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

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Jing_2020_On_the_correspondence_RSOS_191483.pdf (350.6Kb)
Date
15/01/2020
Author
Jing, Wei
Papathomas, Michail
Keywords
Contingency table
Generalized linear modelling
Categorical variables
QA Mathematics
T-NDAS
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Abstract
Consider a set of categorical variables P 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 P∖{Y} become obsolete. We illustrate the derived results with the analysis of a real dataset.
Citation
Jing , W & Papathomas , M 2020 , ' On the correspondence of deviances and maximum-likelihood and interval estimates from log-linear to logistic regression modelling ' , Royal Society Open Science , vol. 7 , no. 1 , 191483 . https://doi.org/10.1098/rsos.191483
Publication
Royal Society Open Science
Status
Peer reviewed
DOI
https://doi.org/10.1098/rsos.191483
ISSN
2054-5703
Type
Journal article
Rights
Copyright © 2020 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited.
Description
Funding: The first author would like to acknowledge the support of the School of Mathematics and Statistics, as well as CREEM, at the University of St Andrews, and the University of St Andrews St Leonard’s 7th Century Scholarship.
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  • University of St Andrews Research
URL
http://dx.doi.org/10.1098/rsos.191483
URI
http://hdl.handle.net/10023/19286

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