Monotone Bounds in Logistic Regression

• Activity Number: 86
• Type: Contributed
• Date/Time: Monday, August 4, 2003 : 8:30 AM to 10:20 AM
• Sponsor: Section on Physical and Engineering Sciences
• Author(s): Friedrich W. Scholz*+
• Companies: Boeing Phantom Works
• Keywords: logistic regression ; confidence bounds ; monotone ; inversion ; bootstrap

Abstract

We consider the following logistic regression model for binary response data: p(t_i)=P(Y_i=1)=1-P(Y_i=0) with \log(p(t_i)/(1-p(t_i)))=a+b t_i for given stimulus levels t_1,\ldots, t_n and unknown parameters a and b. Often it is known that p(t) is an increasing function of t. The corresponding traditional confidence intervals for p(t) do not necessarily share this monotonicity property, depending on the data quality and the confidence level. As a result one cannot invert such bounds to yield the dual confidence bounds for the quantiles associated with p(t) and one is forced into a separate approach for this problem. We present a natural remedy that leads to monotone bounds that can be inverted, i.e., the pointwise lower bound curve for p(t) can dually serve as pointwise upper bound curve for p-quantiles. The remedy is based on a large sample bivariate normal approximations coupled with an exact integration and a root solving step. A bootstrap alternative is offered to improve on the possibly poor bivariate normal approximation. Finally the model is extended to situations where a and b are modeled as linear combinations of known covariates.