The Power Variance Function Copula Model in
Bivariate Survival Analysis: An application to Twin Data
Jose (Pepe) Romeo
University of Santiago, Chile
jose.romeo@usach.cl
Joint work with:
Renate Meyer - University of Auckland, New Zealand
Diego Gallardo - University of Sao Paulo, Brazil
Workshop on Flexible Models for Longitudinal and Survival Data
with Applications in Biostatistics
Coventry – UK, July 2015
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Outline
Introduction, some examples of multivariate lifetimes
Modeling strategies
Copulas
The PVF copula model
I
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Definition, particular and limiting cases and dependence properties
Simulating
Estimation
Simulation study
Application
Final comments and future work
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Introduction
Survival analysis deals with time-to-event data, e.g. time to death or onset of
disease.
In multivariate survival analysis there may be a natural association because
individuals share biological and/or environmental conditions.
Examples:
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Introduction
Survival analysis deals with time-to-event data, e.g. time to death or onset of
disease.
In multivariate survival analysis there may be a natural association because
individuals share biological and/or environmental conditions.
Examples:
I
I
I
I
Groups of individuals under to similar environmental conditions
Time to relapse of a disease and time of death, for a subject
Lifetimes of pairs of human organs (e.g. kidneys, eyes)
Recurrent events: asthma attacks for a subject
Pepe Romeo (USAoCH)
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Introduction
Survival analysis deals with time-to-event data, e.g. time to death or onset of
disease.
In multivariate survival analysis there may be a natural association because
individuals share biological and/or environmental conditions.
Examples:
I
I
I
I
I
Groups of individuals under to similar environmental conditions
Time to relapse of a disease and time of death, for a subject
Lifetimes of pairs of human organs (e.g. kidneys, eyes)
Recurrent events: asthma attacks for a subject
Clustered failure times such as failure times of twins: to investigate whether
the strength of dependence within twin pairs as to the risk of the onset of
various diseases is different for MZ and DZ twins (time to appendectomy)
Pepe Romeo (USAoCH)
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Introduction
Survival analysis deals with time-to-event data, e.g. time to death or onset of
disease.
In multivariate survival analysis there may be a natural association because
individuals share biological and/or environmental conditions.
Examples:
I
I
I
I
I
Groups of individuals under to similar environmental conditions
Time to relapse of a disease and time of death, for a subject
Lifetimes of pairs of human organs (e.g. kidneys, eyes)
Recurrent events: asthma attacks for a subject
Clustered failure times such as failure times of twins: to investigate whether
the strength of dependence within twin pairs as to the risk of the onset of
various diseases is different for MZ and DZ twins (time to appendectomy)
ê The assumption of independence among lifetimes can be unrealistic.
ê It is of interest to estimate and quantify the dependence among the lifetimes
and the effects of covariates under the dependence structure.
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Modeling strategies
Shared Frailty Models (random effect models)
Clayton (1978), Hougaard (2000), Therneau & Grambsch (2000)
Let Tij denote the failure time i-th subject in the j-th cluster, the conditional
hazard function is given by
iid
h(t|wj , xi ) = h0 (t) exp(β 0 xi )wj , where e.g., Wj ∼ Gamma, Log Normal, . . .
Conditional on the frailty term w , the lifetimes are assumed independent,
S(t1 , . . . , tp |w ) = S(t1 |w ) · · · S(tp |w )
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Modeling strategies
Shared Frailty Models (random effect models)
Clayton (1978), Hougaard (2000), Therneau & Grambsch (2000)
Let Tij denote the failure time i-th subject in the j-th cluster, the conditional
hazard function is given by
iid
h(t|wj , xi ) = h0 (t) exp(β 0 xi )wj , where e.g., Wj ∼ Gamma, Log Normal, . . .
Conditional on the frailty term w , the lifetimes are assumed independent,
S(t1 , . . . , tp |w ) = S(t1 |w ) · · · S(tp |w )
Copulas Models (marginal models)
Shih & Louis (1995), Duchateau & Janssen (2008), Wienke (2010)
F (t1 , . . . , tp ) = Cα (F1 (t1 ), . . . , Fp (tp ))
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Copulas
Schweizer & Sklar (1983), Joe (1997), Owzar & Sen (2003), Nelsen (2006)
Let T1 , . . . , Tp r.v.’s with Tj ∼ Fj , j = 1, . . . , p, and joint cdf
F (t1 , . . . , tp ) = Pr{T1 ≤ t1 , . . . , Tp ≤ tp }
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Copulas
Schweizer & Sklar (1983), Joe (1997), Owzar & Sen (2003), Nelsen (2006)
Let T1 , . . . , Tp r.v.’s with Tj ∼ Fj , j = 1, . . . , p, and joint cdf
F (t1 , . . . , tp ) = Pr{T1 ≤ t1 , . . . , Tp ≤ tp }
Therefore, considering Uj = Fj (Tj ) ∼ Uniform(0, 1),
F (t1 , . . . , tp ) = Pr{U1 ≤ F1 (t1 ), . . . , Up ≤ Fp (tp )}
= Cα (F1 (t1 ), . . . , Fp (tp )),
(1)
is called the Copula of the vector (T1 , . . . , Tp ), it is a multivariate cdf defined on
[0, 1]p with uniformly distributed marginals and α ∈ A is a dependence parameter.
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Copulas
Schweizer & Sklar (1983), Joe (1997), Owzar & Sen (2003), Nelsen (2006)
Let T1 , . . . , Tp r.v.’s with Tj ∼ Fj , j = 1, . . . , p, and joint cdf
F (t1 , . . . , tp ) = Pr{T1 ≤ t1 , . . . , Tp ≤ tp }
Therefore, considering Uj = Fj (Tj ) ∼ Uniform(0, 1),
F (t1 , . . . , tp ) = Pr{U1 ≤ F1 (t1 ), . . . , Up ≤ Fp (tp )}
= Cα (F1 (t1 ), . . . , Fp (tp )),
(1)
is called the Copula of the vector (T1 , . . . , Tp ), it is a multivariate cdf defined on
[0, 1]p with uniformly distributed marginals and α ∈ A is a dependence parameter.
From (1) the joint pdf is given by
f (t1 , . . . , tp ) = cα (F1 (t1 ), . . . , Fp (tp ))
p
Y
fj (tj )
j=1
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Archimedean Copulas
Genest & MacKay (1986), Joe (1997), Nelsen (2006).
A copula Cα is called Archimedean if there exists a convex function
ϕ−1
α : [0, ∞] 7→ [0, 1], so that the copula Cα can be written as
Cα (u1 , u2 ) = ϕ−1
α (ϕα (u1 ) + ϕα (u2 )),
for all (u1 , u2 ) ∈ [0, 1]2 and α ∈ A.
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Archimedean Copulas
Genest & MacKay (1986), Joe (1997), Nelsen (2006).
A copula Cα is called Archimedean if there exists a convex function
ϕ−1
α : [0, ∞] 7→ [0, 1], so that the copula Cα can be written as
Cα (u1 , u2 ) = ϕ−1
α (ϕα (u1 ) + ϕα (u2 )),
for all (u1 , u2 ) ∈ [0, 1]2 and α ∈ A.
ê when ϕ−1 corresponds to the Laplace transform of W in a multiplicative
frailty model
h(t|w ) = h0 (t)w ,
copula models induced by frailty models are Archimedean copulas (Oakes,
1989), e.g:
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Archimedean Copulas
Genest & MacKay (1986), Joe (1997), Nelsen (2006).
A copula Cα is called Archimedean if there exists a convex function
ϕ−1
α : [0, ∞] 7→ [0, 1], so that the copula Cα can be written as
Cα (u1 , u2 ) = ϕ−1
α (ϕα (u1 ) + ϕα (u2 )),
for all (u1 , u2 ) ∈ [0, 1]2 and α ∈ A.
ê when ϕ−1 corresponds to the Laplace transform of W in a multiplicative
frailty model
h(t|w ) = h0 (t)w ,
copula models induced by frailty models are Archimedean copulas (Oakes,
1989), e.g:
I
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W ∼ Gamma then (U1 , U2 ) ∼ Clayton copula
W ∼ Positive Stable then (U1 , U2 ) ∼ Gumbel copula
W ∼ Inverse Gaussian then (U1 , U2 ) ∼ Inverse Gaussian copula
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Copulas
Kendall’s tau
Let (T1 , T2 ) a continuous r.v. with copula Cα , then the Kendall’s tau
coefficient for (T1 , T2 ) is given by
ZZ
τα = 4
Cα (u1 , u2 )dCα (u1 , u2 ) − 1
[0,1]2
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Copulas
Kendall’s tau
Let (T1 , T2 ) a continuous r.v. with copula Cα , then the Kendall’s tau
coefficient for (T1 , T2 ) is given by
ZZ
τα = 4
Cα (u1 , u2 )dCα (u1 , u2 ) − 1
[0,1]2
For Archimedean copulas, Kendall’s tau is written as
Z
τα = 4
0
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ϕα (t)
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The PVF (Power variance function) distribution
W ∼ PVF(α, δ, θ) for α ∈ (0, 1), δ > 0, and θ ≥ 0 if for w > 0 the pdf is given by
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The PVF (Power variance function) distribution
W ∼ PVF(α, δ, θ) for α ∈ (0, 1), δ > 0, and θ ≥ 0 if for w > 0 the pdf is given by
f (w |α, δ, θ) = −
∞
k
1
δθα X Γ(kα + 1) −w −α δ
exp −θw +
sin(αkπ)
πw
α
Γ(k + 1)
α
k=1
If θ > 0 all moments exist, E[W ] = δθα−1 and Var[W ] = (1 − α)δθα−2 .
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The PVF (Power variance function) distribution
W ∼ PVF(α, δ, θ) for α ∈ (0, 1), δ > 0, and θ ≥ 0 if for w > 0 the pdf is given by
f (w |α, δ, θ) = −
∞
k
1
δθα X Γ(kα + 1) −w −α δ
exp −θw +
sin(αkπ)
πw
α
Γ(k + 1)
α
k=1
If θ > 0 all moments exist, E[W ] = δθα−1 and Var[W ] = (1 − α)δθα−2 .
Under the reparametrization δ = η 1−α and θ = η, W ∼ PVF(α, η), with
α ∈ (0, 1) and η ≥ 0 and the Laplace transform is given by
1
LW (s) = exp − η 1−α (η + s)α − η
α
Note that E[W ] = 1, Var[W ] = (1 − α)η −1
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The PVF copula
The Laplace transform of the PVF distribution
1
LW (s) = exp − η 1−α (η + s)α − η
α
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The PVF copula
The Laplace transform of the PVF distribution
1
LW (s) = exp − η 1−α (η + s)α − η
= ϕ−1
α,η (s)
α
Following Oakes (1989), the Archimedean copula induced by PVF frailty
distribution is
Cα,η (u1 , u2 )
= ϕ−1
α,η (ϕα,η (u1 ) + ϕα,η (u2 ))
α
i
1
1
1 h 1−α α
α
η
g (u1 ) + g (u2 ) − η − η
= exp −
α
where g (u) = η α − αη α−1 log u.
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The PVF copula
The Laplace transform of the PVF distribution
1
LW (s) = exp − η 1−α (η + s)α − η
= ϕ−1
α,η (s)
α
Following Oakes (1989), the Archimedean copula induced by PVF frailty
distribution is
Cα,η (u1 , u2 )
= ϕ−1
α,η (ϕα,η (u1 ) + ϕα,η (u2 ))
α
i
1
1
1 h 1−α α
α
η
g (u1 ) + g (u2 ) − η − η
= exp −
α
where g (u) = η α − αη α−1 log u.
Note that
if α → 0, (U1 , U2 ) ∼ Clayton copula(η)
if η = 0, (U1 , U2 ) ∼ Gumbel copula(α)
if α = 0.5, (U1 , U2 ) ∼ Inverse Gaussian copula(η)
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The PVF copula - Dependence properties
Kendall’s tau coefficient:
Z
τ =1+4
0
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ϕ(t)
dt
ϕ0 (t)
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The PVF copula - Dependence properties
Kendall’s tau coefficient:
Z
τ =1+4
0
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ϕ(t)
dt ∈ (0, 1)
ϕ0 (t)
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The PVF copula - Dependence properties
Kendall’s tau coefficient:
Z
τ =1+4
0
1
ϕ(t)
dt ∈ (0, 1)
ϕ0 (t)
if α → 1, ∀ η ∈ R + then τ → 0
if η → ∞+ , ∀ α ∈ (0, 1) then τ → 0
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The PVF copula - Dependence properties
Kendall’s tau coefficient:
Z
τ =1+4
0
1
ϕ(t)
dt ∈ (0, 1)
ϕ0 (t)
if α → 1, ∀ η ∈ R + then τ → 0
if η → ∞+ , ∀ α ∈ (0, 1) then τ → 0
if α → 0 and η → 0 then τ → 1
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The PVF copula - Dependence properties
Kendall’s tau coefficient:
Z
τ =1+4
0
1
ϕ(t)
dt ∈ (0, 1)
ϕ0 (t)
if α → 1, ∀ η ∈ R + then τ → 0
if η → ∞+ , ∀ α ∈ (0, 1) then τ → 0
if α → 0 and η → 0 then τ → 1
Copula
PVF
Clayton
Gumbel
Inverse Gaussian
LTDC
0
2−1/η
0
0
UTDC
0
0
2 − 21/α
0
χ(v )
1 − (1 − α)(α log v − η)−1
η −1 + 1
1 − (1 − α)(α log v )−1
1 − (log v − 2η)−1
Table: Lower- and Upper-Tail Dependence and Cross-ratio function χ(v )
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The PVF copula - Simulated data
Scatterplots of 500 samples from PVF copula τ = {0.9, 0.7, 0.5, 0.2}
1.0
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u1
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The PVF copula - Simulated data
Scatterplots of 500 samples from PVF copula τ = {0.9, 0.7, 0.5, 0.2}
1.0
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u1
Pepe Romeo (USAoCH)
0.2
0.4
0.6
0.8
1.0
u1
PVF copula survival analysis
Coventry, July 2015
11 / 22
The PVF copula - Simulated data
Scatterplots of 500 samples from PVF copula τ = {0.9, 0.7, 0.5, 0.2}
1.0
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u1
Pepe Romeo (USAoCH)
0.8
●
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1.0
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u1
●
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●
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0.2
0.4
0.6
0.8
1.0
u1
PVF copula survival analysis
Coventry, July 2015
11 / 22
Simulation of (u1 , u2 ) ∼ PVF(α, η)
Algorithm: The conditional sampling for copulas, see e.g. Nelsen (2006):
Pepe Romeo (USAoCH)
PVF copula survival analysis
Coventry, July 2015
12 / 22
Simulation of (u1 , u2 ) ∼ PVF(α, η)
Algorithm: The conditional sampling for copulas, see e.g. Nelsen (2006):
(i) simulate u2 ∼ Uniform(0, 1) and q ∼ Uniform(0, 1)
∂
(ii) compute FU1 |U2 (u1 |u2 ) := ∂u
C
(u
,
u
)
(q|u2 )
and set u1 = FU−1
1
2
1 |U2
2
u2
(iii) return (u1 , u2 )
Pepe Romeo (USAoCH)
PVF copula survival analysis
Coventry, July 2015
12 / 22
Simulation of (u1 , u2 ) ∼ PVF(α, η)
Algorithm: The conditional sampling for copulas, see e.g. Nelsen (2006):
(i) simulate u2 ∼ Uniform(0, 1) and q ∼ Uniform(0, 1)
∂
(ii) compute FU1 |U2 (u1 |u2 ) := ∂u
C
(u
,
u
)
(q|u2 )
and set u1 = FU−1
1
2
1 |U2
2
u2
(iii) return (u1 , u2 )
FU−1
(q|u2 ) = u1 cannot be solve explicitly in a close form, we solve it numerically
1 |U2
∂
by finding the root of FU1 |U2 (u1 |u2 ) − q = 0, i.e. ∂u
C (u1 , u2 ) − q = 0,
2
Pepe Romeo (USAoCH)
PVF copula survival analysis
Coventry, July 2015
12 / 22
Simulation of (u1 , u2 ) ∼ PVF(α, η)
Algorithm: The conditional sampling for copulas, see e.g. Nelsen (2006):
(i) simulate u2 ∼ Uniform(0, 1) and q ∼ Uniform(0, 1)
∂
(ii) compute FU1 |U2 (u1 |u2 ) := ∂u
C
(u
,
u
)
(q|u2 )
and set u1 = FU−1
1
2
1 |U2
2
u2
(iii) return (u1 , u2 )
FU−1
(q|u2 ) = u1 cannot be solve explicitly in a close form, we solve it numerically
1 |U2
∂
by finding the root of FU1 |U2 (u1 |u2 ) − q = 0, i.e. ∂u
C (u1 , u2 ) − q = 0,
2
(ii) compute FU1 |U2 (u1 |u2 ) =
∂
∂u2 C (u1 , u2 )
u2
and set u1 = FU−1
(q|u2 )
1 |U2
(0)
(ii.a) set an initial value u1 = u0 ∈ (0, 1)
(j+1)
(ii.b) u1
(j+1)
(ii.c) if |u1
(j)
(j)
= u1 − ( ∂u∂ 2 C (u1 , u2 ) − q)
(j)
(j+1)
− u1 | < then return u1 = u1
Pepe Romeo (USAoCH)
PVF copula survival analysis
Coventry, July 2015
12 / 22
Joint Bayesian estimation procedure
(Romeo et al., 2006) Let (T1 , T2 ) ∼ (S1θ1 , S2θ2 ), (f1θ1 , f2θ2 ). For i = 1, . . . , n
suppose that (Ti1 , Ti2 ) and the censoring times (Ci1 , Ci2 ) are independent.
The observed quantities are Zij = min{Tij , Cij } and δij = I [Zij = Tij ], j = 1, 2.
The likelihood function for (α, η, θ1 , θ2 ) is given by
L(α, η, θ1 , θ2 | z1 , δ 1 , z2 , δ 2 ) =
n
Y
(cα,η (S1θ1 (zi1 ), S2θ2 (zi2 ))f1θ1 (zi1 )f2θ2 (zi2 ))
δi1 δi2
i=1
δi1 (1−δi2 )
∂Cα,η (S1θ1 (zi1 ), S2θ2 (zi2 ))
· (−f1θ1 (zi1 ))
∂S1θ1 (zi1 )
(1−δi1 )δi2
∂Cα,η (S1θ1 (zi1 ), S2θ2 (zi2 ))
·
· (−f2θ2 (zi2 ))
∂S2θ2 (zi2 )
·
· Cα,η (S1θ1 (zi1 ), S2θ2 (zi2 ))(1−δi1 )(1−δi2 )
The posterior distribution can be written as
π(α, η, θ1 , θ2 | z1 , δ 1 , z2 , δ 2 ) ∝ L(α, η, θ1 , θ2 | z1 , δ 1 , z2 , δ 2 )π(α)π(η)π(θ1 )π(θ2 )
Pepe Romeo (USAoCH)
PVF copula survival analysis
Coventry, July 2015
13 / 22
Joint Bayesian estimation procedure
(Romeo et al., 2006) Let (T1 , T2 ) ∼ (S1θ1 , S2θ2 ), (f1θ1 , f2θ2 ). For i = 1, . . . , n
suppose that (Ti1 , Ti2 ) and the censoring times (Ci1 , Ci2 ) are independent.
The observed quantities are Zij = min{Tij , Cij } and δij = I [Zij = Tij ], j = 1, 2.
The likelihood function for (α, η, θ1 , θ2 ) is given by
L(α, η, θ1 , θ2 | z1 , δ 1 , z2 , δ 2 ) =
n
Y
(cα,η (S1θ1 (zi1 ), S2θ2 (zi2 ))f1θ1 (zi1 )f2θ2 (zi2 ))
δi1 δi2
i=1
δi1 (1−δi2 )
∂Cα,η (S1θ1 (zi1 ), S2θ2 (zi2 ))
· (−f1θ1 (zi1 ))
∂S1θ1 (zi1 )
(1−δi1 )δi2
∂Cα,η (S1θ1 (zi1 ), S2θ2 (zi2 ))
·
· (−f2θ2 (zi2 ))
∂S2θ2 (zi2 )
·
· Cα,η (S1θ1 (zi1 ), S2θ2 (zi2 ))(1−δi1 )(1−δi2 )
The posterior distribution can be written as
π(α, η, θ1 , θ2 | z1 , δ 1 , z2 , δ 2 ) ∝ L(α, η, θ1 , θ2 | z1 , δ 1 , z2 , δ 2 )π(α)π(η)π(θ1 )π(θ2 )
Posterior computations implemented in SAS - Proc MCMC
Pepe Romeo (USAoCH)
PVF copula survival analysis
Coventry, July 2015
13 / 22
Simulation study I - Small sample properties
We simulate (u1 , u2 ) ∼ PVF(α, η) under the following scheme:
Three sample size: n = {50, 100, 200}
Three level of association: τ = {0.33, 0.50, 0.70}
Three censoring percentages: pc = {5%, 20%, 50%}
500 simulations (replicates)
Pepe Romeo (USAoCH)
PVF copula survival analysis
Coventry, July 2015
14 / 22
Simulation study I - Small sample properties
We simulate (u1 , u2 ) ∼ PVF(α, η) under the following scheme:
Three sample size: n = {50, 100, 200}
Three level of association: τ = {0.33, 0.50, 0.70}
Three censoring percentages: pc = {5%, 20%, 50%}
500 simulations (replicates)
α
η
τ
α
η
τ
α
η
τ
true
0.06
0.90
0.33
0.36
0.10
0.50
0.28
0.01
0.70
estimate
0.404
0.399
0.297
0.403
0.080
0.463
0.282
0.010
0.686
n = 50
bias
0.344
-0.501
-0.034
0.039
-0.020
-0.037
0.002
0.000
-0.015
MSE
0.1182
0.2632
0.0057
0.0090
0.0051
0.0037
0.0018
0.0001
0.0013
estimate
0.269
0.532
0.318
0.377
0.093
0.483
0.277
0.011
0.696
n = 100
bias
0.209
-0.368
-0.013
0.013
-0.007
-0.017
-0.003
0.001
-0.005
MSE
0.0437
0.1380
0.0022
0.0060
0.0027
0.0016
0.0009
0.0000
0.0006
estimate
0.183
0.662
0.326
0.368
0.097
0.491
0.286
0.010
0.693
n = 200
bias
0.123
-0.238
-0.005
0.004
-0.003
-0.009
0.006
0.000
-0.008
MSE
0.0151
0.0603
0.0010
0.0023
0.0012
0.0006
0.0005
0.0000
0.0003
Table: Mean of Bayesian posterior median, bias and MSE for parameters of the PVF
copula model (pc = 20%)
Pepe Romeo (USAoCH)
PVF copula survival analysis
Coventry, July 2015
14 / 22
Simulation study II - Comparison of different copula models
We simulate (u1 , u2 ) ∼ PVF(α, η) under the same previous setup
We fit the PVF copula model and Clayton, Gumbel and Inverse Gaussian
Goodness-of-fit is compared using standard model selection criteria
Pepe Romeo (USAoCH)
PVF copula survival analysis
Coventry, July 2015
15 / 22
Simulation study II - Comparison of different copula models
We simulate (u1 , u2 ) ∼ PVF(α, η) under the same previous setup
We fit the PVF copula model and Clayton, Gumbel and Inverse Gaussian
Goodness-of-fit is compared using standard model selection criteria
n = 50
n = 100
n = 200
Model
PVF
Clayton
Gumbel
InvGaussian
PVF
Clayton
Gumbel
InvGaussian
PVF
Clayton
Gumbel
InvGaussian
AIC
0.102
0.394
0.050
0.454
0.298
0.204
0.022
0.476
0.618
0.042
0.000
0.340
BIC
0.026
0.414
0.068
0.492
0.096
0.266
0.030
0.608
0.288
0.090
0.000
0.622
DIC
0.480
0.208
0.002
0.310
0.602
0.108
0.008
0.282
0.786
0.020
0.000
0.194
LPML
0.462
0.190
0.004
0.344
0.580
0.102
0.012
0.306
0.776
0.018
0.002
0.204
τ -mean
0.489
0.453
0.414
0.449
0.491
0.456
0.417
0.445
0.495
0.454
0.422
0.441
τ -sd
0.084
0.099
0.080
0.051
0.057
0.067
0.056
0.032
0.039
0.047
0.038
0.021
Table: Comparison of different copula models simulating from PVF(0.36, 0.10) copula
through the proportion of times a certain model was chosen (τ = 0.50, pc = 20%)
Pepe Romeo (USAoCH)
PVF copula survival analysis
Coventry, July 2015
15 / 22
Application - Australian NH&MRC Twin data
ê To investigate whether the strength of dependence within twin pairs as to the
risk of the onset of various diseases is different for MZ and DZ twins.
Pepe Romeo (USAoCH)
PVF copula survival analysis
Coventry, July 2015
16 / 22
Application - Australian NH&MRC Twin data
ê To investigate whether the strength of dependence within twin pairs as to the
risk of the onset of various diseases is different for MZ and DZ twins.
A questionnaire was applied for collecting the information, see Duffy et al.
(1990).
Pepe Romeo (USAoCH)
PVF copula survival analysis
Coventry, July 2015
16 / 22
Application - Australian NH&MRC Twin data
ê To investigate whether the strength of dependence within twin pairs as to the
risk of the onset of various diseases is different for MZ and DZ twins.
A questionnaire was applied for collecting the information, see Duffy et al.
(1990).
We analyze time to appendectomy for adult twins comprised of 1798 MZ
(1231 female and 567 male pairs) and 1098 DZ twins (748 female and 350
male pairs)
Pepe Romeo (USAoCH)
PVF copula survival analysis
Coventry, July 2015
16 / 22
Application - Australian NH&MRC Twin data
ê To investigate whether the strength of dependence within twin pairs as to the
risk of the onset of various diseases is different for MZ and DZ twins.
A questionnaire was applied for collecting the information, see Duffy et al.
(1990).
We analyze time to appendectomy for adult twins comprised of 1798 MZ
(1231 female and 567 male pairs) and 1098 DZ twins (748 female and 350
male pairs)
Subjects not undergoing appendectomy prior to survey were considered
censored failure times (approximately 73% for each member of the twins in
both types of zygotes)
Pepe Romeo (USAoCH)
PVF copula survival analysis
Coventry, July 2015
16 / 22
Application - Australian NH&MRC Twin data
ê To investigate whether the strength of dependence within twin pairs as to the
risk of the onset of various diseases is different for MZ and DZ twins.
A questionnaire was applied for collecting the information, see Duffy et al.
(1990).
We analyze time to appendectomy for adult twins comprised of 1798 MZ
(1231 female and 567 male pairs) and 1098 DZ twins (748 female and 350
male pairs)
Subjects not undergoing appendectomy prior to survey were considered
censored failure times (approximately 73% for each member of the twins in
both types of zygotes)
Since any potential effect of a shared environment could be similar for MZ
and DZ twins, a stronger dependence in the risks for appendicitis between
MZ twin pair members would be indicative of a genetic effect and evidence
of heredity in the onset of appendicitis
Pepe Romeo (USAoCH)
PVF copula survival analysis
Coventry, July 2015
16 / 22
Application - Australian NH&MRC Twin data
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Figure: Scatterplot of MZ (left) and DZ (right) twin data. The data points correspond
to non censored times (+), one censored time (4) and both times are censored (◦)
Pepe Romeo (USAoCH)
PVF copula survival analysis
Coventry, July 2015
17 / 22
Application - Australian NH&MRC Twin data
Modelling
Marginal distributions: Tj ∼ Piecewise Exponential(λlj ).
hj (t) = λlj , for t ∈ [al , al+1 ), l = 1, 2, . . . , L,
and prior distribution λlj ∼ Gamma(0.001, 0.001), j = 1, 2.
Pepe Romeo (USAoCH)
PVF copula survival analysis
Coventry, July 2015
18 / 22
Application - Australian NH&MRC Twin data
Modelling
Marginal distributions: Tj ∼ Piecewise Exponential(λlj ).
hj (t) = λlj , for t ∈ [al , al+1 ), l = 1, 2, . . . , L,
and prior distribution λlj ∼ Gamma(0.001, 0.001), j = 1, 2.
Priors for dependence parameters on copula models:
I
I
I
I
PVF α ∼ Beta(1, 1), η ∼ Gamma(0.01, 0.01),
Clayton η ∼ Gamma(0.01, 0.01)
Gumbel α ∼ Beta(1, 1)
Inverse Gaussian η ∼ Gamma(0.01, 0.01)
Pepe Romeo (USAoCH)
PVF copula survival analysis
Coventry, July 2015
18 / 22
Application - Australian NH&MRC Twin data
Modelling
Marginal distributions: Tj ∼ Piecewise Exponential(λlj ).
hj (t) = λlj , for t ∈ [al , al+1 ), l = 1, 2, . . . , L,
and prior distribution λlj ∼ Gamma(0.001, 0.001), j = 1, 2.
Priors for dependence parameters on copula models:
I
I
I
I
PVF α ∼ Beta(1, 1), η ∼ Gamma(0.01, 0.01),
Clayton η ∼ Gamma(0.01, 0.01)
Gumbel α ∼ Beta(1, 1)
Inverse Gaussian η ∼ Gamma(0.01, 0.01)
. . . but, how we can include the gender and type of zygosity in the model?
Pepe Romeo (USAoCH)
PVF copula survival analysis
Coventry, July 2015
18 / 22
Application - Australian NH&MRC Twin data
Modelling
Marginal distributions: Tj ∼ Piecewise Exponential(λlj ).
hj (t) = λlj , for t ∈ [al , al+1 ), l = 1, 2, . . . , L,
and prior distribution λlj ∼ Gamma(0.001, 0.001), j = 1, 2.
Priors for dependence parameters on copula models:
I
I
I
I
PVF α ∼ Beta(1, 1), η ∼ Gamma(0.01, 0.01),
Clayton η ∼ Gamma(0.01, 0.01)
Gumbel α ∼ Beta(1, 1)
Inverse Gaussian η ∼ Gamma(0.01, 0.01)
. . . but, how we can include the gender and type of zygosity in the model?
ê Instead of the conventional approach of analysing MZ and DZ separately,
we allow the association parameter to depend on covariates, i.e.,
including the type of zygosity as a dichotomous covariate as well as the sex of
the twins.
Pepe Romeo (USAoCH)
PVF copula survival analysis
Coventry, July 2015
18 / 22
Application - Australian NH&MRC Twin data
Modelling
Specifically, we include binary covariates x1 , type of zygosity and x2 , sex of
the twins, through the dependence parameter:
I
I
logit(α(x1 , x2 )) = γ0 + γ1 x1 + γ2 x2 , where logit(u) = log(u/(1 − u)) with
u ∈ (0, 1) for α in PVF(α, η) and Gumbel(α)
log(η(x1 , x2 )) = γ0 + γ1 x1 + γ2 x2 , for Clayton(η)
Pepe Romeo (USAoCH)
PVF copula survival analysis
Coventry, July 2015
19 / 22
Application - Australian NH&MRC Twin data
Modelling
Specifically, we include binary covariates x1 , type of zygosity and x2 , sex of
the twins, through the dependence parameter:
I
I
logit(α(x1 , x2 )) = γ0 + γ1 x1 + γ2 x2 , where logit(u) = log(u/(1 − u)) with
u ∈ (0, 1) for α in PVF(α, η) and Gumbel(α)
log(η(x1 , x2 )) = γ0 + γ1 x1 + γ2 x2 , for Clayton(η)
The prior distributions for γ0 , γ1 and γ2 are chosen as Normal(0, 0.0001)
Pepe Romeo (USAoCH)
PVF copula survival analysis
Coventry, July 2015
19 / 22
Application - Australian NH&MRC Twin data
Modelling
Specifically, we include binary covariates x1 , type of zygosity and x2 , sex of
the twins, through the dependence parameter:
I
I
logit(α(x1 , x2 )) = γ0 + γ1 x1 + γ2 x2 , where logit(u) = log(u/(1 − u)) with
u ∈ (0, 1) for α in PVF(α, η) and Gumbel(α)
log(η(x1 , x2 )) = γ0 + γ1 x1 + γ2 x2 , for Clayton(η)
The prior distributions for γ0 , γ1 and γ2 are chosen as Normal(0, 0.0001)
Results
Model
PVF-PWE
Clayton-PWE
Gumbel-PWE
AIC
15122.71
15147.71
15122.61
BIC
15254.07
15273.10
15248.00
DIC
15100.77
15127.03
15101.74
pD
22.06
21.33
21.13
LPML
-7551.83
-7564.39
-7551.68
Table: Model selection criteria for copula models, Australian twin data
Pepe Romeo (USAoCH)
PVF copula survival analysis
Coventry, July 2015
19 / 22
Application - Australian NH&MRC Twin data
Modelling
Specifically, we include binary covariates x1 , type of zygosity and x2 , sex of
the twins, through the dependence parameter:
I
I
logit(α(x1 , x2 )) = γ0 + γ1 x1 + γ2 x2 , where logit(u) = log(u/(1 − u)) with
u ∈ (0, 1) for α in PVF(α, η) and Gumbel(α)
log(η(x1 , x2 )) = γ0 + γ1 x1 + γ2 x2 , for Clayton(η)
The prior distributions for γ0 , γ1 and γ2 are chosen as Normal(0, 0.0001)
Results
Model
PVF-PWE
Clayton-PWE
Gumbel-PWE
AIC
15122.71
15147.71
15122.61
BIC
15254.07
15273.10
15248.00
DIC
15100.77
15127.03
15101.74
pD
22.06
21.33
21.13
LPML
-7551.83
-7564.39
-7551.68
Table: Model selection criteria for copula models, Australian twin data
ê Note that for PVF, the posterior estimate of η is (0.007 ± 0.014)
Pepe Romeo (USAoCH)
PVF copula survival analysis
Coventry, July 2015
19 / 22
Application - Australian NH&MRC Twin data
Results
Parameter
τMZ −Female
τMZ −Male
τDZ −Female
τDZ −Male
Mean
0.229
0.143
0.141
0.085
Median
0.230
0.143
0.141
0.083
SD
0.024
0.031
0.025
0.026
HPD
0.183, 0.276
0.079, 0.197
0.094, 0.193
0.040, 0.137
Table: Posterior Kendall’s τ , Gumbel copula model, Australian twin data
Pepe Romeo (USAoCH)
PVF copula survival analysis
Coventry, July 2015
20 / 22
Final comments and future work
Models based on copulas are considerable flexible respect the marginal
distributions and the dependence structure
PVF copula and particular models
Simulation of (u1 , u2 ) ∼ PVF(α, η)
From simulation study: better performance of the PVF copula model for
n > 100 and τ > 0.33
Pepe Romeo (USAoCH)
PVF copula survival analysis
Coventry, July 2015
21 / 22
Final comments and future work
Models based on copulas are considerable flexible respect the marginal
distributions and the dependence structure
PVF copula and particular models
Simulation of (u1 , u2 ) ∼ PVF(α, η)
From simulation study: better performance of the PVF copula model for
n > 100 and τ > 0.33
ê Application in others areas (finance)
Pepe Romeo (USAoCH)
PVF copula survival analysis
Coventry, July 2015
21 / 22
Final comments and future work
Models based on copulas are considerable flexible respect the marginal
distributions and the dependence structure
PVF copula and particular models
Simulation of (u1 , u2 ) ∼ PVF(α, η)
From simulation study: better performance of the PVF copula model for
n > 100 and τ > 0.33
ê Application in others areas (finance)
ê To implement the case of unbalanced lifetime data (Meyer & Romeo, 2015)
Pepe Romeo (USAoCH)
PVF copula survival analysis
Coventry, July 2015
21 / 22
Final comments and future work
Models based on copulas are considerable flexible respect the marginal
distributions and the dependence structure
PVF copula and particular models
Simulation of (u1 , u2 ) ∼ PVF(α, η)
From simulation study: better performance of the PVF copula model for
n > 100 and τ > 0.33
ê Application in others areas (finance)
ê To implement the case of unbalanced lifetime data (Meyer & Romeo, 2015)
ê Testing precise or sharp hypothesis for PVF copula model:
H0 : α → 0 (Clayton),
H0 : η = 0 (Gumbel)
H0 : α = 0.5 (Inverse Gaussian)
Pepe Romeo (USAoCH)
PVF copula survival analysis
Coventry, July 2015
21 / 22
Final comments and future work
Models based on copulas are considerable flexible respect the marginal
distributions and the dependence structure
PVF copula and particular models
Simulation of (u1 , u2 ) ∼ PVF(α, η)
From simulation study: better performance of the PVF copula model for
n > 100 and τ > 0.33
ê Application in others areas (finance)
ê To implement the case of unbalanced lifetime data (Meyer & Romeo, 2015)
ê Testing precise or sharp hypothesis for PVF copula model:
H0 : α → 0 (Clayton),
H0 : η = 0 (Gumbel)
H0 : α = 0.5 (Inverse Gaussian)
ê Temporal dependence: τ (t)
Pepe Romeo (USAoCH)
PVF copula survival analysis
Coventry, July 2015
21 / 22
Main References
Duchateau, L. and Janssen, P. (2008). The Frailty Model, Springer, NY
Duffy, D.L., Martin, N.G. and Mathews, J.D. (1990). Appendectomy in Australian
twins. American Journal of Human Genetics, 47(3), 590–592.
Hougaard, P. (2000). Analysis of Multivariate Survival Data. Springer, NY
Mai, J. and Scherer, M. (2012). Simulating Copulas: Stochastic Models, Sampling
Algorithms, and Applications. Imperial College, Boca Raton.
Meyer, R. and Romeo, J.S. (2015). Bayesian semi-parametric analysis of recurrent
failure time data using copulas. Biometrical Journal, in press.
Nelsen, R.B. (2006). An Introduction to Copulas, 2nd edition. Springer, NY.
Oakes, D. (1989). Bivariate survival models induced by frailties. Journal of the
American Statistical Association, 84, 487-493.
Romeo, J.S., Tanaka, N.I. and Pedroso de Lima, A.C. (2006). Bivariate survival
modeling: A Bayesian approach based on Copulas. Lifetime Data Analysis, 12,
205-222.
Wienke, A. (2010). Frailty Models in Survival Analysis. Chapman and Hall/CRC,
Boca Raton.
Pepe Romeo (USAoCH)
PVF copula survival analysis
Coventry, July 2015
22 / 22