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Bayesian Semiparametric Multi-State Models
Thomas Kneib & Andrea Hennerfeind
Department of StatisticsLudwig-Maximilians-University Munich
14.10.2006
Thomas Kneib Multi-State Models
Multi-State Models
• Multi-state models form a general class for the description of the evolution of discretephenomena in continuous time.
• We observe paths of a process
X = {X(t), t ≥ 0} with X(t) ∈ {1, . . . , q}.
• Yields a similar data structure as for Markov processes.
• Examples:
– Recurrent events:
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Bayesian Semiparametric Multi-State Models 1
Thomas Kneib Multi-State Models
– Disease progression:
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1 2 3 q-1· · ·
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– Competing risks:
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• (Homogenous) Markov processes can be compactly described in terms of the transitionintensities
λij = lim∆t→0
P (X(t + ∆t) = j|X(t) = i)∆t
Bayesian Semiparametric Multi-State Models 2
Thomas Kneib Multi-State Models
• Often not flexible enough in practice since
– The transition intensities might vary over time.
– The transition intensities might be related to covariates.
– The Markov model implies independent and exponentially distributed waiting times.
Bayesian Semiparametric Multi-State Models 3
Thomas Kneib Human Sleep Data
Human Sleep Data
• Human sleep can be considered an example of a recurrent event type multi-statemodel.
• State Space:
Awake Phases of wakefulnessREM Rapid eye movement phase (dream phase)
Non-REM Non-REM phases (may be further differentiated)
• Aims of sleep research:
– Describe the dynamics underlying the human sleep process.
– Analyse associations between the sleep process and nocturnal hormonal secretion.
– (Compare the sleep process of healthy and diseased persons.)
Bayesian Semiparametric Multi-State Models 4
Thomas Kneib Human Sleep Data
stat
e
awak
eN
on−
RE
MR
EM
time since sleep onset (in hours)
cort
isol
0 2 4 6 8
050
100
150
Bayesian Semiparametric Multi-State Models 5
Thomas Kneib Human Sleep Data
• Data generation:
– Sleep recording based on electroencephalographic (EEG) measures every 30 seconds(afterwards classified into the three sleep stages).
– Measurement of hormonal secretion based on blood samples taken every 10minutes.
– A training night familiarises the participants of the study with the experimentalenvironment.
⇒ Sleep processes of 70 participants.
• Simple parametric approaches are not appropriate in this application due to
– Changing dynamics of human sleep over night.
– The time-varying influence of the hormonal concentration on the transitionintensities.
– Unobserved heterogeneity.
⇒ Model transition intensities nonparametrically.
Bayesian Semiparametric Multi-State Models 6
Thomas Kneib Specification of Transition Intensities
Specification of Transition Intensities
• To reduce complexity, we consider a simplified transition space:
Awake
Non-REM REM
Sleep
?
6
-
¾
λRN(t)
λNR(t)
λAS(t) λSA(t)
Bayesian Semiparametric Multi-State Models 7
Thomas Kneib Specification of Transition Intensities
• Model specification:
λAS,i(t) = exp[γ
(AS)0 (t) + b
(AS)i
]
λSA,i(t) = exp[γ
(SA)0 (t) + b
(SA)i
]
λNR,i(t) = exp[γ
(NR)0 (t) + ci(t)γ
(NR)1 (t) + b
(NR)i
]
λRN,i(t) = exp[γ
(RN)0 (t) + ci(t)γ
(RN)1 (t) + b
(RN)i
]
where
ci(t) =
{1 cortisol > 60 n mol/l at time t
0 cortisol ≤ 60 n mol/l at time t,
b(j)i ∼ N(0, τ2
j ) = transition- and individual-specific frailty terms.
Bayesian Semiparametric Multi-State Models 8
Thomas Kneib Specification of Transition Intensities
• Penalised splines for the baselines and time-varying effects:
– Approximate γ(t) by a weighted sum of B-spline basis functions
γ(t) =∑
j ξjBj(t).
– Employ a large number of basis functions to enable flexibility.
– Penalise k-th order differences between parameters of adjacent basis functions toensure smoothness:
Pen(ξ|τ2) =1
2τ2
∑j(∆kξj)2.
– Bayesian interpretation: Assume a k-th order random walk prior for ξj, e.g.
ξj = 2ξj−1 − ξj−2 + uj, uj ∼ N(0, τ2) (RW2).
– This yields the prior distribution:
p(ξ|τ2) ∝ exp(− 1
2τ2ξ′Kξ
).
Bayesian Semiparametric Multi-State Models 9
Thomas Kneib Counting Process Representation
Counting Process Representation
• A multi-state model with k different types of transitions can be equivalently expressedin terms of k counting processes Nh(t), h = 1, . . . , k counting these transitions.
awak
eN
on−
RE
MR
EM
700 720 740 760 780 800
awake => sleep
700 720 740 760 780 800
2122
2324
Non−REM => REM
700 720 740 760 780 800
67
89
sleep => awake
700 720 740 760 780 800
2223
2425
REM => Non−REM
700 720 740 760 780 800
45
67
Bayesian Semiparametric Multi-State Models 10
Thomas Kneib Counting Process Representation
• From the counting process representation we can derive the likelihood contributionsfor individual i:
li =k∑
h=1
[∫ Ti
0
log(λhi(t))dNhi(t)−∫ Ti
0
λhi(t)Yhi(t)dt
]
=ni∑
j=1
k∑
h=1
[δhi(tij) log(λhi(tij))− Yhi(tij)
∫ tij
ti,j−1
λhi(t)dt
].
k number of possible transitions.
Nhi(t) counting process for type h event and individual i.
Yhi(t) at risk indicator for type h event and individual i.
tij event times of individual i.
ni number of events for individual i.
δhi(tij) transition indicator for type h transition.
Bayesian Semiparametric Multi-State Models 11
Thomas Kneib Counting Process Representation
• The counting process representation also provides a possibility for model validationbased on martingale residuals.
• Every counting process is a submartingale and can therefore (Doob-Meyer-)decomposed as
Nhi(t) = Ahi(t) + Mhi(t)
=∫ t
0
λhi(t)Yhi(t)du + Mhi(t),
where Mhi(t) is a martingale and Ahi(t) is the (predictable) compensator process ofNhi(t).
• The martingales Mhi(t) can be interpreted as continuous-time residuals.
• Plots of Mhi(t) against t can be used to compare models, evaluate the model fit, etc.
Bayesian Semiparametric Multi-State Models 12
Thomas Kneib Bayesian Inference
Bayesian Inference
• In principle, a multi-state model consists of several duration time models
⇒ Adopt methodology developed for nonparametric hazard regression.
• Fully Bayesian inference based on Markov Chain Monte Carlo simulation techniques(Hennerfeind, Brezger & Fahrmeir, 2006):
– Assign inverse gamma priors to the variance and smoothing parameters.
– Metropolis-Hastings update for the regression coefficients (based on IWLS-proposals).
– Gibbs sampler for the variances (inverse gamma with updated parameters).
– Efficient algorithms make use of the sparse matrix structure of the matricesinvolved.
Bayesian Semiparametric Multi-State Models 13
Thomas Kneib Bayesian Inference
• Mixed model based empirical Bayes inference (Kneib & Fahrmeir, 2006):
– Consider the variances and smoothing parameters as unknown constants to beestimated by mixed model methodology.
– Problem: The P-spline priors are partially improper.
– Mixed model representation: Decompose the vector of regression coefficients as
ξ = Xβ + Zb,
wherep(β) ∝ const and b ∼ N(0, τ2I).
⇒ β is a fixed effect and b is an i.i.d. random effect.
– Penalised likelihood estimation of the regression coefficients in the mixed model(posterior modes).
– Marginal likelihood estimation of the variance and smoothing parameters (Laplaceapproximation).
Bayesian Semiparametric Multi-State Models 14
Thomas Kneib Software
Software
• Implemented in BayesX.
• Public domain software package for Bayesian inference in geoadditive and relatedmodels.
• Available from
http://www.stat.uni-muenchen.de/~bayesx
Bayesian Semiparametric Multi-State Models 15
Thomas Kneib Human Sleep Data II
Human Sleep Data II
• Baseline effects I:
0 2 4 6 8
−2.
5−
2.0
−1.
5−
1.0
awake −> sleep (mixed model)
0 2 4 6 8
−2.
5−
2.0
−1.
5−
1.0
awake −> sleep (MCMC)
0 2 4 6 8
−4.
5−
4.0
−3.
5−
3.0
−2.
5−
2.0
−1.
5
sleep −> awake (mixed model)
0 2 4 6 8
−4.
5−
4.0
−3.
5−
3.0
−2.
5−
2.0
−1.
5
sleep −> awake (MCMC)
Bayesian Semiparametric Multi-State Models 16
Thomas Kneib Human Sleep Data II
• Baseline effects II:
0 2 4 6 8
−14
−12
−10
−8
−6
−4
−2
Non−REM −> REM (mixed model)
0 2 4 6 8
−14
−12
−10
−8
−6
−4
−2
Non−REM −> REM (MCMC)
0 2 4 6 8
−4.
5−
4.0
−3.
5−
3.0
−2.
5
REM −> Non−REM (mixed model)
0 2 4 6 8
−4.
5−
4.0
−3.
5−
3.0
−2.
5
REM −> Non−REM (MCMC)
Bayesian Semiparametric Multi-State Models 17
Thomas Kneib Human Sleep Data II
• Time-varying effects for a high level of cortisol:
0 2 4 6 8
−4
−3
−2
−1
01
2
Non−REM −> REM (mixed model)
0 2 4 6 8
−4
−3
−2
−1
01
2
Non−REM −> REM (MCMC)
0 2 4 6 8
−2
−1
01
REM −> Non−REM (mixed model)
0 2 4 6 8
−2
−1
01
REM −> Non−REM (MCMC)
Bayesian Semiparametric Multi-State Models 18
Thomas Kneib Human Sleep Data II
• The fully Bayesian approach detects individual-specific variation for all transitions.
• The empirical Bayes approach only detects individual-specific variation for thetransition between REM and Non-REM.
Bayesian Semiparametric Multi-State Models 19
Thomas Kneib Human Sleep Data II
0 200 400 600 800
−5
05
10
Martingale residuals REM => Non−REMM
arko
v M
odel
0 200 400 600 800
−5
05
10
Mix
ed M
odel
0 200 400 600 800
−5
05
10
MC
MC
Bayesian Semiparametric Multi-State Models 20
Thomas Kneib Human Sleep Data II
0 200 400 600 800
05
1015
2025
3035
Standardised martingale residuals REM => Non−REMM
arko
v M
odel
0 200 400 600 800
05
1015
2025
3035
Mix
ed M
odel
0 200 400 600 800
05
1015
2025
3035
MC
MC
Bayesian Semiparametric Multi-State Models 21
Thomas Kneib Human Sleep Data II
Mar
kov
Mix
ed M
odel
MC
MC
−15 −10 −5 0 5 10
0.0
0.2
0.4
awake => sleep
t = 100−20 −10 0 10
0.00
0.10
awake => sleep
t = 300−20 −10 0 10 20
0.00
0.06
awake => sleep
t = 500−40 −30 −20 −10 0 10 20 30
0.00
0.04
0.08
awake => sleep
t = 700
−5 0 5 10
0.00
0.10
sleep => awake
t = 100−10 0 10 20 30
0.00
0.10
sleep => awake
t = 300−20 −10 0 10 20 30
0.00
0.06
0.12
sleep => awake
t = 500−20 −10 0 10 20 30 40
0.00
0.10
sleep => awake
t = 700
−4 −2 0 2 4 6
0.0
0.2
0.4
Non−REM => REM
t = 200−5 0 5
0.00
0.10
0.20
Non−REM => REM
t = 400−10 −5 0 5 10
0.00
0.10
0.20
Non−REM => REM
t = 600−10 −5 0 5 10 15 20
0.00
0.10
0.20
Non−REM => REM
t = 800
−2 0 2 4
0.0
0.2
0.4
0.6
REM => Non−REM
t = 200−4 −2 0 2 4 6
0.0
0.2
REM => Non−REM
t = 400−5 0 5
0.00
0.10
0.20
REM => Non−REM
t = 600−5 0 5 10
0.00
0.10
0.20
REM => Non−REM
t = 800
Bayesian Semiparametric Multi-State Models 22
Thomas Kneib Human Sleep Data II
Mar
kov
Mix
ed M
odel
MC
MC
−4 −2 0 2 4 6 8
0.0
0.2
0.4
awake => sleep
t = 100−2 0 2 4 6 8
0.0
0.2
0.4
0.6
awake => sleep
t = 300−2 0 2 4 6
0.0
0.4
0.8
awake => sleep
t = 500−2 0 2 4 6
0.0
0.6
1.2
awake => sleep
t = 700
0 5 10 15
0.0
0.2
0.4
sleep => awake
t = 1000 2 4 6 8
0.0
0.4
0.8
sleep => awake
t = 300−1 0 1 2 3 4 5
0.0
0.4
0.8
1.2
sleep => awake
t = 500−1 0 1 2 3 4
0.0
1.0
2.0
sleep => awake
t = 700
0 5 10 15
0.0
0.4
0.8
1.2
Non−REM => REM
t = 200−2 −1 0 1 2 3 4
0.0
0.4
0.8
1.2
Non−REM => REM
t = 400−1 0 1 2
0.0
0.4
0.8
1.2
Non−REM => REM
t = 600−1 0 1 2
0.0
0.5
1.0
1.5
Non−REM => REM
t = 800
0 5 10
0.00
0.15
0.30
REM => Non−REM
t = 2000 5 10
0.0
0.2
0.4
REM => Non−REM
t = 400−2 0 2 4 6
0.0
0.4
0.8
REM => Non−REM
t = 600−1 0 1 2
0.0
0.4
0.8
REM => Non−REM
t = 800
Bayesian Semiparametric Multi-State Models 23
Thomas Kneib Things to remember. . .
Things to remember. . .
• Computationally feasible semiparametric approach for the analysis of multi-statemodels.
• Fully Bayesian and empirical Bayes inference.
• Model validation based on martingale residuals.
• Directly extendable to more complicated models including
– Nonparametric effects of continuous covariates.
– Spatial effects.
– Interaction surfaces and varying coefficients.
• Future work:
– Application to larger data sets and different types of multi-state models.
– Consider coarsened observations, i.e. interval censored multi-state data.
Bayesian Semiparametric Multi-State Models 24
Thomas Kneib References
References
• Brezger, Kneib & Lang (2005): BayesX: Analyzing Bayesian structured additiveregression models. Journal of Statistical Software, 14 (11).
• Hennerfeind, Brezger, and Fahrmeir (2006): Geoadditive survival models.Journal of the American Statistical Association, 101, 1065-1075.
• Kneib & Fahrmeir (2006): A mixed model approach for geoadditive hazardregression. Scandinavian Journal of Statistics, to appear.
• A place called home:
http://www.stat.uni-muenchen.de/~kneib
Bayesian Semiparametric Multi-State Models 25
