Loglinear Model - (More...)

An important constraint on the choice of models to fit, i.e., which terms and which interactions to include, is that whether any marginal totals (row or column) for combinations of terms are fixed in advance. If this applies to any combination of terms, their interaction must be included (the main effects and this particular interaction make up the null model for such data). When only the overall sample size is fixed in advance, there is no constraint on the models that can be fit.
When there are more than two variables, whether any interaction can be omitted from the model can be tested by fitting a model containing all the interactions of a particular order (say, second-order interactions). Then, new models omitting each interaction are fitted. Each new model's regression deviance is compared with the regression deviance of the model containing all interactions. If the resulting difference in regression deviances (and df) results in a large SP, that interaction can be omitted (it is not significantly different but has fewer terms; or it does not significantly contribute to the model).
The loglinear model can be used to find a conditional probability involving the factors in the contingency table for one of the factors chosen as a response variable. When the factor chosen as a response variable is a binomial one, logistic regression can be used
to analyze the same data (a binomial response variable and categorical explanatory variables). Two-factor interactions involving the response factor, in loglinear model, correspond to main effects of the explanatory variables in logistic regression. Logistic regression substitutes the loglinear model with equal results as long as the fitted loglinear model includes the main effects of the response variable and a saturated model for all the explanatory variables. (Online Loglinear Analysis.)