Loglinear model

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Multinomial data (from contingency tables) can be fitted by using a generalized linear model with a Poisson response distribution and a log link function. The resulting models for counts in the cells of a contingency table are known as loglinear models in which the logarithm of the expected value of a count variable is modelled as a linear function of parameters. It allows more than two discrete variables to be analyzed and interactions between any combination of them to be identified. Data sets with a binary response (outcome) variable and a set of explanatory variables that are all categorical can be modelled either by logistic regression or by loglinear modelling.
In loglinear modelling, the counts in the cells of the contingency table are treated as values of the response variable rather than either of the categorical variables defining the rows and columns. A loglinear model relates the distribution of the counts to the values of the explanatory variables (rows and columns), and tests for the presence of an interaction between them. Whether the interaction term is necessary is tested by fitting two models, one without and one with the interaction term, and using the difference between the regression deviance values for the two models (as well as the difference in df). If the SP obtained by the test test is small, the interaction term should be in the model. This corresponds to a significant difference in Fisher's exact test or the -test. For a 2x2 table, the difference between the regression deviances in the two models is
the same as the residual deviance in the model with no interaction term. In loglinear modelling, it makes no sense if any main effect is omitted.
In loglinear modelling (as in other GLMs), the saturated model which includes all interactions fits the data exactly, and the fitted values are exactly equal to the observed values (the residual deviance is zero). The residual deviance of any other model can be
used to test how worse it is compared to the saturated model. A small SP arising from a high residual deviance would mean that it does not fit better than the saturated model (the alternative model is rejected). If the associated SP is not small, then the model is
not significantly worse than the saturated model (i.e., the exact fit).
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