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Efficient logistic regression designs under an imperfect population identifier

Motivated by actual study designs, this article considers efficient logistic regression designs where the population is identified with a binary test that is subject to diagnostic error. We consider the case where the imperfect test is obtained on all participants, while the gold standard test is measured on a small chosen subsample. Under maximum‐likelihood estimation, we evaluate the optimal design in terms of sample selection as well as verification. We show that there may be substantial efficiency gains by choosing a small percentage of individuals who test negative on the imperfect test for inclusion in the sample (e.g., verifying 90% test‐positive cases). We also show that a two‐stage design may be a good practical alternative to a fixed design in some situations. Under optimal and nearly optimal designs, we compare maximum‐likelihood and semi‐parametric efficient estimators under correct and misspecified models with simulations. The methodology is illustrated with an analysis from a diabetes behavioral intervention trial.