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A modified weighted log-rank test for confirmatory trials with a high proportion of treatment switching
TL;DR: A modified weighted log-rank test (mWLR) that aims at balancing these factors by down-weighting events occurring when many patients have switched treatment by predicting the hazard ratio function and using it to compute the weights of the mWLR.
Abstract: In confirmatory cancer clinical trials, overall survival (OS) is normally a primary endpoint in the intention-to-treat (ITT) analysis under regulatory standards. After the tumor progresses, it is common that patients allocated to the control group switch to the experimental treatment, or another drug in the same class. Such treatment switching may dilute the relative efficacy of the new drug compared to the control group, leading to lower statistical power. It would be possible to decrease the estimation bias by shortening the follow-up period but this may lead to a loss of information and power. Instead we propose a modified weighted log-rank test (mWLR) that aims at balancing these factors by down-weighting events occurring when many patients have switched treatment.
As the weighting should be pre-specified and the impact of treatment switching is unknown, we predict the hazard ratio function and use it to compute the weights of the mWLR. The method may incorporate information from previous trials regarding the potential hazard ratio function over time.
We are motivated by the RECORD-1 trial of everolimus against placebo in patients with metastatic renal-cell carcinoma where almost 80\% of the patients in the placebo group received everolimus after disease progression. Extensive simulations show that the new test gives considerably higher efficiency than the standard log-rank test in realistic scenarios.
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TL;DR: In this article, the authors explored the impact of delayed effects on group sequential and adaptive group sequential designs, and made an empirical evaluation in terms of power and type-I error rate of the weighted log-rank test in a simulated scenario with fixed values of the Fleming and Harrington class of weights.
Abstract: Proportional hazards are a common assumption when designing confirmatory clinical trials in oncology. This assumption not only affects the analysis part but also the sample size calculation. The presence of delayed effects causes a change in the hazard ratio while the trial is ongoing since at the beginning we do not observe any difference between treatment arms and after some unknown time point, the differences between treatment arms will start to appear. Hence, the proportional hazards assumption no longer holds and both sample size calculation and analysis methods to be used should be reconsidered. The weighted log-rank test allows a weighting for early, middle and late differences through the Fleming and Harrington class of weights, and is proven to be more efficient when the proportional hazards assumption does not hold. The Fleming and Harrington class of weights, along with the estimated delay, can be incorporated into the sample size calculation in order to maintain the desired power once the treatment arm differences start to appear. In this article, we explore the impact of delayed effects in group sequential and adaptive group sequential designs, and make an empirical evaluation in terms of power and type-I error rate of the of the weighted log-rank test in a simulated scenario with fixed values of the Fleming and Harrington class of weights. We also give some practical recommendations regarding which methodology should be used in the presence of delayed effects depending on certain characteristics of the trial.
11 citations
TL;DR: 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.
Abstract: 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.
6 citations
TL;DR: In this article, the authors proposed a new weight function comprising all numbers at risk, i.e., the overall number at risk and the separate numbers in the groups under study, to detect late differences between survival curves.
Abstract: Background. Survival analysis attracted the attention of different scientists from various domains such as engineering, health, and social sciences. It has been widely exploited in clinical trials when comparing different treatments looking at their survival probabilities. Kaplan–Meier curves plotted from the Kaplan–Meier estimates of survival probabilities are used to depict the general image for such situations. Methods. The weighted log-rank test has been dealt with by suggesting different weight functions which give specific strength in specific situations. In this work, we proposed a new weight function comprising all numbers at risk, i.e., the overall number at risk and the separate numbers at risk in the groups under study, to detect late differences between survival curves. Results. The new test has been found to be a good alternative after the FH (0, 1) test in detecting late differences, and it outperformed all tests in case of small samples and heavy censoring rates according to the simulation studies. The new test kept the same strength when applied to real data where it showed itself to be among the powerful ones or even outperforms all other tests under consideration. Conclusion. As the new test stays stronger in the case of small samples and heavy censoring rates, it may be a better choice whenever targeting the detection of late differences between the survival curves.
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TL;DR: Copyright (©) 1999–2012 R Foundation for Statistical Computing; permission is granted to make and distribute verbatim copies of this manual provided the copyright notice and permission notice are preserved on all copies.
Abstract: Copyright (©) 1999–2012 R Foundation for Statistical Computing. Permission is granted to make and distribute verbatim copies of this manual provided the copyright notice and this permission notice are preserved on all copies. Permission is granted to copy and distribute modified versions of this manual under the conditions for verbatim copying, provided that the entire resulting derived work is distributed under the terms of a permission notice identical to this one. Permission is granted to copy and distribute translations of this manual into another language, under the above conditions for modified versions, except that this permission notice may be stated in a translation approved by the R Core Team.
272,030 citations
"A modified weighted log-rank test f..." refers background in this paper
...025 and M = 10(4) corresponds to the number of simulations implemented implemented in R [36] for each scenario....
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TL;DR: In this paper, the authors show that the limit distribution is normal if n, n$ go to infinity in any arbitrary manner, where n = m = 8 and n = n = 8.
Abstract: Let $x$ and $y$ be two random variables with continuous cumulative distribution functions $f$ and $g$. A statistic $U$ depending on the relative ranks of the $x$'s and $y$'s is proposed for testing the hypothesis $f = g$. Wilcoxon proposed an equivalent test in the Biometrics Bulletin, December, 1945, but gave only a few points of the distribution of his statistic. Under the hypothesis $f = g$ the probability of obtaining a given $U$ in a sample of $n x's$ and $m y's$ is the solution of a certain recurrence relation involving $n$ and $m$. Using this recurrence relation tables have been computed giving the probability of $U$ for samples up to $n = m = 8$. At this point the distribution is almost normal. From the recurrence relation explicit expressions for the mean, variance, and fourth moment are obtained. The 2rth moment is shown to have a certain form which enabled us to prove that the limit distribution is normal if $m, n$ go to infinity in any arbitrary manner. The test is shown to be consistent with respect to the class of alternatives $f(x) > g(x)$ for every $x$.
11,055 citations
"A modified weighted log-rank test f..." refers methods in this paper
...One may view the Wilcoxon test [31] as a weighted version of the LR test, but one could equally well view LR as a weighted version of Wilcoxon....
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01 Mar 1972
3,283 citations
"A modified weighted log-rank test f..." refers background in this paper
...It is well know that under proportional hazards, the log-rank test (LR) is optimal among all tests based on the order of events (and censoring) [35, 43]....
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TL;DR: Treatment with everolimus prolongs progression-free survival relative to placebo in patients with metastatic renal cell carcinoma that had progressed on other targeted therapies, but were mostly mild or moderate in severity.
2,822 citations
"A modified weighted log-rank test f..." refers background in this paper
...In some situations, PFS might be enough to obtain traditional regulatory approval [33, 11, 19, 22]....
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TL;DR: A randomised, double-blind, placebo-controlled, multicentre, international trial to assess tolerability and anticancer efficacy of sunitinib in patients with advanced gastrointestinal stromal tumour, noting significant clinical benefit, including disease control and superior survival.
2,340 citations
"A modified weighted log-rank test f..." refers background in this paper
...In some situations, PFS might be enough to obtain traditional regulatory approval [33, 11, 19, 22]....
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