Confidence bands for the difference of two survival functions under the additive risk model

Abstract

In many clinical studies, a commonly encountered problem is to compare the survival probabilities of two treatments for a given patient with a certain set of covariates, and there is often a need to make adjustments for other covariates that may affect outcomes. One approach is to plot the difference between the two subject-specific predicted survival estimates with a simultaneous confidence band. Such a band will provide useful information about when these two treatments differ and which treatment has a better survival probability. In this paper, we show how to construct such a band based on the additive risk model and we use the martingale central limit theorem to derive its asymptotic distribution. The proposed method is evaluated from a simulation study and is illustrated with two real examples.

A methodology for building a data-enclosing tunnel for automated online-feedback in simulator Training

“…The idea of an enclosing band is also known as confidence band. Two different implementations can be found in Skelton and Willms (2014) and Lee and Hyun (2011).…”

Estimating a Treatment Effect in Residual Time Quantiles Under the Additive Hazards Model

“…This yields the approximation $$\begin{array}{c}left\widehat{\theta }false(t,q|\mathit{Z}false)-\theta false(t,q|\mathit{Z}false)=\frac{1}{{\hat{\lambda }}_{0}false\{\theta false(t,q|\mathit{Z}false)+tfalse\}+{\mathrm{bold-italic\beta }}^{\mathrm{normalT}}\mathit{Z}}\{-{\displaystyle {\int }_t^{\theta false(t,q|\mathit{Z}false)+t}\frac{true{\sum }_{i=1}^{n}d{M}_i(t+v)}{true{\sum }_{i=1}^n{Y}_i(t+v)}}+\\ {\left[-\hat{\theta }(t,qfalse|\mathrm{bold-italicZ})\mathrm{bold-italicZ}+true{\int }_0^{\theta (t,qfalse|\mathrm{bold-italicZ})}\stackrel{bold-italic¯}{\mathrm{bold-italicZ}}false(t+vfalse)du\right]}^{\mathrm{normalT}}false(\stackrel{bold-italic^}{\mathrm{bold-italic\beta }}-\mathit{\beta }false)\}\\ +o_{p}false(\widehat{\theta }false(t,q|\mathit{Z}false)-\theta false(t,q|\mathit{Z}false)false)\end{array}$$Applying the martingale central limit theorem as in Lin and Ying [16] and Lee and Hyun [13], it follows that the variance function can be consistently estimated by $$\begin{array}{c}left{[{\widehat{\lambda }}_0\{\hat{\theta }(t,qfalse|\mathrm{bold-italicZ})+t}^{}\end{array}$$…”

A martingale approach to estimating confidence band with censored data

“…Parzen, Wei and Ying [18] quanti ed the treatment e ect over time in terms of the con dence band on the di erence of two survival functions. Another example, among others, is Lee and Hyun [14] which studied the con dence band for the di erence of two survival functions under the additive risk model. In this paper, following Parzen, Wei and Ying [18], with randomly right-censored data, we present a simulation-based method to obtain simultaneous con dence bands of the survival function which has wide applicability for other probability functions.…”