How to cite this WIREs title:
WIREs Comp Stat

Hypothesis tests with precedence probabilities and precedence‐type tests

Overview

Rajarshi Dey

Published Online: Sep 28 2017

DOI: 10.1002/wics.1417

Full article on Wiley Online Library: HTML PDF

Can't access this content? Tell your librarian.

Precedence probabilities are important tools in a statistician's toolkit. Precedence probabilities can be defined as the probability of observing single samples from K populations in a particular order. Noting that there are K ! possible orders of K populations; these K ! parameters are a useful way to measure the effectiveness of a classifier (AUC/VUS/HUM). Receiver operating characteristic (ROC) curve/surface/manifold, which can be generated by any classifier leads to calculation of the area under curve (AUC)/volume under surface (VUS)/hyper‐volume under manifold (HUM) can be approximated by a single precedence probability and can be nonparametrically estimated via rank‐based U‐statistic. Precedence probabilities can also be used to test equality of K > 2 distribution functions. Hypothesis tests related to both these problems mentioned above are discussed. On the other hand, when we are interested in testing if the K distributions are stochastically ordered, we perform a precedence‐type test. Different nonparametric tests are also discussed in relation to precedence‐type tests. WIREs Comput Stat 2018, 10:e1417. doi: 10.1002/wics.1417 This article is categorized under: Statistical and Graphical Methods of Data Analysis > Nonparametric Methods

References

Chakraborti, S, van der Laan, P. Precedence probability and prediction intervals. J R Stat Soc Series D (The Statistician) 2000, 49:219–228.
Ellis, PD. The Essential Guide to Effect Sizes. Cambridge, UK: Cambridge University Press; 2010.
Dey, R. Inference for the K‐sample problem based on precedence probabilities. Kansas State University, 2011.
Coles, S. An Introduction to Statistical Modeling of Extreme Values. London, UK: Springer; 2001.
Patil, GP, Boswell, MT. A characteristic property of the multivariate normal density function and some of its applications. Ann Math Stat 1970, 41:1970–1977.
Hoeffding, W. A class of statistic with asymptotically normal distributions. Ann Math Stat 1948, 19:293–325.
Randles, RH, Wolfe, DA. Introduction to the Theory of Nonparametric Statistics. New York, NY: John Wiley %26 Sons; 1979.
Green, D, Swets, J. Signal Detection Theory and Psychophysics. New York, NY: John Wiley %26 Sons; 1966.
Hanley, JA, McNeil, BJ. The meaning and use of the area under a receiver operating characteristic (ROC) curve. Radiology 1982, 143:29–36.
Pepe, MS. The Statistical Evaluation of Medical Tests for Classification and Prediction. Oxford: xxx; 2003.
Spackman, KA. Signal Detection Theory: Valuable Tools for Evaluating Inductive Learning. Proceedings of the Sixth International Workshop on Machine Learning. San Mateo, CA: Morgan Kaufmann; 1989.
Zahiri, J, Bozorgmehr, JH, Masoudi‐Nejad, A. Computational prediction of protein‐protein interaction networks: Algorithms and resources. Curr Genomics 2013, 14:397–414. https://doi.org/10.2174/1389202911314060004.
Mann, HB, Whitney, DR. On a test of whether one or two random variables is stochastically larger than the other. Ann Math Stat 1947, 18:50–60.
Hodges, JL, Lehmann, EL. The efficiency of some nonparametric competitors of the t‐test. Ann Math Stat 1956, 27:324–335.
Xiong, C, VanBelle, G, Miller, JP, Morris, JC. Measuring and estimating diagnostic accuracy when there are three ordinal diagnostic groups. Stat Med 2006, 25:1251–1273.
Zhang, Y, Li, J. Combining multiple markers for multi‐category classification: an ROC surface approach. Aust N Z J Stat 2011, 53:63–78.
Mossman, D. Three‐way ROCs medical decision making. Med Decis Making 1999, 19:79–89.
Nakas, CT, Yiannoutsos, CT. Ordered multiple‐class ROC analysis with continuous measurements. Stat Med 2004, 23:3437–3449.
Kolmogorov, AN. Sulla determinazione empirica di une legge di distribuzione. Giornale dell`istituto Italiano degli attuari 1933, 4:83–91.
Massey, FJ. The Kolmogorov–Smirnoff test of goodness of fit. J Am Stat Assoc 1951, 46:68–78.
Anderson, TW, Darling, DA. Asymptotic theory of certain goodness‐of‐fit criteria based on stochastic processes. Ann Math Stat 1952, 23:193–212.
Darling, DA. The Kolmogorov–Smirnoff, Cramer–von Mises tests. Ann Math Stat 1957, 28:823–838.
Pettitt, AN. A two‐sample Anderson–Darling rank statistic. Biometrika 1976, 63:161–168.
Cramer, H. On the composition of elementary errors. Scand Actuar J 1928, 1:13–74.
von Mises, RE. Wahrscheinlichkeit Statistik und Wahrheit. Berlin, Germany: Springer; 1928.
Anderson, TW. On the distribution of two‐sample Cramer–von Mises criterion. Ann Math Stat 1962, 33:1148–1159.
Razali, NM, Wah, YB. Power comparisons of Shapiro–Wilk, Kolmogorov–Smirnov, Lilliefors and Anderson–Darling tests. J Stat Model Analy 2011, 2:21–33.
Kruskal, WH, Wallis, WA. Use of ranks in one‐criterion variance analysis. J Am Stat Assoc 1952, 47:583–621.
Brown, GW, Mood, AM. On median tests for linear hypotheses. In: Proceedings of the 2nd Berkeley Symposium on Mathematics, Statistics and Probability, vol. 2, 1951, 159–166.
van der Waerden, BL. Order tests for the two‐sample problem and their power. Indag Math 1952, 14:453–458.
Bland, JM, Altman, DG. The logrank test. BMJ 2004, 328:1073. https://doi.org/10.1136/bmj.328.7447.1073.
Scholz, FW, Stephens, MA. K‐sample Anderson–Darling tests. J Am Stat Assoc 1987, 82:918–924.
Conover, WJ. Several K‐sample Kolmogorov–Smirnov tests. Ann Math Stat 1965, 36:1019–1026.
Rizzo, ML, Szekely, GJ. DISCO analysis: a nonparametric extension of analysis of variance. Ann Appl Stat 2010, 4:1034–1055.
Ross, SM. Stochastic Processes. Toronto, Canada: John Wiley %26 Sons; 1996.
Jonckheere, AR. A distribution‐free K‐sample test against ordered alternatives. Biometrika 1954, 41:134–145.
Cuzick, J. A Wilcoxon‐type test for trend. Stat Med 1985, 4:87–90.
Hollander, M, Wolfe, DA, Chicken, E. Nonparametric Statistical Methods. 3rd ed. Hoboken, NJ: John Wiley %26 Sons; 2014.
Behnen, K, Neuhaus, G. Rank Tests With Estimated Scores and Their Application. Stuttgart, West Germany: B G Teubner; 1989.
Balakrishnan, N, Ng, HKT. Precedence‐type Tests and Applications. Hoboken, NJ: John Wiley %26 Sons; 2006.
Prentice, RL. Linear rank tests with right censored data. Biometrika 1978, 65:167–179.
Arcones, MA, Kvam, PH, Samaniego, FJ. Nonparametric estimation of a distribution subject to a stochastic precedence constraint. J Am Stat Assoc 2002, 97:170–182.

Access to this WIREs title is by subscription only.

Recommend to Your
Librarian Now!

Hypothesis tests with precedence probabilities and precedence‐type tests

Browse by Topic

Access to this WIREs title is by subscription only.

The latest WIREs articles in your inbox