Abstract
An objective Bayesian procedure for testing in the two way analysis of variance is proposed. In the classical methodology the main effects of the two factors and the interaction effect are formulated as linear contrasts between means of normal populations, and hypotheses of the existence of such effects are tested. In this paper, for the first time these hypotheses have been formulated as objective Bayesian model selection problems. Our development is under homoscedasticity and heteroscedasticity, providing exact solutions in both cases. Bayes factors are the key tool to choose between the models under comparison but for the usual default prior distributions they are not well defined. To avoid this difficulty Bayes factors for intrinsic priors are proposed and they are applied in this setting to test the existence of the main effects and the interaction effect. The method has been illustrated with an example and compared with the classical method. For this example, both approaches went in the same direction although the large P value for interaction (0.79) only prevents us against to reject the null, and the posterior probability of the null (0.95) was conclusive.
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References
Berger JO, Pericchi LR (1996) The intrinsic Bayes factor for model selection and prediction. J Am Stat Assoc 91:109–122
Berger JO, Pericchi LR (2015) Bayes factors. Wiley StatsRef: statistics reference online. Wiley, London
Bertolino F, Moreno E, Racugno W (2000) Bayesian model selection approach to analysis of variance under heteroscedasticity. Statistician 49(4):503–517
Cano JA, Kessler M, Moreno E (2004) On intrinsic priors for nonnested models. Test 13:445–463
Cano JA, Carazo C, Salmerón D (2013) Bayesian model selection approach to the one way analysis of variance under homoscedasticity. Comput Stat 28:919–931
Cano JA, Carazo C, Salmerón D (2016) Objective Bayesian model selection approach to linear contrasts for the one way analysis of variance. Stat Prob Lett 109:54–62
Jeffreys H (1961) Theory of probability. Oxford University Press, Oxford
Robert CP, Casella G (2001) Monte Carlo statistical methods. Springer, Berlin
Rohatgi VK (1984) Statistical inference. Wiley, London
Snedecor GW, Cochran WG (1989) Statistical methods, 8th edn. Iowa State University Press, Iowa City
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This research was supported by the Séneca Foundation Programme for the Generation of Excellence Scientific Knowledge under Project 15220/PI/10.
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Cano, J.A., Carazo, C. & Salmerón, D. Objective Bayesian model selection approach to the two way analysis of variance. Comput Stat 33, 235–248 (2018). https://doi.org/10.1007/s00180-017-0727-1
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DOI: https://doi.org/10.1007/s00180-017-0727-1