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.
[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
[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.
Momeni, R., Etminan, J., & Sadegh, M. K. (2019). A random decision for testing of the homogeneity of normal means against the tree order alternative. Hacettepe Journal of Mathematics and Statistics, 48(5), 1547-1559. https://doi.org/10.15672/hujms.481380
AMA
Momeni R, Etminan J, Sadegh MK. A random decision for testing of the homogeneity of normal means against the tree order alternative. Hacettepe Journal of Mathematics and Statistics. October 2019;48(5):1547-1559. doi:10.15672/hujms.481380
Chicago
Momeni, Reza, Javad Etminan, and Mohammad Khanjari Sadegh. “A Random Decision for Testing of the Homogeneity of Normal Means Against the Tree Order Alternative”. Hacettepe Journal of Mathematics and Statistics 48, no. 5 (October 2019): 1547-59. https://doi.org/10.15672/hujms.481380.
EndNote
Momeni R, Etminan J, Sadegh MK (October 1, 2019) A random decision for testing of the homogeneity of normal means against the tree order alternative. Hacettepe Journal of Mathematics and Statistics 48 5 1547–1559.
IEEE
R. Momeni, J. Etminan, and M. K. Sadegh, “A random decision for testing of the homogeneity of normal means against the tree order alternative”, Hacettepe Journal of Mathematics and Statistics, vol. 48, no. 5, pp. 1547–1559, 2019, doi: 10.15672/hujms.481380.
ISNAD
Momeni, Reza et al. “A Random Decision for Testing of the Homogeneity of Normal Means Against the Tree Order Alternative”. Hacettepe Journal of Mathematics and Statistics 48/5 (October 2019), 1547-1559. https://doi.org/10.15672/hujms.481380.
JAMA
Momeni R, Etminan J, Sadegh MK. A random decision for testing of the homogeneity of normal means against the tree order alternative. Hacettepe Journal of Mathematics and Statistics. 2019;48:1547–1559.
MLA
Momeni, Reza et al. “A Random Decision for Testing of the Homogeneity of Normal Means Against the Tree Order Alternative”. Hacettepe Journal of Mathematics and Statistics, vol. 48, no. 5, 2019, pp. 1547-59, doi:10.15672/hujms.481380.
Vancouver
Momeni R, Etminan J, Sadegh MK. A random decision for testing of the homogeneity of normal means against the tree order alternative. Hacettepe Journal of Mathematics and Statistics. 2019;48(5):1547-59.