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.

An outcome-dependent sampling (ODS) design is a retrospective sampling scheme where one observes the primary exposure variables with a probability that depends on the observed value of the outcome variable. When the outcome of interest is failure time, the observed data are often censored. By allowing the selection of the supplemental samples depends on whether the event of interest happens or not and oversampling subjects from the most informative regions, ODS design for the time-to-event data can reduce the cost of the study and improve the efficiency. We review recent progresses and advances in research on ODS designs with failure time data. This includes researches on ODS related designs like case–cohort design, generalized case–cohort design, stratified case–cohort design, general failure-time ODS design, length-biased sampling design and interval sampling design.

In this paper, we study the constrained estimation in Cox's model for the rightcensored survival data and derive asymptotic properties of the constrained estimator by using the Lagrangian method based on Karush-Kuhn-Tucker conditions. A novel minorizationmaximization (MM) algorithm is developed for calculating the maximum likelihood estimates of the regression coefficients subject to box or linear inequality restrictions in the proportional hazards model. The first M-step of the proposed MM algorithm is to construct a surrogate function with a diagonal Hessian matrix, which can be reached by utilizing the convexity of the exponential function and the negative logarithm function. The second M-step is to maximize the surrogate function with a diagonal Hessian matrix subject to box constraints, which is equivalent to separately maximizing several one-dimensional concave functions with a lower bound and an upper bound constraint, resulting in an explicit solution via a median function. The ascent property of the proposed MM algorithm under constraints is theoretically justified. Standard error estimation is also presented via a non-parametric bootstrap approach. Simulation studies are performed to compare the estimations with and without constraints. Two real data sets are used to illustrate the proposed methods.

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An outcome‐dependent selection (ODS) is a retrospective sampling scheme where one observes the environmental exposure/covariates with a probability that depends on the observed value of the outcome variable. It is an alternative to the more standard random sampling design. By allowing the selection probability of each individual in the ODS sample to depend on the outcome, the investigators attempt to enhance the efficiency and reduce the cost of the study. The case‐control study in epidemiology is a well‐known example of an ODS scheme with a binary outcome variable.