Weighted Logrank Tests Research Articles

Weighted Hazard Ratio Estimation for Delayed and Diminishing Treatment Effect

Nonproportional hazards (NPH) have been observed in confirmatory clinical trials with time to event outcomes. Under NPH, the hazard ratio does not stay constant over time and the log rank test is no longer the most powerful test. The weighted log rank test (WLRT) has been introduced to deal with the presence of nonproportionality. We focus our attention on the WLRT and the complementary Cox model based on time varying treatment effect proposed by Lin and León. We investigate whether the proposed weighted hazard ratio (WHR) approach is unbiased in scenarios where the WLRT statistic is the most powerful test. In the diminishing treatment effect scenario where the WLRT statistic would be most optimal, the time varying treatment effect estimated by the Cox model estimates the treatment effect very close to the true one. However, when the true hazard ratio is large the proposed model overestimates the treatment effect and the treatment profile over time. In the delayed treatment scenario, the estimated treatment effect profile over time is typically close to the true profile. For both scenarios, we have demonstrated analytically that the hazard ratio functions are approximately equal under small treatment effects. When the assumed rate of how quickly the treatment effect profile is diminishing or delaying differs in the analysis from that in the true data generating mechanism, the estimated hazard ratio profile from the WHR approach is biased. Since in practice the true HR time profile may differ from that assumed in the WHR analysis, it may be preferable to use alternative approaches for effect estimation.

Sequential monitoring of cancer immunotherapy trial with random delayed treatment effect

ABSTRACT Cancer immunotherapy trials are frequently characterized by a delayed treatment effect that violates the proportional hazards assumption. The log-rank test (LRT) suffers a substantial loss of statistical power under the nonproportional hazards model. Various group sequential designs using weighted LRTs (WLRTs) have been proposed under the fixed delayed treatment effect model. However, patients enrolled in immunotherapy trials are often heterogeneous, and the duration of the delayed treatment effect is a random variable. Therefore, we propose group sequential designs under the random delayed effect model using the random delayed distribution WLRT. The proposed group sequential designs are developed for monitoring the efficacy of the trial using the method of Lan-DeMets alpha-spending function with O’Brien-Fleming stopping boundaries or a gamma family alpha-spending function. The maximum sample size for the group sequential design is obtained by multiplying an inflation factor with the sample size for the fixed sample design. Simulations are conducted to study the operating characteristics of the proposed group sequential designs. The robustness of the proposed group sequential designs for misspecifying random delay time distribution and domain is studied via simulations.

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 nonproportionalhazards.

Weighted Log-Rank Test for Clinical Trials with Delayed Treatment Effect Based on a Novel Hazard Function Family

In clinical trials with delayed treatment effect, the standard log-rank method in testing the difference between survival functions may have problems, including low power and poor robustness, so the method of weighted log-rank test (WLRT) is developed to improve the test performance. In this paper, a hyperbolic-cosine-shaped (CH) hazard function family model is proposed to simulate delayed treatment effect scenarios. Then, based on Fleming and Harrington’s method, this paper derives the corresponding weight function and its regular corrections, which are powerful in test, theoretically. Alternative methods of parameters selection based on potential information are also developed. Further, the simulation study is conducted to compare the power performance between CH WLRT, classical WLRT, modest weighted log-rank test and WLRT with logistic-type weight function under different hazard scenarios and simulation settings. The results indicate that the CH statistics are powerful and robust in testing the late difference, so the CH test is useful and meaningful in practice.

A simulation-free group sequential design with max-combo tests in the presence of non-proportional hazards.

Non-proportional hazards (NPH) have been observed in many immuno-oncology clinical trials. Weighted log-rank tests (WLRT) with suitable weights can be used to improve the power of detecting the difference between survival curves in the presence of NPH. However, it is not easy to choose a proper WLRT in practice. A versatile max-combo test was proposed to achieve the balance of robustness and efficiency, and has received increasing attention recently. Survival trials often warrant interim analyses due to their high cost and long durations. The integration and implementation of max-combo tests in interim analyses often require extensive simulation studies. In this report, we propose a simulation-free approach for group sequential designs with the max-combo test in survival trials. The simulation results support that the proposed method can successfully control the type I error rate and offer excellent accuracy and flexibility in estimating sample sizes, with light computation burden. Notably, our method displays strong robustness towards various model misspecifications and has been implemented in an R package.

Delayed treatment effects, treatment switching and heterogeneous patient populations: How to design and analyze RCTs in oncology.

SUMMARYIn the analysis of survival times, the logrank test and the Cox model have been established as key tools, which do not require specific distributional assumptions. Under the assumption of proportional hazards, they are efficient and their results can be interpreted unambiguously. However, delayed treatment effects, disease progression, treatment switchers or the presence of subgroups with differential treatment effects may challenge the assumption of proportional hazards. In practice, weighted logrank tests emphasizing either early, intermediate or late event times via an appropriate weighting function may be used to accommodate for an expected pattern of non‐proportionality. We model these sources of non‐proportional hazards via a mixture of survival functions with piecewise constant hazard. The model is then applied to study the power of unweighted and weighted log‐rank tests, as well as maximum tests allowing different time dependent weights. Simulation results suggest a robust performance of maximum tests across different scenarios, with little loss in power compared to the most powerful among the considered weighting schemes and huge power gain compared to unfavorable weights. The actual sources of non‐proportional hazards are not obvious from resulting populationwise survival functions, highlighting the importance of detailed simulations in the planning phase of a trial when assuming non‐proportional hazards.We provide the required tools in a software package, allowing to model data generating processes under complex non‐proportional hazard scenarios, to simulate data from these models and to perform the weighted logrank tests.

PET-directed chemoradiation (CRT) with induction FOLFOX compared to induction carboplatin/paclitaxel (CP) in patients with locally advanced esophageal adenocarcinoma (EA).

4554 Background: Induction chemotherapy with PET-directed CRT and surgery is the standard treatment for locally advanced EA at our institution. Following results of the CALGB 80803 trial, FOLFOX has recently replaced CP as the preferred induction regimen. Methods: We retrospectively evaluated patients with locally advanced EA treated with induction CP vs FOLFOX, followed by trimodality therapy between January 2010 and June 2019. Patients treated with CP with RT followed by surgery without induction chemo were also included. We compared pathological complete response (pCR) and near pCR (ypN0 with ≥90% response) rates in the induction FOLFOX group to the induction CP and no-induction groups. Univariable and multivariable analyses were used to adjust for confounding factors. Disease-free survival (DFS) was estimated by the Kaplan-Meier method and compared between groups using max-combo weighted log rank test. Results: 445 patients were included. Patients in the induction FOLFOX group had significantly higher pCR and near pCR rates vs induction CP patients. Notably, pCR rate was 38% among FOLFOX PET responders vs 19% in non-responders. In multivariable analysis, compared to induction CP, induction FOLFOX administration was an independent predictor of near pCR (OR: 2.22, 95%CI: 1.20-4.20, p = 0.012). Compared to 24% pCR rate among no-induction patients, induction FOLFOX pCR rate was slightly higher at 32%. DFS by 2-years was higher in induction FOLFOX compared to no-induction-treated patients (62% vs. 42%, p = 0.05). Postoperative complication rates were similar among the three groups. Conclusions: PET-directed CRT with FOLFOX instead of CP improves pCR and near pCR rates. Improved DFS was observed in the FOLFOX vs no-induction patients. Longer follow-up is needed to confirm any survival benefits. [Table: see text]

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.

Wild bootstrap logrank tests with broader power functions for testing superiority

A novel wild bootstrap procedure is introduced for testing superiority in unpaired two-sample survival data. Combining classical weighted logrank tests yields a procedure with broader power behavior. Right censoring within the data is allowed and may differ between the groups. The tests are shown to be asymptotically exact under the null, consistent for fixed alternatives and admissible for a larger set of local alternatives. Beside these asymptotic properties, the procedures’ strengths are also illustrated in simulations for finite sample sizes. The tests are implemented in the novel R-package mdir.logrank and its application is demonstrated in an empirical example.

Designing cancer immunotherapy trials with random treatment time-lag effect.

In some clinical settings such as the cancer immunotherapy trials, a treatment time-lag effect may be present and the lag duration possibly vary from subject to subject. An efficient study design and analysis procedure should not only take into account the time-lag effect but also consider the individual heterogeneity in the lag duration. In this paper, we present a Generalized Piecewise Weighted Logrank (GPW-Logrank) test, designed to account for the random time-lag effect while maximizing the study power with respect to the weights. Based on the proposed test, both analytic and numeric approaches are developed for the sample size and power calculation. Asymptotic properties are derived and finite sample efficiency is evaluated in simulations. Compared with the standard practice ignoring the delayed effect, the proposed design and analysis procedures are substantially more efficient when a random lag is expected; further, compared with the existing methods by Xu et al considering the fixed time-lag effect, the proposed approaches are significantly more robust when the lag model is misspecified. An R package (DelayedEffect.Design) is developed for implementation.

Do immune checkpoint inhibitors need new studies methodology?

Immune checkpoint inhibitors (ICI) have widely reshaped the treatment paradigm of advanced cancer patients. Although multiple studies are currently evaluating these drugs as monotherapies or in combination, the choice of the most accurate statistical methods, endpoints and clinical trial designs to estimate the benefit of ICI remains an unsolved methodological issue. Considering the unconventional patterns of response or progression [i.e., pseudoprogression, hyperprogression (HPD)] observed with ICI, the application in clinical trials of novel response assessment tools (i.e., iRECIST) able to capture delayed benefit of immunotherapies and/or to quantify tumor dynamics and kinetics over time is an unmet clinical need. In addition, the proportional hazard model and the conventional measures of survival [i.e., median overall or progression free survival (PFS) and hazard ratios (HR)] might usually result inadequate in the estimation of the long-term benefit observed with ICI. For this reason, innovative methodologies such as milestone analysis, restricted mean survival time (RMST), parametric models (i.e., Weibull distribution, weighted log rank test), should be systematically investigated in clinical trials in order to adequately quantify the fraction of patients who are "cured", represented by the tails of the survival curves. Regarding predictive biomarkers, in particular PD-L1 expression, the integration and harmonization of the existing assays are urgently needed to provide clinicians with reliable diagnostic tests and to improve patient selection for immunotherapy. Finally, developing original and high-quality study designs, such as adaptive or basket biomarker enriched clinical trials, included in large collaborative platforms with multiple active sites and cross-sector collaboration, represents the successful strategy to optimally assess the benefit of ICI in the next future.

Assessment of post-switch endpoints in randomized clinical trials

Background In randomized controlled clinical trials (RCT) in oncology, the switch from reference treatment arm to another randomization arm after observation of the primary endpoint is sometimes considered. The comparability between arms for the assessment of post switch endpoints like overall survival (OS) is then compromised. The Intention-To-Treat (ITT) approach, currently the gold standard for the analysis of such trials, is biased to the null. Other methods accounting for switch to experimental arm have been developed like Inverse Probability of Censoring Weighted log rank test (IPCW), Rank Preserving Survival Failure Time models (RPSFT) or Iterative Parameter Estimation (IPE). The aim of this paper is to evaluate the behavior of each method on treatment effect on real data of a RCT and on simulated data. Methods Avastin Use in Platinum-Resistant Epithelial Ovarian Cancer (AURELIA) trial was a two-arms RCT where treatment switch in the reference arm was allowed. The OS results, estimated with an ITT approach, were compared in terms of hazard ratio (HR) with those of RPSFT, IPE and IPCW. Chosen model for IPCW implementation was a multiple logistic regression model with 10 baseline and time varying factors. Trials were simulated by resampling the AURELIA dataset, according to factors assumed to influence the methods’ evaluations, like switch-rate, time to switch, or sample size and censor rate. Variations of those parameters were explored. Results were compared between methods for each scenario. Results All the methods aimed to correct the OS in the control arm. Whatever the method used, the AURELIA trial results were confirmed. However, the correction was more important with the three methods investigated than with the ITT approach [HR = 0.85 (0.66 ; 1.08)]. The most optimistic correction was obtained with the IPCW method [HR = 0.74 (0.56 ; 0.97)], RPSFT and IPE gave similar results [0.78 (0.61 ; 0.99)]. The factor that had the strongest impact on the HR estimation was the switch rate : a low switch rate was associated with results close to the ITT approach. A high switch rate produced a control group comparable to the experimental group in terms of survival and was thus associated with a HR close to 1. Time to switch had small influence on the HR, a long delay between switch and OS led to an HR estimation near 1, indeed this is an unfavorable case to show a treatment effect. Event rate and sample size factors had no significant influence on the HR evaluations. Globally the simulations showed that all methods gave consistent results, no major differences were observed compared to the ITT approach. Conclusions Even if the treatment switch is an interesting option to allow early access to treatments expected as the most effective, biases on the secondary criteria estimation are important and existing statistical methods to correct these biases are complex and have some limitations. At the design stage, it is thus preferable not to include switch from a data interpretation perspective. If a switch is scheduled, the planned methods should be specify in the protocol and the statistical analysis plan. Therefore data needed for the methods implementation should be carefully collected. Results should always be presented using the ITT approach, the other methods should be considered as sensitivity analyses.

A flexible and coherent test/estimation procedure based on restricted mean survival times for censored time-to-event data in randomized clinical trials.

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

Testing causal effects in observational survival data using propensity score matching design.

Time-to-event data are very common in observational studies. Unlike randomized experiments, observational studies suffer from both observed and unobserved confounding biases. To adjust for observed confounding in survival analysis, the commonly used methods are the Cox proportional hazards (PH) model, the weighted logrank test, and the inverse probability of treatment weighted Cox PH model. These methods do not rely on fully parametric models, but their practical performances are highly influenced by the validity of the PH assumption. Also, there are few methods addressing the hidden bias in causal survival analysis. We propose a strategy to test for survival function differences based on the matching design and explore sensitivity of the P-values to assumptions about unmeasured confounding. Specifically, we apply the paired Prentice-Wilcoxon (PPW) test or the modified PPW test to the propensity score matched data. Simulation studies show that the PPW-type test has higher power in situations when the PH assumption fails. For potential hidden bias, we develop a sensitivity analysis based on the matched pairs to assess the robustness of our finding, following Rosenbaum's idea for nonsurvival data. For a real data illustration, we apply our method to an observational cohort of chronic liver disease patients from a Mayo Clinic study. The PPW test based on observed data initially shows evidence of a significant treatment effect. But this finding is not robust, as the sensitivity analysis reveals that the P-value becomes nonsignificant if there exists an unmeasured confounder with a small impact.

A rank test for bivariate time-to-event outcomes when one event is a surrogate.

In many clinical settings, improving patient survival is of interest but a practical surrogate, such as time to disease progression, is instead used as a clinical trial's primary endpoint. A time-to-first endpoint (e.g., death or disease progression) is commonly analyzed but may not be adequate to summarize patient outcomes if a subsequent event contains important additional information. We consider a surrogate outcome very generally as one correlated with the true endpoint of interest. Settings of interest include those where the surrogate indicates a beneficial outcome so that the usual time-to-first endpoint of death or surrogate event is nonsensical. We present a new two-sample test for bivariate, interval-censored time-to-event data, where one endpoint is a surrogate for the second, less frequently observed endpoint of true interest. This test examines whether patient groups have equal clinical severity. If the true endpoint rarely occurs, the proposed test acts like a weighted logrank test on the surrogate; if it occurs for most individuals, then our test acts like a weighted logrank test on the true endpoint. If the surrogate is a useful statistical surrogate, our test can have better power than tests based on the surrogate that naively handles the true endpoint. In settings where the surrogate is not valid (treatment affects the surrogate but not the true endpoint), our test incorporates the information regarding the lack of treatment effect from the observed true endpoints and hence is expected to have a dampened treatment effect compared with tests based on the surrogate alone. Published 2016. This article is a U.S. Government work and is in the public domain in the USA.

Gaining power and precision by using model-based weights in the analysis of late stage cancer trials with substantial treatment switching.

In randomised controlled trials of treatments for late‐stage cancer, it is common for control arm patients to receive the experimental treatment around the point of disease progression. This treatment switching can dilute the estimated treatment effect on overall survival and impact the assessment of a treatment's benefit on health economic evaluations. The rank‐preserving structural failure time model of Robins and Tsiatis (Comm. Stat., 20:2609–2631) offers a potential solution to this problem and is typically implemented using the logrank test. However, in the presence of substantial switching, this test can have low power because the hazard ratio is not constant over time. Schoenfeld (Biometrika, 68:316–319) showed that when the hazard ratio is not constant, weighted versions of the logrank test become optimal. We present a weighted logrank test statistic for the late stage cancer trial context given the treatment switching pattern and working assumptions about the underlying hazard function in the population. Simulations suggest that the weighted approach can lead to large efficiency gains in either an intention‐to‐treat or a causal rank‐preserving structural failure time model analysis compared with the unweighted approach. Furthermore, violation of the working assumptions used in the derivation of the weights only affects the efficiency of the estimates and does not induce bias or inflate the type I error rate. The weighted logrank test statistic should therefore be considered for use as part of a careful secondary, exploratory analysis of trial data affected by substantial treatment switching. ©©2015 The Authors. Statistics inMedicine Published by John Wiley & Sons Ltd.

이질적인 중도절단분포 하에서 생존분포의 동일성 검정법 비교연구

세 개 이상의 집단에 대한 생존분포의 비교를 위해 가중 로그순위 검정법(Weighted Logrank test)과 그의 특별한 경우인 로그순위 검정법(Logrank test)이 널리 쓰인다. 그러나 이 방법은 근사적인 분포를 이용한 방법이므로 표본 크기가 작은 경우에는 유효하지 못할 수 있으며, 각 집단의 중도절단 분포가 동일하다는 가정 또한 충족되어야 하기 때문에 이 가정이 충족되지 못할 경우에도 검정법의 유효성을 장담할 수 없다. 표본 크기가 작은 경우에 대한 대안으로, 분포에 대한 가정이 없이 관찰된 자료만으로 검정통계량의 분포를 추정하고 그 분포를 이용해 검정하는 순열 검정법(Permutation test)이 제안되었으나, 순열 검정법 또한 각 집단의 중도절단 분포가 동일하다는 가정이 충족되어야 한다. 따라서 순열 검정법을 향상시킨 순열-대치 검정법(Permutation-Imputation test)이 대안이 될 수 있는데, 이는 대치 단계(Imputation step)에서 귀무가설 하에서의 생존확률이 집단에 의존하지 않도록 자료를 조정한 후 순열 검정 단계(Permutation step)를 통해 검정하는 방법이다. 본 논문에서는 근사적 방법, 순열 검정법, 순열-대치 검정법을 로그순위 검정법과 가중 로그순위 검정법의 한 형태인 Prentice-Wilcoxon 검정법에 적용해 각 검정법의 유효성과 검정력을 비교하였다. The Weighted Logrank test and its special case, Logrank test are widely used to compare survival distributions; however, these methods are inappropriate when the sample size is small or censoring distributions are not equal since they use test statistics from approximate distributions. A permutation test can be an alternative for small sample cases; however, this should be used only when censoring distributions are equal. To handle cases with small sample size and unequal censoring distributions, the permutation-imputation method was developed to compare two survival distributions. In this paper, approximate method, permutation method and permutation-imputation method were compared using a Logrank test and Prentice-Wilcoxon test for three or more survival distributions comparison.

Weighted logrank tests for interval censored data when assessment times depend on treatment

We consider weighted logrank tests for interval censored data when assessment times may depend on treatment, and for each individual, we only use the two assessment times that bracket the event of interest. It is known that treating finite right endpoints as observed events can substantially inflate the type I error rate under assessment-treatment dependence (ATD), but the validity of several other implementations of weighted logrank tests (score tests, permutation tests, multiple imputation tests) has not been studied in this situation. With a bounded number of unique assessment times, the score test under the grouped continuous model retains the type I error rate asymptotically under ATD; however, although the approximate permutation test based on the permutation central limit theorem is not asymptotically valid under every ATD scenario, we show through simulation that in many ATD scenarios, it retains the type I error rate better than the score test. We show a case where the approximate permutation test retains the type I error rate when the exact permutation test does not. We study and modify the multiple imputation logrank tests of Huang, Lee, and Yu (2008, Statistics in Medicine, 27: 3217-3226), showing that the distribution of the rank-like scores asymptotically does not depend on the assessment times. We show through simulations that our modifications of the multiple imputation logrank tests retain the type I error rate in all cases studied, even with ATD and a small number of individuals in each treatment group. Simulations were performed using the interval R package.

Midterm Benefits of Preoperative Statin Therapy in Patients Undergoing Isolated Valve Surgery

Recent data have suggested that statins are associated with reduced early mortality and cardiovascular events after valvular heart surgery. The midterm effects of preoperative statin therapy in the setting of valvular heart surgery are presently unclear. All patients (n=2,120) who underwent a valvular procedure between April 2004 and April 2010 were identified. Patients undergoing concomitant coronary artery bypass graft surgery were excluded. Two patient groups were studied: those who received preoperative statin therapy (n=663; 31.3%) and those who did not (n=1,457; 68.7%). Propensity score matching resulted in 381 matched pairs, thus addressing baseline risk imbalances. Thirty-day mortality, readmission rates, postoperative complications, and length of stay were analyzed. Late survival was ascertained by the Social Security Death Index. In the matched group, 30-day mortality was 1.3% (5 of 381) for statin-treated patients versus 4.2% (16 of 381) for statin-untreated patients (p=0.03). After a mean follow-up of 33±23 months, statin therapy was associated with significantly reduced mortality (hazard ratio 0.63, 95% confidence interval: 0.43 to 0.93, p=0.019), independent of known cardiac risk factors. Weighted log rank tests revealed that the mortality difference between the two cohorts occurred early after surgery (p=0.015). Statin users were less likely to be readmitted to the intensive care unit (3.4% versus 8.1%, p=0.01). There were no other significant differences between the two groups in terms of postoperative complications and length of stay. Preoperative statin administration is associated with early reductions in mortality among patients undergoing isolated valvular heart surgery, leading to improved late survival. Future prospective analyses are warranted to optimize statin therapy in this patient population.

A jackknife-based versatile test for two-sample problems with right-censored data

For testing the equality of two survival functions, the weighted logrank test and the weighted Kaplan–Meier test are the two most widely used methods. Actually, each of these tests has advantages and defects against various alternatives, while we cannot specify in advance the possible types of the survival differences. Hence, how to choose a single test or combine a number of competitive tests for indicating the diversities of two survival functions without suffering a substantial loss in power is an important issue. Instead of directly using a particular test which generally performs well in some situations and poorly in others, we further consider a class of tests indexed by a weighted parameter for testing the equality of two survival functions in this paper. A delete-1 jackknife method is implemented for selecting weights such that the variance of the test is minimized. Some numerical experiments are performed under various alternatives for illustrating the superiority of the proposed method. Finally, the proposed testing procedure is applied to two real-data examples as well.