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2401:'s assumed shape, and can be shown to be biased. A simple improvement for such applications, named centered isotonic regression (CIR), was developed by Oron and Flournoy and shown to substantially reduce estimation error for both dose-response and dose-finding applications. Both CIR and the standard isotonic regression for the univariate, simply ordered case, are implemented in the R package "cir". This package also provides analytical confidence-interval estimates.
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484:. For example, one might use it to fit an isotonic curve to the means of some set of experimental results when an increase in those means according to some particular ordering is expected. A benefit of isotonic regression is that it is not constrained by any functional form, such as the linearity imposed by
2323:{\displaystyle f(x)={\begin{cases}{\hat {y}}_{1}&{\text{if }}x\leq x_{1}\\{\hat {y}}_{i}+{\frac {x-x_{i}}{x_{i+1}-x_{i}}}({\hat {y}}_{i+1}-{\hat {y}}_{i})&{\text{if }}x_{i}\leq x\leq x_{i+1}\\{\hat {y}}_{n}&{\text{if }}x\geq x_{n}\end{cases}}}
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has been applied to estimating continuous dose-response relationships in fields such as anesthesiology and toxicology. Narrowly speaking, isotonic regression only provides point estimates at observed values of
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As this article's first figure shows, in the presence of monotonicity violations the resulting interpolated curve will have flat (constant) intervals. In dose-response applications it is usually known that
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Estimation of the complete dose-response curve without any additional assumptions is usually done via linear interpolation between the point estimates.
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identification problem, and proposed a primal algorithm. These two algorithms can be seen as each other's dual, and both have a
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Predicting good probabilities with supervised learning | Proceedings of the 22nd international conference on
Machine learning
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An example of isotonic regression (solid red line) compared to linear regression on the same data, both fit to minimize the
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between points. Isotonic regression is used iteratively to fit ideal distances to preserve relative dissimilarity order.
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Shively, T.S., Sager, T.W., Walker, S.G. (2009). "A Bayesian approach to non-parametric monotone function estimation".
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Oron, AP; Flournoy, N (2017). "Centered
Isotonic Regression: Point and Interval Estimation for Dose-Response Studies".
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Stylianou, MP; Flournoy, N (2002). "Dose finding using the biased coin up-and-down design and isotonic regression".
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is the technique of fitting a free-form line to a sequence of observations such that the fitted line is
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Statistical inference under order restrictions; the theory and application of isotonic regression
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for data points is sought such that order of distances between points in the embedding matches
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2599:"Isotone Optimization in R: Pool-Adjacent-Violators Algorithm (PAVA) and Active Set Methods"
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To complete the isotonic regression task, we may then choose any non-decreasing function
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1436:). Problems of this form may be solved by generic quadratic programming techniques.
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2902:; Mentz, G. (2001). "Isotonic regression: Another look at the changepoint problem".
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472:(or non-increasing) everywhere, and lies as close to the observations as possible.
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Pedregosa, Fabian; et al. (2011). "Scikit-learn:Machine learning in Python".
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2018:, as illustrated in the figure, yielding a continuous piecewise linear function:
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1290:{\displaystyle {\hat {y}}_{i}\leq {\hat {y}}_{j}{\text{ for all }}(i,j)\in E}
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Software for computing isotone (monotonic) regression has been developed for
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Barlow, R. E.; Bartholomew, D. J.; Bremner, J. M.; Brunk, H. D. (1972).
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2710:"Active set algorithms for isotonic regression; A unifying framework"
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1206:{\displaystyle \min \sum _{i=1}^{n}w_{i}({\hat {y}}_{i}-y_{i})^{2}}
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2421:(1964). "Nonmetric Multidimensional Scaling: A numerical method".
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4872:
1857:{\displaystyle \min _{f}\sum _{i=1}^{n}w_{i}(f(x_{i})-y_{i})^{2}}
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Isotonic regression for the simply ordered case with univariate
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1631:. Conversely, Best and Chakravarti studied the problem as an
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specifies the partial ordering of the observed inputs
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Robertson, T.; Wright, F. T.; Dykstra, R. L. (1988).
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Autoregressive conditional heteroskedasticity (ARCH)
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would be to interpolate linearly between the points
2597:Leeuw, Jan de; Hornik, Kurt; Mair, Patrick (2009).
667:{\displaystyle (x_{1},y_{1}),\ldots ,(x_{n},y_{n})}
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488:, as long as the function is monotonic increasing.
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1397:(and may be regarded as the set of edges of some
999:{\displaystyle {\hat {y}}_{i}\leq {\hat {y}}_{j}}
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2708:Best, Michael J.; Chakravarti, Nilotpal (1990).
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4383:Multivariate adaptive regression splines (MARS)
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2707:
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1758:for all i. Any such function obviously solves
1496:that the observations have been sorted so that
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2638:Xu, Zhipeng; Sun, Chenkai; Karunakaran, Aman.
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2886:: CS1 maint: multiple names: authors list (
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510:to calibrate the predicted probabilities of
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2372:. The flat intervals are incompatible with
1363:{\displaystyle E=\{(i,j):x_{i}\leq x_{j}\}}
921:{\displaystyle {\hat {y}}_{i}\approx y_{i}}
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1616:{\displaystyle E=\{(i,i+1):1\leq i<n\}}
674:be a given set of observations, where the
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1949:
1627:for solving the quadratic program is the
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695:
2757:Statistics in Biopharmaceutical Research
2649:. R Foundation for Statistical Computing
2578:. R Foundation for Statistical Computing
480:Isotonic regression has applications in
18:
1751:{\displaystyle f(x_{i})={\hat {y}}_{i}}
5251:
4909:KaplanâMeier estimator (product limit)
2806:Order restricted statistical inference
2011:{\displaystyle (x_{i},{\hat {y}}_{i})}
4982:
4549:
4296:
3595:
3365:
2982:
2926:
2565:
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1956:{\displaystyle x_{i}\in \mathbb {R} }
875:Isotonic regression seeks a weighted
702:{\displaystyle y_{i}\in \mathbb {R} }
5219:
4919:Accelerated failure time (AFT) model
2670:Journal of Machine Learning Research
740:. For generality, each observation
506:Isotonic regression is also used in
5231:
4514:Analysis of variance (ANOVA, anova)
3366:
2461:. 7 August 2005. pp. 625â632.
13:
4609:CochranâMantelâHaenszel statistics
3235:Pearson product-moment correlation
2797:
2560:
14:
5280:
5264:Nonparametric Bayesian statistics
1629:pool adjacent violators algorithm
948:, subject to the constraint that
491:Another application is nonmetric
5230:
5218:
5206:
5193:
5192:
4983:
2866:10.1111/j.1467-9868.2008.00677.x
2569:
2538:10.1111/j.0006-341x.2002.00171.x
586:Problem statement and algorithms
404:
4868:Least-squares spectral analysis
2603:Journal of Statistical Software
1888:and can be used to predict the
1439:In the usual setting where the
1039:{\displaystyle x_{i}\leq x_{j}}
475:
352:Least-squares spectral analysis
290:Generalized estimating equation
110:Multinomial logistic regression
85:Vector generalized linear model
3849:Mean-unbiased minimum-variance
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2368:is not only monotone but also
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5162:Geographic information system
4378:Simultaneous equations models
2779:10.1080/19466315.2017.1286256
2404:
779:{\displaystyle (x_{i},y_{i})}
171:Nonlinear mixed-effects model
4345:Coefficient of determination
3956:Uniformly most powerful test
2334:Centered isotonic regression
1485:{\displaystyle \mathbb {R} }
1429:{\displaystyle 1,2,\ldots n}
1046:. This gives the following
508:probabilistic classification
7:
4914:Proportional hazards models
4858:Spectral density estimation
4840:Vector autoregression (VAR)
4274:Maximum posterior estimator
3506:Randomized controlled trial
812:{\displaystyle w_{i}\geq 0}
512:supervised machine learning
373:Mean and predicted response
10:
5285:
4674:Multivariate distributions
3094:Average absolute deviation
2640:"Package UniIsoRegression"
1623:. In this case, a simple
495:, where a low-dimensional
166:Linear mixed-effects model
16:Type of numerical analysis
5188:
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4662:Structural equation model
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3413:Sample size determination
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1908:values for new values of
332:Least absolute deviations
5259:Nonparametric regression
5157:Environmental statistics
4679:Elliptical distributions
4472:Generalized linear model
4401:Simple linear regression
4171:HodgesâLehmann estimator
3628:Probability distribution
3537:Stochastic approximation
3099:Coefficient of variation
2714:Mathematical Programming
1928:. A common choice when
1668:on already sorted data.
1637:computational complexity
493:multidimensional scaling
80:Generalized linear model
4817:Cross-correlation (XCF)
4425:Non-standard predictors
3859:LehmannâScheffĂ© theorem
3532:Adaptive clinical trial
2916:10.1093/biomet/88.3.793
2467:10.1145/1102351.1102430
845:{\displaystyle w_{i}=1}
5213:Mathematics portal
5034:Engineering statistics
4942:NelsonâAalen estimator
4519:Analysis of covariance
4406:Ordinary least squares
4330:Pearson product-moment
3734:Statistical functional
3645:Empirical distribution
3478:Controlled experiments
3207:Frequency distribution
2985:Descriptive statistics
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2012:
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1050:(QP) in the variables
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786:may be given a weight
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501:order of dissimilarity
411:Mathematics portal
337:Iteratively reweighted
28:
5129:Population statistics
5071:System identification
4805:Autocorrelation (ACF)
4733:Exponential smoothing
4647:Discriminant analysis
4642:Canonical correlation
4506:Partition of variance
4368:Regression validation
4212:(JonckheereâTerpstra)
4111:Likelihood-ratio test
3800:Frequentist inference
3712:Locationâscale family
3633:Sampling distribution
3598:Statistical inference
3565:Cross-sectional study
3552:Observational studies
3511:Randomized experiment
3340:Stem-and-leaf display
3142:Central limit theorem
2616:10.18637/jss.v032.i05
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2013:
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1459:{\displaystyle x_{i}}
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1390:{\displaystyle x_{i}}
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738:partially ordered set
731:
729:{\displaystyle x_{i}}
704:
669:
562:
538:
482:statistical inference
368:Regression validation
347:Bayesian multivariate
64:Polynomial regression
22:
5052:Probabilistic design
4637:Principal components
4480:Exponential families
4432:Nonlinear regression
4411:General linear model
4373:Mixed effects models
4363:Errors and residuals
4340:Confounding variable
4242:Bayesian probability
4220:Van der Waerden test
4210:Ordered alternative
3975:Multiple comparisons
3854:RaoâBlackwellization
3817:Estimating equations
3773:Statistical distance
3491:Factorial experiment
3024:Arithmetic-Geometric
2394:{\displaystyle f(x)}
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2361:{\displaystyle f(x)}
2343:
2025:
1967:
1932:
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1693:{\displaystyle f(x)}
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1661:{\displaystyle O(n)}
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1401:(dag) with vertices
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819:, although commonly
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393:GaussâMarkov theorem
388:Studentized residual
378:Errors and residuals
212:Principal components
182:Nonlinear regression
69:General linear model
5124:Official statistics
5047:Methods engineering
4728:Seasonal adjustment
4496:Poisson regressions
4416:Bayesian regression
4355:Regression analysis
4335:Partial correlation
4307:Regression analysis
3906:Prediction interval
3901:Likelihood interval
3891:Confidence interval
3883:Interval estimation
3844:Unbiased estimators
3662:Model specification
3542:Up-and-down designs
3230:Partial correlation
3186:Index of dispersion
3104:Interquartile range
2827:. New York: Wiley.
2808:. New York: Wiley.
2692:2011JMLR...12.2825P
1884:being nondecreasing
1625:iterative algorithm
1468:totally ordered set
1263: for all
536:{\displaystyle x,y}
459:isotonic regression
238:Errors-in-variables
105:Logistic regression
95:Binomial regression
40:Regression analysis
34:Part of a series on
5269:Numerical analysis
5144:Spatial statistics
5024:Medical statistics
4924:First hitting time
4878:Whittle likelihood
4529:Degrees of freedom
4524:Multivariate ANOVA
4457:Heteroscedasticity
4269:Bayesian estimator
4234:Bayesian inference
4083:KolmogorovâSmirnov
3968:Randomization test
3938:Testing hypotheses
3911:Tolerance interval
3822:Maximum likelihood
3717:Exponential family
3650:Density estimation
3610:Statistical theory
3570:Natural experiment
3516:Scientific control
3433:Survey methodology
3119:Standard deviation
2726:10.1007/bf01580873
2435:10.1007/BF02289694
2391:
2358:
2320:
2315:
2008:
1953:
1918:
1898:
1874:
1854:
1777:
1748:
1690:
1658:
1613:
1545:
1482:
1456:
1426:
1387:
1360:
1287:
1203:
1104:
1036:
996:
938:
918:
862:
842:
809:
776:
726:
699:
664:
560:{\displaystyle x.}
557:
533:
455:numerical analysis
125:Multinomial probit
29:
25:mean squared error
5246:
5245:
5184:
5183:
5180:
5179:
5119:National accounts
5089:Actuarial science
5081:Social statistics
4974:
4973:
4970:
4969:
4966:
4965:
4901:Survival function
4886:
4885:
4748:Granger causality
4589:Contingency table
4564:Survival analysis
4541:
4540:
4537:
4536:
4393:Linear regression
4288:
4287:
4284:
4283:
4259:Credible interval
4228:
4227:
4011:
4010:
3827:Method of moments
3696:Parametric family
3657:Statistical model
3587:
3586:
3583:
3582:
3501:Random assignment
3423:Statistical power
3357:
3356:
3353:
3352:
3202:Contingency table
3172:
3171:
3039:Generalized/power
2834:978-0-471-04970-8
2815:978-0-471-91787-8
2476:978-1-59593-180-1
2295:
2281:
2230:
2213:
2185:
2170:
2107:
2075:
2061:
1996:
1921:{\displaystyle x}
1901:{\displaystyle y}
1877:{\displaystyle f}
1768:
1739:
1466:values fall in a
1264:
1252:
1230:
1171:
1095:
1067:
1048:quadratic program
987:
965:
941:{\displaystyle i}
896:
865:{\displaystyle i}
486:linear regression
447:
446:
100:Binary regression
59:Simple regression
54:Linear regression
5276:
5234:
5233:
5222:
5221:
5211:
5210:
5196:
5195:
5099:Crime statistics
4993:
4992:
4980:
4979:
4897:
4896:
4863:Fourier analysis
4850:Frequency domain
4830:
4777:
4743:Structural break
4703:
4702:
4652:Cluster analysis
4599:Log-linear model
4572:
4571:
4547:
4546:
4488:
4462:Homoscedasticity
4318:
4317:
4294:
4293:
4213:
4205:
4197:
4196:(KruskalâWallis)
4181:
4166:
4121:Cross validation
4106:
4088:AndersonâDarling
4035:
4022:
4021:
3993:Likelihood-ratio
3985:Parametric tests
3963:Permutation test
3946:1- & 2-tails
3837:Minimum distance
3809:Point estimation
3805:
3804:
3756:Optimal decision
3707:
3606:
3605:
3593:
3592:
3575:Quasi-experiment
3525:Adaptive designs
3376:
3375:
3363:
3362:
3240:Rank correlation
3002:
3001:
2993:
2992:
2980:
2979:
2947:
2940:
2933:
2924:
2923:
2919:
2891:
2885:
2877:
2859:
2838:
2819:
2791:
2790:
2772:
2752:
2746:
2745:
2720:(1â3): 425â439.
2705:
2696:
2695:
2685:
2665:
2659:
2658:
2656:
2654:
2644:
2635:
2629:
2628:
2618:
2594:
2588:
2587:
2585:
2583:
2567:
2558:
2557:
2521:
2515:
2514:
2508:
2504:
2502:
2494:
2492:
2491:
2453:
2447:
2446:
2415:
2400:
2398:
2397:
2392:
2367:
2365:
2364:
2359:
2329:
2327:
2326:
2321:
2319:
2318:
2312:
2311:
2296:
2293:
2289:
2288:
2283:
2282:
2274:
2266:
2265:
2241:
2240:
2231:
2228:
2221:
2220:
2215:
2214:
2206:
2199:
2198:
2187:
2186:
2178:
2171:
2169:
2168:
2167:
2155:
2154:
2138:
2137:
2136:
2120:
2115:
2114:
2109:
2108:
2100:
2092:
2091:
2076:
2073:
2069:
2068:
2063:
2062:
2054:
2017:
2015:
2014:
2009:
2004:
2003:
1998:
1997:
1989:
1982:
1981:
1962:
1960:
1959:
1954:
1952:
1944:
1943:
1927:
1925:
1924:
1919:
1907:
1905:
1904:
1899:
1883:
1881:
1880:
1875:
1863:
1861:
1860:
1855:
1853:
1852:
1843:
1842:
1827:
1826:
1808:
1807:
1797:
1792:
1776:
1757:
1755:
1754:
1749:
1747:
1746:
1741:
1740:
1732:
1722:
1721:
1699:
1697:
1696:
1691:
1667:
1665:
1664:
1659:
1622:
1620:
1619:
1614:
1554:
1552:
1551:
1546:
1544:
1543:
1525:
1524:
1512:
1511:
1492:, we may assume
1491:
1489:
1488:
1483:
1481:
1465:
1463:
1462:
1457:
1455:
1454:
1435:
1433:
1432:
1427:
1396:
1394:
1393:
1388:
1386:
1385:
1369:
1367:
1366:
1361:
1356:
1355:
1343:
1342:
1296:
1294:
1293:
1288:
1265:
1262:
1260:
1259:
1254:
1253:
1245:
1238:
1237:
1232:
1231:
1223:
1212:
1210:
1209:
1204:
1202:
1201:
1192:
1191:
1179:
1178:
1173:
1172:
1164:
1157:
1156:
1146:
1141:
1113:
1111:
1110:
1105:
1103:
1102:
1097:
1096:
1088:
1075:
1074:
1069:
1068:
1060:
1045:
1043:
1042:
1037:
1035:
1034:
1022:
1021:
1005:
1003:
1002:
997:
995:
994:
989:
988:
980:
973:
972:
967:
966:
958:
947:
945:
944:
939:
927:
925:
924:
919:
917:
916:
904:
903:
898:
897:
889:
871:
869:
868:
863:
851:
849:
848:
843:
835:
834:
818:
816:
815:
810:
802:
801:
785:
783:
782:
777:
772:
771:
759:
758:
735:
733:
732:
727:
725:
724:
708:
706:
705:
700:
698:
690:
689:
673:
671:
670:
665:
660:
659:
647:
646:
622:
621:
609:
608:
566:
564:
563:
558:
542:
540:
539:
534:
439:
432:
425:
409:
408:
316:Ridge regression
151:Multilevel model
31:
30:
5284:
5283:
5279:
5278:
5277:
5275:
5274:
5273:
5249:
5248:
5247:
5242:
5205:
5176:
5138:
5075:
5061:quality control
5028:
5010:Clinical trials
4987:
4962:
4946:
4934:Hazard function
4928:
4882:
4844:
4828:
4791:
4787:BreuschâGodfrey
4775:
4752:
4692:
4667:Factor analysis
4613:
4594:Graphical model
4566:
4533:
4500:
4486:
4466:
4420:
4387:
4349:
4312:
4311:
4280:
4224:
4211:
4203:
4195:
4179:
4164:
4143:Rank statistics
4137:
4116:Model selection
4104:
4062:Goodness of fit
4056:
4033:
4007:
3979:
3932:
3877:
3866:Median unbiased
3794:
3705:
3638:Order statistic
3600:
3579:
3546:
3520:
3472:
3427:
3370:
3368:Data collection
3349:
3261:
3216:
3190:
3168:
3128:
3080:
2997:Continuous data
2987:
2974:
2956:
2951:
2879:
2878:
2857:10.1.1.338.3846
2835:
2816:
2800:
2798:Further reading
2795:
2794:
2753:
2749:
2706:
2699:
2666:
2662:
2652:
2650:
2642:
2636:
2632:
2595:
2591:
2581:
2579:
2572:"Package 'cir'"
2568:
2561:
2522:
2518:
2506:
2505:
2496:
2495:
2489:
2487:
2477:
2455:
2454:
2450:
2416:
2412:
2407:
2377:
2374:
2373:
2344:
2341:
2340:
2336:
2314:
2313:
2307:
2303:
2292:
2290:
2284:
2273:
2272:
2271:
2268:
2267:
2255:
2251:
2236:
2232:
2227:
2225:
2216:
2205:
2204:
2203:
2188:
2177:
2176:
2175:
2163:
2159:
2144:
2140:
2139:
2132:
2128:
2121:
2119:
2110:
2099:
2098:
2097:
2094:
2093:
2087:
2083:
2072:
2070:
2064:
2053:
2052:
2051:
2044:
2043:
2026:
2023:
2022:
1999:
1988:
1987:
1986:
1977:
1973:
1968:
1965:
1964:
1948:
1939:
1935:
1933:
1930:
1929:
1913:
1910:
1909:
1893:
1890:
1889:
1869:
1866:
1865:
1848:
1844:
1838:
1834:
1822:
1818:
1803:
1799:
1793:
1782:
1772:
1766:
1763:
1762:
1742:
1731:
1730:
1729:
1717:
1713:
1705:
1702:
1701:
1676:
1673:
1672:
1644:
1641:
1640:
1560:
1557:
1556:
1539:
1535:
1520:
1516:
1507:
1503:
1501:
1498:
1497:
1477:
1475:
1472:
1471:
1450:
1446:
1444:
1441:
1440:
1406:
1403:
1402:
1381:
1377:
1375:
1372:
1371:
1351:
1347:
1338:
1334:
1305:
1302:
1301:
1261:
1255:
1244:
1243:
1242:
1233:
1222:
1221:
1220:
1218:
1215:
1214:
1197:
1193:
1187:
1183:
1174:
1163:
1162:
1161:
1152:
1148:
1142:
1131:
1122:
1119:
1118:
1098:
1087:
1086:
1085:
1070:
1059:
1058:
1057:
1055:
1052:
1051:
1030:
1026:
1017:
1013:
1011:
1008:
1007:
990:
979:
978:
977:
968:
957:
956:
955:
953:
950:
949:
933:
930:
929:
912:
908:
899:
888:
887:
886:
884:
881:
880:
857:
854:
853:
830:
826:
824:
821:
820:
797:
793:
791:
788:
787:
767:
763:
754:
750:
745:
742:
741:
720:
716:
714:
711:
710:
694:
685:
681:
679:
676:
675:
655:
651:
642:
638:
617:
613:
604:
600:
595:
592:
591:
588:
549:
546:
545:
522:
519:
518:
478:
443:
403:
383:Goodness of fit
90:Discrete choice
17:
12:
11:
5:
5282:
5272:
5271:
5266:
5261:
5244:
5243:
5241:
5240:
5228:
5216:
5202:
5189:
5186:
5185:
5182:
5181:
5178:
5177:
5175:
5174:
5169:
5164:
5159:
5154:
5148:
5146:
5140:
5139:
5137:
5136:
5131:
5126:
5121:
5116:
5111:
5106:
5101:
5096:
5091:
5085:
5083:
5077:
5076:
5074:
5073:
5068:
5063:
5054:
5049:
5044:
5038:
5036:
5030:
5029:
5027:
5026:
5021:
5016:
5007:
5005:Bioinformatics
5001:
4999:
4989:
4988:
4976:
4975:
4972:
4971:
4968:
4967:
4964:
4963:
4961:
4960:
4954:
4952:
4948:
4947:
4945:
4944:
4938:
4936:
4930:
4929:
4927:
4926:
4921:
4916:
4911:
4905:
4903:
4894:
4888:
4887:
4884:
4883:
4881:
4880:
4875:
4870:
4865:
4860:
4854:
4852:
4846:
4845:
4843:
4842:
4837:
4832:
4824:
4819:
4814:
4813:
4812:
4810:partial (PACF)
4801:
4799:
4793:
4792:
4790:
4789:
4784:
4779:
4771:
4766:
4760:
4758:
4757:Specific tests
4754:
4753:
4751:
4750:
4745:
4740:
4735:
4730:
4725:
4720:
4715:
4709:
4707:
4700:
4694:
4693:
4691:
4690:
4689:
4688:
4687:
4686:
4671:
4670:
4669:
4659:
4657:Classification
4654:
4649:
4644:
4639:
4634:
4629:
4623:
4621:
4615:
4614:
4612:
4611:
4606:
4604:McNemar's test
4601:
4596:
4591:
4586:
4580:
4578:
4568:
4567:
4543:
4542:
4539:
4538:
4535:
4534:
4532:
4531:
4526:
4521:
4516:
4510:
4508:
4502:
4501:
4499:
4498:
4482:
4476:
4474:
4468:
4467:
4465:
4464:
4459:
4454:
4449:
4444:
4442:Semiparametric
4439:
4434:
4428:
4426:
4422:
4421:
4419:
4418:
4413:
4408:
4403:
4397:
4395:
4389:
4388:
4386:
4385:
4380:
4375:
4370:
4365:
4359:
4357:
4351:
4350:
4348:
4347:
4342:
4337:
4332:
4326:
4324:
4314:
4313:
4310:
4309:
4304:
4298:
4290:
4289:
4286:
4285:
4282:
4281:
4279:
4278:
4277:
4276:
4266:
4261:
4256:
4255:
4254:
4249:
4238:
4236:
4230:
4229:
4226:
4225:
4223:
4222:
4217:
4216:
4215:
4207:
4199:
4183:
4180:(MannâWhitney)
4175:
4174:
4173:
4160:
4159:
4158:
4147:
4145:
4139:
4138:
4136:
4135:
4134:
4133:
4128:
4123:
4113:
4108:
4105:(ShapiroâWilk)
4100:
4095:
4090:
4085:
4080:
4072:
4066:
4064:
4058:
4057:
4055:
4054:
4046:
4037:
4025:
4019:
4017:Specific tests
4013:
4012:
4009:
4008:
4006:
4005:
4000:
3995:
3989:
3987:
3981:
3980:
3978:
3977:
3972:
3971:
3970:
3960:
3959:
3958:
3948:
3942:
3940:
3934:
3933:
3931:
3930:
3929:
3928:
3923:
3913:
3908:
3903:
3898:
3893:
3887:
3885:
3879:
3878:
3876:
3875:
3870:
3869:
3868:
3863:
3862:
3861:
3856:
3841:
3840:
3839:
3834:
3829:
3824:
3813:
3811:
3802:
3796:
3795:
3793:
3792:
3787:
3782:
3781:
3780:
3770:
3765:
3764:
3763:
3753:
3752:
3751:
3746:
3741:
3731:
3726:
3721:
3720:
3719:
3714:
3709:
3693:
3692:
3691:
3686:
3681:
3671:
3670:
3669:
3664:
3654:
3653:
3652:
3642:
3641:
3640:
3630:
3625:
3620:
3614:
3612:
3602:
3601:
3589:
3588:
3585:
3584:
3581:
3580:
3578:
3577:
3572:
3567:
3562:
3556:
3554:
3548:
3547:
3545:
3544:
3539:
3534:
3528:
3526:
3522:
3521:
3519:
3518:
3513:
3508:
3503:
3498:
3493:
3488:
3482:
3480:
3474:
3473:
3471:
3470:
3468:Standard error
3465:
3460:
3455:
3454:
3453:
3448:
3437:
3435:
3429:
3428:
3426:
3425:
3420:
3415:
3410:
3405:
3400:
3398:Optimal design
3395:
3390:
3384:
3382:
3372:
3371:
3359:
3358:
3355:
3354:
3351:
3350:
3348:
3347:
3342:
3337:
3332:
3327:
3322:
3317:
3312:
3307:
3302:
3297:
3292:
3287:
3282:
3277:
3271:
3269:
3263:
3262:
3260:
3259:
3254:
3253:
3252:
3247:
3237:
3232:
3226:
3224:
3218:
3217:
3215:
3214:
3209:
3204:
3198:
3196:
3195:Summary tables
3192:
3191:
3189:
3188:
3182:
3180:
3174:
3173:
3170:
3169:
3167:
3166:
3165:
3164:
3159:
3154:
3144:
3138:
3136:
3130:
3129:
3127:
3126:
3121:
3116:
3111:
3106:
3101:
3096:
3090:
3088:
3082:
3081:
3079:
3078:
3073:
3068:
3067:
3066:
3061:
3056:
3051:
3046:
3041:
3036:
3031:
3029:Contraharmonic
3026:
3021:
3010:
3008:
2999:
2989:
2988:
2976:
2975:
2973:
2972:
2967:
2961:
2958:
2957:
2950:
2949:
2942:
2935:
2927:
2921:
2920:
2910:(3): 793â804.
2892:
2850:(1): 159â175.
2839:
2833:
2820:
2814:
2799:
2796:
2793:
2792:
2763:(3): 258â267.
2747:
2697:
2660:
2630:
2589:
2559:
2532:(1): 171â177.
2516:
2507:|website=
2475:
2448:
2429:(2): 115â129.
2419:Kruskal, J. B.
2409:
2408:
2406:
2403:
2390:
2387:
2384:
2381:
2357:
2354:
2351:
2348:
2335:
2332:
2331:
2330:
2317:
2310:
2306:
2302:
2299:
2291:
2287:
2280:
2277:
2270:
2269:
2264:
2261:
2258:
2254:
2250:
2247:
2244:
2239:
2235:
2226:
2224:
2219:
2212:
2209:
2202:
2197:
2194:
2191:
2184:
2181:
2174:
2166:
2162:
2158:
2153:
2150:
2147:
2143:
2135:
2131:
2127:
2124:
2118:
2113:
2106:
2103:
2096:
2095:
2090:
2086:
2082:
2079:
2071:
2067:
2060:
2057:
2050:
2049:
2047:
2042:
2039:
2036:
2033:
2030:
2007:
2002:
1995:
1992:
1985:
1980:
1976:
1972:
1951:
1947:
1942:
1938:
1917:
1897:
1886:
1885:
1873:
1851:
1847:
1841:
1837:
1833:
1830:
1825:
1821:
1817:
1814:
1811:
1806:
1802:
1796:
1791:
1788:
1785:
1781:
1775:
1771:
1745:
1738:
1735:
1728:
1725:
1720:
1716:
1712:
1709:
1689:
1686:
1683:
1680:
1657:
1654:
1651:
1648:
1612:
1609:
1606:
1603:
1600:
1597:
1594:
1591:
1588:
1585:
1582:
1579:
1576:
1573:
1570:
1567:
1564:
1542:
1538:
1534:
1531:
1528:
1523:
1519:
1515:
1510:
1506:
1480:
1453:
1449:
1425:
1422:
1419:
1416:
1413:
1410:
1384:
1380:
1359:
1354:
1350:
1346:
1341:
1337:
1333:
1330:
1327:
1324:
1321:
1318:
1315:
1312:
1309:
1298:
1297:
1286:
1283:
1280:
1277:
1274:
1271:
1268:
1258:
1251:
1248:
1241:
1236:
1229:
1226:
1200:
1196:
1190:
1186:
1182:
1177:
1170:
1167:
1160:
1155:
1151:
1145:
1140:
1137:
1134:
1130:
1126:
1101:
1094:
1091:
1084:
1081:
1078:
1073:
1066:
1063:
1033:
1029:
1025:
1020:
1016:
993:
986:
983:
976:
971:
964:
961:
937:
915:
911:
907:
902:
895:
892:
861:
841:
838:
833:
829:
808:
805:
800:
796:
775:
770:
766:
762:
757:
753:
749:
723:
719:
697:
693:
688:
684:
663:
658:
654:
650:
645:
641:
637:
634:
631:
628:
625:
620:
616:
612:
607:
603:
599:
587:
584:
556:
553:
532:
529:
526:
477:
474:
470:non-decreasing
445:
444:
442:
441:
434:
427:
419:
416:
415:
414:
413:
398:
397:
396:
395:
390:
385:
380:
375:
370:
362:
361:
357:
356:
355:
354:
349:
344:
339:
334:
326:
325:
324:
323:
318:
313:
308:
303:
295:
294:
293:
292:
287:
282:
277:
269:
268:
267:
266:
261:
256:
248:
247:
243:
242:
241:
240:
232:
231:
230:
229:
224:
219:
214:
209:
204:
199:
194:
192:Semiparametric
189:
184:
176:
175:
174:
173:
168:
163:
161:Random effects
158:
153:
145:
144:
143:
142:
137:
135:Ordered probit
132:
127:
122:
117:
112:
107:
102:
97:
92:
87:
82:
74:
73:
72:
71:
66:
61:
56:
48:
47:
43:
42:
36:
35:
15:
9:
6:
4:
3:
2:
5281:
5270:
5267:
5265:
5262:
5260:
5257:
5256:
5254:
5239:
5238:
5229:
5227:
5226:
5217:
5215:
5214:
5209:
5203:
5201:
5200:
5191:
5190:
5187:
5173:
5170:
5168:
5167:Geostatistics
5165:
5163:
5160:
5158:
5155:
5153:
5150:
5149:
5147:
5145:
5141:
5135:
5134:Psychometrics
5132:
5130:
5127:
5125:
5122:
5120:
5117:
5115:
5112:
5110:
5107:
5105:
5102:
5100:
5097:
5095:
5092:
5090:
5087:
5086:
5084:
5082:
5078:
5072:
5069:
5067:
5064:
5062:
5058:
5055:
5053:
5050:
5048:
5045:
5043:
5040:
5039:
5037:
5035:
5031:
5025:
5022:
5020:
5017:
5015:
5011:
5008:
5006:
5003:
5002:
5000:
4998:
4997:Biostatistics
4994:
4990:
4986:
4981:
4977:
4959:
4958:Log-rank test
4956:
4955:
4953:
4949:
4943:
4940:
4939:
4937:
4935:
4931:
4925:
4922:
4920:
4917:
4915:
4912:
4910:
4907:
4906:
4904:
4902:
4898:
4895:
4893:
4889:
4879:
4876:
4874:
4871:
4869:
4866:
4864:
4861:
4859:
4856:
4855:
4853:
4851:
4847:
4841:
4838:
4836:
4833:
4831:
4829:(BoxâJenkins)
4825:
4823:
4820:
4818:
4815:
4811:
4808:
4807:
4806:
4803:
4802:
4800:
4798:
4794:
4788:
4785:
4783:
4782:DurbinâWatson
4780:
4778:
4772:
4770:
4767:
4765:
4764:DickeyâFuller
4762:
4761:
4759:
4755:
4749:
4746:
4744:
4741:
4739:
4738:Cointegration
4736:
4734:
4731:
4729:
4726:
4724:
4721:
4719:
4716:
4714:
4713:Decomposition
4711:
4710:
4708:
4704:
4701:
4699:
4695:
4685:
4682:
4681:
4680:
4677:
4676:
4675:
4672:
4668:
4665:
4664:
4663:
4660:
4658:
4655:
4653:
4650:
4648:
4645:
4643:
4640:
4638:
4635:
4633:
4630:
4628:
4625:
4624:
4622:
4620:
4616:
4610:
4607:
4605:
4602:
4600:
4597:
4595:
4592:
4590:
4587:
4585:
4584:Cohen's kappa
4582:
4581:
4579:
4577:
4573:
4569:
4565:
4561:
4557:
4553:
4548:
4544:
4530:
4527:
4525:
4522:
4520:
4517:
4515:
4512:
4511:
4509:
4507:
4503:
4497:
4493:
4489:
4483:
4481:
4478:
4477:
4475:
4473:
4469:
4463:
4460:
4458:
4455:
4453:
4450:
4448:
4445:
4443:
4440:
4438:
4437:Nonparametric
4435:
4433:
4430:
4429:
4427:
4423:
4417:
4414:
4412:
4409:
4407:
4404:
4402:
4399:
4398:
4396:
4394:
4390:
4384:
4381:
4379:
4376:
4374:
4371:
4369:
4366:
4364:
4361:
4360:
4358:
4356:
4352:
4346:
4343:
4341:
4338:
4336:
4333:
4331:
4328:
4327:
4325:
4323:
4319:
4315:
4308:
4305:
4303:
4300:
4299:
4295:
4291:
4275:
4272:
4271:
4270:
4267:
4265:
4262:
4260:
4257:
4253:
4250:
4248:
4245:
4244:
4243:
4240:
4239:
4237:
4235:
4231:
4221:
4218:
4214:
4208:
4206:
4200:
4198:
4192:
4191:
4190:
4187:
4186:Nonparametric
4184:
4182:
4176:
4172:
4169:
4168:
4167:
4161:
4157:
4156:Sample median
4154:
4153:
4152:
4149:
4148:
4146:
4144:
4140:
4132:
4129:
4127:
4124:
4122:
4119:
4118:
4117:
4114:
4112:
4109:
4107:
4101:
4099:
4096:
4094:
4091:
4089:
4086:
4084:
4081:
4079:
4077:
4073:
4071:
4068:
4067:
4065:
4063:
4059:
4053:
4051:
4047:
4045:
4043:
4038:
4036:
4031:
4027:
4026:
4023:
4020:
4018:
4014:
4004:
4001:
3999:
3996:
3994:
3991:
3990:
3988:
3986:
3982:
3976:
3973:
3969:
3966:
3965:
3964:
3961:
3957:
3954:
3953:
3952:
3949:
3947:
3944:
3943:
3941:
3939:
3935:
3927:
3924:
3922:
3919:
3918:
3917:
3914:
3912:
3909:
3907:
3904:
3902:
3899:
3897:
3894:
3892:
3889:
3888:
3886:
3884:
3880:
3874:
3871:
3867:
3864:
3860:
3857:
3855:
3852:
3851:
3850:
3847:
3846:
3845:
3842:
3838:
3835:
3833:
3830:
3828:
3825:
3823:
3820:
3819:
3818:
3815:
3814:
3812:
3810:
3806:
3803:
3801:
3797:
3791:
3788:
3786:
3783:
3779:
3776:
3775:
3774:
3771:
3769:
3766:
3762:
3761:loss function
3759:
3758:
3757:
3754:
3750:
3747:
3745:
3742:
3740:
3737:
3736:
3735:
3732:
3730:
3727:
3725:
3722:
3718:
3715:
3713:
3710:
3708:
3702:
3699:
3698:
3697:
3694:
3690:
3687:
3685:
3682:
3680:
3677:
3676:
3675:
3672:
3668:
3665:
3663:
3660:
3659:
3658:
3655:
3651:
3648:
3647:
3646:
3643:
3639:
3636:
3635:
3634:
3631:
3629:
3626:
3624:
3621:
3619:
3616:
3615:
3613:
3611:
3607:
3603:
3599:
3594:
3590:
3576:
3573:
3571:
3568:
3566:
3563:
3561:
3558:
3557:
3555:
3553:
3549:
3543:
3540:
3538:
3535:
3533:
3530:
3529:
3527:
3523:
3517:
3514:
3512:
3509:
3507:
3504:
3502:
3499:
3497:
3494:
3492:
3489:
3487:
3484:
3483:
3481:
3479:
3475:
3469:
3466:
3464:
3463:Questionnaire
3461:
3459:
3456:
3452:
3449:
3447:
3444:
3443:
3442:
3439:
3438:
3436:
3434:
3430:
3424:
3421:
3419:
3416:
3414:
3411:
3409:
3406:
3404:
3401:
3399:
3396:
3394:
3391:
3389:
3386:
3385:
3383:
3381:
3377:
3373:
3369:
3364:
3360:
3346:
3343:
3341:
3338:
3336:
3333:
3331:
3328:
3326:
3323:
3321:
3318:
3316:
3313:
3311:
3308:
3306:
3303:
3301:
3298:
3296:
3293:
3291:
3290:Control chart
3288:
3286:
3283:
3281:
3278:
3276:
3273:
3272:
3270:
3268:
3264:
3258:
3255:
3251:
3248:
3246:
3243:
3242:
3241:
3238:
3236:
3233:
3231:
3228:
3227:
3225:
3223:
3219:
3213:
3210:
3208:
3205:
3203:
3200:
3199:
3197:
3193:
3187:
3184:
3183:
3181:
3179:
3175:
3163:
3160:
3158:
3155:
3153:
3150:
3149:
3148:
3145:
3143:
3140:
3139:
3137:
3135:
3131:
3125:
3122:
3120:
3117:
3115:
3112:
3110:
3107:
3105:
3102:
3100:
3097:
3095:
3092:
3091:
3089:
3087:
3083:
3077:
3074:
3072:
3069:
3065:
3062:
3060:
3057:
3055:
3052:
3050:
3047:
3045:
3042:
3040:
3037:
3035:
3032:
3030:
3027:
3025:
3022:
3020:
3017:
3016:
3015:
3012:
3011:
3009:
3007:
3003:
3000:
2998:
2994:
2990:
2986:
2981:
2977:
2971:
2968:
2966:
2963:
2962:
2959:
2955:
2948:
2943:
2941:
2936:
2934:
2929:
2928:
2925:
2917:
2913:
2909:
2905:
2901:
2900:Woodroofe, M.
2897:
2893:
2889:
2883:
2875:
2871:
2867:
2863:
2858:
2853:
2849:
2845:
2840:
2836:
2830:
2826:
2821:
2817:
2811:
2807:
2802:
2801:
2788:
2784:
2780:
2776:
2771:
2766:
2762:
2758:
2751:
2743:
2739:
2735:
2731:
2727:
2723:
2719:
2715:
2711:
2704:
2702:
2693:
2689:
2684:
2679:
2676:: 2825â2830.
2675:
2671:
2664:
2648:
2641:
2634:
2626:
2622:
2617:
2612:
2608:
2604:
2600:
2593:
2577:
2573:
2570:Oron, Assaf.
2566:
2564:
2555:
2551:
2547:
2543:
2539:
2535:
2531:
2527:
2520:
2512:
2500:
2486:
2482:
2478:
2472:
2468:
2464:
2460:
2459:
2452:
2444:
2440:
2436:
2432:
2428:
2424:
2423:Psychometrika
2420:
2414:
2410:
2402:
2385:
2379:
2371:
2352:
2346:
2308:
2304:
2300:
2297:
2285:
2275:
2262:
2259:
2256:
2252:
2248:
2245:
2242:
2237:
2233:
2217:
2207:
2200:
2195:
2192:
2189:
2179:
2164:
2160:
2156:
2151:
2148:
2145:
2141:
2133:
2129:
2125:
2122:
2116:
2111:
2101:
2088:
2084:
2080:
2077:
2065:
2055:
2045:
2040:
2034:
2028:
2021:
2020:
2019:
2000:
1990:
1983:
1978:
1974:
1945:
1940:
1936:
1915:
1895:
1871:
1849:
1839:
1835:
1831:
1823:
1819:
1812:
1804:
1800:
1794:
1789:
1786:
1783:
1779:
1773:
1761:
1760:
1759:
1743:
1733:
1726:
1718:
1714:
1707:
1684:
1678:
1669:
1652:
1646:
1638:
1634:
1630:
1626:
1607:
1604:
1601:
1598:
1595:
1592:
1586:
1583:
1580:
1577:
1574:
1565:
1562:
1540:
1536:
1532:
1529:
1526:
1521:
1517:
1513:
1508:
1504:
1495:
1469:
1451:
1447:
1437:
1423:
1420:
1417:
1414:
1411:
1408:
1400:
1382:
1378:
1352:
1348:
1344:
1339:
1335:
1331:
1325:
1322:
1319:
1310:
1307:
1284:
1281:
1275:
1272:
1269:
1256:
1246:
1239:
1234:
1224:
1198:
1188:
1184:
1180:
1175:
1165:
1153:
1149:
1143:
1138:
1135:
1132:
1128:
1117:
1116:
1115:
1099:
1089:
1082:
1079:
1076:
1071:
1061:
1049:
1031:
1027:
1023:
1018:
1014:
991:
981:
974:
969:
959:
935:
913:
909:
905:
900:
890:
878:
877:least-squares
873:
859:
839:
836:
831:
827:
806:
803:
798:
794:
768:
764:
760:
755:
751:
739:
736:fall in some
721:
717:
691:
686:
682:
656:
652:
648:
643:
639:
632:
629:
626:
618:
614:
610:
605:
601:
583:
581:
577:
573:
568:
554:
551:
530:
527:
524:
515:
513:
509:
504:
502:
498:
494:
489:
487:
483:
473:
471:
467:
465:
460:
456:
452:
440:
435:
433:
428:
426:
421:
420:
418:
417:
412:
407:
402:
401:
400:
399:
394:
391:
389:
386:
384:
381:
379:
376:
374:
371:
369:
366:
365:
364:
363:
359:
358:
353:
350:
348:
345:
343:
340:
338:
335:
333:
330:
329:
328:
327:
322:
319:
317:
314:
312:
309:
307:
304:
302:
299:
298:
297:
296:
291:
288:
286:
283:
281:
278:
276:
273:
272:
271:
270:
265:
262:
260:
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130:Ordered logit
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5109:Econometrics
5059: /
5042:Chemometrics
5019:Epidemiology
5012: /
4985:Applications
4827:ARIMA model
4774:Q-statistic
4723:Stationarity
4619:Multivariate
4562: /
4558: /
4556:Multivariate
4554: /
4494: /
4490: /
4446:
4264:Bayes factor
4163:Signed rank
4075:
4049:
4041:
4029:
3724:Completeness
3560:Cohort study
3458:Opinion poll
3393:Missing data
3380:Study design
3335:Scatter plot
3257:Scatter plot
3250:Spearman's Ï
3212:Grouped data
2907:
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2882:cite journal
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5237:WikiProject
5152:Cartography
5114:Jurimetrics
5066:Reliability
4797:Time domain
4776:(LjungâBox)
4698:Time-series
4576:Categorical
4560:Time-series
4552:Categorical
4487:(Bernoulli)
4322:Correlation
4302:Correlation
4098:JarqueâBera
4070:Chi-squared
3832:M-estimator
3785:Asymptotics
3729:Sufficiency
3496:Interaction
3408:Replication
3388:Effect size
3345:Violin plot
3325:Radar chart
3305:Forest plot
3295:Correlogram
3245:Kendall's Ï
2609:(5): 1â24.
2582:26 December
1864:subject to
1555:, and take
1213:subject to
321:Regularized
285:Generalized
217:Least angle
115:Mixed logit
5253:Categories
5104:Demography
4822:ARMA model
4627:Regression
4204:(Friedman)
4165:(Wilcoxon)
4103:Normality
4093:Lilliefors
4040:Student's
3916:Resampling
3790:Robustness
3778:divergence
3768:Efficiency
3706:(monotone)
3701:Likelihood
3618:Population
3451:Stratified
3403:Population
3222:Dependence
3178:Count data
3109:Percentile
3086:Dispersion
3019:Arithmetic
2954:Statistics
2904:Biometrika
2770:1701.05964
2653:29 October
2526:Biometrics
2490:2020-07-07
2405:References
1700:such that
1633:active set
466:regression
451:statistics
360:Background
264:Non-linear
246:Estimation
4485:Logistic
4252:posterior
4178:Rank sum
3926:Jackknife
3921:Bootstrap
3739:Bootstrap
3674:Parameter
3623:Statistic
3418:Statistic
3330:Run chart
3315:Pie chart
3310:Histogram
3300:Fan chart
3275:Bar chart
3157:L-moments
3044:Geometric
2896:Wu, W. B.
2874:119761196
2852:CiteSeerX
2734:0025-5610
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464:monotonic
227:Segmented
5199:Category
4892:Survival
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4492:Binomial
4447:Isotonic
4034:(normal)
3679:location
3486:Blocking
3441:Sampling
3320:QâQ plot
3285:Box plot
3267:Graphics
3162:Skewness
3152:Kurtosis
3124:Variance
3054:Heronian
3049:Harmonic
2787:88521189
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2443:11709679
2294:if
2229:if
2074:if
1470:such as
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852:for all
709:and the
514:models.
342:Bayesian
280:Weighted
275:Ordinary
207:Isotonic
202:Quantile
5225:Commons
5172:Kriging
5057:Process
5014:studies
4873:Wavelet
4706:General
3873:Plug-in
3667:L space
3446:Cluster
3147:Moments
2965:Outline
2688:Bibcode
2554:8743090
301:Partial
140:Poisson
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4452:Robust
4202:2-way
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3134:Shape
3114:Range
3059:Heinz
3034:Cubic
2970:Index
2870:S2CID
2783:S2CID
2765:arXiv
2738:S2CID
2678:arXiv
2643:(PDF)
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576:Stata
306:Total
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4951:Test
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2888:link
2829:ISBN
2810:ISBN
2730:ISSN
2655:2021
2647:CRAN
2621:ISSN
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