An Equivalence of Conditional and Unconditional Maximum Likelihood Estimators via Infinite Replication of Observations

Motivated by an application with complex survey data, we show that for logistic regression with a simple matched-pairs design, infinitely replicating observations and maximizing the conditional likelihood results in an estimator exactly identical to the unconditional maximum likelihood estimator based on the original sample, which is inconsistent. Therefore, applying conditional likelihood methods to a pseudosample with observations replicated a large number of times can lead to an inconsistent estimator; this casts doubt on one possible approach to conditional logistic regression with complex survey data. We speculate that for more general designs, an asymptotic equivalence holds.