Bayesian analysis of a two-way categorical table incorporating intraclass correlation
It is straightforward to analyze data from a single multinomial table. Specifically, for the analysis of a two-way categorical table, the common chi-squared test of independence between the two variables and maximum likelihood estimators of the cell probabilities are readily available. When the counts in the two-way categorical table are formed from familial data (clusters of correlated data), the common chi-squared test no longer applies. We note that there are several approximate adjustments to the common chi-squared test. For example, Choi and McHugh (Choi, J.W. and McHugh, R.B., 1989, A reduction factor in goodness-of-fit and independence tests for clustered and weighted observations. Biometrics , 45, 979-996.) showed how to adjust the chi-squared statistic for clustered and weighted data. However, our main contribution is the construction and analysis of a Bayesian model which removes all analytical approximations. This is an extension of a standard multinomial-Dirichlet model to include the intraclass correlation associated with the individuals within a cluster. We have used a key formula described by Altham (Altham, P.M., 1976, Discrete variable analysis for individuals grouped into families. Biometrika , 63, 263-269.) to incorporate the intraclass correlation. This intraclass correlation varies with the size of the cluster, but we assume that it is the same for all clusters of the same size for the same variable. We use Markov chain Monte Carlo methods to fit our model, and to make posterior inference about the intraclass correlations and the cell probabilities. Also, using Monte Carlo integration with a binomial importance function, we obtain the Bayes factor for a test of no association. To demonstrate the performance of the alternative test and estimation procedure, we have used data on activity limitation status and age from the National Health Interview Survey and a simulation study.
multinomial-dirichlet bayes factor clustered data gibbs sampler