Estimation of the survival function for Gray's piecewise-constant time-varying coefficients model

Gray's extension of Cox's proportional hazards (PH) model for right-censored survival data allows for a departure from the PH assumption via introduction of time-varying regression coefficients (TVC). For this model estimation of the conditional hazard rate relies on the inclusion of penalized splines. Cubic penalized splines tend to be unstable in the right tail of the distribution and thus quadratic, linear and piecewise-constant penalized splines may be a favourable choice. We derive a survival function estimator for one important member of the class of TVC models--a piecewise-constant time-varying coefficients (PC-TVC) model. Using the first-order Taylor series approximation we also derive an estimate for the variance of the log-transformed and log(-log)-transformed survival function, which in turn leads to estimated confidence limits on the corresponding scales of the survival function. Accuracy in estimating underlying survival times and survival quantiles is assessed for both Cox's and Gray's PC-TVC model using a simulation study featuring scenarios violating the PH assumption. Finally, an example of the estimated survival functions and the corresponding confidence limits derived from Cox's PH and Gray's PC-TVC model, respectively, is presented for a liver transplant data set.