Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical...

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Approximate Bayesian Inference Approximate Bayesian Inference for Survival Models Rupali Akerkar 1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1 With H.Rue/S.Martino

Transcript of Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical...

Page 1: Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1With H.Rue/S.Martino. Approximate Bayesian Inference

Approximate Bayesian Inference

Approximate Bayesian Inference for SurvivalModels

Rupali Akerkar1

Department of Mathematical SciencesNTNU,Norway

15 May 2009

1With H.Rue/S.Martino

Page 2: Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1With H.Rue/S.Martino. Approximate Bayesian Inference

Approximate Bayesian Inference

Introduction

Outline

Basic idea

I Survival model

I Present survival model as a latent Gaussian model

I Apply INLA

I Verify results with MCMC results

Page 3: Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1With H.Rue/S.Martino. Approximate Bayesian Inference

Approximate Bayesian Inference

Introduction

Outline

Basic idea

I Survival model

I Present survival model as a latent Gaussian model

I Apply INLA

I Verify results with MCMC results

Page 4: Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1With H.Rue/S.Martino. Approximate Bayesian Inference

Approximate Bayesian Inference

Introduction

Outline

Basic idea

I Survival model

I Present survival model as a latent Gaussian model

I Apply INLA

I Verify results with MCMC results

Page 5: Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1With H.Rue/S.Martino. Approximate Bayesian Inference

Approximate Bayesian Inference

Introduction

Outline

Basic idea

I Survival model

I Present survival model as a latent Gaussian model

I Apply INLA

I Verify results with MCMC results

Page 6: Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1With H.Rue/S.Martino. Approximate Bayesian Inference

Approximate Bayesian Inference

Introduction

Outline

Outline

I Survival Analysis

I Some definitionsI CensoringI Likelihood

I Survival Models

I Parametric modelsI Semiparametric models for hazard

I piecewise constant modelsI piecewise linear models

I Spatial model

Page 7: Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1With H.Rue/S.Martino. Approximate Bayesian Inference

Approximate Bayesian Inference

Survival Data

Censoring

Let T be a random survival time, the following functions aredefined:

I Density function:

T ∼ f (t)

I Survival function:

S(t) = 1− F (t) =

∫ ∞t

f (u)du

I Hazard function:

h(t) = limδt→0

1

δtP(t < T < t + δt | T > t)

thus

h(t) = limδt→0

S(t)− S(t + δt)

S(t)=

f (t)

S(t)

Page 8: Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1With H.Rue/S.Martino. Approximate Bayesian Inference

Approximate Bayesian Inference

Survival Data

Censoring

Censoring

I Uncensored observation: The failure time is recorded

I Right censored observation: The censoring time C < T isrecorded

I Interval censored observation: The failure time is notobserved exactly but it is known that Tlo < T < Tup

Page 9: Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1With H.Rue/S.Martino. Approximate Bayesian Inference

Approximate Bayesian Inference

Survival Data

Censoring

Each observation is described by a triple(Tlo , Tup, δ) with

Tlo = Tup = T , δ = 1 if the obs. is uncensored

Tlo = Tup = C , δ = 0 if the obs. is right censored

Tlo < Tup, δ = 0 if the obs. is interval censored

Page 10: Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1With H.Rue/S.Martino. Approximate Bayesian Inference

Approximate Bayesian Inference

Survival Data

Likelihood

Likelihood

The likelihood function is:

L =∏

Li

where:

Page 11: Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1With H.Rue/S.Martino. Approximate Bayesian Inference

Approximate Bayesian Inference

Survival Data

Likelihood

I if i is uncensored

Li = h(T )S(T ) = h(T )exp{−∫ T

0h(u)du}

I if i is censored

Li = S(C ) = exp{−∫ C

0h(u)du}

Page 12: Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1With H.Rue/S.Martino. Approximate Bayesian Inference

Approximate Bayesian Inference

Survival Data

Likelihood

I if i is interval censored

Li = S(Tlo)− S(Tup)

= exp{−∫ Tl0

0h(u)du}{1− exp(−

∫ Tup

Tlo

h(u)du)}

Page 13: Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1With H.Rue/S.Martino. Approximate Bayesian Inference

Approximate Bayesian Inference

Survival Data

Likelihood

The contribution to the log-likelihood of data i , (Tlo ,Tup,δi ), is ingeneral

li = δih(Tup,i )−∫ Tup,i

0h(u)du + log{1− exp(−

∫ Tup,i

Tlo,i

h(u)du)}

Page 14: Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1With H.Rue/S.Martino. Approximate Bayesian Inference

Approximate Bayesian Inference

Survival Data

Likelihood

The contribution to the log-likelihood of data i , (Tlo ,Tup,δi ), is ingeneral

li = δih(Tup,i )−∫ Tup,i

0h(u)du + log{1− exp(−

∫ Tup,i

Tlo,i

h(u)du)}

Only included for uncensored data

Page 15: Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1With H.Rue/S.Martino. Approximate Bayesian Inference

Approximate Bayesian Inference

Survival Data

Likelihood

The contribution to the log-likelihood of data i , (Tlo ,Tup,δi ), is ingeneral

li = δih(Tup,i )−∫ Tup,i

0h(u)du + log{1− exp(−

∫ Tup,i

Tlo,i

h(u)du)}

Included for all data

Page 16: Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1With H.Rue/S.Martino. Approximate Bayesian Inference

Approximate Bayesian Inference

Survival Data

Likelihood

The contribution to the log-likelihood of data i , (Tlo ,Tup,δi ), is ingeneral

li = δih(Tup,i )−∫ Tup,i

0h(u)du + log{1− exp(−

∫ Tup,i

Tlo,i

h(u)du)}

Only included for interval censored data

Page 17: Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1With H.Rue/S.Martino. Approximate Bayesian Inference

Approximate Bayesian Inference

Survival Models

Model

Cox Model

The model proposed by Cox in 1972 is

h(t|z1, ...., zp) = h0(t) exp(z1β1 + ....+ zpβp)

I h0 = baseline hazard

I zi = covariates

I βi = regression parameters

For this model, the covariates are assumed to have fixed effects onfailure pattern.But the covariates effects do change with time.

Page 18: Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1With H.Rue/S.Martino. Approximate Bayesian Inference

Approximate Bayesian Inference

Survival Models

Model

More comprehensive model

A more comprehensive model is achieved by assuming

h(t) = exp(z0β0(t) + ...+ z1βp(t))

where β0 = log(h0).

Page 19: Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1With H.Rue/S.Martino. Approximate Bayesian Inference

Approximate Bayesian Inference

Survival Models

Model

I Parametric modelsI ExponentialI Weibull

I Parametric models with frailtyI Semiparametric models

I Piecewise-constant baseline hazardI Piecewise-linear baseline hazard

Page 20: Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1With H.Rue/S.Martino. Approximate Bayesian Inference

Approximate Bayesian Inference

Survival Models

Parametric models

The Weibull modelI The data : (Tlo,i , Tup,i , δi ), i = 1,...,nd

I The hazards rate :

h(u; z , α) = αuα−1exp(η)

with η = zTβI the log-likelihood :

l = δh(Tup)−∫ Tup

0h(u)du + log{1− exp(−

∫ Tup

Tlo

h(u)du)}

= δ[logα + (α− 1)logTup + η]− eηTαup + log{1− e−eη(Tα

up−Tαlo )}

I Priors for parameters :

β ∼ N(0, τβI )

α ∼ π(α)

Page 21: Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1With H.Rue/S.Martino. Approximate Bayesian Inference

Approximate Bayesian Inference

Survival Models

Parametric models

The Weibull modelI The data : (Tlo,i , Tup,i , δi ), i = 1,...,nd

I The hazards rate :

h(u; z , α) = αuα−1exp(η)

with η = zTβI the log-likelihood :

l = δh(Tup)−∫ Tup

0h(u)du + log{1− exp(−

∫ Tup

Tlo

h(u)du)}

= δ[logα + (α− 1)logTup + η]− eηTαup + log{1− e−eη(Tα

up−Tαlo )}

I Priors for parameters :

β ∼ N(0, τβI )

α ∼ π(α)

Page 22: Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1With H.Rue/S.Martino. Approximate Bayesian Inference

Approximate Bayesian Inference

Survival Models

Parametric models

The Weibull modelI The data : (Tlo,i , Tup,i , δi ), i = 1,...,nd

I The hazards rate :

h(u; z , α) = αuα−1exp(η)

with η = zTβI the log-likelihood :

l = δh(Tup)−∫ Tup

0h(u)du + log{1− exp(−

∫ Tup

Tlo

h(u)du)}

= δ[logα + (α− 1)logTup + η]− eηTαup + log{1− e−eη(Tα

up−Tαlo )}

I Priors for parameters :

β ∼ N(0, τβI )

α ∼ π(α)

Page 23: Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1With H.Rue/S.Martino. Approximate Bayesian Inference

Approximate Bayesian Inference

Survival Models

Parametric models

The Weibull modelI The data : (Tlo,i , Tup,i , δi ), i = 1,...,nd

I The hazards rate :

h(u; z , α) = αuα−1exp(η)

with η = zTβI the log-likelihood :

l = δh(Tup)−∫ Tup

0h(u)du + log{1− exp(−

∫ Tup

Tlo

h(u)du)}

= δ[logα + (α− 1)logTup + η]− eηTαup + log{1− e−eη(Tα

up−Tαlo )}

I Priors for parameters :

β ∼ N(0, τβI )

α ∼ π(α)

Page 24: Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1With H.Rue/S.Martino. Approximate Bayesian Inference

Approximate Bayesian Inference

Survival Models

Parametric models

The Weibull model as a latent Gaussian model

I The latent Gaussian field

x = {η1, ..., ηnd, β} ∼ N(0,Q−1)

I The hyperparametersθ = α

I The likelihood

π(data|x,θ) =

nd∏i=1

π(datai |ηi ,θ)

We can apply INLA to this model without problems!!

Page 25: Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1With H.Rue/S.Martino. Approximate Bayesian Inference

Approximate Bayesian Inference

Survival Models

Parametric models

Note

In more general term we can have

η =∑

j

fj(zj) +∑k

βj z̃j + ε

where fj(zj) can represent

I smooth effect of covariate

I time varying effect of covarite

I space effect

As long as the prior for f () is Gaussian we are still in the latentGaussian model framework!

Page 26: Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1With H.Rue/S.Martino. Approximate Bayesian Inference

Approximate Bayesian Inference

Survival Models

The Weibull model

Example1 - Kidney data

(Nahman et al.,1992) The time to the first infection for kidneydialysis patients is analysed. The data are right censored. Onebinary covariate z is catheter placement.The model is

h(t; z) = αtα−1exp{β0 + β1z}

we assumeβ0, β1 ∼ N(0, 0.001)

α ∼ Γ(1, 0.001)

Page 27: Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1With H.Rue/S.Martino. Approximate Bayesian Inference

Approximate Bayesian Inference

Survival Models

The Weibull model

Example

−3 −2 −1 0 1 2

0.0

0.1

0.2

0.3

0.4

0.5

0.6

posterior marginals of intercept

−2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0

0.0

0.2

0.4

0.6

0.8

1.0

posterior marginal densities of fixed effect

0.6 0.8 1.0 1.2 1.4

0.0

0.5

1.0

1.5

2.0

2.5

Alpha.parameter.for.Weibull

dens

Solid line from INLA (0.165sec and 0.0235 for inla.hyperpar),histogram from 3x106 samples from Winbugs(1900 sec)

Page 28: Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1With H.Rue/S.Martino. Approximate Bayesian Inference

Approximate Bayesian Inference

Survival Models

Parametric models with frailty

Parametric models with frailty - Hazard function

Let tij be the survival times for the j th subject in the i th cluster,i=1,...,n,j=1,...,mi , the hazard function is given as:

h(tij ; zij ,wi , α) = αtα−1ij wiexp(zT

ij β)

= αtα−1ij exp(ηij)

withηij = zT

ij β + log(wi )

Page 29: Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1With H.Rue/S.Martino. Approximate Bayesian Inference

Approximate Bayesian Inference

Survival Models

Parametric models with frailty

Parametric models with frailty - Log-likekihood function

The likelihood function for the generic data (Tlo ,Tup, δ) is then

l = δh(Tup)−∫ Tup

0h(u)du + log{1− exp(−

∫ Tup

Tlo

h(u)du)}

= δ[logα + (α− 1)logTup + η]− eηTαup + log{1− e−eη(Tα

up−Tαlo )}

Page 30: Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1With H.Rue/S.Martino. Approximate Bayesian Inference

Approximate Bayesian Inference

Survival Models

Parametric models with frailty

Log-normal model for frailty

If we assume log(wi ) = εi to have a Gaussian prior N(0, τw ), theparametric frailty model falls into the latent Gaussian family.

I The latent Gaussian field

x = {η11, ..., ηnmn , ε1, ...εn, β} ∼ N(0,Q−1)

I the hyperparametersθ = (α, τw )

I The likelihood

π(data|x,θ) =

nd∏i=1

π(data|ηi ,θ)

...and so no problem to apply INLA!!

Page 31: Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1With H.Rue/S.Martino. Approximate Bayesian Inference

Approximate Bayesian Inference

Survival Models

Parametric models with frailty

Example2 - Rat data

(Mantel et al.,1977) study time till tumor development in rats from50 distinct litters, the covarite z is a treatment( drug or placebo),w is the frailty variable (litter/cluster). The model is

h(tij ; zij ,wi , α) = αtα−1exp{β0 + β1z + log(wi )}

We assumeβ0, β1 ∼ N(0, 0.001)

α, τw ∼ Γ(1, 0.001)

Page 32: Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1With H.Rue/S.Martino. Approximate Bayesian Inference

Approximate Bayesian Inference

Survival Models

Parametric models with frailty

Example2 - Rat data

−3.0 −2.5 −2.0 −1.5 −1.0 −0.5 0.0

0.0

0.5

1.0

1.5

posterior marginal densities of intercept

−1 0 1 2 3

0.0

0.2

0.4

0.6

0.8

1.0

1.2

posterior marginal densities of fixed effect

Solid line from INLA(1.422 sec) and histogram from 106 samplesfrom Winbugs(1687 sec)

Page 33: Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1With H.Rue/S.Martino. Approximate Bayesian Inference

Approximate Bayesian Inference

Survival Models

Parametric models with frailty

Example - Rat data

2.5 3.0 3.5 4.0 4.5 5.0 5.5

0.0

0.2

0.4

0.6

posterior marginals of Weibull parameterx

y

0 1000 2000 3000 4000

0.00

000.

0005

0.00

100.

0015

0.00

20

posterior marginals of group−effectx

y

Solid line from INLA(1.422 sec) and histogram from 106 samplesfrom Winbugs(1687 sec)

Page 34: Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1With H.Rue/S.Martino. Approximate Bayesian Inference

Approximate Bayesian Inference

Survival Models

Semiparametric model for hazard

Piecewise constant model for h0(t)

Divide the time line into J predefined intervals, Ik = (Sk ,Sk+1] fork = 1, ..., J with 0 = s1 < ... < sJ <∞, we define the baselinehazard as:

h0(t) = λk if t ∈ Ik = (sk , sk+1]

and the baseline survival is then:

S0(t) = exp{−∫ t

0h0(u)du} = exp{

k−1∑j=1

(sj+1 − sj)λj − (t − sk)λk}

Page 35: Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1With H.Rue/S.Martino. Approximate Bayesian Inference

Approximate Bayesian Inference

Survival Models

Semiparametric model for hazard

Piecewise constant model for h0(t)

In general, let the hazard rate be

h(t; .) = h0(t)exp(zTβ) = exp{zTβ + logh0(t)}= exp{zTβ + logλk}; t ∈ Ik

and assume a RW prior for the piecewise baseline hazard

logλ1, ..., logλJ | τλ ∼ RW (τλ)

thenηk = zTβ + logλk | ... ∼ Gaussian

Page 36: Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1With H.Rue/S.Martino. Approximate Bayesian Inference

Approximate Bayesian Inference

Survival Models

Semiparametric model for hazard

Log-likelihood for right censored data

The log-likelihood contribution for a (possibly) right censoredobservation t ∈ Ik is :

log [h(t; .)δS(t; .)] = δηk − {k−1∑j=1

(sj+1 − sj)eηj + (t − sk)eηk}

= δηk − (t − sk)eηk −k−1∑j=1

(sj+1 − sj)eηj

I This can be seen as the log likelihood from a Poisson withmean (t − sk)eηk observed to be 0 or 1 according to δ

I This can be seen as the likelihood from k − 1 Poisson withmean (sj+1 − sj)e

ηj observed to be 0

Page 37: Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1With H.Rue/S.Martino. Approximate Bayesian Inference

Approximate Bayesian Inference

Survival Models

Semiparametric model for hazard

Log-likelihood for right censored data

The log-likelihood contribution for a (possibly) right censoredobservation t ∈ Ik is :

log [h(t; .)δS(t; .)] = δηk − {k−1∑j=1

(sj+1 − sj)eηj + (t − sk)eηk}

= δηk − (t − sk)eηk −k−1∑j=1

(sj+1 − sj)eηj

I This can be seen as the log likelihood from a Poisson withmean (t − sk)eηk observed to be 0 or 1 according to δ

I This can be seen as the likelihood from k − 1 Poisson withmean (sj+1 − sj)e

ηj observed to be 0

Page 38: Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1With H.Rue/S.Martino. Approximate Bayesian Inference

Approximate Bayesian Inference

Survival Models

Semiparametric model for hazard

Log-likelihood for right censored data

The log-likelihood contribution for a (possibly) right censoredobservation t ∈ Ik is :

log [h(t; .)δS(t; .)] = δηk − {k−1∑j=1

(sj+1 − sj)eηj + (t − sk)eηk}

= δηk − (t − sk)eηk −k−1∑j=1

(sj+1 − sj)eηj

I This can be seen as the log likelihood from a Poisson withmean (t − sk)eηk observed to be 0 or 1 according to δ

I This can be seen as the likelihood from k − 1 Poisson withmean (sj+1 − sj)e

ηj observed to be 0

Page 39: Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1With H.Rue/S.Martino. Approximate Bayesian Inference

Approximate Bayesian Inference

Survival Models

Semiparametric model for hazard

Log-likelihood for right censored data

Each data point ti is written as k ”augmented data points”yi1, ...., yik coming from Poisson distribution

Page 40: Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1With H.Rue/S.Martino. Approximate Bayesian Inference

Approximate Bayesian Inference

Survival Models

Semiparametric models for hazard

Piecewise constant model for h0(t) as latent Gaussian field

I The latent Gaussian field

x = {logλ1, ..., logλJ , β, η11...}

I The hyperparametersθ

I The ”augmented” Poisson data

Page 41: Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1With H.Rue/S.Martino. Approximate Bayesian Inference

Approximate Bayesian Inference

Survival Models

Semiparametric models for hazard

Example3 -Times to death for a Breast-cancer trial

(Sedmak et al., 1989) The time to death of 45 breast cancerpatients is analysed. The data are right censored. One binarycovariate z(immunohistochemical response) is also recorded.The model is:

h(t; z) = h0(t)exp{β0 + β1z}

Moreover we divide the time line into 2 equal intervals

h0(t) = λk t ∈ Ik

and assumelogλ1, logλ2 ∼ RW 1(τλ)

β0, β1 ∼ N(0, 0.001)

Page 42: Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1With H.Rue/S.Martino. Approximate Bayesian Inference

Approximate Bayesian Inference

Survival Models

Semiparametric models for hazard

Example3 -Times to death for a Breast-cancer trial

−1.5 −1.0 −0.5 0.0 0.5 1.0

0.0

0.5

1.0

1.5

posterior marginals of fixed effect

−7.0 −6.5 −6.0 −5.5

0.0

0.5

1.0

1.5

2.0

2.5

posterior marginals of intercept

The solid line is the INLA approximations(0.090sec) and thehistogram is from 5x106 Winbug samples(685sec)

Page 43: Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1With H.Rue/S.Martino. Approximate Bayesian Inference

Approximate Bayesian Inference

Survival Models

Semiparametric models for hazard

Example3 -Times to death for a Breast-cancer trial

−0.15 −0.10 −0.05 0.00 0.05 0.10 0.15

05

1015

20

posterior marginals of hazard for first interval

−10 −5 0 5 10

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

posterior marginals of logprecision

The solid line is the INLA approximations(0.090sec) and thehistogram is from 5x106 Winbug samples(685sec)

Page 44: Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1With H.Rue/S.Martino. Approximate Bayesian Inference

Approximate Bayesian Inference

Survival Models

Semiparametric model for hazard

Piecewise linear model for h0(t)

Divide the time line into J predefined intervals, Ik = (Sk ,Sk+1] fork = 1, ..., J with 0 = s1 < ... < sJ <∞, we define the baselinehazard as:

h0(t) = λj +λj+1 − λj

sj+1 − sjt if t ∈ Ij = (sj , sj+1]

Page 45: Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1With H.Rue/S.Martino. Approximate Bayesian Inference

Approximate Bayesian Inference

Survival Models

Semiparametric model for hazard

Log-likelihood for right censored data

The log-likelihood contribution for a (possibly) right censoredobservation t ∈ Ik

log [h(t; .)S(t; .)] = δηk − w1eη1 −

k−1∑j=2

wjeηj − wkeηk − wk+1e

ηk+1

where, w ’s, the weights are

w1 =s2 − s1

2, wj =

sj+1 − sj−1

2

wk = t − sk + sk−1

2− (t − sk)2

2(sk+1 − sk)and wk+1 =

(t − sk)2

2(sk+ − sk)

Page 46: Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1With H.Rue/S.Martino. Approximate Bayesian Inference

Approximate Bayesian Inference

Survival Models

Semiparametric model for hazard

Example4-Times to death for a Breast-cancer trial

500 1000 1500 2000

−1.

5−

1.0

−0.

50.

00.

51.

0

baseline.hazard.linear

Posterior mean together with 0.025quant 0.975quant

500 1000 1500 2000

−1.

5−

1.0

−0.

50.

00.

51.

0

baseline.hazard.constant

Posterior mean together with 0.025quant 0.975quant

Piecewise linear and constant baseline hazard

Page 47: Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1With H.Rue/S.Martino. Approximate Bayesian Inference

Approximate Bayesian Inference

Survival Models

Semiparametric model with spatial effect

Example5- Leukemia survival data

(Henderson et al.,2002) We study time to death for 1043leukaemia patients.The covariates included are age, wbc, tpi, sex and spatialinformation on district level. Here η is

ηij = α+β.hazj +β.sex ∗ sexi +β.agei +β.tpii +β.wbci +β.spatiali

Page 48: Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1With H.Rue/S.Martino. Approximate Bayesian Inference

Approximate Bayesian Inference

Survival Models

Semiparametric model with spatial effect

Example5- Leukemia survival data

Assumeα, β.sex ∼ N(0, 0.001)

β.haz , β.age, β.tpi , β.wbc ∼ RW 1(τ ′s)

β.spatial ∼ besag

Page 49: Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1With H.Rue/S.Martino. Approximate Bayesian Inference

Approximate Bayesian Inference

Survival Models

Semiparametric model with spatial effect

Example5- Leukemia survival data

2 4 6 8 10

−0.

50.

00.

51.

0

posterior mean and quantiles for age using inla(line) and winbugs(dotted)l

a1

5 10 15

−0.

6−

0.4

−0.

20.

00.

20.

4

posterior mean and quantiles for tpi using inla(line) and winbugs(dotted)1:19

tpi1

5 10 15 20

−0.

4−

0.2

0.0

0.2

district means for inla(red colour) and winbugsx

d1

Posterior means of age,tpi and spatial effect, Solid line(red) is theINLA approximations and dotted line from Winbugs samples.

Page 50: Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1With H.Rue/S.Martino. Approximate Bayesian Inference

Approximate Bayesian Inference

Survival Models

Semiparametric model with spatial effect

Example5-Leukemia survival data

20 30 40 50 60 70 80 90

−1.

00.

01.

0

age1

Posterior mean together with 0.025quant 0.975quant

0 100 200 300 400 500

−1.

00.

01.

0

wbc1

Posterior mean together with 0.025quant 0.975quant

−5 0 5

−0.

6−

0.2

0.2

tpi1

Posterior mean together with 0.025quant 0.975quant

0 5 10 15 20

−0.

6−

0.2

0.2

0.6

district

Posterior mean together with 0.025quant 0.975quant

0 1000 2000 3000 4000 5000

−2

01

2

hazard

Posterior mean together with 0.025quant 0.975quant

Compared these with results by Kneib et al.,2007.

Page 51: Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1With H.Rue/S.Martino. Approximate Bayesian Inference

Approximate Bayesian Inference

Demo

Semiparametric model with spatial effect

Example5-Leukemia survival data-Demo

demo(Leuk)

Page 52: Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1With H.Rue/S.Martino. Approximate Bayesian Inference

Approximate Bayesian Inference

Demo

New inla input

New inla input

data=read.table(”data1.txt”, header = T) this is some datan=length(data$time)surv.time = list(truncation=rep(0,n), event =data$event,lower=data$time,upper = rep(0,n), time=data$time)d = c(as.list(data), surv.time = surv.time)formula = surv.time ∼ placementmodel=inla(formula,family=”weibull”, data= d, verbose=TRUE,keep=TRUE )h=inla.hyperpar(model)

Page 53: Approximate Bayesian Inference for Survival Models · Rupali Akerkar1 Department of Mathematical Sciences NTNU,Norway 15 May 2009 1With H.Rue/S.Martino. Approximate Bayesian Inference

Approximate Bayesian Inference

Summary

Summary

I Many survival models fall in the latent Gaussian models family

I For such models INLA is a fast and reliable tool for estimate

I ... there is still lot to do to make INLA a more general tool forsurvival models