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?” Because 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 2×2 table under the alternative hypothesis, together with an induced prior under the null. With a lack of specific prior information, a diffuse (e.g., uniform) distribution may be used.We claim that this approach is not satisfactory, because in a testing problem one should make sure that the prior under the alternative is adequately centered around the region specified by the null, in order to obtain a fairer comparison between the two hypotheses, especially when the data are in reasonable agreement with the null. 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.
- Bayes factor
- Intrinsic prior