Examples of joint models for multivariate
longitudinal and multistate processes in chronic
diseases
Cécile Proust-Lima
works with Loı̈c Ferrer, Anaı̈s Rouanet
and Hélène Jacqmin-Gadda
INSERM U897, Epidemiology and Biostatistics, Bordeaux, France
Univ. Bordeaux, ISPED, Bordeaux, France
cecile.proust-lima@inserm.fr
Workshop on Flexible Models for Longitudinal and Survival Data with Applications in Biostatistics
Cécile Proust-Lima (INSERM)
Joint models for multiple outcomes
July 2015
1 / 45
Joint modelling principle
Simultaneous modelling of correlated longitudinal and survival data
latent
structure
time to
event
6
1.00
4
0.75
Survival probability
log(PSA+0.1)
longitudinal
marker
2
0
−2
0.50
0.25
0.00
0
5
10
15
0
Years since the end of EBRT
Cécile Proust-Lima (INSERM)
5
10
15
Years since the end of RT
Joint models for multiple outcomes
July 2015
2 / 45
Joint modelling principle
Simultaneous modelling of correlated longitudinal and survival data
latent
structure
longitudinal
marker
time to
event
Objectives :
I
I
I
describe the longitudinal process stopped by the event
predict the risk of event ajusted for the longitudinal process
explore the association between the two processes
Cécile Proust-Lima (INSERM)
Joint models for multiple outcomes
July 2015
2 / 45
2 main families of joint models
latent
structure
longitudinal
marker
time to
event
Mixed model
(usually linear)
Survival model
(usually proportional hazards)
Link with the latent structure :
I
random effects from the mixed model
(shared random effect models)
I
latent class structure
(joint latent class models)
Cécile Proust-Lima (INSERM)
Joint models for multiple outcomes
July 2015
3 / 45
Shared random-effect model (SREM) (Rizopoulos, 2012)
latent
structure
longitudinal
marker
time to
event
Shared random-effects distribution : bi ∼ N (µ, B)
Linear mixed model for the biomarker trajectory :
Yi (tij ) = Yi (tij )∗ + ij = Zi (tij )T bi + XLi (tij )T β + ij with ij ∼ N 0, σ2
Proportional hazard model including marker trajectory characteristics :
T
λ(t | bi ) = λ0 (t)eXSi (t)
→
δ+Wi (bi ,β,t)T η
JM, JMBayes in R, stjm in Stata, JMFit in SAS
Cécile Proust-Lima (INSERM)
Joint models for multiple outcomes
July 2015
4 / 45
Joint latent class model (JLCM) (Proust-Lima et al., 2014)
latent
structure
longitudinal
marker
time to
event
Shared latent class (ci ) membership :
>
eξ0g +XCi ξ1g
with ξ0G = 0 & ξ1G = 0
πig = P(ci = g|Xpi ) = PG
ξ0l +XCi / topξ1l
l=1 e
Class-specific linear mixed model for the biomarker trajectory :
Yi (tij ) |ci =g = Zi (tij )T big + XLi (tij )> βg + ij with big ∼ N (µg , Bg ) , ij ∼ N 0, σ2
Class-specific proportional hazard model :
λ(t | ci = g) = λ0g (t)eXTi (t)δg
→
lcmm in R
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Joint models for multiple outcomes
July 2015
5 / 45
Remarks (Proust-Lima et al., 2014)
Shared random effect models :
I
I
I
extension of the standard time-to-event models
assessment of specific associations (surrogacy)
quantification of the association
Joint latent class models :
I
I
I
heterogeneous population
no assumption on the association
useful for predictive tools
Cécile Proust-Lima (INSERM)
Joint models for multiple outcomes
July 2015
6 / 45
Remarks (Proust-Lima et al., 2014)
Shared random effect models :
I
I
I
extension of the standard time-to-event models
assessment of specific associations (surrogacy)
quantification of the association
Joint latent class models :
I
I
I
heterogeneous population
no assumption on the association
useful for predictive tools
In any case, most developments for :
I
I
→
a Gaussian longitudinal marker
a right-censored time to event
but more complex data in most cohort studies on chronic diseases
Cécile Proust-Lima (INSERM)
Joint models for multiple outcomes
July 2015
6 / 45
In chronic diseases
Longitudinal part :
I
I
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I
I
multiple markers of progression
markers of different nature
Gaussian, binary, poisson
ordinal
continuous but non Gaussian
Survival part :
I
I
I
I
competing risks
recurrent events
multiple events
succession of different events
Cécile Proust-Lima (INSERM)
Joint models for multiple outcomes
July 2015
7 / 45
In chronic diseases
Longitudinal part :
I
I
I
I
I
multiple markers of progression
markers of different nature
Gaussian, binary, poisson
ordinal
continuous but non Gaussian
Survival part :
I
I
I
I
competing risks
recurrent events
multiple events
succession of different events
3 examples of developments through the study of
I
I
progression of localized Prostate cancer after treatment
natural history of Alzheimer’s disease
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Joint models for multiple outcomes
July 2015
7 / 45
Progression of localized Prostate Cancer
after a treatment by radiation therapy
Cécile Proust-Lima (INSERM)
Joint models for multiple outcomes
July 2015
8 / 45
Localized prostate cancer
2
1
0
repeated measures of
PSA (prostate specific
antigen) collected in
routine
−1
I
prognostic factors at
diagnosis (T-stage,
Gleason, dose of RT, ...)
−2
I
log(PSA+0.1)
3
4
Monitoring of patients after radiation therapy for a localized Prostate
cancer :
0
2
4
6
8
10
time since end of RT
Interest in predicting the risk of progression
I
I
multiple types : local recurrence, distant recurrence, death
problem of initiation of new treatment : hormonal treatment
Cécile Proust-Lima (INSERM)
Joint models for multiple outcomes
July 2015
9 / 45
Dynamic prediction of clinical recurrence of any type
(Sène et al., SMMR 2014) :
Individualized probability of clinical recurrence :
1.0 0.0 1.5 0.5 2.0 1.0 2.5 1.5 3.0 2.0
0.2
0.0
0.2
log(PSA + 0.1)
1.0
0.5
0.0
0.5
From M4b
From M2c
2.5
3.0
Init. now
In 1 year
In 2Init.
years
now If In
no1HT
year
In 2 years
1.0
From M4b
From M2c
0.4
0.6
0.8
Probability of recurrence
PSA measures
time of prediction
0.0
PSA measures
time of prediction
1.5
2.0
I
in the next three years
for a man naive of HT
according to hypothetical times of initiation of HT (time-dependent covariate)
1.0
I
0.4
0.6
0.8
Probability of recurrence
I
If no HT
Time (years) since end
Time
of EBRT
(years) since end of EBRT
Cécile Proust-Lima (INSERM)
Joint models for multiple outcomes
July 2015
10 / 45
Dynamic prediction of clinical recurrence of any type
(Sène et al., SMMR 2014) :
Individualized probability of clinical recurrence :
1.0 0.0 1.5 0.5 2.0 1.0 2.5 1.5 3.0 2.0
0.2
0.0
0.2
log(PSA + 0.1)
1.0
0.5
0.0
0.5
From M4b
From M2c
2.5
3.0
Init. now
In 1 year
In 2Init.
years
now If In
no1HT
year
In 2 years
1.0
From M4b
From M2c
0.4
0.6
0.8
Probability of recurrence
PSA measures
time of prediction
0.0
PSA measures
time of prediction
1.5
2.0
I
in the next three years
for a man naive of HT
according to hypothetical times of initiation of HT (time-dependent covariate)
1.0
I
0.4
0.6
0.8
Probability of recurrence
I
If no HT
Time (years) since end
Time
of EBRT
(years) since end of EBRT
Cécile Proust-Lima (INSERM)
Joint models for multiple outcomes
July 2015
10 / 45
Dynamic prediction of clinical recurrence of any type
(Sène et al., SMMR 2014) :
Individualized probability of clinical recurrence :
1.0 0.0 1.5 0.5 2.0 1.0 2.5 1.5 3.0 2.0
0.2
0.0
0.2
log(PSA + 0.1)
1.0
0.5
0.0
0.5
From M4b
From M2c
2.5
3.0
Init. now
In 1 year
In 2Init.
years
now If In
no1HT
year
In 2 years
1.0
From M4b
From M2c
0.4
0.6
0.8
Probability of recurrence
PSA measures
time of prediction
0.0
PSA measures
time of prediction
1.5
2.0
I
in the next three years
for a man naive of HT
according to hypothetical times of initiation of HT (time-dependent covariate)
1.0
I
0.4
0.6
0.8
Probability of recurrence
I
If no HT
Time (years) since end
Time
of EBRT
(years) since end of EBRT
Cécile Proust-Lima (INSERM)
Joint models for multiple outcomes
July 2015
10 / 45
Multiple types of progression
logPSA
Clinical progression is a multistate process :
hormon.
therapy
years
end
treatment
local
recurr.
death
distant
recurr.
Importance of distinguishing the different types to :
I
I
clarify the impact of PSA dynamics and other prognostic factors on each
transition
predict type-specific progressions
Cécile Proust-Lima (INSERM)
Joint models for multiple outcomes
July 2015
11 / 45
The joint longitudinal and multistate model (Ferrer et al., arXiv
2015)
Notations :
multi-state process
I
Ei = {Ei (t), Ti0 ≤ t ≤ Ci } is a non-homogeneous Markov process
F
F
Ei (t) takes values in the finite state space S = {0, 1, . . . , M}
Ti0 the left truncation time, Ci the right censoring time
I
Ti = (Ti1 , . . . , Timi )> the mi observed times with Tir < Ti(r+1) , ∀r ∈ S
I
δi = (δi1 , . . . , δimi )> the vector of observed transition indicators
longitudinal process
I
Yi = (Yi1 , . . . , Yini )> the ni measures of the marker collected at times
ti1 , . . . , tini , with tini ≤ Ti1
Cécile Proust-Lima (INSERM)
Joint models for multiple outcomes
July 2015
12 / 45
The joint longitudinal and multistate model (Ferrer et al., arXiv
2015) (cont’d)
Longitudinal part : mixed model
I
Yi∗ (tij ) + ij
XLi (tij )> β + Zi (tij )> bi + ij
bi ∼ N (0, D), i = (i1 , . . . , ini )> ∼ N (0, σ 2 Ini ), bi
|=
=
=
Yij
i
Survival part : multistate model
λihk (t|bi )
Pr(Ei (t + dt) = k|Ei (t) = h; bi )
dt
>
= λhk,0 (t) exp(XThk,i
γhk + Whk,i (bi , t)> ηhk ), for h, k ∈ S,
= limdt→0
I
λhk,0 (t) parametric baseline intensity, XThk,i prognostic factors
I
Whk,i (bi , t) the dependence structure (Sène et al., J sfds 2014)
F
F
F
F
Whk,i (bi , t) = Yi∗ (t)
Whk,i (bi , t) = ∂Yi∗ (t)/∂t
Whk,i (bi , t) = Yi∗ (t), ∂Yi∗ (t)/∂t
...
Cécile Proust-Lima (INSERM)
>
(true current level)
−→
−→
(true current slope)
−→
(both)
Joint models for multiple outcomes
July 2015
13 / 45
Likelihood function using Yi
L(θ) =
N Z
Y
i=1
|=
Maximum Likelihood Estimation
bi
Ti ,
fY (Yi |bi ; θ) fT (Ti , δi |bi ; θ) fb (bi ; θ) dbi
Rq
with :
I
Random effects part : bi ∼ Nq (0, D)
I
>
Longitudinal part : Yi |bi ∼ Nni (XLi
β + Zi> bi , σ 2 Ini )
I
Multi-state part :
mi −1 fT (Ti , δi |bi ; θ) =
Y
PiEi (Tir ),Ei (Tir ) (Tir , Ti(r+1) |bi )×
r=0
λiEi (Tir ),Ei (Ti(r+1) ) (Ti(r+1) |bi )
with Phh (s, t) = exp
Cécile Proust-Lima (INSERM)
Rt
s
δi(r+1)
P R
t
λhh (u) du = exp − k6=h s λhk (u) du
Joint models for multiple outcomes
July 2015
14 / 45
Implementation under R
Relies on JM package
Implementation procedure decomposed into four steps :
1. lme() function (nlme package) to initialise the parameters in the
longitudinal sub-model ;
2. mstate package to adapt the data to the multi-state framework ;
3. coxph() function (survival package) to initialise the parameters in the
multi-state sub-model ;
4. JMstateModel() function (extension of jointModel()) to estimate all
the parameters of the joint multi-state model.
Likelihood computed and optimised using :
I
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numerical integration algorithms (Gaussian quadratures)
optimisation algorithms (EM + quasi-Newton)
Cécile Proust-Lima (INSERM)
Joint models for multiple outcomes
July 2015
15 / 45
Application
2 cohorts of men with localized prostate cancer treated by radiotherapy
(N=1474)
Repeated measures of PSA
Multi-state representation of the
clinical progressions
Hormonal
Therapy
6
λ02 (t)
λ12 (t)
2
λ23 (t)
λ24 (t)
log(PSA+0.1)
4
End
EBRT
2
λ01 (t)
0
λ03 (t)
−2
0
5
10
λ04 (t)
15
Local
Recurrence
λ14 (t)
Death
4
1
0
λ13 (t)
λ34 (t)
Distant
Recurrence
3
Years since the end of EBRT
10 (3, 21) measurements per patient
(50th, (5th, 95th) %iles)
Cécile Proust-Lima (INSERM)

533 144 227
20
90
 0

0
106
Υ= 0
 0
0
0
0
0
0
matrix of direct transitions
Joint models for multiple outcomes
47
10
33
13
0

523
24 

178
77 
802
July 2015
16 / 45
Specification of the joint model (1/2)
Longitudinal sub-model specification
Yij
=
=
Yi∗ (tij ) + ij
>
β0 + XL0i
β0,cov + bi0 +
>
β1 + XL1i
β1,cov + bi1 × f1 (tij ) +
>
β2 + XL2i
β2,cov + bi2 × f2 (tij ) + ij
I
f1 (t) = (1 + t)−1.2 − 1 and f2 (t) = t
I
bi = (bi0 , bi1 , bi2 )> ∼ N (0, D), D unstructured, i ∼ N (0, σ 2 Ini )
I
XL0i , XL1i and XL2i were obtained using a backward stepwise procedure.
Cécile Proust-Lima (INSERM)
Joint models for multiple outcomes
July 2015
17 / 45
Specification of the joint model (2/2)
Multi-state sub-model specification
λihk (t|bi )
= λhk,0 (t) exp
>
XT,hk,i
γhk
Yi∗ (t)
+
∂Yi∗ (t)/∂t
> ηhk,level
ηhk,slope
I
Log-baseline intensities approximated by B-splines
I
Proportionality assumptions between several baseline intensities
I
Backward stepwise procedure to select the prognostic factors
I
Dependence function chosen by Wald tests
Cécile Proust-Lima (INSERM)
Joint models for multiple outcomes
!
July 2015
18 / 45
Results
Multi-state process
Estimates of the association parameters between
the longitudinal and multi-state processes
Level : 01
Level : 02
Level : 03
Level : 04
Level : 12
Level : 13
Level : 14
Level : 23
Level : 24
Level : 34
Slope : 01
Slope : 02
Slope : 03
Slope : 04
Slope : 12
Slope : 13
Slope : 14
Slope : 23
Slope : 24
Slope : 34
Value
StdErr
0.37
0.09
0.51
0.07
0.45
0.11
−0.17
0.05
−0.16
0.10
−0.41
0.20
0.10
0.14
−0.15
0.09
0.00
0.05
0.04
0.08
.....................
2.54
0.31
3.04
0.25
2.43
0.49
1.03
0.32
2.01
0.61
3.18
0.80
−0.20
1.27
0.97
0.67
0.29
0.52
−0.79
0.78
p-value
< 0.001
< 0.001
< 0.001
0.001
0.110
0.042
0.487
0.120
0.412
0.609
< 0.001
< 0.001
< 0.001
0.001
0.001
< 0.001
0.873
0.150
0.583
0.313
Hormonal
Therapy
λ02 (t)
End
EBRT
λ01 (t)
λ12 (t)
Local
Recurrence
2
λ23 (t)
λ24 (t)
λ14 (t)
Death
4
1
0
λ03 (t)
λ13 (t)
λ04 (t)
λ34 (t)
Distant
Recurrence
3
Prognostic factors : advanced initial stage
not always associated with intensities of
transitions between health states after
adjustment on PSA
Cécile Proust-Lima (INSERM)
Joint models for multiple outcomes
July 2015
19 / 45
Results
Multi-state process
Estimates of the association parameters between
the longitudinal and multi-state processes
Level : 01
Level : 02
Level : 03
Level : 04
Level : 12
Level : 13
Level : 14
Level : 23
Level : 24
Level : 34
Slope : 01
Slope : 02
Slope : 03
Slope : 04
Slope : 12
Slope : 13
Slope : 14
Slope : 23
Slope : 24
Slope : 34
Value
StdErr
0.37
0.09
0.51
0.07
0.45
0.11
−0.17
0.05
−0.16
0.10
−0.41
0.20
0.10
0.14
−0.15
0.09
0.00
0.05
0.04
0.08
.....................
2.54
0.31
3.04
0.25
2.43
0.49
1.03
0.32
2.01
0.61
3.18
0.80
−0.20
1.27
0.97
0.67
0.29
0.52
−0.79
0.78
p-value
< 0.001
< 0.001
< 0.001
0.001
0.110
0.042
0.487
0.120
0.412
0.609
< 0.001
< 0.001
< 0.001
0.001
0.001
< 0.001
0.873
0.150
0.583
0.313
Hormonal
Therapy
λ02 (t)
End
EBRT
λ01 (t)
λ12 (t)
Local
Recurrence
2
λ23 (t)
λ24 (t)
λ14 (t)
Death
4
1
0
λ03 (t)
λ13 (t)
λ04 (t)
λ34 (t)
Distant
Recurrence
3
Prognostic factors : advanced initial stage
not always associated with intensities of
transitions between health states after
adjustment on PSA
Cécile Proust-Lima (INSERM)
Joint models for multiple outcomes
July 2015
19 / 45
Results
Multi-state process
Estimates of the association parameters between
the longitudinal and multi-state processes
Level : 01
Level : 02
Level : 03
Level : 04
Level : 12
Level : 13
Level : 14
Level : 23
Level : 24
Level : 34
Slope : 01
Slope : 02
Slope : 03
Slope : 04
Slope : 12
Slope : 13
Slope : 14
Slope : 23
Slope : 24
Slope : 34
Value
StdErr
0.37
0.09
0.51
0.07
0.45
0.11
−0.17
0.05
−0.16
0.10
−0.41
0.20
0.10
0.14
−0.15
0.09
0.00
0.05
0.04
0.08
.....................
2.54
0.31
3.04
0.25
2.43
0.49
1.03
0.32
2.01
0.61
3.18
0.80
−0.20
1.27
0.97
0.67
0.29
0.52
−0.79
0.78
p-value
< 0.001
< 0.001
< 0.001
0.001
0.110
0.042
0.487
0.120
0.412
0.609
< 0.001
< 0.001
< 0.001
0.001
0.001
< 0.001
0.873
0.150
0.583
0.313
Hormonal
Therapy
λ02 (t)
End
EBRT
λ01 (t)
λ12 (t)
Local
Recurrence
2
λ23 (t)
λ24 (t)
λ14 (t)
Death
4
1
0
λ03 (t)
λ13 (t)
λ04 (t)
λ34 (t)
Distant
Recurrence
3
Prognostic factors : advanced initial stage
not always associated with intensities of
transitions between health states after
adjustment on PSA
Cécile Proust-Lima (INSERM)
Joint models for multiple outcomes
July 2015
19 / 45
Results
Multi-state process
Estimates of the association parameters between
the longitudinal and multi-state processes
Level : 01
Level : 02
Level : 03
Level : 04
Level : 12
Level : 13
Level : 14
Level : 23
Level : 24
Level : 34
Slope : 01
Slope : 02
Slope : 03
Slope : 04
Slope : 12
Slope : 13
Slope : 14
Slope : 23
Slope : 24
Slope : 34
Value
StdErr
0.37
0.09
0.51
0.07
0.45
0.11
−0.17
0.05
−0.16
0.10
−0.41
0.20
0.10
0.14
−0.15
0.09
0.00
0.05
0.04
0.08
.....................
2.54
0.31
3.04
0.25
2.43
0.49
1.03
0.32
2.01
0.61
3.18
0.80
−0.20
1.27
0.97
0.67
0.29
0.52
−0.79
0.78
p-value
< 0.001
< 0.001
< 0.001
0.001
0.110
0.042
0.487
0.120
0.412
0.609
< 0.001
< 0.001
< 0.001
0.001
0.001
< 0.001
0.873
0.150
0.583
0.313
Hormonal
Therapy
λ02 (t)
End
EBRT
λ01 (t)
λ12 (t)
Local
Recurrence
2
λ23 (t)
λ24 (t)
λ14 (t)
Death
4
1
0
λ03 (t)
λ13 (t)
λ04 (t)
λ34 (t)
Distant
Recurrence
3
Prognostic factors : advanced initial stage
not always associated with intensities of
transitions between health states after
adjustment on PSA
Cécile Proust-Lima (INSERM)
Joint models for multiple outcomes
July 2015
19 / 45
Diagnostics for the parametric assumptions
Goodness-of-fit plots for the longitudinal process
Conditional standardized residuals
versus fitted values
I
3
Observed and predicted values of
the biomarker
1.00
2
log(PSA+0.1)
Conditional Standardized Residuals
I
1
0
Observed
Predicted
0.75
0.50
0.25
−1
0.00
−2
−2
0
2
4
0.0
Fitted Values
Cécile Proust-Lima (INSERM)
2.5
5.0
7.5
10.0
12.5
Years since the end of EBRT
Joint models for multiple outcomes
July 2015
20 / 45
Diagnostics for the parametric assumptions
Goodness-of-fit plots for the longitudinal process
Goodness-of-fit plots for the multi-state process
I
Predicted transition probabilities from the joint multi-state model and
non-parametric probability transitions
From state 0
From state 1
1.00
0.75
To state
0.50
Transition Probabilities
1
0.25
2
3
0.00
4
From state 2
From state 3
1.00
Estimators
Par.
0.75
Non−par.
95% CI
0.50
0.25
0.00
0
5
10
15
0
5
10
15
Years since the end of EBRT
Cécile Proust-Lima (INSERM)
Joint models for multiple outcomes
July 2015
20 / 45
Natural history of Alzheimer’s disease,
dementias and cognitive aging
Cécile Proust-Lima (INSERM)
Joint models for multiple outcomes
July 2015
21 / 45
Cognitive aging and dementia
Dementia (e.g. Alzheimer’s disease) characterized by a progressive
decline of cognition
Most interest in
I
I
I
the natural history of dementia
risk factors of cognitive decline and dementia
dynamic individual prediction of dementia
Cécile Proust-Lima (INSERM)
Joint models for multiple outcomes
July 2015
22 / 45
Cognitive aging and dementia
Dementia (e.g. Alzheimer’s disease) characterized by a progressive
decline of cognition
Most interest in
I
I
I
the natural history of dementia
risk factors of cognitive decline and dementia
dynamic individual prediction of dementia
→ 1st complexity : elderly pathology
I
I
I
delayed entry
dementia in competition with death
diagnosis at pre-established visit times
→ 2nd complexity : cognition is not directly observed
I
I
cognitive process (trait) defined in continuous time
repeated psychometric tests measured in discrete times
F
F
F
multiple cognitive functions (langage, memory, attention,...)
noisy measures of overall cognition
limited statistical properties
Cécile Proust-Lima (INSERM)
Joint models for multiple outcomes
July 2015
22 / 45
Latent process mixed model
covariates X(t)
time t
underlying
true PSA level
Y*(t)
PSA
Y at T1
Cécile Proust-Lima (INSERM)
Joint models for multiple outcomes
July 2015
23 / 45
Latent process mixed model
covariates X(t)
time t
cognitive
process
Λ(t)
test 1
Y1 at T11
Cécile Proust-Lima (INSERM)
Joint models for multiple outcomes
July 2015
23 / 45
Latent process mixed model
covariates X(t)
time t
cognitive
process
Λ(t)
...
test 1
Y1 at T11
test k
Yk at T1k
Cécile Proust-Lima (INSERM)
...
test K
YK at T1K
Joint models for multiple outcomes
July 2015
23 / 45
Latent process mixed model
← Λi (t) = XL1i (t)> β + Zi (t)> bi + wi (t)
covariates X(t)
time t
I
I
I
cognitive
process
Λ(t)
...
test 1
Y1 at T11
test k
Yk at T1k
Cécile Proust-Lima (INSERM)
bi ∼ MVN(µ, B)
wi (t) autocorrelated process
bi0 ∼ N(0, 1) for identifiability
...
test K
YK at T1K
Joint models for multiple outcomes
July 2015
23 / 45
Latent process mixed model
← Λi (t) = XL1i (t)> β + Zi (t)> bi + wi (t)
covariates X(t)
time t
I
I
I
cognitive
process
Λ(t)
bi ∼ MVN(µ, B)
wi (t) autocorrelated process
bi0 ∼ N(0, 1) for identifiability
← Ykij = ζ1k + ζ2k Ỹkij
...
test 1
Y1 at T11
test k
Yk at T1k
Cécile Proust-Lima (INSERM)
Ỹkij = Λi (tijk ) +
...
test K
YK at T1K
I
kij
kij ∼ N (0, σ2k )
Joint models for multiple outcomes
July 2015
23 / 45
Latent process mixed model
← Λi (t) = XL1i (t)> β + Zi (t)> bi + wi (t)
covariates X(t)
time t
I
I
I
cognitive
process
Λ(t)
bi ∼ MVN(µ, B)
wi (t) autocorrelated process
bi0 ∼ N(0, 1) for identifiability
← Ykij = ζ1k + ζ2k Ỹkij
...
test 1
Y1 at T11
test k
Yk at T1k
Cécile Proust-Lima (INSERM)
Ỹkij = Λi (tijk ) + XL2i (t)> γ k + αki + kij
...
test K
YK at T1K
I
I
2
αki ∼ N (0, σα
)
k
2
kij ∼ N (0, σk )
Joint models for multiple outcomes
July 2015
23 / 45
Latent process mixed model
← Λi (t) = XL1i (t)> β + Zi (t)> bi + wi (t)
covariates X(t)
time t
I
I
I
cognitive
process
Λ(t)
...
test 1
Y1 at T11
test k
Yk at T1k
bi ∼ MVN(µ, B)
wi (t) autocorrelated process
bi0 ∼ N(0, 1) for identifiability
← linear link function
Y
...
test K
YK at T1K
noisy latent process
Cécile Proust-Lima (INSERM)
Joint models for multiple outcomes
July 2015
23 / 45
Latent process mixed model involving nonlinear link
functions (Proust-Lima et al., 2015)
← Λi (t) = XL1i (t)> β + Zi (t)> bi + wi (t)
covariates X(t)
time t
I
I
I
cognitive
process
Λ(t)
...
test 1
Y1 at T11
test k
Yk at T1k
bi ∼ MVN(µ, B)
wi (t) autocorrelated process
bi0 ∼ N(0, 1) for identifiability
← nonlinear link function
Y
...
test K
YK at T1K
noisy latent process
Cécile Proust-Lima (INSERM)
Joint models for multiple outcomes
July 2015
24 / 45
Latent process mixed model involving nonlinear link
functions (Proust-Lima et al., 2015)
← Λi (t) = XL1i (t)> β + Zi (t)> bi + wi (t)
covariates X(t)
time t
I
I
cognitive
process
Λ(t)
I
← Ykij = Hk ( Ỹkij ; ζk )
parameterized link functions
...
...
test 1
Y1 at T11
test k
Yk at T1k
bi ∼ MVN(µ, B)
wi (t) autocorrelated process
bi0 ∼ N(0, 1) for identifiability
Ỹkij = Λi (tijk ) + XL2i (t)> γ k + αki + kij
test K
YK at T1K
I
I
2
αki ∼ N (0, σα
)
k
2
kij ∼ N (0, σk )
Hk = flexible parameterized transformation for outcome k
→ linear, standardised Beta CDF, quadratic I-splines, thresholds, ...
Cécile Proust-Lima (INSERM)
Joint models for multiple outcomes
July 2015
24 / 45
Joint model for multivariate cognitive measures,
dementia and death
random
effects
u
latent
process
( Λ (t))
...
outcome
Y1 at t j1
outcome
Yk at t jk
...
outcome
YK at t jK
cognitive measures :
I latent process mixed
model
Cécile Proust-Lima (INSERM)
Joint models for multiple outcomes
July 2015
25 / 45
Joint model for multivariate cognitive measures,
dementia and death
random
effects
u
latent
process
( Λ (t))
...
outcome
Y1 at t j1
outcome
Yk at t jk
...
outcome
YK at t jK
cognitive measures :
I latent process mixed
model
Cécile Proust-Lima (INSERM)
times to event
(Τ, δ )
dementia and death ?
I option 1 : first event in competing setting
I option 2 : multistate model
Joint models for multiple outcomes
July 2015
25 / 45
Joint model for multivariate cognitive measures,
dementia and death
latent
classes
c
shared latent quantity = latent classes
I heterogeneous trajectories
I no assumption on the association
random
effects
u
latent
process
( Λ (t))
...
outcome
Y1 at t j1
outcome
Yk at t jk
...
outcome
YK at t jK
cognitive measures :
I latent process mixed
model
Cécile Proust-Lima (INSERM)
times to event
(Τ, δ )
dementia and death ?
I option 1 : first event in competing setting
I option 2 : multistate model
Joint models for multiple outcomes
July 2015
25 / 45
Option 1 : competing setting (Proust-Lima et al., ArXiv 2014)
Times to events = Time to P competing events
I
I
∗
Ti = min(censoring T̃i , and cause-spec. times Ti1∗ , ..., TiP
),
δi = 0 for censored, δi = p otherwise
class-specific cause-specific proportional hazard models
>
λp (t)|ci =g = λ0p (t; νpg ) exp(XTi
ζpg )
I
λ0p parametric (splines, Gompertz, Weibull,...)
Cécile Proust-Lima (INSERM)
Joint models for multiple outcomes
July 2015
26 / 45
Likelihood function using Yi
L(θ) =
N X
G
Y
|=
Maximum Likelihood Estimation
ci
Ti ,
fY (Yi |XLi , Zi , ci = g; θ) fT (Ti , δi |XTi , ci = g; θ) P(ci = g|XCi ; θ)
i=1 g=1
with :
I f (Y |X , Z , c = g; θ) from the latent process mixed model
Y
i Li
i i
F
F
I
closed form if only continuous markers :
= multivariate normal × Jacobian of (Hk )k=1,...,K
by numerical integration otherwise ...
fT (Ti , δi |XTi , ci = g; θ) from the cause-specific model
= overall survival Sig × instantaneous risk for cause p in g if δi = p
I
P(ci = g|XCi ; θ) from a multinomial logistic model
Left truncation (entry at T0i ) : l (θ) = log
L(θ)
T0
QN
i=1
Cécile Proust-Lima (INSERM)
Joint models for multiple outcomes
!
Si (T0i ; θ)
July 2015
27 / 45
Implementation, model selection and evaluation
Iterative (Marquardt) algorithm for a given G
I
I
I
implemented in HETMIXSURV V2 parallel Fortran90 program
validated in simulation studies (with for instance splines and threshold link
functions)
implemented in Jointlcmm (R) for 1 marker
Posterior selection of the optimal number G of latent classes
I
I
Information measures : AIC, BIC
Score Test for conditional independence assumption :
→ longitudinal and survival parts are independent
conditionally on the latent classes
Further evaluation of the model using :
I
I
Posterior classification
stemmed from P(ci = g|Xi , Yi , (Ti , δi ); θ̂)
Longitudinal/Survival predictions versus observations
→ posterior-probability-weighted means over time intervals
Cécile Proust-Lima (INSERM)
Joint models for multiple outcomes
July 2015
28 / 45
Conditional independence assumption :
Class-specific cause-specific proportional hazard model
λp (t)
>
= λ0p (t; νpg ) exp(XTi
ζpg )
latent
classes
c
random
effects
u
latent
process
( Λ (t))
...
outcome
Y1 at t j1
Cécile Proust-Lima (INSERM)
...
outcome
Yk at t jk
outcome
YK at t jK
Joint models for multiple outcomes
multiple cause
time to event
(Τ, δ )
July 2015
29 / 45
Conditional independence assumption : alternative
Class-specific cause-specific proportional hazard model
>
ζpg + uig κp )
λp (t)H1 = λ0p (t; νpg ) exp(XTi
→
> >
Score test for H0 : κp = 0 and H0 : κ = (κ>
1 , ..., κP ) = 0
latent
classes
c
random
effects
u
latent
process
( Λ (t))
...
outcome
Y1 at t j1
Cécile Proust-Lima (INSERM)
...
outcome
Yk at t jk
outcome
YK at t jK
Joint models for multiple outcomes
multiple cause
time to event
(Τ, δ )
July 2015
29 / 45
Application : Semantic memory decline in the elderly
Clinical background
I
I
semantic memory (verbal fluency, ...) affected long before dementia
diagnosis
could play a role for early prediction of dementia
Objective
I
I
describe profiles of semantic memory decline in association with dementia
and death
predict the risk of dementia from semantic memory history
PAQUID cohort data :
I
I
I
I
→
population-based cohort on cerebral aging
65 years and older
22 years of follow-up every 2 or 3 years
subpopulation with genetic information : ApoE4
588 subjects
Cécile Proust-Lima (INSERM)
Joint models for multiple outcomes
July 2015
30 / 45
Dynamics of semantic memory
2 longitudinal measures :
I
I
Isaacs Set Test (IST15) (discrete quantitative in {0-40})
Wechsler similarities test (WST) (ordinal in {0-10})
Trajectory according to age (natural history)
I
age at entry (ageT0), sex (sex), education (EL), apoE4 (E4)
In each latent class g :
Λ(age)|ci =g = b0ig + β1 sex + β2 EL + β3 E4 + β4 ageT0+
(b1ig + β5 sex + β6 EL + β7 E4)×age
(b2ig + β8 sex + β9 EL + β10 E4)×age2 + wi (age)
IST15ij = H1 ( Λ(age1ij ) + α1i + 1ij ; η1 )
WSTij = H2 ( Λ(age2ij ) + α2i + 2ij ; η2 )
with big ∼ N (µg , B), wi ∼ Brownian motion
2
α2i ∼ N (0, σα
), ki ∼ N (0, σk 2 ), k = 1, 2
Cécile Proust-Lima (INSERM)
Joint models for multiple outcomes
July 2015
31 / 45
Preliminary analysis : link functions H1 and H2
In separated analyses, Splines with 3 internal knots =
0
−10
−5
0
5
IST underlying latent process
Cécile Proust-Lima (INSERM)
Wechsler Similarities Test (WST)
2
4
6
8
10
linear
AICd=13866.2
splines (23,27,31)
AICd=13842.4
thresholds
AICd=13857.7
linear
AICd=10738.3
splines(6,8,9)
AICd=10448.3
thresholds
AICd=10409.4
0
Isaacs Set Test (IST)
10
20
30
40
good balance between estimation difficulty and goodness-of-fit
−4
−2
0
2
WST underlying latent process
Joint models for multiple outcomes
July 2015
32 / 45
Risk of dementia in presence of death
2-cause censored time-to-event with delayed entry :
age at entry in the cohort
age at dementia (in between a negative and positive diagnosis)
age at death in the two years after a negative dementia diagnosis
In each latent class g :
λp (t)|ci =g = λ0pg (t; )eζ1p sex+ζ2p EL+ζ3p E4 ,
p = 1, 2
λ0pg (t; ) parametric hazards among Gompertz, Weibull, M-splines (5 knots)
and piecewise constant (5 knots)
In separated analyses, Weibull hazards =
good balance between estimation difficulty and goodness-of-fit
Cécile Proust-Lima (INSERM)
Joint models for multiple outcomes
July 2015
33 / 45
5% significance level
Score Test Statistic
20 40 60 80
27160
120
Selection of the number of classes in the joint model
Global score test
0
27080
BIC
27120
BIC
2
3
4
number of latent classes
5
1
2
3
4
number of latent classes
5
50
120
1
5% significance level
Score Test Statistic
10 20 30 40
Score Test Statistic
20 40 60 80
5% significance level
Death without dementia score test
0
0
Dementia score test
1
2
3
4
number of latent classes
Cécile Proust-Lima (INSERM)
5
1
Joint models for multiple outcomes
2
3
4
number of latent classes
5
July 2015
34 / 45
0
0
Isaacs Set Test
10
20
30
40
Wechsler Similarities Test
2
4
6
8
10
4 profiles of semantic decline, dementia & death
75
80
85
age in years
90
95
65
70
75
80
85
age in years
90
95
90
95
Class 1 (12.1%)
Class 2 (52.2%)
Class 3 (11.2%)
Class 4 (24.5%)
0.0
0.0
cumulative incidence
of free−dementia death
0.2 0.4 0.6 0.8 1.0
70
cumulative incidence
of dementia
0.2 0.4 0.6 0.8 1.0
65
65
70
75
80
85
age in years
Cécile Proust-Lima (INSERM)
90
95
65
70
Joint models for multiple outcomes
75
80
85
age in years
July 2015
35 / 45
Isaacs Set Test
10
20
30
40
Wechsler Similarities Test
2
4
6
8
10
Weighted predictions versus weighted observations
0
0
mean observation
mean subject−specific prediction
70
75
80
85
age in years
90
95
65
cumulative incidence
of free−dementia death
0.2 0.4 0.6 0.8 1.0
cumulative incidence
of dementia
0.2 0.4 0.6 0.8 1.0
65
75
80
85
age in years
90
95
90
95
Class 1 (12.1%)
Class 2 (52.2%)
Class 3 (11.2%)
Class 4 (24.5%)
0.0
0.0
non−parametric estimate
95% bands
prediction
70
65
70
75
80
85
age in years
Cécile Proust-Lima (INSERM)
90
95
65
70
Joint models for multiple outcomes
75
80
85
age in years
July 2015
36 / 45
Individual dynamic prediction : the principle
From a landmark age s
I
I
I
(s)
history of the markers Yi = {Ykij such as tkij ≤ s}
(s)
history of the covariates Xi = {XL1i (tkij ), Zi (tkij ), XL2i (tkij ) such as tkij ≤ s}
other time-independent covariates Xi
Cumulative incidence for cause p at an horizon of t years
(s)
(s)
Ppi (s, t) = P(Ti ≤ s + t, δi = p|Ti > s, Yi , Xi , XTi , Xci ; θ)
I
I
(1)
computed using the Bayes theorem
and the conditional independence assumption
with class-specific and cause-specific cumulative incidence
approximated by Gauss-Legendre
Monte-Carlo approximation of the posterior distribution of Ppi (s, t)
Cécile Proust-Lima (INSERM)
Joint models for multiple outcomes
July 2015
37 / 45
40
Example of individual dynamic prediction
0
cognitive measures
10
20
30
IST
WST
65
Cécile Proust-Lima (INSERM)
70
75
80
Joint models for multiple outcomes
85
90
July 2015
38 / 45
0
0
cognitive measures
10
20
30
IST
WST
dementia
death
0.2
0.4
0.6
0.8
probabilities of event
1
40
Example of individual dynamic prediction
65
70
75
5-year probability of dementia (%) :
5-year probability of death (%) :
Cécile Proust-Lima (INSERM)
80
Age in years
85
90
at 80 years old
13.0 [7.7,21.0]
25.1 [18.1,36.5]
Joint models for multiple outcomes
July 2015
38 / 45
0
0
cognitive measures
10
20
30
IST
WST
dementia
death
0.2
0.4
0.6
0.8
probabilities of event
1
40
Example of individual dynamic prediction
65
70
75
5-year probability of dementia (%) :
5-year probability of death (%) :
Cécile Proust-Lima (INSERM)
80
Age in years
85
at 80 years old
13.0 [7.7,21.0]
25.1 [18.1,36.5]
Joint models for multiple outcomes
90
at 85 years old
16.4 [9.1,29.4]
36.0 [25.9,46.3]
July 2015
38 / 45
Option2 : multistate model with interval censoring
(Rouanet et al., ArXiv 2015)
Transition intensity from state k to state
l for subject i in class g :
latent
classes
c
0
αklig (t) = αklg
(t) eXTi
random
effects
u
γklg class-specific regression
parameters
healthy
...
outcome
Yk at t jk
Cécile Proust-Lima (INSERM)
γklg
0
αklg
class-specific baseline intensity
latent
process
( Λ (t))
outcome
Y1 at t j1
>
ill
...
outcome
YK at t jK
dead
Joint models for multiple outcomes
July 2015
39 / 45
Likelihood function using Yi
L(θ) =
N X
G
Y
|=
Maximum Likelihood Estimation
ci
Ti ,
fY (Yi |ci = g; θ) fD (Di , δi |ci = g; θ) P(ci = g|Xi ; θ)
i=1 g=1
I
P(ci = g|Xi ; θ) from a multinomial logistic model
I
fY (Yi |ci = g; θ) from the latent process mixed model
I
fD (Di , δi |ci = g; θ) from the multistate model with interval censoring
A
D
D>
i = (T0 i , Li , Ri , δi , Ti , δi )
A
with Ri = +∞ if δi = 0.
= e−A01g (T)−A02g (T) α02g (T) +
Z
>
e−A01g (u)−A02g (u) α01g (u) e−(A12g (T)−A12g (u)) α12g (T)du
Li
Cécile Proust-Lima (INSERM)
Joint models for multiple outcomes
July 2015
40 / 45
Application
From PAQUID study (N=3777)
I
I
Change over time of Isaacs set test (verbal fluency)
in association with dementia and death
Mixed model :
Λ(t)|ci =g = b0ig + β1g sex + β2g EL+
(b1ig + β3g EL)×t
(b2ig + β4g EL)×t2
Yij = Λi (tij ) + ij
with
big ∼ N (µg , B)&i ∼ N (0, σ 2 )
Proportional transition intensities of the illness-death model :
0
αklig (t) = αklg
(t) eγkls
Cécile Proust-Lima (INSERM)
sexi +γkle ELi
Joint models for multiple outcomes
July 2015
41 / 45
Four-class Markovian model : poorly educated men
Class 1
Class 2
Class 3
Class 4
Cécile Proust-Lima (INSERM)
N
7.3%
8.3%
34.2%
50.5%
EL=0
42.4
25.0
29.3
37.6
Joint models for multiple outcomes
EL=1
57.6
75.0
70.7
62.4
men
42.8
43.1
39.4
43.9
women
57.2
56.9
60.6
56.1
July 2015
42 / 45
Goodness-of-fit assessment
Class-specific weighted predicted
trajectories vs. observed
predicted class-specific
cumulative incidences vs.
semi-parametric estimator
Cécile Proust-Lima (INSERM)
Joint models for multiple outcomes
July 2015
43 / 45
Concluding remarks
Joint model methodology
I
I
→
extended to multivariate longitudinal markers
extended to multistate process for events
useful for different purposes in chronic diseases
Different assumptions for the shared quantity
I
I
depends on the data
depends on the objective
Parametric assumptions
I
I
I
flexible and/or selected distributions according to the data
progressive construction of the models, goodness-of-fit (graphs, measures,
tests) :
... score tests for conditional independence assumptions (between events
and marker, between events)
Cécile Proust-Lima (INSERM)
Joint models for multiple outcomes
July 2015
44 / 45
Funding and references
Fundings : INCa/Inserm MULTIPLE, Inserm/Region PhD grant
Further details in :
Ferrer Loı̈c et al. (2015). arXiv :1506.07496 [stat].
Rouanet Anaı̈s et al. (2015). arXiv :1506.07415 [stat].
Proust-Lima Cécile et al. (2014). arXiv :1409.7598 [stat].
Proust-Lima Cécile et al. (2014). Statistical Methods in Medical Research, 23(1), 74-90.
Sène Mbéry et al. (2014). Journal de la Société Française de Statistique, 155(1), 134-155. (in english)
Sène Mbéry et al. (2014). Statistical Methods in Medical Research. (online)
Cécile Proust-Lima (INSERM)
Joint models for multiple outcomes
July 2015
45 / 45