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Consider a set of categorical variables $\mathcal{P}$ where at least one,denoted by $Y$, is binary. The log-linear model that describes the counts inthe resulting contingency table implies a specific logistic regression model,with the binary variable as the outcome. Extending results in Christensen(1997), by also considering the case where factors present in the contingencytable disappear from the logistic regression model, we prove that the MaximumLikelihood Estimate (MLE) for the parameters of the logistic regression equalsthe MLE for the corresponding parameters of the log-linear model. We provethat, asymptotically, standard errors for the two sets of parameters are alsoequal. Subsequently, Wald confidence intervals are asymptotically equal. Theseresults demonstrate the extent to which inferences from the log-linearframework can be translated to inferences within the logistic regressionframework, on the magnitude of main effects and interactions. Finally, we provethat the deviance of the log-linear model is equal to the deviance of thecorresponding logistic regression, provided that the latter is fitted to adataset where no cell observations are merged when one or more factors in$\mathcal{P} \setminus \{ Y \}$ become obsolete. We illustrate the derivedresults with the analysis of a real dataset.

On the correspondence of deviances and maximum likelihood and interval estimates from log-linear to logistic regression modelling