Research Article

### A random decision for testing of the homogeneity of normal means against the tree order alternative

Year 2019, Volume 48, Issue 5, 1547 - 1559, 08.10.2019

### Abstract

A typical problem of interest is to compare the $k+1$ normal means under the tree order restriction $\theta_0\leq \theta_i$ for $i=1,\ldots,k$. In this paper, we propose new multiple comparisons procedures for testing of the tree order constraint. New test procedures along with the corresponding simultaneous confidence intervals are motivated by some new estimation methods which are constructed based on a random decision and the Bayesian approach. Also, these procedures are developed for two-sided tree order alternatives. We compare the performance of the proposed methods with some existing test procedures, such as likelihood ratio test and some multiple comparisons tests for the tree order constraint. In some cases, the gains in power due to the proposed procedures are substantial. The results for two sided alternative are similar to the one-sided hypotheses and new procedures perform well for almost every configuration. We illustrate the efficiency of the proposed methods by analyzing of the two bioassay numerical examples.

### References

• [1] R. E. Barlow, D. J. Bartholomew, J. M. Bremner and H. D. Brunk, Statistical Inference under Order Restrictions: The Theory and Application of Isotonic Regression, Wiley, New York, NY, 1972.
• [2] D. J. Bartholomew, A test of homogeneity of means under restricted alternatives, J. R. Stat. Soc. Ser. B. Stat. Methodol., 23, 239-281, 1961.
• [3] J. O. Berger, Statistical Decision Theory and Bayesian Analysis. Second edition. New York: Springer, 1985.
• [4] J. Betcher and S. D. Peddada, Statistical inference under order restrictions in analysis of covariance using a modified restricted maximum likelihood estimator, Sankhya A, Volume 71-B, Part 1, 79-96, 2009.
• [5] A. Cohen, J. H. B. Kemperman and H. B. Sackrowitz, Properties of likelihood inference for order restricted models, J. Multivariate Anal., 72, 50-77, 2000.
• [6] S. Dunbar, M. Conaway and S. D. Peddada, On improved estimation of parameters subject to order restrictions, Statist. Applic., 3, 121-128, 2001.
• [7] C. W. Dunnett, A multiple comparison procedure for comparing several treatments with a control, J. Amer. Statist. Assoc., 50, 1096-1121, 1955.
• [8] J. T. G. Hwang and S. D. Peddada, Confidence interval estimation subject to order restrictions, Ann. Statist., 22, 67-93, 1994.
• [9] J. Kanno, L. Onyon, S. D. Peddada, J. Ashby, E. Jacob and W. Owens, The OECD program to validate the rat uterotrophic bioassay: Phase two-coded single dose studies, Env. Health Persp., 111, 1550-1558, 2002b.
• [10] C. I. C. Lee, The quadratic loss of order restricted estimators for several treatment means and a control mean, Ann. Statist., 16, 751-758, 1988.
• [11] R. Marcus and H. Talpaz, Further results on testing homogeneity of normal means against simple tree alternatives, Comm. Statist. Theory Methods, 21, 2135-2149, 1992.
• [12] R. Momeni, J. Etminan and M. Khanjari Sadegh, Estimation of normal means in the tree order by the weighting methods, Comm. Statist. Simulation Comput., DOI: 10.1080/03610918.2018.1554115. 2018.
• [13] R. Momeni, J. Etminan and M. Khanjari Sadegh, Estimation of parameters in the tree order restriction by a randomized decision, Revised manuscript in J. Stat. Comput. Simul., 2019.
• [14] National Toxicology Program. Toxicology and carcinogenesis studies of anthraquinone in F344/N rats and B6C3F1 mice. Technical Report 494. National Toxicology Program, Research Triangle Park, 1999a.
• [15] S. D. Peddada, J. Haseman, X. Tan and G. Travlos, Tests for simple tree order restriction with application to dose-response studies, J. Roy. Statist. Soc., Ser. C, 55, 493-506, 2006.
• [16] T. Robertson, F. T. Wright and R. L. Dykstra, Order Restricted Statistical Inference. New York: John Wiley, 1988.
• [17] M. J. Silvapulle and P. K. Sen, Constrained Statistical Inference: Inequality, Order and Shape Restrictions, Wiley, New York, NY, 2005.
Year 2019, Volume 48, Issue 5, 1547 - 1559, 08.10.2019

### References

• [1] R. E. Barlow, D. J. Bartholomew, J. M. Bremner and H. D. Brunk, Statistical Inference under Order Restrictions: The Theory and Application of Isotonic Regression, Wiley, New York, NY, 1972.
• [2] D. J. Bartholomew, A test of homogeneity of means under restricted alternatives, J. R. Stat. Soc. Ser. B. Stat. Methodol., 23, 239-281, 1961.
• [3] J. O. Berger, Statistical Decision Theory and Bayesian Analysis. Second edition. New York: Springer, 1985.
• [4] J. Betcher and S. D. Peddada, Statistical inference under order restrictions in analysis of covariance using a modified restricted maximum likelihood estimator, Sankhya A, Volume 71-B, Part 1, 79-96, 2009.
• [5] A. Cohen, J. H. B. Kemperman and H. B. Sackrowitz, Properties of likelihood inference for order restricted models, J. Multivariate Anal., 72, 50-77, 2000.
• [6] S. Dunbar, M. Conaway and S. D. Peddada, On improved estimation of parameters subject to order restrictions, Statist. Applic., 3, 121-128, 2001.
• [7] C. W. Dunnett, A multiple comparison procedure for comparing several treatments with a control, J. Amer. Statist. Assoc., 50, 1096-1121, 1955.
• [8] J. T. G. Hwang and S. D. Peddada, Confidence interval estimation subject to order restrictions, Ann. Statist., 22, 67-93, 1994.
• [9] J. Kanno, L. Onyon, S. D. Peddada, J. Ashby, E. Jacob and W. Owens, The OECD program to validate the rat uterotrophic bioassay: Phase two-coded single dose studies, Env. Health Persp., 111, 1550-1558, 2002b.
• [10] C. I. C. Lee, The quadratic loss of order restricted estimators for several treatment means and a control mean, Ann. Statist., 16, 751-758, 1988.
• [11] R. Marcus and H. Talpaz, Further results on testing homogeneity of normal means against simple tree alternatives, Comm. Statist. Theory Methods, 21, 2135-2149, 1992.
• [12] R. Momeni, J. Etminan and M. Khanjari Sadegh, Estimation of normal means in the tree order by the weighting methods, Comm. Statist. Simulation Comput., DOI: 10.1080/03610918.2018.1554115. 2018.
• [13] R. Momeni, J. Etminan and M. Khanjari Sadegh, Estimation of parameters in the tree order restriction by a randomized decision, Revised manuscript in J. Stat. Comput. Simul., 2019.
• [14] National Toxicology Program. Toxicology and carcinogenesis studies of anthraquinone in F344/N rats and B6C3F1 mice. Technical Report 494. National Toxicology Program, Research Triangle Park, 1999a.
• [15] S. D. Peddada, J. Haseman, X. Tan and G. Travlos, Tests for simple tree order restriction with application to dose-response studies, J. Roy. Statist. Soc., Ser. C, 55, 493-506, 2006.
• [16] T. Robertson, F. T. Wright and R. L. Dykstra, Order Restricted Statistical Inference. New York: John Wiley, 1988.
• [17] M. J. Silvapulle and P. K. Sen, Constrained Statistical Inference: Inequality, Order and Shape Restrictions, Wiley, New York, NY, 2005.

### Details

Primary Language English Statistics and Probability Statistics Reza MOMENİ (Primary Author) 0000-0001-9879-9946 Javad ETMİNAN This is me 0000-0003-0069-5407 Mohammad Khanjari SADEGH This is me 0000-0002-2647-5632 October 8, 2019 Year 2019, Volume 48, Issue 5