Empirical Bayes methods for two-way contingency tables

SUMMARY An empirical Bayes method for smoothing two-way tables is presented. It is based on the use of the log linear model and a normal prior distribution. Estimation of the variance com- ponent in the prior is discussed, and two approximations are considered, one which utilizes the EM algorithm. This paper proposes estimates of contingency table cell probabilities based on combining the log linear model with normal prior distributions. Attention is mainly confined to the two-way table, with possible extensions discussed in the last section. The approach taken is empirical Bayes, but there are obvious parallels with variance component models for con- tingency tables, which are considered in the last section. Our approach draws from many sources. Good (1956) proposed a normal prior model quite similar to ours for the two-way table used in our example. Good's approach is to use a prior distribution on the probabilities of the multinomial to give smoothed cell probabilities. Expanding upon Good's method, Fienberg & Holland (1970, 1973) proposed empirical Bayes estimates for two-way tables with a Dirichlet prior for the cell probabilities. They compared various methods for estimating the parameters of the Dirichlet distribution. We contrast their estimates with ours in our example. Our multinomial-normal model is a minor variant of that introduced by Leonard (1975). Our approach differs chiefly in the handling of the parameters of the normal distribution. Because our models are so similar, we defer discussion of Leonard's work to the next section, where we introduce our sampling model and prior distribution, and discuss estimation of the cell probabilities assuming the parameters in the prior are known. Then we discuss estimation of the prior parameters, give an example, and finally consider advantages, disadvantages and extensions of our approach.