Statistical models for composite endpoints of death and non-fatal events: a review

Short Course Antibiotic Therapy for Catheter-Related Septic Thrombosis: "Caveat Emptor!": Duration of Therapy Should Not Be Set a Priori

There is a growing body of evidence showing no significant difference in clinical outcomes in patients with uncomplicated Gram-negative bloodstream infections (BSIs) receiving 7 or 14 days of therapy. However, the scenario may differ when complicated forms of BSI, such as catheter-related BSIs (CRBSIs) burdened by septic thrombosis (ST), are considered. A recent study showed that a short course of antimicrobial therapy (≤3 weeks) had similar outcomes to a prolonged course on CRBSI-ST. From this perspective, starting from the desirable goal of shortening the treatment duration, we discuss how the path to the correct diagnosis and management of CRBSI-ST may be paved with several challenges. Indeed, patients with ST due to Gram-negative bacteria display prolonged bacteremia despite an indolent clinical course, requiring an extended course of antibiotic treatment guided by negative FUBCs results, which should be considered the real driver of the decision-making process establishing the length of antibiotic therapy in CRBSI-ST. Shortening treatment of complicated CRBSIs burdened by ST is ambitious and advisable; however, a dynamic and tailored approach driven by a tangible outcome such as negative FUBCs rather than a fixed-duration paradigm should be implemented for the optimal antimicrobial duration.

Multiply robust estimation of principal causal effects with noncompliance and survival outcomes

Treatment noncompliance and censoring are two common complications in clinical trials. Motivated by the ADAPTABLE pragmatic clinical trial, we develop methods for assessing treatment effects in the presence of treatment noncompliance with a right-censored survival outcome. We classify the participants into principal strata, defined by their joint potential compliance status under treatment and control. We propose a multiply robust estimator for the causal effects on the survival probability scale within each principal stratum. This estimator is consistent even if one, sometimes two, of the four working models-on the treatment assignment, the principal strata, censoring, and the outcome-is misspecified. A sensitivity analysis strategy is developed to address violations of key identification assumptions, the principal ignorability and monotonicity. We apply the proposed approach to the ADAPTABLE trial to study the causal effect of taking low- versus high-dosage aspirin on all-cause mortality and hospitalization from cardiovascular diseases.

Dissecting the restricted mean time in favor of treatment

The restricted mean time in favor (RMT-IF) summarizes the treatment effect on a hierarchical composite endpoint with mortality at the top. Its crude decomposition into "stage-wise effects," i.e., the net average time gained by the treatment prior to each component event, does not reveal the patient state in which the extra time is spent. To obtain this information, we break each stage-wise effect into subcomponents according to the specific state to which the reference condition is improved. After re-expressing the subcomponents as functionals of the marginal survival functions of outcome events, we estimate them conveniently by plugging in the Kaplan -- Meier estimators. Their robust variance matrices allow us to construct joint tests on the decomposed units, which are particularly powerful against component-wise differential treatment effects. By reanalyzing a cancer trial and a cardiovascular trial, we acquire new insights into the quality and composition of the extra survival times, as well as the extra time with fewer hospitalizations, gained by the treatment in question. The proposed methods are implemented in the rmt package freely available on the Comprehensive R Archive Network (CRAN).

Adaptive clinical trial designs with blinded selection of binary composite endpoints and sample size reassessment

For randomized clinical trials where a single, primary, binary endpoint would require unfeasibly large sample sizes, composite endpoints (CEs) are widely chosen as the primary endpoint. Despite being commonly used, CEs entail challenges in designing and interpreting results. Given that the components may be of different relevance and have different effect sizes, the choice of components must be made carefully. Especially, sample size calculations for composite binary endpoints depend not only on the anticipated effect sizes and event probabilities of the composite components but also on the correlation between them. However, information on the correlation between endpoints is usually not reported in the literature which can be an obstacle for designing future sound trials. We consider two-arm randomized controlled trials with a primary composite binary endpoint and an endpoint that consists only of the clinically more important component of the CE. We propose a trial design that allows an adaptive modification of the primary endpoint based on blinded information obtained at an interim analysis. Especially, we consider a decision rule to select between a CE and its most relevant component as primary endpoint. The decision rule chooses the endpoint with the lower estimated required sample size. Additionally, the sample size is reassessed using the estimated event probabilities and correlation, and the expected effect sizes of the composite components. We investigate the statistical power and significance level under the proposed design through simulations. We show that the adaptive design is equally or more powerful than designs without adaptive modification on the primary endpoint. Besides, the targeted power is achieved even if the correlation is misspecified at the planning stage while maintaining the type 1 error. All the computations are implemented in R and illustrated by means of a peritoneal dialysis trial.

Study design for restricted mean time analysis of recurrent events and death

The restricted mean time in favor (RMT-IF) of treatment has just been added to the analytic toolbox for composite endpoints of recurrent events and death. To help practitioners design new trials based on this method, we develop tools to calculate the sample size and power. Specifically, we formulate the outcomes as a multistate Markov process with a sequence of transient states for recurrent events and an absorbing state for death. The transition intensities, in this case the instantaneous risks of another nonfatal event or death, are assumed to be time-homogeneous but nonetheless allowed to depend on the number of past events. Using the properties of Coxian distributions, we derive the RMT-IF effect size under the alternative hypothesis as a function of the treatment-to-control intensity ratios along with the baseline intensities, the latter of which can be easily estimated from historical data. We also reduce the variance of the nonparametric RMT-IF estimator to calculable terms under a standard set-up for censoring. Simulation studies show that the resulting formulas provide accurate approximation to the sample size and power in realistic settings. For illustration, a past cardiovascular trial with recurrent-hospitalization and mortality outcomes is analyzed to generate the parameters needed to design a future trial. The procedures are incorporated into the rmt package along with the original methodology on the Comprehensive R Archive Network (CRAN).

Reflections on estimands for patient-reported outcomes in cancer clinical trials

It is common and important to include the patient's perspective of the impact of treatment on health-related quality of life (HRQoL) outcomes. In this commentary, we focus on applying the new addendum to ICH E9 guideline E9 (R1) relating to the estimand framework to Patient Reported Outcomes (PROs) collected in cancer clinical trials, from a statistician's viewpoint. Currently, common practice for statistical analysis of PRO endpoints of published cancer clinical trials demonstrates ambiguity, leaving critical questions unspecified, hindering conclusions about the effect of treatment on PRO endpoints as well as comparability between clinical trials. To avoid this scenario, we advocate the systematic use of the estimand framework which requires the prospective definition of clear PRO research questions. Among the five attributes of the estimands framework, the definition of the endpoint (what is the right PRO measure and timeframe to target and why?), the intercurrent event identification and management (what happens with PRO data post-disease progression, what is the impact of death?) and the population-level summary (what is an acceptable statistical summary for PRO data?) require the most attention for PRO estimands. We identify good practice and highlight discussion points including the challenges of statistical analysis in the presence of missing and/or unobservable data and in relation to death. Through this discussion we highlight that there is no "statistical magic", but that the estimand framework will help you find out what you really want to know when quantifying the benefit of treatments from the patients' perspective.

Nonparametric inference of general while-alive estimands for recurrent events

Measuring the treatment effect on recurrent events like hospitalization in the presence of death has long challenged statisticians and clinicians alike. Traditional inference on the cumulative frequency unjustly penalizes survivorship as longer survivors also tend to experience more adverse events. Expanding a recently suggested idea of the "while-alive" event rate, we consider a general class of such estimands that adjust for the length of survival without losing causal interpretation. Given a user-specified loss function that allows for arbitrary weighting, we define as estimand the average loss experienced per unit time alive within a target period and use the ratio of this loss rate to measure the effect size. Scaling the loss rate by the width of the corresponding time window gives us an alternative, and sometimes more photogenic, way of showing the data. To make inferences, we construct a nonparametric estimator for the loss rate through the cumulative loss and the restricted mean survival time and derive its influence function in closed form for variance estimation and testing. As simulations and analysis of real data from a heart failure trial both show, the while-alive approach corrects for the false attenuation of treatment effect due to patients living longer under treatment, with increased statistical power as a result. The proposed methods are implemented in the R-package WA, which is publicly available from the Comprehensive R Archive Network (CRAN).

The Impact of Glucagon-Like Peptide-1 Receptor Agonist on the Cardiovascular Outcomes in Patients With Type 2 Diabetes Mellitus: A Meta-Analysis and Systematic Review

Background

Since 2005, the cardioprotective effects of glucagon-like peptide 1 receptor agonists (GLP-1 RAs) have garnered attention. The cardioprotective effect could be an added benefit to the use of GLP-1 RA. This systematic review and meta-analysis aimed at summarizing observational studies that recruited type 2 diabetes individuals with fewer cardiovascular (CV) events before enrolling in the research.

Methods

Systematically, the databases were searched for observational studies reporting compound CV events and deaths in type 2 diabetics without having the risk of cardiovascular diseases (CVDs) compared to other glucose-lowering agents. A meta-analysis was carried out using random effects model to estimate the overall hazard ratio (HR) with a 95% confidence interval (CI). Five studies were found eligible for the systematic review including a total of 64,452 patients receiving either liraglutide (three studies) or exenatide (two studies).

Results

The pooled HR for major adverse cardiac event (MACE) and extended MACE was 0.72 (95% CI: 0.65 - 0.93, I2 = 68%) and 0.93 (95% CI: 0.89 - 0.98, I2 = 29%), respectively. The pooled HR for hospitalization due to heart failure (HHF) and occurrence of HF was 0.84 (95% CI: 0.77 - 0.91, I2 = 79%) and 0.83 (95% CI: 0.75 - 0.94, I2 = 95%), respectively. For stroke, GLP-1 RA was associated with a significant risk reduction of 0.86 (95% CI: 0.75 - 0.98, I2 = 81%). There was no significant myocardial infarction (MI) risk reduction with GLP-1 RA. As for all-cause mortality, the pooled HR for the occurrence of all-cause mortality was 0.82 (95% CI: 0.76 - 0.88, I2 = 0%). The pooled HR for the occurrence of CV death was 0.75 (95% CI: 0.65 - 0.85, I2 = 38%). GLP-1 RA therapy was associated with a significantly low risk of MACE, extended MACE, all-cause mortality, and CV mortality. Except for MACE, the heterogenicity among the studies was low.

Conclusion

We conclude that GLP-1 RA is associated with a low risk of CV events composites and mortality. The findings support the cardioprotective effect of GLP-1 RA.

On restricted mean time in favor of treatment

The restricted mean time in favor (RMT-IF) of treatment is a nonparametric effect size for complex life history data. It is defined as the net average time the treated spend in a more favorable state than the untreated over a prespecified time window. It generalizes the familiar restricted mean survival time (RMST) from the two-state life-death model to account for intermediate stages in disease progression. The overall estimand can be additively decomposed into stage-wise effects, with the standard RMST as a component. Alternate expressions of the overall and stage-wise estimands as integrals of the marginal survival functions for a sequence of landmark transitioning events allow them to be easily estimated by plug-in Kaplan-Meier estimators. The dynamic profile of the estimated treatment effects as a function of follow-up time can be visualized using a multilayer, cone-shaped "bouquet plot." Simulation studies under realistic settings show that the RMT-IF meaningfully and accurately quantifies the treatment effect and outperforms traditional tests on time to the first event in statistical efficiency thanks to its fuller utilization of patient data. The new methods are illustrated on a colon cancer trial with relapse and death as outcomes and a cardiovascular trial with recurrent hospitalizations and death as outcomes. The R-package rmt implements the proposed methodology and is publicly available from the Comprehensive R Archive Network (CRAN).

The empirical estimate of the survival and variance using a weighted composite endpoint

BACKGROUND

Composite endpoints for estimating treatment efficacy are routinely used in several therapeutic areas and have become complex in the number and types of component outcomes included. It is assumed that its components are of similar asperity and chronology between both treatment arms as well as uniform in magnitude of the treatment effect. However, these assumptions are rarely satisfied. Understanding this heterogeneity is important in developing a meaningful assessment of the treatment effect.

METHODS

We developed the Weighted Composite Endpoint (WCE) method which uses weights derived from stakeholder values for each event type in the composite endpoint. The derivation for the product limit estimator and the variance of the estimate are presented. The method was then tested using data simulated from parameters based on a large cardiovascular trial. Variances from the estimated and traditional approach are compared through increasing sample size.

RESULTS

The WCE method used all of the events through follow-up and generated a multiple recurrent event survival. The treatment effect was measured as the difference in mean survivals between two treatment arms and corresponding 95% confidence interval, providing a less conservative estimate of survival and variance, giving a higher survival with a narrower confidence interval compared to the traditional time-to-first-event analysis.

CONCLUSIONS

The WCE method embraces the clinical texture of events types by incorporating stakeholder values as well as all events during follow-up. While the effective number of events is lower in the WCE analysis, the reduction in variance enhances the ability to detect a treatment effect in clinical trials.

Win statistics (win ratio, win odds, and net benefit) can complement one another to show the strength of the treatment effect on time-to-event outcomes

Conventional analyses of a composite of multiple time-to-event outcomes use the time to the first event. However, the first event may not be the most important outcome. To address this limitation, generalized pairwise comparisons and win statistics (win ratio, win odds, and net benefit) have become popular and have been applied to clinical trial practice. However, win ratio, win odds, and net benefit have typically been used separately. In this article, we examine the use of these three win statistics jointly for time-to-event outcomes. First, we explain the relation of point estimates and variances among the three win statistics, and the relation between the net benefit and the Mann-Whitney U statistic. Then we explain that the three win statistics are based on the same win proportions, and they test the same null hypothesis of equal win probabilities in two groups. We show theoretically that the Z-values of the corresponding statistical tests are approximately equal; therefore, the three win statistics provide very similar p-values and statistical powers. Finally, using simulation studies and data from a clinical trial, we demonstrate that, when there is no (or little) censoring, the three win statistics can complement one another to show the strength of the treatment effect. However, when the amount of censoring is not small, and without adjustment for censoring, the win odds and the net benefit may have an advantage for interpreting the treatment effect; with adjustment (e.g., IPCW adjustment) for censoring, the three win statistics can complement one another to show the strength of the treatment effect. For calculations we use the R package WINS, available on the CRAN (Comprehensive R Archive Network).

Stratified proportional win-fractions regression analysis

The recently proposed proportional win-fractions (PW) model extends the two-sample win ratio analysis of prioritized composite endpoints to regression. Its proportionality assumption ensures that the covariate-specific win ratios are invariant to the follow-up time. However, this assumption is strong and may not be satisfied by every covariate in the model. We develop a stratified PW model that adjusts for certain prognostic factors without setting them as covariates, thus bypassing the proportionality requirement. We formulate the stratified model based on pairwise comparisons within each stratum, with a common win ratio across strata modeled as a multiplicative function of the covariates. Correspondingly, we construct an estimating function for the regression coefficients in the form of an incompleteU $$ U $$ -statistic consisting of within-stratum pairs. Two types of asymptotic variance estimators are developed depending on the number of strata relative to the sample size. This in particular allows valid inference even when the strata are extremely small, such as with matched pairs. Simulation studies in realistic settings show that the stratified model outperforms the unstratified version in robustness and efficiency. Finally, real data from a major cardiovascular trial are analyzed to illustrate the potential benefits of stratification. The proposed methods are implemented in the R package WR, publicly available on the Comprehensive R Archive Network (CRAN).

Power and Sample Size Calculations for the Restricted Mean Time Analysis of Prioritized Composite Endpoints

As a new way of reporting treatment effect, the restricted mean time in favor (RMT-IF) of treatment measures the net average time the treated have had a less serious outcome than the untreated over a specified time window. With multiple outcomes of differing severity, this offers a more interpretable and data-efficient alternative to the prototypical restricted mean (event-free) survival time. To facilitate its adoption in actual trials, we develop simple approaches to power and sample size calculations and implement them in user-friendly R programs. In doing so we model the bivariate outcomes of death and a nonfatal event using a Gumbel-Hougaard copula with component-wise proportional hazards structures, under which the RMT-IF estimand is derived in closed form. In a standard set-up for censoring, the variance of the nonparametric effect-size estimator is simplified and computed via a hybrid of numerical and Monte Carlo integrations, allowing us to compute the power and sample size as functions of component-wise hazard ratios. Simulation studies show that these formulas provide accurate approximations in realistic settings. To illustrate our methods, we consider designing a new trial to evaluate treatment effect on the composite outcomes of death and cancer relapse in lymph node-positive breast cancer patients, with baseline parameters calculated from a previous study.

On recurrent-event win ratio

The win ratio approach proposed by Pocock et al. (2012) has become a popular tool for analyzing composite endpoints of death and non-fatal events like hospitalization. Its standard version, however, draws on the non-fatal event only through the first occurrence. For statistical efficiency and clinical interpretability, we construct and compare different win ratio variants that make fuller use of recurrent events. We pay special attention to a variant called last-event-assisted win ratio, which compares two patients on the cumulative frequency of the non-fatal event, with ties broken by the time of its latest episode. It is shown that last-event-assisted win ratio uses more data than the standard win ratio does but reduces to the latter when the non-fatal event occurs at most once. We further prove that last-event-assisted win ratio rejects the null hypothesis with large probability if the treatment stochastically delays all events. Simulations under realistic settings show that the last-event-assisted win ratio test consistently enjoys higher power than the standard win ratio and other competitors. Analysis of a real cardiovascular trial provides further evidence for the practical advantages of the last-event-assisted win ratio. Finally, we discuss future work to develop meaningful effect size estimands based on the extended rules of comparison. The R-code for the proposed methods is included in the package WR openly available on the Comprehensive R Archive Network.