A note on semiparametric efficient inference for two-stage outcome-dependent sampling with a continuous outcome

Outcome-dependent sampling designs have been shown to be a cost-effective way to enhance study efficiency. We show that the outcome-dependent sampling design with a continuous outcome can be viewed as an extension of the two-stage case-control designs to the continuous-outcome case. We further show that the two-stage outcome-dependent sampling has a natural link with the missing-data and biased-sampling frameworks. Through the use of semiparametric inference and missing-data techniques, we show that a certain semiparametric maximum-likelihood estimator is computationally convenient and achieves the semiparametric efficient information bound. We demonstrate this both theoretically and through simulation.