Inference on the probability P(T1<T2) as a measurement of treatment effect under a density ratio model and random censoring

Let T1 and T2 be the survival times of patients randomized to, respectively, two treatment groups. The probability P(T1<T2) may be used as a measure on the effect of treatments in a randomized clinical trial. With potentially censored data observed and assuming that the density functions of survival times satisfy a semiparametric density ratio model, two estimates of this probability based on, respectively, the conditional and weighted empirical likelihood are proposed. Associated confidence intervals are also derived based on the bootstrap resampling. The assessment on the goodness of fit of the density ratio model is also discussed. The proposed inference procedures are evaluated by Monte Carlo simulations and applied to the analysis of data from a clinical trial on early breast cancer.