Download Current Estimates of the Cure Fraction: A

DOI:10.1093/jncimonographs/lgu015
Published by Oxford University Press 2014.
Current Estimates of the Cure Fraction: A Feasibility Study of
Statistical Cure for Breast and Colorectal Cancer
Margaret R. Stedman, Eric J. Feuer, Angela B. Mariotto
Correspondence to: Margaret R. Stedman, PhD, MPH, National Cancer Institute, Division of Cancer Control and Population Sciences, Surveillance Research
Program, 9609 Medical Center Drive, Bethesda, MD 20892-9765 (e-mail: [email protected]).
Background
The probability of cure is a long-term prognostic measure of cancer survival. Estimates of the cure fraction, the
proportion of patients “cured” of the disease, are based on extrapolating survival models beyond the range of
data. The objective of this work is to evaluate the sensitivity of cure fraction estimates to model choice and study
design.
Methods
Data were obtained from the Surveillance, Epidemiology, and End Results (SEER)-9 registries to construct a
cohort of breast and colorectal cancer patients diagnosed from 1975 to 1985. In a sensitivity analysis, cure fraction estimates are compared from different study designs with short- and long-term follow-up. Methods tested
include: cause-specific and relative survival, parametric mixture, and flexible models. In a separate analysis,
estimates are projected for 2008 diagnoses using study designs including the full cohort (1975–2008 diagnoses)
and restricted to recent diagnoses (1998–2008) with follow-up to 2009.
Results
We show that flexible models often provide higher estimates of the cure fraction compared to parametric mixture models. Log normal models generate lower estimates than Weibull parametric models. In general, 12 years
is enough follow-up time to estimate the cure fraction for regional and distant stage colorectal cancer but not
for breast cancer. 2008 colorectal cure projections show a 15% increase in the cure fraction since 1985.
Discussion
Estimates of the cure fraction are model and study design dependent. It is best to compare results from multiple
models and examine model fit to determine the reliability of the estimate. Early-stage cancers are sensitive to
survival type and follow-up time because of their longer survival. More flexible models are susceptible to slight
fluctuations in the shape of the survival curve which can influence the stability of the estimate; however, stability
may be improved by lengthening follow-up and restricting the cohort to reduce heterogeneity in the data.
J Natl Cancer Inst Monogr 2014;49:244–254
Cure is difficult to identify at the individual level because for certain cancers, there can be reoccurrences many years after periods of
being 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 general population after a long
follow-up period. This concept of cure, known as statistical cure,
represents a long-term prognostic measure of the chances of being
cured, and it is a desirable statistic for the patient and clinician.
Several models exist to estimate statistical cure [eg, mixture (1),
nonmixture (2), and flexible (3) models] and all of these models rely
on the length of follow-up to attain a stable estimate of statistical cure (4). Although cancer registries may have long histories of
incidence and follow-up data, recent advances in treatment, screening, and changes in staging criteria impact survival and complicate
the estimation of statistical cure. Furthermore, there is interest in
predicting the cure fraction for patients recently diagnosed with
minimal follow-up information.
There has been some research to determine the minimal length
of follow-up needed to obtain a stable cure estimate. Yu and colleagues (5) found that length of follow-up time needs to be at least
244
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 using median
survival time and found that the minimal length of follow-up is
site specific. For example, colon cancer was estimated to require a
minimum of 12.2 years and breast cancer, a minimum of 36.2 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 varying model
assumptions.
In this paper, we use the Surveillance Epidemiology and End
Results (SEER) registry data to perform a sensitivity analysis of
the cure fraction estimate to model choice, survival outcome,
and study design. 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. Using
the most recent SEER data, we project the cure fraction by
stage and site for breast and colorectal cancer patients diagnosed
in 2008.
Journal of the National Cancer Institute Monographs, No. 49, 2014
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 using SEER breast and colorectal
cancer data. Next, we provide results from the sensitivity analysis
and future projections of the cure fraction for patients diagnosed
in 2008. Finally, we conclude with some guidance on projecting the
cure fraction using these models.
Data and Methods
Data and Cohort Definitions
We used the SEER site recode variables to identify patients diagnosed at ages 45–74 with colorectal cancer (SEER Site Recode
International Classification of Diseases for Oncology [ICD-O]-3
definition: C180-C189, C260, C199, and C209) and at ages 45–64
with female breast cancer (SEER Site Recode ICD-O-3 definition:
C500-C509) within the SEER-9 Registries. 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 age range was included to increase
the size of the colorectal cohort. We selected malignant cases and
excluded patients diagnosed at death and secondary tumors. All
data were obtained from the November 2011 submission, using
SEER*Stat software, version 8.02, available at our website: http://
seer.cancer.gov/seerstat.
We constructed four cohorts of breast cancer and colorectal cancer patients. The first two cohorts included female breast
and colorectal cancer patients diagnosed between 1975 and 1985
and two different follow-up periods: 12 years (follow-up through
December 1986) and 35 years (follow-up through December
2009). Presuming that having a longer follow-up time would optimize accuracy of the cure estimate, data from the 12- 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 diagnosed patients (diagnosed in 2008),
we constructed two other cohorts: patients diagnosed with female
breast and colorectal cancer from 1975 to 2008 (“full cohort”) and
patients diagnosed with breast and colorectal cancer from 1998 to
2008 (“recent cohort”). All patients were followed from diagnosis
until December 2009. Since entrance into the registry and length
of follow-up depend upon the date of diagnosis (longer follow-up
for the earlier diagnoses), there is confounding between the length
of follow-up and the diagnosis year. To reduce confounding and
also project the cure fraction for patients diagnosed in the most
recent years, we adjusted for diagnosis 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 examining a plot of cancer survival, the cure fraction is the
point in the survival curve where the curve begins to level off and
the slope is predicted to be zero. Sometimes, this occurs well beyond
the observed data (see Figure 1). From a relative survival perspective, cure occurs when the observed survival of the cancer population matches the expected survival of the general population. From
a cause-specific perspective, cure occurs when the cancer population
Figure 1. A 60% cure fraction estimated from the cancer survival curve.
Example results from CanSurv software.
is no longer dying from cancer and the hazard of dying 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 diagnosis year
and SEER historic stage (localized, regional, distant, and all stages
combined). Cause of death is based on the SEER cause-specific
death classification, which classifies deaths related to the cancer
diagnosis using patient death certificate information, tumor location, and comorbidities (http://seer.cancer.gov/causespecific/) (8).
Relative survival can give biased estimates for localized stage breast
cancer because expected survival estimated from the US population
life tables underestimate all-cause mortality in this subgroup (8).
Statistical Cure: Parametric Mixture Cure Model
Although 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, Sc(t), as a mixture of the cure fraction, c,
and the uncured fraction, (1 − c):
Sc (t ) = c + (1 − c ) G (t ; ξ,θ )
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, including the Weibull, log normal, log logistic, 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 log 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 log 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).
Journal of the National Cancer Institute Monographs, No. 49, 2014245
Table 1. Survival functions and their reparameterizations for the parametric mixture cure model
Distribution
Conventional survival
function G(t)
Log normal
Log logistic
Weibull
G(t; µ, σ) = 1 − Φ[(ln{t} − µ)/σ]
G(t; λ, ρ) = [1 − (λt)ρ]−1
G(t; λ, ρ) = exp[−(λt)ρ]
Reparameterization
Transformed G(t)
for CanSurv
Median survival
time in the uncured
ξ = µ, θ = σ
ξ = −lnλ, θ =1/ρ
ξ = −lnλ, θ =1/ρ
G(t; ξ, θ) = 1 − Φ[(ln{t} − ξ)/θ]
G(t; ξ, θ) = [1 + exp{(ln(t) − ξ)/θ}]−1
G(t; ξ, θ) = exp{−exp[(ln{t} − ξ)/θ]}
exp(ξ)
exp(ξ)
The parameters, c, ξ, and θ, may depend on covariates, X, such as
diagnosis year. Covariates are introduced into the models with the
parameters: βc, βξ, and βθ.
c = [1 + exp( − β c 0 − β c1 X )]−1
ξ = β ξ 0 + βξ1 X (ln(2))θ
exp(−ξ)
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 log cumulative excess hazard, ln(Λ(t)), using restricted cubic splines, s(ln(t); γ0),
of the log survival time, t, where γ0 contains the parameters of the
spline function:
ln( Λ (( t )) = s (ln( t ); γ 0 ) This is adapted to predict relative survival, R(t),
θ = exp[ βθ 0 + βθ1 X ]
Accuracy of the cure estimate is improved by adjusting for covariates
but it is also influenced by choice of parametric distribution and overall
fit to the data. As in Huang et al. (11), we allowed the cure fraction, c,
and the median survival in the uncured, ξ, to change simultaneously
as a function of diagnosis year, however additional covariates could
be included in the model. We estimated the parametric mixture cure
model by maximum likelihood estimation methods using CanSurv software (12) available at http://surveillance.cancer.gov/cansurv/. 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 diagnosed in a given year, Y, can be estimated as
c = [1 + exp( − β c 0 − β c1 (Y ))]−1
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 general population and excess mortality is zero. In flexible parametric
survival models, this can be accomplished by constraining the log
cumulative excess hazard function to have a zero slope after a certain point in time.
This is implemented by specifying the K knots from the spline
function in reverse order (kK, …, k1) where the last spline parameter
is restricted to be zero (γ01 = 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:
[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 stage and
adjusted for diagnosis year (11), a continuous covariate, so that we
could obtain site-, stage-, and year-specific estimates. We estimated
median survival of the uncured assuming either a Weibull or log
normal survival function. Parameter estimates were exported to SAS
software to calculate projections of the cure fraction and respective
confidence intervals (CIs) for patients diagnosed in a given 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 modeling the log 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). Integrating these hazards gives more stable estimates of the cumulative mortality, H(t), the cumulative expected
246
ln[ − ln( R( t ))] = s (ln( t ); γ ↓ 0 )
R( t ) = exp[ −exp(γ
+γ
00
+γ
v (ln( t )) + γ
01 1
v (ln( t )) + ... + γ
03 3
v (ln( t ))
02 2
v (ln( t )))]
0 k −1 k −1
In the backwards spline function, γ01, …, γ0k − 1 are the parameters and v1(ln(t)), …, vk − 1(ln(t)) are the basis functions defined as:
v1 (ln( t )) = ln(t ),and
v j (ln(t)) = ( kK − j − ln(t ))3+ − λ j ( kmax − ln(t ))3+ − (1 − λ j )( kmin − ln(t ))3+ , for j = 2, ... , K − 1
where,
λj =
kK − j − k1
kK − k1
Cure, c, is then estimated as,
c = exp[ − exp(γ 00 )]
Journal of the National Cancer Institute Monographs, No. 49, 2014
To adjust for a covariate such as diagnosis year (Y) in the model, we
added the covariate, Y, as follows:
R( t ) = exp[ −exp(γ
+γ
00
+ β (Y ) + γ
v (ln( t )) + ... + γ
03 3
v (ln( t )) + γ
01 1
v (ln( t )))]
0 k −1 k −1
v (ln( t ))
02 2
[2]
so that cure may be predicted for any given diagnosis year, Y:
c = exp[ − exp(γ 00 + 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 varying effects may also be
added to more complex models (3).
The flexible cure model can be estimated using the stpm2 package (13) available in STATA v12. The package is designed to model
relative survival. Although 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 length 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 stpm2 package 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)12] in the flexible model.
Sensitivity Analyses to Length of Follow-up, Model, and
Survival Type
We investigated the sensitivity of cure fraction estimates to length
of follow-up by comparing 1985 cure fraction estimates from the
12-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
significantly above the 25-year observed survival estimates were
considered poor fits to the data.
Sensitivity of 2008 Projected Cure Fraction to Study
Design
Since the cure fraction is a function of diagnosis year in the parametric mixture and flexible parametric models, the predicted cure
fraction was extrapolated to 2008 by setting Y = 2008 in equations
1 and 3. Two different study designs were tested. In one design,
patients were selected from diagnosis years 1998–2008 (recent
cohorts). However, in these cohorts, the lengths of follow-up may
not be long enough to provide stable estimates of the cure fraction.
So, in a second design, the number of diagnosis years was expanded
to include 1975–2008 (full cohorts). While the full cohort provides
sufficient follow-up, a linear trend in diagnosis year over this long
interval may not be appropriate. 2008 projections were compared
from both study designs for cause-specific and relative survival
parametric mixture and flexible models.
Results
Sensitivity to Length of Follow-up: Comparison of 1985
Estimates From the Test and Optimal Cohorts
The 1985 cure fraction for colorectal cancer ranged between 55%
and 78% for localized stage, 41% and 48% for regional stage, 3%
for distant stage, and 42% and 50% for all stages combined (Table 2,
optimal cohort). With the exception of localized stage, cure estimates
are fairly stable from the optimal cohort and similar to estimates
obtained using only 12 years of data (test cohort). For localized stage,
there is more variation across models and the estimates have wider
CIs than other stages. For regional stage, there is less variation across
models and tighter confidence intervals; however, the Weibull and
flexible models from the test cohort tend to slightly overestimate
the cure fraction compared to the 25-year observed cancer survival
(Weibull = 48%, Flexible = 49%, 25-year relative survival = 44%).
For distant stage disease, two of the models did not converge. The
cure fraction was very small, slightly above zero.
The 1985 breast cancer cure fraction estimates showed more
variability and ranged between 57% and 67% (cause-specific) for
localized stage, 35% and 43% for regional stage, 3% and 6% for
distant stage, and 43% and 57% for all stages combined (Table 2,
optimal cohort). Test cohort estimates for breast were less reliable and less precise than the colorectal estimates. In general, test
estimates exceeded the optimal estimates, indicating that 12 years
of data is not enough to obtain an accurate prediction of the cure
fraction. Weibull and flexible estimates tended to be higher than
lognormal estimates and all cases of the flexible model for the test
cohort exceeded the 25-year cancer survival estimate. Distant stage
results were similarly influenced by small numbers of at risk patients
at the end of the survival curve (fewer than 10 survivors 25 years
postdiagnosis). Combining all stages decreased the stability of the
estimate. Although most models converged, test estimates fluctuated widely between 23% and 64%. Relative survival estimates
were higher than cause-specific estimates for the optimal cohort
and these differences were most apparent for localized stage.
2008 Projections of the Cure Fraction: Sensitivity to
Diagnosis Year and Length of Follow-up
To illustrate modeling the cure fraction as a function of calendar
time, we plotted the cure fraction and median survival time estimates
by diagnosis year from each survival model using the test cohort of
patients diagnosed with regional colorectal cancer in 1975–1985 and
followed through 1986. For the parametric mixture models, there is
compensation between the cure fraction estimates and median survival
times, where models with higher cure fractions have lower median
survival times (Figure 2). In the case of the flexible model, there is a
comparatively steep increase in the cure fraction (slope = 0.0053) and
a comparatively flat trend in the median survival (slope = 0.0073) over
time. The cause-specific Weibull model shows a similar increase in the
cure fraction (slope = 0.0058) compared to the flexible model, but with
a slightly steeper trend in median survival over time (slope = 0.0185).
In Figure 3, we compare the change over time in the cure fraction
and median survival time by cohort for the Weibull model with relative survival for regional stage 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 Monographs, No. 49, 2014247
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Journal of the National Cancer Institute Monographs, No. 49, 2014
All
D
R
L
All
D
R
L
Stage
72%
(70%, 74%)§
43%
(41%, 45%)
3%
(2%, 5%)§
47%
(45%, 48%)
74%
(72%, 76%)
42%
(39%, 44%)
7%
(4%, 10%)§
56%
(55%, 58%)
53%
(40%, 67%)
44%
(40%, 47%)
3%
(2%, 4%)
44%
(42%, 46%)
66%
(52%, 80%)
40%
(31%, 49%)
0%
(0%, 0%)
31%
(20%, 41%)
Test cohort
55%
(51%, 59%)
41%
(40%, 42%)
3%
(3%, 4%)
42%
(41%, 43%)
57%
(53%, 60%)
35%
(34%, 37%)
4%
(2%, 5%)
43%
(41%, 45%)
Optimal
cohort
Log normal
77%
65%
(71%, 82%) (62%, 67%)
50%
42%
(47%, 53%)‡ (41%, 44%)
4%
3%
(3%, 6%)
(3%, 4%)
49%
44%
(47%, 51%)‡ (43%, 45%)
—†
67%
(65%, 69%)
62%
40%
(57%, 67%)‡ (39%, 42%)
4%
5%
(0%, 10%)
(3%, 6%)
61%
51%
(55%, 66%) (50%, 52%)
§ 25-year net survival estimate is based on less than 25 years of data.
Optimal
cohort
Weibull
Test cohort
‡ Cure estimate is significantly above net survival (one-sided Z-test, P < .05).
† Cure model did not fully converge.
* Plotted values are in bold print. D = distant; L = local; R = regional.
Breast cancer
Colorectal
cancer
Site
25-year
cancer
survival
Cause-specific survival
77%
(70%, 82%)
44%
(39%, 48%)
2%
(1%, 4%)§
48%
(45%, 51%)
80%
(76%, 83%)
41%
(38%, 45%)
5%
(3%, 9%)§
58%
(56%, 61%)
Optimal
cohort
Test cohort
Test cohort
Optimal
cohort
Flexible
73%
78%
78%
(69%, 77%) (76%, 79%) (76%, 80%)
48%
49%
46%
(47%, 50%)‡ (47%, 50%)‡ (44%, 48%)
—†
5%
3%
(4%, 5%)‡ (3%, 4%)
50%
51%
49%
(49%, 51%) (50%, 52%) (48%, 50%)
78%
82%
77%
(76%, 80%) (81%, 83%)‡ (75%, 78%)
43%
53%
41%
(41%, 45%) (52%, 55%)‡ (40%, 43%)
4%
10%
6%
(3%, 6%)
(8%, 12%)‡ (5%, 8%)
57%
64%
56%
(55%, 58%) (63%, 65%)‡ (54%, 57%)
Optimal
cohort
Weibull
58%
50%
(49%, 68%) (26%, 75%)
41%
44%
48%
(37%, 45%) (43%, 46%) (44%, 51%)
3%
—†
4%
(2%, 4%)
(3%, 5%)‡
41%
46%
47%
(39%, 44%) (45%, 47%) (45%, 49%)
72%
73%
84%
(57%, 87%) (70%, 76%) (77%, 91%)
35%
37%
59%
(25%, 46%) (34%, 39%) (53%, 65%)‡
0%
3%
4%
(0%, 1%)
(1%, 4%)
(0%, 9%)
23%
49%
58%
(11%, 35%) (47%, 51%) (52%, 64%)
—†
25-year cancer
survival
Test cohort
Log normal
Relative survival
Table 2. 1985 cure fraction (95% confidence interval) projections by stage, years follow-up, cancer survival type, and survival function for colorectal and breast cancer patients,
diagnosed between 1975 and 1985 and followed up through 1986 (test cohort) and 2009 (optimal cohort)*
Figure 2. Cure fraction and median survival time for the uncured by diagnosis year and model (relative [rel], cause-specific [cs], flexible [flex], log
normal [ln], weibull); an example of regional stage colorectal cancer from the test cohort (diagnosis years 1975–1985 with follow-up to 1986).
Figure 3. Cure fraction and median survival time for the uncured by
diagnosis year and cohort*, an example from the Weibull relative
model of regional stage colorectal cancer. *Test cohort includes diagnosis years 1975–1985 with follow-up to December 1986. Optimal cohort
includes diagnosis years 1975–1985 with follow-up to December 2009.
Recent cohort includes diagnosis years 1988–2008 with follow-up to
December 2009. Full cohort includes diagnosis years 1975–2008 with
follow-up to December 2009.
survival and a steep trajectory for the cure fraction. The optimal and
recent cohorts are both subsets of the full cohort, however modeling
the full cohort does not match results from the more restricted cohorts
(test and recent). These plots demonstrate how cure, median survival,
and diagnosis year are interconnected, however these results are limited to a few models of regional stage colorectal cancer.
We then investigated results from the full and recent cohorts
by model and study design. The 2008 cure projections for colorectal cancer ranged 75%–92% for localized stage, 54%–72% for
regional stage, 4%–15% for distant stage, and 53%–67% for all
stages combined (Table 3, recent cohort). For breast cancer, the
2008 cure projections were 90%–96% (cause-specific) for localized stage, 65%–96% for regional stage, 1%–26% for distant stage,
and 66%–88% for all stages combined based on results from the
recent cohort. There is much more variability in the 2008 cure
fraction estimates compared to the 1985 cure fraction estimates.
Cure fraction estimates from the full cohort tended to be higher
than estimates from the recent cohort, as displayed in Figure 3.
Cohort differences were greatest for the lognormal 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 converge for most cases of the full cohort.
When all stages were combined, stability improved across models.
Discussion
Cure fractions are difficult to estimate because they imply extrapolations beyond the observed survival time. They vary depending on
cancer site, stage, survival outcome type, length of follow-up, and
Journal of the National Cancer Institute Monographs, No. 49, 2014249
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Journal of the National Cancer Institute Monographs, No. 49, 2014
All
D
R
L
All
D
75%
(55%, 95%)
56%
(49%, 62%)
4%
(2%, 6%)
53%
(50%, 56%)
90%
(84%, 96%)
65%
(53%, 77%)
2%
(0%, 6%)
66%
(57%, 75%)
76%
(74%, 78%)
61%
(61%, 62%)
9%
(9%, 10%)
59%
(59%, 60%)
88%
(88%, 89%)
72%
(71%, 72%)
15%
(13%, 16%)
80%
(80%, 81%)
Full cohort
† Cure model did not fully converge.
* Plotted values are in bold print. D = distant; L = local; R = regional.
Breast cancer
L
Colorectal cancer
R
Stage
Site
Recent
cohort
Log normal
84%
(83%, 84%)
64%
(64%, 65%)
10%
(9%, 10%)
62%
(62%, 63%)
92%
(92%, 92%)
75%
(75%, 76%)
19%
(17%, 21%)
84%
(84%, 84%)
Full cohort
Weibull
83%
(66%, 99%)
65%
(61%, 70%)
7%
(5%, 9%)
60%
(58%, 63%)
96%
(95%, 98%)
96%
(96%, 97%)
17%
(7%, 27%)
85%
(82%, 87%)
Recent
cohort
Cause-specific survival
80%
(58%, 100%)
54%
(44%, 63%)
4%
(2%, 6%)
53%
(49%, 57%)
95%
(92%, 99%)
67%
(55%, 79%)
1%
(0%, 4%)
70%
(60%, 79%)
77%
(73%, 81%)
64%
(63%, 65%)
9%
(8%, 10%)
61%
(61%, 62%)
95%
(95%, 96%)
75%
(74%, 76%)
14%
(12%, 16%)
85%
(85%, 86%)
Full cohort
Log normal
Recent
cohort
85%
(80%, 89%)
13%
(4%, 23%)
86%
(83%, 89%)
92%
(83%, 100%)
63%
(57%, 70%)
7%
(5%, 9%)
60%
(57%, 63%)
—†
Recent cohort
86%
(85%, 87%)
67%
(66%, 67%)
10%
(9%, 10%)
64%
(64%, 65%)
96%
(96%, 96%)
77%
(77%, 78%)
17%
(16%, 19%)
88%
(87%, 88%)
Full cohort
Weibull
Relative survival
43%
(41%, 46%)
—†
—†
—†
30%
(29%, 32%)
—†
—†
—†
Full cohort
Flexible
92%
(90%, 93%)
72%
(70%, 74%)
15%
(14%, 17%)
67%
(66%, 68%)
97%
(96%, 98%)
84%
(82%, 86%)
26%
(23%, 30%)
88%
(87%, 89%)
Recent
cohort
Table 3. 2008 cure (95% confidence interval) projections by stage, years follow-up, cancer survival type, and survival function for colorectal and breast cancer patients diagnosed
between 1998 and 2008 (recent cohort) or 1975–2008 (full cohort) and followed up through 2009
diagnosis 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 lengths of follow-up. For
colorectal cancer, we attained a stable 1985 cure fraction estimate
for regional stage. For localized stage, cure estimates are too inconsistent across models to be conclusive. It is possible that additional
heterogeneity due to unmeasured confounding contributed to the
instability of localized cure fraction estimates, or that additional
follow-up time is needed to attain cure. The distant stage subgroup
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 England (15) and Japan (16). Breast cancer cure fractions are
difficult to estimate (17) mainly because patients may have a reoccurrence after many years of being symptom free. Cure fraction
estimates from only 12 years of data are significantly higher than
cure fraction estimates obtained from 35 years of data, indicating
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 stage.
Log normal models with cause-specific survival are more conservative than other models. Others have proposed models allowing for
more flexibility or gradual decline in the survival curve (18,19).
After evaluating the stability of the estimate (using the test and
optimal cohorts), we extrapolated trends in the cure fraction to make
predictions for patients diagnosed in 2008 and compared results
from the recent and the full cohorts. The full cohort tended to have
higher estimates of the cure fraction than the recent cohort and
have more convergence problems. Convergence improved when
we restricted the analysis to the most recent 12 years of data. The
restricted cohorts reduce the heterogeneity in the data that may
be due to changes in treatment and diagnosis over time. Models
for both the optimal and recent cohorts were adjusted for diagnosis year as a continuous covariate; however the “recent cohort”
demonstrates how limiting the number of diagnosis years and having a more homogenous cohort gives potentially more accurate
estimates of the cure fraction. It is also possible that the year of
diagnosis could be modeled by joinpoint or as a nonlinear covariate
to improve fit of the full cohort. Still, there was greater variability
in the 2008 projections than observed in 1985. 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 diagnosis years, then the model can borrow strength from past years to
improve stability of the cure estimate. However, if a major change
in treatment occurs that dramatically improves survival outcomes,
then past years may not be helpful to improving 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 give higher cure fraction estimates when fit by flexible
and Weibull models. Flexible models consistently produced higher
estimates of the cure fraction compared to parametric models. This
could be due to the cure fraction overcompensating for the effect of
diagnosis year, since the trend in median survival showed minimal
change over time. It may be possible to adjust the baseline hazard
for diagnosis year by adding a time varying covariate to the spline
function (3), however interactions between two continuous predictors are complex to implement and interpret. Simplified models
with a categorical time varying covariate for diagnosis year show
only slight modifications to the cure estimate and small adjustments to the median survival time across diagnosis 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 using these models where cure is observed within the given
follow-up time. An advantage to the flexible model is that one does
not have to define a specific distribution for the survival curve. An
advantage 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 algorithm to solve for parameter estimates. This
method occasionally fails to converge in cases where there is a flat
likelihood or where the data are multimodal. We found our models
did not converge for some cases of localized and distant stage disease and for the full cohort where the flexible model was applied.
For the parametric mixture model, we eased the convergence criteria (12) and for the flexible model, starting values were estimated
from a linear function of log time (13); however, neither approach
improved convergence 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 large cohorts with long follow-up data. Sites
prone to flat likelihood estimates (eg, models that fail to converge)
may be stratified or adjusted to encourage greater homogeneity in the data. An advantage to the data in the SEER program
is the availability of large cohorts of data with long-term followup. With the optimal cohort including 35 years of data, stratified
by stage, specific in age and gender, we improved the stability of
the 1985 cure estimates for breast cancer. Although stage 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 histology and
estrogen receptor (ER) status. There is some general skepticism of
cure estimates in breast cancer patients because breast cancer can
relapse decades after diagnosis 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 evaluating the presence of cure
in these more difficult cases.
In these analyses, we compared study designs 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, 12 years was considered adequate to achieve a cure estimate, however we found a
slight overestimate in cure for most cases with shorter follow-up
time indicating that we may be missing a few events in the 12-year
follow-up period. For breast cancer, shorter follow-up time exaggerated the overestimation of cure due increased numbers of unobserved cancer deaths.
Journal of the National Cancer Institute Monographs, No. 49, 2014251
Yu et al. (20) compared several cure models including the mixture,
nonmixture, and flexible models for relative survival using grouped
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 agree with this assessment although
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 findings indicated
that the flexible model always produce higher estimates of the cure
fraction and we do not recommend reporting cure where there is
instability in the results from parametric models. Instead we prefer
to improve stability of the parametric mixture model through stratification and adjustment for covariates, such as diagnosis year. The
authors also noted that differences in scale of the time interval can
influence estimates from grouped survival data. We did not test this
in our models, as all survival time was specified in yearly intervals.
Based on our findings, we recommend: running multiple models of different parametric families to determine the stability of
the cure fraction estimate, examining plots for fit of the mixture
cure model and for a leveling off of the survival curve tail to support existence of cure, and stratifying and adjusting the models for
important covariates to improve homogeneity in the data. Cohorts
should span a limited time frame to avoid major changes in screening or treatment of the disease that could introduce heterogeneity
in the data. Choice of cause-specific or relative survival outcomes
should be determined by weighing prior knowledge of the matching life table to the accuracy of cause of death information to avoid
introducing bias (8). Length 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 projecting long-term
survival estimates for cancer populations at the time of diagnosis.
However, modelers should be cautious in interpreting these estimates under less ideal conditions, including 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 design. For example, we see stable cases, such as
regional stage colorectal cancer: between 40% and 50% in 1985
increasing to 55%–65% by 2008. 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 c 0 and β c1, consider the transformation g(β c 0 , β c1). The approximate variance of g(β c 0 , β c1) can be
found by the Delta Method:
g ( β c 0 , β c1 )] = GVG’ ,
var[
g12 =
∂ g ( β c 0 , β c1 )
∂ β c1
β c 1 = β c 1
=
−X exp( −β c 0 − β c1 X )
,
[1 + exp( −β c 0 − β c1 X )]2
and V is the variance-covariance matrix of (β c 0 , β c1). Then,
g ( β c 0 , β c1 )] ≈ [ g ]2 var( β c 0 ) + [ g ]2
var[
11
12
var( β c1 ) + 2[ g ][ g ]cov( β c 0 , β c1 )
11
12
2
2
 −A 
 −XA 

 var( β c 0 ) + 
=
2
2
 (1 + A) 
 (1 + A) 
 A2 
 cov( β c 0 , β c1 ),
var( β c1 ) + 2X 
 (1 + A)4 
where,
A = exp( −β c 0 − β c1 X ).
The Wald test statistic for testing H0:g(β c 0 , β c1) = 0 vs the two-sided
alternative is
Z=
g ( β c 0 , β c1 )
,
g ( β c 0 , β c1 )]
var[
where Z is approximately Standard Normal Distribution. A (1 − α)
× 100% CI for g(βc0, βc1) = c is given by
g ( β c 0 , β c1 )],
g ( β c 0 , β c1 ) ± Z a var[
2
where Z a is the upper a -th percentile of the Standard Normal
2
2
Distribution.
Appendix B: Variance and CI for Statistical
Cure From Flexible Parametric Model

Given the parameter estimates γ 00 and β , consider the transforma

tion g(γ 00 , β ). The approximate variance of g(γ 00 , β ) can be found
by the Delta Method:
g (γ , β )] = GVG’ ,
var[
00
where G = [g11 g12] so that
g11 =
∂ g (γ 00 , β )
= exp[ −exp(γ 00 + β X )]
γ 00 = γ 00 , β = β
∂ γ 00
[ −exp(γ + β X )],
00
where G = [g11 g12] so that
252
∂ g ( β c 0 , β c1 )
g11 =
∂ β c0
β c 0 = β c 0
−exp( −β c 0 − β c1 X )
=
,
[1 + exp( −β c 0 − β c1 X )]2
g12 =
∂ g (γ 00 , β )
= exp[ −exp(γ 00 + β X )]
γ 00 = γ 00 , β = β
∂β
[ −exp(γ + β X )][ X ],
00
Journal of the National Cancer Institute Monographs, No. 49, 2014
so that the uncured fraction is assumed to follow a survival distriFz ( t )
− c . To estimate the median survival in the
bution function, c
1− c
Fz ( t )
− c = 0.5 and solve for t.
uncured fraction, we set c
1− c
Note that in the STATA output spline parameters γ02, …, γ0k−1
are orthogonalized and the notation is reversed from the written
formulas, so that γ0j, λj, and vj are γ0K − j, λK − j, and vK − j, respectively.

and V is the variance-covariance matrix of (γ 00 , β ). Then,
g (γ , β )] ≈ [ g ]2 var(γ ) + [ g ]2
var[
11
12
00
00
var( β ) + 2[ g ][ g ]cov(γ , β )
11
12
00
= [ B 2 exp( 2B )]var(γ 00 ) + [( BX )2 exp( 2B )]var( β )
+ 2[ B 2 X exp( 2B )]cov(γ , β ),
00
References
where
B = −exp(γ 00 + β X ).

The Wald test statistic for testing H0:g(γ 00 , β ) = 0 vs the two-sided
alternative is
Z=
g (γ 00 , β )
,
g (γ , β )]
var[
00
where Z is approximately Standard Normal Distribution. A (1 − α)
× 100% CI for g(γ00, β) = c is given by
g (γ , β )],
g (γ 00 , β ) ± Z a var[
00
2
where Z a is the upper a -th percentile of the Standard Normal
2
2
Distribution.
Appendix C: Median Survival in the
Uncured Fraction for the Flexible
Parametric Model without Time Dependent
Covariates
The flexible parametric model adjusting for diagnosis year, Y, is as
follows:
R( t ) = exp[ −exp(γ
+γ
00
+ β (Y ) + γ
v (ln( t )) + ... + γ
03 3
v (ln( t )) + γ
01 1
v (ln( t )))]
0 k k −1
v (ln( t ))
02 2
[2]
where β, γ00,…, γ0k − 1, and v1(ln(t)), …, vk−1ln(t)) are as defined in
“Data and Methods.” We rewrite this as a mixture model (3):
R( t ) = c + (1 − c )
c Fz ( t ) − c
1− c where,
c = exp[ − exp(γ 00 + β (Y ))]
and
Fz ( t ) = exp[γ 02v2 (ln( t )) + γ 03v2 (ln( t )) + ... + γ 0 k −1vk −1 (ln( t ))]
1. Gamel JW, Weller EA, Wesley MN, Feuer EJ. Parametric cure models
of relative and cause-specific survival for grouped survival times. Comput
Methods Programs Biomed. 2000;61(2):99–110.
2. Lambert PC, Thompson JR, Weston CL, Dickman PW. Estimating and
modeling the cure fraction in population-based cancer survival analysis.
Biostatistics. 2007;8(3):576–594.
3. Andersson TM, Dickman PW, Eloranta S, Lambert PC. Estimating and
modelling cure in population-based cancer studies within the framework of flexible parametric survival models. BMC Med Res Methodol.
2011;11:96.
4. Li CS, Taylor JMG, Sy JP. Identifiability of cure models. Stat Probabil Lett.
2001;54(4):389–395.
5.Yu B, Tiwari RC, Cronin KA, Feuer EJ. Cure fraction estimation
from the mixture cure models for grouped survival data. Stat Med.
2004;23(11):1733–1747.
6. Tai P, Yu E, Cserni G, et al. Minimum follow-up time required for the
estimation of statistical cure of cancer patients: verification using data from
42 cancer sites in the SEER database. BMC Cancer. 2005;5:48.
7.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 using data from the Finnish
Cancer Registry. Int J Cancer. 2007;121(9):2052–2059.
8. 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. 2010;102(20):1584–1598.
9. Haybittle JL. Curability of breast cancer. Br Med Bull. 1991;47(2):
319–323.
10. Cho H, Howlader N, Mariotto AB, Cronin KA. Estimating Relative Survival
for Cancer Patients from the SEER Program Using Expected Rates Based on
Ederer I Versus Ederer II Method. Bethesda, MD: National Cancer Institute;
2011. Technical Report No. 2011–01.
11. Huang 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-stage colorectal, ovarian, and testicular cancer? Cancer. 2008;112(10):2289–2300.
12. Yu B, Tiwari RC, Cronin KA, McDonald C, Feuer EJ. CANSURV: a
Windows program for population-based cancer survival analysis. Comput
Methods Programs Biomed. 2005;80(3):195–203.
13. Lambert PC, Royston P. Further development of flexible parametric models for survival analysis. Stata J. 2009;9(2):265–290.
14.Royston P, Parmar MK. Flexible parametric proportional-hazards and
proportional-odds models for censored survival data, with application
to prognostic modelling and estimation of treatment effects. Stat Med.
2002;21(15):2175–2197.
15. Shack LG, Shah A, Lambert PC, Rachet B. Cure by age and stage
at diagnosis for colorectal cancer patients in North West England,
1997–2004: a population-based study. Cancer Epidemiol. 2012;36(6):
548–553.
16. Ito Y, Nakayama T, Miyashiro I, et al. Trends in cure’ fraction from colorectal cancer by age and tumour stage between 1975 and 2000, using population-based data, Osaka, Japan. Jpn J Clin Oncol. 2012;42(10):974–983.
17. Woods LM, Rachet B, Lambert PC, Coleman MP. ‘Cure’ from breast cancer among two populations of women followed for 23 years after diagnosis.
Ann Oncol. 2009;20(8):1331–1336.
Journal of the National Cancer Institute Monographs, No. 49, 2014253
18. Zhang JJ, Peng YW. Accelerated hazards mixture cure model. Lifetime Data
Anal. Dec 2009;15(4):455–467.
19. Lambert PC, Dickman PW, Weston CL, Thompson JR. Estimating the
cure fraction in population-based cancer studies by using finite mixture
models. J Roy Stat Soc C-App. 2010;59(1):35–55.
20.Yu XQ, De Angelis R, Andersson TM, Lambert PC, O’Connell DL,
Dickman PW. Estimating the proportion cured of cancer: some practical
advice for users. Cancer Epidemiol. 2013;37(6):836–42.
254
Notes
We thank Dr Therese Andersson for her helpful explanation of the flexible
model and the STATA stpm2 software package.
Affiliations of authors: Surveillance Research Program, Division of Cancer
Control and Population Sciences, National Cancer Institute, National
Institutes of Health, Bethesda, MD (MRS, EJF, ABM).
Journal of the National Cancer Institute Monographs, No. 49, 2014