Estimation of delay time in survival data with delayed treatment effect

In randomized controlled trials with delayed treatment effect, there is a delay period before the experimental therapy starts to exhibit a beneficial effect. The phenomenon of delayed treatment effect is often observed in the emerging and important field of immuno-oncology. It is important to estimate the duration of delay as this information helps in characterizing the pattern of comparative treatment effect, understanding the mechanism of action of the experimental therapy, and forming optimal treatment strategies. For a fixed delay time, we propose a maximum likelihood estimator and evaluate its asymptotic properties via simulation. We further evaluate two functions that link the pre- and postdelay hazard ratios to the average hazard ratio given a fixed delay time. For the case of random delay time, where the delay time may vary from patient to patient, we propose a semiparametric joint survival model for delay time and event time to estimate the mean delay time and the postdelay hazard ratio, assuming a Beta distribution for the delay time. We describe an extension of the model to estimate subgroup-specific mean delay times. Simulation study and application to data from a clinical trial in colon cancer patients demonstrate the robustness of the proposed model.