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The proportional hazards assumption in the commonly used Cox model forcensored failure time data is often violated in scientific studies. Yang andPrentice (2005) proposed a novel semiparametric two-sample model that includesthe proportional hazards model and the proportional odds model as sub-models,and accommodates crossing survival curves. The model leaves the baseline hazardunspecified and the two model parameters can be interpreted as the short-termand long-term hazard ratios. Inference procedures were developed based on apseudo score approach. Although extension to accommodate covariates wasmentioned, no formal procedures have been provided or proved. Furthermore, thepseudo score approach may not be asymptotically efficient. We study theextension of the short-term and long-term hazard ratio model of Yang andPrentice (2005) to accommodate potentially time-dependent covariates. Wedevelop efficient likelihood-based estimation and inference procedures. Thenonparametric maximum likelihood estimators are shown to be consistent,asymptotically normal, and asymptotically efficient. Extensive simulationstudies demonstrate that the proposed methods perform well in practicalsettings. The proposed method captured the phenomenon of crossing hazards in acancer clinical trial and identified a genetic marker with significantlong-term effect missed by using the proportional hazards model on age-at-onsetof alcoholism in a genetic study.

Efficient Semiparametric Estimation of Short-term and Long-term Hazard Ratios with Right-Censored Data