bayesian structured hazard regression · { only a small number of covariates is available...
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Bayesian Structured Hazard Regression
Andrea Hennerfeind, Ludwig Fahrmeir & Thomas KneibDepartment of Statistics
Ludwig-Maximilians-University Munich
29.05.2006
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Thomas Kneib Outline
Outline
• Leukemia survival data.
• Structured hazard regression for survival times.
• Bayesian inference in structured hazard regression.
– Full Bayesian inference based on MCMC.
– Empirical Bayes inference using mixed model methodology.
• Multi-state models for the analysis of human sleep.
Bayesian Structured Hazard Regression 1
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Thomas Kneib Leukemia Survival Data
Leukemia Survival Data
• Survival time of adults after diagnosis of acute myeloid leukemia.
• 1,043 cases diagnosed between 1982 and 1998 in Northwest England.
• 16 % (right) censored.
• Continuous and categorical covariates:
age age at diagnosis,wbc white blood cell count at diagnosis,sex sex of the patient,tpi Townsend deprivation index.
• Spatial information in different resolution.
Bayesian Structured Hazard Regression 2
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Thomas Kneib Leukemia Survival Data
• Classical Cox proportional hazards model:
λ(t; x) = λ0(t) exp(x′γ).
• Baseline hazard λ0(t) is a nuisance parameter and remains unspecified.
• Estimate γ based on the partial likelihood.
• Questions / Limitations:
– Simultaneous estimation of baseline hazard rate and covariate effects.
– Flexible modelling of covariate effects (e.g. nonlinear effects, interactions).
– Spatially correlated survival times.
– Non-proportional hazards models / time-varying effects.
⇒ Structured hazard regression models.
Bayesian Structured Hazard Regression 3
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Thomas Kneib Leukemia Survival Data
• Replace usual parametric predictor with a flexible semiparametric predictor
λ(t; ·) = λ0(t) exp[f1(age) + f2(wbc) + f3(tpi) + fspat(si) + β1sex]
and absorb the baseline
λ(t; ·) = exp[g0(t) + f1(age) + f2(wbc) + f3(tpi) + fspat(si) + β1sex]
where
– g0(t) = log(λ0(t)) is the log-baseline hazard,
– f1, f2, f3 are nonparametric functions of age, white blood cell count anddeprivation, and
– fspat is a spatial function.
Bayesian Structured Hazard Regression 4
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Thomas Kneib Leukemia Survival Data
−5
−2
14
0 7 14time in years
(a) log(baseline) Log-baseline hazard.
Effect of age at diagnosis.
−1.5
−.7
50
.75
1.5
14 27 40 53 66 79 92age in years
(b) effect of age
Bayesian Structured Hazard Regression 5
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Thomas Kneib Leukemia Survival Data
−.5
0.5
11.5
2
0 125 250 375 500white blood cell count
(c) effect of white blood cell count Effect of white blood cell count.
Effect of deprivation.
−.5
−.2
50
.25
.5
−6 −2 2 6 10townsend deprivation index
(d) effect of townsend deprivation index
Bayesian Structured Hazard Regression 6
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Thomas Kneib Leukemia Survival Data
-0.44 0 0.3
District-level analysis−0.44 0.3
Individual-level analysis
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Thomas Kneib Structured Hazard Regression
Structured Hazard Regression
• A general structured hazard regression model consists of an arbitrary combination ofthe following model terms:
– Log baseline hazard g0(t) = log(λ0(t)).
– Time-varying effects gl(t)ul of covariates ul.
– Nonparametric effects fj(xj) of continuous covariates xj.
– Spatial effects fspat(s) of a spatial location variable s.
– Interaction surfaces fj,k(xj, xk) of two continuous covariates.
– Varying coefficient interactions ujfk(xk) or ujfspat(s).
– Frailty terms bg (random intercept) or xjbg (random slopes).
Bayesian Structured Hazard Regression 8
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Thomas Kneib Structured Hazard Regression
• Penalised splines for the baseline effect, time-varying effects, and nonparametriceffects:
– Approximate f(x) (or g(t)) by a weighted sum of B-spline basis functions
f(x) =∑
ξjBj(x).
– Employ a large number of basis functions to enable flexibility.
– Penalise differences between parameters of adjacent basis functions to ensuresmoothness:
Pen(ξ|τ2) =1
2τ2
∑(∆kξj)2.
– Bayesian interpretation: Assume a k-th order random walk prior for ξj, e.g.
ξj = ξj−1 + uj, uj ∼ N(0, τ2) (RW1).
ξj = 2ξj−1 − ξj−2 + uj, uj ∼ N(0, τ2) (RW2).
Bayesian Structured Hazard Regression 9
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Thomas Kneib Structured Hazard Regression
−2
−1
01
2
−3 −1.5 0 1.5 3
−2
−1
01
2
−3 −1.5 0 1.5 3
−2
−1
01
2
−3 −1.5 0 1.5 3
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Thomas Kneib Structured Hazard Regression
• Bivariate Tensor product P-splines for interaction surfaces:
– Define bivariate basis functions (Tensor products of univariate basis functions).
– Extend random walks on the line to random walks on a regular grid.
f f f f f
f f f f f
f f f f f
f f f f f
f f f f f
v v
v
v
Bayesian Structured Hazard Regression 11
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Thomas Kneib Structured Hazard Regression
• Spatial effects for regional data s ∈ {1, . . . , S}: Markov random fields.
– Bivariate extension of a first order random walk on the real line.
– Define appropriate neighbourhoods for the regions.
– Assume that the expected value of fspat(s) = ξs is the average of the functionevaluations of adjacent sites:
ξs|ξs′, s′ 6= s, τ2 ∼ N
1
Ns
∑
s′∈∂s
ξs′,τ2
Ns
.
τ2
2
t−1 t t+1
f(t−1)
E[f(t)|f(t−1),f(t+1)]
f(t+1)
Bayesian Structured Hazard Regression 12
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Thomas Kneib Structured Hazard Regression
• Spatial effects for point-referenced data: Stationary Gaussian random fields.
– Well-known as Kriging in the geostatistics literature.
– Spatial effect follows a zero mean stationary Gaussian stochastic process.
– Correlation of two arbitrary sites is defined by an intrinsic correlation function.
– Can be interpreted as a basis function approach with radial basis functions.
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Thomas Kneib Structured Hazard Regression
• Cluster-specific frailty terms:
– Account for unobserved heterogeneity.
– Easiest case: i.i.d Gaussian frailty.
• All covariates in the discussed model terms are allowed to be piecewise constanttime-varying.
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Thomas Kneib Bayesian Inference
Bayesian Inference
• Generic representation of structured hazard regression models:
λ(t) = exp [x(t)′γ + f1(z1(t)) + . . . + fp(zp(t))]
• For example:
f(z(t)) = g(t) z(t) = t log-baseline effect,
f(z(t)) = u(t)g(t) z(t) = (u, t) time-varying effect of u(t),f(z(t)) = f(x(t)) z(t) = x(t) smooth function of a continuous
covariate x(t),f(z(t)) = fspat(s) z(t) = s spatial effect,
f(z(t)) = f(x1(t), x2(t)) z(t) = (x1(t), x2(t)) interaction surface,
f(z(t)) = bg z(t) = g i.i.d. frailty bg, g is a groupingindex.
• The generic representation facilitates description of inferential details.
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Thomas Kneib Bayesian Inference
• All vectors of function evaluations fj can be expressed as
fj = Zjξj
with design matrix Zj, constructed from zj(t), and regression coefficients ξj.
• Generic form of the prior for ξj:
p(ξj|τ2j ) ∝ (τ2
j )−kj2 exp
(− 1
2τ2j
ξ′jKjξj
)
• Kj ≥ 0 acts as a penalty matrix, rank(Kj) = kj ≤ dj = dim(ξj).
• τ2j ≥ 0 can be interpreted as a variance or (inverse) smoothness parameter.
• Relation to penalized likelihood: Penalty terms
Pλj(ξj) = log[p(ξj|τ2
j )] = −12λjξ
′jKjξj, λj =
1τ2j
.
Bayesian Structured Hazard Regression 16
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Thomas Kneib Bayesian Inference
• Likelihood for right censored survival times under the assumption of noninformativecensoring:
n∏
i=1
λi(Ti)δi exp
(−
∫ Ti
0
λi(t)dt
),
where δi is the censoring indicator.
• In general, numerical integration has to be used to evaluate the cumulative hazardrate (e.g. the trapezoidal rule).
Bayesian Structured Hazard Regression 17
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Thomas Kneib Fully Bayesian inference based on MCMC
Fully Bayesian inference based on MCMC
• Assign inverse gamma prior to τ2j :
p(τ2j ) ∝ 1
(τ2j )aj+1 exp
(− bj
τ2j
).
Proper for aj > 0, bj > 0 Common choice: aj = bj = ε small.
Improper for bj = 0, aj = −1 Flat prior for variance τ2j ,
bj = 0, aj = −12 Flat prior for standard deviation τj.
• Conditions for proper posteriors in structured hazard regression: Enough uncensoredobservations and either
– proper priors for the variances or
– aj < bj = 0 and rank deficiency in the prior for ξj not too large.
Bayesian Structured Hazard Regression 18
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Thomas Kneib Fully Bayesian inference based on MCMC
• MCMC sampling scheme:
– Metropolis-Hastings update for ξj|·:Propose new state from a multivariate Gaussian distribution with precision matrixand mean
Pj = Z ′jWZj +1τ2j
Kj and mj = P−1j Z ′jW (y − η−j).
IWLS-Proposal with appropriately defined working weights W and workingobservations y.
– Gibbs sampler for τ2j |·:
Sample from an inverse Gamma distribution with parameters
a′j = aj +12rank(Kj) and b′j = bj +
12ξ′jKjξj.
• Efficient algorithms make use of the sparse matrix structure of Pj and Kj.
Bayesian Structured Hazard Regression 19
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Thomas Kneib Empirical Bayes inference based on mixed model methodology
Empirical Bayes inference based on mixed model methodology
• Consider the variances τ2j as unknown constants to be estimated.
• Idea: Consider ξj a correlated random effect with multivariate Gaussian distributionand use mixed model methodology.
• Problem: In most cases partially improper random effects distribution.
• Mixed model representation: Decompose
ξj = Xjβj + Zjbj,
wherep(βj) ∝ const and bj ∼ N(0, τ2
j Ikj).
⇒ βj is a fixed effect and bj is an i.i.d. random effect.
Bayesian Structured Hazard Regression 20
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Thomas Kneib Empirical Bayes inference based on mixed model methodology
• This yields the variance components model
λ(t; ·) = exp [x′β + z′b] ,
where in turnp(β) ∝ const and b ∼ N(0, Q).
• Obtain empirical Bayes estimates / penalized likelihood estimates via iterating
– Penalized maximum likelihood for the regression coefficients β and b.
– Restricted Maximum / Marginal likelihood for the variance parameters in Q:
L(Q) =∫
L(β, b,Q)p(b)dβdb → maxQ
.
• Involves Laplace approximation to the marginal likelihood (similar as in the previoustalk by Havard Rue).
• Corresponds to REML estimation of variances in Gaussian mixed models.
Bayesian Structured Hazard Regression 21
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Thomas Kneib Human Sleep Data
Human Sleep Data
• Consider individual human sleep data as independent realisations of time-continuousstochastic processes with discrete state space {awake, REM, non-REM}.
• Compact description of such a process in terms of transition intensities between thesestates.
• Simple approaches: Markov or Semi-Markov processes.
• Limitations / Questions:
– Changing dynamics of human sleep over night.
– Individual sleeping habits to be described by covariates.
– Only a small number of covariates is available (unobserved heterogeneity).
⇒ Model the transition intensities in analogy to survival models.
Bayesian Structured Hazard Regression 22
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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
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Thomas Kneib Human Sleep Data
• Simplified structure for the transitions:
Awake
Non-REM REM
Sleep
?
6
-
¾
λRN(t)
λNR(t)
λAS(t) λSA(t)
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Thomas Kneib Human Sleep Data
• Model for the transitions:
λAS,i(t) = exp[γ
(AS)0 (t) + siβ
(AS) + b(AS)i
]
λSA,i(t) = exp[γ
(SA)0 (t) + siβ
(SA) + b(SA)i
]
λNR,i(t) = exp[γ
(NR)0 (t) + ci(t)γ
(NR)1 (t) + siβ
(NR) + b(NR)i
]
λRN,i(t) = exp[γ
(RN)0 (t) + ci(t)γ
(RN)1 (t) + siβ
(RN) + b(RN)i
]
where
ci(t) =
{1 cortisol > 60 n mol/l at time t
0 cortisol ≤ 60 n mol/l at time t,
si =
{1 male
0 female,
b(j)i = transition- and individual-specific frailty.
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Thomas Kneib Human Sleep Data
• Use penalized splines for the baseline and time-varying effects.
• I.i.d. Gaussian priors for the frailty terms (with transition-specific variances).
• The likelihood contribution for individual i can be constructed based on a countingprocess formulation of the model:
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
].
Nhi(t) counting process for type h event.
Yhi(t) at risk indicator for type h event.
tij event times of individual i.
ni number of events for individal i.
δhi(tij) transition indicator for type h transition.
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Thomas Kneib Human Sleep Data
⇒ Concepts from survival analysis can be adapted.
• In particular:
– Fully Bayesian inference based on MCMC and
– Mixed model based empirical Bayes inference.
Bayesian Structured Hazard Regression 27
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Thomas Kneib Human Sleep Data
• Baseline effects:
0 2 4 6 8
−2.
4−
2.0
−1.
6−
1.2
awake −> sleep
0 2 4 6 8
−4.
5−
4.0
−3.
5−
3.0
−2.
5−
2.0
−1.
5
sleep −> awake
0 2 4 6 8
−15
−10
−5
0
Non−REM −> REM
0 2 4 6 8
−5.
5−
5.0
−4.
5−
4.0
−3.
5−
3.0
−2.
5
REM −> Non−REM
Bayesian Structured Hazard Regression 28
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Thomas Kneib Human Sleep Data
• Time-varying effects for a high level of cortisol:
0 2 4 6 8
−10
−5
05
Non−REM −> REM
0 2 4 6 8
−3
−2
−1
01
2
REM −> Non−REM
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Thomas Kneib Software
Software
• Estimation was carried out using BayesX.
• Public domain software package for Bayesian inference in geoadditive and relatedmodels.
• Available from
http://www.stat.uni-muenchen.de/~bayesx
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Thomas Kneib Conclusions
Conclusions
• Unified framework for general regression models describing the hazard rate of survivalmodels.
• Bayesian inference based on MCMC or mixed model methodology.
• Extendable to models for transition intensities in multi state models.
• Future work:
– More general censoring mechanisms.
– Conditions for propriety of posteriors.
– Joint modelling of covariates and duration times.
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Thomas Kneib References
References
• Brezger, Kneib & Lang (2005). BayesX: Analyzing Bayesian structured additiveregression models. Journal of Statistical Software, 14 (11).
• Fahrmeir, Kneib & Lang (2004): Penalized structured additive regression forspace-time data: a Bayesian perspective. Statistica Sinica 14, 731–761.
• Hennerfeind, Brezger, and Fahrmeir (2006): Geoadditive Survival Models.Journal of the American Statistical Association, to appear.
• Kneib & Fahrmeir (2006): A mixed model approach for geoadditive hazardregression. Scandinavian Journal of Statistics, to appear.
• Kneib (2006): Geoadditive hazard regression for interval censored survival times.Computational Statistics and Data Analysis, conditionally accepted.
Bayesian Structured Hazard Regression 32