Morris, TP

Quartagno, M

White, IR

English

UCL Discovery

Research LetterCLINICALTRIALSClinical Trials1–3 The Author(s) 2021Article reuse guidelines:sagepub.com/journals-permissionsDOI: 10.1177/17407745211045124journals.sagepub.com/home/ctjWhy restricted mean survival timemethods are especially useful fornon-inferiority trialsMatteo Quartagno , Tim P Morris and Ian R WhiteOur attention was recently captured by the paper fromFreidlin et al.,1 in which the authors investigate theconditions under which testing non-inferiority withtime-to-event data defining the margin as a differencein restricted mean survival time (DRMST) leads tohigher power than defining it as a hazard ratio (HR).We agree with the authors that there is no magic inDRMST, and that it is important to clarify when, andwhy, one method is advantageous over the other. Theauthors have addressed the when, by providing a simu-lation study that indicates that DRMST is more power-ful with low event rates, limited follow-up and largenon-inferiority margins. This is a welcome addition toprevious simulation studies that had either suggested ageneralised power advantage of DRMST2 or shown dif-ferences but without investigating them further;3 how-ever, the issue of clarifying why such differences ariseremains.There are few reasons why using HR may be advan-tageous over DRMST: first, DRMST discards data onfollow-up after t, as one of the scenarios in Freidlinet al.1 was designed to show. Second, it is compromisedby loss to follow-up before t. More generally, whenestimated non-parametrically, DRMST is less efficientthan HR, which is generally estimated through semi orfully parametric models under the proportional hazardsassumption. It is therefore not surprising to see anadvantage in terms of power for HR in certain settings.However, in several scenarios, and in particular withlarge non-inferiority margins and low event rates, con-clusions are reversed so that DRMST has a poweradvantage, and the reasons for this phenomenon arenot well understood.We believe the concept of non-inferiority frontiers,which we introduced in a recent paper,4 helps to explainthe why. The fundamental reason for the difference inpower between DRMST and HR methods is that thenull hypotheses are not the same, even if we make thenon-inferiority margins match, as was done in Freidlinet al.1 and Weir and Trinqart.3 This is because the nullhypotheses are actually curves in space or, as we calledthem, frontiers, rather than single points, which are sim-ply used as assumptions for the purpose of designing afrequentist trial.Figure 1 gives graphical intuition for this point.Figure 1(a) shows the non-inferiority frontiers corre-sponding to DRMST- and HR-based tests in a simula-tion scenario similar to the first in Freidlin et al.1 Thedashed line represents the line of treatment equality,the hollow dot represents the expected control eventrate, and the cross is the corresponding frontier point,that is, the non-inferiority margin if the expected pointwas correct. The turquoise (HR) frontier passes closerto the expected point than the navy (DRMST) frontier,and hence requires a larger sample size to concludenon-inferiority. A similar phenomenon happens withbinary outcomes where, for low event rates, the frontiercorresponding to a risk ratio margin passes closer tothe expected point than one based on a risk differencemargin, and hence implies larger sample sizes.A larger event rate or a smaller margin changes thegraph, as shown in Figure 1(b) and (c), respectively, sothat the different frontiers are much more similar nearthe expected point and the other differences we listedabove give HR the edge over DRMST. EstimatingDRMST by fitting a Cox model could eliminate theremaining differences in favour of HR in these settings,making DRMST always at least as powerful as HR.Nevertheless, this should not be taken to mean that allnon-inferiority trials should be designed usingDRMST.Since different population-level summary measuresimply different null hypotheses, we believe the choiceshould be driven initially by clinical considerations andMRC Clinical Trials Unit, Institute for Clinical Trials and Methodology,University College London, London, UKCorresponding author:Matteo Quartagno, MRC Clinical Trials Unit, Institute for Clinical Trialsand Methodology, University College London, 90 High Holborn, SecondFloor, London WC1V 6LJ, UK.Email: m.quartagno@ucl.ac.uklater tempered by statistical considerations. This is wellrecognised in certain areas: for example, for vaccinenon-inferiority trials, even though, for low infectionrisk, defining the non-inferiority margin as a risk differ-ence would give much greater power, the margin is usu-ally defined as a ratio, because a relative populationsummary is more meaningful (and transportable) insituations where baseline risk varies. The rest of this let-ter details the methods we used to produce the figure.DetailsLet l1 and l0 be the unknown event rates in the experi-mental and control arms, assumed constant. Let le bethe expected event rate in the sample size calculation,assumed the same in both arms. Let lf be the event ratein the experimental arm used to specify the non-inferiority margin: that is, if l0 = le, then non-inferiority means l1 \ lf. Thus, on the HR scale, thenon-inferiority margin is lf / le. Allowing the controlarm event rate to be unknown, this margin impliesnon-inferiority if l1 / l0 \ l f/ le. This equation relat-ing the unknown l1 and l0 is the non-inferiority fron-tier on the constant-HR scale.For constant event rate l and fixed horizon t, therestricted mean survival time (RMST) to time t isr(l) = (1 2 e2lt) / l. The non-inferiority frontier onthis scale is r(l1) 2 r(l0) = r(lf) 2 r(le), where r(lf) 2r(le) is the non-inferiority margin on the DRMST scale.We draw the non-inferiority frontier for three set-tings. As a base case to illustrate the settings whereDRMST shows benefit, we specify moderate eventfractions and a large margin by le = 0.5, lf = 0.75and t = 1. This implies the non-inferiority margin is1.5 on the HR scale and 20.083 on the DRMST scale.We plot the non-inferiority frontiers for l0 rangingfrom 0 to 1, with values of l1 found iteratively in eachcase. To illustrate the settings with large event frac-tions, we change t to 3. To illustrate the setting with asmall margin, we change lf to 0.55. To graphicallyexplain the greater distance of the DRMST frontier, weadditionally show a possible confidence region for con-trol and experimental rate from a hypothetical trialwhere the observed control rate matches exactly theexpected one and the results are on the border of signif-icance on the HR scale, that is, where the confidenceregion just touches the HR frontier; the region doesnot reach the DRMST frontier, hence, non-inferioritycould be concluded on the DRMST scale, but not usingHR.Declaration of conflicting interestsThe author(s) declared no potential conflicts of interest withrespect to the research, authorship and/or publication of thisarticle.FundingThe author(s) disclosed receipt of the following financial sup-port for the research, authorship and/or publication of thisarticle: This work was supported by the Medical ResearchCouncil (MC_UU_12023/29).Figure 1. L’Abbé plot showing the non-inferiority frontiers on the DRMST and HR scale for three hypothetical trial scenarios ((a)base: moderate event fractions; large margin, (b) large event fractions and (c) small margin). The circle represents a possibleconfidence region for the joint distribution of control and experimental event rates for a trial for which the estimated control ratematches the expected one and the estimated active rate is as in the alternative hypothesis.2 Clinical Trials 00(0)ORCID iDsMatteo Quartagno https://orcid.org/0000-0003-4446-0730Tim P Morris https://orcid.org/0000-0001-5850-3610Referenes1. Freidlin B, Hu C and Korn EL. Are restricted mean sur-vival time methods especially useful for noninferioritytrials? Clin Trials 2021; 18: 188–196.2. Uno H, Wittes J, Fu H, et al. Alternatives to hazard ratiosfor comparing efficacy or safety of therapies in noninfer-iority studies. Ann Intern Med 2015; 163: 127–134.3. Weir IR and Trinquart L. Design of non-inferiority rando-mized trials using the difference in restricted mean survivaltimes. Clin Trials 2018; 15: 499–508.4. Quartagno M, Walker AS, Babiker AG, et al. Handlingan uncertain control group event risk in non-inferioritytrials: non-inferiority frontiers and the power-stabilisingtransformation. Trials 2020; 21: 145.Quartagno et al. 3

Why restricted mean survival time methods are especially useful for non-inferiority trials

https://discovery.ucl.ac.uk/10135064/3/2021%20-%20Quartagno%20-%20why%20rmst%20are%20especially%20useful%20for%20ni%20trials%20lte%20-%20clin%20trials.pdf

Why restricted mean survival time methods are especially useful for non-inferiority trials

Abstract

Similar works

Full text

Available Versions

UCL Discovery