4637: 4636: 4631: 4566: 4564: 4563: 4558: 4532: 4530: 4529: 4524: 4509: 4507: 4506: 4501: 4499: 4498: 4486: 4485: 4473: 4472: 4453: 4451: 4450: 4445: 4443: 4441: 4431: 4430: 4414: 4412: 4407: 4397: 4392: 4373: 4362: 4360: 4350: 4332: 4330: 4325: 4315: 4310: 4294: 4267: 4265: 4264: 4259: 4257: 4250: 4249: 4231: 4217: 4216: 4204: 4187: 4180: 4179: 4169: 4164: 4149: 4141: 4133: 4132: 4122: 4117: 4102: 4094: 4082: 4077: 4067: 4062: 4047: 4039: 4028: 4021: 4020: 4011: 4010: 3994: 3989: 3976: 3971: 3958: 3953: 3926: 3924: 3923: 3911: 3902: 3901: 3892: 3891: 3879: 3878: 3865: 3860: 3839: 3837: 3836: 3824: 3805: 3803: 3802: 3797: 3795: 3790: 3785: 3775: 3770: 3755: 3747: 3739: 3738: 3720: 3706: 3705: 3693: 3676: 3672: 3671: 3653: 3638: 3633: 3623: 3618: 3603: 3595: 3589: 3584: 3574: 3569: 3554: 3546: 3540: 3535: 3525: 3520: 3505: 3497: 3489: 3485: 3484: 3466: 3448: 3443: 3430: 3425: 3412: 3407: 3394: 3389: 3374: 3366: 3358: 3354: 3350: 3346: 3345: 3335: 3330: 3315: 3307: 3299: 3298: 3288: 3283: 3268: 3260: 3252: 3251: 3242: 3241: 3231: 3226: 3205: 3203: 3202: 3190: 3170: 3169: 3157: 3156: 3141: 3140: 3128: 3127: 3114: 3109: 3088: 3086: 3085: 3073: 3057: 3055: 3054: 3049: 3037: 3035: 3034: 3029: 3024: 3019: 3012: 3011: 3001: 2995: 2990: 2958: 2952: 2947: 2912: 2897: 2879: 2877: 2876: 2871: 2868: 2863: 2828: 2819: 2818: 2806: 2794: 2792: 2791: 2786: 2783: 2778: 2767: 2752: 2740: 2738: 2737: 2732: 2716: 2714: 2713: 2708: 2696: 2694: 2693: 2688: 2658: 2656: 2655: 2650: 2632: 2630: 2629: 2624: 2588: 2586: 2585: 2580: 2568: 2566: 2565: 2560: 2548: 2546: 2545: 2540: 2538: 2537: 2525: 2524: 2512: 2511: 2492: 2490: 2489: 2484: 2482: 2480: 2479: 2478: 2469: 2468: 2456: 2455: 2439: 2435: 2434: 2422: 2421: 2406: 2405: 2393: 2392: 2376: 2358: 2356: 2355: 2350: 2338: 2336: 2335: 2330: 2318: 2316: 2315: 2310: 2308: 2307: 2291: 2289: 2288: 2283: 2281: 2280: 2264: 2262: 2261: 2256: 2253: 2248: 2226: 2224: 2223: 2218: 2215: 2210: 2185: 2183: 2182: 2177: 2175: 2174: 2162: 2161: 2149: 2148: 2128: 2126: 2125: 2120: 2118: 2117: 2105: 2104: 2092: 2091: 2069: 2067: 2066: 2061: 2056: 2014: 2012: 2011: 2006: 1994: 1992: 1991: 1986: 1974: 1972: 1971: 1966: 1954: 1952: 1951: 1946: 1944: 1943: 1927: 1925: 1924: 1919: 1917: 1916: 1892: 1890: 1889: 1884: 1879: 1876: 1871: 1869: 1849: 1842: 1839: 1828: 1825: 1812: 1794: 1792: 1791: 1786: 1783: 1778: 1756: 1754: 1753: 1748: 1746: 1745: 1726: 1724: 1723: 1718: 1713: 1683: 1681: 1680: 1675: 1672: 1667: 1641: 1639: 1638: 1633: 1631: 1630: 1618: 1617: 1592: 1590: 1589: 1584: 1576: 1575: 1563: 1562: 1541: 1540: 1521: 1520: 1508: 1507: 1486: 1485: 1463: 1461: 1460: 1455: 1443: 1441: 1440: 1435: 1423: 1421: 1420: 1415: 1403: 1401: 1400: 1395: 1393: 1392: 1376: 1374: 1373: 1368: 1366: 1365: 1335: 1333: 1332: 1327: 1315: 1313: 1312: 1307: 1291: 1289: 1288: 1283: 1281: 1279: 1278: 1277: 1255: 1250: 1249: 1248: 1221: 1219: 1218: 1213: 1211: 1210: 1209: 1177: 1175: 1174: 1169: 1167: 1165: 1164: 1163: 1162: 1146: 1145: 1144: 1127: 1126: 1125: 1124: 1101: 1083: 1081: 1080: 1075: 1064: 1063: 1062: 1045: 1043: 1042: 1037: 1026: 1025: 1024: 1007: 1005: 1004: 999: 994: 993: 965: 963: 962: 957: 952: 951: 920: 918: 917: 912: 910: 906: 901: 888: 883: 861: 856: 843: 838: 817: 816: 815: 814: 802: 801: 788: 783: 761: 743: 741: 740: 735: 723: 721: 720: 715: 697: 695: 694: 689: 681: 680: 665: 664: 645: 643: 642: 637: 635: 634: 616: 615: 596: 594: 593: 588: 586: 585: 567: 566: 547: 545: 544: 539: 537: 536: 517: 515: 514: 509: 497: 495: 494: 489: 487: 486: 467: 465: 464: 459: 448:-th we assign a 447: 445: 444: 439: 427: 425: 424: 419: 407: 405: 404: 399: 397: 396: 381: 380: 361: 359: 358: 353: 351: 350: 335: 334: 315: 313: 312: 307: 295: 293: 292: 287: 275: 273: 272: 267: 252: 250: 249: 244: 232: 230: 229: 224: 36:rank correlation 21: 7275: 7274: 7270: 7269: 7268: 7266: 7265: 7264: 7240: 7239: 7238: 7233: 7196: 7167: 7129: 7066: 7052:quality control 7019: 7001:Clinical trials 6978: 6953: 6937: 6925:Hazard function 6919: 6873: 6835: 6819: 6782: 6778:Breusch–Godfrey 6766: 6743: 6683: 6658:Factor analysis 6604: 6585:Graphical model 6557: 6524: 6491: 6477: 6457: 6411: 6378: 6340: 6303: 6302: 6271: 6215: 6202: 6194: 6186: 6170: 6155: 6134:Rank statistics 6128: 6107:Model selection 6095: 6053:Goodness of fit 6047: 6024: 5998: 5970: 5923: 5868: 5857:Median unbiased 5785: 5696: 5629:Order statistic 5591: 5570: 5537: 5511: 5463: 5418: 5361: 5359:Data collection 5340: 5252: 5207: 5181: 5159: 5119: 5071: 4988:Continuous data 4978: 4965: 4947: 4942: 4904: 4885: 4884: 4858: 4819: 4801: 4754: 4752:Further reading 4749: 4748: 4741: 4727: 4723: 4708:10.2307/2281954 4687: 4683: 4678: 4646: 4613: 4610: 4609: 4586: 4552: 4549: 4548: 4545: 4539: 4518: 4515: 4514: 4494: 4490: 4481: 4477: 4468: 4464: 4462: 4459: 4458: 4426: 4422: 4415: 4408: 4403: 4393: 4382: 4374: 4372: 4340: 4333: 4326: 4321: 4311: 4300: 4295: 4293: 4279: 4276: 4275: 4255: 4254: 4245: 4241: 4227: 4212: 4208: 4200: 4185: 4184: 4175: 4171: 4165: 4154: 4140: 4128: 4124: 4118: 4107: 4093: 4078: 4073: 4063: 4052: 4038: 4026: 4025: 4016: 4012: 4006: 4002: 3990: 3985: 3972: 3967: 3954: 3937: 3919: 3915: 3910: 3903: 3897: 3893: 3887: 3883: 3874: 3870: 3861: 3844: 3832: 3828: 3823: 3819: 3817: 3814: 3813: 3793: 3792: 3786: 3781: 3771: 3760: 3746: 3734: 3730: 3716: 3701: 3697: 3689: 3674: 3673: 3667: 3663: 3649: 3634: 3629: 3619: 3608: 3594: 3585: 3580: 3570: 3559: 3545: 3536: 3531: 3521: 3510: 3496: 3487: 3486: 3480: 3476: 3462: 3444: 3439: 3426: 3421: 3408: 3403: 3390: 3379: 3365: 3356: 3355: 3341: 3337: 3331: 3320: 3306: 3294: 3290: 3284: 3273: 3259: 3247: 3243: 3237: 3233: 3227: 3216: 3198: 3194: 3189: 3188: 3184: 3174: 3165: 3161: 3152: 3148: 3136: 3132: 3123: 3119: 3110: 3093: 3081: 3077: 3072: 3068: 3066: 3063: 3062: 3043: 3040: 3039: 3007: 3003: 3002: 3000: 2959: 2957: 2913: 2911: 2887: 2885: 2882: 2881: 2829: 2827: 2814: 2810: 2802: 2800: 2797: 2796: 2768: 2766: 2748: 2746: 2743: 2742: 2726: 2723: 2722: 2702: 2699: 2698: 2664: 2661: 2660: 2638: 2635: 2634: 2594: 2591: 2590: 2574: 2571: 2570: 2569:. Further, let 2554: 2551: 2550: 2533: 2529: 2520: 2516: 2507: 2503: 2501: 2498: 2497: 2474: 2470: 2464: 2460: 2451: 2447: 2440: 2430: 2426: 2417: 2413: 2401: 2397: 2388: 2384: 2377: 2375: 2367: 2364: 2363: 2344: 2341: 2340: 2324: 2321: 2320: 2303: 2299: 2297: 2294: 2293: 2276: 2272: 2270: 2267: 2266: 2249: 2241: 2232: 2229: 2228: 2211: 2203: 2194: 2191: 2190: 2170: 2166: 2157: 2153: 2141: 2137: 2135: 2132: 2131: 2113: 2109: 2100: 2096: 2084: 2080: 2078: 2075: 2074: 2052: 2020: 2017: 2016: 2000: 1997: 1996: 1980: 1977: 1976: 1960: 1957: 1956: 1939: 1935: 1933: 1930: 1929: 1912: 1908: 1906: 1903: 1902: 1899: 1877:Kendall's  1875: 1850: 1838: 1824: 1813: 1811: 1803: 1800: 1799: 1779: 1771: 1762: 1759: 1758: 1738: 1734: 1732: 1729: 1728: 1709: 1689: 1686: 1685: 1668: 1660: 1651: 1648: 1647: 1623: 1619: 1610: 1606: 1601: 1598: 1597: 1571: 1567: 1558: 1554: 1533: 1529: 1516: 1512: 1503: 1499: 1478: 1474: 1472: 1469: 1468: 1449: 1446: 1445: 1429: 1426: 1425: 1409: 1406: 1405: 1388: 1384: 1382: 1379: 1378: 1361: 1357: 1355: 1352: 1351: 1348: 1343: 1321: 1318: 1317: 1301: 1298: 1297: 1273: 1272: 1268: 1254: 1244: 1243: 1239: 1231: 1228: 1227: 1205: 1204: 1200: 1186: 1183: 1182: 1158: 1157: 1153: 1140: 1139: 1135: 1128: 1120: 1119: 1115: 1102: 1100: 1092: 1089: 1088: 1058: 1057: 1053: 1051: 1048: 1047: 1020: 1019: 1015: 1013: 1010: 1009: 986: 982: 971: 968: 967: 944: 940: 929: 926: 925: 902: 894: 884: 867: 857: 849: 839: 822: 807: 803: 794: 790: 784: 767: 762: 760: 752: 749: 748: 729: 726: 725: 703: 700: 699: 673: 669: 657: 653: 651: 648: 647: 627: 623: 608: 604: 602: 599: 598: 578: 574: 559: 555: 553: 550: 549: 529: 525: 523: 520: 519: 503: 500: 499: 479: 475: 473: 470: 469: 453: 450: 449: 433: 430: 429: 413: 410: 409: 386: 382: 376: 372: 367: 364: 363: 340: 336: 330: 326: 321: 318: 317: 301: 298: 297: 281: 278: 277: 261: 258: 257: 238: 235: 234: 218: 215: 214: 211: 195:symmetric group 183:Diaconis (1988) 120: 76: 28: 23: 22: 15: 12: 11: 5: 7273: 7263: 7262: 7257: 7252: 7235: 7234: 7232: 7231: 7219: 7207: 7193: 7180: 7177: 7176: 7173: 7172: 7169: 7168: 7166: 7165: 7160: 7155: 7150: 7145: 7139: 7137: 7131: 7130: 7128: 7127: 7122: 7117: 7112: 7107: 7102: 7097: 7092: 7087: 7082: 7076: 7074: 7068: 7067: 7065: 7064: 7059: 7054: 7045: 7040: 7035: 7029: 7027: 7021: 7020: 7018: 7017: 7012: 7007: 6998: 6996:Bioinformatics 6992: 6990: 6980: 6979: 6967: 6966: 6963: 6962: 6959: 6958: 6955: 6954: 6952: 6951: 6945: 6943: 6939: 6938: 6936: 6935: 6929: 6927: 6921: 6920: 6918: 6917: 6912: 6907: 6902: 6896: 6894: 6885: 6879: 6878: 6875: 6874: 6872: 6871: 6866: 6861: 6856: 6851: 6845: 6843: 6837: 6836: 6834: 6833: 6828: 6823: 6815: 6810: 6805: 6804: 6803: 6801:partial (PACF) 6792: 6790: 6784: 6783: 6781: 6780: 6775: 6770: 6762: 6757: 6751: 6749: 6748:Specific tests 6745: 6744: 6742: 6741: 6736: 6731: 6726: 6721: 6716: 6711: 6706: 6700: 6698: 6691: 6685: 6684: 6682: 6681: 6680: 6679: 6678: 6677: 6662: 6661: 6660: 6650: 6648:Classification 6645: 6640: 6635: 6630: 6625: 6620: 6614: 6612: 6606: 6605: 6603: 6602: 6597: 6595:McNemar's test 6592: 6587: 6582: 6577: 6571: 6569: 6559: 6558: 6534: 6533: 6530: 6529: 6526: 6525: 6523: 6522: 6517: 6512: 6507: 6501: 6499: 6493: 6492: 6490: 6489: 6473: 6467: 6465: 6459: 6458: 6456: 6455: 6450: 6445: 6440: 6435: 6433:Semiparametric 6430: 6425: 6419: 6417: 6413: 6412: 6410: 6409: 6404: 6399: 6394: 6388: 6386: 6380: 6379: 6377: 6376: 6371: 6366: 6361: 6356: 6350: 6348: 6342: 6341: 6339: 6338: 6333: 6328: 6323: 6317: 6315: 6305: 6304: 6301: 6300: 6295: 6289: 6281: 6280: 6277: 6276: 6273: 6272: 6270: 6269: 6268: 6267: 6257: 6252: 6247: 6246: 6245: 6240: 6229: 6227: 6221: 6220: 6217: 6216: 6214: 6213: 6208: 6207: 6206: 6198: 6190: 6174: 6171:(Mann–Whitney) 6166: 6165: 6164: 6151: 6150: 6149: 6138: 6136: 6130: 6129: 6127: 6126: 6125: 6124: 6119: 6114: 6104: 6099: 6096:(Shapiro–Wilk) 6091: 6086: 6081: 6076: 6071: 6063: 6057: 6055: 6049: 6048: 6046: 6045: 6037: 6028: 6016: 6010: 6008:Specific tests 6004: 6003: 6000: 5999: 5997: 5996: 5991: 5986: 5980: 5978: 5972: 5971: 5969: 5968: 5963: 5962: 5961: 5951: 5950: 5949: 5939: 5933: 5931: 5925: 5924: 5922: 5921: 5920: 5919: 5914: 5904: 5899: 5894: 5889: 5884: 5878: 5876: 5870: 5869: 5867: 5866: 5861: 5860: 5859: 5854: 5853: 5852: 5847: 5832: 5831: 5830: 5825: 5820: 5815: 5804: 5802: 5793: 5787: 5786: 5784: 5783: 5778: 5773: 5772: 5771: 5761: 5756: 5755: 5754: 5744: 5743: 5742: 5737: 5732: 5722: 5717: 5712: 5711: 5710: 5705: 5700: 5684: 5683: 5682: 5677: 5672: 5662: 5661: 5660: 5655: 5645: 5644: 5643: 5633: 5632: 5631: 5621: 5616: 5611: 5605: 5603: 5593: 5592: 5580: 5579: 5576: 5575: 5572: 5571: 5569: 5568: 5563: 5558: 5553: 5547: 5545: 5539: 5538: 5536: 5535: 5530: 5525: 5519: 5517: 5513: 5512: 5510: 5509: 5504: 5499: 5494: 5489: 5484: 5479: 5473: 5471: 5465: 5464: 5462: 5461: 5459:Standard error 5456: 5451: 5446: 5445: 5444: 5439: 5428: 5426: 5420: 5419: 5417: 5416: 5411: 5406: 5401: 5396: 5391: 5389:Optimal design 5386: 5381: 5375: 5373: 5363: 5362: 5350: 5349: 5346: 5345: 5342: 5341: 5339: 5338: 5333: 5328: 5323: 5318: 5313: 5308: 5303: 5298: 5293: 5288: 5283: 5278: 5273: 5268: 5262: 5260: 5254: 5253: 5251: 5250: 5245: 5244: 5243: 5238: 5228: 5223: 5217: 5215: 5209: 5208: 5206: 5205: 5200: 5195: 5189: 5187: 5186:Summary tables 5183: 5182: 5180: 5179: 5173: 5171: 5165: 5164: 5161: 5160: 5158: 5157: 5156: 5155: 5150: 5145: 5135: 5129: 5127: 5121: 5120: 5118: 5117: 5112: 5107: 5102: 5097: 5092: 5087: 5081: 5079: 5073: 5072: 5070: 5069: 5064: 5059: 5058: 5057: 5052: 5047: 5042: 5037: 5032: 5027: 5022: 5020:Contraharmonic 5017: 5012: 5001: 4999: 4990: 4980: 4979: 4967: 4966: 4964: 4963: 4958: 4952: 4949: 4948: 4941: 4940: 4933: 4926: 4918: 4912: 4911: 4903: 4902:External links 4900: 4899: 4898: 4861: 4856: 4843: 4822: 4817: 4804: 4799: 4786: 4768:(3): 287–290. 4753: 4750: 4747: 4746: 4739: 4721: 4680: 4679: 4677: 4674: 4645: 4642: 4641: 4640: 4629: 4626: 4623: 4620: 4617: 4585: 4582: 4556: 4541:Main article: 4538: 4535: 4522: 4497: 4493: 4489: 4484: 4480: 4476: 4471: 4467: 4455: 4454: 4440: 4437: 4434: 4429: 4425: 4421: 4418: 4411: 4406: 4402: 4396: 4391: 4388: 4385: 4381: 4377: 4371: 4368: 4365: 4359: 4356: 4353: 4349: 4346: 4343: 4339: 4336: 4329: 4324: 4320: 4314: 4309: 4306: 4303: 4299: 4292: 4289: 4286: 4283: 4269: 4268: 4253: 4248: 4244: 4240: 4237: 4234: 4230: 4226: 4223: 4220: 4215: 4211: 4207: 4203: 4199: 4196: 4193: 4190: 4188: 4186: 4183: 4178: 4174: 4168: 4163: 4160: 4157: 4153: 4147: 4144: 4139: 4136: 4131: 4127: 4121: 4116: 4113: 4110: 4106: 4100: 4097: 4092: 4089: 4086: 4081: 4076: 4072: 4066: 4061: 4058: 4055: 4051: 4045: 4042: 4037: 4034: 4031: 4029: 4027: 4024: 4019: 4015: 4009: 4005: 4001: 3998: 3993: 3988: 3984: 3980: 3975: 3970: 3966: 3962: 3957: 3952: 3949: 3946: 3943: 3940: 3936: 3932: 3929: 3922: 3918: 3914: 3909: 3906: 3904: 3900: 3896: 3890: 3886: 3882: 3877: 3873: 3869: 3864: 3859: 3856: 3853: 3850: 3847: 3843: 3835: 3831: 3827: 3822: 3821: 3807: 3806: 3789: 3784: 3780: 3774: 3769: 3766: 3763: 3759: 3753: 3750: 3745: 3742: 3737: 3733: 3729: 3726: 3723: 3719: 3715: 3712: 3709: 3704: 3700: 3696: 3692: 3688: 3685: 3682: 3679: 3677: 3675: 3670: 3666: 3662: 3659: 3656: 3652: 3648: 3645: 3642: 3637: 3632: 3628: 3622: 3617: 3614: 3611: 3607: 3601: 3598: 3593: 3588: 3583: 3579: 3573: 3568: 3565: 3562: 3558: 3552: 3549: 3544: 3539: 3534: 3530: 3524: 3519: 3516: 3513: 3509: 3503: 3500: 3495: 3492: 3490: 3488: 3483: 3479: 3475: 3472: 3469: 3465: 3461: 3458: 3455: 3452: 3447: 3442: 3438: 3434: 3429: 3424: 3420: 3416: 3411: 3406: 3402: 3398: 3393: 3388: 3385: 3382: 3378: 3372: 3369: 3364: 3361: 3359: 3357: 3353: 3349: 3344: 3340: 3334: 3329: 3326: 3323: 3319: 3313: 3310: 3305: 3302: 3297: 3293: 3287: 3282: 3279: 3276: 3272: 3266: 3263: 3258: 3255: 3250: 3246: 3240: 3236: 3230: 3225: 3222: 3219: 3215: 3211: 3208: 3201: 3197: 3193: 3187: 3183: 3180: 3177: 3175: 3173: 3168: 3164: 3160: 3155: 3151: 3147: 3144: 3139: 3135: 3131: 3126: 3122: 3118: 3113: 3108: 3105: 3102: 3099: 3096: 3092: 3084: 3080: 3076: 3071: 3070: 3047: 3022: 3018: 3015: 3010: 3006: 2998: 2993: 2989: 2986: 2983: 2980: 2977: 2974: 2971: 2968: 2965: 2962: 2955: 2950: 2946: 2943: 2940: 2937: 2934: 2931: 2928: 2925: 2922: 2919: 2916: 2909: 2906: 2903: 2900: 2896: 2893: 2890: 2866: 2862: 2859: 2856: 2853: 2850: 2847: 2844: 2841: 2838: 2835: 2832: 2825: 2822: 2817: 2813: 2809: 2805: 2781: 2777: 2774: 2771: 2764: 2761: 2758: 2755: 2751: 2730: 2717:. Using basic 2706: 2686: 2683: 2680: 2677: 2674: 2671: 2668: 2648: 2645: 2642: 2622: 2619: 2616: 2613: 2610: 2607: 2604: 2601: 2598: 2578: 2558: 2536: 2532: 2528: 2523: 2519: 2515: 2510: 2506: 2494: 2493: 2477: 2473: 2467: 2463: 2459: 2454: 2450: 2446: 2443: 2438: 2433: 2429: 2425: 2420: 2416: 2412: 2409: 2404: 2400: 2396: 2391: 2387: 2383: 2380: 2374: 2371: 2348: 2328: 2306: 2302: 2279: 2275: 2252: 2247: 2244: 2240: 2236: 2214: 2209: 2206: 2202: 2198: 2187: 2186: 2173: 2169: 2165: 2160: 2156: 2152: 2147: 2144: 2140: 2129: 2116: 2112: 2108: 2103: 2099: 2095: 2090: 2087: 2083: 2059: 2055: 2051: 2048: 2045: 2042: 2039: 2036: 2033: 2030: 2027: 2024: 2004: 1984: 1964: 1942: 1938: 1915: 1911: 1898: 1895: 1894: 1893: 1882: 1874: 1868: 1865: 1862: 1859: 1856: 1853: 1848: 1845: 1837: 1834: 1831: 1823: 1820: 1816: 1810: 1807: 1782: 1777: 1774: 1770: 1766: 1744: 1741: 1737: 1716: 1712: 1708: 1705: 1702: 1699: 1696: 1693: 1671: 1666: 1663: 1659: 1655: 1629: 1626: 1622: 1616: 1613: 1609: 1605: 1594: 1593: 1582: 1579: 1574: 1570: 1566: 1561: 1557: 1553: 1550: 1547: 1544: 1539: 1536: 1532: 1527: 1524: 1519: 1515: 1511: 1506: 1502: 1498: 1495: 1492: 1489: 1484: 1481: 1477: 1453: 1433: 1413: 1391: 1387: 1364: 1360: 1347: 1344: 1325: 1305: 1294:Frobenius norm 1276: 1271: 1267: 1264: 1261: 1258: 1253: 1247: 1242: 1238: 1235: 1208: 1203: 1199: 1196: 1193: 1190: 1179: 1178: 1161: 1156: 1152: 1149: 1143: 1138: 1134: 1131: 1123: 1118: 1114: 1111: 1108: 1105: 1099: 1096: 1073: 1070: 1067: 1056: 1035: 1032: 1029: 1018: 997: 992: 989: 985: 981: 978: 975: 955: 950: 947: 943: 939: 936: 933: 922: 921: 905: 900: 897: 893: 887: 882: 879: 876: 873: 870: 866: 860: 855: 852: 848: 842: 837: 834: 831: 828: 825: 821: 813: 810: 806: 800: 797: 793: 787: 782: 779: 776: 773: 770: 766: 759: 756: 744:is defined as 733: 713: 710: 707: 687: 684: 679: 676: 672: 668: 663: 660: 656: 633: 630: 626: 622: 619: 614: 611: 607: 584: 581: 577: 573: 570: 565: 562: 558: 535: 532: 528: 507: 485: 482: 478: 457: 437: 417: 395: 392: 389: 385: 379: 375: 371: 349: 346: 343: 339: 333: 329: 325: 305: 285: 265: 242: 222: 210: 207: 179: 178: 175: 172: 161: 160: 152: 144: 136: 119: 116: 104:no high school 75: 72: 26: 9: 6: 4: 3: 2: 7272: 7261: 7258: 7256: 7253: 7251: 7248: 7247: 7245: 7230: 7229: 7220: 7218: 7217: 7208: 7206: 7205: 7200: 7194: 7192: 7191: 7182: 7181: 7178: 7164: 7161: 7159: 7158:Geostatistics 7156: 7154: 7151: 7149: 7146: 7144: 7141: 7140: 7138: 7136: 7132: 7126: 7125:Psychometrics 7123: 7121: 7118: 7116: 7113: 7111: 7108: 7106: 7103: 7101: 7098: 7096: 7093: 7091: 7088: 7086: 7083: 7081: 7078: 7077: 7075: 7073: 7069: 7063: 7060: 7058: 7055: 7053: 7049: 7046: 7044: 7041: 7039: 7036: 7034: 7031: 7030: 7028: 7026: 7022: 7016: 7013: 7011: 7008: 7006: 7002: 6999: 6997: 6994: 6993: 6991: 6989: 6988:Biostatistics 6985: 6981: 6977: 6972: 6968: 6950: 6949:Log-rank test 6947: 6946: 6944: 6940: 6934: 6931: 6930: 6928: 6926: 6922: 6916: 6913: 6911: 6908: 6906: 6903: 6901: 6898: 6897: 6895: 6893: 6889: 6886: 6884: 6880: 6870: 6867: 6865: 6862: 6860: 6857: 6855: 6852: 6850: 6847: 6846: 6844: 6842: 6838: 6832: 6829: 6827: 6824: 6822: 6820:(Box–Jenkins) 6816: 6814: 6811: 6809: 6806: 6802: 6799: 6798: 6797: 6794: 6793: 6791: 6789: 6785: 6779: 6776: 6774: 6773:Durbin–Watson 6771: 6769: 6763: 6761: 6758: 6756: 6755:Dickey–Fuller 6753: 6752: 6750: 6746: 6740: 6737: 6735: 6732: 6730: 6729:Cointegration 6727: 6725: 6722: 6720: 6717: 6715: 6712: 6710: 6707: 6705: 6704:Decomposition 6702: 6701: 6699: 6695: 6692: 6690: 6686: 6676: 6673: 6672: 6671: 6668: 6667: 6666: 6663: 6659: 6656: 6655: 6654: 6651: 6649: 6646: 6644: 6641: 6639: 6636: 6634: 6631: 6629: 6626: 6624: 6621: 6619: 6616: 6615: 6613: 6611: 6607: 6601: 6598: 6596: 6593: 6591: 6588: 6586: 6583: 6581: 6578: 6576: 6575:Cohen's kappa 6573: 6572: 6570: 6568: 6564: 6560: 6556: 6552: 6548: 6544: 6539: 6535: 6521: 6518: 6516: 6513: 6511: 6508: 6506: 6503: 6502: 6500: 6498: 6494: 6488: 6484: 6480: 6474: 6472: 6469: 6468: 6466: 6464: 6460: 6454: 6451: 6449: 6446: 6444: 6441: 6439: 6436: 6434: 6431: 6429: 6428:Nonparametric 6426: 6424: 6421: 6420: 6418: 6414: 6408: 6405: 6403: 6400: 6398: 6395: 6393: 6390: 6389: 6387: 6385: 6381: 6375: 6372: 6370: 6367: 6365: 6362: 6360: 6357: 6355: 6352: 6351: 6349: 6347: 6343: 6337: 6334: 6332: 6329: 6327: 6324: 6322: 6319: 6318: 6316: 6314: 6310: 6306: 6299: 6296: 6294: 6291: 6290: 6286: 6282: 6266: 6263: 6262: 6261: 6258: 6256: 6253: 6251: 6248: 6244: 6241: 6239: 6236: 6235: 6234: 6231: 6230: 6228: 6226: 6222: 6212: 6209: 6205: 6199: 6197: 6191: 6189: 6183: 6182: 6181: 6178: 6177:Nonparametric 6175: 6173: 6167: 6163: 6160: 6159: 6158: 6152: 6148: 6147:Sample median 6145: 6144: 6143: 6140: 6139: 6137: 6135: 6131: 6123: 6120: 6118: 6115: 6113: 6110: 6109: 6108: 6105: 6103: 6100: 6098: 6092: 6090: 6087: 6085: 6082: 6080: 6077: 6075: 6072: 6070: 6068: 6064: 6062: 6059: 6058: 6056: 6054: 6050: 6044: 6042: 6038: 6036: 6034: 6029: 6027: 6022: 6018: 6017: 6014: 6011: 6009: 6005: 5995: 5992: 5990: 5987: 5985: 5982: 5981: 5979: 5977: 5973: 5967: 5964: 5960: 5957: 5956: 5955: 5952: 5948: 5945: 5944: 5943: 5940: 5938: 5935: 5934: 5932: 5930: 5926: 5918: 5915: 5913: 5910: 5909: 5908: 5905: 5903: 5900: 5898: 5895: 5893: 5890: 5888: 5885: 5883: 5880: 5879: 5877: 5875: 5871: 5865: 5862: 5858: 5855: 5851: 5848: 5846: 5843: 5842: 5841: 5838: 5837: 5836: 5833: 5829: 5826: 5824: 5821: 5819: 5816: 5814: 5811: 5810: 5809: 5806: 5805: 5803: 5801: 5797: 5794: 5792: 5788: 5782: 5779: 5777: 5774: 5770: 5767: 5766: 5765: 5762: 5760: 5757: 5753: 5752:loss function 5750: 5749: 5748: 5745: 5741: 5738: 5736: 5733: 5731: 5728: 5727: 5726: 5723: 5721: 5718: 5716: 5713: 5709: 5706: 5704: 5701: 5699: 5693: 5690: 5689: 5688: 5685: 5681: 5678: 5676: 5673: 5671: 5668: 5667: 5666: 5663: 5659: 5656: 5654: 5651: 5650: 5649: 5646: 5642: 5639: 5638: 5637: 5634: 5630: 5627: 5626: 5625: 5622: 5620: 5617: 5615: 5612: 5610: 5607: 5606: 5604: 5602: 5598: 5594: 5590: 5585: 5581: 5567: 5564: 5562: 5559: 5557: 5554: 5552: 5549: 5548: 5546: 5544: 5540: 5534: 5531: 5529: 5526: 5524: 5521: 5520: 5518: 5514: 5508: 5505: 5503: 5500: 5498: 5495: 5493: 5490: 5488: 5485: 5483: 5480: 5478: 5475: 5474: 5472: 5470: 5466: 5460: 5457: 5455: 5454:Questionnaire 5452: 5450: 5447: 5443: 5440: 5438: 5435: 5434: 5433: 5430: 5429: 5427: 5425: 5421: 5415: 5412: 5410: 5407: 5405: 5402: 5400: 5397: 5395: 5392: 5390: 5387: 5385: 5382: 5380: 5377: 5376: 5374: 5372: 5368: 5364: 5360: 5355: 5351: 5337: 5334: 5332: 5329: 5327: 5324: 5322: 5319: 5317: 5314: 5312: 5309: 5307: 5304: 5302: 5299: 5297: 5294: 5292: 5289: 5287: 5284: 5282: 5281:Control chart 5279: 5277: 5274: 5272: 5269: 5267: 5264: 5263: 5261: 5259: 5255: 5249: 5246: 5242: 5239: 5237: 5234: 5233: 5232: 5229: 5227: 5224: 5222: 5219: 5218: 5216: 5214: 5210: 5204: 5201: 5199: 5196: 5194: 5191: 5190: 5188: 5184: 5178: 5175: 5174: 5172: 5170: 5166: 5154: 5151: 5149: 5146: 5144: 5141: 5140: 5139: 5136: 5134: 5131: 5130: 5128: 5126: 5122: 5116: 5113: 5111: 5108: 5106: 5103: 5101: 5098: 5096: 5093: 5091: 5088: 5086: 5083: 5082: 5080: 5078: 5074: 5068: 5065: 5063: 5060: 5056: 5053: 5051: 5048: 5046: 5043: 5041: 5038: 5036: 5033: 5031: 5028: 5026: 5023: 5021: 5018: 5016: 5013: 5011: 5008: 5007: 5006: 5003: 5002: 5000: 4998: 4994: 4991: 4989: 4985: 4981: 4977: 4972: 4968: 4962: 4959: 4957: 4954: 4953: 4950: 4946: 4939: 4934: 4932: 4927: 4925: 4920: 4919: 4916: 4909: 4906: 4905: 4895: 4889: 4880: 4875: 4871: 4867: 4862: 4859: 4857:0-85264-199-0 4853: 4849: 4844: 4840: 4836: 4832: 4828: 4823: 4820: 4818:0-940600-14-5 4814: 4810: 4805: 4802: 4800:0-521-81099-X 4796: 4792: 4787: 4783: 4779: 4775: 4771: 4767: 4763: 4762: 4761:Psychometrika 4756: 4755: 4742: 4740:9780852641996 4736: 4732: 4725: 4717: 4713: 4709: 4705: 4701: 4697: 4696: 4691: 4685: 4681: 4673: 4671: 4667: 4663: 4658: 4656: 4650: 4627: 4624: 4621: 4618: 4615: 4608: 4607: 4606: 4604: 4600: 4594: 4592: 4581: 4578: 4574: 4570: 4554: 4544: 4534: 4520: 4513: 4495: 4491: 4487: 4482: 4478: 4474: 4469: 4465: 4435: 4432: 4427: 4423: 4416: 4409: 4404: 4400: 4394: 4389: 4386: 4383: 4379: 4375: 4369: 4366: 4363: 4354: 4337: 4334: 4327: 4322: 4318: 4312: 4307: 4304: 4301: 4297: 4290: 4287: 4284: 4274: 4273: 4272: 4246: 4235: 4221: 4213: 4209: 4194: 4191: 4189: 4176: 4172: 4166: 4161: 4158: 4155: 4151: 4145: 4142: 4129: 4125: 4119: 4114: 4111: 4108: 4104: 4098: 4095: 4087: 4084: 4079: 4074: 4070: 4064: 4059: 4056: 4053: 4049: 4043: 4040: 4035: 4032: 4030: 4017: 4013: 4007: 4003: 3999: 3996: 3991: 3986: 3982: 3978: 3973: 3968: 3964: 3955: 3950: 3947: 3944: 3941: 3938: 3934: 3930: 3927: 3920: 3916: 3912: 3907: 3905: 3898: 3888: 3884: 3880: 3875: 3871: 3862: 3857: 3854: 3851: 3848: 3845: 3841: 3833: 3829: 3825: 3812: 3811: 3810: 3787: 3782: 3778: 3772: 3767: 3764: 3761: 3757: 3751: 3748: 3743: 3735: 3724: 3710: 3702: 3698: 3683: 3680: 3678: 3668: 3657: 3643: 3640: 3635: 3630: 3626: 3620: 3615: 3612: 3609: 3605: 3599: 3596: 3591: 3586: 3581: 3577: 3571: 3566: 3563: 3560: 3556: 3550: 3547: 3542: 3537: 3532: 3528: 3522: 3517: 3514: 3511: 3507: 3501: 3498: 3493: 3491: 3481: 3470: 3456: 3453: 3445: 3440: 3436: 3432: 3427: 3422: 3418: 3414: 3409: 3404: 3400: 3391: 3386: 3383: 3380: 3376: 3370: 3367: 3362: 3360: 3351: 3342: 3338: 3332: 3327: 3324: 3321: 3317: 3311: 3308: 3295: 3291: 3285: 3280: 3277: 3274: 3270: 3264: 3261: 3253: 3248: 3244: 3238: 3234: 3228: 3223: 3220: 3217: 3213: 3209: 3206: 3199: 3195: 3191: 3185: 3181: 3178: 3176: 3166: 3162: 3158: 3153: 3149: 3137: 3133: 3129: 3124: 3120: 3111: 3106: 3103: 3100: 3097: 3094: 3090: 3082: 3078: 3074: 3061: 3060: 3059: 3020: 3016: 3013: 3008: 3004: 2996: 2991: 2984: 2981: 2978: 2969: 2966: 2963: 2953: 2948: 2941: 2938: 2935: 2932: 2923: 2920: 2917: 2907: 2901: 2864: 2857: 2854: 2851: 2848: 2839: 2836: 2833: 2823: 2815: 2811: 2779: 2775: 2772: 2769: 2762: 2756: 2728: 2720: 2704: 2684: 2681: 2678: 2675: 2672: 2669: 2666: 2646: 2643: 2640: 2617: 2614: 2611: 2608: 2605: 2602: 2599: 2576: 2556: 2534: 2530: 2526: 2521: 2517: 2513: 2508: 2504: 2475: 2465: 2461: 2457: 2452: 2448: 2441: 2431: 2427: 2423: 2418: 2414: 2402: 2398: 2394: 2389: 2385: 2378: 2372: 2362: 2361: 2360: 2346: 2326: 2304: 2300: 2277: 2273: 2250: 2245: 2242: 2238: 2234: 2212: 2207: 2204: 2200: 2196: 2171: 2167: 2163: 2158: 2154: 2150: 2145: 2142: 2138: 2130: 2114: 2110: 2106: 2101: 2097: 2093: 2088: 2085: 2081: 2073: 2072: 2071: 2049: 2046: 2043: 2040: 2034: 2031: 2028: 2025: 2022: 2002: 1982: 1962: 1940: 1936: 1913: 1909: 1880: 1872: 1863: 1860: 1857: 1851: 1832: 1814: 1808: 1798: 1797: 1796: 1780: 1775: 1772: 1768: 1764: 1742: 1739: 1735: 1714: 1710: 1703: 1700: 1697: 1691: 1669: 1664: 1661: 1657: 1653: 1645: 1627: 1624: 1620: 1614: 1611: 1607: 1603: 1580: 1572: 1568: 1564: 1559: 1555: 1548: 1545: 1542: 1537: 1534: 1530: 1525: 1517: 1513: 1509: 1504: 1500: 1493: 1490: 1487: 1482: 1479: 1475: 1467: 1466: 1465: 1451: 1444:-quality and 1431: 1411: 1389: 1385: 1362: 1358: 1342: 1337: 1323: 1303: 1295: 1265: 1262: 1259: 1251: 1236: 1225: 1197: 1194: 1191: 1150: 1132: 1112: 1109: 1106: 1097: 1087: 1086: 1085: 1071: 1068: 1065: 1054: 1033: 1030: 1027: 1016: 990: 987: 983: 976: 973: 948: 945: 941: 934: 931: 903: 898: 895: 891: 885: 880: 877: 874: 871: 868: 864: 858: 853: 850: 846: 840: 835: 832: 829: 826: 823: 819: 811: 808: 804: 798: 795: 791: 785: 780: 777: 774: 771: 768: 764: 757: 747: 746: 745: 711: 708: 705: 685: 682: 677: 674: 670: 666: 661: 658: 654: 631: 628: 624: 620: 617: 612: 609: 605: 582: 579: 575: 571: 568: 563: 560: 556: 533: 530: 526: 505: 483: 480: 476: 455: 435: 415: 393: 390: 387: 377: 373: 347: 344: 341: 331: 327: 303: 283: 263: 254: 240: 220: 206: 204: 200: 196: 192: 188: 184: 176: 173: 170: 169: 168: 166: 159: 158: 153: 151: 150: 145: 143: 142: 137: 135: 134: 129: 128: 127: 125: 115: 113: 109: 105: 101: 97: 96:medium income 93: 89: 84: 80: 71: 69: 65: 61: 60:nonparametric 57: 53: 49: 46:of different 45: 41: 37: 33: 19: 7226: 7214: 7195: 7188: 7100:Econometrics 7050: / 7033:Chemometrics 7010:Epidemiology 7003: / 6976:Applications 6818:ARIMA model 6765:Q-statistic 6714:Stationarity 6610:Multivariate 6553: / 6549: / 6547:Multivariate 6545: / 6485: / 6481: / 6255:Bayes factor 6154:Signed rank 6066: 6040: 6032: 6020: 5715:Completeness 5551:Cohort study 5449:Opinion poll 5384:Missing data 5371:Study design 5326:Scatter plot 5248:Scatter plot 5241:Spearman's ρ 5230: 5203:Grouped data 4888:cite journal 4869: 4865: 4847: 4833:(1): 91–95. 4830: 4826: 4808: 4790: 4765: 4759: 4730: 4724: 4699: 4693: 4684: 4669: 4665: 4661: 4659: 4654: 4651: 4647: 4602: 4598: 4595: 4587: 4576: 4572: 4568: 4546: 4456: 4270: 3808: 3058:as follows: 2495: 2188: 1900: 1595: 1349: 1180: 923: 428:-th and the 255: 212: 203:metric space 180: 162: 156: 148: 140: 132: 121: 111: 107: 103: 99: 95: 91: 85: 81: 77: 56:significance 51: 39: 35: 29: 7228:WikiProject 7143:Cartography 7105:Jurimetrics 7057:Reliability 6788:Time domain 6767:(Ljung–Box) 6689:Time-series 6567:Categorical 6551:Time-series 6543:Categorical 6478:(Bernoulli) 6313:Correlation 6293:Correlation 6089:Jarque–Bera 6061:Chi-squared 5823:M-estimator 5776:Asymptotics 5720:Sufficiency 5487:Interaction 5399:Replication 5379:Effect size 5336:Violin plot 5316:Radar chart 5296:Forest plot 5286:Correlogram 5236:Kendall's τ 2319:range from 2070:defined by 1646:). The sum 187:permutation 165:coefficient 131:Spearman's 124:correlation 108:high school 100:high income 7244:Categories 7095:Demography 6813:ARMA model 6618:Regression 6195:(Friedman) 6156:(Wilcoxon) 6094:Normality 6084:Lilliefors 6031:Student's 5907:Resampling 5781:Robustness 5769:divergence 5759:Efficiency 5697:(monotone) 5692:Likelihood 5609:Population 5442:Stratified 5394:Population 5213:Dependence 5169:Count data 5100:Percentile 5077:Dispersion 5010:Arithmetic 4945:Statistics 4676:References 2741:, we have 1339:See also: 181:Following 139:Kendall's 112:university 92:low income 32:statistics 6476:Logistic 6243:posterior 6169:Rank sum 5917:Jackknife 5912:Bootstrap 5730:Bootstrap 5665:Parameter 5614:Statistic 5409:Statistic 5321:Run chart 5306:Pie chart 5301:Histogram 5291:Fan chart 5266:Bar chart 5148:L-moments 5035:Geometric 4782:122500836 4625:− 4555:ρ 4521:ρ 4488:− 4433:− 4380:∑ 4370:− 4298:∑ 4291:− 4282:Γ 4222:− 4152:∑ 4105:∑ 4085:− 4050:∑ 3997:− 3935:∑ 3928:⋅ 3881:− 3842:∑ 3758:∑ 3744:− 3711:− 3641:− 3606:∑ 3592:− 3557:∑ 3508:∑ 3454:− 3433:− 3377:∑ 3318:∑ 3271:∑ 3254:− 3214:∑ 3207:⋅ 3159:− 3130:− 3091:∑ 3046:Γ 3014:− 2954:− 2880:and thus 2679:… 2612:… 2527:− 2458:− 2442:∑ 2424:− 2395:− 2379:∑ 2370:Γ 2235:∑ 2197:∑ 2189:The sums 2164:− 2107:− 2044:× 2032:∈ 1881:τ 1861:− 1833:− 1806:Γ 1765:∑ 1701:− 1654:∑ 1604:∑ 1565:− 1549:⁡ 1510:− 1494:⁡ 1270:⟩ 1257:⟨ 1241:‖ 1234:‖ 1202:⟩ 1189:⟨ 1155:‖ 1148:‖ 1137:‖ 1130:‖ 1117:⟩ 1104:⟨ 1095:Γ 1069:− 1031:− 865:∑ 820:∑ 765:∑ 755:Γ 732:Γ 621:− 572:− 391:≤ 345:≤ 241:ρ 221:τ 7260:Rankings 7190:Category 6883:Survival 6760:Johansen 6483:Binomial 6438:Isotonic 6025:(normal) 5670:location 5477:Blocking 5432:Sampling 5311:Q–Q plot 5276:Box plot 5258:Graphics 5153:Skewness 5143:Kurtosis 5115:Variance 5045:Heronian 5040:Harmonic 2359:. Hence 1995:and the 1757:, as is 1684:is just 1596:The sum 498:, and a 155:Somers' 66:and the 44:rankings 7216:Commons 7163:Kriging 7048:Process 7005:studies 6864:Wavelet 6697:General 5864:Plug-in 5658:L space 5437:Cluster 5138:Moments 4956:Outline 4716:2281954 1222:is the 1084:, then 1008:, with 74:Context 48:ordinal 7085:Census 6675:Normal 6623:Manova 6443:Robust 6193:2-way 6185:1-way 6023:-test 5694:  5271:Biplot 5062:Median 5055:Lehmer 4997:Center 4854:  4815:  4797:  4780:  4737:  4714:  4457:where 4271:Hence 1181:where 199:metric 98:, and 6709:Trend 6238:prior 6180:anova 6069:-test 6043:-test 6035:-test 5942:Power 5887:Pivot 5680:shape 5675:scale 5125:Shape 5105:Range 5050:Heinz 5025:Cubic 4961:Index 4778:S2CID 4712:JSTOR 4605:). 189:of a 90:with 6942:Test 6142:Sign 5994:Wald 5067:Mode 5005:Mean 4894:link 4852:ISBN 4813:ISBN 4795:ISBN 4735:ISBN 3809:and 2795:and 2292:and 2227:and 1316:and 1292:the 1226:and 1046:and 966:and 597:and 362:and 296:and 34:, a 6122:BIC 6117:AIC 4874:doi 4835:doi 4770:doi 4704:doi 2339:to 1901:If 1546:sgn 1491:sgn 1350:If 698:if 191:set 30:In 7246:: 4890:}} 4886:{{ 4868:. 4829:. 4776:. 4766:21 4764:. 4710:. 4700:53 4698:. 4533:. 3021:12 2514::= 2151::= 2094::= 1928:, 1377:, 1336:. 110:, 106:, 94:, 70:. 6067:G 6041:F 6033:t 6021:Z 5740:V 5735:U 4937:e 4930:t 4923:v 4896:) 4876:: 4870:3 4841:. 4837:: 4831:2 4784:. 4772:: 4743:. 4718:. 4706:: 4670:r 4666:r 4662:r 4655:r 4628:u 4622:f 4619:= 4616:r 4603:u 4599:f 4577:r 4573:Y 4569:X 4496:i 4492:s 4483:i 4479:r 4475:= 4470:i 4466:d 4439:) 4436:1 4428:2 4424:n 4420:( 4417:n 4410:2 4405:i 4401:d 4395:n 4390:1 4387:= 4384:i 4376:6 4367:1 4364:= 4358:) 4355:U 4352:( 4348:r 4345:a 4342:V 4338:n 4335:2 4328:2 4323:i 4319:d 4313:n 4308:1 4305:= 4302:i 4288:1 4285:= 4252:) 4247:2 4243:) 4239:] 4236:U 4233:[ 4229:E 4225:( 4219:] 4214:2 4210:U 4206:[ 4202:E 4198:( 4195:2 4192:= 4182:) 4177:j 4173:r 4167:n 4162:1 4159:= 4156:j 4146:n 4143:1 4138:( 4135:) 4130:i 4126:r 4120:n 4115:1 4112:= 4109:i 4099:n 4096:1 4091:( 4088:2 4080:2 4075:i 4071:r 4065:n 4060:1 4057:= 4054:i 4044:n 4041:1 4036:2 4033:= 4023:) 4018:j 4014:r 4008:i 4004:r 4000:2 3992:2 3987:j 3983:r 3979:+ 3974:2 3969:i 3965:r 3961:( 3956:n 3951:1 3948:= 3945:j 3942:, 3939:i 3931:n 3921:2 3917:n 3913:1 3908:= 3899:2 3895:) 3889:i 3885:r 3876:j 3872:r 3868:( 3863:n 3858:1 3855:= 3852:j 3849:, 3846:i 3834:2 3830:n 3826:1 3788:2 3783:i 3779:d 3773:n 3768:1 3765:= 3762:i 3752:n 3749:1 3741:) 3736:2 3732:) 3728:] 3725:U 3722:[ 3718:E 3714:( 3708:] 3703:2 3699:U 3695:[ 3691:E 3687:( 3684:2 3681:= 3669:2 3665:) 3661:] 3658:U 3655:[ 3651:E 3647:( 3644:2 3636:2 3631:i 3627:d 3621:n 3616:1 3613:= 3610:i 3600:n 3597:1 3587:2 3582:i 3578:s 3572:n 3567:1 3564:= 3561:i 3551:n 3548:1 3543:+ 3538:2 3533:i 3529:r 3523:n 3518:1 3515:= 3512:i 3502:n 3499:1 3494:= 3482:2 3478:) 3474:] 3471:U 3468:[ 3464:E 3460:( 3457:2 3451:) 3446:2 3441:i 3437:d 3428:2 3423:i 3419:s 3415:+ 3410:2 3405:i 3401:r 3397:( 3392:n 3387:1 3384:= 3381:i 3371:n 3368:1 3363:= 3352:) 3348:) 3343:j 3339:s 3333:n 3328:1 3325:= 3322:j 3312:n 3309:1 3304:( 3301:) 3296:i 3292:r 3286:n 3281:1 3278:= 3275:i 3265:n 3262:1 3257:( 3249:i 3245:s 3239:i 3235:r 3229:n 3224:1 3221:= 3218:i 3210:n 3200:2 3196:n 3192:1 3186:( 3182:2 3179:= 3172:) 3167:i 3163:s 3154:j 3150:s 3146:( 3143:) 3138:i 3134:r 3125:j 3121:r 3117:( 3112:n 3107:1 3104:= 3101:j 3098:, 3095:i 3083:2 3079:n 3075:1 3017:1 3009:2 3005:n 2997:= 2992:4 2988:) 2985:1 2982:+ 2979:n 2976:( 2973:) 2970:1 2967:+ 2964:n 2961:( 2949:6 2945:) 2942:1 2939:+ 2936:n 2933:2 2930:( 2927:) 2924:1 2921:+ 2918:n 2915:( 2908:= 2905:) 2902:U 2899:( 2895:r 2892:a 2889:V 2865:6 2861:) 2858:1 2855:+ 2852:n 2849:2 2846:( 2843:) 2840:1 2837:+ 2834:n 2831:( 2824:= 2821:] 2816:2 2812:U 2808:[ 2804:E 2780:2 2776:1 2773:+ 2770:n 2763:= 2760:] 2757:U 2754:[ 2750:E 2729:U 2705:U 2685:n 2682:, 2676:, 2673:2 2670:, 2667:1 2647:s 2644:, 2641:r 2621:} 2618:n 2615:, 2609:, 2606:2 2603:, 2600:1 2597:{ 2577:U 2557:i 2535:i 2531:s 2522:i 2518:r 2509:i 2505:d 2476:2 2472:) 2466:i 2462:r 2453:j 2449:r 2445:( 2437:) 2432:i 2428:s 2419:j 2415:s 2411:( 2408:) 2403:i 2399:r 2390:j 2386:r 2382:( 2373:= 2347:n 2327:1 2305:i 2301:s 2278:i 2274:r 2251:2 2246:j 2243:i 2239:b 2213:2 2208:j 2205:i 2201:a 2172:i 2168:s 2159:j 2155:s 2146:j 2143:i 2139:b 2115:i 2111:r 2102:j 2098:r 2089:j 2086:i 2082:a 2058:) 2054:R 2050:; 2047:n 2041:n 2038:( 2035:M 2029:b 2026:, 2023:a 2003:y 1983:x 1963:i 1941:i 1937:s 1914:i 1910:r 1873:= 1867:) 1864:1 1858:n 1855:( 1852:n 1847:) 1844:) 1836:( 1830:) 1822:( 1819:( 1815:2 1809:= 1781:2 1776:j 1773:i 1769:b 1743:j 1740:i 1736:a 1715:2 1711:/ 1707:) 1704:1 1698:n 1695:( 1692:n 1670:2 1665:j 1662:i 1658:a 1628:j 1625:i 1621:b 1615:j 1612:i 1608:a 1581:. 1578:) 1573:i 1569:s 1560:j 1556:s 1552:( 1543:= 1538:j 1535:i 1531:b 1526:, 1523:) 1518:i 1514:r 1505:j 1501:r 1497:( 1488:= 1483:j 1480:i 1476:a 1452:y 1432:x 1412:i 1390:i 1386:s 1363:i 1359:r 1324:B 1304:A 1275:F 1266:A 1263:, 1260:A 1252:= 1246:F 1237:A 1207:F 1198:B 1195:, 1192:A 1160:F 1151:B 1142:F 1133:A 1122:F 1113:B 1110:, 1107:A 1098:= 1072:B 1066:= 1060:T 1055:B 1034:A 1028:= 1022:T 1017:A 996:) 991:j 988:i 984:b 980:( 977:= 974:B 954:) 949:j 946:i 942:a 938:( 935:= 932:A 904:2 899:j 896:i 892:b 886:n 881:1 878:= 875:j 872:, 869:i 859:2 854:j 851:i 847:a 841:n 836:1 833:= 830:j 827:, 824:i 812:j 809:i 805:b 799:j 796:i 792:a 786:n 781:1 778:= 775:j 772:, 769:i 758:= 712:j 709:= 706:i 686:0 683:= 678:j 675:i 671:b 667:= 662:j 659:i 655:a 632:i 629:j 625:b 618:= 613:j 610:i 606:b 583:i 580:j 576:a 569:= 564:j 561:i 557:a 534:j 531:i 527:b 506:y 484:j 481:i 477:a 456:x 436:j 416:i 394:n 388:i 384:} 378:i 374:y 370:{ 348:n 342:i 338:} 332:i 328:x 324:{ 304:y 284:x 264:n 157:D 149:Îł 141:τ 133:ρ 20:)

Index

Ordinal association
statistics
rankings
ordinal
significance
nonparametric
Mann–Whitney U test
Wilcoxon signed-rank test
contingency table
correlation
Spearman's ρ
Kendall's τ
Goodman and Kruskal's Îł
Somers' D
coefficient
Diaconis (1988)
permutation
set
symmetric group
metric
metric space
Frobenius inner product
Frobenius norm
Inner product space § Norms on inner product spaces
Kendall tau rank correlation coefficient
summation results
Spearman's rank correlation coefficient
Mann–Whitney_U_test § Rank-biserial_correlation
Mann–Whitney U test
Kruskal, William H.

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