Abstract
Consider a set of categorical variables where at least one, denoted by Y, is binary. The log-linear model that describes the contingency table counts implies a logistic regression model, with outcome Y. Extending results from Christensen (1997, Log-linear models and logistic regression, 2nd edn. New York, NY, Springer), we prove that the maximum-likelihood estimates (MLE) of the logistic regression parameters equals the MLE for the corresponding log-linear model parameters, also considering the case where contingency table factors are not present in the corresponding logistic regression and some of the contingency table cells are collapsed together. We prove that, asymptotically, standard errors are also equal. These results demonstrate the extent to which inferences from the log-linear framework translate 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 no cell observations are collapsed together when one or more factors in become obsolete. We illustrate the derived results with the analysis of a real dataset.
Highlights
Let v = {v1, ..., vn} denote a set of observations, θ = {θ1, ..., θn} a set of parameters, and consider known or nuisance quantities φ = {φ1, ..., φn}
By considering the case where factors present in the contingency table and log-linear model are not present in the corresponding logistic regression model, and some of the contingency table cells are collapsed together
When factors are not present in the logistic regression, one may choose to collapse the counts in the contingency table cells that are only discriminated by the obsolete variables x.2, ..., x.q
Summary
1⁄4 log(m jYþ1,jX,jZ ) À log(m jY,jX,jZ ): For more details, see [1, Section 4.6] where, in addition to the above approach, the alternative of constructing a multinomial model to model the log-odds of an observation at level jY, jY = 1, ..., JY − 1, relative to one at fixed level JY is considered In this manuscript, we focus on the association between log-linear modelling and binary logistic regression. 11], results on the equivalence between MLE and confidence intervals were derived We extend these results, by considering the case where factors present in the contingency table and log-linear model are not present in the corresponding logistic regression model, and some of the contingency table cells are collapsed together. We conclude with a discussion, where we consider possible practical implications of our results
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