Semiparametric analysis of short-term and long-term hazard ratios with two-sample survival data
1 Office of Biostatistics Research, National Heart, Lung, and Blood Institute, 6701 Rockledge Drive, MSC 7938, Bethesda, Maryland 20892, U.S.A. yangso{at}nhlbi.nih.gov, 2 Fred Hutchinson Cancer Research Center, 1100 Fairview Avenue N., MP-1002, Seattle, Washington 98109, U.S.A. rprentic{at}whi.org
Standard approaches to semiparametric modelling of two-sample survival data are not appropriate when the two survival curves cross. We introduce a two-sample model that accommodates crossing survival curves. The two scalar parameters of the model have the interpretations of being the short-term and long-term hazard ratios respectively. The time-varying hazard ratio is expressed semiparametrically by the two scalar parameters and an unspecified baseline distribution. The new model includes the Cox model and the proportional odds model as submodels. For inference we use a pseudo maximum likelihood approach that can be expressed via some simple estimating equations, analogous to that for the maximum partial likelihood estimator of the Cox model, that provide consistent and asymptotically normal estimators. Simulation studies show that the estimators perform well for moderate sample sizes. We also illustrate the methods with a real-data example. The new model can be extended easily to the regression setting.
Key Words: Clinical trial; Crossing survival curves; Martingale; Pseudo maximum likelihood estimator; Semiparametric inference; Survival analysis; Time-varying hazard ratio
Received August 2003. Revised April 2004.