Combined estimating equation approaches for semiparametric transformation models with length-biased survival data

Survival data are subject to length-biased sampling when the survival times are left-truncated and the underlying truncation time random variable is uniformly distributed. Substantial efficiency gains can be achieved by incorporating the information about the truncation time distribution in the estimation procedure [Wang (1989) Journal of the American Statistical Association 84, 742-748; Wang (1996) Biometrika 83, 343-354]. Under the semiparametric transformation models, the maximum likelihood method is expected to be fully efficient, yet it is difficult to implement because the full likelihood depends on the nonparametric component in a complicated way. Moreover, its asymptotic properties have not been established. In this article, we extend the martingale estimating equation approach [Chen et al. (2002) Biometrika 89, 659-668; Kim et al. (2013) Journal of the American Statistical Association 108, 217-227] and the pseudo-partial likelihood approach [Severini and Wong (1992) The Annals of Statistics 4, 1768-1802; Zucker (2005) Journal of the American Statistical Association 100, 1264-1277] for semiparametric transformation models with right-censored data to handle left-truncated and right-censored data. In the same spirit of the composite likelihood method [Huang and Qin (2012) Journal of the American Statistical Association 107, 946-957], we further construct another set of unbiased estimating equations by exploiting the special probability structure of length-biased sampling. Thus the number of estimating equations exceeds the number of parameters, and efficiency gains can be achieved by solving a simple combination of these estimating equations. The proposed methods are easy to implement as they do not require additional programming efforts. Moreover, they are shown to be consistent and asymptotically normally distributed. A data analysis of a dementia study illustrates the methods.