Residual-based sieve maximum full likelihood estimation for the proportional hazards model

The partial likelihood approach is the gold standard approach for estimating regression parameters in proportional hazards model, largely because it avoids specifying the baseline hazard function. However, this approach can lack robustness in the presence of model mis-specification, outliers, or high censoring rates. To address these limitations, we propose a sieve maximum full likelihood estimation method for the proportional hazards model. Our two-step procedure approximates the cumulative hazard function using piecewise polynomials, enabling simultaneous estimation of all model parameters within a single likelihood framework. Unlike the partial likelihood, our method employs the full likelihood to enhance robustness and efficiency, particularly when assumptions are violated or data are heavily censored. We use the profile likelihood to estimate variance and introduce an efficient information criterion for selecting the optimal degree of freedom for the sieve space. Extensive simulations demonstrate that our approach outperforms both partial and existing full likelihood methods under the proportional and non-proportional hazards settings. We illustrate the usefulness of the method using data from real applications in bladder and breast cancer. This approach provides a flexible alternative for survival analysis, particularly when standard methods may be insufficient due to data outliers or violation of model assumptions.