An asymptotic result of conditional logistic regression estimator

• Published in: Communications in statistics. Theory and methods Vol. 52; no. 13; pp. 4729 - 4740
• Main Authors: He, Zhulin, Ouyang, Yuyuan
• Format: Journal Article
• Language: English
• Published: Philadelphia Taylor & Francis 03.07.2023; Taylor & Francis Ltd
• Subjects: Asymptotic properties; Cauchy's differentiation formula; Clusters; conditional logistic regression; Data points; Maximum likelihood estimators; method of steepest descent; Ordinary logistic regression; Polynomials; Regression; Steepest descent method; survey sampling
• ISSN: 0361-0926; 1532-415X
• DOI: 10.1080/03610926.2021.1999978

In cluster-specific studies, ordinary logistic regression and conditional logistic regression for binary outcomes provide maximum likelihood estimator (MLE) and conditional maximum likelihood estimator (CMLE), respectively. In this paper, we show that CMLE is approaching to MLE asymptotically when each individual data point is replicated infinitely many times. Our theoretical derivation is based on the observation that a term appearing in the conditional average log-likelihood function is the coefficient of a polynomial, and hence can be transformed to a complex integral by Cauchy's differentiation formula. The asymptotic analysis of the complex integral can then be performed using the classical method of steepest descent. Our result implies that CMLE can be biased if individual weights are multiplied with a constant, and that we should be cautious when assigning weights to cluster-specific studies.