  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.
