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

TL;DR

This paper establishes the theoretical equivalence between log-linear and logistic regression models in terms of maximum likelihood estimates, standard errors, confidence intervals, and deviance, enabling translation of inferences between these frameworks.

Contribution

It extends Christensen's results by considering disappearing factors, proving the asymptotic equality of estimates, standard errors, confidence intervals, and deviances between log-linear and logistic models.

Findings

MLEs of parameters are equal in both models.

Asymptotic standard errors and Wald confidence intervals are equal.

Deviances of the models are equal under specified conditions.

Abstract

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.