Survival estimation through the cumulative hazard with monotone natural cubic splines using convex optimization-the HCNS approach

Abstract

Background and objectivesIn survival analysis both the Kaplan-Meier estimate and the Cox model enjoy a broad acceptance. We present an improved spline-based survival estimate and offer a fully automated software for its implementation. We explore the use of natural cubic splines that are constrained to be monotone. Apart from its superiority over the Kaplan Meier estimator our approach overcomes limitations of other known smoothing approaches and can accommodate covariates. Unlike other spline methods, concerns of computational problems and issues of overfitting are resolved since no attempt is made to maximize a likelihood once the Kaplan-Meier estimator is obtained. An application to laryngeal cancer data, a simulation study and illustrations of the broad application of the method and its software are provided. In addition to presenting our approaches, this work contributes to bridging a communication gap between clinicians and statisticians that is often apparent in the medical literature. MethodsWe employ a two-stage approach: first obtain the stepwise cumulative hazard and then consider a natural cubic spline to smooth its steps under restrictions of monotonicity between any consecutive knots. The underlying region of monotonicity corresponds to a non-linear region that encompasses the full family of monotone third-degree polynomials. We approximate it linearly and reduce the problem to a restricted least squares one under linear restrictions. This ensures convexity. We evaluate our method through simulations against competitive traditional approaches. ResultsOur method is compared to the popular Kaplan Meier estimate both in terms of mean squared error and in terms of coverage. Over-fitting is avoided by construction, as our spline attempts to approximate the empirical estimate of the cumulative hazard itself, and is not fitted directly on the data. ConclusionsThe proposed approach will enable clinical researchers to obtain improved survival estimates and valid confidence intervals over the full spectrum of the range of the survival data. Our methods outperform conventional approaches and can be readily utilized in settings beyond survival analysis such as diagnostic testing.