Estimating effect of environmental contaminants on women's subfecundity for the MoBa study data with an outcome-dependent sampling scheme

Abstract

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 (ODS) scheme 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. 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 MoBa study, we found a significant effect of an environmental contaminant on fecundability.

Recent progresses in outcome-dependent sampling with failure time data

“…To take advantage of the ODS scheme for right-censored data to yield more powerful and efficient inferences, Ding et al (2014) proposed a general failure-time ODS sampling design . In such a general failure-time ODS design, a random sample (SRS) from the full cohort is selected.…”

“…The likelihood function based on the observed data from the general failure-time ODS design is proportional to $$L_4(\beta ,Q_Z,{\Lambda }_0,S_C)=true[\prod _{i\in S_0}{\left\{f_{\beta ,{\Lambda }_0}(T_i\mid Z_i)\right\}}^{{\Delta }_i}{\left\{{\overline{F}}_{\beta ,{\Lambda }_0}(T_i\mid Z_i)\right\}}^{1-{\Delta }_i}true]\times true[\prod _{k=1}^K\prod _{i\in S_k}{f}_{\beta ,{\Lambda }_0}(T_i\mid Z_i)true]\times true[\prod _{k=0}^K\prod _{i\in S_k}{q}_Z\left(Z_i\right)true]\times true[\prod _{k=1}^K{\left\{{\int }_{\mathcal{Z}}{\int }_{A_k}{f}_{\beta ,{\Lambda }_0}(t\mid Z)S_C\left(t\right)dtd{Q}_Z\left(Z\right)\right\}}^{-n_k}true],$$ where f β , Λ 0 ( t | Z ) and F̄ β , Λ 0 ( t | Z ) are the conditional density function and survival function of T̃ given Z with the baseline cumulative hazard function Λ 0 ( t ), respectively, Q Z (·) and q Z (·) denote the cumulative distribution and density function of Z , respectively, and S C ( t ) are the survival function of the censoring time C . Because the nonparametric portion ( Q Z , Λ 0 , S C ) cannot be separated from the above likelihood function (19) that combines both the conditional parametric likelihood and the marginal semiparametric likelihood, Ding et al (2014) developed an estimated maximum semiparametric empirical likelihood approach for estimation of the regression parameter.…”

“…The general failure-time ODS design proposed by Ding et al (2014) is an improvement over the case–cohort design and the generalized case–cohort design, because the general failure-time ODS design allows the sample selection of cases to depend on the timing of disease endpoints, i.e., by oversampling subjects from the most informative regions. To reap in the benefit of such a general failure-time ODS design, Ding et al (2014) developed a new inferential method and provided an estimated maximum semiparametric empirical likelihood estimator for the parameters of primary interest under the proportional hazards model.…”

“…To reap in the benefit of such a general failure-time ODS design, Ding et al (2014) developed a new inferential method and provided an estimated maximum semiparametric empirical likelihood estimator for the parameters of primary interest under the proportional hazards model. For the additive hazards model, which focuses on risk differences rather than risk ratios, Yu et al (2015) studied a weighted pseudo-score estimating procedure for estimation of regression parameter.…”

Time‐dependent classification accuracy curve under marker‐dependent sampling

“…In biomedical literature, similar ideas have been successfully used to improve efficiency of evaluating the accuracy of a biomarker in classifying disease status (e.g. Morra et al, 2007; Strauss et al, 2010; Williams et al, 2014; Selen et al, 2014; Ding et al, 2014; Schildcrout et al, 2015). …”