  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. Extending results in Christensen
(1997), by also considering the case where factors present in the contingency
table disappear from the logistic regression model, we prove that 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 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.
