Simple polynomial multiplication algorithms for exact conditional tests of linearity in a logistic model

Abstract

The linear logistic model is often employed in the analysis of binary response data. The well-known asymptotic chi-square and likelihood ratio tests are usually used to detect the assumption of linearity in such a model. For small, sparse, or skewed data, the asymptotic theory is however dubious and exact conditional chi-square and likelihood ratio tests may provide reliable alternatives. In this article, we propose efficient polynomial multiplication algorithms to compute exact significance levels as well as exact powers of these tests. Two options, namely the cell- and stage-wise approaches, in implementing these algorithms will be discussed. When sample sizes are large, we propose an efficient Monte Carlo method for estimating the exact significance levels and exact powers. Real data are used to demonstrate the performance with an application of the proposed algorithms.