It is straight forward 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 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. 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 intra-class correlation associated with the individuals within a cluster. This intra-class 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 intra-class correlations and the cell probabilities. We use data from the National Health Interview Survey to show how our alternative test performs and to obtain the posterior density of the cell probabilities. Also, using Monte Carlo integration with a binomial importance function, we obtain the Bayes factor for a test of no association.