Fitting additive risk models using auxiliary information

There has been a growing interest in incorporating auxiliary summary information from external studies into the analysis of internal individual-level data. In this paper, we propose an adaptive estimation procedure for an additive risk model to integrate auxiliary subgroup survival information via a penalized method of moments technique. Our approach can accommodate information from heterogeneous data. Parameters to quantify the magnitude of potential incomparability between internal data and external auxiliary information are introduced in our framework while nonzero components of these parameters suggest a violation of the homogeneity assumption. We further develop an efficient computational algorithm to solve the numerical optimization problem by profiling out the nuisance parameters. In an asymptotic sense, our method can be as efficient as if all the incomparable auxiliary information is accurately acknowledged and has been automatically excluded from consideration. The asymptotic normality of the proposed estimator of the regression coefficients is established, with an explicit formula for the asymptotic variance-covariance matrix that can be consistently estimated from the data. Simulation studies show that the proposed method yields a substantial gain in statistical efficiency over the conventional method using the internal data only, and reduces estimation biases when the given auxiliary survival information is incomparable. We illustrate the proposed method with a lung cancer survival study.