In epidemiological cohort studies, the outcomes of interest are often times to failure events, such as cancer, heart disease and HIV infection, which are relatively rare even after a long period of follow-up; the study cohorts are usually chosen to be very large so as to yield reliable information about the effect of exposure variables on these rare failure times. In many cases, the exposure variables of interest are difficult or expensive to collect or measure. With limited funds, it could be impossible to obtain these variables for all subjects in a large cohort. Prentice (1986) proposed the case-cohort design, where the expensive exposure variables are obtained only for a random sample, called the subcohort, from the study cohort, as well as for subjects who have experienced the failure event during the follow-up period. Extensive research has been done on this design. Under the proportional hazards model, Prentice (1986) and Self & Prentice (1988) proposed pseudolikelihood approaches; Chen & Lo (1999) and Chen (2001) developed estimating equation methods; Marti & Chavance (2011) and Keogh & White (2013) proposed multiple imputation approaches; Scheike & Martinussen (2004) and Zeng & Lin (2014) considered maximum likelihood estimation; and Kang & Cai (2009) and Kim et al. (2013) developed weighted estimating equation methods for case-cohort studies with multiple outcomes. Other related cost-effective sampling schemes include outcome-dependent sampling designs (Zhou et al., 2002; Ding et al., 2014). All of these designs and methods are primarily focused on right-censored data, where the failure time of interest is either exactly observed or right-censored. In practice, however, the occurrences of some failure events, such as HIV infection and diabetes, are not accompanied by any symptoms and their determination relies on laboratory tests or physician diagnosis; the exact times to these failure events are not available.

the case-cohort study design for interval-censored failure time data, which arise when the failure time of interest is observed or known only to belong to a random time interval (Sun, 2006). Such data are often produced in epidemiological studies, biomedical follow-up studies, demographic studies and the social sciences, where the study subjects are examined for occurrence of the failure event only at discrete visits, rather than being continuously monitored. One example is the Atherosclerosis Risk in Communities study, a longitudinal epidemiological cohort study, where the participants’ health status was scheduled to be examined every three years on average. In this study, the occurrence of a disease such as diabetes was known only between two consecutive examinations, so only interval-censored data on time to disease were available. Interval-censoring is a general type of censoring that includes left- and right-censoring as special cases. If a participant had developed the disease at the first follow-up examination , we would have a left-censored observation . Here we consider the interval-censored case-cohort design in which the expensive exposure variables are obtained only for a subcohort that is a simple random sample of the study cohort, as well as for subjects who are known to have experienced the failure event, i.e., for whom the right endpoint of the observed interval is finite.

To the best of our knowledge, there is no published method to date that deals with the general interval-censored case-cohort design described above, although several papers discuss related issues. Gilbert et al. (2005) considered the case-cohort design for a HIV vaccine trial where they treated the midpoint of the finite observed interval as the exact HIV infection time and then employed the method that Self & Prentice (1988) developed for right-censored case-cohort data to perform the analysis. Li et al. (2008) presented a special interval-censored case-cohort design by assuming that the inspection time intervals are fixed and the same for all study subjects and that the number of time intervals does not change with the sample size. Li & Nan (2011) considered fitting the relative risk regression model to the case-cohort sampled current status data, a special case of interval-censored data that arise when each study subject is examined only once for the occurrence of the failure event, so that the failure time is either left- or right-censored at the only examination. In this paper, we consider the case-cohort study design for general interval-censored failure time and develop a novel semiparametric method for fitting the proportional hazards model to data arising from this design.

Many authors have studied regression analysis of interval-censored data, obtained by simple random sampling, under the proportional hazards model. Among others, Finkelstein (1986) considered maximum likelihood estimation with a discrete hazard assumption; Huang (1996) and Zeng et al. (2016) studied fully semiparametric maximum likelihood estimation for current status data and mixed-case interval-censored data, respectively; Satten (1996) proposed a marginal likelihood approach which avoids estimating the baseline hazard function but remains computationally intensive; Satten et al. (1998) developed a rank-based procedure using imputed failure times, where a parametric baseline hazard is assumed; Pan (2000) suggested a multiple imputation approach which is semiparametric, but did not provide a theoretical justification; Lin et al. (2015) and Wang et al. (2016) represented the cumulative baseline hazard function as a monotone spline and then developed methods from Bayesian and frequentist perspectives via two-stage Poisson data augmentations; and Zhang et al. (2010) proposed a spline-based sieve semiparametric maximum likelihood method and proved that the resulting regression parameter estimator is asymptotically normal and efficient. They also provided motivation for the sieve method, reasoning about the choice of basis functions, a theoretical framework, and rigorous proofs based on empirical process theory. Apart from having attractive asymptotic properties under various scenarios (see, e.g., Huang & Rossini, 1997; Shen, 1998; Xue et al., 2004), the sieve method is easy to implement and computationally fast as it usually involves many fewer parameters than a fully semiparametric method. In this paper, we focus on fitting the proportional hazards model to interval censored data from the case-cohort design. We employ inverse probability weighting to construct likelihood and then, following Zhang et al. (2010), develop a Bernstein polynomial-based sieve estimation method. We also present a weighted bootstrap procedure for variance estimation.

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