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# what is restricted mean survival time

J Am Stat Assoc. The restricted mean survival time (RMST) is an alternative treatment outcome measure that can be estimated as the area under the survival curve up to a prespecified time horizon and hence can account for all survival information before that time horizon. For each replicate, the mean incremental cost, the rmstD (for each survival analysis … The aim of this article is to summarize the role of restricted mean survival time (RMST) analysis in oncology. An introduction to the bootstrap. Toh S, Hernández-Díaz S, Logan R, Robins JM, M a H. Estimating absolute risks in the presence of nonadherence: an application to a follow-up study with baseline randomization. Background: Restricted mean survival time is a measure of average survival time up to a specified time point. 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. This is an Open Access article distributed under the terms of the 2015 Feb 15;181(4):238–45. Restricted Mean Survival Time as a Measure to Interpret Clinical Trial Results JAMA Cardiol. Abstract. Prevalence and predictors of highly active antiretroviral therapy use in patients with HIV infection in the united states HCSUS Consortium HIV Cost and Services Utilization J Acquir Immune Defic Syndr 1999. We begin by examining all-cause mortality because we anticipate mortality to be an important competing event for ART initiation and to serve as an example of the implementation of the use of inverse probability-weighted RMST in a setting with no competing events. 2012;21(1):55–75. With this method, the treatment effect is defined as the difference in restricted mean between the trial arms. Three kinds of between-group constrast metrics (i.e. The subdistribution hazard ratio is closer to the null than the cause-specific. Royston P, Parmar MK. CAS  2001;286(20):2568–77. Time to initiating highly active antiretroviral therapy among HIV-infected injection drug users. Google Scholar. https://doi.org/10.1186/s12874-018-0484-z, DOI: https://doi.org/10.1186/s12874-018-0484-z. Therefore, an alternative approach known as the Restricted Mean Survival Time (RMST) or τ-year mean survival time is presented, and its ability to overcome interpretation challenges with the hazard ratio discussed. 7,17 RMST is defined as the area under the KM curve up to a specific time point (t), which is the follow-up time of the study. We consider the design of such trials according to a wide range of possible survival distributions in the control and research arm(s). 2013;32(30):5278–85. The first objective of this paper is to promote the use of the RMST in the epidemiology and medical literature by briefly reviewing its calculation, interpretation, strengths, and limitations. Article  Stat Methods Med Res. KC and CC conceived of the methodological application, performed the analysis and co-wrote the manuscript. We also found little difference in time to ART initiation while alive and retained in clinic among PWID compared to people who did not inject drugs. Invited commentary invited commentary : positivity in practice. For all estimates, we calculated 95% confidence intervals (CI) using the 2.5th and 97.5th percentiles of 10,000 non-parametric bootstrap estimates based on unrestricted random samples from the original data [44]. We further illustrate use of RMST to compare time to initiation of antiretroviral therapy (ART) among persons who inject drugs (PWID) and persons who did not inject drugs, properly accounting for persons who experienced a competing event (death or loss-to-clinic). We present the expected time not on ART while alive and enrolled in the clinic which is, $${\widehat{A}}_{J= ART\ initiation}(t)-{\widehat{L}}_{J\ne ART\ initiation}(t)$$, where $${\widehat{A}}_{J= ART\ initiation}(t)$$ is the area under the survival curve for time to ART initiation and $${\widehat{L}}_{J\ne ART\ initiation}(t)$$ is the area under the CIF for death or loss-to-clinic prior to ART initiation. © 2013 Royston and Parmar; licensee BioMed Central Ltd. the difference in RMST, the ratio of RMST and the ratio of the restricted mean time lost (RMTL)) are computed. CAS  Contains the R code used to set up the analysis, as well as an outline of how the various tables in results section were generated. The authors declare that they have no competing interests. Epidemiology. Using the restricted mean survival time approach, we evaluated the expected time to all-cause mortality and ART initiation by history of IDU among a cohort of HIV-infected patients receiving treatment in Baltimore, MD. Methods: 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. We define an individual’s weight as $${\widehat{W}}_i=\widehat{P\left({X}_i\right)}/\widehat{P\left({X}_i|{Z}_i\right)}$$, where Z is set of covariates sufficient to satisfy the assumption of conditional exchangeability between two groups, X = 1 and X = 0. There are several options for generating covariate-adjusted curves to account for confounding or non-random censoring [20]. Despite its attractive properties, epidemiological studies utilizing RMST remain rare [11,12,13,14], particularly in an observational setting (which would require adjustment of RMST for potential confounding), or in the presence of competing risks (events that preclude the event of interest from occurring). Methods: 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. PubMed Central  The restricted mean survival time is used often in the comparison of cancer treatments when the proportional hazards assumption is in doubt. Following two years of follow-up, PWID begin to experience a shorter time to all-cause mortality than people who did not inject drugs, resulting in an increasingly negative difference in RMST over time. Baseline covariate values were those measured closest to enrollment date within a window of 6 months before to 6 months after enrollment and prior to ART initiation. Everyone is familiar with the use of median survival, or more generally with median time to event (where the event could be progression or treatment failure), to … 1987 Dec;82(400):1169–76. 4. The definite integral of the survival curve from the time origin to some specified follow-up time t, where t is less than or equal to the maximum observed follow-up time, provides an estimate of the expected event-free time from 0 to t [15]. The final analytic sample contained 3044 patients. Herein, we highlight its strengths by comparing time to (1) all-cause mortality and (2) initiation of antiretroviral therapy (ART) for HIV-infected persons who inject drugs (PWID) and persons who do not inject drugs. Rogawski ET, Westreich DJ, Kang G, Ward HD, Cole SR. Brief report: estimating differences and ratios in median times to event. While ART initiation impacts all-cause mortality, it is on the causal pathway between injection drug use and mortality; because this analysis focuses on the total effect of history of injection drug use on mortality, we do not account for ART. The idea is to compare the mean survival only on a limited time horizon, like 0 to 85 years, etc. Briefly, one sufficient set of assumptions includes, in addition to assuming conditional exchangeability between the exposed and unexposed: 1) positivity (everyone has a non-zero probability of being exposed) [27, 28]; 2) treatment version irrelevance [29,30,31]; 3) no interference (one person’s exposure does not affect another’s outcome) [32, 33]; 4) no measurement error [26, 34]; and 5) correct model(s) specification. Restricted mean survival time, a robust measure that represents the mean event-free survival time in a prespecified period, may provide useful information on treatment effect that complements conventional measures of relative and absolute risk reductions. 2004;160:301–5. Lancet. The complement of the CIF provides the survival function for the jth event, S The use of RMST in our analysis to describe the difference in time not on ART while alive and in clinic facilitated the interpretation of our results by providing a clinically meaningful measure. Westreich D, Cole SR. Both the cause-specific (Fig. History of IDU may be under-reported due to social desirability bias. One could easily take the ratio rather than the difference. We consider the design of such trials according to a wide range of possible survival distributions in the control and research arm (s). The restricted mean survival time (RMST) is an alternative robust and clinically interpretable summary measure that does not rely on the PH assumption. Prognosis of HIV-1-infected patients starting highly active antiretroviral therapy: a collaborative analysis of prospective studies. A longitudinal, HIV care continuum: 10-year restricted mean time in each care continuum stage after enrollment in care, by history of injection drug use. Am J Epidemiol. J Epidemiol Community Health. Clin Infect Dis. Andersen PK. The non-parametric bootstrap was performed using 1,000 replicates and was stratified by trial to take into account data clustering. Ghali WA, Quan H, Brant R, van Melle G, Norris CM, Faris PD, et al. Stat Med. However, this includes being in a non-j event state. R: a language and environment for statistical computing. As opposed to the median, the RMST has the advantage of capturing the overall shape of the survival curve, including the so-called “right tail.” One limitation of RMST lies in the mathematical complexity of its calculation (model-dependent analysis). Herein, we highlight its strengths by comparing time to (1) all-cause mortality and (2) initiation of antiretroviral therapy (ART) for HIV-infected persons who inject drugs (PWID) and persons who do not inject drugs. Three kinds of between-group contrast metrics (i.e., the difference in RMST, the ratio of RMST and the ratio of the restricted mean time lost (RMTL)) are computed. Results: J Acquir Immune Defic Syndr 2005;38(1):96–103. However, the results of some recent trials indicate that there is no guarantee that the assumption will hold. You can get the restricted mean survival time with print(km, print.rmean=TRUE). To overcome this in a recent submission, we successfully performed the Restricted Mean Survival Time (RMST). 1,2 In this article, we discuss the advantages of an alternative analytical procedure based on the restricted mean survival time (RMST) 1,2 via 3 examples. New York: Chapman & Hall; 1993. We consider the design of such trials according to a wide range of possible survival distributions in the control and research arm (s). (3) Susarla and Ryzin’s variable upper limit … Irwin JO. Time-to-event data, Randomized controlled trials, Hazard ratio, Non-proportional hazards, Logrank test, Restricted mean survival time, Piecewise exponential distribution. Other assumptions are necessary to interpret contrasts between RMST for X = x causally, but a full discussion of causal inference is beyond scope of this paper; see [23, 25, 26] for more details. The lowest value $${\widehat{S}}_j(t)$$ may attain is bounded by the sum of the cumulative incidences of the competing events, i.e., $${\sum}_{J\ne j}{\widehat{F}}_J(t)$$, since $${\widehat{F}}_j(t)+\sum \limits_{J\ne j}{\widehat{F}}_J(t)+{\widehat{S}}^{\widehat{W}(t)}(t)=1$$. By using inverse probability weights to standardize cohorts of PWID and persons who did not inject drugs to have the same distribution of baseline covariates, we were able to estimate restricted mean times to ART initiation that are not confounded by different clinical indications for treatment. The occurrence of an event in two or more groups is often compared using a risk difference, risk ratio, or hazard ratio [1,2,3,4,5,6,7]. Hernán MA, Cole SR. The baseline median CD4 cell count was 279 (IQR = 94–480) cells/μL and the baseline median log10 HIV RNA was 4.6 (IQR = 3.9–5.3) copies/mL. Here, we describe the use of the restricted mean survival time as a possible alternative tool in the design and analysis of these trials. Restricted mean survival time: an alternative to the hazard ratio for the design and analysis of randomized trials with a time-to-event outcome. Cox DR, Oakes D. Analysis of survival data. As opposed to the median, the RMST has the advantage of capturing the overall shape of the survival curve, including the so-called “right tail.” Hogg RS, Yip B, Chan KJ, Wood E, Craib KJ, O’Shaughnessy MV, et al. Kaplan EL, Meier P. Nonparametric estimation from incomplete observations. (t) = 1 − F The National Institute of Allergy and Infection Diseases at the National Institutes of Health (Grant Numbers T32-AI102623 and P30-AI094189). The numerators of the weights were the marginal probability of being in the exposure group that was observed for that individual. Comparison of 2 methods for calculating adjusted survival curves from proportional hazards models. PubMed Central  For each trial, the ratio of restricted mean survival time (RMST) between the arms was based on reconstructed individual patient data for overall survival. Restricted Mean Survival Time – The Basic Idea. Karrison T. Restricted mean life with adjustment for covariates. Epidemiol Camb Mass. 2015;70(1):23–32. (t) or S The difference between two arms in the restricted mean survival time is an alternative to the hazard ratio. 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. 3. Egger M, May M, Chêne G, Phillips AN, Ledergerber B, Dabis F, et al. When we examine all-cause mortality by era of clinic enrollment (Table 2), the difference in time to death comparing PWID to persons who did not inject drugs in the first five years after clinic enrollment is larger in magnitude in the more recent time periods. The “restricted” component of the mean survival calculation avoids extrapolating the integration beyond the last observed time point. d The solid black line on panels b, c, and d is the lowess smoother to show the overall trend of the estimands. Illustration of a measure to combine viral suppression and viral rebound in studies of HIV therapy. 3, panel C) are variable early into follow-up but level off and are consistently below 1. Restricted mean survival time: an alternative to the hazard ratio for the design and analysis of randomized trials with a time-to-event outcome. Google Scholar. This is especially useful when comparing survival functions that diverge and later converge or cross. An application of restricted mean survival time in a competing risks setting: comparing time to ART initiation by injection drug use. $$A(t)=E\left[\min \left(T,{t}^{\ast}\right)\right]={\int}_0^{t^{\ast }}S(t) dt$$, $$\widehat{A}(t)={\sum}_k\widehat{S}\left({t}_{k-1}\left)\right({t}_k-{t}_{k-1}\right)$$, $${\widehat{W}}_i=\widehat{P\left({X}_i\right)}/\widehat{P\left({X}_i|{Z}_i\right)}$$, $${\widehat{F}}_j(t)=\sum \limits_{t_k\le t}\left[{\widehat{S}}^{\widehat{W}(t)}\left({t}_{k-1}\right)\right]\left[\frac{d_j^{\widehat{W}(t)}\left({t}_k\right)}{n^{\widehat{W}(t)}\left({t}_k\right)}\right]$$, $${\widehat{S}}^{\widehat{W}(t)}(t)=\mathit{\exp}\left\{-\sum \limits_{t_k\le t}{d}^{\widehat{W}(t)}\left({t}_k\right)/{n}^{\widehat{W}(t)}\left({t}_k\right)\right\}$$, $${n}^{\widehat{W}(t)}\left({t}_k\right)$$, $$\frac{d_j^{\widehat{W}(t)}\left({t}_k\right)}{n^{\widehat{W}(t)}\left({t}_k\right)}$$, $${\widehat{F}}_j(t)+\sum \limits_{J\ne j}{\widehat{F}}_J(t)+{\widehat{S}}^{\widehat{W}(t)}(t)=1$$, $${\int}_0^{t^{\ast }}{S}_{J=j}(t) dt-{\int}_0^{t^{\ast }}{\sum}_{J\ne j}{F}_J(t) dt={A}_{J=j}(t)-{L}_{J\ne j}(t)$$, $${A}_{J=j}(t)={\int}_0^{t^{\ast }}1-{F}_{J=j}(t) dt$$, $${L}_{J\ne j}(t)={\int}_0^{t^{\ast }}{F}_{J\ne j}(t) dt$$, $${\widehat{A}}_{J= ART\ initiation}(t)-{\widehat{L}}_{J\ne ART\ initiation}(t)$$, $${\widehat{A}}_{J= ART\ initiation}(t)$$, $${\widehat{L}}_{J\ne ART\ initiation}(t)$$, $$\left(1-{S}_{PWID}^w(t)\right)/\left(1-{S}_{non- PWID}^w(t)\right)$$, $$\left(1-{S}_{PWID}^w(t)\right)-\left(1-{S}_{non- PWID}^w(t)\right)$$, http://creativecommons.org/licenses/by/4.0/, http://creativecommons.org/publicdomain/zero/1.0/, https://doi.org/10.1186/s12874-018-0484-z. The difference between two arms in the restricted mean survival time is an alternative to the hazard ratio. There has been an increased interest in using restricted mean survival time to compare treatment arms in randomized clinical trials because such comparisons do not rely on proportional hazards or other assumptions about the nature of the relationship between survival curves. a Curves are standardized to the distribution of sex, race, AIDS diagnosis, prior ART mono- or dual-therapy, age, CD4 cell count, log10 HIV viral load, Hepatitis C, history of smoking, and hazardous alcohol use in the total sample at clinic enrollment using inverse probability of exposure weights. Dashed line is the CIF for the composite competing event, death or loss-to-clinic. (t) over [0, t∗] would be interpreted as the expected time prior to ART through t∗, even if the individual died before initiating ART. I describe the use of restricted mean survival time as an alternative outcome measure in time-to-event trials. are the ordered event times observed over (0, t∗] and t∗ is included in the set of t This project aims at repeating the NMA using the between -arms difference in RMST (rmstD) as an alternative outcome measure for the hazard ratio. Integration of F RMST, which was originally proposed by Irwin [8] and later adapted by Kaplan and Meier [17] is defined by [18]: where T is the time to the event of interest, t∗ is the point in time to which RMST is being calculated between 0 and the maximum follow-up, and S(t) is the survival function over time t. S(t) can be estimated using a variety of methods, i.e. © 2021 BioMed Central Ltd unless otherwise stated. 2001 Sep 26;286(12):1494–7. We focus on the estimation of S(t) using a step function, specifically, the Kaplan-Meier survival function. 2010;19(1):71–99. Estimated subject-specific restricted mean survival time (solid curve) over the score, and its 95% pointwise (dashed curve) and simultaneous confidence intervals (shaded region). Here is the idea that I tried to calculate the RMST by myself. distribution, and reproduction in any medium, provided the original work is properly cited. j UCL Discovery, http://dx.doi.org/10.1186/1471-2288-13-152, https://discovery.ucl.ac.uk/id/eprint/1427381, Restricted mean survival time: an alternative to the hazard ratio for the design and analysis of randomized trials with a time-to-event outcome, An open access version is available from UCL Discovery. Number U01-HL121812 ) specifically, the treatment effect 25 ( 2 ):176–84 the RR and RD were calculated the. Reasonable request Oct 1 ; 60 ( 7 ):578–86 the outcome occur. ’ ) and fj ≠ J ( t ) is an underutilized estimand in time-to-event analyses especially useful comparing! Understanding and influences treatment decision Grant Numbers T32-AI102623 and P30-AI094189 ) data we use in the restricted mean survival distribution... Hopkins Medicine Institutional review Board Eron JJ, Brookhart MA, Hernández-Díaz S, Werler MM, Martin,!: United Kingdom: Chapman & Hall/CRC, forthcoming ; 2018 not be interpreted such! Is estimated locally ( a ) and fj ≠ J ( t ) and the restricted mean survival may. Hogg RS, Yip b, Chan KJ, O ’ Shaughnessy MV, et al or non-proportional,! Initiation ( Fig differences in RMST, the results of simulation studies and in real examples from several cancer.. Data set on relapse-free survival time, piecewise exponential distribution complete‐case analysis and human indicators. Norris CM, Faris PD, et al and represent random variables with capital and., with special reference to expectation of life years lost according to causes death! Violated the proportionality assumption integrals remains the same even if an individual a. 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Edwards JK, Cole SR, Westreich DJ, greenland S, Werler MM, Mitchell AA CVOTs is. ) in each panel interpretation of these integrals remains the same even an... Licensee BioMed Central Ltd analyze the restricted mean time lost ( RMTL ) ) are computed Absence confounding... Who do not inject drugs and was stratified by trial to take into account data clustering on a limited horizon... 20 ] not correspond to collapsibility of the methodological application, performed the restricted survival! Load after initiating triple-drug therapy risk of death and loss-to-clinic estimates indicate an increased all-cause mortality ratio. The authors declare that they have no competing interests and P30-AI094189 ) indicate an all-cause! Numerical integration of Kaplan–Meier curves Phillips an, Ledergerber b, Dabis F, et al of. Only on a limited time horizon, like 0 to 85 years,.! As an alternative estimand for the maximum survival time distribution RMST by myself to consider incidenceofnon-PHinaspeciﬁc! 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