Supplementary Material for “Estimating the correlation between
semi-competing risk survival endpoints”
By Lexy Sorrell, Yinghui Wei, Ma lgorzata Wojtyś and Peter Rowe
In the supplementary material we consider the Archimedean copulas and derive
their derivatives for use in the likelihood function of equation (8) in the main text.
We provide results from data analysis corresponding to Section 4 in the main text
in Tables S1 to S4. Table S5 details the true values for the parameters used in
the simulation studies involving the Exponential, Weibull and Gompertz survival
distributions with the Normal, Clayton, Frank and Gumbel copulas. Simulation
results for Section 5 of the main text investigating misspecification of the copula
function in each of the three scenarios are provided in Tables S6 to S20. Finally,
the contour plots from Section 4 of the main text are given in Figures S1 and S2.
1 Archimedean copulas
For a 2-dimensional random vector (U, V ), uniformly distributed on the unit cube
[0, 1]2 with observed values u and v, we consider the bivariate copula
C(u, v) = P (U ≤ u, V ≤ v).
Archimedean copulas can be written in the form,
Cθ(u, v) = ψ
−1(ψ(u) + ψ(v)), (1)
where θ is the association parameter and ψ is the generator function of the
Archimedean copula that depends on θ, that is ψ(u) = ψ(u; θ).
Using known formula,
df−1(y) 1
= , (2)
dy f ′(x)|x=f−1(y)
we can use the generator functions of copulas to derive the first and second deriva-
tives to use in the likelihood functions. First, we differentiate the copula function,
equation (1), by u, using equation (2),
dCθ(u, v) d
′
= (ψ−1
1 ψ (u)
(ψ(u)+ψ(v))) = ′ ×ψ
′(u) =
du du ψ (x)| ′
.
x=ψ−1(ψ(u)+ψ(v)) ψ (Cθ(u, v))
(3)
1
By symmetry, differentiating equation (1) by v,
dCθ(u, v) ψ
′(v)
=
dv ψ′
. (4)
(Cθ(u, v))
Finally, the second derivative of equation (1) is found by taking the derivative of
equation (3) with respect to v,
d2Cθ(u, v) d ψ
′(u) ′ d 1=
dudv dv ψ′
= ψ (u)
(Cθ(u, v)) dv ψ′(Cθ(u, v))
′ −1 d= ψ (u) ψ′(Cθ(u, v))
(ψ′(Cθ(u, v)))2 dv
−ψ′(u)
= ψ′′
d
′ (Cθ(u, v)) C2 θ(u, v) (5)(ψ (Cθ(u, v))) dv
−ψ′(u) ′
= ψ′′
ψ (v)
′ (Cθ(u, v))(ψ (Cθ(u, v)))2 ψ′(Cθ(u, v))
−ψ′(u)ψ′(v)ψ′′(Cθ(u, v))
= ′ .(ψ (C 3θ(u, v)))
Clayton Copula
The generator function for the Clayton copula is given by,
t−θ − 1
ψ(t) = . (6)
θ
The first derivative of equation (6) is given by,
−θ−1
ψ′
−θt
(t) = = −t−(θ+1). (7)
θ
The second derivative of equation (6), by differentiating equation (7), is given by,
ψ′′(t) = (θ + 1)t−(θ+2). (8)
Substituting equations (6) and (7) into equations (3) and (4) we find the first
derivatives of the Clayton copula function with respect to u and v to be,
∂C (u, v) −u−(θ+1)θ Cθ(u, v)θ+1
= = , (9)
∂u −Cθ(u, v)−(θ+1) uθ+1
∂C (u, v) −v−(θ+1)θ Cθ(u, v)θ+1
= = . (10)
∂v −C (u, v)−(θ+1)θ vθ+1
2
Substituting equations (6), (7) and (8) into equation (5), we find the second deriva-
tive of the Clayton copula function to be,
∂2C (u, v) u−(θ+1)v−(θ+1)(θ + 1)C (u, v)−(θ+2)θ θ (θ + 1)Cθ(u, v)
2θ+1
= − = . (11)∂u∂v C (u, v) 3(θ+1) uθ+1vθ+1θ
Frank copula
The generator function for the Frank cop(ula is give)n by,−θt
ψ(t) = − e − 1log − . (12)e θ − 1
Therefore, the first derivative of equation (12) is given by,
−θe−θt e−θ −θt′ − − 1 θe θψ (t) =
e−θ − 1 e−
= = . (13)
θt − 1 e−θt − 1 1− eθt
Using the quotient rule, the second derivative of equation (12), by differentiating
equation (13), is given by,
′′ 0(1− eθt)− (−θ2eθt) θ2eθtψ (t) = = . (14)
(1− eθt)2 (1− eθt)2
Substituting equations (12) and (13) into equations (3) and (4) we find the first
derivatives of the Frank copula function with respect to u and v to be,
∂C (u, v) θ 1− eθCθ(u,v)θ 1− eθCθ(u,v)
= = , (15)
∂u 1− eθu θ 1− eθu
∂Cθ(u, v) θ 1− eθCθ(u,v) 1− eθCθ(u,v)
= = . (16)
∂v 1− eθv θ 1− eθv
Substituting equations (12), (13) and (14) into equation (5), we find the second
derivative of the Frank copula function to be,
∂2C (u, v) −θ2 θ2eθCθ(u,v) (1− eθCθ(u,v))3θ
=
∂u∂v (1− eθu)(1− eθv) (1− eθCθ(u,v))2 θ3
−θ4eθCθ(u,v)(1− eθCθ)
= (17)
θ3(1− eθu)(1− eθv)
θeθCθ(u,v)(eθCθ(u,v) − 1)
= .
(eθu − 1)(eθv − 1)
3
Gumbel copula
The generator function for the Gumbel copula is given by the following,
ψ(t) = (− log(t))θ. (18)
The first derivative of equation (18) is(given)by,
−1 −θ(− log(t))θ−1
ψ′(t) = θ(− log(t))θ−1 = . (19)
t t
The second derivative of equation (18) is given by,
′′ θ(θ − 1)(− log(t))θ−2 − (−θ(− log(t))θ−1)ψ (t) =
t2
θ(− log(t))θ−2((θ − 1) + (− log(t)))
= (20)
t2
θ(− log(t))θ−2(θ − 1− log(t))
= .
t2
Substituting equations (18) and (19) into equations (3) and (4) we find the first
derivatives of the Gumbel copula with respect to u and v to be,
∂C (u, v) θ(− log(u))θ−1θ Cθ(u, v) Cθ(u, v)(− log(u))θ−1
= = , (21)
∂u θu(− log(C (u, v)))θ−1 u(− log(C (u, v)))θ−1θ θ
∂Cθ(u, v) θ(− log(v))θ−1Cθ(u, v) Cθ(u, v)(− log(v))θ−1
= = . (22)
∂v θv(− log(Cθ(u, v)))θ−1 v(− log(C (u, v)))θ−1θ
Substituting equations (18), (19) and (20) into equation (5) we find the second
derivative of the Gumbel copula function to be,
∂2Cθ(u, v) −θ2(− log(u))θ−1(− log(v))θ−1 θ(− log(Cθ(u, v)))θ−2(θ − 1− log(Cθ(u, v)))
=
∂u∂v uv (Cθ(u, v))2
(Cθ(u, v))
3
−θ3(− log(Cθ(u, v)))3θ−3
Cθ(− log(u))θ−1(− log(v))θ−1(θ − 1− log(Cθ(u, v)))
= .
uv(− log(Cθ(u, v)))2θ−1
(23)
4
5
2 Tables and Figures
Table S1: Results from renal transplant data analysis, using the Weibull survival distribution and the Normal,
Clayton, Frank and Gumbel copula functions to describe the association between graft failure and death. The shape
and scale parameters, α̂1 and β̂1 for the endpoint graft failure and α̂2 and β̂2 for the endpoint death, respectively, are
given alongside estimated Spearman’s rank correlation coefficient, ρ̂. The AIC is provided for each copula model.
Method α̂1 (95% CI) β̂1 (95% CI) α̂2 (95% CI) β̂2 (95% CI) ρ̂ (95% CI) AIC
Normal 0.610 0.063 0.864 0.039 0.450 4299.5
(0.539, 0.681) (0.050, 0.075) (0.776, 0.953) (0.030, 0.049) (0.348, 0.544)
Clayton 0.633 0.062 0.864 0.040 0.678 4309.4
(0.557, 0.709) (0.049, 0.074) (0.776, 0.953) (0.031, 0.050) (0.528, 0.766)
Frank 0.630 0.063 0.863 0.041 0.540 4300.5
(0.555, 0.704) (0.050, 0.076) (0.775,0.952) (0.031, 0.051) (0.411, 0.640)
Gumbel 0.600 0.062 0.841 0.041 0.301 4322.6
(0.532, 0.668) (0.050, 0.074) (0.756, 0.926) (0.032, 0.051) (0.199, 0.382)
6
Table S2: Results from the renal transplant data analysis, using the Gompertz survival distribution and the Normal,
Clayton, Frank and Gumbel copula functions to describe the association between graft failure and death. The shape
and rate parameters, γ̂1 and λ̂1 for the endpoint graft failure and γ̂2 and λ̂2 for the endpoint death, respectively, are
given alongside estimated Spearman’s rank correlation coefficient, ρ̂. The AIC is provided for each copula model.
Method γ̂1 (95% CI) λ̂1 (95% CI) γ̂2 (95% CI) λ̂2 (95% CI) ρ̂ (95% CI) AIC
Normal -0.083 0.040 -0.007 0.029 0.388 4364.5
(-0.118, -0.047) (0.032, 0.048) (-0.033, 0.019) (0.024, 0.035) (0.304, 0.473)
Clayton -0.063 0.039 -0.010 0.032 0.703 4370.2
(-0.099, -0.027) (0.031, 0.047) (-0.036, 0.015) (0.026, 0.038) (0.573, 0.781)
Frank -0.069 0.040 -0.010 0.032 0.559 4360.8
(-0.104, -0.034) (0.032, 0.048) (-0.036, 0.015) (0.026, 0.038) (0.441, 0.651)
Gumbel -0.091 0.039 -0.004 0.028 0.217 4366.2
(-0.126, -0.055) (0.032, 0.047) (-0.029, 0.022) (0.023, 0.034) (0.137, 0.287)
7
Table S3: Results from the Amsterdam Cohort Study data analysis, using the Weibull survival distribution and
the Normal, Clayton, Frank and Gumbel copula functions to describe the association between virus phenotype
switching from non-syncytium-inducing to syncytium-inducing (SI switch) and death from AIDS. The shape and
scale parameters, α̂1 and β̂1 for the endpoint SI switch and α̂2 and β̂2 for the endpoint death from AIDS, respectively,
are given alongside estimated Spearman’s rank correlation coefficient, ρ̂. The AIC is provided for each copula model.
Method α̂1 (95% CI) β̂1 (95% CI) α̂2 (95% CI) β̂2 (95% CI) ρ̂ (95% CI) AIC
Normal 1.208 0.043 1.957 0.009 0.511 2051.6
(1.032, 1.383) (0.026, 0.059) (1.751, 2.163) (0.005, 0.013) (0.399, 0.624)
Clayton 1.125 0.054 1.952 0.009 0.639 2051.6
(0.954, 1.296) (0.033, 0.074) (1.741, 2.164) (0.005, 0.013) (0.494, 0.730)
Frank 1.061 0.064 1.903 0.010 0.674 2048.2
(0.901, 1.221) (0.040, 0.087) (1.694, 2.112) (0.005, 0.015) (0.564, 0.752)
Gumbel 0.914 0.075 1.933 0.009 0.414 2070.4
(0.773, 1.055) (0.051, 0.100) (1.721, 2.146) (0.005, 0.014) (0.273, 0.518)
8
Table S4: Results from the Amsterdam Cohort Study data analysis, using the Gompertz survival distribution and
the Normal, Clayton, Frank and Gumbel copula functions to describe the association between SI switch and death
from AIDS. The shape and rate parameters, γ̂1 and λ̂1 for the endpoint SI switch and γ̂2 and λ̂2 for the endpoint
death from AIDS, respectively, are given alongside estimated Spearman’s rank correlation coefficient, ρ̂. The AIC
is provided for each copula model.
Method γ̂1 (95% CI) λ̂1 (95% CI) γ̂2 (95% CI) λ̂2 (95% CI) ρ̂ (95% CI) AIC
Normal 0.054 0.052 0.228 0.020 0.567 2053.4
(-0.004, 0.112) (0.036, 0.069) (0.182, 0.274) (0.013, 0.027) (0.466, 0.670)
Clayton 0.072 0.050 0.213 0.022 0.613 2060.5
(0.013, 0.130) (0.034, 0.066) (0.167, 0.259) (0.015, 0.030) (0.458, 0.677)
Frank 0.054 0.054 0.218 0.021 0.583 2053.1
(-0.004, 0.111) (0.037, 0.071) (0.171, 0.264) (0.014, 0.029) (0.458, 0.677)
Gumbel 0.030 0.056 0.231 0.019 0.438 2070.7
(-0.027, 0.087) (0.040, 0.073) (0.185, 0.278) (0.013, 0.026) (0.314, 0.534)
9
Table S5: True values used for simulating data from the Exponential, Weibull and Gompertz survival distributions
used to mimic the Renal transplant and Amsterdam Cohort Study data sets for the simulations investigating the
effects of misspecification. Presented are values of the hazard rates of the non-terminal and terminal events, λ1 and
λ2 for the Exponential distribution. For the Weibull distribution the shape and scale parameters are given by, α1 and
β1 for the non-terminal events, and α2 and β2 for the terminal events, respectively. For the Gompertz distribution,
the shape and rate parameters are given by γ1 and λ1 for the non-terminal events, and γ2 and λ2 for the terminal
events, respectively.
Exponential Weibull Gompertz
Data set Underlying copula
λ1 λ2 ρ α1 β1 α2 β2 ρ γ1 λ1 γ2 λ2 ρ
Normal 0.03 0.03 0.39 0.61 0.06 0.86 0.04 0.45 0.001 0.04 0.001 0.03 0.39
Clayton 0.03 0.03 0.72 0.63 0.06 0.86 0.04 0.68 0.001 0.04 0.001 0.03 0.70
Renal transplant
Frank 0.03 0.03 0.58 0.63 0.06 0.86 0.04 0.54 0.001 0.04 0.001 0.03 0.56
Gumbel 0.03 0.03 0.21 0.60 0.06 0.84 0.04 0.30 0.001 0.04 0.001 0.03 0.22
Normal 0.07 0.07 0.63 1.21 0.04 1.96 0.01 0.51 0.05 0.05 0.23 0.02 0.64
Amsterdam Cohort Clayton 0.07 0.07 0.75 1.13 0.05 1.95 0.01 0.64 0.07 0.05 0.21 0.02 0.61
Study Frank 0.08 0.07 0.70 1.06 0.06 1.90 0.01 0.67 0.05 0.05 0.22 0.02 0.58
Gumbel 0.07 0.07 0.50 0.91 0.08 1.93 0.01 0.41 0.03 0.06 0.23 0.02 0.44
10
2.1 Misspecification of the copula function simulation results with the renal trans-
plant data
Table S6: Simulation results investigating the effect of misspecification of the copula function with
assumed Spearman’s rank correlation coefficient, ρ = 0. The simulated data mimics the hazard rates
for the endpoints of graft failure and death from the renal transplant data set. The data is generated
from the Exponential survival distributions and the Independence copula function. The Exponential
survival distribution along with the Normal, Clayton, Frank and Gumbel copula functions are used to
compare the performance metrics of the hazard rates and correlation estimates under misspecification
of the copula function. The number of simulated data sets is 1000 with 1199 individuals in each. MAE
refers to the mean absolute error and the coverage probability is given as a percentage.
Assumed copula distribution
Underlying copula Normal Clayton Frank Gumbel
MAE Coverage MAE Coverage MAE Coverage MAE Coverage Boundary1
Independence λ1 0.001 95.9 0.001 95.5 0.001 95.9 0.001 95.9
λ2 0.001 94.7 0.001 94.8 0.001 95.2 0.001 95.4
ρ 0.025 97.2 0.029 98.9 0.020 97.4 0.020 99.0 989
1 Boundary refers to the number of times the lower confidence interval for θ is less than or equal to 1, the boundary
for the Gumbel copula.
11
Table S7: Simulation results investigating the effect of misspecification of the copula function with assumed Spear-
man’s rank correlation coefficient, ρ = 0.25. The simulated data mimics the hazard rates for the endpoints of graft
failure and death from the renal transplant data set. The data is generated from the Exponential survival distribu-
tions and the Normal, Clayton, Frank and Gumbel copula functions. The Exponential survival distribution along
with the Normal, Clayton, Frank and Gumbel copula functions are used to compare the performance metrics of the
hazard rates and correlation estimates under misspecification of the copula function. The number of simulated data
sets is 1000 with 1199 individuals in each. MAE refers to the mean absolute error and the coverage probability is
given as a percentage.
Assumed copula distribution
Underlying copula Normal Clayton Frank Gumbel
MAE Coverage MAE Coverage MAE Coverage MAE Coverage Boundary
Normal λ1 0.001 95.5 0.001 93.7 0.001 94.5 0.001 92.3
λ2 0.001 95.2 0.001 95.0 0.001 94.3 0.001 94.7
ρ 0.039 95.0 0.050 94.1 0.040 94.0 0.074 60.0 34
Clayton λ1 0.001 92.6 0.001 93.4 0.001 94.0 0.001 88.5
λ2 0.001 95.3 0.001 95.8 0.001 95.2 0.001 95.2 1
ρ 0.066 77.1 0.047 94.6 0.060 80.6 0.140 12.4 392
Frank λ1 0.001 93.5 0.001 93.9 0.001 95.3 0.001 89.8
λ2 0.001 93.4 0.001 94.9 0.001 95.5 0.001 94.7
ρ 0.041 93.6 0.058 89.2 0.039 95.1 0.088 47.5 68
Gumbel λ1 0.001 93.8 0.001 92.6 0.001 93.6 0.001 95.1
λ2 0.001 94.2 0.001 95.5 0.001 94.2 0.001 94.2
ρ 0.063 76.8 0.056 91.8 0.046 91.4 0.035 93.8
12
Table S8: Simulation results investigating the effect of misspecification of the copula function with assumed Spear-
man’s rank correlation coefficient, ρ = 0.5. The simulated data mimics the hazard rates for the endpoints of graft
failure and death from the renal transplant data set. The data is generated from the Exponential survival distribu-
tions and the Normal, Clayton, Frank and Gumbel copula functions. The Exponential survival distribution along
with the Normal, Clayton, Frank and Gumbel copula functions are used to compare the performance metrics of the
hazard rates and correlation estimates under misspecification of the copula function. The number of simulated data
sets is 1000 with 1199 individuals in each. MAE refers to the mean absolute error and the coverage probability is
given as a percentage.
Assumed copula distribution
Underlying copula Normal Clayton Frank Gumbel
MAE Coverage MAE Coverage MAE Coverage MAE Coverage Boundary
Normal λ1 0.001 94.7 0.001 94.0 0.001 93.9 0.001 90.9
λ2 0.001 95.0 0.001 94.3 0.001 94.4 0.001 92.5
ρ 0.032 95.1 0.054 84.3 0.034 95.0 0.105 14.8
Clayton λ1 0.002 62.7 0.001 95.4 0.001 84.1 0.002 43.9
λ2 0.001 91.2 0.001 96.0 0.001 84.3 0.001 79.2
ρ 0.141 0.0 0.017 95.0 0.070 9.9 0.257 0.0
Frank λ1 0.001 94.8 0.001 94.7 0.001 94.2 0.001 91.2
λ2 0.001 94.9 0.001 95.0 0.001 95.0 0.001 95.2
ρ 0.040 94.5 0.055 93.0 0.040 94.6 0.070 62.8 356
Gumbel λ1 0.001 93.6 0.001 93.0 0.001 91.1 0.001 95.5
λ2 0.001 94.2 0.001 95.1 0.001 91.1 0.001 94.8
ρ 0.084 36.3 0.076 68.9 0.055 74.9 0.029 95.5
13
Table S9: Simulation results investigating the effect of misspecification of the copula function with assumed Spear-
man’s rank correlation coefficient, ρ = 0.75. The simulated data mimics the hazard rates for the endpoints of graft
failure and death from the renal transplant data set. The data is generated from the Exponential survival distribu-
tions and the Normal, Clayton, Frank and Gumbel copula functions. The Exponential survival distribution along
with the Normal, Clayton, Frank and Gumbel copula functions are used to compare the performance metrics of the
hazard rates and correlation estimates under misspecification of the copula function. The number of simulated data
sets is 1000 with 1199 individuals in each. MAE refers to the mean absolute error and the coverage probability is
given as a percentage.
Assumed copula distribution
Underlying copula Normal Clayton Frank Gumbel
MAE Coverage MAE Coverage MAE Coverage MAE Coverage
Normal λ1 0.001 93.6 0.001 92.7 0.001 88.0 0.001 91.7
λ2 0.001 94.6 0.001 95.4 0.001 91.3 0.001 91.3
ρ 0.018 95.0 0.046 59.4 0.019 96.2 0.102 0.9
Clayton λ1 0.001 71.2 0.001 96.4 0.001 87.9 0.002 57.6
λ2 0.001 90.6 0.001 96.0 0.001 89.1 0.001 84.1
ρ 0.143 0.1 0.021 93.9 0.083 13.7 0.262 0.0
Frank λ1 0.001 79.4 0.001 92.2 0.001 93.8 0.002 65.4
λ2 0.001 93.0 0.001 95.0 0.001 95.2 0.001 85.7
ρ 0.054 45.4 0.056 41.0 0.021 95.1 0.160 0.0
Gumbel λ1 0.001 94.3 0.001 89.5 0.002 77.6 0.001 93.0
λ2 0.001 94.0 0.001 93.7 0.002 76.5 0.001 93.6
ρ 0.065 11.0 0.073 20.6 0.046 53.1 0.018 95.0
14
Table S10: Simulation results investigating the effect of misspecification of the copula function with assumed Spear-
man’s rank correlation coefficient, ρ = 0.9. The simulated data mimics the hazard rates for the endpoints of graft
failure and death from the renal transplant data set. The data is generated from the Exponential survival distribu-
tions and the Normal, Clayton, Frank and Gumbel copula functions. The Exponential survival distribution along
with the Normal, Clayton, Frank and Gumbel copula functions are used to compare the performance metrics of the
hazard rates and correlation estimates under misspecification of the copula function. The number of simulated data
sets is 1000 with 1199 individuals in each. MAE refers to the mean absolute error and the coverage probability is
given as a percentage.
Assumed copula distribution
Underlying copula Normal Clayton Frank Gumbel
MAE Coverage MAE Coverage MAE Coverage MAE Coverage
Normal λ1 0.001 94.8 0.001 85.9 0.001 77.8 0.001 93.0
λ2 0.001 94.6 0.001 94.4 0.001 87.3 0.001 92.2
ρ 0.008 94.2 0.024 41.4 0.009 96.1 0.063 0.0
Clayton λ1 0.002 39.8 0.001 94.9 0.001 78.2 0.003 22.5
λ2 0.001 88.9 0.001 94.4 0.001 84.9 0.002 66.6
ρ 0.128 0.0 0.009 95.7 0.047 6.9 0.232 0.0
Frank λ1 0.022 60.5 0.001 91.8 0.001 94.6 0.002 40.6
λ2 0.001 92.7 0.001 94.3 0.001 94.0 0.001 77.5
ρ 0.060 1.2 0.029 25.2 0.009 94.6 0.148 0.0
Gumbel λ1 0.001 95.3 0.001 89.1 0.002 61.9 0.001 95.1
λ2 0.001 94.4 0.001 94.4 0.002 64.9 0.001 94.6
ρ 0.032 4.7 0.038 7.7 0.025 33.9 0.001 93.9
15
2.2 Misspecification of the copula function simulation results with the Amsterdam
Cohort Study data
Table S11: Simulation results investigating the effect of misspecification of the copula function with assumed Spear-
man’s rank correlation coefficient, ρ = 0. The simulated data mimics the hazard rates for the endpoints of SI switch
and death from AIDS from the Amsterdam Cohort Study data set. The data is generated from the Exponential
survival distributions and the Independence copula function. The Exponential survival distribution along with the
Normal, Clayton, Frank and Gumbel copula functions are used to compare the performance metrics of the hazard
rates and correlation estimates under misspecification of the copula function. The number of simulated data sets is
1000 with 329 individuals in each. MAE refers to the mean absolute error and the coverage probability is given as
a percentage.
Assumed copula distribution
Underlying copula Normal Clayton Frank Gumbel
MAE Coverage MAE Coverage MAE Coverage MAE Coverage Boundary
Independence λ1 0.004 97.3 0.005 96.5 0.005 97.3 0.005 96.1
λ2 0.004 95.8 0.004 95.4 0.004 96.1 0.004 95.1
ρ 0.039 97.6 0.042 98.8 0.036 97.6 0.031 99.7 997
16
Table S12: Simulation results investigating the effect of misspecification of the copula function with assumed Spear-
man’s rank correlation coefficient, ρ = 0.25. The simulated data mimics the hazard rates for the endpoints of SI
switch and death from AIDS from the Amsterdam Cohort Study data set. The data is generated from the Ex-
ponential survival distributions and the Normal, Clayton, Frank and Gumbel copula functions. The Exponential
survival distribution along with the Normal, Clayton, Frank and Gumbel copula functions are used to compare the
performance metrics of the hazard rates and correlation estimates under misspecification of the copula function.
The number of simulated data sets is 1000 with 329 individuals in each. MAE refers to the mean absolute error and
the coverage probability is given as a percentage.
Assumed copula distribution
Underlying copula Normal Clayton Frank Gumbel
MAE Coverage MAE Coverage MAE Coverage MAE Coverage Boundary
Normal λ1 0.005 94.4 0.005 93.9 0.005 94.2 0.005 94.2
λ2 0.004 94.3 0.004 94.3 0.004 94.6 0.004 95.4
ρ 0.072 93.8 0.078 91.2 0.070 96.3 0.081 86.0 508
Clayton λ1 0.005 93.4 0.005 94.8 0.004 95.3 0.005 92.7
λ2 0.004 93.5 0.004 95.4 0.004 94.6 0.004 94.4
ρ 0.076 92.4 0.069 94.4 0.072 95.1 0.120 67.6 759
Frank λ1 0.005 94.3 0.005 93.8 0.005 94.1 0.005 93.4
λ2 0.004 94.2 0.004 94.5 0.004 94.0 0.004 94.9
ρ 0.072 94.2 0.084 90.0 0.072 94.8 0.093 82.0 582
Gumbel λ1 0.005 94.2 0.005 90.9 0.005 94.9 0.005 95.0
λ2 0.004 94.4 0.004 94.1 0.004 94.8 0.004 94.4
ρ 0.083 88.1 0.093 85.1 0.076 93.8 0.066 94.9 265
17
Table S13: Simulation results investigating the effect of misspecification of the copula function with assumed Spear-
man’s rank correlation coefficient, ρ = 0.5. The simulated data mimics the hazard rates for the endpoints of SI
switch and death from AIDS from the Amsterdam Cohort Study data set. The data is generated from the Ex-
ponential survival distributions and the Normal, Clayton, Frank and Gumbel copula functions. The Exponential
survival distribution along with the Normal, Clayton, Frank and Gumbel copula functions are used to compare the
performance metrics of the hazard rates and correlation estimates under misspecification of the copula function.
The number of simulated data sets is 1000 with 329 individuals in each. MAE refers to the mean absolute error and
the coverage probability is given as a percentage.
Assumed copula distribution
Underlying copula Normal Clayton Frank Gumbel
MAE Coverage MAE Coverage MAE Coverage MAE Coverage Boundary
Normal λ1 0.005 94.8 0.005 93.3 0.005 94.6 0.004 94.1
λ2 0.004 95.1 0.004 95.9 0.004 94.5 0.004 94.3
ρ 0.056 93.5 0.071 91.3 0.060 95.8 0.086 74.7
Clayton λ1 0.008 66.6 0.004 95.9 0.005 80.8 0.001 61.3
λ2 0.004 92.9 0.004 95.1 0.005 86.1 0.005 90.4
ρ 0.122 24.4 0.029 95.1 0.049 78.1 0.221 0.3
Frank λ1 0.005 93.8 0.005 94.2 0.005 95.3 0.005 95.4
λ2 0.004 95.4 0.004 95.2 0.004 94.9 0.004 95.6
ρ 0.072 96.8 0.074 96.9 0.069 97.3 0.081 91.3 815
Gumbel λ1 0.005 94.2 0.006 88.6 0.005 94.3 0.004 95.0
λ2 0.004 94.5 0.004 94.9 0.004 95.2 0.004 94.3
ρ 0.074 81.4 0.091 83.5 0.066 92.9 0.057 94.2
18
Table S14: Simulation results investigating the effect of misspecification of the copula function with assumed Spear-
man’s rank correlation coefficient, ρ = 0.75. The simulated data mimics the hazard rates for the endpoints of SI
switch and death from AIDS from the Amsterdam Cohort Study data set. The data is generated from the Ex-
ponential survival distributions and the Normal, Clayton, Frank and Gumbel copula functions. The Exponential
survival distribution along with the Normal, Clayton, Frank and Gumbel copula functions are used to compare the
performance metrics of the hazard rates and correlation estimates under misspecification of the copula function.
The number of simulated data sets is 1000 with 329 individuals in each. MAE refers to the mean absolute error and
the coverage probability is given as a percentage.
Assumed copula distribution
Underlying copula Normal Clayton Frank Gumbel
MAE Coverage MAE Coverage MAE Coverage MAE Coverage
Normal λ1 0.004 94.9 0.005 91.7 0.005 93.0 0.004 93.5
λ2 0.004 95.0 0.004 94.8 0.004 95.2 0.004 93.5
ρ 0.033 94.2 0.043 92.9 0.037 95.5 0.076 58.1
Clayton λ1 0.007 74.8 0.004 94.1 0.005 85.6 0.007 67.7
λ2 0.004 93.9 0.004 94.8 0.005 89.7 0.005 87.8
ρ 0.115 43.8 0.036 95.1 0.063 75.7 0.217 2.1
Frank λ1 0.005 85.8 0.005 93.6 0.004 95.1 0.006 83.2
λ2 0.004 96.2 0.004 94.5 0.004 95.0 0.004 92.1
ρ 0.065 80.6 0.042 93.1 0.036 95.9 0.140 16.3
Gumbel λ1 0.004 94.3 0.006 88.2 0.005 91.9 0.004 94.3
λ2 0.004 95.2 0.004 95.9 0.005 91.4 0.004 95.1
ρ 0.045 77.1 0.062 83.1 0.043 91.9 0.035 95.9
19
Table S15: Simulation results investigating the effect of misspecification of the copula function with assumed Spear-
man’s rank correlation coefficient, ρ = 0.9. The simulated data mimics the hazard rates for the endpoints of SI
switch and death from AIDS from the Amsterdam Cohort Study data set. The data is generated from the Ex-
ponential survival distributions and the Normal, Clayton, Frank and Gumbel copula functions. The Exponential
survival distribution along with the Normal, Clayton, Frank and Gumbel copula functions are used to compare the
performance metrics of the hazard rates and correlation estimates under misspecification of the copula function.
The number of simulated data sets is 1000 with 329 individuals in each. MAE refers to the mean absolute error and
the coverage probability is given as a percentage.
Assumed copula distribution
Underlying copula Normal Clayton Frank Gumbel
MAE Coverage MAE Coverage MAE Coverage MAE Coverage
Normal λ1 0.004 95.0 0.005 89.9 0.005 89.0 0.004 95.6
λ2 0.004 94.6 0.004 94.6 0.004 94.9 0.004 93.0
ρ 0.015 94.8 0.022 92.4 0.018 94.3 0.046 48.8
Clayton λ1 0.011 40.6 0.004 95.3 0.006 72.1 0.011 36.3
λ2 0.004 94.0 0.004 95.5 0.005 85.1 0.005 86.2
ρ 0.127 3.8 0.017 95.8 0.039 66.3 0.217 0.1
Frank λ1 0.007 68.0 0.005 92.4 0.004 96.5 0.008 62.5
λ2 0.004 95.6 0.004 94.3 0.004 94.8 0.004 91.9
ρ 0.069 39.0 0.022 91.9 0.019 94.1 0.143 1.2
Gumbel λ1 0.004 93.3 0.006 90.1 0.005 88.4 0.004 95.1
λ2 0.004 94.9 0.004 94.1 0.005 88.8 0.004 94.8
ρ 0.021 78.7 0.042 76.2 0.024 90.0 0.019 94.4
20
2.3 Misspecification of the copula function with data mimicking the simulation
study in Fu et al. (2013)
Table S16: Simulation results investigating the effect of misspecification of the copula function with assumed Spear-
man’s rank correlation coefficient, ρ = 0. The simulated data mimics the hazard rates from the simulation study in
Fu et al. (2013). The data is generated from the Exponential survival distributions and the Independence copula
function. The Exponential survival distribution along with the Normal, Clayton, Frank and Gumbel copula functions
are used to compare the performance metrics of the hazard rates and correlation estimates under misspecification
of the copula function. The number of simulated data sets is 1000 with 100 individuals in each. MAE refers to the
mean absolute error and the coverage probability is given as a percentage.
Assumed copula distribution
Underlying copula Normal Clayton Frank Gumbel
MAE Coverage MAE Coverage MAE Coverage MAE Coverage Boundary
Independence λ1 0.015 95.6 0.014 96.4 0.014 95.2 0.014 95.7
λ2 0.006 95.2 0.006 93.8 0.006 94.8 0.006 94.8
ρ 0.057 96.3 0.047 98.9 0.054 96.4 0.045 99.1 990
21
Table S17: Simulation results investigating the effect of misspecification of the copula function with assumed Spear-
man’s rank correlation coefficient, ρ = 0.25. The simulated data mimics the hazard rates from the simulation study
in Fu et al. (2013). The data is generated from the Exponential survival distributions and the Normal, Clayton,
Frank and Gumbel copula functions. The Exponential survival distribution along with the Normal, Clayton, Frank
and Gumbel copula functions are used to compare the performance metrics of the hazard rates and correlation esti-
mates under misspecification of the copula function. The number of simulated data sets is 1000 with 100 individuals
in each. MAE refers to the mean absolute error and the coverage probability is given as a percentage.
Assumed copula distribution
Underlying copula Normal Clayton Frank Gumbel
MAE Coverage MAE Coverage MAE Coverage MAE Coverage Boundary
Normal λ1 0.013 94.7 0.014 94.7 0.014 95.2 0.014 92.6
λ2 0.006 94.8 0.006 95.2 0.006 94.7 0.006 94.4
ρ 0.099 94.5 0.103 91.1 0.094 97.3 0.102 91.4 748
Clayton λ1 0.014 95.4 0.013 95.5 0.014 92.4 0.014 92.6
λ2 0.006 95.0 0.006 95.0 0.006 94.1 0.006 93.9
ρ 0.095 95.2 0.097 93.1 0.100 96.4 0.115 90.1 816
Frank λ1 0.013 95.6 0.014 94.5 0.013 96.1 0.014 94.2
λ2 0.006 94.7 0.006 93.6 0.006 95.7 0.006 94.1
ρ 0.097 95.0 0.099 92.2 0.093 96.6 0.105 91.9 750
Gumbel λ1 0.014 95.6 0.014 95.4 0.014 95.7 0.014 94.7
λ2 0.006 94.2 0.006 95.1 0.006 95.2 0.006 94.8
ρ 0.105 90.2 0.110 89.2 0.097 96.7 0.086 94.6 551
22
Table S18: Simulation results investigating the effect of misspecification of the copula function with assumed Spear-
man’s rank correlation coefficient, ρ = 0.5. The simulated data mimics the hazard rates from the simulation study in
Fu et al. (2013). The data is generated from the Exponential survival distributions and the Normal, Clayton, Frank
and Gumbel copula functions. The Exponential survival distribution along with the Normal, Clayton, Frank and
Gumbel copula functions are used to compare the performance metrics of the hazard rates and correlation estimates
under misspecification of the copula function. The number of simulated data sets is 1000 with 100 individuals in
each. MAE refers to the mean absolute error and the coverage probability is given as a percentage.
Assumed copula distribution
Underlying copula Normal Clayton Frank Gumbel
MAE Coverage MAE Coverage MAE Coverage MAE Coverage Boundary
Normal λ1 0.013 94.0 0.013 94.3 0.013 93.0 0.013 91.7 8
λ2 0.006 94.9 0.006 93.2 0.006 93.4 0.006 93.3
ρ 0.073 92.4 0.103 80.9 0.076 94.4 0.091 82.7 51
Clayton λ1 0.013 94.4 0.012 96.5 0.015 80.9 0.013 94.1
λ2 0.006 94.2 0.006 95.4 0.007 79.0 0.006 95.0
ρ 0.049 83.9 0.032 96.0 0.035 92.8 0.125 22.4
Frank λ1 0.013 95.4 0.014 95.1 0.015 93.8 0.013 94.4
λ2 0.006 94.1 0.006 95.1 0.006 95.9 0.006 95.6
ρ 0.094 95.4 0.095 98.6 0.094 97.7 0.095 98.6 874
Gumbel λ1 0.013 95.2 0.014 95.2 0.013 94.3 0.012 96.5
λ2 0.006 94.7 0.006 94.4 0.006 94.8 0.006 95.6
ρ 0.083 83.6 0.100 84.0 0.074 94.2 0.068 93.7 12
23
Table S19: Simulation results investigating the effect of misspecification of the copula function with assumed Spear-
man’s rank correlation coefficient, ρ = 0.75. The simulated data mimics the hazard rates from the simulation study
in Fu et al. (2013). The data is generated from the Exponential survival distributions and the Normal, Clayton,
Frank and Gumbel copula functions. The Exponential survival distribution along with the Normal, Clayton, Frank
and Gumbel copula functions are used to compare the performance metrics of the hazard rates and correlation esti-
mates under misspecification of the copula function. The number of simulated data sets is 1000 with 100 individuals
in each. MAE refers to the mean absolute error and the coverage probability is given as a percentage.
Assumed copula distribution
Underlying copula Normal Clayton Frank Gumbel
MAE Coverage MAE Coverage MAE Coverage MAE Coverage
Normal λ1 0.012 95.2 0.012 94.4 0.012 93.8 0.012 95.3
λ2 0.006 95.3 0.006 94.7 0.006 92.5 0.005 95.0
ρ 0.037 91.5 0.088 62.2 0.042 94.5 0.074 66.7
Clayton λ1 0.012 95.9 0.012 94.9 0.015 84.6 0.012 94.3
λ2 0.006 94.6 0.006 94.9 0.007 82.4 0.006 94.6
ρ 0.054 89.1 0.041 94.4 0.048 92.6 0.134 34.0
Frank λ1 0.012 95.2 0.012 93.6 0.011 95.4 0.012 94.7
λ2 0.006 95.5 0.006 94.9 0.006 95.7 0.006 94.5
ρ 0.046 92.0 0.093 57.5 0.041 94.8 0.100 48.5
Gumbel λ1 0.012 95.5 0.012 95.7 0.014 92.5 0.012 96.4
λ2 0.006 94.9 0.006 94.6 0.006 92.9 0.006 95.3
ρ 0.048 73.6 0.089 59.4 0.042 93.4 0.035 95.1
24
Table S20: Simulation results investigating the effect of misspecification of the copula function with assumed Spear-
man’s rank correlation coefficient, ρ = 0.9. The simulated data mimics the hazard rates from the simulation study in
Fu et al. (2013). The data is generated from the Exponential survival distributions and the Normal, Clayton, Frank
and Gumbel copula functions. The Exponential survival distribution along with the Normal, Clayton, Frank and
Gumbel copula functions are used to compare the performance metrics of the hazard rates and correlation estimates
under misspecification of the copula function. The number of simulated data sets is 1000 with 100 individuals in
each. MAE refers to the mean absolute error and the coverage probability is given as a percentage.
Assumed copula distribution
Underlying copula Normal Clayton Frank Gumbel
MAE Coverage MAE Coverage MAE Coverage MAE Coverage
Normal λ1 0.012 96.0 0.012 95.0 0.012 92.2 0.012 95.4
λ2 0.006 95.0 0.006 94.7 0.006 92.9 0.006 93.8
ρ 0.014 93.6 0.056 43.6 0.017 94.1 0.040 49.5
Clayton λ1 0.014 93.5 0.012 94.7 0.018 72.0 0.013 92.8
λ2 0.006 94.7 0.005 95.3 0.009 69.1 0.006 92.0
ρ 0.040 72.2 0.019 95.8 0.020 91.2 0.105 8.6
Frank λ1 0.013 94.8 0.012 94.5 0.011 95.2 0.012 95.5
λ2 0.006 95.8 0.006 94.0 0.005 94.6 0.006 94.8
ρ 0.029 84.7 0.069 34.2 0.016 95.3 0.074 21.4
Gumbel λ1 0.012 94.8 0.012 95.4 0.015 89.5 0.011 95.7
λ2 0.054 94.9 0.006 95.1 0.007 89.5 0.005 95.4
ρ 0.020 69.4 0.061 42.3 0.018 91.1 0.014 94.7
Figure S1: Normal, Clayton, Frank and Gumbel contour plots with their estimated
association parameter from Table 3 of the main text, for the renal transplant data
set.
25
Figure S2: Normal, Clayton, Frank and Gumbel contour plots with their estimated
association parameter from Table 4 of the main text, for the Amsterdam Cohort
Study data set.
26
References
Fu, H., Wang, Y., Liu, J., Kulkarni, P., and Melemed, A. (2013). Joint modeling
of progression-free survival and overall survival by a Bayesian normal induced
copula estimation model. Statistics Medicine, 32:240–254.
27