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598 CH A P T E R 22: Contingency Table Analysis have no need to compute chi-square values (except for comparison with the Bayesian conclusions). There are many advantages of a Bayesian approach. As usual, a significant advantage is never having to compute a p value. Bet- ter yet is that the Bayesian analysis provides credible intervals on the conjoint probabilities and on any desired comparison of conditions. I will call our modeling framework Poisson exponential ANOVA because it uses a Poisson likelihood distribution with an exponential link function from an underlying ANOVA model. This terminology is not conventional, and it might even be misleading if readers mistakenly infer from the term "ANOVA" that there is a metric predicted variable involved. Nevertheless, the terminology is highly descriptive of the structural elements of the model. The model appears in the lower right cell of Table 14.1, p. 385, where its relation to other cases of the generalized linear model is evident. 22.1 POISSON EXPONENTIAL ANOVA 22.1.1 What the Data Look Like To motivate the model, we need first to understand the structure of the data. An example of the sort of data we'll be dealing with is shown in Table 22.1. The data come from a classroom poll of students at the University of Delaware (Snee, 1974). Respondents reported their hair color and eye color, with each variable split into four nominal levels as indicated in Table 22.1. The cells of the table indicate the frequency with which each combination occurred in the sample. Each respondent falls in one and only one cell of the table. The data to be predicted are the cell frequencies. The predictors are the nominal variables. This situation is analogous to two-way ANOVA, which also had two nominal predictors but had several metric values in each cell instead of a single frequency. For data like these, we can ask a number of questions. We could wonder about one variable at a time and ask questions such as "Are there more brown-eyed Table 22.1 Frequencies of Different Combinations of Hair Color and Eye Color (Data from Snee, 1974.) Eye Color Hair Color Black Blond Brunette Red Blue 20 94 84 17 Brown 68 7 119 26 Green 5 16 29 14 Hazel 15 10 54 14