1231482

Modèles de regression en présence de compétition
Aurélien Latouche
To cite this version:
Aurélien Latouche. Modèles de regression en présence de compétition. Sciences du Vivant [q-bio].
Université Pierre et Marie Curie - Paris VI, 2004. Français. �tel-00129238�
HAL Id: tel-00129238
https://tel.archives-ouvertes.fr/tel-00129238
Submitted on 7 Feb 2007
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λ
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1
2
1
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1
2
1
1
1
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F1 (t) = P (T ≤ t, T1 ≤ T2 ) = P (T ≤ t)P (T1 − T2 ≤ 0) =
λ1 (t) =
λ11 (t)
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aa1
,
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a1 + a12 − a12 exp(−a2 t)
=a
,
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aa1
α1 (t) =
.
a1 + a2 exp(at)
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1000
1500
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500
t
t
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(d)
1500
1000
1500
15
5
10
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20
25
30
1000
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a12=0
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a12=0.001
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0
500
1000
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t
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1
1
2
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a
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500
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STATISTICS IN MEDICINE
Statist. Med. 2004; 23:3263–3274
Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/sim.1915
Sample size formula for proportional hazards modelling
of competing risks
Aurelien Latouche∗; † , Raphael Porcher and Sylvie Chevret
Departement de Biostatistique et Informatique Medicale; Hôpital Saint-Louis; Universite Paris 7;
Inserm Erm 321; Paris; France
SUMMARY
To test the eect of a therapeutic or prognostic factor on the occurrence of a particular cause of failure
in the presence of other causes, the interest has shifted in some studies from the modelling of the causespecic hazard to that of the subdistribution hazard. We present approximate sample size formulas for
the proportional hazards modelling of competing risk subdistribution, considering either independent
or correlated covariates. The validity of these approximate formulas is investigated through numerical
simulations. Two illustrations are provided, a randomized clinical trial, and a prospective prognostic
study. Copyright ? 2004 John Wiley & Sons, Ltd.
KEY WORDS:
competing risks; sample size; subdistribution hazard; cumulative incidence
1. INTRODUCTION
In cohort studies, patients are often observed to fail from several distinct causes, the eventual
failure being attributed to one cause exclusively to the others, which denes a competing risks
situation [1]. In this setting, we are often interested in testing the eect of some covariate on
the risk of a specic cause. For example in cancer studies, one could wish to assess the eect
of a new treatment or age on delaying relapse, while some patients will die before relapse.
The eect of such a covariate on specic failure types is usually analysed through the
modelling of the cause-specic hazard function [2]. However, the interpretation of the eects
of the covariate on the specic types of failure is restricted to actual study conditions, and
there is no implication that the same eect would be observed under a new set of conditions,
notably when certain causes of failure would have been eliminated. Except in circumstances of
complete biologic independence among the biological mechanisms giving rise of the various
failure types, it is unrealistic to suppose that general statistical methods can be put forward
that will encompass all possible mechanisms for cause removal [2]. Notably, the instantaneous
∗ Correspondence
to: A. Latouche, Departement de Biostatistique et Informatique Medicale, Hôpital Saint-Louis,
1 avenue Claude Vellefaux, 75475 Paris Cedex 10, France.
† E-mail: [email protected]
Copyright ? 2004 John Wiley & Sons, Ltd.
Received December 2003
Accepted May 2004
3264
A. LATOUCHE, R. PORCHER AND S. CHEVRET
risk of specic failure type is sometimes of less interest than the overall probability of this
specic failure. Such a probability could be formulated as either the marginal distribution
of the specic failure type, or the cumulative incidence function, i.e. the overall probability
of the specic type of failure in the presence of competing causes [3–6]. In some situations,
the marginal distribution could be the relevant target of estimation. However, this marginal
distribution is not identiable from available data without additional assumptions, such as
statistical independence between competing failure types (in which case the marginal hazard
equals the cause-specic hazard). In the situation where the competing risks arise from the
underlying biology of the disease, and not from the observation process, such an assumption is
not clinically relevant. This will be exemplied below on real data (see Section 5). Moreover,
it has been proven that this assumption cannot be veried [7]. Therefore, the cumulative
incidence functions may appear more relevant than marginal probabilities. However, no oneto-one relationship exists between the cause-specic hazard and the cumulative incidence
function. Therefore, in such cases, the emphasis has shifted from the conventional modelling
of the cause-specic hazard function to the modelling of quantities directly tractable to the
cumulative incidence function [8–10].
Let T be the time of failure, the cause of failure ( = 1, denoting the failure cause of
interest) and F1 (:) the cumulative incidence function for failure from cause of interest, i.e.
F1 (t) = Pr (T 6t; = 1). Gray [8] dened the subdistribution hazard for cause 1 as:
1 (t) = lim
dt→0
1
Pr{t 6T 6t + d t; = 1|T ¿t ∪ (T 6t ∩ = 1)}
dt
by contrast to the cause-specic hazard:
1 (t) = lim
dt→0
1
dt
Pr{t 6T 6t + d t; = 1|T ¿t }
By construction, the subdistribution hazard 1 (t) is explicitely related to the cumulative incidence function for failure from cause 1, F1 (t), through
1 (t) = − d log{1 − F1 (t)}= d t
(1)
while the relation between the cause-specic hazard 1 (t) and F1 (t) involves the cause-specic
hazards of all failures types [11].
A semi-parametric proportional hazards model was proposed to test covariate eects on
the subdistribution hazard [9]. Using the partial likelihood principle and weighted estimated
equations, consistent estimators of the covariate eects were derived, either in the absence of
censoring (referred as ‘complete data’), or in the presence of administrative censoring, i.e.
when the potential censoring time is observed on all individuals (‘censoring complete data’).
A weighted estimator for right-censored data (‘incomplete data’) was also proposed, properties
of which were investigated through numerical simulations.
This paper focuses on computing sample size for the Fine and Gray’s model [9], with two
main goals. First, we provide a sample size formula to design a parallel arm randomized
clinical trial with a right-censored competing endpoint, contrasting this computation with the
standard Cox modelling of the cause-specic hazard [12]. Secondly, we compute the required
sample size (or statistical power) to detect a relevant eect in a prognostic study, when dealing
with a competing risks outcome.
Copyright ? 2004 John Wiley & Sons, Ltd.
Statist. Med. 2004; 23:3263–3274
SAMPLE SIZES IN THE COMPETING RISKS SETTING
3265
Section 2 of this paper describes the sample size formula for evaluation of a therapeutical
eect. In Section 3, we extend this sample size formula to the prognostic situation, where
the prognostic factor of interest is possibly correlated with another factor. The validity of the
resulting approximate asymptotic formulas is investigated with respect to their nite sample
behaviour by numerical simulations in Section 4. Our approach is illustrated by two examples,
one is the randomized clinical trial, and the other is a prognostic study. We close the paper
with some discussion.
2. SAMPLE SIZES FOR EVALUATION OF THERAPEUTIC EFFECT IN THE
COMPETING RISK SETTING
Assume a randomized clinical trial is designed to compare two treatments with respect to a
competing risks endpoint. Usually, one treatment is a standard treatment or a placebo, whereas
the other treatment is an experimental treatment.
Let X be a binary covariate representing the treatment group (with X = 1 denoting the experimental group, and X = 0 the standard group) and Y , any additional covariate, independent
of X . Let p, be the proportion of patients randomly allocated to the experimental treatment
group, and n the required sample size of the trial. We wish to test the benet of the experimental group over the standard group with regards to the occurrence of the failure of interest,
either roughly or adjusted on Y . We extend Schoenfeld’s sample size formula [13], developed
in the conventional survival case, to the competing risks setting.
We assume the Fine and Gray’s model [9] for the subdistribution hazard of failure times
of interest,
1 (t; X; Y ) = 0 (t) exp(aX + bY )
In the case of ‘complete data’, the partial likelihood of the model is:
L() =
n
i
exp(axi + byi )
j∈Ri exp(axj + byj )
I (i = 1)
which, besides the denition of the risk-set Ri = { j : (Ti 6Tj ) ∪ (Tj 6Ti ∩ j = 1)}, is similar to
the Cox partial likelihood. Fine and√Gray [9] have shown that the Wald (partial) statistic to
test the null hypothesis {a = 0} is nâ where â is the maximum partial likelihood estimate
(MPLE) of a, which is asymptotically Gaussian with zero mean and estimated variance V0 .
Using the same terminology as in Reference [13, p. 502], this variance expresses:
V0 =
i∈E
Mi (x; b̂){1−Mi (x; b̂)}−{
with Mi (x; b) =
of b.
j∈Ri xj
n
2
i∈E Mi (xy; b̂)−Mi (x; b̂)×Mi (y; b̂)} =
i∈E Mi (y; b̂){1−Mi (y; b̂)}
exp(byj )=
Copyright ? 2004 John Wiley & Sons, Ltd.
j∈Ri
exp(byj ) and E = {i : i = 1} and b̂ is the MPLE
Statist. Med. 2004; 23:3263–3274
3266
A. LATOUCHE, R. PORCHER AND S. CHEVRET
Assuming that the variance of the Wald statistic under the alternative hypothesis is approximately equal to V0 , as in Reference [14, p. 442], it is straightforward that:
V0 ≈
n
ep(1 − p)
(2)
where e is the number of failures of interest.
Of note, the subdistribution hazard ratio, = exp(a), can be expressed as
=
log{1 − F1 (t; X = 0; Y)}
log{1 − F1 (t; X = 1; Y)}
To determine the sample size for this trial, we rst calculate the number e of failures of
interest required to control for both type I and type II error rates of and , respectively, as
follows:
e=
(u=2 + u )2
(log )2 p(1 − p)
(3)
where u denotes the (1 − )-quantile of the standard Gaussian distribution.
Since the ‘censoring complete data’ analysis relies on a proper partial likelihood, previous
results are readily inherited from classical Cox model with censoring [9, p. 499]. Formula (3)
thus holds in the cases of ‘complete data’ and ‘censoring complete data’.
The total required sample size is,
n=
(u=2 + u )2
(log )2 p(1 − p)
(4)
where is the proportion of failures of interest at the time of analysis, Ta .
Formula (4) looks similar to that derived by Schoenfeld [13] for the survival Cox regression
model, with , the subdistribution hazard ratio instead of the hazard ratio. Nevertheless, since
the eect of a covariate on the cause-specic hazard for a particular failure can be quite
dierent than its eect on the subdistribution function [8, 15], actually, formulas are dierent.
Finally, note that, in the absence of any censoring, the proportion of failures of interest
(that is identical to the proportion of non-censored observations in the Schoenfeld’s formula
[13]) reduces to the value at the time of analysis of the subdistribution function for the
failure of interest in the whole cohort, F1 (Ta ). In the case of ‘censoring complete data’,
can be estimated roughly by (1 − c)F1 (Ta ), where c is the expected proportion of censored
observations.
3. SAMPLE SIZES FOR THE EVALUATION OF PROGNOSTIC FACTORS FOR
COMPETING RISKS OUTCOMES
We now consider the planning of a prospective cohort study aiming at assessing whether
or not a particular exposure is associated with the subsequent occurrence of the failure of
interest. Let X be a binary covariate representing the exposure (with X = 1 if exposed and
X = 0 otherwise), and Y another prognostic covariate. For simplicity, we assume that Y is
Copyright ? 2004 John Wiley & Sons, Ltd.
Statist. Med. 2004; 23:3263–3274
SAMPLE SIZES IN THE COMPETING RISKS SETTING
3267
binary. Let p = Pr (X = 1); q = Pr (Y = 1); p0 = Pr (X = 1 | Y = 0) and p1 = Pr (X = 1 | Y = 1).
The correlation coecient of X and Y; , is
Cov(X; Y )
q(1 − q)
= (p1 − p0 ) ×
= p(1
− p)
p(1 − p)q(1 − q)
Using the same approximation as Schmoor et al. [14], we obtain,
n=
(u=2 + u )2
(log )2 p(1 − p) (1 − 2 )
(5)
Mathematical details are given in Appendix A. Formula (5) is similar to formulas derived
by Schmoor et al. [14] for the Cox model, and by Hsieh et al. [16] for linear and logistic
regression models, with the same variance ination factor 1=(1 − 2 ).
As in the independent case, formula (5) holds in the cases of ‘complete data’ and ‘censoring
complete data’. In the case of ‘incomplete data’ (i.e. in case of right censoring), Fine and Gray
[9] showed that the variance of the estimator was very close to that based on the ‘censoring
complete data’, suggesting that both our proposed formulas (4) and (5) could apply in such
a case. Hence, we decided to perform a simulation study to assess the validity of the sample
size formulas for right-censored data in nite samples.
4. SIMULATION STUDY
In this section, we present the results of numerical investigations. In each set of simulations,
we considered a failure cause of interest and a competing cause of failure, and two possibly
correlated binary covariates (X; Y ).
Failure times data were generated through the method described by Fine and Gray [9].
Briey, the subdistribution for the failures of interest are given by,
Pr (Ti 6t; i = 1; Xi ; Yi ) = 1 − [1 − Pr ( = 1; Xi = 0; Yi = 0)(1 − exp(−t))]exp(aXi +bYi )
which is a unit exponential mixture with mass 1 − Pr ( = 1; X; Y ) at ∞ when (X; Y ) = (0; 0),
and uses the proportional subdistribution hazards model to obtain the subdistribution for nonzero covariate values. The subdistribution for the competing risks failure cause was obtained
using an exponential distribution with rate exp(a2 Xi + b2 Yi ). Covariate values of (X; Y ) were
generated from a bivariate Bernoulli process with parameters p; q, and dened above.
We rst assumed that there was no censoring. Next, we considered ‘incomplete data’ with
right-censoring times generated from Uniform distribution, to reach an average of 30 per cent
of censored observations. In each simulated set, we generated an n-sample of subjects, where
n was computed according to equation (5) for predetermined = 0:05 and = 0:20. We took
Pr ( = 1; (X; Y ) = (0; 0)) = 0:5; = (1 − c)F1 (Ta) = 0:2, ∈ {1; 1:5; 2; 3; 4}; b = 1; a2 = b2 = 1,
p = q = 0:5, and ∈ {0; 0:2; 0:3; 0:4}.
In each situation, a total of 10 000 independent data sets were generated, with estimation of
the actual level and power of the Wald test for H0 : {a = 0}. Moreover, the respective eects
of , and b with other parameters xed were investigated.
Table I displays the observed level and power as obtained from the 10 000 simulations,
together with computed sample size, corresponding to several combinations of parameters ,
Copyright ? 2004 John Wiley & Sons, Ltd.
Statist. Med. 2004; 23:3263–3274
3268
A. LATOUCHE, R. PORCHER AND S. CHEVRET
Table I. Sample size for nominal type I error rate 0.05 and power 0.80, observed
level and power of the partial Wald test according to the correlation between both
covariates, the subdistribution hazard ratio and the censoring rate.
= exp(a)
No censoring
0
0.2
0.4
0
0.2
0.4
0
0.2
0.4
0
0.2
0.4
1.5
2
3
4
30 per cent censoring
0
1.5
0.2
0.4
0
2
0.2
0.4
0
3
0.2
0.4
0
4
0.2
0.4
Observed level
Observed power
n
0.0525
0.0520
0.0519
0.0445
0.0461
0.0483
0.0512
0.0498
0.0535
0.0520
0.0530
0.0640
0.8010
0.8083
0.8032
0.8311
0.8322
0.8302
0.8624
0.8652
0.8619
0.8771
0.8868
0.8729
382
398
455
131
137
156
53
55
62
33
35
39
0.0488
0.0496
0.0506
0.0461
0.0519
0.0502
0.0485
0.0495
0.0500
0.0417
0.0531
0.0560
0.7848
0.7804
0.7853
0.8169
0.8175
0.8242
0.8629
0.8555
0.8541
0.8683
0.8728
0.8715
546
569
650
187
195
223
75
78
89
47
49
56
, in the uncensored and 30 per cent-censored cases. In the uncensored setting, results show
that our formula performs well for small values of the subdistribution hazard ratio , while
it overpowers the study for increased values of . The level of the Wald test was found to
be approximately equal to the nominal level of 5 per cent, except for small sample sizes
corresponding to high values of . Results were very similar in the censored case, with a
relatively slight decrease in power as compared to the uncensored case.
The more specic eects of the formula’s parameters ( and a) as well as that of the other
regression parameter b are detailed in Figures 1 and 2. For xed covariates eect (a = log(2)
and b = − a), Figure 1 illustrates that the formula accounts well for a non-zero correlation .
Both the observed level and power remained stable around their nominal values for a range
of correlation coecient values ranging from −0:9 to 0:9.
The left plot of Figure 2 represents the observed level and power when a varies, while b
is set to zero. Results show that our formula correctly estimates the sample size for small
(absolute) values of a. For large negative values of a, however, the computed sample size
leads to under-powered studies whereas for large positive values of a, the studies are overpowered. The right plot of Figure 2 displays observed level and power when b varies, when
a = log(2) and = 0:2. As expected, when the eect of Y is in the same direction as that
Copyright ? 2004 John Wiley & Sons, Ltd.
Statist. Med. 2004; 23:3263–3274
3269
SAMPLE SIZES IN THE COMPETING RISKS SETTING
700
0.8
alpha
power
sample size
0.6
600
400
0.4
n
500
300
0.2
200
0.0
100
-1.0
-0.5
0.0
ρ
0.5
1.0
Figure 1. Sample size, observed level and power, according to the correlation
coecient between X and Y .
1.0
1.0
0.8
0.8
0.6
0.6
alpha
power
alpha
power
0.4
0.4
0.2
0.2
0.0
0.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
-2
a
-1
0
1
2
b
Figure 2. Observed level and power according to the values of the regression coecients a and b
associated with the covariate of interest and the other covariate, respectively.
of X , the power of the trial is higher than its nominal value, and opposite eects of both
correlated variables (a situation which is however rather theoretical) lead to a decrease in
statistical power.
5. EXAMPLES
The sample size formulas presented in the previous sections are illustrated by two examples.
Copyright ? 2004 John Wiley & Sons, Ltd.
Statist. Med. 2004; 23:3263–3274
3270
A. LATOUCHE, R. PORCHER AND S. CHEVRET
Table II. Comparison of sample size computation for a clinical trial comparing an experimental (E)
.
or CIF
and a standard (S) treatment groups, based on estimates of F1 computed either from 1 − KM
Ta (h)
12
24
48
Estimates of F1
S
E
1 − KM
CIF
1 − KM
CIF
1 − KM
CIF
0.20
0.20
0.60
0.60
0.865
0.77
0.30
0.30
0.75
0.67
0.90
0.78
=
log(1−FˆE (Ta ))
log(1−FˆS (Ta ))
1.6
1.6
1.5
1.2
1.2
1.03
e
191
191
245
1160
1357
47339
(per cent)
25.0
67.5
87.75
n
218
218
364
1596
1546
70131
5.1. Planning a randomized clinical trial
We retrospectively redesigned a phase III randomized clinical trial, to illustrate the use of
our proposed sample size formula contrasting with the use of the approach based on the
cause-specic hazard ratio.
This double-blind randomised clinical trial was conducted to compare the ecacy in the
induction of labour of vaginal misoprostol (experimental arm, X = 1) with vaginal dinoprosone
(standard arm, X = 0) [17]. The primary outcome of the trial was vaginal delivery within
24 h while time to vaginal delivery dened a secondary outcome. A total of 370 women were
enrolled.
Of note, in this setting, caesarian sections act as competing risks outcomes. To measure
the eect of misoprostol, the cause-specic hazard, that is the instantaneous risk of vaginal
delivery, is of less interest than the overall probability of vaginal delivery, which quanties
the overall success rate of the method used for labour induction.
Analysis of the trial was based on the complement of the Kaplan–Meier survival estimates
) of the outcome in each treatment group. Since it has been shown that
(denoted 1 − KM
caeserian sections are related to prolonged labour, i.e. when vaginal birth is not possible or
not safe for the mother or the child, both risks are obviously not independent, and marginal
probabilities could not be validly estimated. We thus computed nonparametric estimates of the
), treating caesarian sections as competing risks
cumulative incidence functions (denoted CIF
outcomes. We then attempted to show how the use of either approach would have modied
the computed sample size of the trial.
Therefore, on the basis of these estimates, we computed the sample size in each setting
accordingly, using three possible time points for the outcome that could have been chosen
by the investigator to plan a new trial (vaginal delivery at either Ta = 12; 24 or 48 h). Table
II displays the main results of these computations. The cause-specic hazard ratio diered
(Figure 3).
and CIF
from the subdistribution hazard ratio due to dierences between 1 − KM
These dierences result in dierences in sample size computations. For instance, as reported
in Table II, estimates of vaginal delivery rate at 24 h yield to a sample size of 364 based on
.
and 1596 based on CIF
1 − KM
5.2. Posterior power assessement in a cohort study
The second example is based on a cohort study involving 107 patients with acute or chronic
leukemia who underwent allogeneic stem cell transplantation, that was conducted to identify
Copyright ? 2004 John Wiley & Sons, Ltd.
Statist. Med. 2004; 23:3263–3274
SAMPLE SIZES IN THE COMPETING RISKS SETTING
3271
1.0
Probability of vaginal delivery
0.8
0.6
0.4
1 KM (Standard)
1 KM (Experimental)
CIF(Standard)
CIF(Experimental)
0.2
0.0
0
10
20
30
40
50
60
Time (hours)
Figure 3. Estimation of the cumulative incidence of vaginal delivery in both randomized
groups according to the statistical handling of caesarian sections: either censored using the 1
), or considered as a competing event in cumulative
minus Kaplan–Meier estimate (1 − KM
).
incidence function estimation (CIF
prognostic factors for early outcomes including infections, hematological recovery, acute graft
versus host disease (aGVHD) and survival [18]. We focused on one particular endpoint, that
is aGVHD.
Statistical analysis was initially based on the Fine and Gray model, considering death prior
to aGVHD as a competing risk event. Indeed, from a clinical point of view, the overall
probability of developing aGVHD was found more meaningful than the instantaneous risk
of aGVHD, owing to the short exposure period for aGVHD (restricted to the rst 100 days
post-transplant). Marginal probability of aGVHD could have been of interest if the removal
of aGVHD was expected to have no eect on failure rates for the remaining causes such as
death prior to aGVHD. In our setting, this was unlikely and we focused on the cumulative
incidence functions, and subdistribution hazards.
The study identied both female donor to male recipient and a particular gene polymorphism for the donor interleukin-1 (IL-1) cytokine as predictive of aGVHD. More precisely,
the estimated subdistribution hazard ratio for both variables was 1.91 (95 per cent condence interval 1.05–3.47) and 2.07 (95 per cent condence interval 1.09–3.91), respectively.
Otherwise, the analysis failed to identify any additional prognostic factor for aGVHD.
However, investigators would have expected a prognostic eect of interleukin-6 (IL-6) gene
polymorphism, approximately of the same magnitude as that of IL-1 gene polymorphism. We
wondered whether this could not be explained by a lack of statistical power.
Let X be a binary covariate, denoting the presence of donor IL-6 gene polymorphism, and
Y be a discrete score (taking four distinct values) dened as a linear combination of the two
Copyright ? 2004 John Wiley & Sons, Ltd.
Statist. Med. 2004; 23:3263–3274
3272
A. LATOUCHE, R. PORCHER AND S. CHEVRET
binary variables female donor to male recipient and donor IL-1 gene polymorphism. Since
the variance ination factor still holds in the Cox model with non-binary covariates [19], we
decided to apply formula (5) and compute statistical power to detect a subdistribution hazard
ratio of 2. The correlation coecient , between X and Y in equation (5) using estimated
regression coecients was estimated at 0.132 from the data.
Given the proportion of subjects with IL-6 gene polymorphism of p = 0:39, = 0:505,
inverting formula (5) yielded that the study had a 69 per cent power to detect a subdistribution
hazard ratio of 2.
In other words, if one wish to plan a future study, a sample size of n = 139 (resp, n = 186)
patients would be necessary to reach a power of 80 per cent (resp, 90 per cent).
6. DISCUSSION
Computation of sample size is important in designing experiments. In cohort studies, analysis
is frequently complicated by the presence of competing risks. However, despite the increasing
literature devoted to competing risks [1, 3–6, 8, 9, 15, 20], no specic method for sample size
computation has been proposed, besides that based on cause-specic hazards [21]. As mentioned above, the use of cause-specic hazards in the competing risks setting can be restrictive,
so that developing sample size formulas based on comparison of subdistribution hazards was
of prime interest. Two situations of planning were considered, a randomized clinical trial and
a prognostic cohort study, since both may have to deal with competing risks outcomes and
to focus on cumulative incidence functions rather than cause-specic hazards.
In both situations, the main objective of the study was to estimate the eect of a therapeutic
or prognostic factor, respectively. We proposed to test such eects on the subdistribution
hazard, using the proportional hazards model proposed by Fine and Gray [9]. Estimation
in this model from uncensored data is easily provided by using a Cox model where all
competing risks failures have been censored at +∞ [11]. Moreover, due to the structure of
the partial likelihood for the subdistribution, the results provided in the setting of the standard
Cox model [13, 14] were adapted in the current set-up. Of note, several assumptions were
required. Notably, formulas (4) and (5) only hold for contiguous alternatives. Nevertheless,
similar results have been stressed for the standard Cox model [13, 14]. Finally, the formulas
only apply without right censoring at least theoretically. Indeed, as reported in the simulation
study, the sample size formula for right-censored data could be reasonably approximated by
that for the ‘censoring complete data’. This allows a greater applicability of the formulas.
Actually, when computing sample size for time-to-failure data in the presence of competing
risks, a meaningful expected covariate eect on the subdistribution hazard ratio must be specied. Nevertheless, the subdistribution hazard does not have a natural interpretation. Hence,
in the examples, we formulated the eect size directly in terms of cumulative incidences
functions instead of subdistribution hazards, which may be easier to interpret. This also exemplies the fundamental dierence between both hazard ratios, and the reasons why sample
sizes computation may be aected by the model choice. As illustrated in the rst example,
we showed that a covariate may have a non-negligible inuence on the cause-specic hazard
(with estimated = 1:2), but no eect on the cause-specic subdistribution (with estimated
subdistribution hazard ratio = 1:03), as previously reported [9].
Copyright ? 2004 John Wiley & Sons, Ltd.
Statist. Med. 2004; 23:3263–3274
SAMPLE SIZES IN THE COMPETING RISKS SETTING
3273
Since the rst aim of the paper was to provide a sample size formula for clinical trials,
we investigated the situation of two independent binary factors. The second objective of the
paper was to deal with the planning of prognostic studies, which is often overlooked leading
to inconsistency of prognostic studies [14]. Therefore, we extended sample size formulas to
handle for multiple models with correlated covariates. For simplicity, we considered binary
covariates and we ignored the accrual pattern. Nevertheless, this could be easily extended to
more general situations, as previously done by Hsieh and Lavori [19] and Lachin and Foulkes
[22], respectively.
APPENDIX A
Under H0 : {a = 0}, it is possible to approximate Mi (x) by p1 qi + p0 (1 − qi )=qi + (1 − qi ),
where = exp(b) and qi is the probability of Y equal to 1 at time ti . Similarly, we approximate
Mi (y) as qi =qi +(1 − qi ) and Mi (xy) as p1 qi =qi +(1 − qi ). This yields to Mi (xy) = p1 Mi (y)
and Mi (x) = p1 Mi (y)+p0 (1 − Mi (y)). Thus, Mi (xy) − Mi (x)Mi (y) = (p1 − p0 )Mi (y)[1 − Mi (y)]
and
n
V0 = i∈E [Mi (y)(p1 − p0 )(1 − p1 − p0 ) + p0 (1 − p0 )]
As i∈E Mi (y) approximately equals to q × e, V0 reduces to V0 = n=ep(1 − p)(1 − 2 ). The
latter is equal to the variance in the independent case multiplied by the variance ination
factor 1=(1 − 2 ). To illustrate that i∈E Mi (y) ≈ q × e, we considered that time to failures from the cause
of interest were exponentially distributed, i.e. 10 (t) = , as did Schmoor et al. [14]. We also
assumed that there is no censoring. Under this assumption, the subdistribution function is
F1 (t) = 1 − q exp(−t) − (1 − q) exp(−t)
with derivative with respect to t, f(t) = [q exp(−t) + (1 − q) exp(−t)]
It follows that,
qi =
q exp(−ti )
q exp(−ti ) + (1 − q) exp(−ti )
and
e
−1 i∈E
leading to
i∈E
Mi (y) →
0
∞
q exp(−t)
f(t) d t
q exp(−t) + (1 − q) exp(−t)
Mi (y) → q × e.
ACKNOWLEDGEMENTS
We are grateful to Doctor Patrick Rozenberg and Professor Eliane Gluckman for providing access to
data. We also want to thank the referees for their helpful comments that led to important improvements.
Copyright ? 2004 John Wiley & Sons, Ltd.
Statist. Med. 2004; 23:3263–3274
3274
A. LATOUCHE, R. PORCHER AND S. CHEVRET
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Misspecified regression model for the cumulative
incidence function
A. Latouche,1,∗ V. Boisson,2 S. Chevret1 and R. Porcher1
1
Département de Biostatistique et Informatique Médicale, Hôpital Saint-Louis,
Université Denis Diderot, Inserm Erm 321, Paris, France
2
Laboratoire de Statistique Théorique et Appliquée
Université Pierre et Marie Curie, Paris, France
Summary. We considered a competing risks setting, when evaluating the prognostic
influence of an exposure on a specific cause of failure. Two main regression models are
used in such analyses, the Cox cause-specific proportional hazards model and the subdistribution proportional hazards model. We examine the properties of the estimator
based on the latter model when the true model is the former. An explicit relationship between subdistribution hazards ratio and cause-specific hazards ratio is derived,
assuming a parametric distribution for latent failure times.
Key words: Model misspecification; Cumulative incidence; Proportional hazards.
1.
Introduction
In the competing risks setting, subjects may fail from distinct and exclusive causes.
Thus, observed data typically consist in both a failure time T ≥ 0 and a failure cause
∈ {1, . . . , K}, which is unobserved if T is right-censored (this will be denoted further
= 0). Consider that we are interested in estimating the effect of an exposure, Z, on
a particular cause of failure, identified by = 1.
To examine the effect of an exposure on a particular cause of failure in the setting
∗
email: [email protected]
1
of competing risks, two main approaches are used (Andersen et al., 2002). The most
common approach is to focus on the modelling of the cause-specific hazard of this failure
cause (Prentice et al., 1978), widely through the use of a Cox model. The second
approach is to compare the cumulative incidence functions (CIF) of this failure cause
between exposed and unexposed groups, either directly (Gray, 1988; Pepe, 1991), or by
modelling the hazard function associated with the CIF, the so-called subdistribution
hazard (Fine and Gray, 1999; Fine, 2001).
Let us denote the CIF for failure of cause k by:
Fk (t) = Pr(T ≤ t, = k),
(1)
and the marginal survival function:
S(t) = Pr(T > t) = 1 −
X
Fk (t).
k
For simplicity, the Fk (t) are assumed to be continuous with subdensities fk (t) (with
respect to Lebesgue measure). The cause-specific hazard function is defined by:
1
Pr(t ≤ T < t + δt, = k|T ≥ t) = fk (t)/S(t),
δt→0 δt
λk (t) = lim
(2)
while the subdistribution hazard (Gray, 1988) is defined by:
1
Pr(t ≤ T < t+δt, = k|T ≥ t∪(T < t∩ 6= k)) = fk (t)/[1−Fk (t)]. (3)
δt→0 δt
αk (t) = lim
All these functions are identifiable from observations (Tsiatis, 1975; Prentice et al.,
1978). To relate the cause-specific hazard (Prentice et al., 1978) on the exposure covariate Z, the Cox proportional hazards model (Cox, 1972) is often used while a similar
model was proposed for the subdistribution hazard (Fine and Gray, 1999). From equations (2) and (3), it is clear that the subdistribution hazard can be directly obtained
2
from the CIF, whereas the relationship between the cause-specific hazard and the CIF
involves the marginal survival function, i.e., the competing risks. One further consequence of the use of these two different approaches is that the effect of a covariate on
either function can be different, as illustrated by the example given in Gray (1988, p
1142).
In practice, when reporting analysis of competing risks failure time data, a graphical
display of the probabilities of failure causes against time is useful. These probabilities
can be defined in two ways: the ”crude” probability, which corresponds to the CIF
and the ”net” probability, which corresponds to the probability of the failure cause
of interest in the hypothetical situation where it is the only cause of failure acting on
the population (Tsiatis, 1975; Tsiatis, 1998). If ”crude” probabilities can be estimated
using observed data, ”net” probabilities are not identifiable from observations, unless
additional assumptions on the mechanism underlying the competing risks situation,
such as the so-called independent competing risks framework. In this case, the failure time T is assumed to be the minimum of K mutually independent latent failure
times, each corresponding to a failure of a particular cause (Sampford, 1952). This
assumption is however unverifiable (Tsiatis, 1975), and strong evidence of complete
biologic independence among the physiological mechanisms giving rise to the various
failure causes would be required to justify such an assumption (Prentice et al., 1978).
Cause-specific hazards could also be used for graphical representation, but they are
less interpretable, particulary in terms of magnitude of the proportion of patients failing from the different failure causes (Pepe and Mori, 1993). Therefore, CIFs are well
suited to summarize competing risks failure time data and a direct modelling of such
quantities seems relevant to avoid presenting conflicting or contradictory results.
However, when assessing the prognostic value of an exposure on a specific failure
3
cause, a multiplicative effect of the exposure on the cause-specific hazard appears more
clinically understandable. First, as exposed above, the cause-specific hazard can be
expressed using the Cox model, which is widely used in the medical literature. Secondly, it seems natural that the physiological effect of a treatment or any prognostic
exposure would be to reduce or increase the probability of the failure cause of interest
at any time, conditionally on being still alive at that time. On the contrary, a decrease
(resp. increase) in the cumulative incidence function could be due either to a physiological effect of the exposure or to an increase (resp. decrease) in the probability of the
competing failure causes.
This is first exemplified on a real data example. Then, we place ourselves in a the setting where the true model for the covariate effect is a Cox proportional (cause-specific)
hazards model, and derive asymptotic properties of regression parameter estimates
reached by fitting a regression model for the subdistribution hazard.
2.
A real example
In this section, we reanalyzed the mgus data set presented in Therneau (2000, pp 175–
177), using the two regression approaches described above. This data set represents
observations from 241 patients with monoclonal gammopathy of undetermined significance (MGUS) identified at the Mayo Clinic before Jan. 1, 1971, with a 20 to 35 years
follow-up for each patient (Kyle, 1993). MGUS is considered as a potential precursor
to several plasma cell malignancies. From a competing risks point of view, several
competing causes of failure were reported: Failure from multiple myeloma (n = 39),
amyloidosis (n = 8), macroglobulineamia (n = 7), other lymphoproliferative diseases
(n = 5) or first death (n = 130). Due to the rather small number of failures, we considered two main failure causes, namely ”death”, and ”plasma cell malignancy”. We
4
focused on the estimation of the effect on failure of age, distinguishing two categories
according to the sample’s median, that is 64 years. The Figure 1 displays the estimated CIF of the two failure causes in each age category. Clearly, patients aged 64 ys
or more were less likely to develop plasma cell malignancy than those aged 64 years
or less. This is in accordance with the estimated subdistribution hazard ratio of 0.43
(95% confidence interval: 0.25-0.74) in this age category (Table 1). By contrast, the
cause-specific hazard ratio of age above 64 years category was estimated at 0.80 (95%
confidence interval: 0.45-1.40). Actually, as depicted in Figure 1, those patients aged
64 years or more had a much higher cumulative incidence of first death. At least, as
exposed above, this highlights the necessity to display the probability of the competing failure causes across the covariate groups before any interpretation of the effect of
that covariate on the CIF of the failure cause of interest. Of note, whatever the model
used, assumption of proportional hazards was roughly checked through the display of
cumulative hazards in each age category (Figure 2).
[Figure 1 about here.]
[Table 1 about here.]
[Figure 2 about here.]
3.
Misspecified model for the cumulative incidence function
For simplicity, we considered two failure causes ( = 1, 2), where = 1 denoted the
failure cause of interest, and assumed that the true underlying model generating the
data is a proportional hazards model for the cause-specific hazard function, i.e. :
λk (t; Z) = λk0 (t) exp(βk Z) k = 1, 2
5
(4)
where λk0 (t) are unspecified continuous positive functions and Z is a binary covariate
representing the exposure. We wished to fit a proportional subdistribution hazards
model for failure of cause 1:
α1 (t; Z) = α10 (t) exp(γZ).
We examined the properties of the estimator of γ under the condition given above.
Because of the nice structure of the partial likelihood for the subdistribution, results
in Solomon (1984) are somewhat straightforward. Let us recall the particularity, in
terms of counting process, of the Fine and Gray model, in the presence of administrative
censoring, i.e., when the potential censoring time is observed on all individuals (referred
as “censoring complete data” in Fine and Gray, 1999).
Suppose a sample of size n and define the process Ni (t) = 1(Ti ≤t,εi =1) , which takes
value zero until individual i fails from cause 1. Let Yi (t) = 1 − Ni (t−), be the indicator
of individual i being at risk of failure before time t. We suppose that Ci is independent of (Ni , Yi , Zi ), where the subscript i corresponds to the individual, and that
(Ni , Yi , Zi , Ci ) are independent and identically distributed replicates of (N, Y, Z, C). In
case of censoring, Yi∗ (t) = {1 − Ni (t−)}1(Ci ≥t) = Yi (t−)1(Ci ≥t) .
Let γ̂ be the MPLE of γ. Using the theorem of Struthers and Kalbfleisch (1986, pp
365), γ̂ is a consistent estimator of γ ∗ , where γ ∗ is the solution of
(
"
#)
Pn
Z ∞ X
n
∗
E[Z
Y
(t)
exp(γZ
)]
1
j
j
j
j=1
E Yi∗ (t)α1 (t; Zi ) Zi − Pn
dt = 0
∗
n
0
j=1 E[Yj (t) exp(γZj )]
i=1
or equivalently
I(γ, β1 ) =
Z
∞
E f1 (t; Z)1(C≥t)
0
E[Z exp(γZ)(1 − F1 (t; Z))1(C≥t) ]
Z−
E[exp(γZ)(1 − F1 (t; Z))1(C≥t) ]
where f1 (t; Z) = dF1 (t; Z)/dt.
6
dt = 0,
No explicit solution for γ ∗ is available, but a Taylor expansion around (0, 0) provides
the following approximation:
γ∗ ' h
−1
∂
I(γ, β1 )
∂γ
i
(
I(0, 0) + β1
(0,0)
∂
I(γ, β1 )
∂β1
(0,0)
)
(5)
with I(0, 0) depending on β2 , the covariate distribution, and the censoring distribution.
4.
Illustration: Absolutely continuous bivariate exponential model
To illustrate the application of formula (5), we exemplified these results using a parametric model, for which explicit solutions for γ ∗ are obtained. Additionally, a simulation
study was performed to investigate the relevance of the approximation in small samples.
4.1 Parametric setting
For illustration, we used a parametric latent failure times model. In such a model,
each possible cause of failure is represented by a latent failure time, Tk , k = 1, 2,
while the observable variables introduced above are defined as T = min(T1 , T2 ) and
= 2 − 1(T1 ≤T2 ) .
We considered the case where (T1 , T2 ) has an absolutely continuous bivariate exponential distribution as introduced by Block and Basu (1974), denoted (T1 , T2 ) ∼
ACBV E(a1 , a2 , a12 ), where a1 , a2 , and a12 are the distribution parameters. Accordingly, the joint survival function of (T1 ,T2 ) is given by:
S(t1 , t2 ) = Pr(T1 > t1 , T2 > t2 ) = a/(a1 + a2 ) exp[−a1 t1 − a2 t2 − a12 max(t1 , t2 )]
−a12 /(a1 + a2 ) exp[−a max(t1 , t2 )],
where a = a1 + a2 + a12 , with the cause-specific and subdistribution hazards for failure
of cause 1:
λ10 (t) = λ10 =
7
aa1
a1 + a 2
and
α10 (t) =
aa1
,
a1 + a2 exp(at)
respectively.
Such a choice allows to consider both independent (a12 = 0) and dependent (a12 6=
0) latent failure times. Moreover, when a12 = 0, both T1 and T2 have a marginal
exponential distribution of parameter a1 and a2 , respectively. Thus, this distribution
can also accomodate with the classical independent exponential latent failure times
approach.
Let Z be a binary exposure of interest with p = Pr(Z = 1). According to the
regression model (4), the cause-specific hazard, the subdistribution function and the
subdistribution hazard function of failure of cause 1 express as:
λ1 (t; Z) =
F1 (t; Z) =
aa1
exp(β1 Z),
a1 + a 2
a1 exp(β1 Z)
[1 − exp[−Ψ(Z)t]],
a1 exp(β1 Z) + a2 exp(β2 Z)
and
α1 (t; Z) =
a1 exp(β1 Z)Ψ(Z) exp{−Ψ(Z)t}
,
a2 exp(β2 Z) + a1 exp(β1 Z) exp{−Ψ(Z)t}
respectively, where Ψ(Z) = λ10 exp(β1 Z) + λ20 exp(β2 Z).
In case of no censoring (referred as “complete data”), equation (5) reduces to:
γ∗ ' c0 + c1 β1
(6)
where c0 = c0 (β2 , V ar(Z)) and c1 = c1 (β2 , p). Calculation details are given in the
Appendix.
4.2 Numerical Studies
Validity of formula (6) was investigated by performing a series of simulation studies.
Each simulated data set of size n = 400 consisted of two balanced groups (exposed or
8
unexposed) of observations, each corresponding to a different ACBVE distribution. For
the unexposed group (Z = 0), the (a1 , a2 ) parameters were set to {(0.5, 1), (1, 0.5)},
while a12 ranged in {0, 0.5, 1, 1.5, 2, 2.5}. For the exposed group (Z = 1), an ACBVE
distribution was used with parameter (a01 , a02 , a012 ). To reach proportional cause-specific
hazards (4), the following ACBVE parameters a01 , a02 were derived:

−1


aa2 exp(β1 )(A+1)
0

a2 = a12 −A + (a1 +a2 ) exp(β1 −β2 ) − 1
where A =
a1
a2



a01 = a02 A
(7)
exp(β1 − β2 ) + 1, while a012 = a12 for simplicity.
ACBVE failure times were generated using the method proposed in Friday and Patil
(1977). For any data set, γ
b was computed by using the cmprsk package of R (R Devel-
opment Core Team, 2004), while γ ∗ was computed from equation (5), with numerical
integrations based on the integrate function of R. For each configuration of the simulation parameters, N = 5000 independent data sets were generated to estimate the
mean (and standard error) of the N values of γ
b. Table (2) reports the results of the
simulations.
[Table 2 about here.]
Expectedly, formula (6) provides a good approximation of γ̂ for small values of β1
in reasonable sample sizes. However, departures of β1 from zero led to a biased value
of approximation of γ ∗ . By contrast, dependence between latent failure times had no
effect on both γ ∗ and γ̂. This is likely due to the exp(−at) term, which is negligible in
the evaluation of the integral (see Appendix).
Additionally, we studied the influence of β2 and p on γ ∗ . Results are displayed in
Figure (3).
9
[Figure 3 about here.]
It seems that c1 has a little influence on the value of γ ∗ , whereas the shape of the
calculated regression coefficient is rather driven by c0 . This comment applied in both
settings, i.e., when β2 , the regression coefficient for the competing failure cause, varied
from -1 to 1, and for value of p varying from 10 % to 90%.
5.
Concluding Remarks
In a competing risks setting, when estimating the effect of an exposure on a specific
failure time, there is still an open choice (dilemma) between the cause-specific and
cumulative incidence approaches. The two models share the same proportional hazards
assumptions but for two quantities that differ (greatly). Therefore, this can lead to
model misspecification.
Actually, we showed that, when an exposure acts on a failure cause of interest
through a multiplicative effect on the cause-specific hazard, analysis based on a proportional hazards model for the subdistribution hazard achieved a different effect, depending on the cause-specific effect on the failure cause of interest, but also on the
cause-specific effect on the competing failure causes. This was exemplified on a real
data set in MGUS, where a protective effect of advanced age on the subdistribution
hazard of plasma cell malignancy was found, without any cause-specific effect. Of note,
fitting a parametric exponential model to the MGUS data (as a12 is of little influence
on γ ∗ ), we found, under the assumption of β1 = 0 and β2 = −1.4 that γ ∗ was -1.10,
which is not too far from the estimated value -0.84 (se 0.278).
An explicit relationship between both effects was derived assuming a parametric
ACBVE model of the latent failure times. This relates closely to papers from Solomon
(1984) and Struthers and Kalbfleisch (1986). They considered misspecification with
10
regards to missing covariates, functional form misspecification for the covariates and
accelerated failure times model. By contrast, we considered misspecification with regards to the the hazard function for the failure cause of interest. Through simulation
studies, we found that this approximate works well for small values of β1 , whatever the
dependence between the latent failure times. The converse analysis was not conducted,
because it appeared less plausible.
References
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multi-state model. Statistical Methods in Medical Research 11, 203–215.
Block, H. W. and Basu, A. P. (1974). A continuous bivariate exponential extension.
Journal of the American Statistical Association 69, 1031–1037.
Cox, D. R. (1972). Regression models and life tables. Journal of the Royal Statistical
Society, Series B 34, 187–220.
Fine, J. P. (2001). Regression modelling of competing crude failure probabilities. Biostatistics 2, 85–97.
Fine, J. P. and Gray, R. J. (1999). A proportional hazards model for subdistribution
of a competing risk. Journal of the American Statistical Association 94, 496–509.
Friday, D. and Patil, G. (1977). A bivariate exponential model with applications to
reliability and computer generation of random variables. In Tsokos, C. and Shimi,
I., editors, The Theory and Applications of Reliability, pages 527–549. Academic
Press, New York.
Gray, R. J. (1988). A class of k-sample tests for comparing the cumulative incidence of
11
a competing risk. The Annals of Statistics 116, 1141–1154.
Kyle, R. (1993). ”benign” monoclonal gammopathy - after 20 to 35 years of follow-up.
In Mayo Clinic Proceedings, volume 68, pages 26–36.
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studies. Journal of the American Statistical Association 86, 770–778.
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12
Encyclopedia of Biostatistics, pages 824–834. John Wiley & Sons, New York.
Appendix A
Expections are taken relative to the covariate Z. The following development heavily
relies on Solomon (1984) and Struthers and Kalbfleisch (1986) apart from the counting
process involved in the partial likelihood.
A Taylor expension around the neigbourhood of (γ, β1 ) = (0, 0) leads to:
Z ∞ p[l2 (1) + a1 exp(−Ψ0 (1)t)]
Γ−1
dt
E (a1 + a2 ) exp(−Ψ0 (Z)t) Z −
γ∗ '
V ar(Z) 0
(a1 + l2 (1))E[(1 − F1 (t; Z))|(0,0) ]
Z ∞
(1 − λ1 t)[a2 + a1 exp(−at)] exp(−Ψ0 (1)t)
−1
+ β1 Γ
E[(1 − F1 (t; Z))|(0,0) ]
0
E[exp(−Ψ0 (Z)t)]a1 [a2 + a1 exp(−at)]
+
{l2 (1) + [a1 λ1 t + (λ1 t − 1)l2 (1)] exp(−Ψ0 (1)t)}dt,
(a1 + l2 (1))E 2 [(1 − F1 (t; Z))|(0,0) ]
where Ψ0 (Z)) = Ψ(Z)|(β1 =0) and li (Z) = ai exp(βi Z), i ∈ (1, 2)
Γ=
Z
∞
0
[a2 exp(β2 ) + a1 exp(−Ψ0 (1)t)][a2 + a1 exp(−at)]
E[exp(−Ψ0 (Z)t)]
×
dt
− F1 (t; Z))|(0,0) ]
a1 + a2 exp(β2 )
E 2 [(1
E[exp(−Ψ0 (Z)t)] = p exp[−(λ10 + λ20 exp(β2 ))t] + (1 − p) exp(−at)
E[(1 − F1 (t; Z))|(0,0) ] = [(pa1 + a2 )l2 (1) + (1 − p)a1 (l2 (1) + a1 ) exp(−at)
+ pa1 (a1 + a2 ) exp(−Ψ0 (1)t) + (1 − p)a1 a2 ]
/ (a1 + a2 )(a1 + a2 exp(β2 ))
and p = Pr(Z = 1).
13
Death
1.0
1.0
Plasma cell malignancy
0.6
0.8
age < 64 ys
age >= 64 ys
0.0
0.2
0.4
Probability
0.6
0.4
0.0
0.2
Probability
0.8
age < 64 ys
age >= 64 ys
0
5
10
15
20
25
30
35
0
Years
5
10
15
20
25
30
35
Years
Figure 1. Cumulative incidence of first death or plasma cells malignancy according to
age category for the MGUS data.
14
0
−2
−6
−4
log(− log(1 − F1(t)))
0
−2
−4
−6
log(Λ1(t))
1.0
1.5
2.0
2.5
3.0
3.5
1.0
log(t)
1.5
2.0
2.5
3.0
3.5
log(t)
Figure 2. Cumulative hazard of occurrence of a plasma cell malignancy in MGUS data,
expressed either as the cumulative cause-specific hazard, log Λ1 (t), or the cumulative
subdistribution hazard, log[− log(1 − F1 (t))].
15
1.0
1.0
0.5
0.5
0.0
0.0
−0.5
−0.5
−1.0
−0.5
0.0
0.5
1.0
0.2
0.4
0.6
0.8
Pr(Z = 1)
β2
Figure 3. c0 (dotted line), c1 (dashed line), γ ∗ (solid line), while β2 varying with
(β1 , p, a1 , a2 , a12 ) = (0.1, 0.5, 0.5, 1, 0)(left plot), while p varying (right plot) with
(β1 , β2 , a1 , a2 , a12 ) = (0.1, 0.5, 0.5, 1, 0)
16
Table 1
Estimated effects of the covariate ”age ≥ 64 ys” on the cause-specific and
subdistribution hazards of developping a plasma cell malignancy and dying without
developping a malignancy.
Effet of age
Failure cause
Plasma cell malignacy
Death as first event
Estimated regression parameter
(Standard Error)
Cause-specific Subdistribution
hazard
hazard
-0.225 (0.285)
-0.842 (0.278)
1.39 (0.198)
1.31 (0.192)
17
Table 2
Simulation Results
a1
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
1
1
1
1
1
a2
1
1
1
1
1
1
1
1
1
1
0.5
0.5
0.5
0.5
0.5
a12
0
0
0
0
0
0.5
1
1.5
2
2.5
0.5
1
1.5
2
2.5
β1
0.1
0.2
0.3
0.4
0.5
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
β2
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
18
γ∗
-0.296
-0.205
-0.114
-0.022
0.068
-0.296
-0.296
-0.296
-0.296
-0.296
-0.152
-0.152
-0.152
-0.152
-0.152
Mean(γ̂)(se)
-0.301 (0.186)
-0.213 (0.181)
-0.128 (0.183)
-0.038 (0.177)
0.049 (0.174)
-0.308 (0.188)
-0.306 (0.189)
-0.307 (0.189)
-0.300 (0.189)
-0.304 (0.186)
-0.174 (0.128)
-0.173 (0.129)
-0.173 (0.129)
-0.173 (0.128)
-0.179 (0.129)
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bimj header will be provided by the publisher
A note on including time-dependent covariate in regression
model for competing risks data
A. Latouche∗1 , R. Porcher1 , and S. Chevret1
1
Département de Biostatistique et Informatique Médicale
Hôpital Saint-Louis, AP-HP,
Université Paris 7 and INSERM ERM 0321
F-75010, Paris, France
Summary
Recently, regression analysis of the cumulative incidence function has gained interest in competing risks
data analysis, through the model proposed by Fine and Gray (JASA 1999;94:496–509). In this note, we
point out that inclusion of time-dependent covariates in this model can lead to serious bias. We illustrate the
problems arising in such a context, using bone marrow transplant data as a working example and numerical
simulations. Practical advices are given, preventing the misuse of this model.
Key words: time-dependent covariate, subdistribution, competing risk
1 Introduction
In longitudinal cohort studies, competing risks failure time data are commonly encountered. For instance,
after allogeneic bone-marrow transplantation (aBMT) for leukemic patients in complete remission, deaths
in remission compete with relapse. This will be our working example. To isolate the effect of covariates
∗
Corresponding author: e-mail: [email protected], Phone: +33 142 499 742, Fax: +33 142 499 745
Copyright line will be provided by the publisher
2
A. Latouche: Time-dependent covariate in the competing risk setting
on these risks, several regression models can be used. Actually, regression analysis of competing risks
failure time can be performed either by modelling the cause specific hazard function or the cumulative
incidence function (also known as the subdistribution function). The former approach is commonly used
in this setting (Rosenberg et al., 2004; Cornelissen et al., 2001). However, the instantaneous risk of specific failure cause is sometimes of less interest than the overall probability of this specific failure. In our
working example, actually, the overall probability of death in remission, often referred as “treatment related mortality” appears more interesting than the instantaneous risk of dying in remission. Otherwise, the
overall probability of relapse is also of interest to quantify the outcomes in the population of transplanted
patients.
Such a probability of failure could be formulated as either the marginal distribution (of the specific failure cause), that is the probability of this failure cause in a population where only this failure cause acts, or
the cumulative incidence function, i.e., the overall probability of the specific cause of failure in the presence
of the competing failure causes. However, the marginal distribution is not identifiable from available data
without additional assumptions, such as independence between competing failure causes (Tsiatis, 1998).
Therefore, cumulative incidence functions may appear more relevant than marginal probabilities (Pepe and
Mori, 1993; Korn and Dorey, 1992; Gaynor et al., 1993).
To assess the effect of a covariate on the cumulative incidence of a competing risk, Fine and Gray (1999)
proposed a regression model. It has been recently used to model clinical data in cancer (Colleoni et al.,
2000; Robson et al., 2004) or hematology (Rocha et al., 2001, 2002). It allows to estimate the effect of
constant (time-fixed) covariates on the subdistribution hazard of specific failure causes. Time-by-covariate
interaction is handled by this model, but most of time-dependent covariates such as “one time jumps”
(taking 0 value unless the outcome of interest is observed, and 1 thereafter) are not. For instance, in the
context of our working example, patients with leukemia frequently develop after aBMT acute graft versus
host disease (aGvHD) wherin the transplanted immune cells attack the host tissues. Some evidence exists
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3
to consider that occurrence of aGvHD modifies patients’ outcome as it increases risk of mortality but decreases risk of relapse. One could be interested in estimating the effect of such a time-dependent covariate
(taking zero values unless the aGvHD is observed and 1 thereafter) on the occurrence of failures of interest
(death or relapse).
We show that inclusion of such internal time-dependent covariate is not relevant when modelling the subdistribution hazards as it implies conditioning on the future. This article should be considered as a guideline
for preventing the misuse of the model in this setting. In Section 2, we present the Fine and Gray regression
model. A real data example is proposed in Section 3. In Section 4, we present a Monte Carlo simulation
study to assess the resulting bias in estimating the effect of a time-dependent covariate using the Fine and
Gray model. Concluding remarks are presented in Section 5.
2 Models
Let T be the failure time, ² the cause of failure, where ² = 1 denotes the cause of interest and ² = 2
the competing cause (considering, without loss of generality, a single competing failure cause), and Fi =
Pr(T ≤ t, ² = i) the cumulative incidence function of failure from the cause i (= 1, 2). Gray (1988)
defined the subdistribution hazard for cause i as:
λi (t) = lim
dt→0
1
Pr {t ≤ T ≤ t + dt, ² = i|T ≥ t ∪ (T ≤ t ∩ ² 6= i)} ,
dt
by contrast to the cause-specific hazard:
αi (t) = lim
dt→0
1
Pr {t ≤ T ≤ t + dt, ² = i|T ≥ t} .
dt
Similarly to the Cox model for the cause-specific hazard, αi (t; X(t)) = αi0 (t) exp{bi X(t)}, where αi0 (t)
is a non specified baseline hazard function, and bi is the regression parameter, Fine and Gray (1999) proposed a regression model for the subdistribution hazard: λi (t; X(t)) = λi0 (t) exp{βi X(t)}. By construction, the subdistribution hazard is explicitely related to the cumulative incidence function of failure from
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4
A. Latouche: Time-dependent covariate in the competing risk setting
cause i, by λi (t) = −d log{1 − Fi (t)}/dt, while the relation between the cause-specific hazard and the cumulative incidence function is less straightforward, and involves the cause-specific hazard of failure from
other causes.
For inference in this model from j = 1, . . . , N individuals, the risk set at time t expresses as R(t) =
{j : (t ≤ Tj ) ∪ (Tj ≤ t ∩ ²j 6= i)}. This includes individuals who have not failed from any cause by t,
like in the Cox model for the cause specific hazard (with risk set at time t defined by {j : t ≤ Tj )}), and,
in addition, those who have previously failed from the competing cause before t.
Let T be the time of failure of the individual, and Z be the time to occurrence of any event of interest.
Suppose that we wish to estimate the effect of X(t) = 1{Z≤t} on the subdistribution hazard λ1 (t) at a
particular time τ . If T ≤ τ and the cause of failure is not that of interest (² = 2), the risk set comprises
individuals who have not experienced any failure, and those who have previously failed from the competing
cause. Moreover, in the case of an absorbing competing cause of failure such as death, the covariate value
of a patient who dies cannot be observed anymore while the patient is still considered to be at risk until the
maximum observation time of the cohort. This is illustrated in Figure 1, in absence of censoring.
[Fig. 1 about here.]
Let “non-identifiable path” denote further those observations, in opposition to “identifiable path” where
the occurrence of the competing cause of failure does not avoid the observation of X(t).
3 A clinical example
We illustrated estimation of the effect of such a time-dependent covariate on a specific failure cause on real
data. Data consist in a sample of 180 children with acute leukemia who underwent aBMT between 1994
and 1998 (Rocha et al., 2001). Of these 180 patients, 34 developed aGvHD followed by either relapse for 6
patients or death in remission for 22. Among the 146 patients who did not experience aGvHD, there were
60 relapses and 22 deaths in remission (Figure 2). No patient was loss to follow-up. We were concerned
by estimating the effect of aGvHD on the occurrence of relapse (² = 1).
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5
[Fig. 2 about here.]
Estimation of β1 was carried out using the survival package of R Team (2004) with competing
failure observations censored at their follow-up time (difference between the reference date and the entry
date), as censoring only resulted from administrative loss to follow-up. Estimation of b1 was performed
by using a standard Cox model, where deaths in remission were censored at the time of death. The timedependent covariate, aGvHD, was considered as a one-time jump, taking the value 0 unless aGvHD is
observed. Of note, for the Fine and Gray model, the last value of the jump was carried out forward after
the competing failure time of death in remission.
The estimated effect of aGvHD on the hazard of relapse, with death in remission defining the competing
cause of failure, was statistically significant, with β̂1 = −0.975 (SE = 0.429, p = 0.023). By contrast,
the effect of aGvHD on the cause-specific hazard of relapse was not, with b̂1 = −0.404 (SE = 0.43,
p = 0.322).
In the next Section, a simulation study will exhibit the fact that the former estimate have no sense as we
are obviously in a “non-identifiable path” setting.
4 Simulation
We conducted a simulation study to numerically illustrate probles arising when using the Fine and Gray
model to estimate the effect of a time-dependent covariate on the subdistribution hazard of failure. Specifically, we were interested in examining the bias in estimating β1 when the competing cause of failure is
either non absorbing or absorbing for the covariate process. For the time dependent covariate, we considered a one jump process as defined by X(t) = 1{Z≤t} , where Z is the time to occurrence of some event
that could be related to the outcome. We attempted to mimic the data example exposed above.
All simulations were based on 1,000 realizations with sample sizes of 250. For simplicity, we supposed
the absence of right censoring. The occurrence of jump in the covariate process was generated from a
Bernoulli distribution with parameter q = 0.6.
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6
A. Latouche: Time-dependent covariate in the competing risk setting
Then, the time to occurrence of aGvHD, Z, was chosen to reach a probability near 1 of aGvHD at time
100 (days), as aGvHD is defined only within the first 100 days post-transplant, with a shape similar to
that observed on real data sample. Thus, the individual times Z were generated from a random variable
40 × W , where W has Weibull distribution with shape parameter of 2 and scale parameter of 1.
Generating failure times was complicated by the presence of the time-dependent covariate. It was based
on inversion of the cumulative subdistribution hazard functions, adapting the method proposed by Leemis
et al. (1990) in the case of survival data. Lifetime data from the cause of interest were generated as
described by Fine and Gray (1999). Details of the failure times generation is presented in the Appendix.
Simulation codes are available upon request to the corresponding author.
We simulated two types of covariate paths: (i) identifiable paths, when the covariate process of patients
who experienced the competing failure cause can still be observed, and (ii) non identifiable paths, when
the time-dependent covariate X(t) cannot be observed after occurrence of the competing failure cause. In
this latter case we used the value of X(t) at the time of failure throughout the risk set. Parameters (β1 , β2 )
were set at (0.5, 0.5) in (i), and at (−0.5, 0.5) in (ii).
From the 1,000 simulations, we computed the mean estimate of β1 (E(β̂1 )) and of the proportion γ of
patients who experienced the competing cause of failure before any jump of X(t), for values of p ranging
from 0.1 to 1 and values of K = r2 /r1 ranging from 0.1 to 2.
We begin by presenting simulation results from model with identifiable paths. Figure 3 displays the
mean estimate of β̂1 against K (Figure 3a) and p (Figure 3c). Whatever the value of K and of p, E(β̂1 ) was
close to its nominal value. This exemplifies the ability of the model to estimate the regression coefficient
when the entire covariate path is known.
[Fig. 3 about here.]
Figure 4 displays simulations results when the occurrence of the competing cause of failure avoids the
observation of the jump process (non identifiable paths). Contrarily to the previous observable case, β̂1 was
systematically biased, with bias increasing with K (Figure 4a). Interestingly, the shape of the estimated
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7
β1 against K was very similar to that of γ (Figure 4b). Next, for K = 1, we computed E(β̂1 ) for values
of parameter p from 0.1 to 1 (Figure 4c). It appears that the estimates β̂1 are biased, except in the case of
p = 1, i.e., when all individuals fail from the cause of interest. In this case, γ is obviously null, as shown
on Figure 4d. When p is close to zero, F1 (t) ≈ 0, and the model is “ill-posed”, so that computing β̂1 does
not make any sense. Similar shapes were observed for values of r1 = 0.005, 0.01, 0.02, with an increase
in the bias of E(β̂1 ) as r1 increases (or equivalently an increase in γ as shown on Figure 4b). Of note, a
linear decrease of γ with p was observed (Figure 4d), whereas such pattern was not found between E(β̂1 )
and p.
[Fig. 4 about here.]
Moreover in our simulation setting, one can show that: γ = (1 − p){q + (1 − q) × C}, where C is the
probability of jump after failure, conditional on failure from competing cause, and is therefore independent
of p and q. As a result, γ is indeed a decreasing linear function of p as shown in Figures 3d and 4d.
5 Discussion
In this paper, we showed, on the basis of a working example and a simulation study, that the Fine and
Gray model is not appropriate for estimating the effect of any time-dependent covariate unless the entire
covariate path is observable. Otherwise, i.e., in the case of so-called “internal” time-dependent covariate
using the terminology of Kalbfleisch and Prentice (1980), the use of the Fine and Gray model can lead to
a serious bias in estimate, even in the simple studied case of a one time jump process, which is actually
often observed in clinical epidemiology data.
Since the Fine and Gray model can only be used if the entire path of the time-dependent covariate
is known, obviously, this prohibits the introduction of any time-dependent covariate in the model when
death is a competing cause of failure. For instance, in our working example, no valid estimation of the
effect of aGvHD on the subdistribution hazard of relapse could be obtained, due to deaths in remission.
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8
A. Latouche: Time-dependent covariate in the competing risk setting
Nevertheless, besides death, non fatal competing events could also be considered similarly, unless checking
carefully that the observation period does not end with the occurrence of the competing event.
Our main concern was to prevent the misuse of the Fine and Gray model with time-dependent explanatory variables. Our simulation studies also provide a better understanding of the structure of the “unnatural”
risk set of the Fine and Gray model, pointing out that competing failures stay in the risk set until censoring
time.
To cope with estimation of the effect of time-dependent covariates, other statistical models should be
proposed. Multistate models with cause-specific transition rate have already been used (Andersen et al.,
2002; Hougaard, 1999). Further work is needeed to estimate time-dependent transition (non-homogeneous
markov process) in this setting.
Appendix
Briefly, the subdistribution of failure times from the cause of interest (² = 1) is given by F1 (t, X(t)) =
1 − [1 − p{1 − exp(−r1 t)}]exp(β1 X(t)) , which is a unit exponential mixture with mass 1 − p at +∞, where
p is the proportion of failures from the cause of interest, and uses the proportional subdistribution hazards
model to obtain the subdistribution for nonzero covariate values. Let ψ(X(t)) be the link function relating
the covariate process to the subdistribution hazard function, and Ψ(.) the cumulative link function i.e.
Ψ(t) =
Rt
0
ψ(X(u))du. Let Λ1 (.) be the cumulative subdistribution hazard function, Λ1 (t) =
Rt
0
λ1 (u)du.
As a result, Λ1 (t) = − log S10 (t) for t ≤ τ and Λ1 (t) = − log S10 (τ ) + exp(β1 ) × {log S10 (τ ) −
log S10 (t)} otherwise, where S10 (t) = 1 − F1 (t, X(t) = 0). Failure times from the cause of interest were
thus generated through, t ← Ψ−1 [Λ−1
1 {− log(1 − u)}], where u is taken from a uniform distribution on
[0, 1] and ψ(X(t)) = exp{β1 X(t)}.
Since the subdistribution for the competing failure cause was considered exponentially distributed with
rate r2 , we directly used the non-modified algorithm of Leemis et al. (1990) to generate corresponding
competing failure times, with the link function ψ(X(t)) = exp{β2 X(t)}.
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References
P. K. Andersen, S.Z. Abildstrom, and S. Rosthoj. Competing risks as a multi-state model. Statistical
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A. Latouche: Time-dependent covariate in the competing risk setting
=1
X(τ )
X(τ )
1+
1+
0
0
0
+
+
+
Z
T
T∗ τ
0
+
+
T
T∗ τ
=2
X(τ )
X(τ )
1+
1+
0
0
0
+
+
+
Z
T
T∗ τ
0
+
+
T
T∗ τ
Fig. 1 Illustration of the covariate path, X(τ ) overtime τ according to the experienced events. T denotes the failure
time and Z denotes the time to occurrence of the event of interest. Upper plots concern patients who failed from the
cause of interest (² = 1) while lower plots concern patients who failed from the competing failure cause (² = 2). Left
plots concerns patients who experienced the event of interest, and right plots concern patients who did not. T ∗ denotes
the maximal failure time among the sample.
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13
1.0
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0.6
0.4
0.2
0.0
Cumulative incidence function
0.8
Relapse
Death in remission
0
10
20
30
40
50
60
Months
Fig. 2 Estimated cumulative incidences of relapse and death in remission
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14
A. Latouche: Time-dependent covariate in the competing risk setting
(b)
0.7
0.35
(a)
r1=0.005
+ + +
0.30
+
+ + + + + β1
+
+
+
+
+
0.25
+
+
+
γ
0.5
r1=0.02
+
+
+
+
+
0.3
0.20
0.4
^
E(β1)
0.6
r1=0.01
0.50
1.00
2.00
0.10
0.25
0.50
r2 r1
r2 r1
(c)
(d)
1.00
2.00
0.8
0.25
0.7
0.10
r1=0.005
+
+
0.6
0.6
r1=0.01
r1=0.02
+
+ + + + β1
+ + +
+
0.4
+
+
γ
0.5
^
E(β1)
+
+
+
0.2
0.4
+
+
0.0
0.3
+
0.10
0.40
0.70
p
1.00
+
0.10
0.40
0.70
1.00
p
Fig. 3 Simulation results in the case of identifiable path: Mean value of β̂1 (a) and the proportion γ of patients who
experience the competing failure cause before any jump (b) against the ratio r2 /r1 of the rates of failures from cause
2 and 1. Mean value of β̂1 (c) and γ (d) against the proportion p of failure from cause 1.
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15
+
r1=0.005
r1=0.02
0.00
+
+
+
+ +
+
0.30
γ
+
+
+
+
0.25
−0.25
+
β1
+ +
+
0.20
+
0.50
1.00
2.00
0.10
0.25
0.50
r2 r1
r2 r1
(c)
(d)
r1=0.005
1.00
2.00
+
r1=0.02
0.6
+
+
+
+
−0.25
γ
+
+
0.4
+ + +
+ +
+
0.00
0.25
0.8
0.50
0.25
r1=0.01
+
+ β1
0.2
+
−0.50
+
+
+
0.10
0.0
−0.75
+
+
+
0.10
^
E(β1)
+
0.35
r1=0.01
0.25
+
+
−0.75
−0.50
^
E(β1)
(b)
0.40
0.50
(a)
0.40
0.70
p
1.00
+
0.10
0.40
0.70
1.00
p
Fig. 4 Simulation results in the case of non-identifiable path: Mean value of β̂1 (a) and the proportion γ of patients
who experience the competing failure cause before any jump (b) against the ratio r2 /r1 of the rates of failures from
cause 2 and 1. Mean value of β̂1 (c) and γ (d) against the proportion p of failure from cause 1.
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How to improve our understanding of competing risks analysis?
Aurélien Latouche, Matthieu Resche-Rigon, Sylvie Chevret, Raphaël Porcher
Affiliation: Département de Biostatistique et Informatique Médicale, ERM 0321-INSERM,
Hôpital Saint Louis, Université Paris 7, AP-HP, Paris
Acknowledgments of research support: We wish to acknowledge Pr Sylvie Castaigne and Pr
Hervé Dombret for the availability of the ALFA 9000 trial data
Corresponding author : Aurélien Latouche, Département de Biostatistique et Informatique
Médicale, Hôpital Saint Louis. 1, avenue Claude Vellefaux, 75475 Paris cedex 10, France.
Phone number : 33 1 42 49 97 42
Fax number : 33 1 42 49 97 45
Email address: [email protected]
Running head: Understanding competing risks
Key Words : competing risks; cumulative incidence; survival models
1
Abstract (234 words)
-
PURPOSE: In the analysis of competing risks outcomes, there is still an open debate
with regards to the use of a regression model, either the cause-specific Cox model or
the model proposed by Fine and Gray for the hazard associated with the cumulative
incidence function. Our purpose was to further detail the implications of either choice
when estimating the benefit of a new treatment on delaying the outcome of interest in
this competing risks setting.
-
PATIENTS AND METHODS: Differences are exemplified first through the risk set
associated with each model, using an easily understandable graphical representation.
Then, differences are illustrated using a real data sample from the randomized clinical
trial ALFA 9000 conducted in acute myeloid leukemia to assess the benefit on
relapse-free interval of timed-sequential induction over the standard treatment.
-
RESULTS: Estimated adjusted treatment benefit could be modified by the handling of
competing risks outcomes, with disappearance of statistical significance when using a
cause-specific model (p= 0.13) as compared to the use of Fine and Gray model (p=
0.05).
-
CONCLUSION: Differences in quantities of interest could help in selecting the
regression model to be used. If effect on instantaneous hazard is that of interest, the
cause-specific model is to be used, while if we are only interested in modeling the
effect of covariate on the probability of a particular outcome, the Fine and Gray model
is that of choice.
2
Introduction
In evaluating the clinical benefit of treatments in oncology, the gold standard is to
demonstrate a benefit in terms of overall survival (OS). Nevertheless, a recent analysis of
oncology drugs approved in the USA through the regular process in the last 13 years showed
that two thirds of them were based on other endpoints than survival1. This is also true with
regards to evaluation of new treatment strategies in malignant hematological diseases. In this
setting, indeed, expected benefits in survival are likely to be small, requiring the conduct of
large randomized clinical trials2. Therefore, when such large trials appear unrealistic,
alternative end points are more and more used, also justified by the complex course of these
patients. Actually, since these patients can fail from many causes such as treatment-related
complications, relapse, or death, we are also interested in estimating the effect of the new
treatment on either cause of failure. Accordingly, besides overall survival, one of the most
used end points in evaluating new therapies in malignant hematological diseases is time to
relapse, also called the relapse-free interval (RFI). Nevertheless, statistical analysis of RFI
should be handled differently from that of OS.
The paper is organized as follows. First, we present a rapid overview of competing risks
analysis, exemplified on the analysis of RFI by contrast to OS. Then, in order to understand
the fundamental differences between available regression models, we present a graphical
representation of the risk set associated with each model. Third, we illustrated, using a real
sample data, the differences reached by using both approaches in the estimation of the benefit
on RFI of timed-sequential induction in acute myeloid leukemia. Finally, we provide some
help for practitioners in choosing either modeling approach.
3
An overview of methods for competing risks
There are several reasons why the statistical analyses of OS and of RFI are actually different.
First, while the observation process of death can be disturbed only by the interruption of
follow-up (generating “censored observations”), the occurrence of relapse itself, and not only
the observation process, can be also suppressed by the previous occurrence of death in
remission. In other words, in the analysis of OS, there is only one risk (death) acting on the
population, whereas in the analysis of RFI, there is a so-called “competing risks” setting,
defined by the simultaneous action of two risks, namely death in remission and relapse.
Secondly, when reporting analysis of the potential benefit of treatment on the outcome, a
graphical display of the probabilities of failure causes against time is useful. Therefore, to
display cumulative probability of the outcome over time from the sample of enrolled patients,
nonparametric methods are commonly used in both settings. Nevertheless, while the
worldwide known Kaplan-Meier method is used to estimate the OS curve, the literature has
emphasized the inappropriateness of this method when estimating the cumulative probability
of relapse3-6. Actually, the cumulative incidence function (CIF) of relapse appears more
attractive, in the sense that it takes into account the competing risk of death in remission. In
other words, it is interpretable as the “crude” probability of relapse when at least two
competing risks act on the population (without further assumptions about the dependence
among the failure times), while the Kaplan-Meier approach achieves an estimate of the “net”
probability of relapse, i.e., in the hypothetical setting where the risk of relapse is the only one
acting on the population, assuming independent competing risks7. Indeed, without such an
assumption, the Kaplan-Meier estimate has no probability interpretation8.
Finally, when evaluating the treatment benefit adjusting for potential confounders, regression
models have to be used. Despite numerous available models, the Cox semi-parametric
proportional hazards model is the quasi-unique way to perform prognostic analyses for OS,
4
whatever the causes, in routine practice9. It expresses the probability of death in an
infinitesimal interval of time, conditionally on being alive at the onset of the time interval, as
a multiplicative function of covariates. By contrast, in the competing risks setting provided by
the analysis of RFI, two major tools are available. The cause-specific hazard function for
relapse is the instantaneous risk of occurrence of relapse, conditional on being alive and free
of relapse. It can be directly modeled, as previously, by using a Cox model, where the deaths
in remission are censored at the time of their death10.
Otherwise, some authors have proposed to use regression models for the hazard associated
with the CIF6, such as the semi-parametric model developed by Fine and Gray11. The effect of
the covariate in such a model can be directly translated in terms of CIF. It has been yet used in
the prognostic evaluation on neutrophils recovery after bone marrow transplantation12, breast
cancer death13,14 or metastasis15, and even RFI16,17. The general interest in these quantities as
found in papers published in the Journal of Clinical Oncology until November 2004, is
summarized in Table 1, suggesting a clear preference for cause-specific or Cox model over
the Fine and Gray model, though the parallel use of cumulative incidence functions is
frequently reported. Of note, most of the review papers on practical use of competing risks
analysis have emphasized the use of cumulative incidence functions, while overlooking
regression models3,6,18-20. Nevertheless, before claiming that Cox model is misused in this
context and that regression models in competing risks must be based on regression model for
the risk associated with CIF, we aim at further detailing the implications of either choice
when estimating the benefit of a new treatment on delaying relapse.
Illustrative example : Representation of the Risk Sets
Let us consider an illustrative example, where N=16 patients are randomized between two
treatment arms (A and B). Suppose that the main outcome measure is a failure time, while
5
patients are subject to another competing risks outcome that prevents the observation of the
former. For simplicity, assume that there is no censoring, that is, all patients were followed up
until either outcome happens.
Table 2 displays the observed ranks of time to outcomes after randomization (either primary
or competing risks), that are ordered increasingly irrespective of the treatment arm.
Regression analysis is based on iterative computations over time, involving the risk set at
each ordered outcome of interest. Such a risk set is defined at any particular time t, by the set
of individuals who are still exposed to the outcome of interest at t. Nevertheless, there are
some discrepancies between the risk sets computed in the Cox model and the CIF based
model: In the Cox cause-specific hazard model, an individual is at risk at time t only if he has
not experienced any outcome yet. In the CIF-based model, an individual is at risk at time t if
he has not experienced the outcome of interest before t, which also includes individuals
having already experienced the competing risks outcome before t. Despite this unnatural
definition, this risk set accommodates with valid statistical properties11.
The Figure 1 represents the risk sets associated with either regression approach. It is a abacuslike plot of the Table 2. Actually, it exhibits that, when using the Cox cause-specific model,
the risk set decreases more rapidly over time due to the disappearance of outcomes of interest
and competing risks outcomes, than when using the CIF based approach, that is only affected
by the occurrence of the outcomes of interest. This could result in differences in estimates of
treatment effect, due to the fact that patients who experienced the competing risks outcome
will belong to all risk sets of the Fine and Gray regression analysis while they will be
excluded from the risk set at the time they fail in the Cox model. In case of distinct effect of
the treatment under study on the two competing risks, for instance, one could expect a benefit
of the treatment through the use of Cox model while no benefit will be shown through the use
of the CIF based approach, and conversely.
6
Of note, when analyzing the outcome of interest, results are similar whether the competing
causes are distinguished or mixed altogether.
A real data set example: the ALFA 9000 trial
We reanalyzed the previously cited randomized clinical trial16, conducted by the Acute
Leukemia French Association (ALFA) cooperative group, where 592 patients with either de
novo or secondary acute myeloid leukemia (AML) were randomized between one of the three
following induction arms: reinforced standard “3+7” induction (arm A, 197 patients including
110 aged less than 50 years); double induction (arm B, 198 patients including 114 aged less
than 50 years); or timed-sequential induction (arm C, 197 patients including 121 aged less
than 50 years). The main end point was RFI, calculated from the date of the first CR to the
date of the first relapse. In the primary analysis, authors accounted for competing risks deaths
and allogeneic bone-marrow transplantations in first complete remission (CR) using
cumulative incidence curves, then compared by the Gray test while the Fine and Gray model
was used to estimate CIF associated-hazards ratio (or sub-distribution hazard ratio, SHR).
Overall differences in RFI were not significant among the 3 randomization arms (P=0.39 and
0.15 when arm B and C were compared to arm A, respectively, using the Gray test) ; by
contrast, in patients aged 50 years or less, RFI was significantly improved from the arm C as
compared to the control arm A (P=0.038, using the Gray test)16.
We decided to perform separate analyses using either the Cox cause-specific hazards model,
or the Fine and Gray model for the CIF associated hazards. For simplicity, we focused on the
comparison in the AML patients aged less than 50 years who were randomly allocated in the
reinforced standard “3+7” induction (A) and the timed-sequential induction (C) arms.
Of the 231 patients aged 50 years or less allocated in arm A or arm C, 41 (17 in arm A and 24
in arm C) did not achieve CR, due to either resistant disease or induction death. Further
7
analyses will deal with the remaining 190 patients. After CR, 90 relapsed (52 in arm A and 38
in arm C), 35 received allogeneic bone marrow transplantation in first CR (16 in arm A and
19 in arm C) and 16 died in first CR (8 in either arm A or C). Figure 2 displays the schematic
representation of the risks sets according to the model. At the beginning, there are 190
patients in both risk sets.
Table 3 displays the estimation of the estimated hazard ratio of treatment arm and potential
confounders (namely, sex, age, karyotypic abnormalities and de novo AML) using both
regression models. We note that benefit of the timed-sequential induction arm (arm C) is
similarly estimated in both approaches, while discrepancies in estimates were found for the
effect of age and unfavorable karyotype. When multivariable models were fit, results differed
according to the approach: The Cox cause-specific model only retained unfavorable karyotype
as associated with RFI while the CIF based Fine and Gray model retained both treatment arm
and age as associated with RFI (Table 3). This could rely on the differences in competing
risks outcomes according to patient subsets. In fact, while treatment arm has no significant
effect on the CIF of BMT (p= 0.67) nor on the CIF of death in CR (p= 0.95), there was a
significant decreased incidence of death in CR in the subset of patients age 26 or less (p=
0.03) (Figure 3) as well as non significant trends towards increase in CIF or both BMT (p=
0.18) and death in CR (p= 0.30) in the subset of patients with unfavorable karyotype. In other
words, a decrease (resp., increase) in the cumulative incidence function of relapse in one
subset of patients could be due either to a physiological effect of the treatment on that risk, or
to an increase (resp., decrease) in the probability of the competing risks outcomes.
Discussion
We have attempted to clarify the issues raised by the use of regression models in the setting of
competing risks. Actually, while there is a general consensus over the use of cumulative
8
incidence function estimates when displaying outcomes over patients subsets, the choice of
regression modeling in order to estimate the benefit of treatment or prognostic covariates in
such a setting is still an open issue.
In the analysis of competing risks data, the first question should be to address the competing
risks setting itself. Such a setting involves the simultaneous exposure to more than one risk. In
fact, when one exposure is clearly delayed over the other, for instance if no relapse is
observed within the first 100 days while allografted patients are exposed to acute graft versus
host disease, such a competition is questionable. Nevertheless, it is clear from Figure 1 that,
in this case, risks sets will be similar so that both regression analyses will achieve similar
results.
Secondly, the competing risks outcomes should be clearly discussed with regards to the
independence assumption with the process of interest. For instance, when assessing survival
of hepato-cellular carcinoma, hepatitis transplantation could appear as likely related with the
risk of death, so that ignoring the competition and treating them as censored observations is
misleading.
Third, in case of more than one simultaneous risks acting on the population, and contrary to
what is commonly stated, it appears that both approaches can be valuable, since they focus on
distinct quantities of interest. If we are interested in estimating the effect of the treatment on
the hazard of a specific outcome (such as relapse), conditionally on being alive at the time, the
cause-specific approach appears of prime interest. By contrast, if we focus on estimating the
effect of the treatment on the crude probability of experiencing that outcome while other risks
act on the population, the Fine and Gray approach should be used. Differences in risk sets
reached by the use of both models are exemplified on a small data set.
Nevertheless, some could favor the cause-specific approach due to the following points. First,
the cause-specific hazard can be expressed using the Cox model, which is widely used in the
9
medical literature. Moreover, it appears more clinically understandable to assume a
multiplicative effect of the treatment (or any covariate) on the cause-specific hazard, i.e. on
the instantaneous risk of relapse, conditional on being alive and free of relapse. In other
words, it seems natural that the physiological effect of a treatment or any prognostic exposure
would be to reduce or increase the probability of the outcome of interest at any time,
conditionally on being still alive at that time.
On the contrary, a decrease (resp., increase) in the cumulative incidence function could be due
either to a physiological effect of the exposure or to an increase (resp., decrease) in the
probability of the competing failure causes. As illustrated in the AML90 trial analyses, the
overall effect of unfavorable karyotype on CIF associated risk could be “masked” by the
simultaneous increase in competing risks outcomes: Patients with unfavorable karyotype were
more likely to die in CR or being transplanted, so that the resulting effect is to erase the
increase in CIF of relapse. By contrast, young patients were poorly exposed to death, so that
resulting effect on CIF of relapse could be artificially increased. Therefore, to better interpret
and analyze the effect of treatment (or any other covariate) on the cumulative incidence of
relapse using the Fine and Gray model, it is mandatory to display concomitantly the estimated
effects of that covariate on either competing risks outcomes, as pointed out by Pepe3.
Finally, the choice of either approach could rely on the main question of interest. Sometimes,
our primary concern is to model the instantaneous risk of developing the outcome of interest.
For instance, in the previous example, estimating the influence of treatment on the
instantaneous risk of developing relapse could be considered as the most important question.
In this case, one should use the Cox cause-specific model, that allows to estimate the hazard
ratio of covariates as a measure of association between this risk and the covariate. By
contrast, our primary concern can be to model the prevalence of the outcome of interest, as
epidemiologists do. This is notably the case when the occurrence of the outcome of interest is
10
restricted over a short time period (for instance, acute graft versus host disease following
allogeneic bone marrow transplantation) or on a particular exposure (for instance, death in
intensive care unit). In these cases, the model of interest is clearly the CIF-associated model,
such as that proposed by Fine and Gray.
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double induction and timed-sequential induction to a "3 + 7" induction in adults with AML:
long-term analysis of the Acute Leukemia French Association (ALFA) 9000 study. Blood
104:2467-74, 2004
17.
de Botton S, Coiteux V, Chevret S, et al: Outcome of childhood acute
promyelocytic leukemia with all-trans-retinoic acid and chemotherapy. J Clin Oncol 22:140412, 2004
18.
Alberti C, Metivier F, Landais P, et al: Improving estimates of event incidence
over time in populations exposed to other events: application to three large databases. J Clin
Epidemiol 56:536-45, 2003
12
19.
Gooley TA, Leisenring W, Crowley J, et al: Estimation of failure probabilities
in the presence of competing risks: new representations of old estimators. Stat Med 18:695706, 1999
20.
Machin D: On the evolution of statistical methods as applied to clinical trials. J
Intern Med 255:521-8, 2004
13
Figure legends
Figure 1: Fictive data set- Schematic representation of the risk sets of both regression models
Figure 2: ALFA 9000 Trial- Schematic representation of the risk sets of both regression
models
Figure 3: ALFA 9000 Trial- Estimated cumulative incidence functions of relapse and
competing risks outcomes according to arm (C vs. A), and age (<26 vs. > 26)
14
Fine and Gray model
11
9
7
1
3
5
In the risk set
11
9
7
5
1
3
In the risk set
4
6
8
10
12
14
16
16
14
12
10
8
6
Out the risk set
4
2
Time
2
Time
Out the risk set
Failure of interest
No failure
Other failure
13
15
Failure of interest
No failure
Other failure
13
15
Cox model
15
201
Fine and Gray model
16
81
121
Failure of interest
No failure
Other failure
censored
41
81
121
161
201
161
121
81
41
Out the risk set
Time
41
1 1
41
81
121
In the risk set
161
Failure of interest
No failure
Other failure
censored
201
Out the risk set
1 1
In the risk set
161
201
Cox model
Time
1.0
0.8
0.6
0.4
CIF of competing outcomes
BMT in Arm A
BMT in Arm C
Death in Arm A
Death in Arm C
0.0
0.2
1.0
0.6
0.4
0.0
0.2
CIF of relapse
0.8
Arm A
Arm C
0
20
40
60
80
100 120 140
0
20
40
100 120 140
1.0
0.8
0.4
0.6
BMT in age<26
BMT in age>26
Death in age<26
Death in age>26
0.0
0.2
CIF of competing outcomes
0.2
0.4
0.6
0.8
Age<26
Age>26
0.0
CIF of relapse
80
Months
1.0
Months
60
0
20
40
60
80
100 120 140
0
Months
20
40
60
80
Months
17
100 120 140
Table 1: Journal of Clinical Oncology online search in text, abstract, or title, from January
1983 up to November, 2004
Query
Number of references found
“Competing risk(s)”
111
“Competing risk(s)” AND “cause specific”
30
“Competing risk(s)” AND “Cox”
76
“Competing
risk(s)”
AND
“cumulative
72
incidence”
“Competing risk(s)” AND “Fine and Gray”
8
“Competing
53
risk(s)”
AND
“cumulative
incidence” AND “Cox”
18
Table 2: Illustrative example : Fictive data set
Rank of failure
Type of outcome
Treatment arm
1: Outcome of interest
2: Competing risks outcome
1
1
B
2
1
B
3
2
B
4
2
B
5
2
A
6
1
A
7
2
B
8
1
B
9
2
A
10
2
A
11
1
A
12
1
B
13
2
B
14
1
A
15
2
A
16
1
A
19
Table 3: ALFA 9000 Trial Data. Estimation of the benefit of time-sequential induction on
RFI according to the regression model
Measure of treatment benefit Cox cause-specific hazard CIF based hazard Fine and
(95%CI)*
model
Gray model**
Treatment arm C
0.80 (0.64-0.98); p=0.032
0.80 (0.65-0.99); p=0.038
Age ≥ 26 (Q1)
0.72 (0.46-1.15) ; p= 0.17
0.66 (0.42-1.04); p=0.08
De novo AML
0.85 (0.27-2.70); p=0.79
1.74 (0.52-5.83); p=0.37
Female gender
1.07 (0.71-1.62); p=0.75
1.00 (0.66-1.51); p=0.98
Unfavorable karyotype
2.43 (1.26-4.69); p= 0.008 1.43 (0.72-2.85); p= 0.30
Univariable models
Multivariable model
Treatment arm C
0.83 (0.64-1.06); p=0.13
0.79 (0.60-1.00); p=0.05
Age ≥ 26 (Q1)
0.63 (0.36-1.10); p=0.10
0.56 (0.33-0.95); p=0.032
Unfavorable karyotype
2.36 (1.21-4.57); p= 0.01
1.57 (0.81-3.04); p= 0.18
* either cause-specific hazard ratio (HR) or CIF-associated hazard ratio (also called the
subdistribution hazard ratio, SHR)
** with death or BMT in CR as competing risks outcomes
20
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