Ascertainment Conditional Maximum Likelihood for Continuous Outcome Under Two-Phase Response-Selective Design

Data collection procedures are often time-consuming and expensive. An alternative to collecting full information from all subjects enrolled in a study is a two-phase design: Variables that are inexpensive or easy to measure are obtained for the study population, and more specific, expensive, or hard-to-measure variables are collected only for a well-selected sample of individuals. Often, only these subjects that provided full information are used for inference, while those that were partially observed are discarded from the analysis. Recently, semiparametric approaches that use the entire dataset, resulting in fully efficient estimators, have been proposed. These estimators, however, have challenges incorporating multiple covariates, are computationally expensive, and depend on tuning parameters that affect their performance. In this paper, we propose an alternative semiparametric estimator that does not pose any distributional assumptions on the covariates or measurement error mechanism and can be applied to a wider range of settings. Although the proposed estimator is not semiparametric efficient, simulations show that the loss of efficiency to estimate the parameters associated with the partially observed covariates is minimal. We highlight the estimator's applicability to real-world problems, where data structures are complex and rich, and complicated regression models are often necessary.