Biometrika Advance Access originally published online on February 28, 2007
Biometrika 2007 94(1):135-152; doi:10.1093/biomet/asm014
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Copyright © 2007 Biometrika Trust
Articles |
Extending conventional priors for testing general hypotheses in linear models
Department of Statistics and Operations Research, University of Valencia, 46100 Valencia, Spain
Department of Economics and Finance, University of Castilla-La Mancha, 02071 Albacete, Spain
susie.bayarri{at}uv.es
gonzalo.garciadonato{at}uclm.es
Received for publication 1 March 2005.
Revision received 1 July 2006.
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Abstract |
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We consider that observations come from a general normal linear model and that it is desirable to test a simplifying null hypothesis about the parameters. We approach this problem from an objective Bayesian, model-selection perspective. Crucial ingredients for this approach are ‘proper objective priors’ to be used for deriving the Bayes factors. Jeffreys-Zellner-Siow priors have good properties for testing null hypotheses defined by specific values of the parameters in full-rank linear models. We extend these priors to deal with general hypotheses in general linear models, not necessarily of full rank. The resulting priors, which we call ‘conventional priors’, are expressed as a generalization of recently introduced ‘partially informative distributions’. The corresponding Bayes factors are fully automatic, easily computed and very reasonable. The methodology is illustrated for the change-point problem and the equality of treatments effects problem. We compare the conventional priors derived for these problems with other objective Bayesian proposals like the intrinsic priors. It is concluded that both priors behave similarly although interesting subtle differences arise. We adapt the conventional priors to deal with nonnested model selection as well as multiple-model comparison. Finally, we briefly address a generalization of conventional priors to nonnormal scenarios.
Key Words: analysis-of-variance model • changepoint problem • model selection • objective Bayesian methods • partially informative distribution • regression model