Symmetric tests and confidence intervals for survival probabilities and quantiles of censored survival data

Using Bayesian statistics in confirmatory clinical trials in the regulatory setting: a tutorial review

Bayesian statistics plays a pivotal role in advancing medical science by enabling healthcare companies, regulators, and stakeholders to assess the safety and efficacy of new treatments, interventions, and medical procedures. The Bayesian framework offers a unique advantage over the classical framework, especially when incorporating prior information into a new trial with quality external data, such as historical data or another source of co-data. In recent years, there has been a significant increase in regulatory submissions using Bayesian statistics due to its flexibility and ability to provide valuable insights for decision-making, addressing the modern complexity of clinical trials where frequentist trials are inadequate. For regulatory submissions, companies often need to consider the frequentist operating characteristics of the Bayesian analysis strategy, regardless of the design complexity. In particular, the focus is on the frequentist type I error rate and power for all realistic alternatives. This tutorial review aims to provide a comprehensive overview of the use of Bayesian statistics in sample size determination, control of type I error rate, multiplicity adjustments, external data borrowing, etc., in the regulatory environment of clinical trials. Fundamental concepts of Bayesian sample size determination and illustrative examples are provided to serve as a valuable resource for researchers, clinicians, and statisticians seeking to develop more complex and innovative designs.

MOVER confidence intervals for a difference or ratio effect parameter under stratified sampling

Stratification is commonly employed in clinical trials to reduce the chance covariate imbalances and increase the precision of the treatment effect estimate. We propose a general framework for constructing the confidence interval (CI) for a difference or ratio effect parameter under stratified sampling by the method of variance estimates recovery (MOVER). We consider the additive variance and additive CI approaches for the difference, in which either the CI for the weighted difference, or the CI for the weighted effect in each group, or the variance for the weighted difference is calculated as the weighted sum of the corresponding stratum-specific statistics. The CI for the ratio is derived by the Fieller and log-ratio methods. The weights can be random quantities under the assumption of a constant effect across strata, but this assumption is not needed for fixed weights. These methods can be easily applied to different endpoints in that they require only the point estimate, CI, and variance estimate for the measure of interest in each group across strata. The methods are illustrated with two real examples. In one example, we derive the MOVER CIs for the risk difference and risk ratio for binary outcomes. In the other example, we compare the restricted mean survival time and milestone survival in stratified analysis of time-to-event outcomes. Simulations show that the proposed MOVER CIs generally outperform the standard large sample CIs, and that the additive CI approach performs better than the additive variance approach. Sample SAS code is provided in the Supplementary Material.

Some new confidence intervals for Kaplan-Meier based estimators from one and two sample survival data

The restricted mean survival time (RMST) has been popularly used to assess the treatment effect in survival trials. Greenwood's formula is often used to estimate the variance of RMST, and the resulting Wald confidence interval (CI) tends to be liberal in small and moderate samples. We propose the empirical likelihood ratio, score-type, and loglog transformed CIs for RMST in a single sample. The method of variance estimates recovery technique is used to derive the CIs for the difference and ratio parameters in the two sample inference. A variance estimate, which assumes equal survival curves, but possibly different censoring rates in the two groups, is proposed for comparing two groups. The new variance estimate shows excellent performance in testing for superiority, and also works well for a noninferiority test with a small margin, and for the interval estimation when the two survival curves are close. We use similar techniques to construct CIs for comparing two milestone survival probabilities. Numerical examples are used to assess these interval estimation methods.

Confidence intervals for the cumulative incidence function via constrained NPMLE

The cumulative incidence function (CIF) displays key information in the competing risks setting, which is common in medical research. In this article, we introduce two new methods to compute non-parametric confidence intervals for the CIF. First, we introduce non-parametric profile-likelihood confidence intervals. The method builds on constrained non-parametric maximum likelihood estimation (NPMLE), for which we derive closed-form formulas. This method can be seen as an extension of that of Thomas and Grunkemeier (J Am Stat Assoc 70:865-871, 1975) to the competing risks setting, when the CIF is of interest instead of the survival function. Second, we build on constrained NPMLE to introduce constrained bootstrap confidence intervals. This extends an interesting approach introduced by Barber and Jennison (Biometrics 52:430-436, 1999) to the competing risks setting. A simulation study illustrates how these methods can perform as compared to benchmarks implemented in popular software. The results suggest that more accurate confidence intervals than usual Wald-type ones can be obtained in the case of small to moderate sample sizes and few observed events. An application to melanoma data is provided for illustration purpose.

Nonparametric generalized fiducial inference for survival functions under censoring

Since the introduction of fiducial inference by Fisher in the 1930s, its application has been largely confined to relatively simple, parametric problems. In this paper, we present what might be the first time fiducial inference is systematically applied to estimation of a nonparametric survival function under right censoring. We find that the resulting fiducial distribution gives rise to surprisingly good statistical procedures applicable to both one-sample and two-sample problems. In particular, we use the fiducial distribution of a survival function to construct pointwise and curvewise confidence intervals for the survival function, and propose tests based on the curvewise confidence interval. We establish a functional Bernstein–von Mises theorem, and perform thorough simulation studies in scenarios with different levels of censoring. The proposed fiducial-based confidence intervals maintain coverage in situations where asymptotic methods often have substantial coverage problems. Furthermore, the average length of the proposed confidence intervals is often shorter than the length of confidence intervals for competing methods that maintain coverage. Finally, the proposed fiducial test is more powerful than various types of log-rank tests and sup log-rank tests in some scenarios. We illustrate the proposed fiducial test by comparing chemotherapy against chemotherapy combined with radiotherapy, using data from the treatment of locally unresectable gastric cancer.

The Wally plot approach to assess the calibration of clinical prediction models

A prediction model is calibrated if, roughly, for any percentage x we can expect that x subjects out of 100 experience the event among all subjects that have a predicted risk of x%. Typically, the calibration assumption is assessed graphically but in practice it is often challenging to judge whether a "disappointing" calibration plot is the consequence of a departure from the calibration assumption, or alternatively just "bad luck" due to sampling variability. We propose a graphical approach which enables the visualization of how much a calibration plot agrees with the calibration assumption to address this issue. The approach is mainly based on the idea of generating new plots which mimic the available data under the calibration assumption. The method handles the common non-trivial situations in which the data contain censored observations and occurrences of competing events. This is done by building on ideas from constrained non-parametric maximum likelihood estimation methods. Two examples from large cohort data illustrate our proposal. The 'wally' R package is provided to make the methodology easily usable.

Pointwise confidence intervals for a survival distribution with small samples or heavy censoring

We propose a beta product confidence procedure (BPCP) that is a non-parametric confidence procedure for the survival curve at a fixed time for right-censored data assuming independent censoring. In such situations, the Kaplan-Meier estimator is typically used with an asymptotic confidence interval (CI) that can have coverage problems when the number of observed failures is not large, and/or when testing the latter parts of the curve where there are few remaining subjects at risk. The BPCP guarantees central coverage (i.e. ensures that both one-sided error rates are no more than half of the total nominal rate) when there is no censoring (in which case it reduces to the Clopper-Pearson interval) or when there is progressive type II censoring (i.e. when censoring only occurs immediately after failures on fixed proportions of the remaining individuals). For general independent censoring, simulations show that the BPCP maintains central coverage in many situations where competing methods can have very substantial error rate inflation for the lower limit. The BPCP gives asymptotically correct coverage and is asymptotically equivalent to the CI on the Kaplan-Meier estimator using Greenwood's variance. The BPCP may be inverted to create confidence procedures for a quantile of the underlying survival distribution. Because the BPCP is easy to implement, offers protection in settings when other methods fail, and essentially matches other methods when they succeed, it should be the method of choice.

A non-parametric procedure for evaluating treatment effect in the meta-analysis of survival data

This paper addresses the problem of combining information from independent clinical trials which compare survival distributions of two treatment groups. Current meta-analytic methods which take censoring into account are often not feasible for meta-analyses which synthesize summarized results in published (or unpublished) references, as these methods require information usually not reported. The paper presents methodology which uses the log(-log) survival function difference, (i.e. log(-logS2(t))-log(-logS1(t)), as the contrast index to represent the multiplicative treatment effect on survival in independent trials. This article shows by the second mean value theorem for integrals that this contrast index, denoted as theta, is interpretable as a weighted average on a natural logarithmic scale of hazard ratios within the interval [0,t] in a trial. When the within-trial proportional hazards assumption is true, theta is the logarithm of the proportionality constant for the common hazard ratio for the interval considered within the trial. In this situation, an important advantage of using theta as a contrast index in the proposed methodology is that the estimation of theta is not affected by length of follow-up time. Other commonly used indices such as the odds ratio, risk ratio and risk differences do not have this invariance property under the proportional hazard model, since their estimation may be affected by length of follow-up time as a technical artefact. Thus, the proposed methodology obviates problems which often occur in survival meta-analysis because trials do not report survival at the same length of follow-up time. Even when the within-trial proportional hazards assumption is not realistic, the proposed methodology has the capability of testing a global null hypothesis of no multiplicative treatment effect on the survival distributions of two groups for all studies. A discussion of weighting schemes for meta-analysis is provided, in particular, a weighting scheme based on effective sample sizes is suggested for the meta-analysis of time-to-event data which involves censoring. A medical example illustrating the methodology is given. A simulation investigation suggested that the methodology performs well in the presence of moderate censoring.

Long-term survival after falls among the elderly in institutional care

Falls of the elderly are a major problem in institutional care. However, more comprehensive studies concerning the long-time survival of fallen institutionalised elderly are lacking. We investigated the 5-year survival of institutionalised elderly fallers and controls. Data of the patients aged over 60 years, who fell during the 1-year period (n=218) in four institutions were collected prospectively. The controls consisted of patients of the same age who did not fall within the same period (n=632). The survival of both groups was analysed by gender in the total data, and in the short-term (ST) and long-term (LT) patients separately. In addition, the survival of fallers was investigated according to the number of falls per patient. After follow-up, 164 (75%) fallers and 369 (58%) controls were dead. The female controls survived best and the survival of the male fallers was the poorest. The death rate was higher than expected among female fallers and lower than expected among female controls. The survival of the patients who fell twice during the 1-year period was clearly poorer than of those who fell once or three or more times. No difference in the survival rates was found between the injured and non-injured fallers. Falls in institutional care predict poor survival. The first and especially the second fall should be prevented.

Bootstrap-type confidence intervals for quantiles of the survival distribution

In this paper we outline and illustrate an easy to program method for analytically calculating both parametric and non-parametric bootstrap-type confidence intervals for quantiles of the survival distribution based on right censored data. This new approach allows for the incorporation of covariates within the framework of parametric models. The procedure is based upon the notion of fractional order statistics and is carried forth using a simple beta transformation of the estimated survival function (parametric or non-parametric). It is the only direct method currently available in the sense that all other methods are based on inverting test statistics or employing confidence intervals for other survival quantities. We illustrate that the new method has favourable coverage probabilities for median confidence intervals as compared to six other competing methods.