Tests Based on Intrinsic Priors for the Equality of Two Correlated Proportions

Abstract

Correlated proportions arise in longitudinal (panel) studies. A typical example is the "opinion swing" problem: "Has the proportion of people favoring a politician changed after his recent speech to the nation on TV?". Since the same group of individuals is interviewed before and after the speech, the two proportions are correlated. A natural null hypothesis to be tested is whether the corresponding population proportions are equal. A standard Bayesian approach to this problem has already been considered in the literature, based on a Dirichlet prior for the cell-probabilities of the underlying two-by-two table under the alternative hypothesis, together with an induced prior under the null.In lack of specific prior information, a diffuse (e.g. uniform) distribution may be used. We claim that this approach is not satisfactory, since in a testing problem one should make sure that the prior under the alternative be adequately centered around the region specified by the null, in order to obtain a fair comparison between the two hypotheses. Following an intrinsic prior methodology, we develop two strategies for the construction of a collection of objective priors increasingly peaked around the null. We provide a simple interpretation of their structure in terms of weighted imaginary sample scenarios. We illustrate our method by means of three examples, carrying out sensitivity analysis and providing comparison with existing results.

Integrated Objective Bayesian Estimation and Hypothesis Testing

“…When the prior concentrates its mass around the null hypothesis, as the intrinsic priors do with a degree of concentration controlled by the training sample size, the resulting likelihood of the alternative model will be a much more serious competitor of the null likelihood, and in this case the null can be rejected. For the notion of intrinsic priors see Berger and Pericchi (1996) and Moreno et al (1998), and for an analysis of the concentration of the intrinsic prior of the alternative model on the null and its implication on robustness in testing problems see Consonni and La Rocca (2008), and Casella and Moreno (2009).…”

Assessing Robustness of Intrinsic Tests of Independence in Two-Way Contingency Tables

“…(See, e.g., Jeffreys 1961, chap. 5;Gunel and Dickey 1974, who noted that this is the "Savage continuity condition"; Berger and Sellke 1987;Casella and Berger 1987;Morris 1987;Berger 1994;Robert 2001, who also discussed the Jeffreys-Lindley paradox; and Consonni and La Rocca 2008. ) It is important to realize that if a prior on the unrestricted hypothesis H 1 concentrates probability near H 0 , this does not necessarily favor H 0 , but rather focuses the test on model alternatives that are close to H 0 .…”

Objective Bayesian meta‐analysis for sparse discrete data

“…This is a method for converting improper prior distributions, which are not suitable for model selection, into suitable priors for model selection. However, the intrinsic priors also improve the performance of Bayesian model selection procedures with proper priors.…”