Cost-effectiveness study design and proper statistical inference methods allow biomedicine data to be collected with less cost while produce more efficient parameter estimators. Motivated by the need from our on-going environmental study in the Norwegian Mother and Child Cohort (MoBa) study, we consider an outcome-dependent sampling scheme (ODS) for failure time data with censoring. Like the case-cohort design, the ODS design enriches the observed sample by selectively including certain failure subjects. We present an estimated maximum semiparametric empirical likelihood estimation (EMSELE) under the proportional hazards model framework. The asymptotic properties of the proposed estimator were derived. Simulation studies were conducted to evaluate the small-sample performance of our proposed method against other standard designs and estimators. The standard fecundability odds ratio approach in the current epidemiology based on logistic regression yields inconclusive results with different time to pregnancy cutpoints. Our analyses show that the proposed estimator and design is more efficient than the current default approach and other competing approaches. Applying the proposed approach with the data set from the Norwegian Mother and Child Cohort Study, we found a significant effect of perfluorooctanoic acid (PFOA) on fecundability.
