Nonparametric estimation of the concordance
correlation coefficient under univariate censoring
Ying Guo
∗
and Amita K. Manatunga
Department of Biostatistics, Rollins School of Public Health of Emory University,
1518 Clifton RD NE, Atlanta, GA, 30322, USA
email: [email protected]
SUMMARY.
Assessing agreement is often of interest in clinical studies to evaluate the
similarity of measurements produced by different raters or methods on the same subjects.
Lin’s (1989, Biometrics, 255-268) concordance correlation coefficient (CCC) has become a
popular measure of agreement for correlated continuous outcomes. However, commonly used
estimation methods for the CCC do not accommodate censored observations and are therefore not applicable for survival outcomes. In this paper, we propose to estimate the CCC
nonparametrically through the bivariate survival function. The presented estimator of the
CCC is proven to be strongly consistent and asymptotically normal, with consistent bootstrap variance estimator. Furthermore, we propose a time-dependent agreement coefficient
as an extension of Lin’s (1989) CCC for measuring the agreement between survival times
among subjects who survive beyond some time point. A nonparametric estimator is developed for the time-dependent agreement coefficient. It has the same asymptotic properties as
the estimator of the CCC. Simulation studies are conducted to evaluate the performance of
the proposed estimators. A real data example in a prostate cancer study is used to illustrate
the method.
KEY WORDS: Agreement; Concordance correlation coefficient; Time-dependent agreement
measure; Bivariate survival times; Univariate censoring.
1. Introduction
The interest to assess agreement between correlated measurements arises in many
scenarios in biomedical sciences. For example, in clinical practice, the disease status of
1
patients may be evaluated using the same scale by different raters and the reliability of
the scale is determined by measuring the agreement among raters. In other situations,
an event can be measured by two kinds of approaches: a gold standard, and a relatively
simple approximate method. The agreement between the two methods is assessed to
decide whether the simple method can act as a reasonable replacement for the gold
standard (Laurent, 1998).
For data that are measured on a continuous scale, the agreement of correlated outcomes was traditionally assessed by Pearson’s correlation coefficient, the paired t-test,
the coefficient of variation or the intraclass correlation coefficient in its original form.
However, none of these methods were originally developed as agreement measures and
cannot adequately assess the agreement characteristics. Lin (1989) introduced a more
appropriate agreement index for continuous outcomes– the concordance correlation
coefficient (CCC). It is based on the scaled expected squared difference between two
correlated variables and measures the deviation of observations from the 45 degree line
which represents the perfect agreement. The CCC has an appealing interpretation as
being the product of an accuracy coefficient which measures the agreement between the
two marginal distributions and a precision coefficient which measures the association
or correlation between the two measurements. Lin (1989) showed the CCC can be
expressed as a function of the marginal mean, marginal variance and covariance of the
two correlated measurements. To estimate the coefficient, he proposed to substitute
the sample counterparts of the marginal moments and covariance. He also provided
parametric inference for the estimator based on an underlying bivariate normal distribution. Later, other authors relaxed the bivariate normal assumption and developed
alternative estimation methods. For example, King and Chinchilli (2001b) generalized
Lin’s CCC by applying alternative distance functions and proposed robust estimators
based on U-statistics. Barnhart and Williamson (2001) used generalized estimation
2
equations (GEE) approach to estimate and model the CCC.
In biomedical sciences, researchers often are interested in the measuring the agreement between two survival times that are measured on the same subjects. For example,
in depression studies, the time of onset of clinical depression are measured using both
clinician-administered and patient self-reported scales. Evaluating agreement between
the disease onset times is useful in assessing the reliability of patients self-report and
identifying an appropriate instrument for diagnosing depression. Later in this paper,
we will describe another study where the time to the recurrence of prostate cancer is
assessed by two different techniques in the medical practice. To evaluate their comparability, researchers need to assess the agreement between the disease-free survival times
measured on the same subjects using the two techniques. Censored observations occur
frequently in such survival studies. Commonly used estimation methods for the CCC
(Lin, 1989; King and Chinchilli, 2001b; Barnhart and Williamson, 2001) are based on
complete measurements and cannot be directly applied.
Another special feature related to measuring agreement with survival outcomes is
the need to assess the agreement among subjects whose survival times exceed some
time point. For example, in genetic studies, researchers study twin data to investigate
potential genetic influences on life span. But important genetic effects may exist only
in old age (Anderson et al., 1992). An explanation is that if one of the twin died at very
young age due to unnatural causes such as a car accident, his/her time of death does
not provide much information for the expected life span of the other member. Hence,
research interest in these studies only centers on twins with both members surviving
beyond certain age. Anderson et al. (1992) proposed time-dependent association
measures for application to bivariate survival analysis. In this paper, we present a
time-dependent agreement measure which is extended from the CCC. The proposed
time-dependent coefficient is useful in the following ways: it enables measuring the
3
agreement on a subpopulation of interest by conditioning on subjects’ survival status;
and it reveals how the strength of agreement evolves along the time among survivors
remaining in the study.
In this paper, we propose to nonparametrically estimate the CCC for multivariate
survival times through their joint survival function. We use Lin and Ying’s (1993)
estimator of the bivariate survival function under univariate censoring to demonstrate
the method. The proposed method is useful in assessing the agreement between survival
times that are measured on the same subjects by different raters or methods. Our
estimator of the CCC possesses theoretical as well as practical advantages: it does
not require any distributional assumptions on bivariate survival times; it has desirable
asymptotic properties such as strong consistency and asymptotical normality; it is
easy to calculate in practice by exploiting the fact that correlated survival outcomes
in agreement studies are subject to univariate censoring. In Section 2, we present
the nonparametric estimator for the CCC and provide its asymptotic properties. We
also present the multivariate extension of the proposed estimator for multiple raters
or methods. A time-dependent agreement measure is then introduced for describing
the temporal pattern in the agreement structure among survivors along the time. A
nonparametric estimator is developed for the time-dependent coefficient. In Section 3,
we demonstrate the estimation and inference procedure. In Section 4, we present an
application to data from a prostate cancer study. Simulation studies are performed in
Section 5 to evaluate the performance of the method using data generated from various
bivariate survival functions. The results show that the proposed estimators perform
reasonably well in the presence of censoring. We conclude with a discussion in Section
6.
4
2. Methods
2.1 The proposed estimator of the concordance correlation coefficient
Here, we first introduce the concordance correlation coefficient.
2.1.1 The concordance correlation coefficient
Let Y1 and Y2 denote a pair of continuous outcomes of the same individual assessed
by two raters or methods. Lin’s (1989) CCC for Y1 and Y2 is defined as
E[(Y1 − Y2 )2 ]
E[(Y1 − Y2 )2 | Y1 , Y2 are independent]
2cov(Y1 , Y2 )
=
= ρχa .
var(Y1 ) + var(Y2 ) + [E(Y1 ) − E(Y2 )]2
ρc = 1 −
(1)
The CCC is the product of the precision coefficient ρ that measures the strength
of association and the accuracy coefficient χa that measures the agreement between
the two marginal distributions. Here, ρ is the Pearson correlation coefficient and
χa =
2
1
$+ $
+ν 2
with ν 2 =
(µ1 −µ2 )2
σ1 σ2
representing the location shift and $ = σ1 /σ2 repre-
senting the scale shift. The CCC ranges from -1 to 1 with a value of 1 representing a
perfect agreement, a value of -1 representing a perfect disagreement and a value of 0
representing no beyond chance agreement. Lin (1989) proposed an estimator for the
CCC using the sample counterparts of the mean, variance and covariance functions.
Other commonly used estimation methods include the U-statistics approach (King and
Chinchilli, 2001b) and the GEE approach (Barnhart and Williamson, 2001). All these
estimation methods are appropriate for uncensored data.
2.1.2 The proposed estimator for the concordance correlation coefficient
In this paper, we are interested in assessing the agreement between two survival
times, T1 and T2 , that are measured on the same subject by different raters or methods.
5
We propose to estimate the CCC through the bivariate survival function. Denote the
joint survival function of (T1 , T2 ) as S(t1 , t2 ) = Pr(T1 ≥ t1 , T2 ≥ t2 ). The mean,
variance and covariance functions of T1 and T2 can be written as
Z
∞
E(Tj ) =
Sj (t)dt,
0
Z
var(Tj ) = 2
Z
cov(T1 , T2 ) =
∞
0
∞
0
j = 1, 2,
Z
tSj (t)dt − [
∞
Sj (t)dt]2 ,
Z ∞
Z ∞
Z
S(t1 , t2 )dt1 dt2 −
S1 (t)dt
j = 1, 2,
0
0
0
and
∞
S2 (t)dt,
0
where S1 (t) = S(t, 0) and S2 (t) = S(0, t). Therefore, the CCC can be expressed as a
functional, say ϕ, of the bivariate survival function S. It is then natural to construct an
estimator for the coefficient using a nonparametric estimator of the bivariate survival
function.
In agreement studies, survival times are usually measured on the same subject by
different raters or methods. The censoring on correlated survival times happen simultaneously when a subject drops out from the study. In other words, the survival times
measured on the same subject are censored by a single censoring variable. Therefore,
to construct the estimator of the CCC, we use Lin and Ying’s (1993) estimator of the
bivariate survival function under univariate censoring. There are several advantages
related to Lin and Ying’s estimator: it is considerably simpler for practical use than
other nonparametric estimators (Dabrowska, 1988; Prentice and Cai, 1992); it has desirable asymptotic properties as well as good performance for small sample sizes; it
reduces to the usual empirical survival function in the absence of censoring; it is a
natural generalization of the univariate product-limit survival function estimator and
has good efficiency when the two survival times are highly correlated, which is usually
the case in agreement studies.
6
Let (Ti1 , Ti2 ) (i = 1, . . . , n) be n independent and identically distributed pairs of
survival times with survival function S(t1 , t2 ) = Pr(T1 ≥ t1 , T2 ≥ t2 ). Let Ci (i =
1, . . . , n) denote independent and identically distributed censoring times with survival
function G(t) = Pr(C ≥ t). The censoring time C is assumed to be independent of
(T1 , T2 ). The observed data consist of random vectors (Tei1 , Tei2 , δi1 , δi2 ) (i = 1, . . . , n)
where Teij = Tij ∧ Ci and δij = I(Tij ≤ Ci ) for j = 1, 2. Here and in the following, we
define a ∨ b = max(a, b) and a ∧ b = min(a, b).
From the independence of the survival times and censoring time, the observed survival times Tei1 , Tei2 have the survival function S(t1 , t2 )G(t1 ∨ t2 ). Lin and Ying (1993)
then proposed the estimator for S as
−1
b 1 , t2 ) = n
S(t
Pn
i=1
I(Tei1 ≥ t1 , Tei2 ≥ t2 )
.
b 1 ∨ t2 )
G(t
(2)
b 1 ∨ t2 ) is the univariate product-limit estimator based on
Here, the denominator G(t
ei , δ c ) where C
ei = Ci ∧ (Ti1 ∨ Ti2 ) = Tei1 ∨ Tei2 and δ c = I{Ci ≤ (Ti1 ∨ Ti2 )} = 1 − δi1 δi2 .
(C
i
i
When there is no censoring, Lin and Ying’s (1993) estimator reduces to the usual
empirical bivariate survival function.
Lin and Ying (1993) have shown that on D = [0, τ ]2 where (τ, τ ) is a point such
that S(τ, τ )G(τ ) > 0, the estimator Sb is uniformly strongly consistent and has weak
√ b
convergence results. That is, for (t1 , t2 ) ∈ [0, τ ]2 , n{S(t
1 , t2 ) − S(t1 , t2 )} weakly converges to a zero-mean Gaussian process W (t1 , t2 ). The asymptotic covariance of the
Gaussian process W at two time points (t11 , t12 ) and (t21 , t22 ) is
cov{W (t11 , t12 ), W (t21 , t22 )} =
³
− S(t11 , t12 )S(t21 , t22 ) 1 −
S(t11 ∨ t21 , t12 ∨ t22 )
G{(t11 ∨ t12 ) ∧ (t21 ∨ t22 )}
Z (t11 ∨t12 )∧(t21 ∨t22 )
´
dG(u)
.
G2 (u)Pr{(T1 ∨ T2 ) ≥ u}
0
7
(3)
Since the CCC is a function of the bivariate survival function, i.e. ρc = ϕ(S),
we propose the following nonparametric estimator where Sb is Lin and Ying’s (1993)
estimator,
R
R
b t)dt
b 1 , t2 )dt1 dt2 − τ̂1 S(t,
b 0)dt τ̂2 S(0,
S(t
0
0
ρ̂c = R τ̂1 0 0
,
R τ̂2
R τ̂1
R τ̂2
b 0)dt +
b t)dt −
b 0)dt
b t)dt
t
S(t,
t
S(0,
S(t,
S(0,
0
0
0
0
R τ̂1 R τ̂2
(4)
and τ̂j = max(Te1j , · · · , Tenj ), j = 1, 2. We will present the estimation and inference
procedure for ρ̂c later.
In the following, we derive the asymptotic statistical properties of ρ̂c via the Hadamard
b
differentiability of the functional ϕ and the statistical properties of S.
Lemma 1 Let S0 be the collection of bivariate survival functions S on R2 with finite
support. For S ∈ S0 , let τ1 = sup{t : S(t, 0) > 0} and τ2 = sup{t : S(0, t) > 0}.
Here, τ1 and τ2 are arbitrarily large but finite positive real values. Then the functional
ϕ : S0 7→ R
R τ1 R τ2
R τ1
R τ2
S(t
,
t
)dt
dt
−
S(t,
0)dt
S(0, t)dt
1
2
1
2
0 R
0 R
R τ2
ϕ(S) = R τ1 0 0
.
τ1
τ
tS(t, 0)dt + 0 tS(0, t)dt − 0 S(t, 0)dt 0 2 S(0, t)dt
0
is Hadamard-differentiable in S0 .
The proof of Lemma 1 is provided in Appendix A.
The following theorem establishes the asymptotic properties of the estimator ρ̂c .
The proofs are provided in Appendix A.
Theorem 1 Suppose that
Z
τj
p
tSj (t)dt → 0
and
τ̂j
√
Z
τj
n
τ̂j
8
p
Sj (t)dt → 0
for j=1,2. Here, S1 (t) = S(t, 0) and S2 (t) = S(0, t). The proposed estimator ρ̂c has
the following asymptotic properties as n → ∞
(i) The estimator ρ̂c is strongly consistent. That is, |ρ̂c −ρc | → 0
with probability 1.
(ii)The estimator ρ̂c has the following weak convergence result,
√
d
n(ρ̂c − ρc ) → ϕ0S (W ),
where ϕ0S (W ) follows a zero-mean normal distribution. Here, W is the zero-mean Gaussian process with the covariance function defined (3). ϕ0S is the Hadamard derivative
of ϕ at S,
ϕ0S (W ) =
AB − C + (1 − B)E
,
B2
with
Z
τ1
Z
τ2
A =
W (t1 , t2 )dt1 dt2 ,
Z τ1
Z τ2
Z τ1
Z τ2
B =
t S(t, 0)dt +
t S(0, t)dt −
S(t, 0)dt
S(0, t)dt,
0
0
0
0
Z τ1
Z τ2
C =
t W (t, 0)dt +
t W (0, t)dt,
0
0
Z τ1
Z τ2
Z τ1
Z τ2
E =
S(t, 0)dt
W (0, t)dt +
W (t, 0)dt
S(0, t)dt.
0
0
0
0
0
0
(iii) Let (Tei1 , Tei2 , δi1 , δi2 ), i = 1, . . . , n represent the observed data. By randomly sampling with replacement from (Tei1 , Tei2 , δi1 , δi2 ), i = 1, . . . , n, a bootstrap estimator ρ#
c can
√
be obtained based on the bootstrap sample. Then n(ρ#
c − ρ̂c ), given the observed data,
√
weakly converges to the same limit distribution as n(ρ̂c − ρc ) in probability.
2.1.3 Extension to multiple raters
The multivariate generalization of the CCC has been proposed by several authors
(Lin, 1989; King and Chinchilli, 2001a; Barnhart et al., 2002) to measure the agreement
among multiple raters. Suppose the survival time of a subject is assessed by J raters
9
with a continuous scale. Let T1 , . . . , TJ be measurements from the J raters. The overall
CCC for measuring agreement among the J raters is defined as (Barnhart et al., 2002)
o
nP
J−1 PJ
2
E
(T
−
T
)
k
j=1
k=j+1 j
o
ρoc = 1 − n P
J−1 PJ
2
E
j=1
k=j+1 (Tj − Tk ) |T1 , . . . , TJ are independent
PJ−1 PJ
2 j=1 k=j+1 cov(Tj , Tk )
=
.
PJ
P
PJ
2
(J − 1) j=1 var(Tj ) + J−1
j=1
k=j+1 [E(Tj ) − E(Tk )]
(5)
As with the CCC in (1), the overall CCC is defined based on the squared pairwise difference. Let Sjk be the bivariate survival function for (Tj , Tk ) where j < k ∈
{1, . . . , J}, i.e. Sjk (tj , tk ) = Pr(Tj ≥ tj , Tk ≥ tk ). Let Sj be the marginal survival
function for Tj with j = 1, . . . , J. We rewrite the overall CCC as
PJ−1 PJ
ρoc =
j=1
(J − 1)
k=j+1 [
PJ R τj
j=1
0
R τj R τk
R τj
R τk
S
(t
,
t
)dt
dt
−
S
(t
)dt
Sk (tk )dtk ]
jk
j
k
j
k
j
j
j
0
0
0
0
,
R τj
R τk
PJ−1 PJ
tj Sj (tj )dtj − j=1 k=j+1 0 Sj (tj )dtj 0 Sk (tk )dtk
(6)
where τj = sup{t : Sj (t) > 0}, j = 1, . . . , J.
If the J correlated survival times are subject to independent censorship by a single
censoring variable, Lin and Ying’s (1993) estimator of the bivariate survival function
can be extended to estimate the overall CCC. Let (Ti1 , . . . , TiJ ) (i = 1, . . . , n) be independent multivariate survival times for n subjects by the J raters. The censoring
times Ci (i = 1, . . . , n) are independent and identically distributed with survival function G(t). The observed data consist of (Tei1 , . . . , TeiJ , δi1 , . . . , δiJ )(i = 1, . . . , n) with
Teij = Tij ∧ Ci and δij = I(Tij ≤ Ci ). The extended Lin and Ying’s estimator for Sjk is
Sbjk (tj , tk ) = n−1
n
X
b j ∨ tk ),
I(Teij ≥ tj , Teik ≥ tk )/G(t
i=1
ei = Ci ∧
b is the product-limit estimator based on (C
ei , δ c ) (i = 1, . . . , n) with C
where G
i
10
Q
(Ti1 ∨ Ti2 . . . , ∨TiJ ) = Tei1 ∨ . . . , ∨TeiJ and δic = I{Ci ≤ (Ti1 ∨ . . . , ∨TiJ )} = 1 − Jj=1 δij .
2.2 Time-dependent concordance correlation coefficient
In assessing agreement between correlated survival times, researchers may want to
focus on a subpopulation of subjects whose time-to-events are beyond some time point
according to both raters or methods. In other situations, the changes in the strength
of agreement with elapse of time may be of interest. In this section, we propose a timedependent CCC to characterize the agreement between two survival times conditional
on subjects’ survival status,
ρ∗c (t01 , t02 ) = 1 −
E[(T1 − T2 )2 |T1 > t01 , T2 > t02 ]
.
E[(T1 − T2 )2 |T1 > t01 , T2 > t02 , T1 , T2 are independent]
(7)
This time-dependent coefficient measures the agreement between T1 and T2 among
subjects who survive beyond the time point (t01 , t02 ), i.e. T1 > t01 , T2 > t02 . By evaluating
the time-dependent CCC for (t01 , t02 ) on the two dimensional plane, one can describe
how the strength of agreement changes along the time among survivors remaining in
the study.
It is straightforward to show that the time-dependent CCC is a function, say ϕ1 , of
the bivariate survival function S. We propose a nonparametric estimator for the timeb i.e. ρ̂∗c (t01 , t02 ) = ϕ1 (S).
b The functional ϕ1 is
dependent CCC using Lin and Ying’s S,
shown to be Hadamard differentiable. Therefore, for (t01 , t02 ) such that S(t01 , t02 ) > 0, the
time-dependent CCC estimator ρ̂∗c (t01 , t02 ) possesses the same asymptotic properties as
the CCC estimator ρ̂c ; namely, strong consistency, weak convergence to a zero-mean
normal distribution and asymptotic validity of the bootstrap estimates.
3. Estimation
11
When there is no censoring, The proposed estimator of the CCC in (4) is reduced to
Lin’s sample estimator (Lin, 1989). To estimate the CCC in the presence of censored
observations, we define the following two functions,
Z
T
b (T ) =
U
0
Z
1
dt
b
G(t)
T
Vb (T ) =
0
2t
dt.
b
G(t)
(8)
By substituting Lin and Ying’s survival function estimator Ŝ into the proposed CCC
estimator in (4), we can show that the CCC estimator can be written in terms of the
b and Vb evaluated with various elements,
functions of U
i
n h
n
n
P
b (Tei1 ∨ Tei2 ) − U
b (Tei1 ∧ Tei2 )} − n−1 P U
b (Tei1 ) P U
b (Tei2 )
Vb (Tei1 ∧ Tei2 ) + (Tei1 ∧ Tei2 ){U
ρ̂c =
i=1
i=1
1
2
2 P
n
n
n
P
P
b (Tei1 ) P U
b (Tei2 )
Vb (Teij ) − n−1
U
j=1 i=1
i=1
i=1
.
i=1
(9)
b
Due to the complexity of the estimator ρ̂c and of the covariance function of S,
an expression for the variance of ρ̂c is analytically too complicated. To consistently
estimate the variance of ρ̂c , we propose a bootstrap procedure. Bootstrapped samples are drawn randomly with replacement from (Tei1 , Tei2 , δi1 , δi2 )(i = 1, . . . , n). We
consider two kinds of confidence intervals for ρc . The first is the bootstrap percentile
confidence interval which defines confidence limits as percentiles of bootstrap sample
estimates. Alternatively, we construct the confidence interval based on the asymptotic
normality of ρ̂c . The normal approximation of ρ̂c can be improved by using Fisher’s
Z-transformation (Lin, 1989)
1 1 + ρ̂c
Ẑ = tanh−1 (ρ̂c ) = ln
.
2 1 − ρ̂c
Therefore, we first construct the normal approximation confidence interval for Ẑ using
12
the bootstrap standard error of Ẑ. The confidence interval of ρ̂c is then obtained by
back transforming (ẐL , ẐU ),
(
exp(2ẐL ) − 1 exp(2ẐU ) − 1
,
).
exp(2ẐL ) + 1 exp(2ẐU ) + 1
The nonparametric estimator of the time-dependent CCC ρ̂∗c (t01 , t02 ) can also be obb (·) and Vb (·) with arguments defined by (Tei1 , Tei2 ) and
tained through functions of U
(t01 , t02 ). The bootstrap procedure is used to calculate the variance estimator and confidence interval.
4. Prostate Cancer Data
We illustrate the proposed method using the data from a prostate cancer study.
Prostate cancer is the most common cancer in US men. One major difficulty in treating and monitoring prostate cancer is the lack of a standard definition for disease free
state after treatments. Even though there is a consensus that posttreatment disease
status is reflected in the prostate specific antigen (PSA) with high level PSA indicating cancer relapse, there is no universal agreement on the exact pattern of the PSA
level that defines disease recurrence. Various definitions of disease free state had been
proposed for different treatments. The disease-free survival rates are calculated for
treatments based on the corresponding definitions and are used as an important guidance for physicians in treatment selection. Since the survival rates depend heavily
on the definition of disease free state, the potential discrepancies between definitions
may lead to incorrect conclusions regarding the effectiveness of different treatments.
Therefore, it is important to assess the agreement between different definitions before
comparing the disease-free survival rates derived from them.
Radical prostatectomy and irradiation are two commonly used treatments for prostate
13
cancer (Critz, 1995). For radical prostatectomy, disease free state is defined by reaching
and maintaining an undetectable prostate specific antigen (PSA) nadir ranging from
0.2ng/ml to 0.5ng/ml (Critz, 1996). For irradiation, according to the American Society
of Therapeutic Radiation Oncology(ASTRO) consensus criteria (1997), posttreatment
disease free state is represented by a non-rising PSA with a rising PSA defined as three
consecutive PSA increases measured six months apart. For years, there is the controversy regarding the disease free rates between the two treatments. Some researchers
claim the irradiation cures fewer patients than the radical prostatectomy while others
argue the two are equally effective (Critz, 1996). In order to compare the disease-free
survival rates between the two treatments, researchers need to study the agreement
between the two definitions of disease free state. Additionally, the potential difference
in the strength of agreement among various covariate subgroups is also of interest.
In this study, 1369 men with prostate cancer received simultaneous irradiation by
integrating iodine 125 prostate implant with a follow-up external beam radiation. The
disease status of all subjects was evaluated every six months after the treatment of the
external beam radiation and the survival time was defined as the time elapsed from the
end of the irradiation to the recurrence of prostate cancer which is determined based
on two different definitions. T1 is the time when patients’ posttreatment PSA level
exceeded the nadir of 0.2ng/ml. T2 is based on the ASTRO definition and represents
the midpoint between the time when the lowest PSA was achieved after irradiation
and the time when the first of the three consecutive rises in the PSA level occurred.
The two disease recurrence times were subject to independent censoring by the same
censoring variable C which represents the end of the follow-up time on a patient. The
observed times Te1 and Te2 are the minimum of the disease time and the censoring time,
i.e. Tej = min(Tj , C), j = 1, 2. Patient characteristics recorded at baseline include age,
pretreatment PSA level and the grade of prostate cancer.
14
[INSERT FIGURE 1]
Figure 1 presents the subjects’ observed times which range from less than 1 year
to 13.5 years. The plot shows that the observations of most of subjects were on the
45◦ line. For the remaining subjects, the observed disease-free time was longer by the
ASTRO definition than by the nadir 0.2ng/ml definition. The marginal Kaplan-Meier
curves of disease-free rates based on the two definitions are illustrated in Figure 2.
The proposed method was applied to measure the agreement between T1 and T2 .
The estimated CCC was 0.792 and the standard error based on 500 bootstrap samples
was 0.08. The 95% confidence interval based on the 2.5% and 97.5% empirical percentiles of the bootstrap sample estimates was (0.594,0.884). The normal approximation confidence interval for the CCC based on the Z-transformation was (0.607,0.896).
To the best of our knowledge, there is as yet no literature giving a descriptive scale
for the degree of agreement measured by the CCC. Due to the close relationship between
the CCC and the kappa coefficient (King and Chinchilli, 2001a), we adopt Landis &
Koch (1977) scale for interpretation. Hence, there was substantial agreement between
the two survival times based on the ASTRO and nadir 0.2ng/ml definitions although
the disease recurrence time was occasionally longer by the ASTRO definition than by
the nadir definition. We then considered the agreement among subjects who were free
of prostate cancer recurrence for at least two years after the treatment. The estimated
time-dependent CCC given T1 > 2 and T2 > 2 was 0.637 with the 95% bootstrap
percentile confidence interval of (0.351, 0.853). The estimated time-dependent CCC
conditional on survival beyond two years was lower than the estimated CCC for all
subjects. This result is consistent with the observation in Figure 1 that the agreement
between T1 and T2 was weaker if the recurrence occurred after the first two years.
A reviewer pointed out that another reason that may have contributed to the lower
15
time-dependent CCC is that the range of the data is smaller.
Furthermore, we considered the effect of pretreatment PSA level on the strength
of agreement between the two recurrence times. The estimated CCC is 0.715 for
subjects whose pretreatment PSA is greater than the median and 0.761 for those whose
pretreatment PSA is less than or equal to the median. To test whether the strength
of agreement differs significantly between the two pretreatment PSA subgroups, we
constructed an asymptotic normal test statistic based on the Z-transformation
tanh−1 (0.715) − tanh−1 (0.761)
√
.
0.2742 + 0.2362
where 0.274 and 0.236 are the bootstrap standard errors for the transformed CCC. The
test statistic is -0.127 with the p-value of 0.90. Therefore, the pretreatment PSA level
does not have significant effects on the strength of agreement.
The above asymptotic normal test can be applied to compare CCCs from two
different studies. However, caution should be taken in interpreting results when the
two studies have significantly different follow-up period.
5. Simulation Studies
Simulation studies were conducted to assess the performance of the nonparametric estimators of the CCC and time-dependent CCC, and their bootstrap variance
estimators and confidence intervals.
For the CCC estimator, bivariate survival times were generated from the Clayton
model (Clayton, 1978) with the sample sizes of 50 and 100. We assume that T1 and T2
have exponential marginal distributions with the mean of 1 and 1.5, respectively. The
bivariate survival function for Clayton model is therefore
2
S(t1 , t2 ) = (et1 /θ + e 3 t2 /θ − 1)−θ ,
16
(10)
with constant odds ratio of 1 + 1θ , where θ → ∞ gives independence and θ → 0 gives
maximal positive dependence. We considered three sets of simulations with different θ
parameters such that the true CCC of the corresponding Clayton model is 0.553, 0.713
and 0.846, representing moderate, substantial and almost perfect agreement between
the survival times. The bivariate survival times were subject to right censorship by
means of an independent exponentially distributed censoring variable. We considered
the censoring rates of 10%, 50% and 70%.
[INSERT TABLE 1]
Table 1 summarizes the results based on 500 simulation runs under various sampling
configurations. Sample means and standard deviations for ρ̂c are presented, along with
the mean of the standard error estimates based on 200 bootstrap samples. The coverage
probabilities are calculated for the confidence intervals based on the bootstrap sample
percentiles and the normal approximation. As a comparison to the proposed method,
we also perform a crude estimation which only utilizes the complete observations in
the simulated data sets. Lin’s (1989) sample estimator of the CCC and its asymptotic
standard error are obtained based on these complete cases.
As expected, the proposed CCC estimator is the same as Lin’s (1989) estimator
when there is no censoring (Results are not shown in Table 1). In the presence of
censoring, the bias of the proposed CCC estimator is much smaller than that the
crude estimator, especially with medium to heavy censoring. The bias of the proposed
estimator is negligible when there is almost perfect agreement between the two survival
times. For data with moderate and substantial agreement with medium to heavy
censoring, the proposed estimator tends to underestimate the CCC.
The sample standard deviation of the proposed estimator decreases by increasing
the sample size and reducing the censoring proportion. Bootstrap standard errors are
17
close to the standard deviations with low censoring but tend to be biased downwords
with medium to heavy censoring. The coverage probabilities of the proposed estimator
are substantially higher than those of the crude estimator. Between the two methods
for confidence intervals, the one based on bootstrap sample percentiles performs significantly better than that from the normal approximation under the small sample sizes
considered in the simulation study. The coverage of both kinds of confidence intervals
is close to the nominal level with low censoring but is a bit lower with medium to
heavy censoring. However, the bootstrap percentile confidence interval still maintains
a coverage of around 90% when the censoring rate is as high as 70%.
For the time-dependent CCC estimator, data were generated from the three Clayton models described before with the sample sizes of 100 and 200. We evaluated
the time-dependent CCC at (t01 , t02 ) = (0.405, 0.405) where the marginal survival function is approximately 2/3 and 0.76 for T1 and T2 . The bivariate survival function at
(0.405, 0.405) is 0.553, 0.581 and 0.653 for the three Clayton models with moderate,
substantial and almost perfect agreement, respectively.
Table 2 presents the simulation results for the time-dependent CCC estimator. The
bias of the estimator and its bootstrap standard error is negligible with low censoring.
For small sample size with medium to heavy censoring, the proposed estimator and
the bootstrap standard error tend to be biased downwards. However, almost all of the
estimated biases of the proposed estimator are less than 10% for data with substantial
to almost perfect agreement. The coverage probabilities of the 95% bootstrap percentile
confidence interval are somewhat lower than the nominal level but remain around 90%
in most situations.
[INSERT TABLE 2]
6. Discussion
18
This paper presents a new approach to estimate CCC for measuring agreement
among correlated survival outcomes with censored observations. We propose to estimate the CCC through the bivariate survival function since the survival function can be
consistently estimated in the presence of censoring. Among the various survival function estimators, we choose Lin and Ying’s (1993) estimator because it is considerably
simpler and has desirable small sample and asymptotic properties. In literature, other
nonparametric estimators of the bivariate survival function under univariate censoring
have been proposed such as Tsai and Crowley (1998), which has the smallest variance
among all path-dependent estimators including Lin and Ying’s (1993). As a referee
suggested, the asymptotic efficiency of the CCC estimator may be improved by using Tsai and Crowley’s bivariate survival function estimator. Future work are needed
to compare the statistical properties of CCC estimators based on different survival
function estimates.
We present a nonparametric estimator of the CCC under univariate censoring.
Under the same framework, the proposed method could easily be extended to bivariate
censoring. In that case, bivariate survival function estimators such as those proposed
by Dabrowska (1988) and Prentice-Cai (1992) can be used to obtain the nonparametric
estimator for the CCC.
Previous work has shown that the temporal pattern in the dependent structure between correlated survival outcomes is determined by the parametric assumption of the
bivariate survival time distribution (Oakes, 1989; Anderson et al., 1992). For example,
Oakes(1989) showed that the cross ratio, which is a time-dependent association measure, is constant across time for the Clayton model while decreases monotonically along
the time for other bivariate distributions such as the positive stable frailty model. In
this paper, we propose a nonparametric estimator of the time-dependent CCC without
imposing any parametric assumption on the joint distribution of T1 and T2 . There19
fore, the estimated time-dependent CCC is free of undue influences from parametric
assumptions and reflects the empirical agreement structure between T1 and T2 . The
method proposed in this paper is applicable for assessing agreement for continuous
survival data. For discrete survival times, the agreement structure can be measured
through a local kappa coefficient (Guo and Manatunga, 2005).
The proposed method could be extended to estimate the CCC in the case of other
types of censoring such as interval censoring by using nonparametric maximum likelihood estimates of the bivariate survival function proposed by Betensky and Finkelstein
(1999) and Gentleman and Vandal (2001). However, since the theoretical properties
of these NPMLEs remain unsolved, the theoretical properties of the CCC estimator
cannot be established.
Recently, Liu et al. (2005) proposed using a likelihood-based estimation method
to assess the CCC for time to event data subject to censoring. Their method assumes
a parametric distribution model for multivariate survival times so that the likelihood
contribution from censored observations can be calculated. Naturally, the CCC estimator of Liu et al. (2005) is more efficient than the proposed nonparametric estimator
if the correct distribution is assumed. Additionally, it is easier to accommodate left
or interval censoring with the parametric assumption. However, there are certain disadvantages related to the parametric distributional assumptions: the estimated CCC
may vary for the same data when different distributions are used; it is difficult to
formally check the parametric bivariate distribution assumption in the presence of censoring (Liu et al., 2005); transformation of the survival times may be needed to satisfy
the distribution assumption. Consequently, the CCC estimate reflects the agreement
between the transformed survival times and is hard to interpret on the original scale.
Compared to Liu et al. (2005), the proposed nonparametric estimation approach does
not assume any parametric distribution models and is more flexible to use in practice.
20
Appendix A: Proofs
Proof of Lemma 1
We first present the definition for Hadamard differentiability. Let F0 be a collection
of cumulative density functions on Rd , where d is the dimension, and D = {c(H1 −H2 ) :
c ∈ R, H1 , H2 ∈ F0 }. Assume F0 is closed under mixtures so that D is a vector
space. Given a norm % on F0 , induce a norm on D by ||c(H1 − H2 )|| = |c|%(H1 , H2 ).
The functional P : F0 7→ R is Hadamard differentiable at H ∈ F0 iff there exists
a continuous linear functional LP,H on D such that for all real sequences gj → 0,
∆, ∆j ⊂ D with ||∆j − ∆|| → 0 and H + gj ∆j ∈ F0 ∀j,
½
lim
j→∞
¾
P (H + gj ∆j ) − P (H)
− LP,H (∆) = 0,
gj
and the function LP,H is the Hadamard derivative of P at H.
The following Lemma A.1 establishes the product and quotient rules for Hardamard
differentiability. The proof of Lemma A.1 is straightforward.
Lemma A.1: Let P1 and P2 be two Hadamard differentiable functional: F0 7→ R. Denote the Hadamard derivative of Pi at H by LPi ,H for i = 1, 2. Define their product by
P1 P2 with (P1 P2 )(H) = P1 (H)P2 (H). Then, the product P1 P2 is also Hadamard differentiable and the Hadamard derivative of P1 P2 at H is LP1 P2 ,H = P2 (H)LP1 ,H +
³ ´
1
P1 (H)LP2 ,H . Furthermore, define functional P12 as P12 (H) = P2 (H)
. Then P12 is
Hadamard differentiable and the Hadamard derivative of
1
P2
1
at H is L1/P2 ,H = − (P2 (H))
2 LP2 ,H .
Now, Let’s define functionals
Z
τ1
Z
Z
τ2
P1 (S) =
S(t1 , t2 )dt1 dt2 ,
0
P2 (S) =
0
Z
τ1
S1 (t)dt,
0
Z
P4 (S) =
tS1 (t)dt,
τ2
and P5 (S) =
0
tS2 (t)dt.
0
21
S2 (t)dt,
0
Z
τ1
τ2
P3 (S) =
Then
ϕ(S) =
P1 (S) − P2 (S)P3 (S)
.
P4 (S) + P5 (S) − P2 (S)P3 (S)
We first show Hadamard differentiability of P1 . Note
P1 (S + gj ∆j ) − P1 (S)
= lim
lim
j→∞
j→∞
gj
Z
τ1
Z
τ2
∆j (t1 , t2 )dt1 dt2 = P1 (∆).
0
(11)
0
The function P1 is a linear map and is also bounded because sup||S||=1 |P1 (S)| <
R τ1 R τ2
1dt1 dt2 = τ1 τ2 < ∞. Therefore, P1 is a continuous linear map. Thus, equation
0
0
(11) shows P1 is Hadamard differentiable at S with the derivative LP1 ,S = P1 . Similarly,
We can show that functionals Pk : S0 7→ R, k = 2, . . . , 5 are Hadamard differentiable
with the derivatives LPk ,S = Pk . The bivariate survival function ϕ(S) is then Hadamard
differentiable according to Lemma A.1. This completes the proof for Lemma 1.
Proof of Theorem 1
Before proving (i) − (iii) of Theorem 1, we first show the following result
√
p
b →
n [ρ̂c − ϕ(S)]
0.
Consider the difference between the numerators of
√
(12)
nρ̂c and
√
b
nϕ(S),
Z Z
Z τ̂1
Z τ̂2
o
√ ¯¯n τ̂1 τ̂2
b 1 , t2 )dt1 dt2 −
b 0)dt
b t)dt
S(t
S(t,
S(0,
n¯
0
0
0
Z τ1 0 Z τ2
o¯
n Z τ1Z τ2
b t)dt ¯¯
b 0)dt S(0,
b 1 , t2 )dt1 dt2 − S(t,
S(t
−
0
0
0
0
Z
Z τ2
Z τ2
Z τ̂1
Z τ1Z τ̂2
√ ¯¯ τ1
b 0)dt
b t)dt + S(0,
b t)dt S(t,
b 0)dt −
b 1 , t2 )dt1 dt2
= n¯
S(t,
S(0,
S(t
τ̂1
0
τ̂2
0
22
τ̂1
0
Z τ2Z τ1
¯
b 1 , t2 )dt2 dt1 ¯¯
−
S(t
τ̂2
0
Z
Z τ2
Z τ2
Z τ̂1
Z τ1Z τ̂2
√ h τ1
b
b
b 1 , 0)dt1 dt2
≤ n
S(t, 0)dt
1dt +
S(0, t)dt
1dt +
S(t
0
τ̂2
0
τ̂1 0
Z τ̂τ12Z τ1
i
b t2 )dt2 dt1
+
S(0,
τ̂2 0
Z τ1
Z τ2
i
√ h
b
b t)dt .
= n (τ2 + τ̂2 )
S(t, 0)dt + (τ1 + τ̂1 )
S(0,
(13)
τ̂1
Due to the assumption that
τ̂2
√ R τj
p
n τ̂j Sj (t)dt → 0 for j = 1, 2 and the uniform strong
b the right-hand side of the inequality (13)
consistency of Lin and Ying’s estimator S,
√
converges to zero in probability. Therefore, the numerator of nρ̂c converges to the
√
b in probability.
numerator of nϕ(S)
√
Similarly, we can show the denominator of nρ̂c converges to the denominator of
R
√
p
b in probability from the assumption that τj tSj (t)dt →
nϕ(S)
0 for j = 1, 2 and the
τ̂j
b This completes the proof for (12).
uniform strong consistency of S.
Next, we prove (i) − (iii) of the theorem.
p
b → ϕ(S) = ρc follows the continuity of the Hadamard differenThe result of ϕ(S)
tiable function ϕ and the uniform strong consistency of Lin and Ying’s estimator Ŝ.
p
Then ρ̂c → ρc via (12). Thus, the statement (i) is true.
Due to (12), to prove (ii) is equivalent to show that
√
It’s shown that
√
d
b − ϕ(S)) → ϕ0 (Z),
n(ϕ(S)
S
as n → ∞.
(14)
b 1 , t2 ) − S(t1 , t2 )} weakly converges to a tight, zero mean
n{S(t
Gaussian process W (t1 , t2 ) (Lin and Ying, 1993). The functional ϕ is proven to be
Hadamard-differentiable. Then, the statement (14) is true according to the functional
delta method (van der Vaart and Wellner, 1996). Because W is a tight Gaussian
process, the derivative ϕ0S (W ) is normally distributed and is obtained through the
23
product and quotient rules of the Hadamard differentiability presented in Lemma A.1.
Thus, the statement (ii) of the theorem is proven true.
To show the asymptotic result of (iii) for the bootstrap estimate, define S # as
Lin and Ying’s estimator of the joint survival function based on a bootstrap sample
# #
(Tei1# , Tei2# , δi1
, δi2 ), i = 1, . . . , n randomly selected from the observed data. Due to the
Hadamard differentiability of the function ϕ and the functional delta method for boot√
b
strap by van der Vaart and Wellner (1996), the distribution of n{ϕ(S # ) − ϕ(S)},
conditional on the observed data, is asymptotically consistent in probability for estimat√
b − ϕ(S)}. Therefore, √n(ρ#
ing the distribution of n{ϕ(S)
c − ρ̂c ), given the observed
√
data, weakly converges to the same limit distribution as n(ρ̂c − ρc ) in probability.
ACKNOWLEDGEMENTS
We thank Dr Andrew N Hill from Emory University for providing helpful information
regarding the Hadamard differentiability. This work was partially supported by NIH
grants: M01-RR00039 and R01-ES012458-01.
24
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27
∗
10
50
70
10
50
70
100
10
50
70
100
50
10
50
70
10
50
70
50
100
93.8
89.3
89.3
97.7
92.8
89.8
90.8
88.7
83.8
93.2
88.1
83.0
91.0
87.3
88.3
0.841 (0.018)
0.816 (0.031)
0.782 (0.049)
0.841 (0.027)
0.821 (0.046)
0.788 (0.061)
0.689 (0.064)
0.637 (0.085)
0.564 (0.149)
0.688 (0.094)
0.628 (0.151)
0.548 (0.158)
0.523 (0.094)
0.468 (0.117)
0.501 (0.154)
0.022 (0.012)
0.034 (0.006)
0.051 (0.011)
0.022 (0.009)
0.048 (0.012)
0.067 (0.018)
0.048 (0.008)
0.062 (0.011)
0.105 (0.027)
0.068 (0.016)
0.106 (0.032)
0.137 (0.039)
0.067 (0.010)
0.083 (0.012)
0.124 (0.027)
98.4
88.7
71.9
97.9
93.8
88.5
83.6
70.9
60.0
85.2
78.1
70.7
80.1
70.7
87.3
Lin’s method
with complete obs. only
sample estimator Asymptotic Coverage
mean(SD)
SE (SD)
Prob.
0.523 (0.133)
0.096 (0.018)
83.6
0.463 (0.196)
0.140 (0.033)
81.1
0.498 (0.181)
0.155 (0.045)
89.3
Coverage probabilities II: normal approximation confidence interval .
0.844 (0.019) 0.019 (0.006) 94.1
0.835 (0.041) 0.037 (0.011) 91.0
0.852 (0.065) 0.056 (0.018) 91.0
0.844 (0.029) 0.035 (0.013) 97.7
0.838 (0.056) 0.050 (0.016) 92.2
0.851 (0.080) 0.071 (0.024) 93.4
0.706 (0.064) 0.059 (0.016) 92.0
0.690 (0.084) 0.077 (0.024) 90.4
0.663 (0.170) 0.135 (0.049) 89.5
0.699 (0.097) 0.090 (0.028) 92.8
0.669 (0.146) 0.128 (0.047) 89.6
0.631 (0.202) 0.162 (0.061) 89.3
0.540 (0.097) 0.087 (0.021) 93.0
0.508 (0.124) 0.109 (0.031) 89.1
0.518 (0.215) 0.170 (0.057) 90.6
Sample Censoring
ρ̂c
Bootstrap
Coverage Prob.∗
size
%
mean(SD)
SE (SD)
I
II
50
10
0.531 (0.139) 0.121 (0.032) 92.8
92.2
50
0.501 (0.193) 0.166 (0.052) 91.6
88.1
70
0.506 (0.220) 0.194 (0.067) 90.2
89.5
Coverage probabilities I: bootstrap percentile confidence interval.
0.846
0.713
0.553
ρc
Proposed method
Table 1: Summary statistics for the proposed estimator of the CCC and Lin’s sample estimator for various bivariate
survival models with heterogeneous marginal distributions based on 500 simulation runs.
∗
10
50
70
10
50
70
200
10
50
70
200
100
10
50
70
100
10
50
70
0.770 (0.026) 0.025 (0.007)
0.766 (0.039) 0.033 (0.013)
0.791 (0.105) 0.094 (0.030)
0.761 (0.040) 0.037 (0.012)
0.759 (0.056) 0.050 (0.020)
0.732 (0.141) 0.133 (0.048)
0.610 (0.070) 0.067 (0.017)
0.560 (0.132) 0.114 (0.039)
0.589 (0.231) 0.196 (0.067)
0.585 (0.103) 0.096 (0.028)
0.565 (0.138) 0.127 (0.044)
0.532 (0.287) 0.245 (0.094)
90.6
89.6
89.8
89.8
88.9
86.1
91.8
89.1
93.0
90.8
88.9
90.8
92.4
91.2
92.4
Bootstrap
Coverage
SE (SD)
Probability∗
0.135 (0.035)
91.0
0.171 (0.055)
89.5
0.305 (0.116)
91.8
for various bivariate survival models based on 500
0.428 (0.105) 0.099 (0.023)
0.417 (0.188) 0.156 (0.048)
0.418 (0.317) 0.266 (0.087)
For bootstrap percentile confidence interval.
0.772
0.609
200
Table 2: Summary statistics for the time-dependent CCC estimator
simulation runs.
ρ̂c (t01 , t02 )
Sample ρc (t01 , t02 ) Censoring
size
%
mean(SD)
0.430 100
10
0.395 (0.148)
50
0.367 (0.196)
70
0.357 (0.369)
14
8
6
0
2
4
T2: ASTRO (years)
10
12
+
^
^
++
++
+
+
^
++
^
++++
^
^ ^
++++
+
+
+
^
^
^ +++++
^^ ^
++
^ ^
+
^
+
+
^
^
+
^
^
++++++
^
^
^
+
+
+
+
++
+++++
^ ^^
^ ^^
^
+++++
^ ^
^
^^
^ +++++++++++
^
++++++
^
++++++
^ ^ ^ ^
++++++
^
+
+
+
+
^
+
+
+++++++
^
T1-complete, T2-complete
^ ^ ++++++++++++++
^ ^^
+
+
^
+
+
+
^ ++++++
+
+
+
+
^
+
+
+
^
T1-censored, T2-censored
+++++
^
^
^+++++++
++
+++++++
+
+
^
+
+
+
+
+
T1-complete, T2-censored
^ ^ ++++++++++++
+
+++++++
+
>
+
+
+
+
^
T1-censored, T2-complete
+++++
^
++++
+++++
+
+
+
+
+
++
++
++++
^
^
0
2
4
6
8
10
12
14
T1: nadir02 (years)
Figure 1: Prostate cancer disease-free survival time after irradiation based on the two
definitions of disease free state.
30
1.0
0.8
0.6
0.4
0.0
0.2
Estimated disease−free rates
nadir02 definition
ASTRO definition
0
2
4
6
8
10
12
14
Follow−up time (years)
Figure 2: Marginal Kaplan-Meier curves for disease-free survival rates after irradiation
based on the two definitions.
31