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Biostatistics Advance Access published online on September 29, 2008
Biostatistics, doi:10.1093/biostatistics/kxn030
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© The Author 2008. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org.

Generalized linear models with unspecified reference distribution

Paul J. Rathouz*

Department of Health Studies, University of Chicago, 5841 South Maryland Avenue, MC 2007, Chicago, IL 60637, USA prathouz{at}uchicago.edu

Liping Gao

Department of Ecology and Evolution, University of Chicago, 5801 South Ellis Avenue, Chicago, IL 60637, USA

* To whom correspondence should be addressed.

We propose a new class of semiparametric generalized linear models. As with existing models, these models are specified via a linear predictor and a link function for the mean of response Y as a function of predictors X. Here, however, the "baseline" distribution of Y at a given reference mean µ0 is left unspecified and is estimated from the data. The response distribution when the mean differs from µ0 is then generated via exponential tilting of the baseline distribution, yielding a response model that is a natural exponential family, with corresponding canonical link and variance functions. The resulting model has a level of flexibility similar to the popular proportional odds model. Maximum likelihood estimation is developed for response distributions with finite support, and the new model is studied and illustrated through simulations and example analyses from aging research.

Keywords: Baseline distribution; Canonical link; Density ratio model; Exponential tilting; Linear exponential family; Natural exponential family; Quasi-likelihood; Semiparametric model

Received July 20, 2007; revised April 23, 2008; accepted for publication July 24, 2008.


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