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

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 in the resulting contingency table implies a specific logistic regression model, with the binary variable as the outcome. We prove that, when the log-linear model is the largest of all log-linear models that correspond to the logistic regression, the Maximum Likelihood Estimate (MLE) for the parameters of the logistic regression equals the MLE for the corresponding parameters of the log-linear model. We prove that, asymptotically, standard errors for the two sets of parameters are also equal. Subsequently, Wald confidence intervals are asymptotically equal. These results demonstrate the extent to which inferences from the log-linear framework can be translated to inferences within the logistic regression framework, on the magnitude of main effects and interactions. Finally, we prove that when the log-linear model is the largest of all models that correspond to the logistic regression, the deviance of the log-linear model is equal to the deviance of the corresponding logistic regression, provided that the latter is fitted to a dataset where no cell observations are merged when one or more factors in $\mathcal{P} \setminus \{ Y \}$ become obsolete. We illustrate the derived results with the analysis of a real dataset.