An Estimated Likelihood Method for Continuous Outcome Regression Models With Outcome-Dependent Sampling

Abstract

Many biomedical observational studies attempt to relate a continuous outcome to an environmental exposure and other important covariates. If the outcome is easier or cheaper to measure relative to the exposure of interest, then the outcome may be observed for every member of a finite-study population, whereas exposure measurements may be obtained only for a relatively small subsample of this population. Rather than selecting a simple random subsample of individuals for exposure measurement, investigators may attempt to enhance study efficiency by allowing the selection probabilities to depend on the observed outcomes; we refer to such sampling schemes as outcome-dependent sampling (ODS). Standard estimation methods that ignore the ODS design will yield biased and inconsistent parameter estimates. Furthermore, it is generally desirable to use estimators that incorporate all available data as analyses restricted to subjects with complete information are inefficient. To this end, we extend an estimated likelihood method, originally developed for discrete outcome measurement error problems in which accurate exposure measurements are made only for a simple random “validation” sample, to allow for continuous outcomes and ODS designs. We derive the asymptotic properties of the proposed estimator and use simulated data to show that the asymptotic results closely approximate the finite-sample properties in samples of moderate size. We also use simulated data to compare the performance of our proposed estimator with that of existing methods applicable to the ODS problem.