A study on additive hazard model with additive frailty for semi-competing risks data

Abstract

In this study, we proposed an illness-death model with Lin and Ying (1994)''s additive hazard and additive frailty for the regression analysis on semi-competing risks problem in a general morbidity/mortality process.In the competing risks data, the occurrence of any one type of events precludes all other types of events and then prevents observing their occurring times. However, in the general morbidity/mortality process, death can preclude the relapse of colon cancer, but not vice versa. Although relapsed cancer patients are observed in the first event, the time to death for these patients can still be observed after relapse. That is called semi-competing risks data.Comparing with the Cox-type hazard, the additive hazard function is more natural and properly partitions the effect of the covariate on one transition into the other transition, namely, internal consistency in the illness-death model by Klein (2006). In the proposed model, we adapted the additive frailty to describe the association between the covariates and failure time in terms of the risk difference rather than the risk ratio.For the inference, we considered a full maximum likelihood on the complete data and incorporated an Estimation-Maximization algorithm to deal with frailty and Gauss-Laguerre quadrature method for calculating the expectations of the functions of frailty. We used the expected observed information matrix for standard errors of the parameter estimates. The piecewise constant baseline hazards are simply assumed to have different constants in the same intervals on three hazards such as relapse, death without relapse and death with relapse. The frailty was assumed to have a gamma distribution. In the simulation, we found that the regression estimates were robust when the intervals for the baseline hazards was mis-specified or the true frailty distribution deviated from the assumed gamma distribution. The proposed model was applied to the data from a national intergroup trial in the 1980''s to study the effectiveness of two adjuvant therapy regimens for the improvement of surgical cure rates in stage III colon cancer. We compared the group treated with levamisole plus fluorouracil with the untreated group using the semi-competing risks model with cancer recurrence and death.