This paper discovers an inherent relationship between the survival model with covariate measurement error and the frailty model. The discovery motivates our using a frailty-based estimating equation to draw inference for the proportional hazards model with error-prone covariates. Our established framework accommodates general distributional structures for the error-prone covariates, not restricted to a linear additive measurement error model or Gaussian measurement error. When the conditional distribution of the frailty given the surrogate is unknown, it is estimated through a semiparametric copula function. The proposed copula-based approach enables us to fit flexible measurement error models without the curse of dimensionality as in nonparametric approaches, and to be applicable with an external validation study. Large sample properties are derived and finite sample properties are investigated through extensive simulation studies. The methods are applied to a study of physical activity in relation to breast cancer mortality in the Nurses’ Health Study.
Kim, Sehee; Li, Yi; and Spiegelman, Donna, "A Frailty Approach for Survival Analysis with Error-prone Covariate" (January 2013). The University of Michigan Department of Biostatistics Working Paper Series. Working Paper 102.