## Inverse Confidence Limits

In bioassay problems, estimates of the values of the independent
variables that yield a desired response are often needed.
For instance, the value yielding a 50% response rate
(called the ED50 or LD50) is often used.
The INVERSECL option requests that confidence
limits be computed for the value of the independent
variable that yields a specified response.
These limits are computed only for the first
continuous variable effect in the model.
The other variables are set either at their
mean values if they are continuous or at the
reference (last) level if they are discrete variables.
For a discussion of inverse confidence
limits, refer to Hubert, Bohidar, and Peace (1988).
For the PROBIT procedure, the response variable is a probability.
An estimate of the first continuous variable value needed
to achieve a response of *p* is given by

where *F* is the cumulative distribution function used to model the
probability, **x**^{*} is the vector of independent variables
excluding the first one, **b**^{*} is the vector of parameter
estimates excluding the first one, and *b*_{1} is the estimated
regression coefficient for the independent variable of interest.
Note that, for both binary and ordinal models, the INVERSECL
option provides
estimates of the value of *x*_{1}
yielding Pr( first response level) = *p*,
for various values of *p*.
This estimator is given as a ratio of random
variables, for example, *r*=*a*/*b*.
Confidence limits for this ratio can
be computed using Fieller's theorem.
A brief description of this theorem follows.
Refer to Finney (1971) for a more complete
description of Fieller's theorem.

If the random variables *a* and *b* are thought to
be distributed as jointly normal, then for any fixed
value *r* the following probability statement holds
if *z* is an quantile from the
standard normal distribution and **V** is
the variance-covariance matrix of a and *b*.

Usually the inequality can be solved
for *r* to yield a confidence interval.
The PROBIT procedure uses a value of 1.96 for *z*, corresponding
to an value of 0.05, unless the goodness-of-fit
*p*-value is less than the specified
value of the HPROB= option.
When this happens, the covariance matrix
is scaled by the heterogeneity factor, and
a *t* distribution quantile is used for *z*.

It is possible for the roots of the equation
for *r* to be imaginary or for the confidence
interval to be all points outside of an interval.
In these cases, the limits are set
to missing by the PROBIT procedure.

Although the normal and logistic distribution give
comparable fitted values of *p* if the empirically observed
proportions are not too extreme, they can give appreciably
different values when extrapolated into the tails.
Correspondingly, the estimates of the confidence limits
and dose values can be different for the two distributions
even when they agree quite well in the body of the data.
Extrapolation outside of the range of the actual
data is often sensitive to model assumptions,
and caution is advised if extrapolation is necessary.

Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.