Confidence Bands For Survival Function Research Articles

Simultaneous confidence bands for survival functions from twice censorship

When a failure time observation is right censored, it is not observed beyond a random right threshold. When a possibly right censored observation is further susceptible to a random left threshold, a twice censored observation results. Patilea and Rolin have proposed product limit estimators for survival functions from twice censored data. Simultaneous confidence bands (SCBs) for survival functions from twice censored data are constructed in a way that mimics the approach that Hollander, McKeague and Yang pursued for random right censoring. Their nonparametric likelihood ratio function is adjusted, providing the basis for constructing the SCBs. The critical value needed for the SCBs is obtained using the bootstrap, for which asymptotic justification is provided. The SCBs align nicely as “neighborhoods” of the Patilea–Rolin nonparametric survival function estimator, in much the same way the likelihood ratio SCBs under random censoring are the “neighborhoods” of the Kaplan–Meier estimator. A simulation study supports the effectiveness of the proposed method. An illustration is given using synthetic data.

OptBand: optimization-based confidence bands for functions to characterize time-to-event distributions

Classical simultaneous confidence bands for survival functions (i.e., Hall–Wellner, equal precision, and empirical likelihood bands) are derived from transformations of the asymptotic Brownian nature of the Nelson–Aalen or Kaplan–Meier estimators. Due to the properties of Brownian motion, a theoretical derivation of the highest confidence density region cannot be obtained in closed form. Instead, we provide confidence bands derived from a related optimization problem with local time processes. These bands can be applied to the one-sample problem regarding both cumulative hazard and survival functions. In addition, we present a solution to the two-sample problem for testing differences in cumulative hazard functions. The finite sample performance of the proposed method is assessed by Monte Carlo simulation studies. The proposed bands are applied to clinical trial data to assess survival times for primary biliary cirrhosis patients treated with D-penicillamine.

Bootstrap likelihood ratio confidence bands for survival functions under random censorship and its semiparametric extension

Simultaneous confidence bands for survival functions, from randomly right censored data, can be computed by inverting likelihood ratio functions based on appropriate thresholds. Sometimes, however, the requisite asymptotic distributions are intractable, or thresholds based on Brownian bridge approximations are not easy to obtain when simultaneous confidence bands over only sub-regions are possible or desired. We obtain the thresholds by bootstrapping (i) a nonparametric likelihood ratio function via censored data bootstrap and (ii) a semiparametric adjusted likelihood ratio function via a two-stage bootstrap that utilizes a model for the second stage. These two scenarios are grounded respectively in standard random censorship and its semiparametric extension introduced by Dikta. The two bootstraps, which are different in the way resampling is done, are shown to have asymptotic validity. The respective confidence bands are neighborhoods of the well-known Kaplan–Meier estimator and the more recently developed Dikta’s semiparametric counterpart. As evidenced by a validation study, both types of confidence bands provide approximately correct coverage. The model-based confidence bands, however, are tighter than the nonparametric ones. Two sensitivity studies reveal that the model-based method performs well when standard binary regression models are fitted, indicating its robustness to misspecification as well as its practical applicability. An illustration is given using real data.

Model-based confidence bands for survival functions

This paper focuses on a novel method of developing one-sample confidence bands for survival functions from right censored data. The approach is model-based, relying on a parametric model for the conditional expectation of the censoring indicator given the observed minimum, and derives its strength from easy access to a good-fitting model among a plethora of choices available for binary response data. The substantive methodological contribution is in exploiting a semiparametric estimator of the survival function to produce improved simultaneous confidence bands. To obtain critical values for computing the confidence bands, a two-stage bootstrap approach that combines the classical bootstrap with the more recent model-based regeneration of censoring indicators is proposed and a justification of its asymptotic validity is also provided. Several different confidence bands are studied using the proposed approach. Numerical studies, including robustness of the proposed bands to misspecification, are carried out to check efficacy. The method is illustrated using two lung cancer data sets.

Width-scaled confidence bands for survival functions

A popular simultaneous confidence band for survival functions is the equal precision band of Nair [1984. Confidence bands for survival functions with censored data: a comparative study. Technometrics 26, 265–275] This band is found by adjusting the level of Wald-type pointwise confidence intervals centered on the Kaplan–Meier estimator. The present paper develops a complementary method of adjusting pointwise confidence intervals to produce a simultaneous band. Our approach is to scale the width, rather than the level, of the pointwise confidence intervals. The resulting adjustment of the pointwise band, called a width-scaled band, provides an attractive alternative to the equal precision band. Empirical likelihood-based width-scaled bands are studied in the one- and two-sample censored data settings. An example with real data is included.

Appendix A: Principles of information

Simultaneous confidence bands for survival functions, from randomly right censored data, can be computed by inverting likelihood ratio functions based on appropriate thresholds. Sometimes, however, the requisite asymptotic distributions are intractable, or thresholds based on Brownian bridge approximations are not easy to obtain when simultaneous confidence bands over only sub-regions are possible or desired. We obtain the thresholds by bootstrapping (i) a nonparametric likelihood ratio function via censored data bootstrap and (ii) a semiparametric adjusted likelihood ratio function via a two-stage bootstrap that utilizes a model for the second stage. These two scenarios are grounded respectively in standard random censorship and its semiparametric extension introduced by Dikta. The two bootstraps, which are different in the way resampling is done, are shown to have asymptotic validity. The respective confidence bands are neighborhoods of the well-known Kaplan–Meier estimator and the more recently developed Dikta’s semiparametric counterpart. As evidenced by a validation study, both types of confidence bands provide approximately correct coverage. The model-based confidence bands, however, are tighter than the nonparametric ones. Two sensitivity studies reveal that the model-based method performs well when standard binary regression models are fitted, indicating its robustness to misspecification as well as its practical applicability. An illustration is given using real data.

Confidence Bands for Survival Functions with Censored Data: A Comparative Study

Consider the problem of obtaining simultaneous confidence bands for the survival function S(x) when the data are arbitrarily right censored. The usual pointwise confidence intervals based on Greenwood's variance formula can be adapted to yield a large-sample confidence band. This band has, in a certain sense, equal precision at each point of S(x). It is compared with the censored versions of the Kolmogorov band and the Rényi band. The comparisons are made in terms of the widths and the adequacy of large-sample approximations and are carried out under various censoring models and degrees of censoring. The bands are illustrated by applying them to data from a mechanical-switch life test.

Confidence Bands for Survival Functions With Censored Data: A Comparative Study

Consider the problem of obtaining simultaneous confidence bands for the survival function S(x) when the data are arbitrarily right censored. The usual pointwise confidence intervals based on Greenwood's variance formula can be adapted to yield a large-sample confidence band. This band has, in a certain sense, equal precision at each point of S(x). It is compared with the censored versions of the Kolmogorov band and the Renyi band. The comparisons are made in terms of the widths and the adequacy of large-sample approximations and are carried out under various censoring models and degrees of censoring. The bands are illustrated by applying them to data from a mechanical-switch life test.