Empirical likelihood for linear transformation models with interval-censored failure time data

AMS subject classifications: 62G05 62G10 62F12 Keywords: Confidence intervals/regions Coverage probability Interval-censored failure time data Jackknife empirical likelihood Linear transformation models Estimating equations a b s t r a c t For regression analysis of interval-censored failure time data, Zhang et al. (2005) [40] proposed an estimating equation approach to fit linear transformation models. In this paper, we develop two empirical likelihood (EL) inference approaches for the regression parameters based on the generalized estimating equations. The limiting distributions of log-empirical likelihood ratios are derived and empirical likelihood confidence intervals for any specified component of regression parameters are obtained. We carry out extensive simulation studies to compare the proposed methods with the method discussed by Zhang et al. (2005) [40]. The simulation results demonstrate that the EL and jackknife EL methods for linear transformation models have better performance than the existing normal approximation method based on coverage probability of confidence intervals in most cases, and they enable us to overcome an under-coverage problem for the confidence intervals of the regression parameters using a normal approximation when sample sizes are small and right censoring is heavy. Two real data examples are provided to illustrate our procedures. 1. Introduction Let T denote the time of occurrence of an event of interest. We say that T is interval-censored if its true value is not observed but only known to lie in an interval, say, (L, R]. ''Such data are frequently observed in clinical trials or longitudinal studies that entail periodic follow-ups'' (see [40]). For example, after radiotherapy, a cancer patient is often required to visit the physician regularly to examine whether or not there is disease progression. If progression was not observed at, say, the 1-year follow-up but was at the 1.5-year follow-up, then the disease progression time would be known to be in (1, 1.5], contributing an interval-censored failure time. In this paper we focus on regression analysis of interval-censored failure time data. To this end, several methods have been proposed in the literature. For a comprehensive review, see [29,39]. In particular, [40] considered a class of linear transformation models which contain the proportional hazards model and the proportional odds model as special cases. In that paper they proposed an estimating equation approach to estimate the regression parameters and showed that the estimators always exist, are unique and consistent. To draw inference, they also established a normal approximation for the asymptotic distribution of …