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Statistical inference under order restrictions on both rows and columns of a matrix, with an application in toxicology

Eric Teoh, Abraham Nyska, Uri Wormser, and Shyamal D. Peddada

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Abstract

We present a general methodology for performing statistical inference on the components of a real-valued matrix parameter for which rows and columns are subject to order restrictions. The proposed estimation procedure is based on an iterative algorithm developed by Dykstra and Robertson (1982) for simple order restriction on rows and columns of a matrix. For any order restrictions on rows and columns of a matrix, sufficient conditions are derived for the algorithm to converge in a single application of row and column operations. The new algorithm is applicable to a broad collection of order restrictions. In practice, it is easy to design a study such that the sufficient conditions derived in this paper are satisfied. For instance, the sufficient conditions are satisfied in a balanced design. Using the estimation procedure developed in this article, a bootstrap test for order restrictions on rows and columns of a matrix is proposed. Computer simulations for ordinal data were performed to compare the proposed test with some existing test procedures in terms of size and power. The new methodology is illustrated by applying it to a set of ordinal data obtained from a toxicological study.

Chapter information

Source
N. Balakrishnan, Edsel A. Peña and Mervyn J. Silvapulle, eds., Beyond Parametrics in Interdisciplinary Research: Festschrift in Honor of Professor Pranab K. Sen (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2008), 62-77

Dates
First available in Project Euclid: 1 April 2008

Permanent link to this document
https://projecteuclid.org/euclid.imsc/1207058264

Digital Object Identifier
doi:10.1214/193940307000000059

Mathematical Reviews number (MathSciNet)
MR2462197

Subjects
Primary: 62F10: Point estimation
Secondary: 62G09: Resampling methods 62G10: Hypothesis testing

Keywords
linked parameters matrix partial order maximally-linked subgraph order-restriction ordinal data simple order simple tree order umbrella order

Rights
Copyright © 2008, Institute of Mathematical Statistics

Citation

Teoh, Eric; Nyska, Abraham; Wormser, Uri; Peddada, Shyamal D. Statistical inference under order restrictions on both rows and columns of a matrix, with an application in toxicology. Beyond Parametrics in Interdisciplinary Research: Festschrift in Honor of Professor Pranab K. Sen, 62--77, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2008. doi:10.1214/193940307000000059. https://projecteuclid.org/euclid.imsc/1207058264


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References

  • [1] Agresti, A. and Coull, B. A. (2002). The analysis of contingency tables under inequality constraints. J. Statist. Plann. Inference 107 45–73.
  • [2] Barlow, R., Bartholomew, D., Bremmer, J. and Brunk, H. (1972). Statistical Inference Under Order Restrictions, 3rd ed. ed. Wiley, New York.
  • [3] Berger, V., Permutt, T. and Ivanova, A. (1998). Convex hull test for ordered categorical data. Biometrics 54 1541–1550.
  • [4] Cohen, A., Kemperman, J. and Sackrowitz, H. (2000). Properties of likelihood inference for order restricted models. J. Multivariate Anal. 72 50–77.
  • [5] Cohn, S., Schloemann, S., Tessner, T., Seibert, K. and Stenson, W. (1997). Crypt stem cell survival in the mouse intestinal epithelium is regulated by prostaglandins synthesized through cyclooxygenase-1. J. Clinical Investigation 99 1367–1379.
  • [6] Conaway, M. R., Dunbar, S. and Peddada, S. D. (2004). Designs for single- or multiple-agent phase I trials. Biometrics 60 661–669.
  • [7] Dykstra, R. L. and Robertson, T. (1982). An algorithm for isotonic regression for two or more independent variables. Ann. Statist. 10 708–716.
  • [8] Franck, W. E. (1984). A likelihood ratio test for stochastic ordering. J. Amer. Statist. Assoc. 79 686–691.
  • [9] Grove, D. M. (1980). A test of independence against a class of ordered alternatives in a 2×C contingency table. J. Amer. Statist. Assoc. 75 454–459.
  • [10] He, L., Liu, L., Hahn, E. and Gamelli, R. (2001). he expression of cyclooxygenase and the production of prostaglandin E2 in neutrophils after burn injury and infection. J. Burn Care Rehabilitation 22 58–64.
  • [11] Hwang, J. T. G. and Peddada, S. D. (1994). Confidence interval estimation subject to order restrictions. Ann. Statist. 22 67–93.
  • [12] Lee, C.-I. C. (1988). Quadratic loss of order restricted estimators for treatment means with a control. Ann. Statist. 16 751–758.
  • [13] Lee, J., Mukhtar, H., Bickers, D., Kopelovich, L. and Athar, M. (2003). Cyclooxygenases in the skin: Pharmacological and toxicological implications. Toxicology and Applied Pharmacology 192 294–306.
  • [14] Lin, H., Lin, T., Cheung, W., Nian, G., Tseng, P., Chen, S., Chen, J., Shyue, S., Liou, J., Wu, C. and Wu, K. (2002). Cyclooxygenase-1 and bicistronic cyclooxygenase-1/prostacyclin synthase gene transfer protect against ischemic cerebral infarction. Circulation 105 1962–1969.
  • [15] Lehmann, E. L. (1983). Theory of Point Estimation. Wiley, New York.
  • [16] Nair, V. N. (1987). Chi-squared-type tests for ordered alternatives in contingency tables. J. Amer. Statist. Assoc. 82 283–291.
  • [17] Peddada, S. D., Prescott, K. E. and Conaway, M. (2001). Tests for order restrictions in binary data. Biometrics 57 1219–1227.
  • [18] Peddada, S. D., Dunson, D. B. and Tan, X. (2005). Estimation of order-restricted means from correlated data. Biometrika 92 703–715.
  • [19] Præstgaard, J. T. and Huang, J. (1996). Asymptotic theory for nonparametric estimation of survival curves under order restrictions. Ann. Statist. 24 1679–1716.
  • [20] Robertson, T. and Wright, F. T. (1981). Likelihood ratio tests for and against a stochastic ordering between multinomial populations. Ann. Statist. 9 1248–1257.
  • [21] Robertson, T., Wright, F. T. and Dykstra, R. L. (1988). Order Restricted Statistical Inference. Wiley, Chichester.
  • [22] Silvapulle, M. J. (1997). A curious example involving the likelihood ratio test against one-sided hypotheses. American Statistician 51 178–180.
  • [23] Silvapulle, M. J. and Sen, P. K. (2005). Constrained Statistical Inference. Wiley, Hoboken, NJ.
  • [24] Vane, J., Bakhle, Y. and Botting, R. (1998). Cyclooxygenases 1 and 2. Annual Review of Pharmacology and Toxicology 38 97–120.
  • [25] Wang, Y. (1996). A likelihood ratio test against stochastic ordering in several populations. J. Amer. Statist. Assoc. 91 1676–1683.
  • [26] Wormser, U., Langenbach, R., Peddada, S., Sintov, A., Brodsky, B. and Nyska, A. (2004). Reduced sulfur mustard-induced skin toxicity in cyclooxygenase-2 knockout and celecoxib-treated mice. Toxicology and Applied Pharmacology 200 40–47.