A Large-Sample Confidence Interval for the Inverse Prediction of Quantile Differences in Logistic Regression for Two Independent Tests

摘要

Logistic regression is a nonlinear method used for modeling a dichotomous (i.e., binary) response variable as a function of covariates. Such models have wide applicability and have proved especially useful in the health sciences where the question of effective dose (ED) or lethal dose (LD) is a central issue. For example, in toxicology studies, LD_(50) traditionally denotes the dose at which 50% of treated individuals perish. More generally, ED_(100p) is the dose level corresponding to an average given percentage, 100p (i.e., the 100pth quantile) of individuals responding to a treatment. In our field of flight test, logistic regression has a similar wide applicability. It is common to record a response as {hit, miss} or {success, failure} and to count the number of response successes at each level of input; the response is thus a quantal variable. In the example we develop here, the interest is in an application of blip-scan radar where the response is Y- {detect, no detect} and the covariate is range from target. In particular, we are interested in obtaining a confidence interval for the difference in range between two same-percentile values, one from each of two independent flights. The difference may be due to a different radar equipment configuration on each of the two flights and engineers are interested in quantifying the size of this difference in the detection performance. We approach the problem analytically and derive a symmetric confidence interval approximation for the average difference that is straightforward to compute and does not require simulation. Our results are based on the large-sample properties of maximum likelihood estimates and extend a result in nonlinear modeling given by Schwenke and Milliken (1991). The confidence interval so constructed is shown to give good probability coverage. Monte Carlo simulation is used to evaluate the procedure.