White Paper for the Improved Accuracy of Statistical Tests in Quantal Response Using Firth's Penalized Likelihood

Sensitivity testing is a type of testing in which a stressor (independent variable) is continuous, and the response (dependent variable) is binary. Ballistic limit testing is a type of sensitivity testing where the stressor is the velocity of a kinetic energy threat, and the response is penetration (either a partial or complete penetration) of an armor target. During ballistic limit testing, both the threat velocity and the penetration response are recorded for each shot. Then, the data are analyzed to model the probability of complete penetration as a function of threat velocity.In sensitivity testing, it is often desirable to use the location-scale parameterization instead of the linear parameterization. Maximum likelihood estimates are known to have small sample bias. Firths logistic regression may be used to reduce this bias. Another advantage to Firths logistic regression is it can be used to determine a unique solution when there is separation in the data. In ballistic limit testing, separation in the data is often described as there being no zone of mixed results (i.e., no overlap in partial and complete penetrations). Finally, penalized likelihood ratio tests, based on Firths logistic regression, may be used to improve the accuracy of statistical tests which is the focus of this paper.