On selecting the best treatment in a generalized linear model |
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| Institution: | 1. Department of Mathematics, Statistics, and Computer Science, P.O. Box 1881, Marquette University, Milwaukee, WI 53201, USA;2. Department of Mathematics, Statistics, University of Illinois at Chicago, 851 S. Morgan Street, Chicago, IL 60607-7045, USA;1. McMaster University, Hamilton, Canada;2. University of Illinois at Chicago, USA;1. Institute of Statistical Science, Academia Sinica, Taipei, Taiwan, ROC;2. Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur - 208 016, India;3. Decision Sciences Area, Indian Institute of Management Lucknow, Lucknow - 226 013, India;1. School of Astronautics, Beihang University, Beijing 100191, PR China;2. Key Laboratory of Spacecraft Design Optimization and Dynamic Simulation Technologies of Ministry of Education, Beihang University, Beijing 100191, PR China;3. General Design Department of SCAAT, Chengdu 610100, PR China;4. University of Texas at Arlington Research Institute, University of Texas at Arlington, Fort Worth, Texas 76118, USA;1. Department of Diagnostic Sciences; Director, General Practice Residency; Director, Hospital Dentistry, Rutgers School of Dental Medicine, Newark, NJ, USA;2. Director, Pacific Center for Orofacial Disorders, Arthur A. Dugoni School of Dentistry, University of the Pacific, San Francisco, CA, USA;3. Department of Diagnostic Sciences; Assistant Director, General Practice Residency, Rutgers School of Dental Medicine, Newark, NJ, USA;1. Assistant Professor, Department of Advanced General Dentistry, Dental Hospital, Dankook University, Cheonan, South Korea;2. Resident, Department of Advanced General Dentistry, Dental Hospital, Yonsei University, Seoul, South Korea;3. Researcher, Department of Advanced General Dentistry, Dental Hospital, Yonsei University, Seoul, South Korea;4. Clinical Assistant Professor, Department of Advanced General Dentistry, Dental Hospital, Dankook University, Cheonan, South Korea;5. Associate Professor, Department of Advanced General Dentistry, Dental Hospital, Yonsei University, Seoul, South Korea |
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| Abstract: | The problem of selecting the best treatment is studied under generalized linear models. For certain balanced designs, it is shown that simple rules are Bayes with respect to any non-informative prior on the treatment effects under any monotone invariant loss. When the nuisance parameters such as block effects are assumed to follow a uniform (improper) prior or a normal prior, Bayes rules are obtained for the normal linear model under more suitable balanced designs, keeping the generality of the loss and the generality of the non-informativeness on the prior of the treatment effects. These results are extended to certain types of informative priors on the treatment effects. When the designs are unbalanced, algorithms based on the Gibbs sampler and the Laplace method are provided to compute the Bayes rules. |
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