 Methodology
 Open access
 Published:
A simulation study comparing the power of nine tests of the treatment effect in randomized controlled trials with a timetoevent outcome
Trials volumeÂ 21, ArticleÂ number:Â 315 (2020)
Abstract
Background
The logrank test is routinely applied to design and analyse randomized controlled trials (RCTs) with timetoevent 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 nonproportional hazards (nonPH): early, late or nearPH 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 nonPH patterns. Composite tests attempt to improve power across a wider range of nonPH 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 followup times. We systematically investigated the influence of high or low controlarm 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 nearPH, 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 controlarm events considerably affected power.
Conclusions
Assuming ignorance of the likely treatment effect, the best choice is VWLR2. Several nonstandard tests performed well when the correct type of treatment effect was assumed. A low controlarm 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 nonproportional hazards of the treatment effect is warranted.
Background
Randomized controlled trials (RCTs) with a timetoevent outcome are typically designed according to sample size and power calculations using the logrank test. The treatment effect is summarized by the hazard ratio (HR) between the control and research arms, usually estimated with a Cox proportional hazards (PH) model. During the last decade or so, researchers e.g. [1, 2] have demonstrated that nonproportional hazards (nonPH) occur fairly often in trials across a range of medical research areas. NonPH may threaten the power of the logrank test, potentially distorting the findings of a trial and jeopardizing its success. It is therefore important that trial designers take into account the possibility and, if feasible, the probable nature of nonPH in the particular setting of the study.
We assume that HR <1 denotes a reduction in the hazard of an event (e.g. death) in a research arm. NonPH means that the HR varies systematically over followup time. We may usefully distinguish four types of HR patterns: PH, early or diminishing effect, late or delayed effect, other. PH includes the nullhypothesis case of identical survival curves in the trial arms. With an early effect, the HR is <1 in the early followup and increases later. An early effect may, for example, be provoked by â€˜wearing offâ€™ of the effectiveness of a therapy that is administered for a limited period and then stopped. A late or delayed effect may occur in screening or prevention trials or in trials in immunooncology settings, in which the treatment effect is expected to take time to manifest. Subsequently, we refer to such patterns generically as late effects.
The â€˜otherâ€™ type covers all other possibilities, of which the most readily recognizable are crossing survival functions. Our impression is that in real trials, PH, early and late patterns predominate. Other patterns are not necessarily simple to characterize; therefore, only these three are studied in the present paper.
Figure 1 gives examples of pairs of KaplanMeier survival curves illustrating the four types of treatment effects we have discussed.
We created the datasets by simulation to illustrate reasonable and plausible curves. Although the overall HRs, numbers of events and logrank test p values for the treatment comparisons are approximately the same in each case, the survivalcurve comparisons differ considerably. For example, the difference in median survival time (research minus control) is largest with the early and late effects, somewhat smaller with PH and close to zero (and negative) for the â€˜otherâ€™ pattern (d).
The focus of the present paper is on tests of the null hypothesis of identical survival functions against specific alternative hypotheses (PH, early effect, late effect). It is widely recognized that the logrank test may lose power, sometimes severely, in nonPH situations. Making extensive use of simulation, our aim is to identify good candidate(s) for resilience from a set of nine tests that we have selected. A â€˜resilientâ€™ test is one that exhibits acceptable power under PH and also under some common patterns of nonPH, while maintaining the type 1 error rate close to the nominal level. In addition to power, we therefore also assessed the type 1 error of the tests. The tests and the rationale for their selection are described in the next section.
The structure of the article is as follows. In â€˜Methodsâ€™, we describe the tests to be compared and our approach to simulation of the performance (power) of the tests. In â€˜Resultsâ€™ we report our findings on type 1 error and power. This section also includes a comparison of the tests on three selected trials with apparently differing types of treatment effects. We end with a Discussion and our Conclusions.
Methods
Tests to be compared
Many tests of two survival curves have been proposed over the last five decades or so, but very few have found their way into practice in trials. We have focused on the most popular, the logrank, a small number of related tests and particularly on more recent composite tests comprising two or more component tests. The logrank test is the de facto standard for trial design and analysis and is therefore the natural comparator for other tests. Variants of the logrank test are typically weighted in such a way as to be sensitive to particular nonPH patterns. Composite tests are an attempt to improve power across a wider range of nonPH patterns than the logrank test manages.
We have not included tests which require prespecifying a single, fixed time point, t^{âˆ—} say, for their evaluation. Examples are the difference at t^{âˆ—} in KaplanMeier survival estimates or in restricted mean survival times (RMSTs). Although such tests are intuitively simple and appealing, their power with some nonPH patterns is vulnerable to poor choices of t^{âˆ—}.
The nine tests we have included are described briefly below. All computations were performed using Stata 15.1 [3].
Logrank test (LR)
The logrank test is the optimal (most powerful) rank test under PH. The test is also valid under nonPH alternatives, but it may then lack power.
Earlyeffect weighted logrank test (LRE)
LRE is a weighted logrank test with FlemingHarrington weight function (1,0) [4]; that is w_{i}=S(t_{i}âˆ’0). LRE is intended to be sensitive to early effects. It is similar to the PetoPetoPrentice test [5, 6].
Lateeffect weighted logrank test (LRL)
LRL is a weighted logrank test with FlemingHarrington weight function (0,1); that is w_{i}=1âˆ’S(t_{i}âˆ’0). LRL is intended to be sensitive to late effects.
Supremum logrank test (SupLR)
The supremum logrank test [7] is based on the maximal logrank test statistic over the event times t_{1},â€¦,t_{r}. It is calculated by restricting the logrank test to time t_{i} and then taking the supremum test statistic over the t_{i}. Local minima or maxima in the test statistic may be detected by the supremum logrank test which may indicate a nonrandom difference between the survival functions.
Joint test (J)
The joint test [8] combines a Cox test (essentially identical to the logrank test) with a standard test of nonPH, the GrambschTherneau test. Under PH, the two component tests are independent. The joint test statistic is the sum of the two model chisquare values. It has a known distribution under the null and under PH alternatives.
Combined test (C)
The combined test [9] combines a Cox test with a permutation test based on the maximal squared standardized difference in RMST between the control and research arms. Maximization is over a predefined small set of event times (t^{âˆ—}). Royston et al. [2] showed that the combined test outperformed the Cox test (and implicitly the logrank test) in 55 randomized comparisons based on reconstructed data from 50 RCTs in various medical research areas.
Weighted combined test (WC)
The weighted combined test (unpublished, available in Stata from the first author) is an attempt to improve the performance of the combined test when a delayed/late treatment effect is present. The Cox test component is replaced by the LRL test.
Versatile weighted logrank test (VWLR)
A â€˜versatileâ€™ test is one derived by combining several weighted logrank tests in different ways [10, 11]. Like the C and WC tests, a versatile test is designed to be sensitive to different types of departures from the null hypothesis H_{0}:S_{0}(t)=S_{1}(t). Karrisonâ€™s proposal [12] which we use here, is the maximum squareroot chisquare statistic among three correlated logrank tests: unweighted (i.e. standard LR), earlyeffect weights (as in LRE) and lateeffect weights (as in LRL). The asymptotic null distribution of the test statistic is available in closed form.
Modified versatile weighted logrank test (VWLR2)
VWLR2, the modified version of VWLR, is unpublished and is available as a Stata program from the first author. It incorporates a small but potentially important change to one component of the VWLR test. The LRE test with weights w_{i}=S(t_{i}âˆ’0) is replaced by a logrank test with weights given by
where S_{max}=1 and S_{min} is the minimum of the left estimate, S(tâˆ’0), of the KaplanMeier survival function. The support of these weights is the interval [0.001,1]. The aim is to increase power when the data exhibit an early effect with a low event rate. With such data, the coefficient of variation of the original weights is small, and therefore the weighted test (LRE) too closely resembles the standard test (LR). By construction, the issue of toosimilar weights does not arise with the LRL component of VWLR. The null distribution of the test statistic for VWLR2 follows from the general result for the maximum of weighted logrank tests as derived in [4], section 7.5, theorem 7.5.1.
Simulation scenarios
We assessed the power of the nine tests under four alternatives: null case (identical survival distributions in control and research arms), PH with HR =0.75, early effect, late effect. Furthermore, we studied two survival distributions in the control arm: high event rate (S_{min}â‰ƒ0.1) and low event rate (S_{min}â‰ƒ0.9). The goal was to identify, within the constraints of the simulation design, the test(s) which performed best under PH, early effect, late effect and overall across all three patterns.
Approach to simulation
Survival distributions
Survival distributions in the control arms of two real trials were chosen to represent the survival function in the control arm of simulated datasets with low or high event rates, respectively. The first trial [13] had a low event rate and the second [14] a high event rate. The survival functions were approximated using flexible parametric models [15, 16]. A restricted cubic spline with 5 degrees of freedom was used to model the log cumulative hazard function as a function of log time to event in each control arm. Figure 2 shows the observed (KaplanMeier) and fitted survival functions in each of the original datasets.
We used the same approach to approximate the distribution of time to censoring in each dataset (data not shown).
From the fitted flexible parametric models, we obtained small numbers of parameters which describe the survival and timetocensoring distributions in the two datasets. We used the estimated parameter values with suitable sample sizes in subsequent simulations by applying the communitycontributed Stata package stsurvsim [17, 18]. Further details are given below.
Simulating early and late treatment effects
To create simple early and late treatment effects, we chose step functions for the timedependent hazard ratio, HR (t), as shown in Table 1.
The design implements a treatment effect (expressed as HR) that persists over (0,t^{âˆ—}) and then ceases (early effect), or one that is 1.0 when tâ‰¤t^{âˆ—} and <1.0 for t>t^{âˆ—} (late effect).
If the change point t^{âˆ—} is â€˜largeâ€™ in the earlyeffect case, the HR will be nearly constant over the observed followup, and the treatment effect will be close to PH. Vice versa, if t^{âˆ—} is â€˜close to 0â€™ in the lateeffect case, the treatment effect will be close to PH. We expect the logrank test to perform (nearly) optimally in such a situation. In other situations, we would expect tests specifically designed to detect types of nonPH to outperform the LR.
The time scale embodied in t^{âˆ—} is arbitrary. A less scaledependent meaning of t^{âˆ—} being â€˜largeâ€™ and â€˜close to 0â€™ may be attributed to the cumulative proportion of events in the trial before t^{âˆ—}, known as the information fraction (IF). The IF is an important parameter of the alphaspending functions for groupsequential trials. To remove the effect on the IF of the alternative distributions we simulated, we limited the IF to the control arm.
We quantified the performance of the nine tests in the early and lateeffects cases in relation to the control arm IF as follows. We selected seven suitably placed values of t^{âˆ—} for each effect type and event rate (see Table 2).
The t^{âˆ—} values given in Table 2 provide an appropriate spread of IFs in the control arm of the two original datasets.
With each chosen value of t^{âˆ—}, we computed the sample size for each of the four designs (low/high event rate by early/late effect) using the Analysis of Resources for Trials (ART) communitycontributed software package for Stata [19]. The procedure was used to obtain sample sizes that offer a power of 80% or 90% for the LR. We took the LR as the benchmark test for power comparisons with the other eight tests. Based on the benchmark power, the sample sizes used in the simulations varied widely, between about 100 and 18,000 (data not shown).
The â€˜trueâ€™ survival functions used in the simulations for each of the scenarios shown in Table 2 are illustrated in Fig. 3.
Note that in panels c and d of Fig. 3, t^{âˆ—}=0 corresponds to PH. Despite the abrupt, seemingly unrealistic step changes in the HR at t^{âˆ—} used to define nonPH, the survival functions look both reasonable and plausible.
Realistic datasets were simulated for each arm of a hypothetical trial with 1:1 treatment allocation using the flexible parameter estimates and the Stata program stsurvsim cited in the previous section. Five thousand replicates were simulated for each power, event rate, effect type and value of t^{âˆ—}. Power of a given test at the twosided 5% significance level was estimated as the number of replicates in which p<0.05, divided by 5000. Mean observed IF values in the control arm of the simulated datasets were used to define the xaxis in graphs of the power for the various tests and conditions.
Treatment effects under PH
Treatment effects under PH are covered by the special case of late effects with t^{âˆ—}=0 and were not handled separately.
Type 1 error
Treatment effects under the null hypothesis H_{0}:S_{0}(t)=S_{1}(t) were tested using simulation, as with the power studies. For each chosen sample size (n), we simulated 5000 replicate datasets comprising two replicates each with n/2 observations, based on the estimated timetoevent and timetocensoring distributions in the control arm. We took 12 values of n in the range [400,10,000] in the low eventrate scenario and 12 more in the range [40,1000] in the high eventrate scenario. Under PH, the effective sample size is the number of events. With the given sample sizes and event rates, we aimed to cover a wide range of numbers of events.
The empirical type 1 error (size) of a given test at the nominal Î±=0.05 level was estimated as the number of replicates in which p<0.05, divided by 5000. A test with size exceeding 0.05 is termed â€˜anticonservativeâ€™ or â€˜liberalâ€™, whereas one whose size is below 0.05 is deemed â€˜conservativeâ€™.
Results
Type 1 error
We estimated the empirical type 1 error (size) of the nine tests at the Î±=0.05 level by simulation. We pooled the 5000 replicates for each event rate and sample size into datasets, each with 5000Ã—12=60,000 observations. Results by event rate are given in Table 3.
The supremum logrank (SupLR, results shown in bold type) test stands out as it is conservative for both event rates, the size being about 4.5%. For the low event rate, the size of the remaining tests is close to the nominal 5%, whereas for the high event rate, the WC, LRE, LRL, VWLR and VWLR2 tests appear a little anticonservative.
Further investigation (data not shown) revealed that minor size inflation may occur in the high eventrate case when there are fewer than approximately 100 events in the dataset (see Table 4).
The SupLR test is again conservative for both event rates. Inflation of the size of most of the other tests, including the LR test, occurs with the high event rate. Such inflation is not evident when there are more than 100 events (data not shown).
Aside from minor issues, all of the empirical type 1 errors of the nine tests are close to the nominal, twosided 5% significance level. We are therefore justified in performing power assessments of all the tests.
Power
We report power results for an early effect and then for a late effect. Each category is subdivided into low and high event rates.
Early effect
Low event rate
Figure 4 shows the power for eight tests as a function of the mean IF.
We have excluded the LRL test because its power is low in this setting and its results reduce the legibility of the plots. When IF <0.8, five tests are superior to the logrank: J, C, WC, SupLR and VWLR2. When IF >0.8, the treatment effect approaches PH. Here, the WC, J and VWLR tests are weakest; the other five tests perform about the same. Irrespective of the IF, the LRE test is slightly better than the LR.
Figure 5 displays the results in a different way. For each of the 7+7=14 sets of simulated datasets, the test with the largest empirical power is identified. This â€˜bestâ€™ result among all nine tests serves as the benchmark power and constitutes the horizontal axis of each plot. Note that the test which performs best is not necessarily the same for all 14 sets of datasets.
For a given test, we define the deficit to be the mean difference over the 14 datasets between the power of the test and the benchmark. The deficit for each test is plotted as the dashed horizontal line parallel to the solid line of identity representing the benchmark.
According to the deficit metric, the five best tests (with the deficit in parentheses) are C (0.010), SupLR (0.011), VWLR2 (0.017), J (0.018) and WC (0.040). The LRL test is by far the worst performer here (deficit 0.610).
High event rate
Plots for the high event rate equivalent to Figs. 4 and 5 are 6 and 7, respectively.
A slightly different pattern emerges. With IF <0.8, all tests (except LRL, excluded) outperform the LR. With IF >0.8, two tests are worst: J and WC.
All tests except LRL have broadly similar deficits, with that for the LR test being the largest (0.068). The VWLR and VWLR2 tests now perform about the same.
Late effect
Low event rate
Plots equivalent to Figs. 4 and 5 are 8 and 9.
When IF >0.2, five tests dominate: J, WC, LRL, VWLR, VWLR2. The C and SupLR tests are consistently worse than LR. When IF <0.2 (near PH), the worst three tests are J, WC, LRL.
The deficits show a clear picture. The best five tests are J (0.022), WC (0.024), LRL (0.017), VWLR (0.008), J (0.016). The C (0.129), SupLR (0.151) and LRE (0.109) tests are all worse than the LR (0.097).
High event rate
Plots for the high event rate equivalent to Figs. 8 and 9 are 10 and 11, respectively.
The results are broadly similar to those for the low event rate.
Summary
The power results for the four subcases (early/late effects by low/high event rates) are summarized in Fig. 12.
Results for PH treatment effects (represented by late effects with t^{âˆ—}=0) have been extracted separately, and deficits for late effects exclude them. For a given type of treatment effect, we subjectively defined as acceptable a test with a mean deficit across low and high event rates of <0.05. Acceptable tests and their mean deficits are summarized in Table 5.
If an early effect is expected, acceptable tests are SupLR, C, VWLR2, J, LRE and WC. If a late effect is expected, acceptable tests are LRL, VWLR, VWLR2, WC and J. Under PH, acceptable tests are LR, LRE, VWLR, C, VWLR2 and SupLR.
If the expected type of treatment effect is unknown, our preference is for VWLR2, since it is the only test that is acceptable with all three types of treatment effects. Its maximum deficit of 0.031 across the six individual results is the smallest among the nine tests and the only one that is <0.05. VWLR2 also has the smallest overall mean deficit (0.021). On this criterion the secondbest test is J, with a maximum deficit of 0.080 (mean 0.041). However, J performs poorly under PH. See also Fig. 12.
Example
We exemplify the performance of the nine tests with three RCTs, chosen because they appear to show early, late and PH treatment effects. PATCH1 [20] concerns treating cellulitis of the leg. UKCTOCS [21] is a trial of screening for ovarian cancer. For illustration, we have combined the two research arms (different screening modes) into a single arm (screenees). RE01 [22] compares palliative treatments in advanced kidney cancer. Table 6 gives basic information on the studies.
Figure 13 shows KaplanMeier curves for the three datasets.
In Fig. 13 note the large sample size and very low event rate in b, the UKCTOCS ovarian cancer screening trial.
Table 7 shows the p values for the treatment effect on applying the nine tests to the trial datasets.
It is striking that, in each dataset, the tests deemed â€˜acceptableâ€™ for the corresponding type of treatment effect have the lowest p values (shown in bold type) among the nine.
In PATCH1, the â€˜standardâ€™ test, LR, just misses significance at the conventional 0.05 level. All other tests except LRL are significant. In UKCTOCS, only the acceptable tests for a late effect are significant. In RE01, all the tests are significant, while LR has the smallest p value.
Are alternative tests ready for the primary analysis?
Based largely on our simulation results, we have recommended VWLR2 as a good choice of resilient test under prior ignorance of the characteristics of the survival curves. However, as with all simulation studies, only a tiny fraction of possible types of survival curves has been explored. For example, a recent editorial in the context of cancer trials [23] (see their Figure A1) demonstrated anomalous behaviour of the LRL test. The authors showed a constructed example in which the experimental arm survival curve always lay below the controlarm curve, whereas the lateeffect (LRL) test rejected the null hypothesis in favour of the experimental arm. Such a result seems to contradict common sense. However, it can be understood in terms of the conditional survival distributions that manifest after the initial steep drop in survival in the experimental arm. Details will be explored elsewhere.
For weighted logrank tests and versatile tests that include them, several strands of supporting research evidence are lacking before the tests may be regarded as serious candidates for practical use in trial design and analysis. For example, we need more comprehensive examples of their characteristics under different nonPH patterns, possibly including a more detailed and nuanced understanding of the effect of the FlemingHarrington parameter values on test power.
For alternative tests in general, we need to know how to use them in the primary analysis, how to assess data maturity (readiness to analyse accrued data) and also how to perform intermediate analyses for benefit or lack of benefit.
We conclude that although useful progress has been made, much needs to be done before any proposed alternative tests are considered ready for the primary analysis of trial data.
Discussion
Based on our extensive simulation study, we conclude that the modified VWLR test VWLR2 is probably the best general choice among the nine tests we have compared when the form of any difference between survival curves cannot be predicted reliably in advance. We would argue that this would hold true in many cases. The VWLR2 test has an advantage over the combined (C) and weighted combined (WC) tests in that the distribution of the test statistic is known under the null hypothesis S_{0}(t)=S_{1}(t). An important question is whether the simplified type of nonPH we have investigated here (see further remarks below) is general enough to enable a broader recommendation. This question can really only be addressed by both performing further simulation studies with a wider range of alternative hypotheses and comparing the preferred test(s) with others when applied to a varied spectrum of datasets from real RCTs.
In an unpublished research report posted online (see https://arxiv.org/abs/1909.09467v1) after our manuscript was submitted to Trials, Lin and colleagues [24] came to conclusions broadly similar to ours. They also used Monte Carlo simulation to study the performance of nine tests, of which only LR, LRL and LRE overlapped our set. The six additional tests were a weighted logrank test with index (1,1); a versatile weighted logrank test with four components (MaxCombo, identical to the present VWLR test except that it also includes the (1,1) test); difference in restricted mean survival time; Breslowâ€™s test; weighted KaplanMeier test; and Leeâ€™s combination test. Besides PH and a range of earlyeffect and lateeffect nonPH examples, they investigated scenarios with crossing survival curves. Consistent with us, they summarised by stating â€˜There is not a single most powerful test across all scenarios. In the absence of prior knowledge regarding the PH or nonPH patterns, the MaxCombo test is relatively robust across patterns.â€™ In both our and their investigations, a composite weighted logrank test seems to perform well.
In a recent analysis [2], we compared the combined test (C) with the Cox test (very similar to LR). We found results in favour of the combined test in an analysis of datasets reconstructed from the published KaplanMeier survival curves in 50 phase III RCTs. The trials, which were reported in four leading medical journals in 2013, were in a variety of medical research areas. In this particular sample of trials, graphical analysis suggested that significant treatment effects were mostly nearPH or early in nature, clear late effects being rare. However, in some areas of medical research, for example immunooncology and screening and prevention trials, late effects are often anticipated. Thus, the potential to detect late effects remains important.
As with all simulation studies, due to the inevitable restriction on the numbers and types of scenarios that may be investigated, interpretation and generalizability of results require caution. We have limited our early and late scenarios to piecewise constant HRs with a single change point placed at different time points (see Fig. 3 for the corresponding survival functions). The full range of possible early or late effects is not and cannot be represented. However, our approach allowed us to study how the position of the change point in the HR in relation to the controlarm information fraction affected power, taking the LR test as the benchmark. This turned out to be an important consideration (see Figs. 4, 6, 8, 10). The five tests we identified as most powerful for an early effect were superior only when the change point was at IF âˆ¼0.8 or smaller. The equivalent condition for detecting a late effect was IF âˆ¼0.2 or larger. Whether such a characterization is of practical help in selecting a test prospectively when designing a new trial needs further exploration.
A major issue we have not considered here is how best to describe and estimate treatment effects under nonPH. Hitherto, standard practice has been to use a test and an estimate of the treatment effect, together with its confidence interval (CI), that are coherent. This is perfectly reasonable under PH, when the null hypothesis concerns the HR, and the latter is a design parameter which is meaningful and independent of followup time. Many earlier trials, for example some in oncology with simpler research regimens, were reasonable candidates for PH and were possibly too small to detect important nonPH except in rare cases. Today, treatments are more complex, sample sizes are often large and followup is sometimes of necessity long (e.g. in screening trials for relatively rare conditions). Consequently, the chance of encountering nonPH is much larger than before. It may be argued that what is needed is a resilient test and, not necessarily coherent with it, relevant measures to help describe and interpret the treatment effect. Of the tests we have studied here, several are constructed from more than one component and therefore have no obvious associated estimate.
In the case of nonPH (and, arguably, even of PH), no single summary measure can adequately capture the treatment effect. One is left with careful inspection of the estimated survival curves in order to judge the clinically relevant nature and magnitude of the treatment effect. Investigation of the related topics of estimation and interpretation is beyond the scope of the present paper. We shall discuss these topics in a later paper.
We also note that some people object to tests that place more weight at certain times compared with others, for example by placing more weight on a later event. Such weighting schemes may imply that having a late event is worse than having an early event. When the survival curves cross, it can even happen that LRL rejects in favour of one treatment arm and LRE in favour of the other arm. Our view is that we are testing whether the two survival curves are equal. If we conclude that they are not, and the curves cross, the preferred treatment will depend on individual preferences regarding the tradeoff between early versus later risks.
A key question potential users will ask is how high a cost (i.e. increase in sample size) is incurred under PH when using a test other than the standard logrank. Our results on power deficit (see Fig. 12) do address this issue, but further simulations, for example with different controlarm survival distributions and event rates, would certainly be desirable.
Where do we go from here? The only test that has been extensively researched, implemented, validated and used in a multitude of trials is the logrank. Furthermore, monitoring trial maturity and hence determining when the trial is ready to analyse is straightforward, requiring only the cumulative number of events. Under nonPH and using a different test, how best to assess maturity is an open issue. Further experience with the power of a test in different nonPH situations is needed.
How would a test be used in practice? Stata software is (or will soon be) freely available to perform all nine tests investigated here, and power/sample size calculations have been implemented for some of the nonstandard tests, e.g. those in [25, 26] for the combined test (C), and will soon be made available for the preferred test, Ka2. Clearly, a preferred test would have to be specified up front in the study protocol for use in the sample size calculations and in the primary analysis. We stress the need to avoid defective statistical practice, such as performing a logrank test first and finding it to be â€˜nearlyâ€™ significant, followed up by (say) a combined test to try to obtain more power and â€˜achieveâ€™ the magic p<0.05. How to implement appropriate guidance (e.g. stopping rules) for benefit or lack of benefit at interim analyses when using an alternative test also requires investigation.
Conclusions
On present evidence, our test of choice is VWLR2. The recommendation assumes ignorance of the type of treatment effect to be expected. Several tests performed well when the correct type of treatment effect was assumed: SupLR, C, VWLR2, J, LRE and WC with an early effect; LRL, VWLR, VWLR2, WC and J with a late effect; and LR with a PH or nearPH effect. A low controlarm event rate reduced the power of weighted logrank tests targeting early effects. Test size was somewhat inflated with a high event rate and less than about 100 events in the dataset. The results must be regarded as initial. Further investigation of test characteristics with different types of nonproportional hazards of the treatment effect may be required.
Availability of data and materials
The datasets used to provide controlarm survival distributions as the basis of simulation studies are available from the corresponding author on reasonable request.
Abbreviations
 C:

Combined
 HR:

Hazard ratio
 J:

Joint
 LR:

Logrank
 LRE:

Logrank (early)
 LRL:

Logrank (late)
 MRC:

Medical Research Council
 NonPH:

Nonproportional hazards
 PH:

Proportional hazards
 RCT:

Randomized controlled trial
 SupLR:

Supremum logrank
 UCL:

University College London
 VWLR2:

Modified versatile weighted logrank
 VWLR:

Versatile weighted logrank
 WC:

Weighted combined
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Acknowledgements
We thank Ian White for helpful comments on the manuscript.
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The research was supported by the Medical Research Council programmes MC_UU_12023/21 and MC_UU_12023/29.
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PR and MKBP drafted the article. PR devised and performed the statistical analysis and prepared the manuscript, including figures and tables. Both authors read and approved the final manuscript.
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Both authors are biostatisticians. MKBP is the director of the Medical Research Council (MRC) Clinical Trials Unit at University College London (UCL) and of the UCL Institute for Clinical Trials Methodology. PR is a senior scientist in the same unit and a professor of statistics at UCL.
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Royston, P., B. Parmar, M.K. A simulation study comparing the power of nine tests of the treatment effect in randomized controlled trials with a timetoevent outcome. Trials 21, 315 (2020). https://doi.org/10.1186/s1306302041532
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DOI: https://doi.org/10.1186/s1306302041532