Designing superiority trials with window mean survival time as a primary endpoint.

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

Sample size calculation for the combination test under nonproportional hazards.

For sample size calculation in clinical trials with survival endpoints, the logrank test, which is the optimal method under the proportional hazard (PH) assumption, is predominantly used. In reality, the PH assumption may not hold. For example, in immuno-oncology trials, delayed treatment effects are often expected. The sample size without considering the potential violation of the PH assumption may lead to an underpowered study. In recent years, combination tests such as the maximum weighted logrank test have received great attention because of their robust performance in various hazards scenarios. In this paper, we propose a flexible simulation-free procedure to calculate the sample size using combination tests. The procedure extends the Lakatos' Markov model and allows for complex situations encountered in a clinical trial, like staggered entry, dropouts, etc. We evaluate the procedure using two maximum weighted logrank tests, one projection-type test, and three other commonly used tests under various hazards scenarios. The simulation studies show that the proposed method can achieve the target power for all compared tests in most scenarios. The combination tests exhibit robust performance under correct specification and misspecification scenarios and are highly recommended when the hazard-changing patterns are unknown beforehand. Finally, we demonstrate our method using two clinical trial examples and provide suggestions about the sample size calculations under nonproportional hazards.

A survival tree based on stabilized score tests for high-dimensional covariates.

A survival tree can classify subjects into different survival prognostic groups. However, when data contains high-dimensional covariates, the two popular classification trees exhibit fatal drawbacks. The logrank tree is unstable and tends to have false nodes; the conditional inference tree is difficult to interpret the adjusted P-value for high-dimensional tests. Motivated by these problems, we propose a new survival tree based on the stabilized score tests. We propose a novel matrix-based algorithm in order to tests a number of nodes simultaneously via stabilized score tests. We propose a recursive partitioning algorithm to construct a survival tree and develop our original R package uni.survival.tree (https://cran.r-project.org/package=uni.survival.tree) for implementation. Simulations are performed to demonstrate the superiority of the proposed method over the existing methods. The lung cancer data analysis demonstrates the usefulness of the proposed method.

On assessing survival benefit of immunotherapy using long-term restricted mean survival time.

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

Versatile tests for window mean survival time.

Window mean survival time (WMST) evaluates the mean survival between a lower time horizon, τ 0 $$ {\tau}_0 $$ , and an upper time horizon, τ 1 $$ {\tau}_1 $$ . As a flexible extension of restricted mean survival time, specific clinically relevant windows of time can be assessed for survival difference accompanied by a communicable interpretation of estimates and tests. In its original application, WMST required the pre-specification of a window through the selection of appropriate window bounds, τ 0 $$ {\tau}_0 $$ and τ 1 $$ {\tau}_1 $$ . In the instance of severe window misspecification of τ 0 $$ {\tau}_0 $$ and τ 1 $$ {\tau}_1 $$ , the analysis may suffer from low power and a less meaningful interpretation. In this article, we introduce versatile tests whose procedures are based on the simultaneous use of multiple WMST test statistics that are asymptotically normal under the null hypothesis of no difference between two groups. Simulations are performed to examine the power of the tests in moderate sample sizes when the data are uncensored to heavily censored with a ramp-up enrollment period. The survival scenarios chosen for simulation are intended to imitate those which are commonly encountered in oncology, especially in trials involving immunotherapies. Implementation of the procedures is discussed in two real data examples for illustration. Functions for performing versatile WMST tests are provided in the survWMST package in R.

Survival analysis of nonperiodic stimulation (NPS) performance.

Electrical Stimulation (ES) of the nervous system is a promising alternative to treat refractory epilepsy. Recent developments in the area have led to a novel method involving a non-standard form of electrical stimulation with randomized inter-pulse intervals called non-periodic stimulation (NPS). Although it is an interesting approach, there is limited statistical proof to confirm its effectiveness. Therefore this brief communication presents a survival analysis of a pre-clinical trial to assess the significance of NPS therapy. The experiment comprised four groups of rats that have been compared: two with and two without NPS treatment. ES was applied bilaterally to the amygdala in animals subjected to the pentylenetetrazole continuous infusion (10 mg/ml/min) model, myoclonic or tonic-clonic generalized seizures were triggered. The Kaplan-Meier estimator was used to develop survival functions and the Logrank test was carried out to check the differences among groups. The first comparison was made between two groups of rats that developed generalized tonic-clonic seizures (GTC groups), those who received NPS treatment took longer to develop epileptic seizures. The logrank test proved statistical difference due to reaching a p-value of 7%. The second comparison was performed between two groups of rats that developed myoclonic seizures (MYO groups), and once again better survival probabilities were observed for the NPS group. The Logrank test revealed a p-value of 0.5% thereof. Thus, a survival analysis of NPS treatment proved effectiveness against seizures by promoting an anticonvulsant effect. By comparing the groups selected for this study, it was found that the NPS treatment yielded better results, mainly against myoclonic seizures.

Impact of unequal censoring and insufficient follow-up on comparing survival outcomes: Applications to clinical studies.

Clinical trials with survival endpoints are typically designed to enroll patients for a specified number of years, (usually 2-3 years) with another specified duration of follow-up (usually 2-3 years). Under this scheme, patients who are alive or free of the event of interest at the termination of the study are censored. Consequently, a patient may be censored due to insufficient follow-up duration or due to being lost to follow-up. Potentially, this process could lead to unequal censoring in the treatment arms and lead to inaccurate and adverse conclusions about treatment effects. In this article, using extensive simulation studies, we assess the impact of such censorings on statistical procedures (the generalized logrank tests) for comparing two survival distributions and illustrate our observations by revisiting Mukherjee et al.'s1 findings of cardiovascular events in patients who took Rofecoxib (Vioxx).

Window mean survival time.

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

Sample size calculation for logrank test and prediction of number of events over time.

We review and compare existing methods for sample size calculation based on the logrank statistic and recommend the method of Lakatos for its accuracy and flexibility in allowing time-dependent rates of event, loss to follow-up, and noncompliance. We extend the Lakatos method to allow a general follow-up scheme, to handle non-inferiority tests, and to predict the number of events over calendar time. We apply the Lakatos method to the simple nonproportional hazard situation of delayed treatment effect to facilitate the comparison of different weighting methods and to evaluate the performance of the maximum combination tests. We use simulation studies to confirm the validity of the Lakatos method and its extensions.

On selecting the critical boundary functions in group-sequential trials with two time-to-event outcomes.

We investigate selection of critical boundary functions for testing the hypotheses of two time-to-event outcomes as both primary endpoints or a primary and a secondary endpoint in group-sequential clinical trials, where (1) the effect sizes of endpoints are unequal, or (2) one endpoint is for short-term evaluation and the other for long-term evaluation. Bonferroni-Holm and fixed-sequence procedures are considered. We assess the effects of the magnitudes of the hazard ratios and the correlation between the endpoints on statistical powers and provide guidance for consideration.

Statistical adaptation to oncology drug development evolution.

Cancer treatment started with surgery at least three thousand years ago. Radiation therapy was added in 1896 with chemotherapy started 50 years later. These "cut, burn, and poison" techniques try to kill cancer cells directly and have been the main approaches in treating cancer until recently. In the past few years, immunotherapies have revolutionized cancer treatment. Instead of treating the disease, immunotherapies treat the patient with the disease; more precisely, correct the patient's immune system so that it can fight cancer in a long term, which makes the cure of metastatic cancers a real possibility. To adapt to the evolution of oncology treatment, clinical trial designs and statistical analysis methodologies are required to change accordingly in order to efficiently bring novel oncology medicines to cancer patients. For example, one of the major differences between immunotherapies and chemotherapies is that immunotherapies may take longer to have an effect but generally last longer with some patients cured. Trial design assumptions and adaptation rules (if adaptive design is used) need to take account of this delayed effect and long-term cure effect phenomenon. At the same time, more efficient statistical tests such as Fleming-Harrington test and Zmax test can be used to improve statistical power over the conventional logrank test for the analyses of time-to-event data that often exhibit non-proportional hazards. This article intends to describe how oncology drug development evolves over time and how statistical methods change accordingly.

Comparison of survival distributions in clinical trials: A practical guidance.

BACKGROUND: In randomized clinical trials with censored time-to-event outcomes, the logrank test is known to have substantial statistical power under the proportional hazards assumption and is widely adopted as a tool to compare two survival distributions. However, the proportional hazards assumption is impossible to validate in practice until the data are unblinded. However, the statistical analysis plan of a randomized clinical trial and in particular its primary analysis method must be pre-specified before any unblinded information may be reviewed. PURPOSE: The purpose of this article is to guide applied biostatisticians in the prespecification of a desired primary analysis method when a treatment effect with nonproportional hazards is anticipated. While articles proposing alternate statistical tests are aplenty, to the best of our knowledge, there is no article available that attempts to simplify the choice and prespecification of a primary statistical test under specific expected patterns on nonproportional hazards. We provide such guidance by reviewing various tests proposed as more powerful alternatives to the standard logrank test under nonproportional hazards and simultaneously comparing their performance under a wide variety of nonproportional hazards scenarios to elucidate their advantages and disadvantages. METHOD: In order to select the most preferable test for detecting specific differences between survival distributions of interest while controlling false positive rates, we review and assess the performance of weighted and adaptively weighted logrank tests, weighted and adaptively weighted Kaplan-Meier tests and versatile tests under various patterns of nonproportional hazards treatment effects through simulation. CONCLUSION: We validate some of the claimed properties of the proposed extensions and identify tests that may be more preferable under specific expected pattern of nonproportional hazards when such knowledge is available. We show that versatile tests, while achieving robustness to departures from proportional hazards, may lose interpretation of directionality (superiority or inferiority) and can only be seen to test departures from equality. Detailed summary and discussion of the performance of each test in terms of type I error rate and power are provided to formulate specific guidance about their applicability and use.

A simulation study comparing the power of nine tests of the treatment effect in randomized controlled trials with a time-to-event outcome.

BACKGROUND: The logrank test is routinely applied to design and analyse randomized controlled trials (RCTs) with time-to-event outcomes. Sample size and power calculations assume the treatment effect follows proportional hazards (PH). If the PH assumption is false, power is reduced and interpretation of the hazard ratio (HR) as the estimated treatment effect is compromised. Using statistical simulation, we investigated the type 1 error and power of the logrank (LR)test and eight alternatives. We aimed to identify test(s) that improve power with three types of non-proportional hazards (non-PH): early, late or near-PH treatment effects. METHODS: We investigated weighted logrank tests (early, LRE; late, LRL), the supremum logrank test (SupLR) and composite tests (joint, J; combined, C; weighted combined, WC; versatile and modified versatile weighted logrank, VWLR, VWLR2) with two or more components. Weighted logrank tests are intended to be sensitive to particular non-PH patterns. Composite tests attempt to improve power across a wider range of non-PH patterns. Using extensive simulations based on real trials, we studied test size and power under PH and under simple departures from PH comprising pointwise constant HRs with a single change point at various follow-up times. We systematically investigated the influence of high or low control-arm event rates on power. RESULTS: With no preconceived type of treatment effect, the preferred test is VWLR2. Expecting an early effect, tests with acceptable power are SupLR, C, VWLR2, J, LRE and WC. Expecting a late effect, acceptable tests are LRL, VWLR, VWLR2, WC and J. Under near-PH, acceptable tests are LR, LRE, VWLR, C, VWLR2 and SupLR. Type 1 error was well controlled for all tests, showing only minor deviations from the nominal 5%. The location of the HR change point relative to the cumulative proportion of control-arm events considerably affected power. CONCLUSIONS: Assuming ignorance of the likely treatment effect, the best choice is VWLR2. Several non-standard tests performed well when the correct type of treatment effect was assumed. A low control-arm event rate reduced the power of weighted logrank tests targeting early effects. Test size was generally well controlled. Further investigation of test characteristics with different types of non-proportional hazards of the treatment effect is warranted.

On the empirical choice of the time window for restricted mean survival time.

The t-year mean survival or restricted mean survival time (RMST) has been used as an appealing summary of the survival distribution within a time window [0, t]. RMST is the patient's life expectancy until time t and can be estimated nonparametrically by the area under the Kaplan-Meier curve up to t. In a comparative study, the difference or ratio of two RMSTs has been utilized to quantify the between-group-difference as a clinically interpretable alternative summary to the hazard ratio. The choice of the time window [0, t] may be prespecified at the design stage of the study based on clinical considerations. On the other hand, after the survival data have been collected, the choice of time point t could be data-dependent. The standard inferential procedures for the corresponding RMST, which is also data-dependent, ignore this subtle yet important issue. In this paper, we clarify how to make inference about a random "parameter." Moreover, we demonstrate that under a rather mild condition on the censoring distribution, one can make inference about the RMST up to t, where t is less than or even equal to the largest follow-up time (either observed or censored) in the study. This finding reduces the subjectivity of the choice of t empirically. The proposal is illustrated with the survival data from a primary biliary cirrhosis study, and its finite sample properties are investigated via an extensive simulation study.

Using Cox regression to develop linear rank tests with zero-inflated clustered data.

Zero-inflated data arise in many fields of study. When comparing zero-inflated data between two groups with independent subjects, a two degree-of-freedom test has been developed, which is the sum of a 1 degree-of-freedom Pearson chi-square test for the 2×2 table of group vs dichotomized outcome (0,> 0) and a 1 degree-of-freedom Wilcoxon rank-sum test for the values of the outcome > 0. Here, we extend this 2 degree-of-freedom test to clustered data settings. We first propose using an estimating equations score statistic from a time-varying weighted Cox regression model under naive independence, with a robust sandwich variance estimator to account for clustering. Since our proposed test statistics can be put in the framework of a Cox model, to gain efficiency over naive independence, we apply a generalized estimating equations (GEE) Cox model with a non-independence 'working correlation' between observations in a cluster. The proposed methods are applied to a General Social Survey study of days with mental health problems in a month, in which 52.3% of subjects report they have no days with problems, a zero-inflated outcome. A simulation study is used to compare our proposed test statistics to previously proposed zero-inflated test statistics.

Bootstrap and permutation rank tests for proportional hazards under right censoring.

We address the testing problem of proportional hazards in the two-sample survival setting allowing right censoring, i.e., we check whether the famous Cox model is underlying. Although there are many test proposals for this problem, only a few papers suggest how to improve the performance for small sample sizes. In this paper, we do exactly this by carrying out our test as a permutation as well as a wild bootstrap test. The asymptotic properties of our test, namely asymptotic exactness under the null and consistency, can be transferred to both resampling versions. Various simulations for small sample sizes reveal an actual improvement of the empirical size and a reasonable power performance when using the resampling versions. Moreover, the resampling tests perform better than the existing tests of Gill and Schumacher and Grambsch and Therneau . The tests' practical applicability is illustrated by discussing real data examples.

Planning and analyzing clinical trials with competing risks: Recommendations for choosing appropriate statistical methodology.

In the analysis of time-to-event data, competing risks occur when multiple event types are possible, and the occurrence of a competing event precludes the occurrence of the event of interest. In this situation, statistical methods that ignore competing risks can result in biased inference regarding the event of interest. We review the mechanisms that lead to bias and describe several statistical methods that have been proposed to avoid bias by formally accounting for competing risks in the analyses of the event of interest. Through simulation, we illustrate that Gray's test should be used in lieu of the logrank test for nonparametric hypothesis testing. We also compare the two most popular models for semiparametric modelling: the cause-specific hazards (CSH) model and Fine-Gray (F-G) model. We explain how to interpret estimates obtained from each model and identify conditions under which the estimates of the hazard ratio and subhazard ratio differ numerically. Finally, we evaluate several model diagnostic methods with respect to their sensitivity to detect lack of fit when the CSH model holds, but the F-G model is misspecified and vice versa. Our results illustrate that adequacy of model fit can strongly impact the validity of statistical inference. We recommend analysts incorporate a model diagnostic procedure and contingency to explore other appropriate models when designing trials in which competing risks are anticipated.

Modestly weighted logrank tests.

We propose a new class of weighted logrank tests (WLRTs) that control the risk of concluding that a new drug is more efficacious than standard of care, when, in fact, it is uniformly inferior. Perhaps surprisingly, this risk is not controlled for WLRT in general. Tests from this new class can be constructed to have high power under a delayed-onset treatment effect scenario, as well as being almost as efficient as the standard logrank test under proportional hazards.

Combined test versus logrank/Cox test in 50 randomised trials.

BACKGROUND: The logrank test and the Cox proportional hazards model are routinely applied in the design and analysis of randomised controlled trials (RCTs) with time-to-event outcomes. Usually, sample size and power calculations assume proportional hazards (PH) of the treatment effect, i.e. the hazard ratio is constant over the entire follow-up period. If the PH assumption fails, the power of the logrank/Cox test may be reduced, sometimes severely. It is, therefore, important to understand how serious this can become in real trials, and for a proven, alternative test to be available to increase the robustness of the primary test. METHODS: We performed a systematic search to identify relevant articles in four leading medical journals that publish results of phase 3 clinical trials. Altogether, 50 articles satisfied our inclusion criteria. We digitised published Kaplan-Meier curves and created approximations to the original times to event or censoring at the individual patient level. Using the reconstructed data, we tested for non-PH in all 50 trials. We compared the results from the logrank/Cox test with those from the combined test recently proposed by Royston and Parmar. RESULTS: The PH assumption was checked and reported only in 28% of the studies. Evidence of non-PH at the 0.10 level was detected in 31% of comparisons. The Cox test of the treatment effect was significant at the 0.05 level in 49% of comparisons, and the combined test in 55%. In four of five trials with discordant results, the interpretation would have changed had the combined test been used. The degree of non-PH and the dominance of the p value for the combined test were strongly associated. Graphical investigation suggested that non-PH was mostly due to a treatment effect manifesting in an early follow-up and disappearing later. CONCLUSIONS: The evidence for non-PH is checked (and, hence, identified) in only a small minority of RCTs, but non-PH may be present in a substantial fraction of such trials. In our reanalysis of the reconstructed data from 50 trials, the combined test outperformed the Cox test overall. The combined test is a promising approach to making trial design and analysis more robust.

Multivariate two-sample permutation tests for trials with multiple time-to-event outcomes.

Clinical trials involving multiple time-to-event outcomes are increasingly common. In this paper, permutation tests for testing for group differences in multivariate time-to-event data are proposed. Unlike other two-sample tests for multivariate survival data, the proposed tests attain the nominal type I error rate. A simulation study shows that the proposed tests outperform their competitors when the degree of censored observations is sufficiently high. When the degree of censoring is low, it is seen that naive tests such as Hotelling's T2 outperform tests tailored to survival data. Computational and practical aspects of the proposed tests are discussed, and their use is illustrated by analyses of three publicly available datasets. Implementations of the proposed tests are available in an accompanying R package.