On quantifying expert opinion about multinomial models that contain covariates

This paper addresses the task of forming a prior distribution to represent expert opinion about a multinomial model that contains covariates. The task has not previously been addressed. We suppose the sampling model is a multinomial logistic regression and represent expert opinion about the regression coefficients by a multivariate normal distribution. This logistic-normal model gives a flexible prior distribution that can capture a broad variety of expert opinion. The challenge is to (i) find meaningful assessment tasks that an expert can perform and which should yield appropriate information to determine the values of parameters in the prior distribution, and (ii) develop theory for determining the parameter values from the assessments. A method is proposed that meets this challenge. The method is implemented in interactive user-friendly software that is freely available. It provides a graphical interface that the expert uses to assess quartiles of sets of proportions and the method determines a mean vector and a positive-definite covariance matrix to represent the expert's opinions. The chosen assessment tasks yield parameter values that satisfy the usual laws of probability without the expert being aware of the constraints this imposes. Special attention is given to feedback that encourages the expert to consider his/her opinions from a different perspective. The method is illustrated in an example that shows its viability and usefulness