Semi-parametric Survival Analysis with Time Dependent Covariates · Background • Models/methods...
-
Upload
nguyenkiet -
Category
Documents
-
view
220 -
download
0
Transcript of Semi-parametric Survival Analysis with Time Dependent Covariates · Background • Models/methods...
Semi-parametric SurvivalAnalysis with Time Dependent
CovariatesAdam Branscum Tim Hanson
University of Kentucky University of Minnesota
and
Wesley Johnson Prakash Laud
UC Irvine Medical College of Wisconsin
Cambridge: Semi-parametric Survival Analysis
Survival Analysis Background• Cox Ph model is flexible but often fails to fit
• Semiparametric versions of AFT and proportional oddsmodels are competitors
• Ancient attempts to provide semi-parametric approachesto the AFT model met with limited success (Miller,1976; Buckley and James, 1979; Koul Susarla and VanRyzin, 1981; Christensen and Johnson, 1988
• Recent Bayesian approaches are promising (Kuo andMallick, 1997; Kottas and Gelfand, 2001; Gelfand andKottas, 2002; Walker and Mallick, 1999, 2001; Hansonand Johnson, 2002, 2004.
• Recent frequentist approaches tend to focus onasymptotics for regression coefficients (Lin and Ying,1995; Tseng, Wang and Hsieh, 2005)
Cambridge: Semi-parametric Survival Analysis
Background• Models/methods for Time Dependent Covariates are
more sparse (Cox, 1972; Cox and Oakes, 1984; Robinsand Tsiatis, 1992; Lin and Ying, 1995; Shyer et al.,1999)
• Frequentist joint modeling (Davidian and Tsiatis, 2003;Tseng, Hsieh and Wang, 2005)
• Bayesian approaches to joint modeling (Law, Taylor andSandler, 2002; Brown and Ibrahim, 2003; Brown,Ibrahim and DeGruttola, 2005)
• We develop Bayesian semi-parametric approaches for theCox, Cox and Oakes, and AFT models
• We also develop a Bayesian joint-modeling approachusing Cox, Cox and Oakes and Proportional Odds models
Cambridge: Semi-parametric Survival Analysis
The Basics of Survival Modeling• Let T > 0 denote a random survival (event) time.
• S(t) = P (T > t) : Survival Function
• h(t)dt = P (T ∈ [t, t + dt)|T ≥ t) : Hazard Function
• Denote risk factors as x = (x1, . . . , xp). The PH modelrelates covariates to the hazard and survival function as:
h(t|x) = exp(xβ)h0(t)
S(t|x) = S0(t)exβ
• Censored Survival Data:
ti, δi, xi : i = 1, ..., n
Cambridge: Semi-parametric Survival Analysis
Alternative Models• AFT Model:
S(t|x) = S0(exp(xβ)t) ⇔ T = exp(xβ)V :
• Prop. Odds Model:
S(t|x)
1 − S(t|x)= exβ S0(t)
1 − S0(t)
Cambridge: Semi-parametric Survival Analysis
Models for S0
• Dirichlet Process (DP) (Ferguson, 1973)
• Mixtures of Dirichlet Processes (MDP) (Antoniak, 1974)
• Dirichlet Process Mixtures (DPM) (Escobar, 1994)
• Polya Tree and Mixture of PT Process (MPT) (Lavine,1992, 1994; Hanson, 2006)
• Alternative to semi-parametric models
Dependent Dirichlet Process (DDP) regression model(MacEachern, 1999: De Iorio et al. 2004; De Iorio et al.2007)
Cambridge: Semi-parametric Survival Analysis
Polya Trees• Split sample space Ω into two disjoint sets B0 and B1;
further split B0 into B00 and B01, split B1 into B10 andB11:
B0 B1
B00 B01 B10 B11
• Define
Y0 = P (V ∈ B0), Y1 = P (V ∈ B1),
Y00 = P (V ∈ B00|V ∈ B0),
Y01 = P (V ∈ B01|V ∈ B0),
Y10 = P (V ∈ B10|V ∈ B1),
Y11 = P (V ∈ B11|V ∈ B1).
• Then P (V ∈ Bij) = YiYij. Cambridge: Semi-parametric Survival Analysis
• Continue: Let ǫ = ǫ1 · · · ǫm be an arbitrary binarynumber.
• Split Bǫ → Bǫ0, Bǫ1 ∀ǫ.
• Then
Yǫ0 = P (V ∈ Bǫ0|V ∈ Bǫ)
Yǫ1 = P (V ∈ Bǫ1|V ∈ Bǫ)
⇒
P (V ∈ Bǫ1···ǫm) =
m∏
j=1
Yǫ1···ǫj
Cambridge: Semi-parametric Survival Analysis
PT• Create random PM on S0:
(Yǫ1···ǫm0, Yǫ1···ǫm1) ∼ Dir(αǫ1···ǫm0, αǫ1···ǫm1)
• Random S0 specified by
• Π = ∪∞
j=1Bǫ1···ǫj: ǫ1 · · · ǫj ∈ 0, 1j
• A = ∪∞
j=1αǫ1···ǫj: ǫ1 · · · ǫj ∈ 0, 1j
Cambridge: Semi-parametric Survival Analysis
PT• S0|Π,A ∼ PT (Π,A)
• Lavine (1992, 1994) catalogues Polya tree theory
• Conjugacy: V1|S0 ∼ S0 −→
S0|V1,Π,A ∼ PT (Π,A∗), A∗ = αǫ + IBǫ(V1)
• Specify Π and A only to level M −→“partially specified Polya tree”
• S0|ΠM ,AM ∼ FPT (ΠM ,AM )
Cambridge: Semi-parametric Survival Analysis
PT• Ferguson (1974):
αǫ1···ǫm−10 = αǫ1···ǫm−11 = cm2
⇒ S0 absolutely continuous
• Large c results in a parametric analysis, and small cresults in a more non-parametric analysis
Cambridge: Semi-parametric Survival Analysis
Center Process Around Sθ• By definition of the process
ESθ(Bǫ1···ǫm) =
(
αǫ1
α0 + α1
)(
αǫ1ǫ2
αǫ10 + αǫ11
)
· · ·
· · ·
(
αǫ1···ǫm
αǫ1···ǫm−10 + αǫ1···ǫm−11
)
• If αǫ0 = αǫ1 for all ǫ, then ESθ(Bǫ1···ǫm) = 2−m.
• Sθ(Bǫ1···ǫm) = 2−m ⇒ ES0(Bǫ) = Sθ(Bǫ)
10
Cambridge: Semi-parametric Survival Analysis
Predictive Density• Let
Vi ∼ S0, i = 1, ...n + 1
S0|Π,A ∼ PT (Π,A)
V = (V1, . . . , Vn)′
• Define fθ = −S′
θ
Cambridge: Semi-parametric Survival Analysis
Pred Dens and Marg Post for β
fVn+1(w|V ) =
M(w)∏
j=2
cj2 + nǫj(w)(V )
2cj2 + nǫj−1(w)(V )
2M(w)−1fθ(w),
For the AFT model
p(β|data) ∝ p(β)×n
∏
j=1
fVj(Tje
−xjβ |Vi = Tie−xiβ, i < j)e−xjβ
Cambridge: Semi-parametric Survival Analysis
Mixture of Polya Trees• Can make exact inferences for β
• However, choosing Sθ for particular fixed θ, is ad hoc &the partition affects inferences for β
• Solution: Mixture of Polya Trees
S0|Πθ,A ∼ PT (Πθ,A)
θ ∼ p(θ), β ∼ p(β)
• Has the additional nice property of centering on aparametric family, like the family of log normal pdf’s, orWeibull family...
Cambridge: Semi-parametric Survival Analysis
Full AFT Model with MPT Prior
Ti = exp(xiβ)Vi
V1, . . . , Vn|S0iid∼ S0, S0|θ ∼ PT (Πθ,A)
β ∼ p(β), θ ∼ p(θ)
• Predictive density for Tn+1|x, data is differentiableeverywhere; partition effects are“smoothed”
• Exact inference for β, θ|data is possible
• S0 centered on a parametric family of probabilitydistributions
• We set S0(0, 1] = 0.5 with probability one
• Results in median regression eg. med(T )|x = exβ
Cambridge: Semi-parametric Survival Analysis
• Can place prior on c
• Easy to incorporate informative prior information for βas in BCJ (1999) or Ibrahim and Chen (2000)
• Can use output from parametric analysis in constructingcandidate in Metropolis sampler
Cambridge: Semi-parametric Survival Analysis
Time Dependent Covariates• Stanford Heart Transplant Data: Time of HTP is
not known at the beginning of the study.
• Let Z1(t) be zero until the time of HTP and oneafterwards
• Let Z2(t) be the mismatch score between donor andrecipient hearts. Takes the value zero before HTP and aparticular value afterwards
• Goal is to measure effect of HTP and mismatch score onsurvival prospects.
Cambridge: Semi-parametric Survival Analysis
Cerebral Edema (CE)• CE is a complication of diabetic ketoacidosis (DK) in
children
• Children are admitted to the hospital for DK and CEmay or may not occur
• Children are monitored over time. The response is timeto CE after entry into the hospital
• Fixed covariates are age and BUN
• Time Dependent covariates are Sodium administered,fluids administered, and bicarbonate administered
• Goal is to determine if procedures of administeringvarious fluids is hastening the onset of CE
Cambridge: Semi-parametric Survival Analysis
Cox TDC Model (CTD)• Let z(t) : t > 0 be a vector of TDC covariate
processes, which we assume are fixed and known for now
• Define the Cox TD hazard function as
h(t|z, β) = ez(t)βh0(t)
where h0(·) is an arbitrary“baseline”hazard function
• Let rj , j = 0, 1, . . . be the grid of times over whichz(t) : t > 0 is constant, eg. no known changes
• Denote the rj ’s as changepoints for the covariate process
• Relative hazard for any two individuals is constant inbetween each adjacent pair of changepoints
Cambridge: Semi-parametric Survival Analysis
AFT TDC Model (AFTD)• Prentice and Kalbfleisch (1979)
h(t|z, β) = ez(t)βh0(tez(t)β)
• Can show that this model is equivalent to a mixture oftruncated AFT models over each of the adjacentchangepoint intervals, ([rj−1, rj)), where the acceleration
factor (AF) for the jth interval is cj ≡ ez(rj−1)β.
• Both the CTD and AFTD models presume that the riskof failure at time t only depends on the current values ofthe TDC’s, and not their history.
Cambridge: Semi-parametric Survival Analysis
Cox and Oakes TDC Model (COTD)• Model assumes that an individual with covariate z(·)
uses up their time at a rate of ez(t)β relative to“baseline”, namely
T0 =
∫ T
0ez(s)βds.
• The corresponding hazard function is
h(t|z, β) = ez(t)βh0(c(t)t), c(t) =1
t
∫ t
0ez(s)βds
• This model presumes that there is a cumulative effect ofthe covariate process up to time t that will effect thehazard of failure at that time.
20 Cambridge: Semi-parametric Survival Analysis
MFPT Baseline for All Models• Assume the same MFPT prior for all three models, eg.
S0 ∼ PT (AM ,ΠθM ), θ ∼ p(θ)
• Center PT on the family Sθ : θ ∈ Θ
• Assume that, for given θ, the prior on the intervals atthe highest level of the tree is governed by Sθ
• A Lik cont (no marg) for the AFTD model isLz(β,ΞM , θ|T = t) =
m∏
j=1
pj
2MpθNfθ(cm+1t)cm+1
Sθ(cm+1rm|ΞM ),
where pj = S0(cjrj|ΞM )/S0(cjrj−1|ΞM )
Cambridge: Semi-parametric Survival Analysis
Likelihood Functions• The likelihood contribution for an observation
right-censored at time t is Lz(β,ΞM , θ|T > t) =
m∏
j=1
pj
S0(cm+1t|ΞM , θ)
S0(cm+1rm|ΞM , θ)
• The complete data involve n independent event times,ti
ni=1, that are the observed survival times (Ti = ti) or
are right-censoring times (Ti > ti), and
• n covariate processes zi(·)ni=1
• The complete likelihood is
L(β,ΞM , θ) =
n∏
i=1
Li(β,ΞM , θ)
Cambridge: Semi-parametric Survival Analysis
Gibbs Sampling• Alternate between sampling β, θ|ΞM and ΞM |β, θ
• The former can be sampled via Metropolis-Hastingsusing a parametric model in WinBUGS or SAS to obtaina suitable candidate distribution
• Use MH for updating the components (Yǫ0, Yǫ1), withcandidate
(Y ∗
ǫ0, Y∗
ǫ1) ∼ Beta(mYǫ0,mYǫ1)
typically m = 20 or 30
• Can easily handle interval censored data
• Other likelihoods are similarly obtained
Cambridge: Semi-parametric Survival Analysis
Simulated Data• Simulate data from true baseline of log normal(0.69,
0.04) with two distinct TDC’s
• The first TDC is constant at zero, and the second iszero up to one unit of time and is one thereafter.
• Ten data points with TDC 1 and 90 with TDC 2
• The regression coefficient is β = 0.69
• Fit MFPT with c = 1 and M = 4, and with log-logisticfamily as base
• Uniform priors on finite intervals for (θ1, θ2, β)
25
Cambridge: Semi-parametric Survival Analysis
AFTD CO PH
E(ℓn(Lik)) 55 47 49.5
LPML 51 42 46
β .65 1.73 3.17
Prob Interval (.48,.96) (1.34,2.22) (2.23,4.22)
Posterior inferences for simulated data.
Cambridge: Semi-parametric Survival Analysis
Candidate Generating Distributions• If Sθ is exponential with parameter θ, then the AFTD,
COTD, and CTD models are the same
• The likelihood is
L(β, θ) =
n∏
i=1
Ji∏
j=1
e−θ[rij−ri,j−1]exi,j−1β
e[ti−ri,Ji]e
−θxi,Jiβ
θδi
• Readily implemented in SAS, S-plus, WinBUGS... toobtain starting values and covariance matrices for thecandidate generating distribution (CGD)
Cambridge: Semi-parametric Survival Analysis
CGD’s• We generally used the log-logistic to center the three
MPT survival models
• Used WinBUGS fit to get rough candidate generatingcovariance matrix for (β, θ) using random-walk M-Hchain
• Only needed 10,000 iterates in the final runs. Can all beeasily automated
• Jara, DP Package
Cambridge: Semi-parametric Survival Analysis
Stanford Heart Transplant Data• Data on patients admitted to Stanford Program and
analyzed using the Cox model with TDC’s (Crowley andHu, 1977)
• Lin and Ying (1995) use same data to illustrate theirheuristic procedure for COTD justified by asymptoticproperties
• We fit data using CTD, COTD and AFTD models withMFPT prior; M = 5 and c = 1.
Cambridge: Semi-parametric Survival Analysis
Stanford Study
xi1(t) =
0 if t < zi
1 if t ≥ zi
xi2(t) =
0 if t < zi
age at transplant − 35 if t ≥ zi
xi3(t) =
0 if t < zi
mismatch score − 0.5 if t ≥ zi
30
Cambridge: Semi-parametric Survival Analysis
Stanford Study
AFTD COTD CTD
ELL -461 -460 -458
LPML -468 -467 -464
Stat -1.76 -1.10 -1.04
(-3.86,1.57) (-2.70,0.50) (-1.99,-0.17)
Age-35 0.104 0.054 0.058
(-0.020,0.260) (-0.004,0.133) (0.015,0.107)
Mis-0.5 1.63 0.64 0.49
(-0.38,3.89) (-0.30,1.52) (-0.09,1.03)
Cambridge: Semi-parametric Survival Analysis
Stanford Study• The relative hazard (RH), comparing individual w/ no
HTP to an individual how gets one after 6 months
2.83 (1.19, 7.31)
Cambridge: Semi-parametric Survival Analysis
Stanford Study• Parametric exponential yielded posterior median
estimates for (β1, β2, β3)
(−2.74, 0.08, 0.98)
LPML = −486.3
• Integrated Cox-Snell residuals show extreme curvature
• Lin and Ying (1995) semiparametric-partial-likelihoodestimates
(−1.99, 0.096, 0.93)
Closer to exponential than semiparametric
Cambridge: Semi-parametric Survival Analysis
CE Data• Range of LPML’s ranged between -175 to - 176
• AFTD appears to fit the best based on residual plots
Cambridge: Semi-parametric Survival Analysis
Relative hazards in the OR over time
5 15
14
6
14
AF
7.00 8.00
9.00 10.00
11.00 12.00
Cambridge: Semi-parametric Survival Analysis
Joint modeling setting• Longitudinal data associated with terminal event of
interest
• Conditional on longitudinal process, we have survivalanalysis with TDC’s
• Longitudinal process is often observed with error
• With TDC’s, process was assumed constant betweenobservation times
• Can lead to bias (Prentice, 1982)
• Joint modeling is used to make inferences for assessing:
1. Trends in the time course of a longitudinal process
2. The association between de-noised time-dependentprocesses and event prognosis
Cambridge: Semi-parametric Survival Analysis
Alternatives to Joint Modeling• Don’t model the longitudinal data. Survival analysis with
TDCs (subsequently called RAW)
• Two-stage procedures (called Imputation):
Model the observed longitudinal process assuming ithas noise
Impute the de-noised signal process; treat it as aTDC
• Compare joint analyses with these
Cambridge: Semi-parametric Survival Analysis
Joint Modeling• Model the longitudinal data
f(y(·)|γ)
Conditional on that, model the survival time,
f(T |y, ξ)
• Longitudinal process, xi(·), is measured with error so weobserve yi(·) at several time points where
yi(t) = xi(t) + ǫi(t)
xi(t) = f(t)γ + g(t)bi + Ui(t) + ziα
ǫi(t)iid∼ N(0, σ2)
Cambridge: Semi-parametric Survival Analysis
Imputation• Use the longitudinal model to obtain xi(t)
• Use data ti, δi, xi as if xi were observed
• Define the cumulative history Xt = x(s) : s ≤ t
Cambridge: Semi-parametric Survival Analysis
Inferences: Bayesian Joint Modeling• Here, (after some modeling) we obtain,
f(yf , Tf |data) = f(yf |data)f(Tf |yf , data)
yf is a hypothetical observed history
• Prognosis based on their predictive density,f(Tf |yf , data). Compare these for different hypotheticalhistories. Set yf = yi
• Conditional hazards:
h(t|Xt, data) =
∫
h(t|Xt, ξ)p(dξ|data)
for hypothetical Xt.
Cambridge: Semi-parametric Survival Analysis
Models for survival data with TDC’s• Tseng et al (2005) developed a semiparametric
frequentist joint model using the COAFT (Monte CarloEM algorithm with bootstrap se’s for reg coeffs)
• Sundaram (2006) extended the proportional odds modelto allow for TDCs yielding a POTDC model, which isdefined by
d
dt
[
1 − S(t|x(·))
S(t|x(·))
]
= ex(t)β d
dt
[
1 − S0(t)
S0(t)
]
Cambridge: Semi-parametric Survival Analysis
Illustration: Medfly Data• Data from a study on reproductive patterns of 1000
female Mediterranean fruit flies.
• Obtained by recording the number of eggs producedeach day throughout their lifespans
• Goal was to examine the association between eggproduction patterns and lifetime
• Sample size of 251 flies with lifespans ranging from 22to 99 days, and no censored observations
Cambridge: Semi-parametric Survival Analysis
Fitted trajectory: Fly 1• Fitted trajectory for a“typical”medfly. Similar shapes for
PO, PH, CO, and longitudinal only analysis
5 10 15 20 250
1
2
3
4
Cambridge: Semi-parametric Survival Analysis
Model for longitudinal data• Compare with a previous joint analysis (Tseng et al,
2005), so we use their structure
• yi = (yi1, . . . , yini)′ are the ni longitudinal
measurements of subject i at times ti = (ti1, . . . , tini)′
• Model specifies that trajectories satisfy
yij|bi, σ2 ⊥∼ N
(
bi1g1(tij) + bi2g2(tij) + · · · + bidgd(tij), σ2)
• Individual trajectories
bi|µ,Σiid∼ Nd(µ,Σ).
Cambridge: Semi-parametric Survival Analysis
Model fitting• Let xi(t|bi) = bi1g1(t) + · · · + bidgd(t)
• For joint models, survival is specified conditional on
xi(·|bi)ni=1
• S0 modeled with MFPT prior• log-logistic centering family, i.e.
E(S0(t)) = (1 + t1/τe−α/τ )−1
• collection of branch probabilities ΞM
• weight parameter c
• Let θ = (α, τ,ΞM , c)
• A model [Ti|θ, β, xi(·|bi)] is specified as CO, PO, or PH
10
Cambridge: Semi-parametric Survival Analysis
Model fitting• Independent priors:
• p(µ,Σ, β, α, τ) ∝ |Σ|−(d+1)/2
• p(σ−2) ∝ 1/σ−2
• c ∼ Γ(c|ac, bc)
• (Xj,2k−1, Xj,2k) ∼ Dirichlet(cj2, cj2)
• The posterior based on the survival portion, thelongitudinal portion, and the prior is then
p(β,θ,µ,Σ, σ|T,y1:n) =[
n∏
i=1
f(Ti|xi(·|bi),θ, β)δiS(Ti|xi(·|bi),θ, β)1−δi
]
×
[
n∏
i=1
p(yi|bi, σ)p(bi|µ,Σ)
]
p(β,θ,µ,Σ, σ)
Cambridge: Semi-parametric Survival Analysis
Model fitting• The full conditional distributions for µ, Σ, and σ−2 are:
Σ−1|b1:n,µ ∼ Wishart
n,
[
n∑
i=1
(bi − µ)(bi − µ)′
]−1
µ|b1:n,Σ ∼ Nd
(
b•,Σ/n)
σ−2|b1:n ∼ Γ
0.5n
∑
i=1
ni, 0.5∑
i,j
(yij − xi(tij|bi))2
• Metropolis-Hastings steps were used to sample the fullconditionals for the bi’s (random-walk M-H), ΞM (w/beta proposals), c (w/ truncated normal proposal),(α, β, τ) (w/ random walk M-H).
Cambridge: Semi-parametric Survival Analysis
Illustration: Medfly DataResponse
ln(yi(t) + 1)
and
xi(t|bi) = b1i ln(t) + b2i(t − 1)
Cambridge: Semi-parametric Survival Analysis
Model comparison• negative-LPML statistics (smaller is better) comparing
modeling approaches:
Model Method PO PH CO
parametric raw 867 870 937
MPT raw 865 866 938
MPT imputed 947 959 973
parametric joint 947 959 973
MFPT joint 945 956 973
• Summary based on LPML criterion:• Predictively, PO and PH models preferred over CO• Survival with fixed TDC’s preferred over joint• MFPT improves predictive performance only slightly
compared to parametric modelCambridge: Semi-parametric Survival Analysis
Fitted trajectory: Fly 1• Fitted trajectory for a“typical”medfly. Similar shapes for
PO, PH, CO, and longitudinal only analysis
5 10 15 20 250
1
2
3
4
Cambridge: Semi-parametric Survival Analysis
Predictive survival density: Fly 1• Solid is PO, dashed is PH, and dotted is CO
20 40 60 80 100
0.02
0.04
0.06
0.08
Cambridge: Semi-parametric Survival Analysis
Fitted trajectory: Fly 2• Fitted trajectory for another medfly using PO, PH, CO,
and longitudinal only analysis
0 10 20 30 40 500
1
2
3
4
Cambridge: Semi-parametric Survival Analysis
Predictive survival density: Fly 2• PO (solid), PH (dashed) and CO (dotted) analyses using
Raw trajectories.
20 40 60 80 100
0.02
0.04
0.06
0.08
0.1
Cambridge: Semi-parametric Survival Analysis
Predictive survival density: Fly 2• Raw trajectory (dashed line); joint analysis (solid line)
20 40 60 80 100
0.02
0.04
0.06
0.08
Cambridge: Semi-parametric Survival Analysis
Posterior inference for β
Model Method PO PH CO
parametric raw −0.75 −0.65 −0.36 (−0.44,−0.27)
MPT raw −0.74 −0.64 −0.37 (−0.45,−0.29)
MPT imputed −0.74 −0.37 0.16 (−0.01,0.30)
parametric joint −0.78 −0.39 0.19 (0.01,0.33)
MPT joint −0.79 −0.40 0.19 (0.01,0.32)
• Pr(β < 0|T,y1:n) = 1 for PO and PH models⇒ survival prospects are better for the most fertile flies.
• Inferences based on CO are different for joint modelsthan for models based on raw trajectories
Cambridge: Semi-parametric Survival Analysis
Why I like MFPT’s for SA• Prior centered on parametric family; DPM Not
• Easy to place informative prior on reg coeffs; DPM Not
• No need to marginalize over S0
• Inferences on functionals of S0 simple
• Median regression is immediate; DPM not
• No“sticky clusters”
• Hanson (2006, JASA)
• Hanson, T., Branscum, A., and Johnson, W.O. (2005).Bayesian nonparametric modeling and data analysis: anintroduction. In Bayesian Thinking: Modeling andComputation (Handbook of Statistics, volume 25)
Cambridge: Semi-parametric Survival Analysis