The Annals of Statistics

The Robustness and Sensitivity of the Mixed-Dirichlet Bayesian Test for "Independence" in Contingency Tables

I. J. Good and J. F. Crook

Full-text: Open access

Abstract

A mixed-Dirichlet prior was previously used to model the hypotheses of "independence" and "dependence" in contingency tables, thus leading to a Bayesian test for independence. Each Dirichlet has a main hyperparameter $\kappa$ and the mixing is attained by assuming a hyperprior for $\kappa$. This hyperparameter can be regarded as a flattening or shrinking constant. We here review the method, generalize it and check the robustness and sensitivity with respect to variations in the hyperpriors and in their hyperhyperparameters. The hyperpriors examined included generalized log-Students with various numbers of degrees of freedom $\nu$. When $\nu$ is as large as 15 this hyperprior approximates a log-normal distribution and when $\nu = 1$ it is a log-Cauchy. Our experiments caused us to recommend the log-Cauchy hyperprior (or of course any distribution that closely approximates it). The user needs to judge values for the upper and lower quartiles, or any two quantiles, of $\kappa$, but we find that the outcome is robust with respect to fairly wide variations in these judgments.

Article information

Source
Ann. Statist., Volume 15, Number 2 (1987), 670-693.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176350368

Digital Object Identifier
doi:10.1214/aos/1176350368

Mathematical Reviews number (MathSciNet)
MR888433

Zentralblatt MATH identifier
0665.62057

JSTOR
links.jstor.org

Subjects
Primary: 62H17: Contingency tables
Secondary: 62H15: Hypothesis testing 62F15: Bayesian inference

Keywords
Bayes factors against independence Bayesian robustness Bayesians (averaging over) Dirichlet-multinomial distribution flattening constants hierarchical Bayes hyper-hyperparameters log-Cauchy distribution log-normal distribution log-Student distribution mixtures of conjugate priors multinomial signficance tests multivaraite Bayesian methods shrinking constants

Citation

Good, I. J.; Crook, J. F. The Robustness and Sensitivity of the Mixed-Dirichlet Bayesian Test for "Independence" in Contingency Tables. Ann. Statist. 15 (1987), no. 2, 670--693. doi:10.1214/aos/1176350368. https://projecteuclid.org/euclid.aos/1176350368


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