Percent maturity ( pmat ) and power curves as a function of t Sample size calculations are potentially fragile, since they depend strongly on assumptions. k In reality, of course, the change is an illusion caused by ignoring the time element in reporting the results. Section ‘A strategy for design and analysis of clinical trials discusses our proposed strategy for trial design and analysis. We see that t 1 ∗ in treatment group j and n It needs to be accompanied by other statistics, such as the estimated median survival times and/or the survival probabilities at specific time(s), or indeed the RMST. Vertical lines show 1 We finish with a discussion and our conclusions. e 2) and the other ART design parameters, finds j is estimated by the delta method within a flexible parametric model [10, 11], using the stpm2 program [12] for Stata. ∗. ̂ The instantaneous hazard rate is the limit of the number of events per unit time divided by the number at risk, as the time interval approaches 0. ̂ For example, the survival function for t ∈ (0,τ The results are needed in the sample size calculations. We suggest determining the design value, 2002, 21: 2175-2197. As always with trial design, the key assumptions of data structure and relevant parameters critically affect the required sample size, and some kind of informal sensitivity analysis should always done. Owing to the current dominance of the HR and its presumed time independence, trial reports often ignore the possibility of non-PH and typically place little emphasis on the extent of follow-up, which should be a key aspect of the trial design and analysis. 2 For the non-PH designs, the sample size is 27 to 42 percent larger for the logrank than the RMST approach. ∗) (i = 1,…,m), are an independent, identically distributed sample from some distribution.  are known as knots. ∗ Define the allocation ratio as r = n … Mok TS, Wu YL, Thongprasert S, Yang CH, Chu DT, Saijo N, Sunpaweravong P, Han B, Margono B, Ichinose Y, Nishiwaki Y, Ohe Y, Yang JJ, Chewaskulyong B, Jiang H, Duffield EL, Watkins CL, Armour AA, Fukuoka M: Gefitinib or carboplatin - paclitaxel in pulmonary adenocarcinoma. 1996, 334: 1-6. We are grateful to M. Sydes and two reviewers (Y. Wang and S. Wacholder) for helpful comments that have helped us to strengthen the manuscript. It calculates the sample size by simulation according to the methods described in sections ‘Sample size for RMST difference’ and ‘Standard error of RMST in the ART setting’. des If too many knots are specified, the model can be over-fitted and the parameter estimates correspondingly unstable. It has become apparent in some recently reported trials, e.g. EXAMPLE Kaplan-Meier estimates and summary statistics were pre-pared using the following fictitious survival time data, with the does not depend on the treatment effect observed in the data, but on the designed difference in RMST and its observed variance as functions of t The main advantages of our proposed method are interpretability of the RMST difference from a clinical perspective as loss of life expectancy (when the outcome of interest is mortality), and robustness of the estimator to the proportional hazards assumption. e 2], (τ ∗, whereas Δ is a design value. Usually, the HR and its CI are reported, and often, the median survival time and/or estimated survival probabilities at fixed time points are also presented. Stat J. For both logrank and RMST-based sample size calculations, we fix K We also investigate sample size for an alternative design based on non-PH of the treatment effect. , since it does not reflect the increased uncertainty associated with censoring. j+1] (j = 0,1,…,k) is. k Δ ∗ To minimize the sample size, the PH design requires a 2) = (3,8) yr and compute n according to eqn. The y-axis is the proportion of subjects surviving. 2 in the interval (τ Under PH, for example, the HR can usefully be applied to the survival function in the control arm to obtain an impression of the survival curve in the research arm. For OE02, however, the z-statistic from the Cox model is fairly constant over time, whereas for the RMST tests it diminishes steadily. Details of the calculations and the results are given in the Appendix: RMST and RSDST for a piecewise exponential distribution. 2003, 97: 1663-1771. The significance levels are close to nominal for the logrank and RMST tests in both scenarios. The values of RMST in each trial arm are absolute measures of survival time. A graph of power or maturity against t ∗ near , as RSDST/ t The basis of our approach is the familiar comparison of two means using an unpaired t test. ∗, but otherwise ϕ is unknown. Long-term outcomes after percutaneous coronary intervention (PCI) with contemporary drug-eluting stents, as compared with coronary-artery bypass grafting (CABG), in patients with left main coronary artery disease are not clearly established. Lancet. Section ‘Further issues’ makes a qualitative comparison between various measures of a treatment effect and describes results of RMST and logrank analyses in four cancer trials. A central tool in the approach is the realistic representation of the survival function in each trial arm as a piecewise exponential distribution. ∗ The mean time from initial symptoms to diagnosis in the overall cohort (N = 2,635) was 31 months, and the median time was 12.8 months (range, −5.5 to 654.9 months). t 2 (the maximum possible follow-up time of any patient within the design), and computing n by simulation, as described above. ’s The alternative hypothesis is H Let T be a continuous random variable with cumulative distribution function F(t) on the interval [0,∞).Its survival function or reliability function is: = ({>}) = ∫ ∞ = − ().Examples of survival functions. Piecewise exponential models are needed here, since they make it easy to specify the model in terms of survival probabilities and hazards and provide analytic expressions for the RMST and RSDST. Two standard errors of an estimated probability of 90 and 5 percent are 0.85 and 0.62 percent, respectively. This article is published under license to BioMed Central Ltd. In the section ‘Restricted mean survival time (RMST)’, we describe the RMST and the corresponding standard deviation (RSDST) in general terms and specifically for a piecewise exponential distribution. ∗ for PH (solid lines) and non-PH (dashed lines) trial designs. Let h Restricted definition is - subject or subjected to restriction: such as. PubMed  des The integrated survival function from 0 to t 2 = 1 yr. / 2 are measured in trial time, i.e. 1 - μ The logrank and RMST tests of the treatment effect give P-values with a similar interpretation. However, to our knowledge flexible parametric models cannot be used on their own to design a trial. Other than the need to define a suitable time horizon t p Correspondence to z ∗. ∑ 1 : μ j 2 One way is to estimate the underlying true distribution. As more data accumulate, pmat increases; when it reaches 100%, the data are ready for analysis (under the assumptions of the design). des Working off-campus? A test the null hypothesis Δ = 0 is made by comparing Suppose we are sampling at random from the distribution of a positively bound random variable, T. We sample n Article  from trial data. Given estimates of ϕ 0 i 0 : μ For j = 0,1,…,k the interval duration δ The estimated variance of the treatment effect provides a way forward. μ Furthermore, the HR depends on the follow-up time. des In our earlier paper [1], we suggested reporting the RMST and its difference between trial arms, with a CI. ̂ In general, statistically significant differences in RMST from randomized trials may appear ‘small’, but they may be more realistic and clinically meaningful than superficially more impressive relative effects on the hazards. Note that RSDST is a known function of the design parameters (see Appendix: RMST and RSDST for a piecewise exponential distribution) and does not need to be estimated by simulation. 1 . 0 The absolute difference in survival and the difference in median survival time, although often quoted, are weak because they represent only a ‘snapshot’ of the difference in survival functions. Additionally, we discuss how to set the specific time point to define the RMST from two main points of view. Assuming no dropout, therefore individual patients are followed up for at least 3 years and at most 8 years, depending on when they entered the trial. where ϕ is some positive scaling factor. (j = 0,1) may be calculated by the delta method separately in each arm. t 20). Cookies policy. We also address the question of how to assess data maturity, i.e. ∗ belongs to interval (τ For example, for t 1 Andersen PK, Perme MP: Pseudo-observations in survival analysis. If you do not receive an email within 10 minutes, your email address may not be registered, 2 = 3 yr. We vary ,τ μ Further insight into the behaviour of the logrank and RMST tests is provided by Figure 3. 1 This could be regarded as an assumption under the null hypothesis of Δ = 0, since there is then no difference between treatments. Further experience with the RMST measure in a larger number of trials has given us the impression that when the PH assumption is approximately satisfied, the test of the null hypothesis based on RMST difference often has operating characteristics similar to the logrank test. They tell us little about the previous or subsequent survival experiences. σ Estimates were restricted to the median follow-up time of the RECOURSE trial. ∗ values. z Flexible parametric models are unsuitable for exploring hypothetical RMST values associated with a design with given hazard ratio(s) and control arm survival function. Statistical Considerations for Sequential Analysis of the Restricted Mean Survival Time for Randomized Clinical Trials. For a given trial dataset, SE Here, the time to event was reported as the restricted mean survival time. PubMed Google Scholar. j The upper limit of the integral above is often finite because … ∗, the RMST is estimated by integration as in (1) and the RSDST as and SE j final Restricted mean survival time (RMST) for a mortality outcome in a trial may loosely be described as the life expectancy over the restricted period between randomization and a defined, clinically relevant time horizon, usually called t ∗. For the PH designs, the sample size is about 8 percent larger with the RMST approach than with the logrank approach. e However, HR patterns between treatments whose modes of action differ (e.g. A standard approach to analysis would be to assume PH, test the null hypothesis of no treatment effect using the logrank test, and estimate the HR in a Cox model with randomized treatment as the only covariate. ∗ is measured in analysis time, with each patient’s date of entry as the origin (t = 0). Figure 1 shows the resulting sample sizes for both designs. non-PH). We report a small simulation study comparing the significance level and power of the logrank and RMST tests under a piecewise exponential model with non-proportional or proportional hazards, incorporating staggered entry of patients and varying length of recruitment and follow-up. . The alternative hypothesis is that they are as given (implicitly or explicitly) in step 6 above. Stat Med. Schemper M, Wakounig S, Heinze G: The estimation of average hazard ratios by weighted Cox regression. Δ surgery vs chemotherapy, or targeted agent vs conventional therapy) may be hard to predict. We used specially written Stata software that, given (K By using this website, you agree to our Note that in the ASTEC trial, mortality in the research arm is actually non-significantly worse than in the control arm. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Our strategy is to take the standard case described above as our starting point for RMST sample size calculations based on the ART design assumptions, and modify it as necessary. final The set-up is similar to that described in the section ‘Comparing RMST and logrank based sample sizes’, except that we vary the recruitment period (K A simple example of departure from PH occurs when one group is assigned to immediate surgical treatment and the other to medical treatment. k+1 in a categorization (τ estimated from the current data with the target value, Δ It equals the area under the survival curve S (t) from t = 0 to t = t ∗ in 30 equal-sized steps between 1 and 10.1017/S0022172400014443. Number of times cited according to CrossRef: Restricted Mean Survival Time Estimation: Nonparametric and Regression Methods. The ART-based approach to trial design defines a recruitment time (K t - in calendar time whose origins are the dates of randomization of the first and last patient, respectively. (3) implicitly requires the variance of RMST under the piecewise exponential model around which ART is based. ̂ Note that var Although not explicit, the implication is that the main trial result would be reported as the HR with a confidence interval (CI). 1(t) in the control and research arms, respectively, the difference in RMST between arms, Δ, is given by. Restricted mean survival time (analogous to the area under the curve for a survival plot) was assessed with the use of the pseudo-mean values approach. S = 0 Note that the only components of the sample size calculation that change with recruitment (K ∗ = 4 yr, might be preferred; this has power and maturity slightly reduced to about 80 and 81 percent, respectively. ) See also Royston et al’s [21] proposed graphical comparison of observed and imputed times to event between trial arms, which carries a similar message. max. des The restricted standard deviation (RSDST) is Restricted mean survival time: an alternative to the hazard ratio for the design and analysis of randomized trials with a time-to-event outcome. The approach quantifies areas under Kaplan-Meier curves and compares different treatments by using the restricted mean survival time, ... differed with and without toxicity is also worthy of investigation and would provide greater insight into a functional definition of toxicity for integration … At its simplest, the method accepts a single exponential distribution in each of the control and research arms, characterized by a single, constant hazard or equivalently by the median time to event. A hazards model with a time-dependent treatment effect suggests that the hazard ratio is below 1 near t = 0 and tends to approximately 1 over time. Median and mean survival time. We consider determining Let Δ = 0 and Δ = μ We have described the calculation of two primary values of t To be clear, we remind the reader that t 2 = 8 - K The median survival time is calculated as the smallest survival time for which the survivor function is less than or equal to 0.5. where var ∗ is defined as . We do this by Monte Carlo simulation, as follows. Restricted mean survival times were calculated using the standardized survival package “stpm2_standsurv” . We call this value The sample size is designed to give the test of the RMST difference power of 90 percent to reject the null hypothesis at the 5 percent level. and 2008, 100: 92-97. In exploring simulated trial data with staggered entry of patients and a fixed follow-up time, we found that ϕ was very close to 1 when patients were recruited over a relatively short period and followed up for a reasonably long length of time. 0 Google Scholar. Notionally, the RMST μ may be estimated as the sample mean j For example, the assumed survival distribution may be wrong, or the pattern of recruitment and follow-up may be at variance from that expected. Analytic results for RMST and RSDST are available when the survival time has a piecewise exponential distribution. The designs assume recruitment over K For example, the survival curves could cross at the median or at some other t restricted mean survival time is a robust measure that represents the mean event-free survival time in a prespecified period.29,30 The statistical analyses were performed with Stata version 16.1 (StataCorp LP, College Station, TX, USA) and included the use of the stpm2 28 program and R version 3.6.3 (R Foundation for Statistical Computing, Vienna, Austria) and the twang 26 package. 1,…,X ∗. In a previous report [6], we described the implementation of a general method, Assessment of Resources for Trials or ART, for designing a trial allowing for possible non-uniform accrual rates, non-proportional hazards, loss to follow-up and cross-over of patients between treatment arms. j It is data-driven only with respect to the variance of the RMST difference. 3. More technically, we are unconvinced by papers such as Schemper et al [2] where an overall estimate of the HR is regarded as an average of time-dependent HRs over the event times, nor by proposed variants based on different and arbitrary weighting schemes. We demonstrate the use of restricted mean survival time and a test of the difference in restricted means as an alternative measure of treatment effect. The former is driven by the theoretical structure of the design and the latter by the trial data as recorded. (i.e. It may be used to design a trial with two or more parallel groups and a time-to-event outcome. When it comes to the analysis of the trial data, for various reasons the precise data structure that is obtained may differ from the design under which In this paper, we consider replacing a logrank-based sample size calculation and presentation of results with one based on RMST and its difference between trial arms. First, we draw a large random sample of m time-to-event observations from the piecewise exponential distribution of interest and determine. values are 7.5 and 4.3 yr for the PH and non-PH cases, respectively, with corresponding sample sizes of 461 and 326. Princeton's WordNet (4.80 / 5 votes) Rate this … Please check your email for instructions on resetting your password. : 2. final Flexible parametric models are suitable tools for the purpose, because, for example, a cumulative hazards model with 3 d.f. We perform a small simulation study to check the power and significance level of the proposed test of RMST difference. m OpenUrl CrossRef PubMed J Clin Oncol. 0 = 0, δ A restricted area is one that you need…. Regression models are used to obtain hazard ratios and their confidence intervals.. The sample size and events for an RMST design based on the same assumptions are 1108 patients and 848 events, with ̂ Values are compared with those from the standard approach which utilizes the logrank test. Tentatively, we believe it is. In a two-arm clinical trial with survival functions S Parmar MK, Ledermann JA, Colombo N: Paclitaxel plus platinum-based chemotherapy versus conventional platinum-based chemotherapy in women with relapsed ovarian cancer: the ICON4/AGO-OVAR-2.2 trial. j = min(T the RMST) is given by, We also need the expectation E Restricted mean survival time: an alternative to the hazard ratio for the design and analysis of randomized trials with a time-to-event outcome. ∗ and the design assumptions (PH or non-PH) both influence the sample size quite markedly. Article  ̂ MRC Clinical Trials Unit at UCL, Aviation House, 125 Kingsway, London, WC2B 6NH, UK, You can also search for this author in 10.1002/cncr.11234. ∗ but still show a substantial difference in RMST at Here, we consider ASTEC vs. OE02 in Table 6. An alternative recommended measure is the restricted mean survival time (RMST),8 which is the area under a survival curve between two time points, typically the time of randomisation and the end of the follow-up period .9 The RMST of an overall survival curve is a measure of the average duration of survival over the follow-up period.6 8 10 A treatment effect can then be quantified as the difference … nnn = ∗ for the final analysis of the data. For example, the sample mean is no longer an unbiased estimator of the population mean (of survival time). j . = , as expected. 1 = 5 and illustrate what happens with K ∗ = 5 years seen in ASTEC is arguably of little practical importance. n The restricted mean is a measure of average survival from time 0 to a specified time point, and may be estimated as the area under the survival curve up to that point. The reason is because, to a good approximation, var (logHR) is proportional to 1/e, so that e is a measure of the amount of information in the data. There are many options for extending the model if non-PH is detected. S ∗. We have chosen these particular trials because the estimated treatment effect is approximately the same (around HR = 0.85), yet they have widely varying mortality rates. ∗ 0, t ∗ 1. However, the value of t 1 The RMST difference of 0.09 years at t ,t By contrast, K A notable feature of Figure 3 is the instability of the z-statistic (hence P-value) for the tests in the truncated data, which seems greater for the Cox test. i Data sharing is not applicable to this article as no new data were created or analyzed in this study. The difference in RMST is determined by the survival functions specified in the control and research arms through piecewise exponential distributions, exactly as in ART. This implies a reduction of 25 percent in the instantaneous mortality rate at all times after randomization. An advantage of the RMST is that it is valid under any distribution of the time to event in the treatment groups, of which PH models are a (small) sub-class. Parameter estimation by maximum likelihood is straightforward. Instructions on resetting your password there is no guarantee that the SE n. Of monitoring for maturity to trials designed with an RMST outcome σ 0 2 n... Are mature enough for the treatment effect ( e.g, elected leaders have a 50 chance. When 1006 patients had been recorded as knots Heinze G: the is. The previous trial, it may not get this far, in case! Dg: Visualizing length of survival in time-to-event studies: a complement to Kaplan-Meier plots Cite this as... Value ( proportional hazards assumption ) or for individual periods ( time-dependent HR ) HR later substantially... Way forward and deserves greater attention events ) be unreliable determine t ∗ > τ K, restricted mean survival time definition )! Size calculations PC, royston P: further development of flexible parametric models are used to the. Atrisk= option assumed magnitude and pattern of the treatment effect can be accessed here: http: for... Ph and non-PH designs ratios and their confidence intervals through the standard case, the definition of piecewise exponential with... 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Target HR is a complete definition of RMST and RSDST and to extensions which time-dependent! Is that an average HR is uninterpretable HR depends on the logrank test, of. Non-Parametric but the drawback of being non-parametric but the drawback of being relatively slow to compute presumably the behaviour the... For Sequential analysis of clinical trials for practical purposes and/or recruit more patients over a period... Out may be seen in the current data assessed using the standardized survival package “ stpm2_standsurv.!, e.g and/or recruit more patients ∗ at or near the maximum permissible is needed: Stata Press element reporting... An absolute measure, is that they are not sufficient, it is data-driven only with respect to the and. Its slowness makes it cumbersome when simulation with many replicates is needed see Reference [ 20 ].. > τ K vector are allowed discusses our proposed strategy for design analysis... Maturity ’ of equal duration, giving an allocation ratio as r = n 0... 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( 2013 ) Cite this article with your friends and colleagues Lambert PC: flexible parametric models survival. Arm as a piecewise exponential distribution the treatment effect ( e.g at t ∗ and the and! Depend strongly on the assumed magnitude and pattern of the 36-month period is fast and efficient queries other. This point is given in the present paper for both estimation and simulation purposes restricted mean survival time for the... Models readily lend itself to estimating the RMST is specifically aligned to a chosen t ∗ at or the... Φ j, τ K are known as knots achieved when var Δ ̂ is familiar! To each treatment arm separately appears to give an adequate fit to a t... A chosen t ∗ the third method, flexible parametric models are suitable tools for the PH,. Assumptions ( PH or not the latter case, at which point 691 events ( deaths ) had been.... Any supporting information supplied by the authors ’ original submitted files for.. 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Trial arm to perform a Kaplan-Meier the ART workup, the change is an important of. Sample variance of the restricted mean survival times were calculated using the ATRISK=.... All equal to 0.5 that σ 0 2 and n have ‘ Monte Carlo simulation, as just.... An assumption under the piecewise exponential model the origin of the logrank test under non-PH underpowered! Methods are broadly in agreement in the non-PH designs further development of flexible parametric models are used to a. Med Res Methodol 13, article number: 152 alternative hypothesis is H 0: μ =! Occurs when one group is assigned to immediate surgical treatment and the other to medical treatment a flexible parametric,! Events ( deaths ) had been recorded Table 6. not calculated to calculate the RMST needs to put...