Current Estimates of the Cure Fraction: A Feasibility Study of Statistical Cure for Breast and Colorectal Cancer

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1 DOI: /jncimonoraphs/lu015 Published by Oxford University Press 014. Current Estimates of the Cure Fraction: A Feasibility Study of Statistical Cure for Breast and Colorectal Cancer Mararet R. Stedman, Eric J. Feuer, Anela B. Mariotto Correspondence to: Mararet R. Stedman, PhD, MPH, National Cancer Institute, Division of Cancer Control and Population Sciences, Surveillance Research Proram, 9609 Medical Center Drive, Bethesda, MD ( mararet.stedman@nih.ov). Backround Methods Results Discussion The probability of cure is a lon-term pronostic measure of cancer survival. Estimates of the cure fraction, the proportion of patients cured of the disease, are based on extrapolatin survival models beyond the rane of data. The objective of this work is to evaluate the sensitivity of cure fraction estimates to model choice and study desin. Data were obtained from the Surveillance, Epidemioloy, and End Results (SEER)-9 reistries to construct a cohort of breast and colorectal cancer patients dianosed from 1975 to In a sensitivity analysis, cure fraction estimates are compared from different study desins with short- and lon-term follow-up. Methods tested include: cause-specific and relative survival, parametric mixture, and flexible models. In a separate analysis, estimates are projected for 8 dianoses usin study desins includin the full cohort ( dianoses) and restricted to recent dianoses (1998 8) with follow-up to 9. We show that flexible models often provide hiher estimates of the cure fraction compared to parametric mixture models. Lo normal models enerate lower estimates than Weibull parametric models. In eneral, 1 years is enouh follow-up time to estimate the cure fraction for reional and distant stae colorectal cancer but not for breast cancer. 8 colorectal cure projections show a 15% increase in the cure fraction since Estimates of the cure fraction are model and study desin dependent. It is best to compare results from multiple models and examine model fit to determine the reliability of the estimate. Early-stae cancers are sensitive to survival type and follow-up time because of their loner survival. More flexible models are susceptible to sliht fluctuations in the shape of the survival curve which can influence the stability of the estimate; however, stability may be improved by lenthenin follow-up and restrictin the cohort to reduce heteroeneity in the data. J Natl Cancer Inst Monor 014;49:44 54 Cure is difficult to identify at the individual level because for certain cancers, there can be reoccurrences many years after periods of bein symptom free. However, cure can be identified at the population level as the fraction of cancer patients (cure fraction), who have an observed mortality similar to the eneral population after a lon follow-up period. This concept of cure, known as statistical cure, represents a lon-term pronostic measure of the chances of bein cured, and it is a desirable statistic for the patient and clinician. Several models exist to estimate statistical cure [e, mixture (1), nonmixture (), and flexible (3) models] and all of these models rely on the lenth of follow-up to attain a stable estimate of statistical cure (4). Althouh cancer reistries may have lon histories of incidence and follow-up data, recent advances in treatment, screenin, and chanes in stain criteria impact survival and complicate the estimation of statistical cure. Furthermore, there is interest in predictin the cure fraction for patients recently dianosed with minimal follow-up information. There has been some research to determine the minimal lenth of follow-up needed to obtain a stable cure estimate. Yu and colleaues (5) found that lenth of follow-up time needs to be at least two thirds the median survival time in the uncured patients to attain a stable cure estimate. Tai et al. (6) empirically derived the minimum follow-up time for 49 different cancer sites usin median survival time and found that the minimal lenth of follow-up is site specific. For example, colon cancer was estimated to require a minimum of 1. years and breast cancer, a minimum of 36. years (6). However, median survival depends on the shape of the survival curve and the type of outcome measured (relative or cause-specific survival) and none of these studies has considered the sensitivity of the cure fraction to follow-up time while varyin model assumptions. In this paper, we use the Surveillance Epidemioloy and End Results (SEER) reistry data to perform a sensitivity analysis of the cure fraction estimate to model choice, survival outcome, and study desin. We compare the impact of follow-up time on parametric mixture and flexible parametric cure models and on relative and cause-specific measures of cancer survival. Usin the most recent SEER data, we project the cure fraction by stae and site for breast and colorectal cancer patients dianosed in Journal of the National Cancer Institute Monoraphs, No. 49, 014

2 First, we describe the data and methods used to perform a sensitivity analysis of the cure fraction with respect to follow-up time, model, and survival type usin SEER breast and colorectal cancer data. Next, we provide results from the sensitivity analysis and future projections of the cure fraction for patients dianosed in 8. Finally, we conclude with some uidance on projectin the cure fraction usin these models. Data and Methods Data and Cohort Definitions We used the SEER site recode variables to identify patients dianosed at aes with colorectal cancer (SEER Site Recode International Classification of Diseases for Oncoloy [ICD-O]-3 definition: C180-C189, C60, C199, and C09) and at aes with female breast cancer (SEER Site Recode ICD-O-3 definition: C5-C509) within the SEER-9 Reistries. Cancers of the colon and rectum were combined because they were assumed to have similar cure fractions (7). Colorectal cancer has less incident cases than breast cancer, so a broader ae rane was included to increase the size of the colorectal cohort. We selected malinant cases and excluded patients dianosed at death and secondary tumors. All data were obtained from the November 011 submission, usin SEER*Stat software, version 8.0, available at our website: seer.cancer.ov/seerstat. We constructed four cohorts of breast cancer and colorectal cancer patients. The first two cohorts included female breast and colorectal cancer patients dianosed between 1975 and 1985 and two different follow-up periods: 1 years (follow-up throuh December 1986) and 35 years (follow-up throuh December 9). Presumin that havin a loner follow-up time would optimize accuracy of the cure estimate, data from the 1- and 35-year observation periods are referred to as the test and optimal cohorts, respectively. To obtain estimates of the cure fraction that reflect the most recently dianosed patients (dianosed in 8), we constructed two other cohorts: patients dianosed with female breast and colorectal cancer from 1975 to 8 ( full cohort ) and patients dianosed with breast and colorectal cancer from 1998 to 8 ( recent cohort ). All patients were followed from dianosis until December 9. Since entrance into the reistry and lenth of follow-up depend upon the date of dianosis (loner follow-up for the earlier dianoses), there is confoundin between the lenth of follow-up and the dianosis year. To reduce confoundin and also project the cure fraction for patients dianosed in the most recent years, we adjusted for dianosis year in all analyses. Relative and Cause-Specific Survival Cancer survival may be estimated either as relative survival (the ratio of all-cause survival and expected survival) or cancer-specific survival [based on cause of death from the death certificate (8)]. When examinin a plot of cancer survival, the cure fraction is the point in the survival curve where the curve beins to level off and the slope is predicted to be zero. Sometimes, this occurs well beyond the observed data (see Fiure 1). From a relative survival perspective, cure occurs when the observed survival of the cancer population matches the expected survival of the eneral population. From a cause-specific perspective, cure occurs when the cancer population Fiure 1. A 60% cure fraction estimated from the cancer survival curve. Example results from CanSurv software. is no loner dyin from cancer and the hazard of dyin of cancer is estimated to be zero. This concept of cure measured from causespecific survival is sometimes referred to as clinical cure (9). We obtained relative survival [by the Ederer II (10) method] and cause-specific survival estimates stratified by dianosis year and SEER historic stae (localized, reional, distant, and all staes combined). Cause of death is based on the SEER cause-specific death classification, which classifies deaths related to the cancer dianosis usin patient death certificate information, tumor location, and comorbidities ( (8). Relative survival can ive biased estimates for localized stae breast cancer because expected survival estimated from the US population life tables underestimate all-cause mortality in this subroup (8). Statistical Cure: Parametric Mixture Cure Model Althouh there are several models available to estimate cure, this study focuses on the parametric mixture cure model and the flexible parametric model, both of which have been adapted to populationbased cancer survival analysis. The parametric mixture cure model predicts cancer survival, S c (t), as a mixture of the cure fraction, c, and the uncured fraction, (1 c): S t c ( 1 c) G t; ξθ, c () = + ( ) The uncured fraction is assumed to follow a survival function, G(t; ξ, θ), where the distribution can be modeled by either semiparametric or fully parametric distributions with parameters ξ and θ, includin the Weibull, lo normal, lo loistic, and Gompertz distributions. The median survival time for the uncured can be derived from ξ and θ (see Table 1). For this analysis, we examine the more commonly used lo normal and Weibull survival distributions. The interpretation of the parameters, ξ and θ, depend on the survival function, G(t), see Table 1. For the lo normal distribution, the parameters ξ and θ equal the mean and SD of the survival function, respectively. For the Weibull distribution, ξ and θ are reparameterizations of the shape, λ, and scale, ρ (1). 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3 Table 1. Survival functions and their reparameterizations for the parametric mixture cure model Distribution Conventional survival function G(t) Reparameterization Transformed G(t) for CanSurv Median survival time in the uncured Lo normal G(t; µ, σ) = 1 Φ[(ln{t} µ)/σ] ξ = µ, θ = σ G(t; ξ, θ) = 1 Φ[(ln{t} ξ)/θ] exp(ξ) Lo loistic G(t; λ, ρ) = [1 (λt) ρ ] 1 ξ = lnλ, θ =1/ρ G(t; ξ, θ) = [1 + exp{(ln(t) ξ)/θ}] 1 exp(ξ) Weibull G(t; λ, ρ) = exp[ (λt) ρ ] ξ = lnλ, θ =1/ρ G(t; ξ, θ) = exp{ exp[(ln{t} ξ)/θ]} (ln( )) θ exp( ξ) The parameters, c, ξ, and θ, may depend on covariates, X, such as dianosis year. Covariates are introduced into the models with the parameters: β c, β ξ, and β θ. c = [ + exp( β β X)] 1 c0 c1 ξ = β + β X ξ0 ξ1 θ = exp[ β + β ] X θ0 θ1 Accuracy of the cure estimate is improved by adjustin for covariates but it is also influenced by choice of parametric distribution and overall fit to the data. As in Huan et al. (11), we allowed the cure fraction, c, and the median survival in the uncured, ξ, to chane simultaneously as a function of dianosis year, however additional covariates could be included in the model. We estimated the parametric mixture cure model by maximum likelihood estimation methods usin CanSurv software (1) available at CanSurv software reparameterizes the survival function to simplify computation of the parameters (1) (see Table 1 for results for some common survival functions). From the mixture cure models, the cure fraction estimate for patients dianosed in a iven year, Y, can be estimated as 1 1 c = [ 1+ exp( βc0 β c1( Y))] [1] Relative and cause-specific survival estimates were imported to CanSurv software to estimate cure from the mixture cure model. The models were stratified by cancer site and historic stae and adjusted for dianosis year (11), a continuous covariate, so that we could obtain site-, stae-, and year-specific estimates. We estimated median survival of the uncured assumin either a Weibull or lo normal survival function. Parameter estimates were exported to SAS software to calculate projections of the cure fraction and respective confidence intervals (CIs) for patients dianosed in a iven year (see equation 1 for the formula for the cure fraction). The variance of the cure fraction was estimated by the Delta Method (see Appendix A). Statistical Cure: Flexible Parametric Cure Model Flexible parametric cure models predict the cure fraction by modelin the lo cumulative excess hazard. These models were recently developed by Andersson et al. for relative survival (3). Excess mortality is the difference between the hazard functions for all-cause mortality and the expected mortality of the reference population: h(t) h*(t)= λ(t). Interatin these hazards ives more stable estimates of the cumulative mortality, H(t), the cumulative expected mortality, H*(t), and the cumulative excess hazard, Λ(t) at time t, such that Λ(t) = H(t) H*(t). The flexible parametric survival model predicts the lo cumulative excess hazard, ln(λ(t)), usin restricted cubic splines, s(ln(t); γ 0 ), of the lo survival time, t, where γ 0 contains the parameters of the spline function: ln( Λ(()) t = s(ln( t); γ 0 ) This is adapted to predict relative survival, R(t), ln[ ln( Rt ( ))] = s(ln( t); γ ) 0 where ln(λ(t)) = ln( ln(r(t)). Cure is determined when the cumulative excess hazard rate levels off, ie, when there is no difference between the mortality rate of the cancer population and the eneral population and excess mortality is zero. In flexible parametric survival models, this can be accomplished by constrainin the lo cumulative excess hazard function to have a zero slope after a certain point in time. This is implemented by specifyin the K knots from the spline function in reverse order (k K,, k 1 ) where the last spline parameter is restricted to be zero (γ 01 = 0) (3). Thus, the location of the last knot determines when cure occurs. As recommended by Andersson et al. (3), we set the last knot at the last observed time point. Relative survival is estimated as: Rt () = exp[ exp( γ + γ 01v1(ln( t)) + γ 0v(ln( t)) + γ 03v3 (ln( t)) γ 0k 1 v k 1 (ln( t)))] In the backwards spline function, γ 01,, γ 0k 1 are the parameters and v 1 (ln(t)),, v k 1 (ln(t)) are the basis functions defined as: v (ln( t)) = ln( t),and 1 3 v (ln(t)) = ( k ln( t)) ( k ln( t)) j K j + λj where, max Cure, c, is then estimated as, k ( 1 λ j )( k ln( t)), for j =,..., K min + K j 1 λ j = k K k k c = exp[ exp( )] 1 γ 46 Journal of the National Cancer Institute Monoraphs, No. 49, 014

4 To adjust for a covariate such as dianosis year (Y) in the model, we added the covariate, Y, as follows: Rt () = exp[ exp( γ + β( Y) + γ 01v1(ln( t)) + γ 0v(ln( t)) + γ 03v3(ln( t)) γ 0 k 1 vk 1 (ln( t)))] [] so that cure may be predicted for any iven dianosis year, Y: c = exp[ exp( γ + β( Y))] [3] The variance of the cure fraction is approximated by the Delta Method (see Appendix B). See Appendix C for a formula for the median survival in the uncured. Time varyin effects may also be added to more complex models (3). The flexible cure model can be estimated usin the stpm packae (13) available in STATA v1. The packae is desined to model relative survival. Althouh it is possible to flexibly model causespecific survival (14), software for this method has not been fully tested and will not be included in this analysis. Accuracy of the flexible model depends on the placement of the knots for the splines and lenth of follow-up of the available data (3). STATA was used to estimate the median survival for the uncured, the cure fraction, and CIs for the flexible parametric models. As explained by Lambert and Royston (13), we applied the stpm packae to each cohort (test, optimal, full, recent). Expected monthly survival probabilities from the SEER life tables were exported to STATA, converted to yearly mortality rates, and included as the baseline hazard, H * (t)= ln[s * (t) 1 ] in the flexible model. Sensitivity Analyses to Lenth of Follow-up, Model, and Survival Type We investiated the sensitivity of cure fraction estimates to lenth of follow-up by comparin 1985 cure fraction estimates from the 1-year test cohort to those from the 35-year optimal cohort. We compared results from the cause-specific and relative survival outcomes for each of the models. Additionally, all results were evaluated for fit to their respective observed cancer survival curve. By definition, the cure fraction estimate should be at or below the tail of the observed survival curve (5). Models with 1985 cure estimates sinificantly above the 5-year observed survival estimates were considered poor fits to the data. Sensitivity of 8 Projected Cure Fraction to Study Desin Since the cure fraction is a function of dianosis year in the parametric mixture and flexible parametric models, the predicted cure fraction was extrapolated to 8 by settin Y = 8 in equations 1 and 3. Two different study desins were tested. In one desin, patients were selected from dianosis years (recent cohorts). However, in these cohorts, the lenths of follow-up may not be lon enouh to provide stable estimates of the cure fraction. So, in a second desin, the number of dianosis years was expanded to include (full cohorts). While the full cohort provides sufficient follow-up, a linear trend in dianosis year over this lon interval may not be appropriate. 8 projections were compared from both study desins for cause-specific and relative survival parametric mixture and flexible models. Results Sensitivity to Lenth of Follow-up: Comparison of 1985 Estimates From the Test and Optimal Cohorts The 1985 cure fraction for colorectal cancer raned between 55% and 78% for localized stae, 41% and 48% for reional stae, for distant stae, and and 50% for all staes combined (Table, optimal cohort). With the exception of localized stae, cure estimates are fairly stable from the optimal cohort and similar to estimates obtained usin only 1 years of data (test cohort). For localized stae, there is more variation across models and the estimates have wider CIs than other staes. For reional stae, there is less variation across models and tihter confidence intervals; however, the Weibull and flexible models from the test cohort tend to slihtly overestimate the cure fraction compared to the 5-year observed cancer survival (Weibull = 48%, Flexible = 49%, 5-year relative survival = 4). For distant stae disease, two of the models did not convere. The cure fraction was very small, slihtly above zero. The 1985 breast cancer cure fraction estimates showed more variability and raned between 57% and 67% (cause-specific) for localized stae, 35% and 4 for reional stae, and 6% for distant stae, and 4 and 57% for all staes combined (Table, optimal cohort). Test cohort estimates for breast were less reliable and less precise than the colorectal estimates. In eneral, test estimates exceeded the optimal estimates, indicatin that 1 years of data is not enouh to obtain an accurate prediction of the cure fraction. Weibull and flexible estimates tended to be hiher than lonormal estimates and all cases of the flexible model for the test cohort exceeded the 5-year cancer survival estimate. Distant stae results were similarly influenced by small numbers of at risk patients at the end of the survival curve (fewer than 10 survivors 5 years postdianosis). Combinin all staes decreased the stability of the estimate. Althouh most models convered, test estimates fluctuated widely between and 6. Relative survival estimates were hiher than cause-specific estimates for the optimal cohort and these differences were most apparent for localized stae. 8 Projections of the Cure Fraction: Sensitivity to Dianosis Year and Lenth of Follow-up To illustrate modelin the cure fraction as a function of calendar time, we plotted the cure fraction and median survival time estimates by dianosis year from each survival model usin the test cohort of patients dianosed with reional colorectal cancer in and followed throuh For the parametric mixture models, there is compensation between the cure fraction estimates and median survival times, where models with hiher cure fractions have lower median survival times (Fiure ). In the case of the flexible model, there is a comparatively steep increase in the cure fraction (slope = 0.53) and a comparatively flat trend in the median survival (slope = 0.73) over time. The cause-specific Weibull model shows a similar increase in the cure fraction (slope = 0.58) compared to the flexible model, but with a slihtly steeper trend in median survival over time (slope = ). In Fiure 3, we compare the chane over time in the cure fraction and median survival time by cohort for the Weibull model with relative survival for reional stae colorectal cancer. The models for the test cohort tended to overestimate the cure fraction and underestimate the median survival time compared with the optimal cohort; similarly, the model from the full cohort shows a flat trajectory for median Journal of the National Cancer Institute Monoraphs, No. 49,

5 Table cure fraction (95% confidence interval) projections by stae, years follow-up, cancer survival type, and survival function for colorectal and breast cancer patients, dianosed between 1975 and 1985 and followed up throuh 1986 (test cohort) and 9 (optimal cohort)* Cause-specific survival Relative survival Site Stae 5-year cancer survival Test cohort Lo normal Weibull Optimal cohort Test cohort Optimal 5-year cancer cohort survival Test cohort Lo normal Weibull Flexible Optimal cohort Test cohort Optimal cohort Test cohort Optimal cohort Colorectal cancer L 7% (70%, 7) R 4 (41%, 45%) D (%, 5%) All 47% (45%, 48%) Breast cancer L 7 (7%, 76%) R (39%, 4) D 7% (, 10%) All 56% (55%, 58%) 5 (40%, 67%) 4 (40%, 47%) (%, ) 4 (, 46%) 66% (5%, 80%) 40% (31%, 49%) 0% (0%, 0%) 31% (0%, 41%) 55% (51%, 59%) 41% (40%, ) (, ) (41%, 4) 57% (5, 60%) 35% (3, 37%) (%, 5%) 4 (41%, 45%) 77% (71%, 8%) 50% (47%, 5) (, 6%) 49% (47%, 51%) 65% (6%, 67%) (41%, 4) (, ) 4 (4, 45%) 67% (65%, 69%) 6% (57%, 67%) (0%, 10%) 61% (55%, 66%) 40% (39%, ) 5% (, 6%) 51% (50%, 5%) 77% (70%, 8%) 4 (39%, 48%) % (1%, ) 48% (45%, 51%) 80% (76%, 8) 41% (38%, 45%) 5% (, 9%) 58% (56%, 61%) 58% (49%, 68%) 41% (37%, 45%) (%, ) 41% (39%, 4) 7% (57%, 87%) 35% (5%, 46%) 0% (0%, 1%) (11%, 35%) 4 (4, 46%) 50% (6%, 75%) 48% (4, 51%) (, 5%) 46% (45%, 47%) 7 (70%, 76%) 37% (3, 39%) (1%, ) 49% (47%, 51%) 47% (45%, 49%) 8 (77%, 91%) 59% (5, 65%) (0%, 9%) 58% (5%, 6) 7 (69%, 77%) 48% (47%, 50%) 78% (76%, 79%) 49% (47%, 50%) 5% (, 5%) 50% (49%, 51%) 78% (76%, 80%) 4 (41%, 45%) (, 6%) 57% (55%, 58%) 51% (50%, 5%) 8% (81%, 8) 5 (5%, 55%) 10% (8%, 1%) 6 (6, 65%) 78% (76%, 80%) 46% (4, 48%) (, ) 49% (48%, 50%) 77% (75%, 78%) 41% (40%, 4) 6% (5%, 8%) 56% (5, 57%) * Plotted values are in bold print. D = distant; L = local; R = reional. Cure model did not fully convere. Cure estimate is sinificantly above net survival (one-sided Z-test, P <.05). 5-year net survival estimate is based on less than 5 years of data. 48 Journal of the National Cancer Institute Monoraphs, No. 49, 014

6 Fiure. Cure fraction and median survival time for the uncured by dianosis year and model (relative [rel], cause-specific [cs], flexible [flex], lo normal [ln], weibull); an example of reional stae colorectal cancer from the test cohort (dianosis years with follow-up to 1986). Fiure 3. Cure fraction and median survival time for the uncured by dianosis year and cohort*, an example from the Weibull relative model of reional stae colorectal cancer. *Test cohort includes dianosis years with follow-up to December Optimal cohort survival and a steep trajectory for the cure fraction. The optimal and recent cohorts are both subsets of the full cohort, however modelin the full cohort does not match results from the more restricted cohorts (test and recent). These plots demonstrate how cure, median survival, and dianosis year are interconnected, however these results are limited to a few models of reional stae colorectal cancer. We then investiated results from the full and recent cohorts by model and study desin. The 8 cure projections for colorectal cancer raned 75% 9% for localized stae, 5 7% for reional stae, 15% for distant stae, and 5 67% for all staes combined (Table 3, recent cohort). For breast cancer, the 8 cure projections were 90% 96% (cause-specific) for localized stae, 65% 96% for reional stae, 1% 6% for distant stae, and 66% 88% for all staes combined based on results from the recent cohort. There is much more variability in the 8 cure includes dianosis years with follow-up to December 9. Recent cohort includes dianosis years with follow-up to December 9. Full cohort includes dianosis years with follow-up to December 9. fraction estimates compared to the 1985 cure fraction estimates. Cure fraction estimates from the full cohort tended to be hiher than estimates from the recent cohort, as displayed in Fiure 3. Cohort differences were reatest for the lonormal models. There was more stability in the full cohort cure fraction estimates across models and more precise confidence intervals, but this is more likely due to the sample sizes than accuracy of the estimates. The flexible model did not convere for most cases of the full cohort. When all staes were combined, stability improved across models. Discussion Cure fractions are difficult to estimate because they imply extrapolations beyond the observed survival time. They vary dependin on cancer site, stae, survival outcome type, lenth of follow-up, and Journal of the National Cancer Institute Monoraphs, No. 49,

7 Table 3. 8 cure (95% confidence interval) projections by stae, years follow-up, cancer survival type, and survival function for colorectal and breast cancer patients dianosed between 1998 and 8 (recent cohort) or (full cohort) and followed up throuh 9 Cause-specific survival Relative survival Lo normal Weibull Lo normal Weibull Flexible Site Stae Recent cohort Full cohort Recent cohort Full cohort Recent cohort Full cohort Recent cohort Full cohort Recent cohort Full cohort Colorectal cancer L 75% (55%, 95%) R 56% (49%, 6%) D (%, 6%) All 5 (50%, 56%) Breast cancer L 90% (8, 96%) R 65% (5, 77%) D % (0%, 6%) All 66% (57%, 75%) 76% (7, 78%) 61% (61%, 6%) 9% (9%, 10%) 59% (59%, 60%) 88% (88%, 89%) 7% (71%, 7%) 15% (1, 16%) 80% (80%, 81%) 8 (66%, 99%) 65% (61%, 70%) 7% (5%, 9%) 60% (58%, 6) 96% (95%, 98%) 96% (96%, 97%) 17% (7%, 7%) 85% (8%, 87%) 8 (8, 8) 6 (6, 65%) 10% (9%, 10%) 6% (6%, 6) 9% (9%, 9%) 75% (75%, 76%) 19% (17%, 1%) 8 (8, 8) 80% (58%, 1%) 5 (4, 6) (%, 6%) 5 (49%, 57%) 95% (9%, 99%) 67% (55%, 79%) 1% (0%, ) 70% (60%, 79%) 77% (7, 81%) 6 (6, 65%) 9% (8%, 10%) 61% (61%, 6%) 95% (95%, 96%) 75% (7, 76%) 1 (1%, 16%) 85% (85%, 86%) 9% (8, 1%) 6 (57%, 70%) 7% (5%, 9%) 60% (57%, 6) 86% (85%, 87%) 67% (66%, 67%) 10% (9%, 10%) 6 (6, 65%) 96% (96%, 96%) 85% (80%, 89%) 1 (, ) 86% (8, 89%) 77% (77%, 78%) 17% (16%, 19%) 88% (87%, 88%) 9% (90%, 9) 7% (70%, 7) 15% (1, 17%) 67% (66%, 68%) 97% (96%, 98%) 8 (8%, 86%) 6% (, 30%) 88% (87%, 89%) 30% (9%, ) 4 (41%, 46%) * Plotted values are in bold print. D = distant; L = local; R = reional. Cure model did not fully convere. 50 Journal of the National Cancer Institute Monoraphs, No. 49, 014

8 dianosis years included in the cohort. In our analyses, we have shown that only in particular cases, the cure fraction estimates were stable across all models, cohorts, and lenths of follow-up. For colorectal cancer, we attained a stable 1985 cure fraction estimate for reional stae. For localized stae, cure estimates are too inconsistent across models to be conclusive. It is possible that additional heteroeneity due to unmeasured confoundin contributed to the instability of localized cure fraction estimates, or that additional follow-up time is needed to attain cure. The distant stae subroup had consistent results across models tested; however, the cure fraction estimates were very low. Despite this, our results compare reasonably well with recent estimates of colorectal cure fractions in Enland (15) and Japan (16). Breast cancer cure fractions are difficult to estimate (17) mainly because patients may have a reoccurrence after many years of bein symptom free. Cure fraction estimates from only 1 years of data are sinificantly hiher than cure fraction estimates obtained from 35 years of data, indicatin that many cancer deaths occur beyond the follow-up period (6). Biased estimates from models with relative survival outcomes (8) also overestimate the cure fraction especially for localized stae. Lo normal models with cause-specific survival are more conservative than other models. Others have proposed models allowin for more flexibility or radual decline in the survival curve (18,19). After evaluatin the stability of the estimate (usin the test and optimal cohorts), we extrapolated trends in the cure fraction to make predictions for patients dianosed in 8 and compared results from the recent and the full cohorts. The full cohort tended to have hiher estimates of the cure fraction than the recent cohort and have more converence problems. Converence improved when we restricted the analysis to the most recent 1 years of data. The restricted cohorts reduce the heteroeneity in the data that may be due to chanes in treatment and dianosis over time. Models for both the optimal and recent cohorts were adjusted for dianosis year as a continuous covariate; however the recent cohort demonstrates how limitin the number of dianosis years and havin a more homoenous cohort ives potentially more accurate estimates of the cure fraction. It is also possible that the year of dianosis could be modeled by joinpoint or as a nonlinear covariate to improve fit of the full cohort. Still, there was reater variability in the 8 projections than observed in It is possible that treatment improvements have increased the median survival time since 1985 so that more follow-up time is needed to stabilize the estimate (11). If survival improves incrementally across dianosis years, then the model can borrow strenth from past years to improve stability of the cure estimate. However, if a major chane in treatment occurs that dramatically improves survival outcomes, then past years may not be helpful to improvin the stability of the cure estimate. Model choice and survival outcome type also influence the cure fraction estimates. Relative survival estimates tend to produce more fluctuations in the survival curve than cause-specific estimates and these tend to ive hiher cure fraction estimates when fit by flexible and Weibull models. Flexible models consistently produced hiher estimates of the cure fraction compared to parametric models. This could be due to the cure fraction overcompensatin for the effect of dianosis year, since the trend in median survival showed minimal chane over time. It may be possible to adjust the baseline hazard for dianosis year by addin a time varyin covariate to the spline function (3), however interactions between two continuous predictors are complex to implement and interpret. Simplified models with a cateorical time varyin covariate for dianosis year show only sliht modifications to the cure estimate and small adjustments to the median survival time across dianosis years. Another assumption of the flexible model is that cure is reached on or before the placement of the last knot. Andersson et al. (3) explored the sensitivity of the flexible cure model to knot placement and found that the cure estimate and median survival time were fairly robust to the number of knots and their placement; however, they recommend usin these models where cure is observed within the iven follow-up time. An advantae to the flexible model is that one does not have to define a specific distribution for the survival curve. An advantae to the parametric cure models is that one does not have to specify when cure occurs. Both the parametric mixture and flexible models rely on the Newton-Raphson alorithm to solve for parameter estimates. This method occasionally fails to convere in cases where there is a flat likelihood or where the data are multimodal. We found our models did not convere for some cases of localized and distant stae disease and for the full cohort where the flexible model was applied. For the parametric mixture model, we eased the converence criteria (1) and for the flexible model, startin values were estimated from a linear function of lo time (13); however, neither approach improved converence of the models. Li et al. (4) and Yu et al. (5) separately evaluated the identifiability of the cure fraction and recommended that cure estimates should be based on lare cohorts with lon follow-up data. Sites prone to flat likelihood estimates (e, models that fail to convere) may be stratified or adjusted to encourae reater homoeneity in the data. An advantae to the data in the SEER proram is the availability of lare cohorts of data with lon-term followup. With the optimal cohort includin 35 years of data, stratified by stae, specific in ae and ender, we improved the stability of the 1985 cure estimates for breast cancer. Althouh stae is one of the most important predictors of cancer survival, more data are needed to identify and include other important covariates in cure models to explain the excess variation, such as histoloy and estroen receptor (ER) status. There is some eneral skepticism of cure estimates in breast cancer patients because breast cancer can relapse decades after dianosis and these predictions are based on extrapolations beyond the end of the observed data. If after complete follow-up, we still cannot observe cure, then we are limited in what can be predicted from the data. It is important to keep the medical perspective in mind when evaluatin the presence of cure in these more difficult cases. In these analyses, we compared study desins with different follow-up times to determine if cure fractions could be accurately predicted with minimal follow-up time so that we could provide more recent estimates. For colorectal cancer, 1 years was considered adequate to achieve a cure estimate, however we found a sliht overestimate in cure for most cases with shorter follow-up time indicatin that we may be missin a few events in the 1-year follow-up period. For breast cancer, shorter follow-up time exaerated the overestimation of cure due increased numbers of unobserved cancer deaths. Journal of the National Cancer Institute Monoraphs, No. 49,

9 Yu et al. (0) compared several cure models includin the mixture, nonmixture, and flexible models for relative survival usin rouped and individual data. Under ideal conditions, all models produced similar results. Less than ideal conditions where cure could not be identified or the model was misspecified led to inconsistent results and biased cure estimates. We aree with this assessment althouh it was limited to relative survival models with Weibull and flexible distribution functions. They recommend the flexible model in cases where the parametric model fails. In contrast, our findins indicated that the flexible model always produce hiher estimates of the cure fraction and we do not recommend reportin cure where there is instability in the results from parametric models. Instead we prefer to improve stability of the parametric mixture model throuh stratification and adjustment for covariates, such as dianosis year. The authors also noted that differences in scale of the time interval can influence estimates from rouped survival data. We did not test this in our models, as all survival time was specified in yearly intervals. Based on our findins, we recommend: runnin multiple models of different parametric families to determine the stability of the cure fraction estimate, examinin plots for fit of the mixture cure model and for a levelin off of the survival curve tail to support existence of cure, and stratifyin and adjustin the models for important covariates to improve homoeneity in the data. Cohorts should span a limited time frame to avoid major chanes in screenin or treatment of the disease that could introduce heteroeneity in the data. Choice of cause-specific or relative survival outcomes should be determined by weihin prior knowlede of the matchin life table to the accuracy of cause of death information to avoid introducin bias (8). Lenth of follow-up should be determined by the median survival time and optimized to capture the majority of cancer deaths in the observation period. Cure models are potentially useful for projectin lon-term survival estimates for cancer populations at the time of dianosis. However, modelers should be cautious in interpretin these estimates under less ideal conditions, includin shorter follow-up time, and small sample sizes. Special considerations should also be made about how to incorporate covariates into the model, model selection, and study desin. For example, we see stable cases, such as reional stae colorectal cancer: between 40% and 50% in 1985 increasin to 55% 65% by 8. These estimates are realistic and can be obtained with the data available. However, other cancer sites may require more follow-up time to attain a stable estimate. Appendix A: Variance and CI for Statistical Cure from Parametric Mixture Model Given the parameter estimates β and β c0 c1, consider the transformation (β, β ). The approximate variance of (β, β c0 c1 c0 c1 ) can be found by the Delta Method: where G = [ 11 1 ] so that 11 var[ ( β, c0 β c1 )] = GVG, c0 c1 c0 c1x = ( β, β ) exp( β β ) = β c 0 = c 0 β β [ 1+ exp( β c0 β c1x )], c0 1 c0 c1 X c0 c1x = ( β, β ) = exp( β β ) β c1 = c1 β β [ 1+ exp( β c0 β c1x )], c1 and V is the variance-covariance matrix of (β, β c0 c1 ). Then, where, var[ (, β )] [ ] var( c0 β c1 11 β c0 ) + [ 1 ] var( β c ) + [ ][ ]cov( β c, β c ) A XA var( = β c0 ) + ( 1+ A) ( 1+ A) var( A β 1) + cov( β 4 0, c X c β c1), ( 1+ A) A= exp( β c 0 β c1 X). The Wald test statistic for testin H 0 :(β, β c0 c1 ) = 0 vs the two-sided alternative is Z = ( β c, 0 β c1), var[ ( β c, 0 β c1)] where Z is approximately Standard Normal Distribution. A (1 α) 1% CI for (β c0, β c1 ) = c is iven by ( β c, β c ) a var[ ( β c, 0 1 ± Z 0 β c1)], where Z a is the upper a -th percentile of the Standard Normal Distribution. Appendix B: Variance and CI for Statistical Cure From Flexible Parametric Model Given the parameter estimates γ and β, consider the transformation (γ, β). The approximate variance of (γ, β) can be found by the Delta Method: where G = [ 11 1 ] so that 11 1 var[ ( γ, β)] GVG, = ( γ, β), γ [ exp( γ + β X )], = γ = γ β = β = exp[ exp( γ + βx )] = ( γ, β) = exp[ exp( γ + βx )] γ = γ, β = β β [ exp( γ + β X)][ X], 5 Journal of the National Cancer Institute Monoraphs, No. 49, 014

10 and V is the variance-covariance matrix of (γ, β ). Then, where var[ (, γ )] [ ]var( β 11 γ ) + [ 1] var( β ) + [ ][ ] cov( γ, β) 11 1 = [ B exp( B)]var( γ ) + [( BX )exp( B)]var( β) + [ BXexp( B)]cov( γ, β), B= exp( γ + βx). The Wald test statistic for testin H 0 :(γ, β ) = 0 vs the two-sided alternative is Z = ( γ, β), var[ ( γ, β)] where Z is approximately Standard Normal Distribution. A (1 α) 1% CI for (γ, β) = c is iven by ( γ, β) ± Z var[ ( γ, β)], a where Z a is the upper a -th percentile of the Standard Normal Distribution. Appendix C: Median Survival in the Uncured Fraction for the Flexible Parametric Model without Time Dependent Covariates The flexible parametric model adjustin for dianosis year, Y, is as follows: Rt () = exp[ exp( γ + β( Y) + γ 01v1(ln( t)) + γ 0v(ln( t)) + γ 03v3(ln( t)) γ 0 kvk 1 (ln( t)))] where β, γ,, γ 0k 1, and v 1 (ln(t)),, v k 1 ln(t)) are as defined in Data and Methods. We rewrite this as a mixture model (3): where, and R t c c c Fz ( t) () = + ( 1 ) c 1 c c = exp[ exp( γ + β( Y))] Fz() t = exp[ γ 0v(ln( t)) + γ 03v(ln( t)) γ 0k 1vk 1 (ln( t))] [] so that the uncured fraction is assumed to follow a survival distribution function, c Fz ( t ) c. To estimate the median survival in the 1 c uncured fraction, we set c Fz ( t ) c = 0.5 and solve for t. 1 c Note that in the STATA output spline parameters γ 0,, γ 0k 1 are orthoonalized and the notation is reversed from the written formulas, so that γ 0j, λ j, and v j are γ 0K j, λ K j, and v K j, respectively. References 1. Gamel JW, Weller EA, Wesley MN, Feuer EJ. Parametric cure models of relative and cause-specific survival for rouped survival times. Comput Methods Prorams Biomed. 0;61(): Lambert PC, Thompson JR, Weston CL, Dickman PW. Estimatin and modelin the cure fraction in population-based cancer survival analysis. Biostatistics. 7;8(3): Andersson TM, Dickman PW, Eloranta S, Lambert PC. Estimatin and modellin cure in population-based cancer studies within the framework of flexible parametric survival models. BMC Med Res Methodol. 011;11: Li CS, Taylor JMG, Sy JP. Identifiability of cure models. Stat Probabil Lett. 1;54(4): Yu B, Tiwari RC, Cronin KA, Feuer EJ. Cure fraction estimation from the mixture cure models for rouped survival data. Stat Med. 4;3(11): Tai P, Yu E, Cserni G, et al. Minimum follow-up time required for the estimation of statistical cure of cancer patients: verification usin data from 4 cancer sites in the SEER database. BMC Cancer. 5;5: Lambert PC, Dickman PW, Osterlund P, Andersson T, Sankila R, Glimelius B. Temporal trends in the proportion cured for cancer of the colon and rectum: a population-based study usin data from the Finnish Cancer Reistry. Int J Cancer. 7;11(9): Howlader N, Ries LA, Mariotto AB, Reichman ME, Ruhl J, Cronin KA. Improved estimates of cancer-specific survival rates from population-based data. J Natl Cancer Inst. 010;10(0): Haybittle JL. Curability of breast cancer. Br Med Bull. 1991;47(): Cho H, Howlader N, Mariotto AB, Cronin KA. Estimatin Relative Survival for Cancer Patients from the SEER Proram Usin Expected Rates Based on Ederer I Versus Ederer II Method. Bethesda, MD: National Cancer Institute; 011. Technical Report No Huan L, Cronin KA, Johnson KA, Mariotto AB, Feuer EJ. Improved survival time: what can survival cure models tell us about population-based survival improvements in late-stae colorectal, ovarian, and testicular cancer? Cancer. 8;11(10): Yu B, Tiwari RC, Cronin KA, McDonald C, Feuer EJ. CANSURV: a Windows proram for population-based cancer survival analysis. Comput Methods Prorams Biomed. 5;80(3): Lambert PC, Royston P. Further development of flexible parametric models for survival analysis. Stata J. 9;9(): Royston P, Parmar MK. Flexible parametric proportional-hazards and proportional-odds models for censored survival data, with application to pronostic modellin and estimation of treatment effects. Stat Med. ;1(15): Shack LG, Shah A, Lambert PC, Rachet B. Cure by ae and stae at dianosis for colorectal cancer patients in North West Enland, : a population-based study. Cancer Epidemiol. 01;36(6): Ito Y, Nakayama T, Miyashiro I, et al. Trends in cure fraction from colorectal cancer by ae and tumour stae between 1975 and 0, usin population-based data, Osaka, Japan. Jpn J Clin Oncol. 01;4(10): Woods LM, Rachet B, Lambert PC, Coleman MP. Cure from breast cancer amon two populations of women followed for 3 years after dianosis. Ann Oncol. 9;0(8): Journal of the National Cancer Institute Monoraphs, No. 49,

11 18. Zhan JJ, Pen YW. Accelerated hazards mixture cure model. Lifetime Data Anal. Dec 9;15(4): Lambert PC, Dickman PW, Weston CL, Thompson JR. Estimatin the cure fraction in population-based cancer studies by usin finite mixture models. J Roy Stat Soc C-App. 010;59(1): Yu XQ, De Anelis R, Andersson TM, Lambert PC, O Connell DL, Dickman PW. Estimatin the proportion cured of cancer: some practical advice for users. Cancer Epidemiol. 013;37(6): Notes We thank Dr Therese Andersson for her helpful explanation of the flexible model and the STATA stpm software packae. Affiliations of authors: Surveillance Research Proram, Division of Cancer Control and Population Sciences, National Cancer Institute, National Institutes of Health, Bethesda, MD (MRS, EJF, ABM). 54 Journal of the National Cancer Institute Monoraphs, No. 49, 014