Statistical Modelling 15 (4) (2015), 345–365

Eliciting prior distributions for extra parameters in some generalized linear models

Fadlalla G. Elfadaly
Department of Mathematics and Statistics,
The Open University,
UK
e-mail: fadlalla.elfadaly@open.ac.uk

and


Department of Statistics,
Cairo University,
Egypt


Paul H. Garthwaite
Department of Mathematics and Statistics,
The Open University,
UK


Abstract:

To elicit an informative prior distribution for a normal linear model or a gamma generalized linear model (GLM), expert opinion must be quantified about both the regression coefficients and the extra parameters of these models. The latter task has attracted comparatively little attention. In this article, we introduce two elicitation methods that aim to complete the prior structure of the normal and gamma GLMs. First, we develop a method of assessing a conjugate prior distribution for the error variance in normal linear models. The method quantifies an expert's opinions through assessments of a median and conditional medians. Second, we propose a novel method for eliciting a lognormal prior distribution for the scale parameter of gamma GLMs. Given the mean value of a gamma distributed response variable, the method is based on conditional quartile assessments. It can also be used to quantify an expert's opinion about the prior distribution for the shape parameter of any gamma random variable, if the mean of the distribution has been elicited or is assumed to be known. In the context of GLMs, the mean value is determined by the regression coefficients. Interactive graphics is the medium through which assessments for the two proposed methods are elicited. Examples illustrating use of the methods are given. Computer programs that implement both methods are available.

Keywords:

Elicitation; Subjective prior distribution; normal linear model; gamma distribution; generalized linear model; interactive graphical software.

Downloads:

Example data and code can be found here.
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