Log-Linear Models: Definition
This chapter introduces log-linear models which are the most widely used simple structures in the analysis of categorical data. Their simplicity comes from a multiplicative structure, where the multipliers depend on subsets of the variables, but not on all variables together. These subsets are the allowed interactions, and larger subsets of variables exhibit no interaction, as measured by the conditional odds ratio. When the logarithm of this multiplicative structure is taken, one obtains a linear structure on the logarithmic scale. More formally, a log-linear model is defined by a mixed parameterization of the distributions on the contingency table, introduced first. It consists of conditional odds ratios on an ascending class of subsets of the variables and of marginal distributions on the complement descending class. The log-linear model is obtained by setting all the conditional odds ratios on the descending class equal to 1. The resulting models include generalizations of independence for three-way tables, which are discussed in detail, like joint, multiple, and conditional independence and also the model of no second-order interaction. Log-linear models may also be equivalently defined through defining a log-linear representation of the cell probabilities and then setting some of the log-linear parameters to zero.
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