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)


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


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


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)


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)


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


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


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 )


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


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


Predictive Density• Let

Vi ∼ S0, i = 1, ...n + 1

S0|Π,A ∼ PT (Π,A)

V = (V1, . . . , Vn)′

• Define fθ = −S′

θ


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


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...


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


• 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


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.


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


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


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.


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 )


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)


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


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


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.


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)


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


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.


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


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)


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)


0 180 360

0.25

0.5

0.75

1survival


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


CE Data• Range of LPML’s ranged between -175 to - 176

• AFTD appears to fit the best based on residual plots


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


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


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


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)


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


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.


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)

]


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


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


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).


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


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).


Illustration: Medfly DataResponse

ln(yi(t) + 1)

and

xi(t|bi) = b1i ln(t) + b2i(t − 1)


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


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


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


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


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


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


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)


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