Ordinal dominance curve based inference for stochastically ordered distributions

Journal of the Royal Statistical Society. Series B: Statistical Methodology

Volumn 74, Issue 5, 2012, Pages 825-847

  Davidov, Ori ^a   Herman, Amir ^a  

^a UNIVERSITY OF HAIFA   (Israel)

Author keywords

Area under the curve; Order restricted statistical inference; Rank tests; Stochastic order

Indexed keywords

EID: 84867657539     PISSN: 13697412     EISSN: 14679868     Source Type: Journal    
DOI: 10.1111/j.1467-9868.2012.01031.x     Document Type: Article

References (41)

1. Arcones, M. A., Kvam, P. H. and Samaniego, F. J. (2002) Nonparametric estimation of a distribution subject to a stochastic precedence constraint. J. Am. Statist. Ass., 97, 170-182.
2. Bailer, J. A. and Oris, J. T. (1994) Assessing toxicity of pollutants in aquatic systems. In Case Studies in Biometry (eds N. Lange, L. Ryan, L. Billard, D. Brillinger, L. Conquest and J. Greenhouse ). New York: Wiley.
3. Bamber, D. (1975) The area above the ordinal dominance graph and the area below the receiver operating characteristic graph. J. Math. Psychol., 12, 387-415.
4. Brunk, H. D., Franck, W. E., Hanson, D. L. and Hogg, R. V. (1966) Maximum likelihood estimation of the distribution of two stochastically ordered random variables. J. Am. Statist. Ass., 61, 1067-1080.
5. Carolan, C. A. and Tebbs, J. M. (2005) Nonparametric tests for and against likelihood ratio ordering in the two-sample problem. Biometrika, 92, 159-171.
6. Davidov, O. (2012) Ordered inference, rank statistics and combining p-values: a new perspective. Statist. Methodol., 9, 456-465.
7. Davidov, O. and Herman, A. (2010) Testing for order among K populations: theory and examples. Can. J. Statist., 38, 97-115.
8. El Barmi, H. and Mukerjee, H. (2005) Inferences under a stochastic ordering constraint: the k-sample case. J. Am. Statist. Ass., 100, 252-261.
9. Finner, H. and Strassburger, K. (2002) The partitioning principle: a powerful tool in multiple decision theory. Ann. Statist., 30, 1194-1213.
10. Franck, W. E. (1984) A likelihood ratio test for stochastic ordering. J. Am. Statist. Ass., 79, 686-691.
11. Hand, D. J. and Till, R. J. (2001) A simple generalization of the area under the curve for multiple class classification problems. Mach. Learn., 45, 171-186.
12. Hauck, W. W., Hyslop, T. and Anderson, S. (2000) Generalized treatment effects for clinical trials. Statist. Med., 19, 887-899.
13. Hsieh, F. and Turnbull, B. W. (1996) Nonparametrics and semiparametric estimation of the receiver operating characteristic curve. Ann. Statist., 24, 25-40.
14. Hsu, J. C. and Berger, R. L. (1999) Stepwise confidence intervals without multiplicity adjustment for dose-response and toxicity studies. J. Am. Statist. Ass., 94, 468-482.
15. Hwang, G. and Peddada, S. D. (1994) Confidence interval estimation subject to order restrictions. Ann. Statist., 22, 67-93.
16. International Conference on Harmonization (1994) Dose response information to support drug registration. Guideline E4. Committee for Proprietary Medical Products, European Agency for Evaluation of Medical Products, London.
17. Joe, E. (1997) Multivariate Models and Dependence Concepts. New York: Chapman and Hall.
18. Jonckheere, A. R. (1954) A distribution free k-sample test against ordered alternatives. Biometrika, 41, 133-145.
19. Kotz, S., Lumelskii, Y. and Pensky, M. (2003) The Stress-Strength Model and Its Generalizations. Singapore: World Scientific Publishing.
20. Lee, Y. J. and Wolfe, D. A. (1976) A distribution-free test for stochastic ordering. J. Am. Statist. Ass., 71, 722-727.
21. Lehmann, E. L. (1966) Some concepts of dependence. Ann. Math. Statist., 37, 1137-1153.
22. Lehmann, E. L. and Rojo, J. (1992) Invariant directional orderings. Ann. Statist., 20, 2100-2110.
23. Lehmann, E. L. and Romano J. P. (2005) Testing Statistical Hypotheses, 3rd edn. New York: Springer.
24. Nakas, C. T. and Yiannoutsos, C. T. (2004) Ordered multiple class ROC analysis with continuous measurements. Statist. Med., 23, 3437-3449.
25. Peddada, S. D., Dinse, G. and Kissling, G. (2007) Incorporating historical control data when comparing tumor incidence rates. J. Am. Statist. Ass., 102, 1212-1220.
26. Peddada, S. D., Prescott, K. and Conaway, M. (2001) Tests for order restrictions in binary data. Biometrics, 57, 1219-1227.
27. Pepe, M. S. (2003) The Statistical Evaluation of Medical Tests for Classification and Predication. New York: Oxford University Press.
28. Perman, M. and Wellner, J. (1996) On the distribution of Brownian area. Ann. Appl. Probab., 6, 1091-1111.
29. Robert, C. P., Hwang, G. and Strawderman, W. (1993) Is Pitman closeness a reasonable criterion? J. Am. Statist. Ass., 88, 57-63.
30. Robertson, T., Wright, F. T. and Dykstra, R. (1988) Order Restricted Statistical Inference. New York: Wiley.
31. Rojo, J. (2004) On the Estimation of Survival Functions under a Stochastic Order Constraint. Hayward: Institute of Mathematical Statistics.
32. Rojo, J. and Ma, Z. (1996) On the estimation of stochastically ordered survival functions. J. Statist. Computn Simuln, 55, 1-21.
33. Shaked, M. and Shanthikumer, J. G. (2007) Stochastic Orders. New York: Springer.
34. Shirley, E. (1977) A non-parametric equivalent of Williams' test for contrasting increasing dose levels of a treatment. Biometrics, 33, 386-389.
35. Tamhane, A. C. and Logan, B. R. (2006) Multiple comparisons procedures in dose response studies. In Dose Finding in Drug Development (ed. N. Ting ). New York: Springer Science+Business Media.
36. Terpstra, T. J. (1952) The asymptotic normality and consistency of Kendall's test against trend when ties are present in one ranking. Indag. Math., 14, 327-333.
37. Terpstra, J. T. and Magel, R. C. (2003) A new nonparametric test for the ordered alternative problem. Nonparam. Statist., 15, 289-301.
38. Tolmatz, L. (2000) Asymptotics of the distribution of the integral of the absolute value of the Brownian bridge for large arguments. Ann. Probab., 28, 132-139.
39. Tryon, V. P. and Hettmansperger, T. P. (1973) A class of non-parametric tests for homogeneity against ordered alternatives. Ann. Statist., 1, 1061-1070.
40. Wang, Y. (1996) A likelihood ratio test against stochastic ordering in several populations. J. Am. Statist. Ass., 91, 1676-1683.
41. Williams, D. A. (1971) A test for differences between treatment means when several doses are compared with a zero dose. Biometrics, 27, 103-117.