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

Abstract

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.

Estimating an Individual’s Probability of Revision Surgery After Knee Replacement: A Comparison of Modeling Approaches Using a National Data Set

“…Here we assumed a proportional-hazards spline model where time-dependent effects were not considered. This may also have caused bias in the risk estimates ( 63 ) of prosthesis revision. The flexible model can be further extended for possible improvement in fit by adding terms for interactions between covariates and the effect of time ( 27 ).…”

Capturing simple and complex time-dependent effects using flexible parametric survival models: A simulation study

“…Although alternative methods exist for analyzing non-proportional time-to-event data, much of which has been done in a Cox setting (Gray 1992;Hastie and Tibshirani 1993;Grambsch and Therneau 1994;Abrahamowicz, MacKenzie, and Esdaile 1996;Abrahamowicz and MacKenzie 2007;Wynant and Abrahamowicz 2015), here we concentrate on assessing the ability of flexible parametric survival models in capturing time-dependent effects. Under proportional hazards, flexible parametric survival models have been shown to be able to more accurately capture complex hazard functions than standard parametric models and provide unbiased estimates of hazard ratios (Rutherford, Crowther, and Lambert 2015), but no assessment under non-proportionality has been done.…”

Assessing the time dependence of prognostic values of cytology and human papillomavirus testing in cervical cancer screening

“…Accordingly, study events were (by definition) interval‐censored. To ensure that our use of the customized flexible modeling program was appropriate in this context, sensitivity analyses comparing estimates from conventional vs . interval‐censored Cox PH models were first conducted and results did not differ materially (data not shown).…”

“…Yet, inappropriate use of this model may lead to biased results, inaccurate risk prediction, and reduced statistical power . Therefore, if one or both of the PH and linearity assumptions were rejected in our analyses, nonproportional (i.e., time‐dependent (TD)) and/or nonlinear (NL) effects were modeled using the flexible generalization of the Cox PH model . The insights that stand to be gained by modeling these effects are of clinical relevance .…”