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The pattern of the difference between two survival curves we often observe in randomized clinical trials for evaluating immunotherapy is not proportional hazards; the treatment effect typically appears several months after the initiation of the treatment (ie, delayed difference pattern). The commonly used logrank test and hazard ratio estimation approach will be suboptimal concerning testing and estimation for those trials. The long-term restricted mean survival time (LT-RMST) approach is a promising alternative for detecting the treatment effect that potentially appears later in the study. A challenge in employing the LT-RMST approach is that it must specify a lower end of the time window in addition to a truncation time point that the RMST requires. There are several investigations and suggestions regarding the choice of the truncation time point for the RMST. However, little has been investigated to address the choice of the lower end of the time window. In this paper, we propose a flexible LT-RMST-based test/estimation approach that does not require users to specify a lower end of the time window. Numerical studies demonstrated that the potential power loss by adopting this flexibility was minimal, compared to the standard LT-RMST approach using a prespecified lower end of the time window. The proposed method is flexible and can offer higher power than the RMST-based approach when the delayed treatment effect is expected. Also, it provides a robust estimate of the magnitude of the treatment effect and its confidence interval that corresponds to the test result., (© 2023 John Wiley & Sons Ltd.)

The hazard ratio (HR) has been the most popular measure to quantify the magnitude of treatment effect on time-to-event outcomes in clinical research. However, the traditional Cox's HR approach has several drawbacks. One major issue is that there is no clear interpretation when the proportional hazards (PH) assumption does not hold, because the estimated HR is affected by study-specific censoring time distribution in non-PH cases. Another major issue is that the lack of a group-specific absolute hazard value in each group obscures the clinical significance of the magnitude of the treatment effect. Given these, we propose average hazard with survival weight (AH-SW) as a summary metric of event time distribution and will use difference in AH-SW (DAH-SW) or ratio of AH-SW (RAH-SW) to quantify the treatment effect magnitude. The AH-SW is interpreted as a person-time incidence rate that does not depend on random censoring. It is defined as the ratio of cumulative incidence probability and restricted mean survival time (RMST), which can be estimated non-parametrically. Numerical studies demonstrate that DAH-SW and RAH-SW offer almost identical power to Cox's HR-based tests under PH scenarios and can be more powerful for delayed-difference patterns often seen in immunotherapy trials. Like median and RMST differences, the proposed approach is a good model-free alternative to the HR-based approach for evaluating the treatment effect magnitude. Such a model-free measure will increase the likelihood that results from clinical studies are correctly interpreted and generalized to future populations., (© 2023 John Wiley & Sons Ltd.)

Between-group comparison based on the restricted mean survival time (RMST) is getting attention as an alternative to the conventional logrank/hazard ratio approach for time-to-event outcomes in randomized controlled trials (RCTs). The validity of the commonly used nonparametric inference procedure for RMST has been well supported by large sample theories. However, we sometimes encounter cases with a small sample size in practice, where we cannot rely on the large sample properties. Generally, the permutation approach can be useful to handle these situations in RCTs. However, a numerical issue arises when implementing permutation tests for difference or ratio of RMST from two groups. In this article, we discuss the numerical issue and consider six permutation methods for comparing survival time distributions between two groups using RMST in RCTs setting. We conducted extensive numerical studies and assessed type I error rates of these methods. Our numerical studies demonstrated that the inflation of the type I error rate of the asymptotic methods is not negligible when sample size is small, and that all of the six permutation methods are workable solutions. Although some permutation methods became a little conservative, no remarkable inflation of the type I error rates were observed. We recommend using permutation tests instead of the asymptotic tests, especially when the sample size is less than 50 per arm., (© 2020 John Wiley & Sons, Ltd.)

For a two-group comparative study, a stratified inference procedure is routinely used to estimate an overall group contrast to increase the precision of the simple two-sample estimator. Unfortunately, most commonly used methods including the Cochran-Mantel-Haenszel statistic for a binary outcome and the stratified Cox procedure for the event time endpoint do not serve this purpose well. In fact, these procedures may be worse than their two-sample counterparts even when the observed treatment allocations are imbalanced across strata. Various procedures beyond the conventional stratified methods have been proposed to increase the precision of estimation when the naive estimator is consistent. In this paper, we are interested in the case when the treatment allocation proportions vary markedly across strata. We study the stochastic properties of the two-sample naive estimator conditional on the ancillary statistics, the observed treatment allocation proportions and/or the stratum sizes, and present a biased-adjusted estimator. This adjusted estimator is asymptotically equivalent to the augmentation estimators proposed under the unconditional setting. Moreover, this consistent estimation procedure is also equivalent to a rather simple procedure, which estimates the mean response of each treatment group first via a stratum-size weighted average and then constructs the group contrast estimate. This simple procedure is flexible and readily applicable to any target patient population by choosing appropriate stratum weights. All the proposals are illustrated with the data from a cardiovascular clinical trial, whose treatment allocations are imbalanced. [ABSTRACT FROM AUTHOR]

Several methods in survival analysis are based on the proportional hazards assumption. However, this assumption is very restrictive and often not justifiable in practice. Therefore, effect estimands that do not rely on the proportional hazards assumption are highly desirable in practical applications. One popular example for this is the restricted mean survival time (RMST). It is defined as the area under the survival curve up to a prespecified time point and, thus, summarizes the survival curve into a meaningful estimand. For two-sample comparisons based on the RMST, previous research found the inflation of the type I error of the asymptotic test for small samples and, therefore, a two-sample permutation test has already been developed. The first goal of the present paper is to further extend the permutation test for general factorial designs and general contrast hypotheses by considering a Wald-type test statistic and its asymptotic behavior. Additionally, a groupwise bootstrap approach is considered. Moreover, when a global test detects a significant difference by comparing the RMSTs of more than two groups, it is of interest which specific RMST differences cause the result. However, global tests do not provide this information. Therefore, multiple tests for the RMST are developed in a second step to infer several null hypotheses simultaneously. Hereby, the asymptotically exact dependence structure between the local test statistics is incorporated to gain more power. Finally, the small sample performance of the proposed global and multiple testing procedures is analyzed in simulations and illustrated in a real data example., (© 2024 The Authors. Statistics in Medicine published by John Wiley & Sons Ltd.)

In randomized clinical trials where time-to-event is the primary outcome, almost routinely, the logrank test is prespecified as the primary test and the hazard ratio is used to quantify treatment effect. If the ratio of 2 hazard functions is not constant, the logrank test is not optimal and the interpretation of hazard ratio is not obvious. When such a nonproportional hazards case is expected at the design stage, the conventional practice is to prespecify another member of weighted logrank tests, eg, Peto-Prentice-Wilcoxon test. Alternatively, one may specify a robust test as the primary test, which can capture various patterns of difference between 2 event time distributions. However, most of those tests do not have companion procedures to quantify the treatment difference, and investigators have fallen back on reporting treatment effect estimates not associated with the primary test. Such incoherence in the "test/estimation" procedure may potentially mislead clinicians/patients who have to balance risk-benefit for treatment decision. To address this, we propose a flexible and coherent test/estimation procedure based on restricted mean survival time, where the truncation time τ is selected data dependently. The proposed procedure is composed of a prespecified test and an estimation of corresponding robust and interpretable quantitative treatment effect. The utility of the new procedure is demonstrated by numerical studies based on 2 randomized cancer clinical trials; the test is dramatically more powerful than the logrank, Wilcoxon tests, and the restricted mean survival time-based test with a fixed τ, for the patterns of difference seen in these cancer clinical trials., (Copyright © 2018 John Wiley & Sons, Ltd.)

With censored event time observations, the logrank test is the most popular tool for testing the equality of two underlying survival distributions. Although this test is asymptotically distribution free, it may not be powerful when the proportional hazards assumption is violated. Various other novel testing procedures have been proposed, which generally are derived by assuming a class of specific alternative hypotheses with respect to the hazard functions. The test considered by Pepe and Fleming (1989) is based on a linear combination of weighted differences of the two Kaplan-Meier curves over time and is a natural tool to assess the difference of two survival functions directly. In this article, we take a similar approach but choose weights that are proportional to the observed standardized difference of the estimated survival curves at each time point. The new proposal automatically makes weighting adjustments empirically. The new test statistic is aimed at a one-sided general alternative hypothesis and is distributed with a short right tail under the null hypothesis but with a heavy tail under the alternative. The results from extensive numerical studies demonstrate that the new procedure performs well under various general alternatives with a caution of a minor inflation of the type I error rate when the sample size is small or the number of observed events is small. The survival data from a recent cancer comparative study are utilized for illustrating the implementation of the process., (Copyright © 2015 John Wiley & Sons, Ltd.)

Window mean survival time (WMST), a simple extension of restricted mean survival time (RMST), allows for clinicians to evaluate the mean survival difference between treatment groups in specific windows of time during the follow-up period of a trial. The advantages of WMST are numerous. Not only does it produce estimates of treatment effect that can be meaningfully interpreted, but also has power advantages over competing methods when hazards are non-proportional (NPH). WMST, like RMST, is currently underutilized due to clinicians' lack of familiarity with tests comparing mean survival times and the lack of tools to facilitate trial design with this endpoint. The aim of this article is to provide investigators with insights and software to design trials with WMST as the primary endpoint. Functions for performing power and sample size calculations are provided in the survWMST package in R available on GitHub., (© 2023 John Wiley & Sons Ltd.)

Risk prediction procedures can be quite useful for the patient's treatment selection, prevention strategy, or disease management in evidence-based medicine. Often, potentially important new predictors are available in addition to the conventional markers. The question is how to quantify the improvement from the new markers for prediction of the patient's risk in order to aid cost-benefit decisions. The standard method, using the area under the receiver operating characteristic curve, to measure the added value may not be sensitive enough to capture incremental improvements from the new markers. Recently, some novel alternatives to area under the receiver operating characteristic curve, such as integrated discrimination improvement and net reclassification improvement, were proposed. In this paper, we consider a class of measures for evaluating the incremental values of new markers, which includes the preceding two as special cases. We present a unified procedure for making inferences about measures in the class with censored event time data. The large sample properties of our procedures are theoretically justified. We illustrate the new proposal with data from a cancer study to evaluate a new gene score for prediction of the patient's survival., (Copyright © 2012 John Wiley & Sons, Ltd.)

For modern evidence-based medicine, a well thought-out risk scoring system for predicting the occurrence of a clinical event plays an important role in selecting prevention and treatment strategies. Such an index system is often established based on the subject's 'baseline' genetic or clinical markers via a working parametric or semi-parametric model. To evaluate the adequacy of such a system, C-statistics are routinely used in the medical literature to quantify the capacity of the estimated risk score in discriminating among subjects with different event times. The C-statistic provides a global assessment of a fitted survival model for the continuous event time rather than focussing on the prediction of bit-year survival for a fixed time. When the event time is possibly censored, however, the population parameters corresponding to the commonly used C-statistics may depend on the study-specific censoring distribution. In this article, we present a simple C-statistic without this shortcoming. The new procedure consistently estimates a conventional concordance measure which is free of censoring. We provide a large sample approximation to the distribution of this estimator for making inferences about the concordance measure. Results from numerical studies suggest that the new procedure performs well in finite sample., (Copyright © 2011 John Wiley & Sons, Ltd.)

Recent observations, especially in cancer immunotherapy clinical trials with time-to-event outcomes, show that the commonly used proportional hazard assumption is often not justifiable, hampering an appropriate analysis of the data by hazard ratios. An attractive alternative advocated is given by the restricted mean survival time (RMST), which does not rely on any model assumption and can always be interpreted intuitively. Since methods for the RMST based on asymptotic theory suffer from inflated type-I error under small sample sizes, a permutation test was proposed recently leading to more convincing results in simulations. However, classical permutation strategies require an exchangeable data setup between comparison groups which may be limiting in practice. Besides, it is not possible to invert related testing procedures to obtain valid confidence intervals, which can provide more in-depth information. In this paper, we address these limitations by proposing a studentized permutation test as well as respective permutation-based confidence intervals. In an extensive simulation study, we demonstrate the advantage of our new method, especially in situations with relatively small sample sizes and unbalanced groups. Finally, we illustrate the application of the proposed method by re-analyzing data from a recent lung cancer clinical trial., (© 2023 The Authors. Statistics in Medicine published by John Wiley & Sons Ltd.)

An assurance calculation is a Bayesian alternative to a power calculation. One may be performed to aid the planning of a clinical trial, specifically setting the sample size or to support decisions about whether or not to perform a study. Immuno-oncology is a rapidly evolving area in the development of anticancer drugs. A common phenomenon that arises in trials of such drugs is one of delayed treatment effects, that is, there is a delay in the separation of the survival curves. To calculate assurance for a trial in which a delayed treatment effect is likely to be present, uncertainty about key parameters needs to be considered. If uncertainty is not considered, the number of patients recruited may not be enough to ensure we have adequate statistical power to detect a clinically relevant treatment effect and the risk of an unsuccessful trial is increased. We present a new elicitation technique for when a delayed treatment effect is likely and show how to compute assurance using these elicited prior distributions. We provide an example to illustrate how this can be used in practice and develop open-source software to implement our methods. Our methodology has the potential to improve the success rate and efficiency of Phase III trials in immuno-oncology and for other treatments where a delayed treatment effect is expected to occur., (© 2024 The Author(s). Statistics in Medicine published by John Wiley & Sons Ltd.)

This article is concerned with sample size determination methodology for prediction models. We propose to combine the individual calculations via learning-type curves. We suggest two distinct ways of doing so, a deterministic skeleton of a learning curve and a Gaussian process centered upon its deterministic counterpart. We employ several learning algorithms for modeling the primary endpoint and distinct measures for trial efficacy. We find that the performance may vary with the sample size, but borrowing information across sample size universally improves the performance of such calculations. The Gaussian process-based learning curve appears more robust and statistically efficient, while computational efficiency is comparable. We suggest that anchoring against historical evidence when extrapolating sample sizes should be adopted when such data are available. The methods are illustrated on binary and survival endpoints., (© 2024 The Author(s). Statistics in Medicine published by John Wiley & Sons Ltd.)

Network meta-analysis (NMA) is essential for clinical decision-making. NMA enables inference for all pair-wise comparisons between interventions available for the same indication, by using both direct evidence and indirect evidence. In randomized trials with time-to event outcome data, such as lung cancer data, conventional NMA methods rely on the hazard ratio and the proportional hazards assumption, and ignore the varying follow-up durations across trials. We introduce a novel multivariate NMA model for the difference in restricted mean survival times (RMST). Our model synthesizes all the available evidence from multiple time points simultaneously and borrows information across time points through within-study covariance and between-study covariance for the differences in RMST. We propose an estimator of the within-study covariance and we then assume it to be known. We estimate the model under the Bayesian framework. We evaluated our model by conducting a simulation study. Our multiple-time-point model yields lower mean squared error over the conventional single-time-point model at all time points, especially when the availability of evidence decreases. We illustrated the model on a network of randomized trials of second-line treatments of advanced non-small-cell lung cancer. Our multiple-time-point model yielded increased precision and detected evidence of benefit at earlier time points as compared to the single-time-point model. Our model has the advantage of providing clinically interpretable measures of treatment effects., (© 2021 John Wiley & Sons Ltd.)

We propose a class of alternative estimates and tests to restricted mean survival time (RMST) which improves power in numerous survival scenarios while maintaining a level of interpretability. The industry standards for interpretable hypothesis tests in survival analysis, RMST and logrank tests (LRTs), can suffer from low power in cases where the proportional hazards assumption fails. In particular, when late differences occur between survival curves, our proposed estimate and class of tests, window mean survival time (WMST), outperforms both RMST and LRT without sacrificing interpretability, unlike weighted rank tests (WRTs). WMST has the added advantage of maintaining high power when the proportional hazards assumption is met, while WRTs do not. With testing methods often being chosen in advance of data collection, WMST can ensure adequate power without distributional assumptions and is robust to the choice of its restriction parameters. Functions for performing WMST analysis are provided in the survWM2 package in R., (© 2021 John Wiley & Sons Ltd.)

The challenges and potential benefits of incorporating biomarkers into clinical trial designs have been increasingly discussed, in particular to develop new agents for immune‐oncology or targeted cancer therapies. To more accurately identify a sensitive subpopulation of patients, in many cases, a larger sample size—and consequently higher development costs and a longer study period—might be required. This article discusses a biomarker‐based Bayesian (BM‐Bay) randomized clinical trial design that incorporates a predictive biomarker measured on a continuous scale with pre‐determined cutoff points or a graded scale to define multiple patient subpopulations. We consider designing interim analyses with suitable decision criteria to achieve correct and efficient identification of a target patient population for developing a new treatment. The proposed decision criteria allow not only the take‐in of sensitive subpopulations but also the ruling‐out of insensitive ones on the basis of the efficacy evaluation of a time‐to‐event outcome. Extensive simulation studies are conducted to evaluate the operating characteristics of the proposed method, including the probability of correct identification of the desired subpopulation and the expected number of patients, under a wide range of clinical scenarios. For illustration purposes, we apply the proposed method to design a randomized phase II immune‐oncology clinical trial. [ABSTRACT FROM AUTHOR]

Patients with cardiovascular diseases who experience disease-related short-term events, such as hospitalizations, often exhibit diverse long-term survival outcomes compared to others. In this study, we aim to improve the prediction of long-term survival probability by incorporating two short-term events using a flexible varying coefficient landmark model. Our objective is to predict the long-term survival among patients who survived up to a pre-specified landmark time since the initial admission. Inverse probability weighting estimation equations are formed based on the information of the short-term outcomes before the landmark time. The kernel smoothing method with the use of cross-validation for bandwidth selection is employed to estimate the time-varying coefficients. The predictive performance of the proposed model is evaluated and compared using predictive measures: area under the receiver operating characteristic curve and Brier score. Simulation studies confirm that parameters under the landmark models can be estimated accurately and the predictive performance of the proposed method consistently outperforms existing methods that either do not incorporate or only partially incorporate information from two short-term events. We demonstrate the practical application of our model using a community-based cohort from the Atherosclerosis Risk in Communities (ARIC) study., (© 2024 John Wiley & Sons Ltd.)

Mammography remains the primary screening strategy for breast cancer, which continues to be the most prevalent cancer diagnosis among women globally. Because screening mammograms capture both the left and right breast, there is a nonnegligible correlation between the pair of images. Previous studies have explored the concept of averaging between the pair of images after proper image registration; however, no comparison has been made in directly utilizing the paired images. In this paper, we extend the bivariate functional principal component analysis over triangulations to jointly characterize the pair of imaging data bounded in an irregular domain and then nest the extracted features within the survival model to predict the onset of breast cancer. The method is applied to our motivating data from the Joanne Knight Breast Health Cohort at Siteman Cancer Center. Our findings indicate that there was no statistically significant difference in model discrimination performance between averaging the pair of images and jointly modeling the two images. Although the breast cancer study did not reveal any significant difference, it is worth noting that the methods proposed here can be readily extended to other studies involving paired or multivariate imaging data., (© 2024 John Wiley & Sons, Ltd.)

Competing risks data are commonly encountered in randomized clinical trials or observational studies. Ignoring competing risks in survival analysis leads to biased risk estimates and improper conclusions. Often, one of the competing events is of primary interest and the rest competing events are handled as nuisances. These approaches can be inadequate when multiple competing events have important clinical interpretations and thus of equal interest. For example, in COVID‐19 in‐patient treatment trials, the outcomes of COVID‐19 related hospitalization are either death or discharge from hospital, which have completely different clinical implications and are of equal interest, especially during the pandemic. In this paper we develop nonparametric estimation and simultaneous inferential methods for multiple cumulative incidence functions (CIFs) and corresponding restricted mean times. Based on Monte Carlo simulations and a data analysis of COVID‐19 in‐patient treatment clinical trial, we demonstrate that the proposed method provides global insights of the treatment effects across multiple endpoints. [ABSTRACT FROM AUTHOR]

Prognostic models are useful tools for assessing a patient's risk of experiencing adverse health events. In practice, these models must be validated before implementation to ensure that they are clinically useful. The concordance index (C‐Index) is a popular statistic that is used for model validation, and it is often applied to models with binary or survival outcome variables. In this paper, we summarize existing criticism of the C‐Index and show that many limitations are accentuated when applied to survival outcomes, and to continuous outcomes more generally. We present several examples that show the challenges in achieving high concordance with survival outcomes, and we argue that the C‐Index is often not clinically meaningful in this setting. We derive a relationship between the concordance probability and the coefficient of determination under an ordinary least squares model with normally distributed predictors, which highlights the limitations of the C‐Index for continuous outcomes. Finally, we recommend existing alternatives that more closely align with common uses of survival models. [ABSTRACT FROM AUTHOR]

Treatment specific survival curves are an important tool to illustrate the treatment effect in studies with time‐to‐event outcomes. In non‐randomized studies, unadjusted estimates can lead to biased depictions due to confounding. Multiple methods to adjust survival curves for confounders exist. However, it is currently unclear which method is the most appropriate in which situation. Our goal is to compare forms of inverse probability of treatment weighting, the G‐Formula, propensity score matching, empirical likelihood estimation and augmented estimators as well as their pseudo‐values based counterparts in different scenarios with a focus on their bias and goodness‐of‐fit. We provide a short review of all methods and illustrate their usage by contrasting the survival of smokers and non‐smokers, using data from the German Epidemiological Trial on Ankle‐Brachial‐Index. Subsequently, we compare the methods using a Monte‐Carlo simulation. We consider scenarios in which correctly or incorrectly specified models for describing the treatment assignment and the time‐to‐event outcome are used with varying sample sizes. The bias and goodness‐of‐fit is determined by taking the entire survival curve into account. When used properly, all methods showed no systematic bias in medium to large samples. Cox regression based methods, however, showed systematic bias in small samples. The goodness‐of‐fit varied greatly between different methods and scenarios. Methods utilizing an outcome model were more efficient than other techniques, while augmented estimators using an additional treatment assignment model were unbiased when either model was correct with a goodness‐of‐fit comparable to other methods. These "doubly‐robust" methods have important advantages in every considered scenario. [ABSTRACT FROM AUTHOR]

Competing risks survival data in the presence of partially masked causes are frequently encountered in medical research or clinical trials. When longitudinal biomarkers are also available, it is of great clinical importance to examine associations between the longitudinal biomarkers and the cause‐specific survival outcomes. In this article, we propose a cause‐specific C‐index for joint models of longitudinal and competing risks survival data accounting for masked causes. We also develop a posterior predictive algorithm for computing the out‐of‐sample cause‐specific C‐index using Markov chain Monte Carlo samples from the joint posterior of the in‐sample longitudinal and competing risks survival data. We further construct the Δ$$ \Delta $$C‐index to quantify the strength of association between the longitudinal and cause‐specific survival data, or between the out‐of‐sample longitudinal and survival data. Empirical performance of the proposed assessment criteria is examined through an extensive simulation study. An in‐depth analysis of the real data from large cancer prevention trials is carried out to demonstrate the usefulness of the proposed methodology. [ABSTRACT FROM AUTHOR]

Randomized clinical trials are often designed to assess whether a test treatment prolongs survival relative to a control treatment. Increased patient heterogeneity, while desirable for generalizability of results, can weaken the ability of common statistical approaches to detect treatment differences, potentially hampering the regulatory approval of safe and efficacious therapies. A novel solution to this problem is proposed. A list of baseline covariates that have the potential to be prognostic for survival under either treatment is pre-specified in the analysis plan. At the analysis stage, using all observed survival times but blinded to patient-level treatment assignment, "noise" covariates are removed with elastic net Cox regression. The shortened covariate list is used by a conditional inference tree algorithm to segment the heterogeneous trial population into subpopulations of prognostically homogeneous patients (risk strata). After patient-level treatment unblinding, a treatment comparison is done within each formed risk stratum and stratum-level results are combined for overall statistical inference. The impressive power-boosting performance of our proposed 5-step stratified testing and amalgamation routine (5-STAR), relative to that of the logrank test and other common approaches that do not leverage inherently structured patient heterogeneity, is illustrated using a hypothetical and two real datasets along with simulation results. Furthermore, the importance of reporting stratum-level comparative treatment effects (time ratios from accelerated failure time model fits in conjunction with model averaging and, as needed, hazard ratios from Cox proportional hazard model fits) is highlighted as a potential enabler of personalized medicine. An R package is available at https://github.com/rmarceauwest/fiveSTAR., (© 2020 John Wiley & Sons, Ltd.)

Restricted mean survival time (RMST) evaluates the mean event-free survival time up to a prespecified time point. It has been used as an alternative measure of treatment effect owing to its model-free structure and clinically meaningful interpretation of treatment benefit for right-censored data. In clinical trials, another type of censoring called interval censoring may occur if subjects are examined at several discrete time points and the survival time falls into an interval rather than being exactly observed. The missingness of exact observations under interval-censored cases makes the nonparametric measure of treatment effect more challenging. Employing the linear smoothing technique to overcome the ambiguity, we propose a new model-free measure for the interval-censored RMST. As an alternative to the commonly used log-rank test, we further construct a hypothesis testing procedure to assess the survival difference between two groups. Simulation studies show that the bias of our proposed interval-censored RMST estimator is negligible and the testing procedure delivers promising performance in detecting between-group difference with regard to size and power under various configurations of survival curves. The proposed method is illustrated by reanalyzing two real datasets containing interval-censored observations., (© 2020 John Wiley & Sons, Ltd.)

Wide heterogeneity exists in cancer patients' survival, ranging from a few months to several decades. To accurately predict clinical outcomes, it is vital to build an accurate predictive model that relates the patients' molecular profiles with the patients' survival. With complex relationships between survival and high-dimensional molecular predictors, it is challenging to conduct nonparametric modeling and irrelevant predictors removing simultaneously. In this article, we build a kernel Cox proportional hazards semi-parametric model and propose a novel regularized garrotized kernel machine (RegGKM) method to fit the model. We use the kernel machine method to describe the complex relationship between survival and predictors, while automatically removing irrelevant parametric and nonparametric predictors through a LASSO penalty. An efficient high-dimensional algorithm is developed for the proposed method. Comparison with other competing methods in simulation shows that the proposed method always has better predictive accuracy. We apply this method to analyze a multiple myeloma dataset and predict the patients' death burden based on their gene expressions. Our results can help classify patients into groups with different death risks, facilitating treatment for better clinical outcomes., (© 2023 The Authors. Statistics in Medicine published by John Wiley & Sons Ltd.)

In clinical or epidemiological follow‐up studies, methods based on time scale indicators such as the restricted mean survival time (RMST) have been developed to some extent. Compared with traditional hazard rate indicator system methods, the RMST is easier to interpret and does not require the proportional hazard assumption. To date, regression models based on the RMST are indirect or direct models of the RMST and baseline covariates. However, time‐dependent covariates are becoming increasingly common in follow‐up studies. Based on the inverse probability of censoring weighting (IPCW) method, we developed a regression model of the RMST and time‐dependent covariates. Through Monte Carlo simulation, we verified the estimation performance of the regression parameters of the proposed model. Compared with the time‐dependent Cox model and the fixed (baseline) covariate RMST model, the time‐dependent RMST model has a better prediction ability. Finally, an example of heart transplantation was used to verify the above conclusions. [ABSTRACT FROM AUTHOR]

With the increasing importance of predictive modeling in health research comes the need for methods to rigorously assess predictive accuracy. We consider the problem of evaluating the accuracy of predictive models for nominal outcomes when outcome data are coarsened at random. We first consider the problem in the context of a multinomial response modeled by polytomous logistic regression. Attention is then directed to the motivating setting in which class membership corresponds to the state occupied in a multistate disease process at a time horizon of interest. Here, class (state) membership may be unknown at the time horizon since disease processes are under intermittent observation. We propose a novel extension to the polytomous discrimination index to address this and evaluate the predictive accuracy of an intensity‐based model in the context of a study involving patients with arthritis from a registry at the University of Toronto Centre for Prognosis Studies in Rheumatic Diseases. [ABSTRACT FROM AUTHOR]

The difficulty in identifying cancer stage in health care claims data has limited oncology quality of care and health outcomes research. We fit prediction algorithms for classifying lung cancer stage into three classes (stages I/II, stage III, and stage IV) using claims data, and then demonstrate a method for incorporating the classification uncertainty in survival estimation. Leveraging set‐valued classification and split conformal inference, we show how a fixed algorithm developed in one cohort of data may be deployed in another, while rigorously accounting for uncertainty from the initial classification step. We demonstrate this process using SEER cancer registry data linked with Medicare claims data. [ABSTRACT FROM AUTHOR]

Window mean survival time (WMST) evaluates the mean survival between a lower time horizon, τ0$$ {\tau}_0 $$, and an upper time horizon, τ1$$ {\tau}_1 $$. As a flexible extension of restricted mean survival time, specific clinically relevant windows of time can be assessed for survival difference accompanied by a communicable interpretation of estimates and tests. In its original application, WMST required the pre‐specification of a window through the selection of appropriate window bounds, τ0$$ {\tau}_0 $$ and τ1$$ {\tau}_1 $$. In the instance of severe window misspecification of τ0$$ {\tau}_0 $$ and τ1$$ {\tau}_1 $$, the analysis may suffer from low power and a less meaningful interpretation. In this article, we introduce versatile tests whose procedures are based on the simultaneous use of multiple WMST test statistics that are asymptotically normal under the null hypothesis of no difference between two groups. Simulations are performed to examine the power of the tests in moderate sample sizes when the data are uncensored to heavily censored with a ramp‐up enrollment period. The survival scenarios chosen for simulation are intended to imitate those which are commonly encountered in oncology, especially in trials involving immunotherapies. Implementation of the procedures is discussed in two real data examples for illustration. Functions for performing versatile WMST tests are provided in the survWMST package in R. [ABSTRACT FROM AUTHOR]

Identifying exceptional responders or nonresponders is an area of increased research interest in precision medicine as these patients may have different biological or molecular features and therefore may respond differently to therapies. Our motivation stems from a real example from a clinical trial where we are interested in characterizing exceptional prostate cancer responders. We investigate the outlier detection and robust regression problem in the sparse proportional hazards model for censored survival outcomes. The main idea is to model the irregularity of each observation by assigning an individual weight to the hazard function. By applying a LASSO‐type penalty on both the model parameters and the log transformation of the weight vector, our proposed method is able to perform variable selection and outlier detection simultaneously. The optimization problem can be transformed to a typical penalized maximum partial likelihood problem and thus it is easy to implement. We further extend the proposed method to deal with the potential outlier masking problem caused by censored outcomes. The performance of the proposed estimator is demonstrated with extensive simulation studies and real data analyses in low‐dimensional and high‐dimensional settings. [ABSTRACT FROM AUTHOR]

In randomized clinical trials, incorporating baseline covariates can improve the power in hypothesis testing for treatment effects. For survival endpoints, the Cox proportional hazards model with baseline covariates as explanatory variables can improve the standard logrank test in power. Although this has long been recognized, this adjustment is not commonly used as the primary analysis and instead the logrank test followed by the estimation of the hazard ratio between treatment groups is often used. By projecting the score function for the Cox proportional hazards model onto a space of covariates, the logrank test can be more powerful. We derive a power formula for this augmented logrank test under the same setting as the widely used power formula for the logrank test and propose a simple strategy for sizing randomized clinical trials utilizing historical data of the control treatment. Through numerical studies, the proposed procedure was found to have the potential to reduce the sample size substantially as compared to the standard logrank test. A concern to utilize historical data is that those might not reflect well the data structure of the study to design and then the sample size calculated might not be accurate. Since our power formula is applicable to datasets pooled across the treatment arms, the validity of the power calculation at the design stage can be checked in blind reviews. [ABSTRACT FROM AUTHOR]

Group sequential design (GSD) has become a popular choice in recent clinical trials as it improves trial efficiency by providing options for early termination. The implementation of traditional tests for survival analysis (eg, the log‐rank test and the Cox proportional hazard (PH) model) in the GSD setting has been widely discussed. The PH assumption is required for conventional (sequential) design, it is, however, often violated in practice. As an alternative, some generalized tests have been proposed (eg, the Max‐Combo test) and their efficacies have been established. In this article, we explore the application of a more flexible, "first hitting time" based threshold regression (TR) model to GSD. TR assumes that subjects' health status is a latent (unobservable) process, and the clinical event of interest occurs when the latent health process hits a pre‐specified boundary. The simulation results supported our findings that, in most cases, this comparable new method can successfully control type I error while providing higher early stopping opportunities in the sequential design, even when non‐proportional hazard presents. [ABSTRACT FROM AUTHOR]

Previous articles in Statistics in Medicine describe how to calculate the sample size required for external validation of prediction models with continuous and binary outcomes. The minimum sample size criteria aim to ensure precise estimation of key measures of a model's predictive performance, including measures of calibration, discrimination, and net benefit. Here, we extend the sample size guidance to prediction models with a time‐to‐event (survival) outcome, to cover external validation in datasets containing censoring. A simulation‐based framework is proposed, which calculates the sample size required to target a particular confidence interval width for the calibration slope measuring the agreement between predicted risks (from the model) and observed risks (derived using pseudo‐observations to account for censoring) on the log cumulative hazard scale. Precise estimation of calibration curves, discrimination, and net‐benefit can also be checked in this framework. The process requires assumptions about the validation population in terms of the (i) distribution of the model's linear predictor and (ii) event and censoring distributions. Existing information can inform this; in particular, the linear predictor distribution can be approximated using the C‐index or Royston's D statistic from the model development article, together with the overall event risk. We demonstrate how the approach can be used to calculate the sample size required to validate a prediction model for recurrent venous thromboembolism. Ideally the sample size should ensure precise calibration across the entire range of predicted risks, but must at least ensure adequate precision in regions important for clinical decision‐making. Stata and R code are provided. [ABSTRACT FROM AUTHOR]

Semiparametric accelerated failure time (AFT) models are a useful alternative to Cox proportional hazards models, especially when the assumption of constant hazard ratios is untenable. However, rank‐based criteria for fitting AFT models are often nondifferentiable, which poses a computational challenge in high‐dimensional settings. In this article, we propose a new alternating direction method of multipliers algorithm for fitting semiparametric AFT models by minimizing a penalized rank‐based loss function. Our algorithm scales well in both the number of subjects and number of predictors, and can easily accommodate a wide range of popular penalties. To improve the selection of tuning parameters, we propose a new criterion which avoids some common problems in cross‐validation with censored responses. Through extensive simulation studies, we show that our algorithm and software is much faster than existing methods (which can only be applied to special cases), and we show that estimators which minimize a penalized rank‐based criterion often outperform alternative estimators which minimize penalized weighted least squares criteria. Application to nine cancer datasets further demonstrates that rank‐based estimators of semiparametric AFT models are competitive with estimators assuming proportional hazards in high‐dimensional settings, whereas weighted least squares estimators are often not. A software package implementing the algorithm, along with a set of auxiliary functions, is available for download at github.com/ajmolstad/penAFT. [ABSTRACT FROM AUTHOR]

Nonproportional hazards (NPHs) are often observed in survival trials such as the immunotherapy cancer trials. Under NPH, the classical log‐rank test can be inefficient, and the estimated hazards ratio from the Cox model is difficult to interpret. The weighted log‐rank test, and the tests for comparing the restricted mean survival time or the milestone survival become increasingly popular in handling NPH. The sample size calculation for these tests may require high‐dimensional numerical integration. We present a sample size determination method for survival trials via product integration on the basis of a continuous‐time multistate Markov model. The main challenge of the method lies in the design of the multistate model under a complex NPH pattern, and this is illustrated for NPH induced by delayed effect with individual heterogeneity in the lag duration, cure fractions, and treatment switching due to disease progression or noncompliance. Numerical examples are presented to demonstrate the accuracy of the proposed method. We obtain the following findings. The powers of the tests for milestone survival and RMST depend on both the trial duration and milestone timepoint, and may not increase as the milestone timepoint increases. If the milestone timepoint is appropriately chosen, the RMST test can be more powerful than the conventional log‐rank test in the presence of diminishing treatment effect or in the proportional hazards cure model. In general, the RMST test yields lower power than a proper Fleming‐Harrington weighted log‐rank test. [ABSTRACT FROM AUTHOR]

Time‐dependent receiver operating characteristic curves are often used to evaluate the classification performance of continuous measures when considering time‐to‐event data. When one is interested in evaluating the predictive performance of multiple covariates, it is common to use the Cox proportional hazards model to obtain risk scores; however, previous work has shown that when the model is mis‐specified, the estimand corresponding to the partial likelihood estimator depends on the censoring distribution. In this manuscript, we show that when the risk score model is mis‐specified, the AUC will also depend on the censoring distribution, leading to either over‐ or under‐estimation of the risk score's predictive performance. We propose the use of censoring‐robust estimators to remove the dependence on the censoring distribution and provide empirical results supporting the use of censoring‐robust risk scores. [ABSTRACT FROM AUTHOR]

High‐dimensional data are becoming increasingly common in the medical field as large volumes of patient information are collected and processed by high‐throughput screening, electronic health records, and comprehensive genomic testing. Statistical models that attempt to study the effects of many predictors on survival typically implement feature selection or penalized methods to mitigate the undesirable consequences of overfitting. In some cases survival data are also left‐truncated which can give rise to an immortal time bias, but penalized survival methods that adjust for left truncation are not commonly implemented. To address these challenges, we apply a penalized Cox proportional hazards model for left‐truncated and right‐censored survival data and assess implications of left truncation adjustment on bias and interpretation. We use simulation studies and a high‐dimensional, real‐world clinico‐genomic database to highlight the pitfalls of failing to account for left truncation in survival modeling. [ABSTRACT FROM AUTHOR]

In prognosis studies to evaluate association between a continuous biomarker and a survival outcome, investigators often classify subjects into two subclasses of the high‐ and low‐expression groups and apply simple survival analysis techniques of the Kaplan‐Meier method and the logrank test. The high‐ and low‐expressions are defined according to whether or not the observation of the biomarker is higher than the cut‐off value, which is heterogeneous across studies. The heterogeneous definitions of the cut‐off value make it difficult to apply the standard meta‐analysis techniques. We propose a method to estimate the concordance index for a survival outcome synthesizing published prognosis studies, in which the Kaplan‐Meier estimates for the high‐ and low‐expression groups are reported. We illustrate our proposed method with a real dataset for meta‐analysis of prognosis studies evaluating Ki‐67 in early breast cancer and evaluate its performance with a simulation study. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In many biomedical studies, participants are monitored at periodic visits until the occurrence of the failure event. Biomarkers are often measured repeatedly during these visits, and such measurements can facilitate updated disease prediction. In this work, we propose a two‐dimensional incident dynamic area under curve (AUC), to capture the variability due to both the biomarker assessment time and the prediction time to comprehensively quantify the predictive performance of a longitudinal biomarker. We propose a pseudo partial‐likelihood to achieve consistent estimation of the AUC under two realistic scenarios of visit schedules. Variance estimation methods are designed to facilitate inferential procedures. We examine the finite‐sample performance of our method through extensive simulations. The methods are applied to a study of chronic myeloid leukemia to evaluate the predictive performance of longitudinally collected gene expression levels. [ABSTRACT FROM AUTHOR]

Methods for estimating heterogeneous treatment effect in observational data have largely focused on continuous or binary outcomes, and have been relatively less vetted with survival outcomes. Using flexible machine learning methods in the counterfactual framework is a promising approach to address challenges due to complex individual characteristics, to which treatments need to be tailored. To evaluate the operating characteristics of recent survival machine learning methods for the estimation of treatment effect heterogeneity and inform better practice, we carry out a comprehensive simulation study presenting a wide range of settings describing confounded heterogeneous survival treatment effects and varying degrees of covariate overlap. Our results suggest that the nonparametric Bayesian Additive Regression Trees within the framework of accelerated failure time model (AFT‐BART‐NP) consistently yields the best performance, in terms of bias, precision, and expected regret. Moreover, the credible interval estimators from AFT‐BART‐NP provide close to nominal frequentist coverage for the individual survival treatment effect when the covariate overlap is at least moderate. Including a nonparametrically estimated propensity score as an additional fixed covariate in the AFT‐BART‐NP model formulation can further improve its efficiency and frequentist coverage. Finally, we demonstrate the application of flexible causal machine learning estimators through a comprehensive case study examining the heterogeneous survival effects of two radiotherapy approaches for localized high‐risk prostate cancer. [ABSTRACT FROM AUTHOR]

Pathologic complete response (pCR) is a common primary endpoint for a phase II trial or even accelerated approval of neoadjuvant cancer therapy. If granted, a two‐arm confirmatory trial is often required to demonstrate the efficacy with a time‐to‐event outcome such as overall survival. However, the design of a subsequent phase III trial based on prior information on the pCR effect is not straightforward. Aiming at designing such phase III trials with overall survival as primary endpoint using pCR information from previous trials, we consider a mixture model that incorporates both the survival and the binary endpoints. We propose to base the comparison between arms on the difference of the restricted mean survival times, and show how the effect size and sample size for overall survival rely on the probability of the binary response and the survival distribution by response status, both for each treatment arm. Moreover, we provide the sample size calculation under different scenarios and accompany them with the R package survmixer where all the computations have been implemented. We evaluate our proposal with a simulation study, and illustrate its application through a neoadjuvant breast cancer trial. [ABSTRACT FROM AUTHOR]

Summary: The nested case‐control (NCC) design has been widely adopted as a cost‐effective sampling design for biomarker research. Under the NCC design, markers are only measured for the NCC subcohort consisting of all cases and a fraction of the controls selected randomly from the matched risk sets of the cases. Robust methods for evaluating prediction performance of risk models have been derived under the inverse probability weighting framework. The probabilities of samples being included in the NCC cohort can be calculated based on the study design ''a previous study'' or estimated non‐parametrically ''a previous study''. Neither strategy works well due to model mis‐specification and the curse of dimensionality in practical settings where the sampling does not entirely follow the study design or depends on many factors. In this paper, we propose an alternative strategy to estimate the sampling probabilities based on a varying coefficient model, which attains a balance between robustness and the curse of dimensionality. The complex correlation structure induced by repeated finite risk set sampling makes the standard resampling procedure for variance estimation fail. We propose a perturbation resampling procedure that provides valid interval estimation for the proposed estimators. Simulation studies show that the proposed method performs well in finite samples. We apply the proposed method to the Nurses' Health Study II to develop and evaluate prediction models using clinical biomarkers for cardiovascular risk. [ABSTRACT FROM AUTHOR]

In randomized clinical trials with survival outcome, there has been an increasing interest in subgroup identification based on baseline genomic, proteomic markers, or clinical characteristics. Some of the existing methods identify subgroups that benefit substantially from the experimental treatment by directly modeling outcomes or treatment effect. When the goal is to find an optimal treatment for a given patient rather than finding the right patient for a given treatment, methods under the individualized treatment regime framework estimate an individualized treatment rule that would lead to the best expected clinical outcome as measured by a value function. Connecting the concept of value function to subgroup identification, we propose a nonparametric method that searches for subgroup membership scores by maximizing a value function that directly reflects the subgroup‐treatment interaction effect based on restricted mean survival time. A gradient tree boosting algorithm is proposed to search for the individual subgroup membership scores. We conduct simulation studies to evaluate the performance of the proposed method and an application to an AIDS clinical trial is performed for illustration. [ABSTRACT FROM AUTHOR]

A typical challenge facing the design and analysis of immuno‐oncology (IO) trials is the prevalence of nonproportional hazards (NPH) patterns manifested in Kaplan‐Meier curves under time‐to‐event endpoints. The NPH patterns would violate the proportional hazards assumption, and yet conventional design and analysis strategies often ignore such a violation, resulting in underpowered or even falsely negative IO studies. In this article, we show, both empirically and analytically, that treating nonresponders in IO studies of inadequate size would give rise to a variety of NPH patterns; we then present a novel design and analysis strategy, P%‐responder information embedded (PRIME), to properly incorporate the dichotomized response incurred from treating nonresponders. Empirical studies demonstrate that the proposed strategy can achieve desirable power, whereas the conventional alternative leads to a severe power loss. The PRIME strategy allows us to quantify the impact of treating nonresponders on study efficiency, thereby enabling a proper design of IO trials with an adequate power. More importantly, it pinpoints a solution to enhance the study efficiency and alleviates the NPH patterns by enrolling more prospective responders. An R package (Immunotherapy.Design) is developed for implementation. [ABSTRACT FROM AUTHOR]

The prediction reliability is of primary concern in many clinical studies when the objective is to develop new predictive models or improve existing risk scores. In fact, before using a model in any clinical decision making, it is very important to check its ability to discriminate between subjects who are at risk of, for example, developing certain disease in a near future from those who will not. To that end, the time‐dependent receiver operating characteristic (ROC) curve is the most commonly used method in practice. Several approaches have been proposed in the literature to estimate the ROC nonparametrically in the context of survival data. But, except one recent approach, all the existing methods provide a nonsmooth ROC estimator whereas, by definition, the ROC curve is smooth. In this article we propose and study a new nonparametric smooth ROC estimator based on a weighted kernel smoother. More precisely, our approach relies on a well‐known kernel method used to estimate cumulative distribution functions of random variables with bounded supports. We derived some asymptotic properties for the proposed estimator. As bandwidth is the main parameter to be set, we present and study different methods to appropriately select one. A simulation study is conducted, under different scenarios, to prove the consistency of the proposed method and to compare its finite sample performance with a competitor. The results show that the proposed method performs better and appear to be quite robust to bandwidth choice. As for inference purposes, our results also reveal the good performances of a proposed nonparametric bootstrap procedure. Furthermore, we illustrate the method using a real data example. [ABSTRACT FROM AUTHOR]

Assessing and comparing the performance of correlated predictive scores are of current interest in precision medicine. Given the limitations of available theoretical approaches for assessing and comparing the predictive accuracy, numerical methods are highly desired which, however, have not been systematically developed due to technical challenges. The main challenges include the lack of a general strategy on effectively simulating many kinds of correlated predictive scores each with some given level of predictive accuracy in either concordance index or the area under a receiver operating characteristic curve area under the curves (AUC). To fill in this important knowledge gap, this paper is to provide a general copula-based numeric framework for assessing and comparing predictive performance of correlated predictive or risk scores. The new algorithms are designed to effectively simulate correlated predictive scores with given levels of predictive accuracy as measured in terms of concordance indices or time-dependent AUC for predicting survival outcomes. The copula-based numerical strategy is convenient for numerically evaluating and comparing multiple measures of predictive accuracy of correlated risk scores and for investigating finite-sample properties of test statistics and confidence intervals as well as assessing for optimism of given performance measures using cross-validation or bootstrap. [ABSTRACT FROM AUTHOR]

Several studies for the clinical validity of circulating tumor cells (CTCs) in metastatic breast cancer were conducted showing that it is a prognostic biomarker of overall survival. In this work, we consider an individual patient data meta-analysis for nonmetastatic breast cancer to assess the discrimination of CTCs regarding the risk of death. Data are collected in several centers and present correlated failure times for subjects of the same center. However, although the covariate-specific time-dependent receiver operating characteristic (ROC) curve has been widely used for assessing the performance of a biomarker, there is no methodology yet that can handle this specific setting with clustered censored failure times. We propose an estimator for the covariate-specific time-dependent ROC curves and area under the ROC curve when clustered failure times are detected. We discuss the assumptions under which the estimators are consistent and their interpretations. We assume a shared frailty model for modeling the effect of the covariates and the biomarker on the outcome in order to account for the cluster effect. A simulation study was conducted and it shows negligible bias for the proposed estimator and a nonparametric one based on inverse probability censoring weighting, while a semiparametric estimator, ignoring the clustering, is markedly biased. Finally, in our application to breast cancer data, the estimation of the covariate-specific area under the curves illustrates that the CTCs discriminate better patients with inflammatory tumor than patients with noninflammatory tumor, with respect to their risk of death. [ABSTRACT FROM AUTHOR]