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

ABSTRACT

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, LD₅₀ traditionally denotes the dose at which 50% of treated individuals perish. More generally, is the dose level corresponding to an average given percentage, 100p (i.e., the 100th 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.

KEYWORDS:

• calibration
• confidence interval
• delta method
• effective dose
• inverse prediction
• LD₅₀
• logistic regression
• percentile
• quantal response
• quantile
• radar
• Wald

ACKNOWLEDGMENT

The authors thank two anonymous reviewers for their valuable comments.

Notes

¹ This is often a particular problem when the number of observations is small or the number of estimated parameters is large. In the present article neither of these two conditions apply.

²

³ Note that we have designed the curves to cross over each other to demonstrate that our method deals with this scenario as well.

⁴ Using the R language function: rbinom(1,1,p[i]).

⁵ Adapted from Kendall and Stuart (1961, §10.6).

Additional information

Notes on contributors

Arnon Hurwitz

Dr. Arnon Hurwitz Statistical Methods Group, Mathematical Statistician, Edwards AFB, CA, 812 TSS/ENT. Dr. Hurwitz earned BS, BS (Hons), PhD – Mathematical Statistics – University of Cape Town, South Africa, MS – Applied Statistics – Oxford University, UK. Dr. Hurwitz has an extensive background in industrial applications of statistics, and was a Lecturer in Economics & Statistics, Graduate School of Business, UCT. E-mail: [email protected].

Todd Remund

Mr. Todd Remund earned Bachelor and Master of Science degrees in statistics from Brigham Young University. Since 2010 he has worked as a mathematical statistician in the statistics department, Edwards Air Force Base. From 2007 to 2010 he worked as a mathematician at Alliant Techsystems in signal processing, data analysis for shuttle flight, and RSRM static test. E-mail: [email protected].