Flexible estimation of survival curves conditional on non-linear and time-dependent predictor effects

Prognostic studies often estimate survival curves for patients with different covariate vectors, but the validity of their results depends largely on the accuracy of the estimated covariate effects. To avoid conventional proportional hazards and linearity assumptions, flexible extensions of Cox's proportional hazards model incorporate non-linear (NL) and/or time-dependent (TD) covariate effects. However, their impact on survival curves estimation is unclear. Our primary goal is to develop and validate a flexible method for estimating individual patients' survival curves, conditional on multiple predictors with possibly NL and/or TD effects. We first obtain maximum partial likelihood estimates of NL and TD effects and use backward elimination to select statistically significant effects into a final multivariable model. We then plug the selected NL and TD estimates in the full likelihood function and estimate the baseline hazard function and the resulting survival curves, conditional on individual covariate vectors. The TD and NL functions and the log hazard are modeled with unpenalized regression B-splines. In simulations, our flexible survival curve estimates were unbiased and had much lower mean square errors than the conventional estimates. In real-life analyses of mortality after a septic shock, our model improved significantly the deviance (likelihood ratio test = 84.8, df = 20, p < 0.0001) and changed substantially the predicted survival for several subjects.