Ann Inst Stat Math (2008) 60:483–498 DOI 10.1007/s10463-007-0117-5 Some remarks on Bayesian inference for one-way ANOVA models Fabrizio Solari · Brunero Liseo · Dongchu Sun Received: 14 July 2005 / Revised: 13 November 2006 / Published online: 11 May 2007 © The Institute of Statistical Mathematics, Tokyo 2007 Abstract We consider the standard one-way ANOVA model; it is well-known that classical statistical procedures are based on a scalar non-centrality parame- ter. In this paper we explore both marginal likelihood and integrated likelihood functions for this parameter and we show that they exactly lead to the same answer. On the other hand, we prove that a fully Bayesian testing procedure may provide different conclusions, depending on what is considered to be the real quantity of interest in the model or, said differently, which are the compet- ing hypotheses. We illustrate these issues via a real data example. Keywords Integrated likelihood · Marginal likelihood · Model choice · Objective Bayes factor · Reference prior 1 Introduction Analysis of variance (ANOVA) is an extremely important method in explor- atory and confirmatory data analysis (Gelman, 2005). Here we focus on the one-way ANOVA, where we assume that data are observed according to the (full) model M F F. Solari ISTAT, Via Magenta 4, Rome 00184, Italy B. Liseo (B ) Dip. di studi geoeconomici, linguistici, statistici e storici, Università di Roma Sapienza, viale del Castro Laurenziano 9, Rome 00161, Italy e-mail: brunero.liseo@uniromal.it D. Sun Department of Statistics, University of Missouri, 146 Middlebush Hall, Columbia, MO 65211-6100, USA