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

An Asymptotic Result of Conditional Logistic Regression Estimator