Confidence intervals in general regression models that utilize uncertain prior information

TL;DR

This paper develops a confidence interval for a parameter in a regression model that effectively incorporates uncertain prior information, ensuring good coverage and reduced expected length when the prior is correct.

Contribution

It introduces a method to construct confidence intervals that utilize uncertain prior information in general regression models, extending previous linear regression approaches.

Findings

The proposed interval maintains good coverage properties.

Expected length is substantially reduced when prior information is correct.

The method performs well in practical bioassay applications.

Abstract

We consider a general regression model, without a scale parameter. Our aim is to construct a confidence interval for a scalar parameter of interest $\theta$ that utilizes the uncertain prior information that a distinct scalar parameter $\tau$ takes the specified value $t$. This confidence interval should have good coverage properties. It should also have scaled expected length, where the scaling is with respect to the usual confidence interval, that (a) is substantially less than 1 when the prior information is correct, (b) has a maximum value that is not too large and (c) is close to 1 when the data and prior information are highly discordant. The asymptotic joint distribution of the maximum likelihood estimators $\theta$ and $\tau$ is similar to the joint distributions of these estimators in the particular case of a linear regression with normally distributed errors having known variance. This similarity is used to construct a confidence interval with the desired properties by using the confidence interval, computed using the R package ciuupi, that utilizes the uncertain prior information in this particular linear regression case. An important practical application of this confidence interval is to a quantal bioassay carried out to compare two similar compounds. In this context, the uncertain prior information is that the hypothesis of "parallelism" holds. We provide extensive numerical results that illustrate the properties of this confidence interval in this context.