A Bayesian joint dispersion model with flexible links

Results

A Bayesian Joint Dispersion Model

With Flexible Links

Background HIV/AIDS Application Results

Summary

1

_Background

Longitudinal studies

Outline

2

HIV/AIDS Application

Data

Exploratory analysis

Joint Model

3

_Results

Results

Questions of interest

Biomarker, Y1

•

e.g. CD4 counts, collected repeatedly over time (longitudinal data)

time to an event of interest, Y2

•

e.g. death from any cause (survival data)

Separate Analysis

•

does treatment affect survival?

•

are the average longitudinal evolutions different between males

and females?

Joint Analysis

Background HIV/AIDS Application Results Longitudinal studies Outline

Questions of interest

Biomarker, Y1

•

e.g. CD4 counts, collected repeatedly over time (longitudinal data)

time to an event of interest, Y2

•

e.g. death from any cause (survival data)

Separate Analysis

•

does treatment affect survival?

•

are the average longitudinal evolutions different between males

and females?

Joint Analysis

•

what is the effect of the missing information due to drop-out in

assessing the trends of the repeated measures?

•

what is the effect of the longitudinal evolution of CD4 cell count in

Results

Questions of interest

Biomarker, Y1

•

e.g. CD4 counts, collected repeatedly over time (longitudinal data)

time to an event of interest, Y2

•

e.g. death from any cause (survival data)

Separate Analysis

•

does treatment affect survival?

•

are the average longitudinal evolutions different between males

and females?

Joint Analysis

Background

HIV/AIDS Application Results

Longitudinal studies Outline

Correlated data in longitudinal studies

0 1 2 3 4 5 0.0 2.5 5.0 7.5 10.0 Time Biomar k er v alues Longitudinal Trajectory 0 0.1 0.2 0.3 0.4 0.5 0.0 2.5 5.0 7.5 10.0 Time hazard Hazard Function Latent Structure

Longitudinal Marker (Y1) Time-To-Event (Y2)

If the two processes are associated ⇒ define a model for their joint probability

distribution:

f (y

, y

)

Results Joint Model

Analysing HIV/AIDS data through

Background HIV/AIDS Application Results Data Exploratory analysis Joint Model

Data

network of 88 laboratories located in every state in Brazil during

Jan 2002 – Dec 2006;

Sample: n = 500 individuals; 2757 repeated measurements;

Outcomes:

CD4

+

_{T lymphocyte counts, Y ,}

_and

_{time-to-death, T}

_;

Covariates:

age

(<50=0, ≥50=1);

sex

(Female=0, Male=1);

PrevOI

(previous opportunistic infection at study entry=1, no

previous infection=0);

measurement times

;

date of diagnosis

;

date of death

;

failure indicator, δ

;

Patients: 34 deaths. 88% between 15 and 49 years old; 60%

males. 61% no previous infection. Initial CD4 median: 269

Results Joint Model

Background HIV/AIDS Application Results Data Exploratory analysis Joint Model

Intra-individual variance (dispersion)

0 100 200 300 400 500 0 10 20 30 40

Ordered subjects by std. deviation

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● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●

lowess smooth of the CD4counts connecting the standard deviations

Individuals values of the

√

CD4 vs Std. Deviation (ordered) suggests considerable

within-subject variance heterogeneity. Individuals with higher

√

CD4 values are

associated with a higher variability.

Results Joint Model

Longitudinal specification

Mixed-effects dispersion model

(McLain et al. 2012)

y

|b

, σ

∼ N m

(t

), σ

,

j = 1, . . . , n

(1)

m

(t

) = β

x

(t

) + b

w

(t

),

(2)

σ

= σ

exp{ β

x

(t

) + b

w

(t

) },

(3)

•

y

= (y

, . . . , y

) → n

observed repeated measures,

√

CD4

•

t

= (t

, . . . , t

) →

visiting times

•

x

, x

, w

and w

→ individual covariates (time-dependent?)

•

β

and β

→ population parameters

Background HIV/AIDS Application Results Data Exploratory analysis Joint Model

Longitudinal specification

Mixed-effects dispersion model

(McLain et al. 2012)

y

|b

, σ

∼ N m

(t

), σ

,

j = 1, . . . , n

(1)

m

(t

) = β

x

(t

) + b

w

(t

),

(2)

σ

= σ

exp{ β

x

(t

) + b

w

(t

) },

(3)

•

y

= (y

, . . . , y

) → n

observed repeated measures,

√

CD4

•

t

= (t

, . . . , t

) →

visiting times

•

x

and w2i

→ individual covariates (time-dependent?)

•

β

and β2

→ population parameters

Results Joint Model

Longitudinal outcome

√

CD4

ij

|b

i

, σ

2

i

∼ N m

i

(t

ij

), σ

i

2

Longitudinal mean

m

(t

) = β

+β

sex

+β

age

+β

PrevOI

+β

t

+b

+b

t

(4)

Dispersion model (3) may assume:

σ

= σ

2

0

exp{β

sex + β

age + β

PrevOI + b

},

(5)

σ

= σ

exp{b

},

(6)

σ

,

(7)

σ

= σ

.

(8)

Background HIV/AIDS Application Results Data Exploratory analysis Joint Model

Longitudinal outcome

√

CD4

ij

|b

i

, σ

2

i

∼ N m

i

(t

ij

), σ

i

2

Longitudinal mean

m

(t

) = β

+β

sex

+β

age

+β

PrevOI

+β

t

+b

+b

t

(4)

Dispersion model (3) may assume:

σ

= σ

2

0

exp{β

sex + β

age + β

PrevOI + b

},

(5)

σ

= σ

exp{b

},

(6)

σ

,

(7)

σ

= σ

.

(8)

Priors

•

β

∼ N (0, 100); p = 0, . . . , 4 and β2q

∼ N (0, 100); q = 1, . . . , 3

•

b

|Σ ∼ N

(0, Σ);

Σ

∼ Wish(R, ξ)

•

log(σ0

) ∼ U (−100, 100); or log(σ

) ∼ U (−100, 100)

Results Joint Model

Survival specification

Time-dependent coefficients (Penalized cubic B-Splines)

h

(t|b

, σ

) = h

(t) exp{β

>

3

x

+ C

{b

, σ

; g(t)}} = h

(t) exp{%

(t)}

(9)

•

C

{.} → specifies which components of the longitudinal process

are related to h

(.)

•

Link → Shared parameters

I

b

, σ

•

x

→ baseline covariates

•

β

→ population parameters

•

h

(t) →

parametric (e.g. Weibull); P-Splines; Piecewise constant

function.

Background HIV/AIDS Application Results Data Exploratory analysis Joint Model

Survival specification

Time-dependent coefficients (Penalized cubic B-Splines)

h

(t|b

, σ

) = h

(t) exp{β

>

3

x

+ C

{

b

, σ

; g(t)}} = h

(t) exp{%

(t)}

(9)

•

C

{.} → specifies which components of the longitudinal process

are related to h

(.)

•

Link → Shared parameters

I

bi, σi

•

x

→ baseline covariates

•

β

→ population parameters

•

h

(t) →

parametric (e.g. Weibull); P-Splines; Piecewise constant

function.

•

g(t) = (g

(t), . . . , g

(t)) →

suitable vector of smooth functions

Results Joint Model

Survival specification

Time-dependent coefficients (Penalized cubic B-Splines)

h

(t|b

, σ

) = h

(t) exp{β

>

3

x

+ C

{b

, σ

; g(t)}} = h

(t) exp{%

(t)}

(9)

•

C

{.} → specifies which components of the longitudinal process

are related to h

(.)

•

Link → Shared parameters

I

b

, σ

•

x

→ baseline covariates

•

β

→ population parameters

•

h

parametric (e.g. Weibull); P-Splines; Piecewise constant

function.

Background HIV/AIDS Application Results Data Exploratory analysis Joint Model

Time-to-death

h

(t|b

, σ

) = h

(t) exp{β

x

+ C

{b

, σ

; g(t)}} = h

(t) exp{%

(t)}

(10)

all models

β

₃

>

x

3i

= β

31sex

i

+ β

32age

i

+ β

33PrevOI

i

(11)

C

i

(.)

may assume:

C

i

(.) = g

1

(t)b

1i,1

+ g

2

(t)b

1i,2

+ g

3

(t)b

2i

(12)

C

i

(.) = g

1

(t)b

1i,1

+ g

2

(t)b

1i,2

+ g

3

(t)σ

i

(13)

C

i

(.) = g

1

(t)b

1i,1

+ g

2

(t)b

1i,2

(14)

g

1

(t), g

2

(t), g

3

(t) →

Penalized Splines with 19 internal knots.

•

g

(t) =

P

γ

B

(t),

l = 1, 2, 3

•

γ

∼ N (0, 1000), γ

|τ

∼ N (γ

, τ

),

q = 2, . . . , 19

•

1/τ

∼ G(0.001, 0.001), l = 0, 1, 2, 3.

3 scenarios for h

0

(t) = log(g

0

(t)) →

Weibull,

Penalized Splines

Results Joint Model

Time-to-death

h

(t|b

, σ

) = h

(t) exp{β

x

+ C

{b

, σ

; g(t)}} = h

(t) exp{%

(t)}

(10)

all models

β

₃

>

x

3i

= β

31

sex

i

+ β

32

age

i

+ β

33

PrevOI

i

(11)

C

i

(.)

may assume:

C

i

(.) = g1(t)b1i,1

+ g2(t)b1i,2

+ g3(t)b2i

(12)

C

i

(.) = g

1

(t)b

1i,1

+ g

2

(t)b

1i,2

+ g

3

(t)σ

i

(13)

C

i

(.) = g1(t)b1i,1

+ g2(t)b1i,2

(14)

g

1

(t), g

2

(t), g

3

(t) →

Penalized Splines with 19 internal knots.

•

g

(t) =

P

19

q=1

γ

B

(t),

l = 1, 2, 3

Background HIV/AIDS Application Results Data Exploratory analysis Joint Model

Time-to-death

h

(t|b

, σ

) = h

(t) exp{β

x

+ C

{b

, σ

; g(t)}} = h

(t) exp{%

(t)}

(10)

all models

β

₃

>

x

3i

= β

31

sex

i

+ β

32

age

i

+ β

33

PrevOI

i

(11)

C

i

(.)

may assume:

C

i

(.) = g

1

(t)b

1i,1

+ g

2

(t)b

1i,2

+ g

3

(t)b

2i

(12)

C

i

(.) = g

1

(t)b

1i,1

+ g

2

(t)b

1i,2

+ g

3

(t)σ

i

(13)

C

i

(.) = g

1

(t)b

1i,1

+ g

2

(t)b

1i,2

(14)

g

1(t), g2(t), g3(t) →

Penalized Splines with 19 internal knots.

•

g

(t) =

P

γ

B

(t),

l = 1, 2, 3

•

γ

∼ N (0, 1000), γ

|τ

∼ N (γ

, τ

),

q = 2, . . . , 19

•

1/τ

∼ G(0.001, 0.001), l = 0, 1, 2, 3.

3 scenarios for h

0

(t) = log(g

0

(t)) →

Weibull,

Penalized Splines

Results Joint Model

Time-to-death

h

(t|b

, σ

) = h

(t) exp{β

x

+ C

{b

, σ

; g(t)}} = h

(t) exp{%

(t)}

(10)

all models

β

₃

>

x

3i

= β

31

sex

i

+ β

32

age

i

+ β

33

PrevOI

i

(11)

C

i

(.)

may assume:

C

i

(.) = g

1

(t)b

1i,1

+ g

2

(t)b

1i,2

+ g

3

(t)b

2i

(12)

C

i

(.) = g

1

(t)b

1i,1

+ g

2

(t)b

1i,2

+ g

3

(t)σ

i

(13)

C

i

(.) = g

1

(t)b

1i,1

+ g

2

(t)b

1i,2

(14)

g

1

(t), g

2

(t), g

3

(t) →

Penalized Splines with 19 internal knots.

•

g

(t) =

P

19

q=1

γ

B

(t),

l = 1, 2, 3

Background HIV/AIDS Application Results Data Exploratory analysis Joint Model

Joint likelihood

We consider:

y

i

|b

i

⊥ T

i

|b

i

;

y

ij

|b

i

⊥ y

il

|b

i

, j 6= l

non-informative right censoring

L(θ, b, σ | D) =

N

Y

i=1





n

Y

j=1

p(y

i

(t

ij

)|θ, b

i

, σ

2

i

)





p(T

i

, δ

i

|θ, b

i

, σ

i

)

where

D = {D

i

}

N

i=1

= {(y

i

, t

i

, T

i

, δ

i

)}

N

i=1

→ observed data for the N

independent individuals

θ →

other parameters;

Results Joint Model

Joint likelihood

We consider:

y

i

|b

i

⊥ T

i

|b

i

;

y

ij

|b

i

⊥ y

il

|b

i

, j 6= l

non-informative right censoring

L(θ, b, σ | D) =

N

Y

i=1





n

Y

j=1

p(y

i

(t

ij

)|θ, b

i

, σ

2

i

)





p(T

i

, δ

i

|θ, b

i

, σ

i

)

where

p(y

i

(t

ij

) | θ, b

i

, σ

i

2

) =

1

p2πσ

2

i

exp

(

−

[y

i

(t

ij

) − m

i

(t

ij

)]

2

2σ

2

i

)

Background HIV/AIDS Application Results Data Exploratory analysis Joint Model

Joint likelihood

We consider:

y

i

|b

i

⊥ T

i

|b

i

;

y

ij

|b

i

⊥ y

il

|b

i

, j 6= l

non-informative right censoring

L(θ, b, σ | D) =

N

Y

i=1





n

Y

j=1

p(y

i

(t

ij

)|θ, b

i

, σ

2

i

)





p(T

i

, δ

i

|θ, b

i

, σ

i

)

where

p(T

i

, δ

i

| θ, b

i

, σ

i

) =h(T

i

|θ, b

i

, σ

i

)

δi

× S(T

i

|θ, b

i

, σ

i

)

=[h

0

(T

i

) exp{β

>

3

x3i

+ C

i

{b

i

, σ

i

; g(t)}}]

δi

×

exp

(

−

Z

Ti

0

h

0

(u) exp{β

3

>

x

3i

+ C

i

{b

i

, σ

i

; g(t)}}du

Results

Models comparison

MCMC simulation within WinBUGS.

Longitudinal model

Survival model

m

σ

%

(t)

h

Weibull P-Spline Piecewise

(4)

(5)

(11) + (12)

14671

12573

14317

(4)

(6)

14700

12848

14483

(4)

(5)

(11) + (13)

14307

12605

13365

(4)

(6)

14452

12917

13571

(4)

(7)

13134

12104 ♣ 12921

(4)

(8)

13956

12887

13533

(4)

(5)

(11) + (14)

14811

13334

14463

(4)

(6)

14923

13688

14599

(4)

(7)

14314

13144

13968

(4)

(8)

14627

13553

14355

(4)

(8)

(11)+g

b

+g

b

16984

15779

16383

Background HIV/AIDS Application

Results

Posterior estimates for the time-dependent coefficients

0 1 2 3 4 5 −3 −2 −1 0 t g0 (t) 0 1 2 3 4 5 −2 −1 0 1 2 3 t g1 (t) 0 1 2 3 4 5 −2 −1 0 1 2 t g2 (t) 0 1 2 3 4 5 −1 0 1 2 3 4 5 t g3 (t)

Figure 1:

Posterior mean estimates, together with the corresponding 95% Credible Bands (CB),

for the selected model ♣. The top left panel shows g

= log (h

)

and the subsequent panels have

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Simultaneous probability statements for bayesian p-splines. Statistical Modelling 8, 2 (2008), 141–168.

FAUCETT, C. L.,ANDTHOMAS, D. C.

Simultaneously modelling censored survival data and repeatedly measured covariates: a gibbs sampling approach. Statistics in Medicine 15, 15 (Aug 1996), 1663–1685.

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Bayesian Survival Analysis. Springer-Verlag, 2001.

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Bayesian p-splines.

Journal of Computational and Graphical Statistics 13, 1 (2004), 183–212.

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Winbugs – a bayesian modelling framework: Concepts, structure, and extensibility. Statistics and Computing 10 (2000), 325–337.

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A joint mixed effects dispersion model for menstrual cycle length and time-to-pregnancy. Biometrics 68, 2 (Feb 2012), 648–656.

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Joint Models for Longitudinal and Time-to-Event Data With Applications in R. Chapman and Hall/CRC, 2012.