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Bias in Mean Survival From Fitting Cure Models With Limited Follow-Up

Open ArchivePublished:July 31, 2020DOI:https://doi.org/10.1016/j.jval.2020.02.015

      Highlights

      • Mixture cure models may reduce bias compared with traditional parametric models when a proportion of patients may be cured, but their accuracy is subject to certain conditions, in particular, adequate follow-up.
      • Using mixture cure models to estimate mean survival with limited follow-up can result in both conservative and anticonservative bias.
      • Caution should be taken when using mixture cure models in scenarios with limited follow-up.

      Abstract

      Objectives

      When populations contain mixtures of cured and uncured patients, the use of traditional parametric approaches to estimate overall survival (OS) can be biased. Mixture cure models may reduce bias compared with traditional parametric models, but their accuracy is subject to certain conditions. Importantly, mixture cure models assume that that there is enough follow-up to identify individuals censored at the end of the follow-up period as cured. The purpose of this article is to describe biases that can occur when mixture cure models are used to estimate mean survival from data with limited follow-up.

      Methods

      We analyzed 6 trials conducted by the SWOG Cancer Research Network Leukemia Committee. For each trial, we analyzed 2 data sets: the data released to the committee when the results of the trial were unblinded and a second data set with additional follow-up. We estimated mean OS using parametric survival models with and without a cure fraction.

      Results

      When using mixture cure models, in 4 trials, estimates of mean OS were higher with the first analysis (with limited follow-up) compared with estimates from data with longer follow-up. In 1 trial, the reverse pattern was observed. In 1 trial, the cure estimate changed little with additional follow-up.

      Conclusions

      Caution should be taken when using mixture cure models in scenarios with limited follow-up. The biases resulting from fitting these models may be exacerbated when the models are being used to extrapolate OS and estimate mean OS.

      Keywords

      Introduction

      In recent years, several trials evaluating novel treatments have suggested that a fraction of patients will survive long-term, potentially exhibiting a risk of death similar to persons without cancer.
      • Hodi F.S.
      • O’Day S.J.
      • McDermott D.F.
      • et al.
      Improved survival with ipilimumab in patients with metastatic melanoma.
      When populations contain mixtures of cured and uncured patients, the use of traditional parametric approaches to estimate overall survival (OS) has been shown to produce biased estimates.
      • Othus M.
      • Bansal A.
      • Koepl L.
      • et al.
      Accounting for cured patients in cost-effectiveness analysis.
      In these situations, mixture cure models are attractive alternatives. Mixture cure models explicitly characterize heterogeneity by estimating the probability a patient is cured (potentially conditional on patient-level covariates), while modeling survival separately for the cured and not cured subpopulations.
      • Berkson J.
      • Gage R.P.
      Survival curve for cancer patients following treatment.
      • Kuk A.Y.
      • Chen C.-H.
      A mixture model combining logistic regression with proportional hazards regression.
      • Peto R.
      • Peto J.
      Asymptotically efficient rank invariant test procedures.
      Although mixture cure models can reduce bias in survival estimation compared with traditional parametric models, their accuracy is subject to certain conditions. Importantly, mixture cure models implicitly assume that that there is enough follow-up to identify “failures” (ie, deaths), such that individuals censored at the end of the follow-up period are in fact cured. Survival curves that represent mixtures of cured and uncured patients will exhibit a plateau at the tail of the Kaplan-Meier curve, indicating that there are no failures after a particular time point.
      • Maller R.A.
      • Zhou S.
      Testing for sufficient follow-up and outliers in survival data.
      Trials with limited follow-up, however, may also exhibit an apparent plateau in the Kaplan-Meier curve because of heavy censoring and small numbers of patients represented in the plateau. We urge caution in these situations. Accordingly, the purpose of this article is to describe biases that can occur when mixture cure models are used to estimate mean survival from data with limited follow-up. In particular, we will show that such models can result in both conservative and anticonservative bias in a manner that is unpredictable. To illustrate our concern, we take advantage of archived data for 6 trials from the SWOG Cancer Research Network Leukemia Committee. Using parametric models that do and do not incorporate a cure fraction, we estimate mean survival using the data initially provided to the study team (exhibiting some plateau in survival) and then reestimate these values using additional years of follow-up.

      Approaches to Cure Models

      Cure models have been an active area of research for more than 50 years. The most popular framework is to assume that the study population is a mixture of patients who are cured and patients who are not cured and to explicitly model this mixture.
      • Berkson J.
      • Gage R.P.
      Survival curve for cancer patients following treatment.
      • Kuk A.Y.
      • Chen C.-H.
      A mixture model combining logistic regression with proportional hazards regression.
      • Peto R.
      • Peto J.
      Asymptotically efficient rank invariant test procedures.
      In this framework, regression models can be used to estimate the probability that a patient is cured and to predict the survival of patients who are not cured.
      There are no diagnostic tests that can assess whether an individual patient is cured of their cancer. Long-term follow-up is the ultimate way to identify a cured subpopulation. Accordingly, mixture cure models do not identify individual patients as cured but rather estimate a probability that a patient is cured. What is considered adequate follow-up to be confident that a cured fraction exists varies across cancers? “Long” follow-up may be less than 5 years for a cancer with high mortality, whereas 20 years may not provide adequate follow-up for low-mortality cancers.
      Logistic regression is a common choice for modeling the probability that a patient is cured.
      • Sy J.P.
      • Taylor J.M.
      Estimation in a Cox proportional hazards cure model.
      Both patients who are cured and patients who are not cured are subject to “background” mortality not related to cancer. Patients who are not cured are subject to additional mortality from their cancer, and parametric survival models are often used to estimate this excess mortality. Mathematically, the survival for a population with a cure fraction described by a mixture model can be written as:
      S(t, x)=SB(t, x)[p(x)+(1-p(x))SE(t, x)],
      (1)


      where S(t, x) denotes the (average) survival function of the whole population (cured and uncured) at time t conditional on covariates x, SB(t, x) denotes the survival function for a population without the disease of interest at time t conditional on covariates x, p(x) denotes the probability of being cured conditional on covariates x, and SE(t, x) denotes the disease-specific (eg, cancer) survival function at time t conditional on covariates x.
      • Lambert P.C.
      • Thompson J.R.
      • Weston C.L.
      • et al.
      Estimating and modeling the cure fraction in population-based cancer survival analysis.
      ,
      • Lambert P.C.
      Modeling of the cure fraction in survival studies.
      We note that SB, p, and SE can be written more generally to depend on different covariates, but we focus on the scenario with shared covariates without loss of generality. SB can be calculated from external data; for our application, we used age- and gender-matched mortality data from US Social Security life tables. We modeled p(x) with logistic regression, and we considered several parametric models for SE.

      Cure Models in Economic Evaluation

      Economic evaluations of competing interventions often estimate mean OS to calculate other statistics such as quality-adjusted life-years and the incremental cost-effectiveness ratio. If observed survival is not zero at the end of the observation period, the mean value cannot be estimated without constructing a model for extrapolation. Parametric models such as the Weibull and log-normal can be used. If a population contains a mixture of cured and not cured patients, the mean survival of the population can be calculated as the weighted average of the mean survival times of the cured and not cured subpopulations, weighted by the relative proportions. In the model in Equation 1, the mean OS of the cured proportion is the mean of the background survival (SB), whereas the mean OS for patients who are not cured is a function of both the background survival (SB) and the disease-related survival (SE). The mean of a random variable with survival function S(t) is equal to 0S(t)dt, so the mean OS for cured patients is equal to 0SB(t)dtand the mean OS for patients who are not cured is equal to 0SB(t)SE(t)dt.

      Study Design

       Patient Populations

      We analyzed 3 phase II, 1 phase II/III, and 2 phase III trials that were conducted by the SWOG Cancer Research Network (SWOG.org) that had primary results released to the SWOG Leukemia Committee between 2009 and 2016.
      • Petersdorf S.H.
      • Kopecky K.J.
      • Slovak M.
      • et al.
      A phase 3 study of gemtuzumab ozogamicin during induction and postconsolidation therapy in younger patients with acute myeloid leukemia.
      • Radich J.P.
      • Kopecky K.J.
      • Appelbaum F.R.
      • et al.
      A randomized trial of dasatinib 100 mg versus imatinib 400 mg in newly diagnosed chronic-phase chronic myeloid leukemia.
      • Nand S.
      • Othus M.
      • Godwin J.E.
      • et al.
      A phase 2 trial of azacitidine and gemtuzumab ozogamicin therapy in older patients with acute myeloid leukemia.
      • Ravandi F.
      • Othus M.
      • O’Brien S.M.
      • et al.
      US intergroup study of chemotherapy plus dasatinib and allogeneic stem cell transplant in Philadelphia chromosome positive ALL.
      • Sekeres M.A.
      • Othus M.
      • List A.F.
      • et al.
      Randomized phase II study of azacitidine alone or in combination with lenalidomide or with vorinostat in higher-risk myelodysplastic syndromes and chronic myelomonocytic leukemia: North American Intergroup Study SWOG S1117.
      • Garcia-Manero G.
      • Othus M.
      • Pagel J.M.
      • et al.
      SWOG S1203: a randomized phase III study of standard cytarabine plus daunorubicin (7+3) therapy versus idarubicin with high dose cytarabine (IA) with or without vorinostat (IA+V) in younger patients with previously untreated acute myeloid leukemia (AML).
      • Deininger M.W.
      • Kopecky K.J.
      • Radich J.P.
      • et al.
      Imatinib 800 mg daily induces deeper molecular responses than imatinib 400 mg daily: results of SWOG S0325, an intergroup randomized PHASE II trial in newly diagnosed chronic phase chronic myeloid leukaemia.
      We selected 2016 as the latest release time to allow for additional follow-up between the original primary results and our follow-up survival analysis (in 2019). We note that for some leukemias, the label long-term survival model may be more appropriate than cure model. For clarity, in the following we use the term cure model, but the term long-term survival model could be used interchangeably as appropriate for the application. Institutional review boards of participating institutions approved all protocols, and patients were treated according to the Declaration of Helsinki.

       Statistical Methods

      Survival was estimated using the Kaplan-Meier method. There were no significant treatment effects on survival endpoints in any of the randomized trials analyzed, and analyses are presented for all arms analyzed together to increase precision in estimates. We fit both models with and without a cure fraction to estimate mean survival for each trial based on the archived outcome data from when the primary results of the trial were released and then again from survival data from January 2019. Parameters for p(x) and SE(t, x) from Equation 1 were estimated using the score equations from the log-likelihood in Lambert.
      • Lambert P.C.
      Modeling of the cure fraction in survival studies.
      Mean survival for patients who were not cured (equal to 0SB(t)SE(t)dt) was calculated by evaluating the numerical integral. Background mortality for each patient was taken from the age- and gender-matched US Social Security area population based on the year the first patient was enrolled on the study. Although background rates from most population-based sources also contain mortality associated with leukemia, in practice this has little effect on the parameter estimates.
      • Ederer F.
      The relative survival rate: a statistical methodology.
      Akaike information criterion (AIC) was estimated for each model. We considered Weibull, log-normal, and log-logistic parametric survival models and used AIC to evaluate model fit. Weibull models fit all the trials best (by AIC) for both cure and noncure models and so are presented here.

      Results

      The characteristics of each trial are summarized in Table 1. In the interest of space, we provide detailed analysis of 2 of the trials, S1117 and S1203. Outcomes of the other 4 trials are summarized briefly below.
      Table 1Characteristics of trials analyzed.
      S0106S0325S0703S0805S1117S1203
      DiseaseAMLCMLAMLALLMDSAML
      PhaseIIIIIIIIIII/IIIIII
      Number of arms231133
      Treatments evaluated7 + 3, 7 + 3 + GOImatinib, dasatinibAza + GOHyper-CVAD + dasatinibAza, Aza + Len, Aza + Vor7 + 3, IA, IA + Vor
      n59839213995277738
      Age, median (range), y47 (18-60)50 (18-90)73 (60-88)44 (20-60)70 (28-93)49 (18-60)
      Percentage female473940553149
      Years enrolled2004 to 20092004 to 20092009 to 20122010 to 20132012 to 20142013 to 2016
      Year primary results released200920122013201620142016
      Protocol maximum follow-up5 years5 years5 years5 years5 years5 years
      Percentage censored when results released vs 201958 vs 4995 vs 9327 vs 771 vs 6071 vs 2669 vs 49
      7 + 3 indicates standard of care AML therapy with cytarabine + daunorubicin; Aza, azacytidine; ALL, acute lymphoblastic leukemia; AML indicates acute myeloid leukemia; CML, chronic myeloid leukemia; GO, gemtuzumab ozogamicin (Mylotarg); Hyper-CVAD, standard-of-care ALL therapy regimen including the drugs cyclophosphamide, vincristine, doxorubicin, dexamethasone, methotrexate, and cytarabine; IA, high-dose cytarabine and idarubicin; Len, lenalidomide; MDS, myelodysplastic syndrome; Vor, vorinostat.
      For S1117, when the data were released to the study team in 2014, there was a small plateau at the tail of the Kapan-Meier curve (Fig. 1A). S1117 was closed soon after the phase II accrual goal was met, so there was heavy censoring along the survival curve. With additional follow-up in 2019 (Fig. 1B), there was no apparent plateau in survival. Fitting a standard Weibull model (without a cure fraction) to each of these curves, the 2019 data Weibull results estimated longer survival compared with the 2014 estimate (Figs. 1C and D). Fitting mixture cure models to both data sets, the 2014 data led to a more optimistic estimate of long-term OS than the 2019 data (Supplementary Fig. 1E and F, found at https://doi.org/10.1016/j.jval.2020.02.015). Table 2 summarizes the mean OS for each of these models. For the standard Weibull model, the mean OS was estimated to be 1.9 years with the 2014 data compared with 2.8 years with the 2019 data; with the mixture cure Weibull model, the mean OS was 7.7 years versus 4.9 years for 2014 and 2019, respectively.
      Figure thumbnail gr1
      Figure 1Overall survival (OS) of S1117. (A) Kaplan-Meier estimate of OS when primary results released in 2014. (B) Kaplan-Meier estimate of OS with data from 2014 and 2019. (C, D) Kaplan-Meier estimates of OS (solid lines) and standard Weibull models (dotted lines).
      Table 2Mean OS in years for standard Weibull and mixture cure Weibull models.


      Trial
      YearWeibullMixture cure Weibull
      Mean OSAICMean OS not curedMean OS curedProportion curedOverall mean OSAIC
      S010620094.0757.42.036.532%13.2975.8
      20199.81611.68.236.4<1%8.41725.8
      S0325201259.8183.421.232.240%25.5179.3
      201967.9204.823.232.130%25.9195.7
      S070320131.6249.41.113.715%3.0475.5
      20191.5334.41.313.75%1.9508.1
      S0805201612.1179.81.739.966%26.9186.7
      201912.3240.410.839.9<1%10.6227.1
      S011720141.9289.41.017.241%7.7442.6
      20192.8820.01.617.221%4.9823.3
      S120320163.7929.01.936.231%12.41419.1
      20196.91813.01.736.143%16.42120.1
      AIC indicates Akaike information criterion; OS, overall survival.
      When the S1203 data were released to the study team in 2016, there was a small plateau at the tail of the Kaplan-Meier curve, with heavy censoring before the plateau (Supplementary Fig. 2A). S1203 was a randomized phase III trials that stopped at an interim analysis because of futility; the interim analysis occurred soon after accrual to the trial completed, leading to heavy censoring over the survival curve. With additional follow-up in 2019 (Supplementary Fig. 2B), there was no evidence of a plateau in survival, as patients continued to die during the ongoing follow-up. Fitting a standard Weibull model to each of these curves, the 2019 data Weibull fit estimated longer survival compared with the 2016 estimates (Supplementary Figs. 2C and D). Fitting mixture cure models to both data sets, the 2016 data lead to a less optimistic estimate of OS (Supplementary Figs. 2E and F). Table 2 summarizes the mean OS for each of these models. For the standard Weibull model, the mean OS was estimated to be 3.7 years with the 2016 data compared with 6.9 years with the 2019 data; with the mixture cure Weibull model, the mean OS was 12.4 years versus 16.4 years for 2014 and 2019, respectively.
      Survival curves for the other trials are provided in the supplemental figures, and mean OS estimates are summarized in Table 2. For most of the cohorts, the AIC values were smaller (smaller values indicating better fit) for the standard Weibull models without a cure fraction compared with the mixture cure models (Table 2).

      Discussion

      We have previously published analyses demonstrating that standard survival modeling techniques may be insufficient to estimate survival for cost-effectiveness studies in situations in which therapies appear to cure a proportion of patients treated.
      • Othus M.
      • Bansal A.
      • Koepl L.
      • et al.
      Accounting for cured patients in cost-effectiveness analysis.
      In our example, we found that a cure model approach yielded substantially different estimates of mean survival compared with traditional modeling approaches, such that the estimate of incremental cost-effectiveness was significantly changed using the cure models. Since publication of our finding, mixture cure models have been used in several economic evaluations,
      • Roth J.A.
      • Sullivan S.D.
      • Lin V.W.
      • et al.
      Cost-effectiveness of axicabtagene ciloleucel for adult patients with relapsed or refractory large B-cell lymphoma in the United States.
      ,
      • Whittington M.D.
      • McQueen R.B.
      • Ollendorf D.A.
      • et al.
      Long-term survival and cost-effectiveness associated with axicabtagene ciloleucel vs chemotherapy for treatment of B-Cell lymphoma.
      perhaps in part due to the expectation that mixture cure modeling will uniformly increase estimates of survival, and therefore cost-effectiveness, compared with traditional modeling. In this article, we demonstrate that early, enthusiastic use of mixture cure models may be a problematic choice, particularly for studies with small sample sizes or immature follow-up data for estimating OS.
      Although we believe that cure models can be a useful tool, we also are aware of the number of assumptions that are required to use such models. Prior to this report, no study has evaluated the consequences of limited follow-up in the applications of cure models in health economic analyses. Our analysis provides empirical evidence on the potential implications in violations in the assumption of adequate follow-up when using cure models. Under an assumption that cured patients will never experience the event of interest, prior work has found that cure fraction estimates can be quite sensitive to model assumptions.
      • Yu B.
      • Tiwari R.C.
      • Cronin K.A.
      • et al.
      Cure fraction estimation from the mixture cure models for grouped survival data.
      Similarly, applying different parametric models to data with limited follow-up can produce substantially different estimates of long-term outcome.
      • Bansal A.
      • Sullivan S.D.
      • Lin V.W.
      • et al.
      Estimating long-term survival for patients with relapsed or refractory large B-cell lymphoma treated with chimeric antigen receptor therapy: a comparison of standard and mixture cure models.
      We found that cure models were sensitive to the plateau of the survival function, even in the presence of limited information. For example, in S1117, there are 12 censored patients in the plateau of the 2014 data, whereas there are 79 patients censored before the plateau, so the plateau estimate is based on data with limited follow-up. Cure models use the plateau as an estimate of the cure fraction,
      • Maller R.A.
      • Zhou S.
      Testing for sufficient follow-up and outliers in survival data.
      and with additional follow-up, it is clear that the 2014 estimates are an overestimate of survival. We note that evidence for a true plateau is dependent on the length of follow-up; the apparent “plateau” in the 2014 analysis of S1117 disappears when the Kaplan-Meier curve is recalculated using additional years of follow-up.
      Although the mixture cure model fit for study S1117 using data available in 2014 estimated a higher mean OS than a model fit to more mature data from 2019 (7.7 versus 4.9 years), we observed the opposite pattern in S1203, in which mean OS reported using 2016 data was less than the more mature data in 2019 (12.4 versus 16.4 years).
      In S0106, S0703, and S0805, the mean OS from the mixture cure model fit on earlier data sets was greater than the mean with additional follow-up, often by more than 50%. In S0325, the results were fairly similar in the 2 analyses, likely because of the small number of events in the trial and limited changes in event rates with additional follow-up.
      In 5 of the 6 trials evaluated, we found that the bias from fitting mixture cure models to data with limited follow-up resulted in both conservative and anticonservative estimates of mean OS compared with estimates with additional follow-up. When bias is predictable, statistical methods can be used to adjust for such bias. The bias from fitting a cure model with limited follow-up does not appear to be predictable and so is unlikely to be able to be remediated through bias-adjustment techniques.
      Perhaps the most important lesson from this exercise is the need to reevaluate the potential “cure fractions” after additional years of follow-up. SWOG trials typically continue to collect outcome data on patients after the primary results of the study are released; data are collected for a protocol-specified amount of time regardless of when the primary outcomes are published. In addition, SWOG archives outcomes of each study twice a year until the primary results are released, and then archives all published analyses of each study. The problem of immature survival data is most acute for treatments that the Food and Drug Administration (FDA) has designated with breakthrough status or accelerated approval. In these situations, the FDA requires demonstration of effect on a surrogate endpoint or intermediate clinical endpoint considered reasonably likely to predict a clinical benefit.

      US Food and Drug Administration. Fast Track, Breakthrough Therapy, Accelerated Approval, Priority Review (content current as of 2/23/2019). https://www.fda.gov/patients/learn-about-drug-and-device-approvals/fast-track-breakthrough-therapy-accelerated-approval-priority-review. Accessed July 28, 2020.

      OS data are typically immature at the time the initial findings are released. Although FDA approval of products with these designations may be sufficient to introduce them into clinical practice, health technology assessments and economic evaluations may be problematic. As we have noted in a previous evaluation of a breakthrough therapy that appears to have a cured fraction, estimates of cost-effectiveness must be considered preliminary until survival data are mature.
      • Roth J.A.
      • Sullivan S.D.
      • Lin V.W.
      • et al.
      Cost-effectiveness of axicabtagene ciloleucel for adult patients with relapsed or refractory large B-cell lymphoma in the United States.
      When considering health economic analyses using potential surrogate endpoints for a cure fraction such as progression-free survival, relapse-free survival, or event-free survival, the same issues with bias associated with limited follow-up arise. Because the event horizon for surrogate endpoints is often significantly shorter than OS, adequate follow-up for fitting a cure model may be reached sooner. We note that because surrogate endpoints such as progression-free survival include additional events beyond death, if a cure fraction exists, the surrogate endpoint may underestimate the true cure fraction. Before using a surrogate endpoint to estimate a cure fraction in a data set, retrospective analyses validating the surrogate endpoint in this specific treatment and disease setting would be required, and future validation with additional follow-up would be warranted.
      There is no formal statistical test to evaluate the adequacy of follow-up to fit cure models. Those that have been proposed can be both conservative and anticonservative.
      • Maller R.A.
      • Zhou S.
      Testing for sufficient follow-up and outliers in survival data.
      Prior authors have suggested examining likelihood plots and parameter estimates for different parametric choices of the survival function for noncured patients; longer follow-up may be needed if the likelihood function is flat or if the estimates show significant variation. Further research is needed to identify tests for adequacy of follow-up.
      When there are limited follow-up data on many patients, even in a setting where there is scientific or clinical expectation of a cure fraction, we do not recommend using cure models in for inference health economic analyses. The bias is potentially too large and unpredictable to allow for reliable inference. In situations in which adequate follow-up is available, we agree with others who recommend fitting multiple parametric models for disease-specific survival function in addition to models without cure fractions as sensitivity analyses.
      • Latimer N.R.
      Survival analysis for economic evaluations alongside clinical trials—extrapolation with patient-level data: inconsistencies, limitations, and a practical guide.
      ,
      • Jackson C.
      • Stevens J.
      • Ren S.
      • et al.
      Extrapolating survival from randomized trials using external data: a review of methods.
      In addition to analyzing the graphical fit of each model, model fit criteria such as the AIC and Bayesian information criterion can be used to guide model selection. In some of our examples, the AIC suggested that alternatives to cure fraction modeling may be preferable.
      We conclude that caution should be taken when using mixture cure models in scenarios in which assumptions about adequate follow-up are not met. The bias resulting from fitting these models may be exacerbated when the models are being used to extrapolate OS and calculate mean OS.

      Supplemental Material

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