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A unified Bayesian inference on treatment means with order constraints

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  • Oh, Man-Suk
  • Shin, Dong Wan

Abstract

In some applications involving comparison of treatment means, it is known a priori that population means are ordered in a certain way. In such situations, imposing constraints on the treatment means can greatly increase the effectiveness of statistical procedures. This paper proposes a unified Bayesian method which performs a simultaneous comparison of treatment means and parameter estimation in ANOVA models with order constraints on the means. A continuous prior restricted to order constraints is employed, and posterior samples of parameters are generated using a Markov chain Monte Carlo method. Posterior probabilities of all possible hypotheses on the equality/inequality of treatment means are obtained using Savage-Dickey density ratios, for which we propose a simple and computationally efficient estimation method. Posterior densities and HPD intervals of parameters of interest are estimated with almost no extra cost, given some by-products from the test procedure. Simulation study results show that the proposed method outperforms the test without constraints and that the method is powerful in detecting the true hypothesis. The method is applied to the ramus bone sizes of 20 boys, which were measured at four time points. The proposed Bayesian test reveals that there are two growth spurts in the ramus bone size during the observed period, which could not be detected by pairwise comparisons of the means.

Suggested Citation

  • Oh, Man-Suk & Shin, Dong Wan, 2011. "A unified Bayesian inference on treatment means with order constraints," Computational Statistics & Data Analysis, Elsevier, vol. 55(1), pages 924-934, January.
    • Handle: RePEc:eee:csdana:v:55:y:2011:i:1:p:924-934
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    References listed on IDEAS

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    1. Park, Soo Jung & Wan Shin, Dong & Uk Park, Byeong & Chul Kim, Woo & Oh, Man-Suk, 2005. "Bayesian test for asymmetry and nonstationarity in MTAR model with possibly incomplete data," Computational Statistics & Data Analysis, Elsevier, vol. 49(4), pages 1192-1204, June.
    2. David B. Dunson & Brian Neelon, 2003. "Bayesian Inference on Order-Constrained Parameters in Generalized Linear Models," Biometrics, The International Biometric Society, vol. 59(2), pages 286-295, June.
    3. Nashimoto, Kane & Wright, F.T., 2005. "Multiple comparison procedures for detecting differences in simply ordered means," Computational Statistics & Data Analysis, Elsevier, vol. 48(2), pages 291-306, February.
    4. Oh, Man-Suk, 1999. "Estimation of posterior density functions from a posterior sample," Computational Statistics & Data Analysis, Elsevier, vol. 29(4), pages 411-427, February.
    5. Nashimoto, Kane & Wright, F.T., 2005. "A note on multiple comparison procedures for detecting differences in simply ordered means," Statistics & Probability Letters, Elsevier, vol. 73(4), pages 393-401, July.
    6. Junfeng Shang & Joseph E. Cavanaugh & Farroll T. Wright, 2008. "A Bayesian Multiple Comparison Procedure for Order‐Restricted Mixed Models," International Statistical Review, International Statistical Institute, vol. 76(2), pages 268-284, August.
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    Cited by:

    1. Sysoev, O. & Burdakov, O. & Grimvall, A., 2011. "A segmentation-based algorithm for large-scale partially ordered monotonic regression," Computational Statistics & Data Analysis, Elsevier, vol. 55(8), pages 2463-2476, August.
    2. Conde David & Salvador Bonifacio & Rueda Cristina & Fernández Miguel A., 2013. "Performance and estimation of the true error rate of classification rules built with additional information. An application to a cancer trial," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 12(5), pages 583-602, October.

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