RMST-based multiple contrast tests in general factorial designs

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.

Twenty-five Years Overall Survival Prognostic Value of the Lymph Node Ratio in De Novo Metastatic Breast Cancer

BACKGROUND/AIM

The lymph node ratio (LNR) indicates the number of involved lymph nodes divided by the number of lymph nodes found during axillary exploration. This study investigated the prognostic value of the LNR in de novo metastatic breast cancer (dnMBC). We hypothesized that LNR might predict long-term survival even in cases where the disease has already disseminated beyond the regional stage.

PATIENTS AND METHODS

Patients with dnMBC were selected from the Surveillance, Epidemiology, and End Results (SEER) 9-registries database 1988-2012. Positive lymph nodes (npos) were categorized as pN0 (npos=0), pN1 (npos=1 to 3), pN2 (npos=4 to 9), and pN3 (npos≥10). The LNR was categorized as Lnr0 (LNR=0), Lnr1 (LNR=0.01 to 0.20), Lnr2 (LNR=0.21 to 0.65), and Lnr3 (LNR≥0.65). The prognostic values were compared using Gini's mean difference Δ of the restricted mean overall survival time (RMST) according to npos versus LNR groups.

RESULTS

A total of 12,085 patients with dnMBC had LNR data. At 25 years follow-up, the npos RMSTs were 10.4, 5.1, 5.8, and 5.0 years, for pN0 to pN3, respectively. The npos Gini's Δ was 2.8 years (standard error ±0.2). The LNR RMSTs were 10.4, 9.9, 7.6, and 4.0 years for Lnr0 to Lnr3, respectively. Δ for LNR was 3.6 (±0.2) years. Among node positive cases, the LNR low-risk group had an RMST of 9.9 years, approaching node-negative cases, while the high-risk group had an RMST of 4.0 years.

CONCLUSION

LNR identified different prognostic groups, suggesting a possible role of lymph node involvement as a marker of lymphangiogenesis or lymphatic changes in the immune microenvironment, which warrants further investigation in dnMBC.

Comparing Survival Extrapolation within All-Cause and Relative Survival Frameworks by Standard Parametric Models and Flexible Parametric Spline Models Using the Swedish Cancer Registry

BACKGROUND

In health technology assessment, restricted mean survival time and life expectancy are commonly evaluated. Parametric models are typically used for extrapolation. Spline models using a relative survival framework have been shown to estimate life expectancy of cancer patients more reliably; however, more research is needed to assess spline models using an all-cause survival framework and standard parametric models using a relative survival framework.

AIM

To assess survival extrapolation using standard parametric models and spline models within relative survival and all-cause survival frameworks.

METHODS

From the Swedish Cancer Registry, we identified patients diagnosed with 5 types of cancer (colon, breast, melanoma, prostate, and chronic myeloid leukemia) between 1981 and 1990 with follow-up until 2020. Patients were categorized into 15 cancer cohorts by cancer and age group (18-59, 60-69, and 70-99 y). We right-censored the follow-up at 2, 3, 5, and 10 y and fitted the parametric models within an all-cause and a relative survival framework to extrapolate to 10 y and lifetime in comparison with the observed Kaplan-Meier survival estimates. All cohorts were modeled with 6 standard parametric models (exponential, Weibull, Gompertz, log-logistic, log-normal, and generalized gamma) and 3 spline models (on hazard, odds, and normal scales).

RESULTS

For predicting 10-y survival, spline models generally performed better than standard parametric models. However, using an all-cause or a relative survival framework did not show any distinct difference. For lifetime survival, extrapolating from a relative survival framework agreed better with the observed survival, particularly using spline models.

CONCLUSIONS

For extrapolation to 10 y, we recommend spline models. For extrapolation to lifetime, we suggest extrapolating in a relative survival framework, especially using spline models.

HIGHLIGHTS

For survival extrapolation to 10 y, spline models generally performed better than standard parametric models did. However, using an all-cause or a relative survival framework showed no distinct difference under the same parametric model.Survival extrapolation to lifetime within a relative survival framework agreed well with the observed data, especially using spline models.Extrapolating parametric models within an all-cause survival framework may overestimate survival proportions at lifetime; models for the relative survival approach may underestimate instead.

Advanced considerations in survival analysis

Investigation of survival during the follow-up period is common in cardiovascular research and has intrinsic issues that require precise knowledge, such as survival or censoring. Besides, as the follow-up period lengthens and events other than mortality are studied, the analysis becomes more complex, so Kaplan-Meier analyses or Cox models are not always sufficient. In this primer, we provide the reader with detailed information on the interpretation of the most common survival analyses and delve into methods to analyse competing risks or alternatives to the conventional methods when the proportional hazards assumption is not met.

An Overview of Introductory and Advanced Survival Analysis Methods in Clinical Applications: Where Have we Come so far?

The time-to-event relationship for survival modeling is considered when designing a study in clinical trials. However, because time-to-event data are mostly not normally distributed, survival analysis uses non-parametric data processing and analysis methods, mainly Kaplan-Meier (KM) estimation models and Cox proportional hazards (CPH) regression models. At the same time, the log-rank test can be applied to compare curves from different groups. However, resorting to conventional survival analysis when fundamental assumptions, such as the Cox PH assumption, are not met can seriously affect the results, rendering them flawed. Consequently, it is necessary to examine and report more sophisticated statistical methods related to the processing of survival data, but at the same time, able to adequately respond to the contemporary real problems of clinical applications. On the other hand, the frequent misinterpretation of survival analysis methodology, combined with the fact that it is a complex statistical tool for clinicians, necessitates a better understanding of the basic principles underlying this analysis to effectively interpret medical studies in making treatment decisions. In this review, we first consider the basic models and mechanisms behind survival analysis. Then, due to common errors arising from the inappropriate application of conventional models, we revise more demanding statistical extensions of survival models related to data manipulation to avoid wrong results. By providing a structured review of the most representative statistical methods and tests covering contemporary survival analysis, we hope this review will assist in solving problems that arise in clinical applications.

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).

Semiparametric estimation of restricted mean survival time as a function of restriction time

The restricted mean survival time (RMST) is an appealing measurement in clinical or epidemiological studies with censored survival outcome and receives a lot of attention in the past decades. It provides a useful alternative to the Cox model for evaluating the covariate effect on survival time. The covariate effect on RMST usually varies with the restriction time. However, existing methods cannot address this problem properly. In this article, we propose a semiparametric framework that directly models RMST as a function of the restriction time. Our proposed model adopts a widely-used proportional form, enabling the estimation of RMST predictions across an interval using a unified model. Furthermore, the covariate effect for multiple restriction time points can be derived simultaneously. We develop estimators based on estimating equations theories and establish the asymptotic properties of the proposed estimators. The finite sample properties of the estimators are evaluated through extensive simulation studies. We further illustrate the application of our proposed method through the analysis of two real data examples. Supplementary Material are available online.

Multiply robust causal inference of the restricted mean survival time difference

The hazard ratio (HR) remains the most frequently employed metric in assessing treatment effects on survival times. However, the difference in restricted mean survival time (RMST) has become a popular alternative to the HR when the proportional hazards assumption is considered untenable. Moreover, independent of the proportional hazards assumption, many comparative effectiveness studies aim to base contrasts on survival probability rather than on the hazard function. Causal effects based on RMST are often estimated via inverse probability of treatment weighting (IPTW). However, this approach generally results in biased results when the assumed propensity score model is misspecified. Motivated by the need for more robust techniques, we propose an empirical likelihood-based weighting approach that allows for specifying a set of propensity score models. The resulting estimator is consistent when the postulated model set contains a correct model; this property has been termed multiple robustness. In this report, we derive and evaluate a multiply robust estimator of the causal between-treatment difference in RMST. Simulation results confirm its robustness. Compared with the IPTW estimator from a correct model, the proposed estimator tends to be less biased and more efficient in finite samples. Additional simulations reveal biased results from a direct application of machine learning estimation of propensity scores. Finally, we apply the proposed method to evaluate the impact of intrapartum group B streptococcus antibiotic prophylaxis on the risk of childhood allergic disorders using data derived from electronic medical records from the Children's Hospital of Philadelphia and census data from the American Community Survey.

A comparison of different population-level summary measures for randomised trials with time-to-event outcomes, with a focus on non-inferiority trials

BACKGROUND

The population-level summary measure is a key component of the estimand for clinical trials with time-to-event outcomes. This is particularly the case for non-inferiority trials, because different summary measures imply different null hypotheses. Most trials are designed using the hazard ratio as summary measure, but recent studies suggested that the difference in restricted mean survival time might be more powerful, at least in certain situations. In a recent letter, we conjectured that differences between summary measures can be explained using the concept of the non-inferiority frontier and that for a fair simulation comparison of summary measures, the same analysis methods, making the same assumptions, should be used to estimate different summary measures. The aim of this article is to make such a comparison between three commonly used summary measures: hazard ratio, difference in restricted mean survival time and difference in survival at a fixed time point. In addition, we aim to investigate the impact of using an analysis method that assumes proportional hazards on the operating characteristics of a trial designed with any of the three summary measures.

METHODS

We conduct a simulation study in the proportional hazards setting. We estimate difference in restricted mean survival time and difference in survival non-parametrically, without assuming proportional hazards. We also estimate all three measures parametrically, using flexible survival regression, under the proportional hazards assumption.

RESULTS

Comparing the hazard ratio assuming proportional hazards with the other summary measures not assuming proportional hazards, relative performance varies substantially depending on the specific scenario. Fixing the summary measure, assuming proportional hazards always leads to substantial power gains compared to using non-parametric methods. Fixing the modelling approach to flexible parametric regression assuming proportional hazards, difference in restricted mean survival time is most often the most powerful summary measure among those considered.

CONCLUSION

When the hazards are likely to be approximately proportional, reflecting this in the analysis can lead to large gains in power for difference in restricted mean survival time and difference in survival. The choice of summary measure for a non-inferiority trial with time-to-event outcomes should be made on clinical grounds; when any of the three summary measures discussed here is equally justifiable, difference in restricted mean survival time is most often associated with the most powerful test, on the condition that it is estimated under proportional hazards.

Two-sample inference procedures under nonproportional hazards

We introduce a new two-sample inference procedure to assess the relative performance of two groups over time. Our model-free method does not assume proportional hazards, making it suitable for scenarios where nonproportional hazards may exist. Our procedure includes a diagnostic tau plot to identify changes in hazard timing and a formal inference procedure. The tau-based measures we develop are clinically meaningful and provide interpretable estimands to summarize the treatment effect over time. Our proposed statistic is a U-statistic and exhibits a martingale structure, allowing us to construct confidence intervals and perform hypothesis testing. Our approach is robust with respect to the censoring distribution. We also demonstrate how our method can be applied for sensitivity analysis in scenarios with missing tail information due to insufficient follow-up. Without censoring, Kendall's tau estimator we propose reduces to the Wilcoxon-Mann-Whitney statistic. We evaluate our method using simulations to compare its performance with the restricted mean survival time and log-rank statistics. We also apply our approach to data from several published oncology clinical trials where nonproportional hazards may exist.

Restricted mean survival time to estimate an intervention effect in a cluster randomized trial

For time-to-event outcomes, the difference in restricted mean survival time is a measure of the intervention effect, an alternative to the hazard ratio, corresponding to the expected survival duration gain due to the intervention up to a predefined time t*. We extended two existing approaches of restricted mean survival time estimation for independent data to clustered data in the framework of cluster randomized trials: one based on the direct integration of Kaplan-Meier curves and the other based on pseudo-values regression. Then, we conducted a simulation study to assess and compare the statistical performance of the proposed methods, varying the number and size of clusters, the degree of clustering, and the magnitude of the intervention effect under proportional and non-proportional hazards assumption. We found that the extended methods well estimated the variance and controlled the type I error if there was a sufficient number of clusters (≥ 50) under both proportional and non-proportional hazards assumption. For cluster randomized trials with a limited number of clusters (< 50), a permutation test for pseudo-values regression was implemented and corrected the type I error. We also provided a procedure to estimate permutation-based confidence intervals which produced adequate coverage. All the extended methods performed similarly, but the pseudo-values regression offered the possibility to adjust for covariates. Finally, we illustrated each considered method with a cluster randomized trial evaluating the effectiveness of an asthma-control education program.

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).

Restricted Mean Survival Time-Can It Be a New Tool in Assessing the Survival of Non-Small Cell Lung Cancer Patients Treated with Immune Checkpoint Inhibitors?

BACKGROUND

Lung cancer (LC) is the first and most lethal cancer in the world; identifying new methods to treat it, such as immune checkpoint inhibitors (ICIs), is needed. ICIs treatment is very effective, but it comes bundled with a series of immune-related adverse events (irAEs). Restricted mean survival time (RMST) is an alternative tool for assessing the patients' survival when the proportional hazard assumption (PH) fails.

METHODS

We included in this analytical cross-sectional observational survey patients with metastatic non-small-cell lung cancer (NSCLC), treated for at least 6 months with ICIs in the first- and second-line settings. Using RMST, we estimated the overall survival (OS) of patients by dividing them into two groups. A multivariate Cox regression analysis was performed to determine the impact of the prognostic factors on OS.

RESULTS

Seventy-nine patients were included (68.4% men, mean age 63.8), and 34/79 (43%) presented irAEs. The OS RMST of the entire group was 30.91 months, with a survival median of 22 months. Thirty-two out of seventy-nine (40.5%) died before we ended our study. The OS RMST and death percentage favored the patients who presented irAEs (long-rank test, p = 0.036). The OS RMST of patients with irAEs was 35.7 months, with a number of deaths of 12/34 (35.29%), while the OS RMST of the patients without irAEs was 17 months, with a number of deaths of 20/45 (44.44%). The OS RMST by the line of treatment favored the first line of treatment. In this group, the presence of irAEs significantly impacted the survival of these patients (p = 0.0083). Moreover, patients that experienced low-grade irAEs had a better OS RMST. This result has to be cautiously regarded because of the small number of patients stratified according to the grades of irAEs. The prognostic factors for the survival were: the presence of irAEs, Eastern Cooperative Oncology Group (ECOG) performance status and the number of organs affected by metastasis. The risk of dying was 2.13 times higher for patients without irAEs than for the patients who presented irAEs, (CI) 95% of 1.03 to 4.39. Moreover, by increasing the ECOG performance status by one point, the risk of death increased by 2.28 times, with a CI 95% of 1.46 to 3.58, while the involvement of more metastatic organs was associated with a 1.60 times increase in the death risk, with a CI 95% of 1.09 to 2.36. Age and the type of tumor were not predictive for this analysis.

CONCLUSIONS

The RMST is a new tool that helps researchers to better address the survival in studies with ICIs treatment where the PH fails, and the long-rank test is less efficient due to the existence of the long-term responses and delayed treatment effects. Patients with irAEs have a better prognosis than those without irAEs in the first-line settings. The ECOG performance status and the number of organs affected by metastasis must be considered when selecting patients for ICIs treatment.

Studentized permutation method for comparing two restricted mean survival times with small sample from randomized trials

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.

Ratio and difference of average hazard with survival weight: New measures to quantify survival benefit of new therapy

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.

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).

Fast approximations of pseudo-observations in the context of right censoring and interval censoring

In the context of right-censored and interval-censored data, we develop asymptotic formulas to compute pseudo-observations for the survival function and the restricted mean survival time (RMST). These formulas are based on the original estimators and do not involve computation of the jackknife estimators. For right-censored data, Von Mises expansions of the Kaplan-Meier estimator are used to derive the pseudo-observations. For interval-censored data, a general class of parametric models for the survival function is studied. An asymptotic representation of the pseudo-observations is derived involving the Hessian matrix and the score vector. Theoretical results that justify the use of pseudo-observations in regression are also derived. The formula is illustrated on the piecewise-constant-hazard model for the RMST. The proposed approximations are extremely accurate, even for small sample sizes, as illustrated by Monte Carlo simulations and real data. We also study the gain in terms of computation time, as compared to the original jackknife method, which can be substantial for a large dataset.

A nonparametric simultaneous confidence band for biomarker effect on the restricted mean survival time

Study of prognostic and predictive biomarkers plays an important role in the design and analysis of clinical trials. The Cox proportional hazards model is often used to study the biomarker main effect and the treatment-biomarker interaction effect for survival data. The estimated effects can be biased if the proportional hazards assumption is violated. The restricted mean survival time is becoming popular in clinical studies for having a clear intuitive interpretation. In this article, we first propose nonparametric methods to make statistical inference for the one-sample problem of the biomarker effect on the restricted mean survival time; we then extend the methods to the two-sample problem for studying the difference in the biomarker effects between treatment groups in clinical trials. For a given biomarker, the restricted mean survival time is estimated by kernel smoothing methods with the inverse probability of censoring weights. We prove the consistency for the estimates and develop simultaneous confidence bands for the biomarker effects on the restricted mean survival time. The simultaneous confidence bands are evaluated in extensive simulation studies and are found to have good finite sample performance. We then apply the proposed methods to a breast cancer study conducted by the Breast International Group (BIG) to illustrate how the Ki67 biomarker, a protein marker of cell proliferation, affects the survival time of patients, compared between the treatment groups.

Restricted mean survival time versus conventional effect summary for treatment decision-making: A mixed-methods study

BACKGROUND

Treatment effect is typically summarized in terms of relative risk reduction or number needed to treat ("conventional effect summary"). Restricted mean survival time (RMST) summarizes treatment effect in terms of a gain or loss in event-free days. Older adults' preference between the two effect summary measures has not been studied.

METHODS

We conducted a mixed methods study using a quantitative survey and qualitative semi-structured interviews. For the survey, we enrolled 102 residents with hypertension at five senior housing facilities (mean age 81.3 years, 82 female, 95 white race). We randomly assigned respondents to either RMST-based (n = 49) or conventional decision aid (n = 53) about the benefits and harms of intensive versus standard blood pressure-lowering strategies and compared decision conflict scale (DCS) responses (range: 0 [no conflict] to 100 [maximum conflict]; <25 is associated with implementing decisions). We used a purposive sample of 23 survey respondents stratified by both their random assignment and DCS from the survey. Inductive qualitative thematic analysis explored complementary perspectives on preferred ways of summarizing treatment effects.

RESULTS

The mean (standard deviation) total DCS was 22.0 (14.3) for the conventional decision aid group and 16.7 (14.1) for the RMST-based decision aid group (p = 0.06), but the proportion of participants with a DCS <25 was higher in the RMST-based group (26 [49.1%] vs 34 [69.4%]; p = 0.04). Qualitative interviews suggested that, regardless of effect summary measure, older individuals' preference depended on their ability to clearly comprehend quantitative information, clarity of presentation in the visual aid, and inclusion of desired information.

CONCLUSIONS

When choosing a blood pressure-lowering strategy, older adults' perceived uncertainty may be reduced with a time-based effect summary, although our study was underpowered to detect a statistically significant difference. Given highly variable individual preferences, it may be useful to present both conventional and RMST-based information in decision aids.

Lymphoepithelial carcinoma of the head and neck: a SEER analysis of prognostic factors for survival

OBJECTIVE

To explore the epidemiological characteristics of patients with lymphoepithelial carcinoma (LEC) of the head and neck and the prognostic factors.

METHODS

We conducted a retrospective cohort study of cases of head and neck LEC retrieved from the Surveillance, Epidemiology and End Results database. Kaplan-Meier survival analysis and the log-rank test were employed to assess overall survival (OS) and cancer-specific survival (CSS). Univariate and multivariate analyses were used to construct Cox regression models. We established nomograms to predict OS and CSS among patients with nasopharyngeal LEC, who were divided into high- and low-risk groups based on the OS nomograms to compare the effects of treatment using the restricted mean survival time (RMST).

RESULTS

The 5-year OS and CSS rates of the cohort were 70.8% and 74.8%, respectively. Advanced age, unmarried status, black race, distant metastasis, and the absence of surgical treatment were significantly associated with decreased survival rates. RMST did not differ between the combined treatment (radiotherapy and chemotherapy) and radiotherapy monotherapy groups, but chemotherapy alone displayed poor efficacy.

CONCLUSIONS

Head and neck LEC is associated with a favorable prognosis. Radiotherapy plays a significant role in managing patients with nasopharyngeal LEC, which is influenced by multiple prognostic factors.

Treatment effect measures under nonproportional hazards

In a clinical trial with a time-to-event endpoint the treatment effect can be measured in various ways. Under proportional hazards all reasonable measures (such as the hazard ratio and the difference in restricted mean survival time) are consistent in the following sense: Take any control group survival distribution such that the hazard rate remains above zero; if there is no benefit by any measure there is no benefit by all measures, and as the magnitude of treatment benefit increases by any measure it increases by all measures. Under nonproportional hazards, however, survival curves can cross, and the direction of the effect for any pair of measures can be inconsistent. In this paper we critically evaluate a variety of treatment effect measures in common use and identify flaws with them. In particular, we demonstrate that a treatment's benefit has two distinct and independent dimensions which can be measured by the difference in the survival rate at the end of follow-up and the difference in restricted mean survival time, and that commonly used measures do not adequately capture both dimensions. We demonstrate that a generalized hazard difference, which can be estimated by the difference in exposure-adjusted subject incidence rates, captures both dimensions, and that its inverse, the number of patient-years of follow-up that results in one fewer event (the NYNT), is an easily interpretable measure of the magnitude of clinical benefit.

Utilizing restricted mean duration of response for efficacy evaluation of cancer treatments

In oncology clinical trials, response-based endpoints (time to response, objective response, duration of response [DOR]) are commonly used to detect therapeutic effect to support proof-of-concept or submission decisions. The restricted mean DOR (RMDOR) was recently proposed as a composite nonparametric method to efficiently quantify the treatment effect related to tumor reductions, which offers an intuitive way to perform statistical inference in cross-arm comparison and has since been applied in some Phase III studies. In this paper, we provide further technical details and asymptotic properties of the RMDOR method and discuss the selection of the truncation time. A simulation study is conducted comparing the performance of the proposed method with existing standard methods in hypothesis testing and quantification of treatment efficacy. We use two oncology Phase III examples to illustrate the method. An R package PBIR and a SAS macro are available to perform statistical inference based on the RMDOR.

Regression modeling of restricted mean survival time for left-truncated right-censored data

The restricted mean survival time (RMST) is a clinically meaningful summary measure in studies with survival outcomes. Statistical methods have been developed for regression analysis of RMST to investigate impacts of covariates on RMST, which is a useful alternative to the Cox regression analysis. However, existing methods for regression modeling of RMST are not applicable to left-truncated right-censored data that arise frequently in prevalent cohort studies, for which the sampling bias due to left truncation and informative censoring induced by the prevalent sampling scheme must be properly addressed. The pseudo-observation (PO) approach has been used in regression modeling of RMST for right-censored data and competing-risks data. For left-truncated right-censored data, we propose to directly model RMST as a function of baseline covariates based on POs under general censoring mechanisms. We adjust for the potential covariate-dependent censoring or dependent censoring by the inverse probability of censoring weighting method. We establish large sample properties of the proposed estimators and assess their finite sample performances by simulation studies under various scenarios. We apply the proposed methods to a prevalent cohort of women diagnosed with stage IV breast cancer identified from surveillance, epidemiology, and end results-medicare linked database.

Validating Restricted Mean Survival Time Estimates From Reconstructed Kaplan-Meier Data Against Original Trial Individual Patient Data From Trials Conducted by the Canadian Cancer Trials Group

OBJECTIVES

The development of novel cancer therapies, including immuno-oncology agents, has increased interest in reconstructed individual patient data (IPD) based restricted mean survival time (RMST) analyses. Additionally, reconstructed IPD-based RMST is recommended in cost-effectiveness analyses when original trial IPD are not available. Nevertheless, recently concerns regarding potential bias of reconstructed-IPD RMST have been presented, because reconstructed-IPD RMSTs have not been validated and previous validation endpoints may not capture the entire Kaplan-Meier (KM) curve, especially the "tail." Our study aims to validate the recommended method of IPD reconstruction by comparing reconstructed IPD- and original trial IPD-based RMST.

METHODS

Canadian Cancer Trials Group trials from 1990 to 2017 were included. Overall survival and progression-free survival IPD were reconstructed based on published KM curves using the Guyot method. Analysts were blinded to original trial IPD. RMST was calculated at 1 year and over the entire KM curve. Reconstructed-IPD and original trial-IPD (gold-standard) RMSTs were compared for accuracy and predictive error via mean deviation, mean absolute error (MAE), mean percentage bias, and Bland-Altman plots and across KM curve quality (vector traced or bitmapped).

RESULTS

We identified 39 trials. The mean deviation, MAE, and mean percentage bias of RMST between the reconstructed IPD and original trial IPD were small. In particular, the mean deviation was -0.01 months and -0.04 months, MAE was 0.19 months and 0.24 months, and mean percentage bias was 0.82% and 0.84% in overall survival KM curves in control and experimental arms, respectively. Accuracy was generally not associated with KM curve quality.

CONCLUSIONS

RMST derived from reconstructed IPD displayed excellent accuracy and predictive error compared with the gold standard. Reconstructed IPD could be used to calculate RMST in lieu of original trial IPD, to facilitate decision making for clinicians, researchers, and policy makers.

Restricted mean survival time regression model with time-dependent covariates

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.

First-line chemoimmunotherapy versus chemotherapy in PD-L1-negative patients with non-small-cell lung cancer

Background: In PD-L1-negative patients with advanced non-small-cell lung cancer (NSCLC), conclusive evidence in support of specific treatments remains lacking. Objectives: The efficacy of first-line chemoimmunotherapy versus chemotherapy alone was compared. Methods: Eligible randomized studies that included patients with advanced NSCLC irrespective of PD-L1 status who were treated with chemoimmunotherapy as the first line were identified. Kaplan-Meier curves were extracted and analyzed using restricted mean survival time (RMST). Patient-level data were reconstructed from progression-free survival (PFS) graphs. A Bayesian network meta-analysis (NETMA) was carried out. Results: In five trials selected, chemoimmunotherapy regimens, compared with chemotherapy alone, resulted in an improvement in PFS without statistical significance. In the NETMA, chemoimmunotherapy was found to slightly improve PFS. Conclusion: This analysis showed that the incremental benefit of chemoimmunotherapy versus chemotherapy is limited.

Conversion of non-inferiority margin from hazard ratio to restricted mean survival time difference using data from multiple historical trials

The restricted mean survival time measure has gained a lot of interests for designing and analyzing oncology trials with time-to-event endpoints due to its intuitive clinical interpretation and potentially high statistical power. In the non-inferiority trial literature, restricted mean survival time has been used as an alternative measure for reanalyzing a completed trial, which was originally designed and analyzed based on traditional proportional hazard model. However, the reanalysis procedure requires a conversion from the non-inferiority margin measured in hazard ratio to a non-inferiority margin measured by restricted mean survival time difference. An existing conversion method assumes a Weibull distribution for the population survival time of the historical active control group under the proportional hazard assumption using data from a single trial. In this article, we develop a methodology for non-inferiority margin conversion when data from multiple historical active control studies are available, and introduce a Kaplan-Meier estimator-based method for the non-inferiority margin conversion to relax the parametric assumption. We report extensive simulation studies to examine the performances of proposed methods under the Weibull data generative models and a piecewise-exponential data generative model that mimic the tumor recurrence and survival characteristics of advanced colon cancer. This work is motivated to achieve non-inferiority margin conversion, using historical patient-level data from a large colon cancer clinical database, to reanalyze an internationally collaborated non-inferiority study that evaluates 6-month versus 3-month duration of adjuvant chemotherapy in stage III colon cancer patients.

The Degree of Programmed Death-Ligand 1 (PD-L1) Positivity as a Determinant of Outcomes in Metastatic Triple-Negative Breast Cancer Treated With First-Line Immune Checkpoint Inhibitors

In metastatic triple-negative breast cancer (TNBC), the efficacy of immune checkpoint inhibitors (ICIs) in combination with chemotherapy has been demonstrated in randomized clinical trials (RCTs). Despite this, an indirect comparison is not yet available. Reconstruction of individual patient data from Kaplan-Meier curves allows the indirect comparison of different treatments. We analyzed six overall survival (OS) curves from three RCTs. In patients with ≥1% positivity, atezolizumab was found to determine a significantly better OS than pembrolizumab. As regards pembrolizumab, adopting a threshold of PD-L1 positivity ≥10% (as opposed to ≥1%) improved median survival to a remarkable extent (23.0 vs 15.5 months).

Waiting Time for Second Kidney Transplantation and Mortality

BACKGROUND AND OBJECTIVES

The median kidney transplant half-life is 10-15 years. Because of the scarcity of donor organs and immunologic sensitization of candidates for retransplantation, there is a need for quantitative information on if and when a second transplantation is no longer associated with a lower risk of mortality compared with waitlisted patients treated by dialysis. Therefore, we investigated the association of time on waiting list with patient survival in patients who received a second transplantation versus remaining on the waiting list.

DESIGN, SETTING, PARTICIPANTS, & MEASUREMENTS

In this retrospective study using target trial emulation, we analyzed data of 2346 patients from the Austrian Dialysis and Transplant Registry and Eurotransplant with a failed first graft, aged over 18 years, and waitlisted for a second kidney transplantation in Austria during the years 1980-2019. The differences in restricted mean survival time and hazard ratios for all-cause mortality comparing the treatment strategies "retransplant" versus "remain waitlisted with maintenance dialysis" are reported for different waiting times after first graft loss.

RESULTS

Second kidney transplantation showed a longer restricted mean survival time at 10 years of follow-up compared with remaining on the waiting list (5.8 life months gained; 95% confidence interval, 0.9 to 11.1). This survival difference was diminished in patients with longer waiting time after loss of the first allograft; restricted mean survival time differences at 10 years were 8.0 (95% confidence interval, 1.9 to 14.0) and 0.1 life months gained (95% confidence interval, -14.3 to 15.2) for patients with waiting time for retransplantation of <1 and 8 years, respectively.

CONCLUSIONS

Second kidney transplant is associated with patient survival compared with remaining waitlisted and treatment by dialysis, but the survival difference diminishes with longer waiting time.

Bayesian multivariate network meta-analysis model for the difference in restricted mean survival times

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.

G-computation and doubly robust standardisation for continuous-time data: A comparison with inverse probability weighting

In time-to-event settings, g-computation and doubly robust estimators are based on discrete-time data. However, many biological processes are evolving continuously over time. In this paper, we extend the g-computation and the doubly robust standardisation procedures to a continuous-time context. We compare their performance to the well-known inverse-probability-weighting estimator for the estimation of the hazard ratio and restricted mean survival times difference, using a simulation study. Under a correct model specification, all methods are unbiased, but g-computation and the doubly robust standardisation are more efficient than inverse-probability-weighting. We also analyse two real-world datasets to illustrate the practical implementation of these approaches. We have updated the R package RISCA to facilitate the use of these methods and their dissemination.

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.

A note on confidence intervals for the restricted mean survival time based on transformations in small sample size

Restricted mean survival time (RMST) is one measure now used to summarize time-to-event type data, but it has been pointed out that the distribution of differences in RMST deviates markedly from a normal distribution for controlled clinical trials with small sample sizes. Therefore, we conducted a numerical simulation of the RMST in which the one-sample survival time follows a Weibull distribution, comparing eight different confidence intervals combining two types of variance with four types of variable transformations, including no transformation. The evaluation items were the coverage probability and the above and below error probabilities for the true value. The variance types were based on Greenwood's formula and its Kaplan-Meier correction. The arcsine square root transformation, logit transformation, and complementary log-log transformation were used as the variable transformations. When the sample size was small and the event rate was low, the confidence interval of the untransformed RMST tended to have a small coverage probability and to be overestimated. Variance by Kaplan-Meier correction improved the coverage. The problems of coverage and overestimation were also improved by variable transformations, and in particular, applying the logit transformation and the complementary log-log transformation both resulted in substantial improvements. Our study suggested that it is preferable to construct the confidence intervals of RMST using the logit transformation for variances based on Greenwood's formula in small sample size trials. The SAS code to replicate the analyses is available at https://github.com/HiroyaHashimoto/SAS-Programs.

Adjusting for covariates in analysis based on restricted mean survival times

We summarize extensions to the analysis of restricted mean survival time (RMST) in the context of time-to-event outcomes. The RMST estimate and its inference are based on the classical Kaplan-Meier curves. When covariate effects are considered, an adjusted RMST (ARMST) estimate can be derived analogously based on adjusted Kaplan-Meier curves. The adjusted Kaplan-Meier Estimator (AKME) was developed to reduce confounding by the method of inverse probability of treatment weighting. We will show how the ARMST method combines the concepts of the RMST and AKME to make inferences. Two regression based methods to adjust for potential covariate effect on the RMST estimates will be compared with the ARMST approach. Simulation studies are performed to compare the different methods with and without covariate adjustments. In addition, we will summarize the extension of RMST and ARMST to the setting with competing risks. The restricted mean time lost (RMTL) and adjusted RMTL (ARMTL) are estimates of interest from cumulative incidence curves. A phase 3 oncology clinical trial example is provided to demonstrate the applications of these methods.

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.

Restricted mean survival time versus conventional measures for treatment decision-making

BACKGROUND/OBJECTIVES

Restricted mean survival time (RMST) summarizes treatment effect in terms of a gain or loss in the event-free days. It remains uncertain whether communicating treatment benefit and harm using RMST-based summary is more effective than conventional summary based on absolute and relative risk reduction. We compared the effect of RMST-based approach and conventional approach on decisional conflict using an example of intensive versus standard blood pressure-lowering strategies.

DESIGN

On-line survey.

SETTING

A convenience sample of patients in the United States.

PARTICIPANTS

Two hundred adults aged 65 and older with hypertension requiring anti-hypertensive treatment (response rate 85.5%).

INTERVENTIONS

Participants were randomly assigned to either RMST-based summary or conventional summary about the benefit and harm of blood pressure-lowering strategies.

MEASUREMENTS

Decisional Conflict Scale (DCS), ranging from 0 (no conflict) to 100 (high conflict), and preference for intensive blood pressure-lowering strategy.

RESULTS

Participants assigned to RMST-based approach (n = 100) and conventional approach (n = 100) had similar age (mean [standard deviation, SD]: 72.3 [5.6] vs 72.8 [5.5] years) and proportions of female (50 [50.0%] vs 61 [61.0%]) and white race (92 [92.0%] vs 92 [92.0%]). The mean (SD) DCS score was 25.2 (15.0) for RMST-based approach and 25.6 (14.1) for conventional approach (p = 0.84). The number (%) of participants who preferred intensive strategy was 10 (10.0%) for RMST-based approach and 14 (14.0%) for conventional approach (p = 0.52). The results were consistent in subgroups defined by age, sex, education level, cardiovascular disease status, and predicted mortality risk categories.

CONCLUSION

In a sample of relatively healthy older adults with hypertension, RMST-based approach was as effective as conventional approach on decisional conflict about choosing a blood pressure-lowering strategy. This study provides proof-of-concept evidence that RMST-based approach can be used in conjunction with absolute and relative risk reduction for communicating treatment benefit and harm in a decision aid.

Weighted estimators of the complier average causal effect on restricted mean survival time with observed instrument-outcome confounders

A major concern in any observational study is unmeasured confounding of the relationship between a treatment and outcome of interest. Instrumental variable (IV) analysis methods are able to control for unmeasured confounding. However, IV analysis methods developed for censored time-to-event data tend to rely on assumptions that may not be reasonable in many practical applications, making them unsuitable for use in observational studies. In this report, we develop weighted estimators of the complier average causal effect (CACE) on the restricted mean survival time in the overall population as well as in an evenly matchable population (CACE-m). Our method is able to accommodate instrument-outcome confounding and adjust for covariate-dependent censoring, making it particularly suited for causal inference from observational studies. We establish the asymptotic properties and derive easily implementable asymptotic variance estimators for the proposed estimators. Through simulation studies, we show that the proposed estimators tend to be more efficient than instrument propensity score matching-based estimators or IPIW estimators. We apply our method to compare dialytic modality-specific survival for end stage renal disease patients using data from the U.S. Renal Data System.

Number needed to treat analysis applied to pembrolizumab plus chemotherapy for first-line treatment of non-squamous non-small cell lung cancer

AIMS

Considering that healthcare systems' financial resources are limited, we aimed to analyze the number needed to treat (NNT) and cost of preventing an event (COPE) related to drug use from Supplementary Health System (SSS) perspective.

METHODS

Data from KEYNOTE-189 (NCT02578680) were considered, comparing pembrolizumab + chemotherapy to chemotherapy alone. A cost-per-responder model was developed considering the 24- and 12-month time horizons for overall survival (OS) and progression-free survival (PFS) endpoints, respectively. Restricted mean survival time (RMST) and restricted mean time-on-treatment (ToT) were determined for NNT and COPE calculation. Costs were reported in American dollars (USD) and reflect those related to drug use. The analysis was conducted for the total indicated population, and an exploratory assessment was carried out for subgroups with different programmed death-ligand 1 (PD-L1) expression levels.

RESULTS

Considering PFS data, the overall population NNT_RMST to prevent a progression event with pembrolizumab + chemotherapy versus chemotherapy was 2.63 (95%CI: 1.90-4.02) with an estimated COPE of 251,038 USD (95%CI: 181,359-383,717) in the 12-months follow-up. Regarding OS endpoint, overall NNT_RMST and COPE were 3.18 (95%CI: 2.20-5.31) and 414,163 (95%CI: 286,528-691,573) USD respectively, in the 24 months follow-up. The PFS NNT was lower with higher levels of PD-L1 expression (1.71, 3.22 and 5.53 for PD-L1 ≥ 50%, PD-L1 1%-49%, and PD-L1 < 1% groups, respectively), while there was no such apparent relationship for OS (3.23, 4.37 and 2.80 for PD-L1 ≥ 50%, PD-L1 1%-49%, and PD-L1 < 1% groups, respectively). The 95%CIs overlapped for PFS and OS NNT across the PD-L1 subgroups.

CONCLUSION

The magnitude of benefit of the pembrolizumab combination used for first-line non-small cell lung cancer (NSCLC) treatment to improve survival compared to chemotherapy alone was confirmed. The exploratory analysis from the SSS perspective suggests no differences among the PDL-1 subgroups in terms of clinical benefit or economic impact.

Extrapolation of Survival Curves Using Standard Parametric Models and Flexible Parametric Spline Models: Comparisons in Large Registry Cohorts with Advanced Cancer

BACKGROUND

It is often important to extrapolate survival estimates beyond the limited follow-up times of clinical trials. Extrapolated survival estimates can be highly sensitive to model choice; thus, appropriate model selection is crucial. Flexible parametric spline models have been suggested as an alternative to standard parametric models; however, their ability to extrapolate is not well understood.

AIM

To determine how well standard parametric and flexible parametric spline models predict survival when fitted to registry cohorts with artificially right-censored follow-up times.

METHODS

Adults with advanced breast, colorectal, small cell lung, non-small cell lung, or pancreatic cancer with a potential follow-up time of 10 y were selected from the SEER 1973-2015 registry data set. Patients were classified into 15 cohorts by cancer and age group at diagnosis (18-59, 60-69, 70+ y). Follow-up times for each cohort were right censored at 20%, 35%, and 50% survival. Standard parametric models (exponential, Weibull, Gompertz, log-logistic, log-normal, generalized gamma) and spline models (proportional hazards, proportional odds, normal/probit) were fitted to the 10-y data set and the 3 right-censored data sets. Predicted 10-y restricted mean survival time and percentage surviving at 10 y were compared with the observed values.

RESULTS

Across all data sets, the spline odds and spline normal models most frequently gave accurate predictions of 10-y survival outcomes. Visually, spline models tended to demonstrate better fit to the observed hazard functions than standard parametric models, both in the censored and 10-y data.

CONCLUSIONS

In these cohorts, where there was little uncertainty in the observed data, the spline models performed well when extrapolating beyond the observed data. Spline models should be routinely included in the set of models that are fitted when extrapolating cancer survival data.

A nonparametric method for value function guided subgroup identification via gradient tree boosting for censored survival data

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.

Restricted mean survival time as a function of restriction time

Restricted mean survival time (RMST) is a clinically interpretable and meaningful survival metric that has gained popularity in recent years. Several methods are available for regression modeling of RMST, most based on pseudo-observations or what is essentially an inverse-weighted complete-case analysis. No existing RMST regression method allows for the covariate effects to be expressed as functions over time. This is a considerable limitation, in light of the many hazard regression methods that do accommodate such effects. To address this void in the literature, we propose RMST methods that permit estimating time-varying effects. In particular, we propose an inference framework for directly modeling RMST as a continuous function of L. Large-sample properties are derived. Simulation studies are performed to evaluate the performance of the methods in finite sample sizes. The proposed framework is applied to kidney transplant data obtained from the Scientific Registry of Transplant Recipients.

Estimating the Lifetime Benefits of Treatments for Heart Failure

OBJECTIVES

This study compared ways of describing treatment effects. The objective was to better explain to clinicians and patients what they might expect from a given treatment, not only in terms of relative and absolute risk reduction, but also in projections of long-term survival.

BACKGROUND

The restricted mean survival time (RMST) can be used to estimate of long-term survival, providing a complementary approach to more conventional metrics (e.g., absolute and relative risk), which may suggest greater benefits of therapy in high-risk patients compared with low-risk patients.

METHODS

Relative and absolute risk, as well as the RMST, were calculated in heart failure with reduced ejection fraction (HFrEF) trials.

RESULTS

As examples, in the RALES trial (more severe HFrEF), the treatment effect metrics for spironolactone versus placebo on heart failure hospitalization and/or cardiovascular death were a hazard ratio (HR) of 0.67 (95% confidence interval [CI]: 0.5 to 0.77), number needed to treat = 9 (7 to 14), and age extension of event-free survival +1.1 years (-0.1 to + 2.3 years). The corresponding metrics for EMPHASIS-HF (eplerenone vs. placebo in less severe HFrEF) were 0.64 (0.54 to 0.75), 14 (1 to 22), and +2.9 (1.2 to 4.5). In patients in PARADIGM-HF aged younger than 65 years, the metrics for sacubitril/valsartan versus enalapril were 0.77 (95% CI: 0.68 to 0.88), 23 (15 to 44), and +1.7 (0.6 to 2.8) years; for those aged 65 years or older, the metrics were 0.83 (95% CI: 0.73 to 0.94), 29 (17 to 83), and +0.9 (0.2 to 1.6) years, which provided evidence of a greater potential life extension in younger patients. Similar observations were found for lower risk patients.

CONCLUSIONS

RMST event-free (and overall) survival estimates provided a complementary means of evaluating the effect of therapy in relation to age and risk. They also provided a clinically useful metric that should be routinely reported and used to explain the potential long-term benefits of a given treatment, especially to younger and less symptomatic patients.

Restricted mean survival time for interval-censored data

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.

Patients With Lung Cancer Have High Susceptibility of COVID-19: A Retrospective Study in Wuhan, China

Patients with lung cancer are presumed to be at high risk from COVID-19 infection due to underlying malignancy. A total of 31 COVID-19 patients with pre-diagnosed lung cancer and 186 age and sex matched COVID-19 patients without cancer in 6 hospitals in Wuhan, China were identified in our study. There was a significantly higher level of IL-6 in lung cancer group showed by multifactorial analysis. The restricted mean survival time in 10, 20, and 53 days in COVID-19 patients with lung cancer were ealier than non-cancer COVID-19 patients in the same observation time (all P values < 0.05). Our results indicated that pre-diagnosed lung cancer was associated with higher morbidity and mortality in COVID-19 patients.

Empirical power comparison of statistical tests in contemporary phase III randomized controlled trials with time-to-event outcomes in oncology

BACKGROUND

More than 95% of recent cancer randomized controlled trials used the log-rank test to detect a treatment difference making it the predominant tool for comparing two survival functions. As with other tests, the log-rank test has both advantages and disadvantages. One advantage is that it offers the highest power against proportional hazards differences, which may be a major reason why alternative methods have rarely been employed in practice. The performance of statistical tests has traditionally been investigated both theoretically and numerically for several patterns of difference between two survival functions. However, to the best of our knowledge, there has been no attempt to compare the performance of various statistical tests using empirical data from past oncology randomized controlled trials. So, it is unknown whether the log-rank test offers a meaningful power advantage over alternative testing methods in contemporary cancer randomized controlled trials. Focusing on recently reported phase III cancer randomized controlled trials, we assessed whether the log-rank test gave meaningfully greater power when compared with five alternative testing methods: generalized Wilcoxon, test based on maximum of test statistics from multiple weighted log-rank tests, difference in t-year event rate, and difference in restricted mean survival time with fixed and adaptive τ.

METHODS

Using manuscripts from cancer randomized controlled trials recently published in high-tier clinical journals, we reconstructed patient-level data for overall survival (69 trials) and progression-free survival (54 trials). For each trial endpoint, we estimated the empirical power of each test. Empirical power was measured as the proportion of trials for which a test would have identified a significant result (p value < .05).

RESULTS

For overall survival, t-year event rate offered the lowest (30.4%) empirical power and restricted mean survival time with fixed τ offered the highest (43.5%). The empirical power of the other types of tests was almost identical (36.2%-37.7%). For progression-free survival, the tests we investigated offered numerically equivalent empirical power (55.6%-61.1%). No single test consistently outperformed any other test.

CONCLUSION

The empirical power assessment with the past cancer randomized controlled trials provided new insights on the performance of statistical tests. Although the log-rank test has been used in almost all trials, our study suggests that the log-rank test is not the only option from an empirical power perspective. Near universal use of the log-rank test is not supported by a meaningful difference in empirical power. Clinical trial investigators could consider alternative methods, beyond the log-rank test, for their primary analysis when designing a cancer randomized controlled trial. Factors other than power (e.g. interpretability of the estimated treatment effect) should garner greater consideration when selecting statistical tests for cancer randomized controlled trials.

Estimation on conditional restricted mean survival time with counting process

In a comparative longitudinal clinical study, multiple clinical events of interest are typically collected in timing and occurrence during the follow-up period. These clinical events are often indicative of disease burden over the study period and provide overall evidence of benefit/risk of one treatment relative to another. While these clinical events are usually used to form a composite endpoint, only the first occurrence of the composite endpoint event is considered in primary efficacy analysis. This type of analysis is commonly performed but it may not be ideal. Most of the existing methods for analyzing multiple event-time data were developed, relying on certain model assumptions. However, the assumptions may greatly affect the inferences for treatment effect. In this paper, we propose a simple, non-parametric estimator of conditional mean survival time for multiple events to quantify treatment effect which has clinically meaningful interpretation. We use simulation studies to evaluate the performance of the new method. Further, we apply this method to analyze the data from a cardiovascular clinical trial as an illustration.

Quantifying treatment differences in confirmatory trials under non-proportional hazards

Proportional hazards are a common assumption when designing confirmatory clinical trials in oncology. With the emergence of immunotherapy and novel targeted therapies, departure from the proportional hazard assumption is not rare in nowadays clinical research. Under non-proportional hazards, the hazard ratio does not have a straightforward clinical interpretation, and the log-rank test is no longer the most powerful statistical test even though it is still valid. Nevertheless, the log-rank test and the hazard ratio are still the primary analysis tools, and traditional approaches such as sample size increase are still proposed to account for the impact of non-proportional hazards. The weighed log-rank test and the test based on the restricted mean survival time (RMST) are receiving a lot of attention as a potential alternative to the log-rank test. We conduct a simulation study comparing the performance and operating characteristics of the log-rank test, the weighted log-rank test and the test based on the RMST, including a treatment effect estimation, under different non-proportional hazards patterns. Results show that, under non-proportional hazards, the hazard ratio and weighted hazard ratio have no straightforward clinical interpretation whereas the RMST ratio can be interpreted regardless of the proportional hazards assumption. In terms of power, the RMST achieves a similar performance when compared to the log-rank test.

Outcomes at 5 years in patients with severe aortic stenosis: reviewing current information using the restricted mean survival time

The information about outcomes at 5 years in patients receiving transcatheter aortic valve replacement (TAVR) has grown. We interpreted the information on this topic using the restricted mean survival time (RMST). The purpose of our study was to summarise the current evidence using an original outcome measure with potential methodological advantages. Four cohorts of patients, previously published in the literature, met our criterion of 5 years of follow-up after the implant; another cohort was identified from a group of controls subjected to surgical replacement of the valve. The estimated values of RMST at 5 years for the 5 patient cohorts were the following (N = number of patients, all time values in years): a) real-world high surgical risk cohort: N = 114, RMST = 3.80; b) real-world cohort treated with Corevalve: N = 309, RMST = 3.79; c) a real-world cohort treated with Sapien: N = 180, RMST = 3.61; d) TAVR arm of a randomized trial in intermediate risk patients: N = 1,011; RMST = 3.73; e) surgical replacement arm of the same trial: N = 1,021, RMST = 3.72. The main result of our analysis based on the RMST is represented by the extreme homogeneity of the outcomes (RMSTs ranging from 3.61 to 3.80 years per patient) that remained virtually constant irrespective of the baseline risk of the patients (intermediate or high risk) and regardless of whether the intervention was transcatheter or by surgical replacement. Last but not least, our analysis showed the good methodological performance of the RMST in this disease condition.

Moving beyond the Cox proportional hazards model in survival data analysis: a cervical cancer study

OBJECTIVES

This study explored the prognostic factors and developed a prediction model for Chinese-American (CA) cervical cancer (CC) patients. We compared two alternative models (the restricted mean survival time (RMST) model and the proportional baselines landmark supermodel (PBLS model, producing dynamic prediction)) versus the Cox proportional hazards model in the context of time-varying effects.

SETTING AND DATA SOURCES

A total of 713 CA women with CC and available covariates (age at diagnosis, International Federation of Gynecology and Obstetrics (FIGO) stage, lymph node metastasis and radiation) from the Surveillance, Epidemiology and End Results database were included.

DESIGN

We applied the Cox proportional hazards model to analyse the all-cause mortality with the proportional hazards assumption. Additionally, we applied two alternative models to analyse covariates with time-varying effects. The performances of the models were compared using the C-index for discrimination and the shrinkage slope for calibration.

RESULTS

Older patients had a worse survival rate than younger patients. Advanced FIGO stage patients showed a relatively poor survival rate and low life expectancy. Lymph node metastasis was an unfavourable prognostic factor in our models. Age at diagnosis, FIGO stage and lymph node metastasis represented time-varying effects from the PBLS model. Additionally, radiation showed no impact on survival in any model. Dynamic prediction presented a better performance for 5-year dynamic death rates than did the Cox proportional hazards model.

CONCLUSIONS

With the time-varying effects, the RMST model was suggested to explore diagnosis factors, and the PBLS model was recommended to predict a patient's w-year dynamic death rate.

Statistical Issues and Lessons Learned From COVID-19 Clinical Trials With Lopinavir-Ritonavir and Remdesivir

Background

Recently, three randomized clinical trials on coronavirus disease (COVID-19) treatments were completed: one for lopinavir-ritonavir and two for remdesivir. One trial reported that remdesivir was superior to placebo in shortening the time to recovery, while the other two showed no benefit of the treatment under investigation.

Objective

The aim of this paper is to, from a statistical perspective, identify several key issues in the design and analysis of three COVID-19 trials and reanalyze the data from the cumulative incidence curves in the three trials using more appropriate statistical methods.

Methods

The lopinavir-ritonavir trial enrolled 39 additional patients due to insignificant results after the sample size reached the planned number, which led to inflation of the type I error rate. The remdesivir trial of Wang et al failed to reach the planned sample size due to a lack of eligible patients, and the bootstrap method was used to predict the quantity of clinical interest conditionally and unconditionally if the trial had continued to reach the originally planned sample size. Moreover, we used a terminal (or cure) rate model and a model-free metric known as the restricted mean survival time or the restricted mean time to improvement (RMTI) to analyze the reconstructed data. The remdesivir trial of Beigel et al reported the median recovery time of the remdesivir and placebo groups, and the rate ratio for recovery, while both quantities depend on a particular time point representing local information. We use the restricted mean time to recovery (RMTR) as a global and robust measure for efficacy.

Results

For the lopinavir-ritonavir trial, with the increase of sample size from 160 to 199, the type I error rate was inflated from 0.05 to 0.071. The difference of RMTIs between the two groups evaluated at day 28 was –1.67 days (95% CI –3.62 to 0.28; P=.09) in favor of lopinavir-ritonavir but not statistically significant. For the remdesivir trial of Wang et al, the difference of RMTIs at day 28 was –0.89 days (95% CI –2.84 to 1.06; P=.37). The planned sample size was 453, yet only 236 patients were enrolled. The conditional prediction shows that the hazard ratio estimates would reach statistical significance if the target sample size had been maintained. For the remdesivir trial of Beigel et al, the difference of RMTRs between the remdesivir and placebo groups at day 30 was –2.7 days (95% CI –4.0 to –1.2; P<.001), confirming the superiority of remdesivir. The difference in the recovery time at the 25th percentile (95% CI –3 to 0; P=.65) was insignificant, while the differences became more statistically significant at larger percentiles.

Conclusions

Based on the statistical issues and lessons learned from the recent three clinical trials on COVID-19 treatments, we suggest more appropriate approaches for the design and analysis of ongoing and future COVID-19 trials.