Joint Modelling of Accelerated Failure Time
and Longitudinal Data
By
Yi-Kuan Tseng
Joint Work With
Professor Jane-Ling Wang
Professor Fushing Hsieh
Tseng Y.K., Hsieh F., and Wang J.L. (2005). 92, pp. 587-603, Biometrika.
CD4 count plot of five patients
60
ID 69
ID 58
ID 62
ID 64
ID 74
50
CD4 count
40
30
20
10
0
0
500
1000
1500
2000
Days
2500
3000
3500
I. Introduction


W (t )  X (t )  e(t )
X (t ) : longitudinal covariates
e(t ) : independent measurement error
{t | X (t )}  0 (t ) exp(  X (t )}
X (t )  { X ( s ) : 0  s  t};
 : regression parameter
0 : unspecified baseline hazard rate function

CD4 counts and time to AIDS (or death)
 X (t )  b0  b1t
Self and Powitan(1992), Degruttola and Tu(1994),
Tsiatis et al.(1995), Faucett and Thomas (1996),
Wulfsohn and Tsiatis(1997) Bycott and Taylor(1998)
Dafni and Tsiatis (1998), Tsiatis and Davidian (2001)
 X (t )  f (t )T b  U (t )
f(t) :a vector of known functions of time t, U (t) : a
stochastic process
Taylor et al.(1994), Lavalley and Degruttola(1996),
Henderson et al.(2000),Wang and Taylor(2001), Xu
and Zeger(2001)

Two-stage partial likelihood approaches
-truncation causes bias

Joint likelihood approaches
-robust to the distribution of random effects
- unbiased
- efficient

Bayesian approaches

Conditional score approaches
 Accelerated
failure time model is an attractive
alternative when the proportional hazard
assumption fails.

For time independent covariates X:
log T    ' X  e
T : survival time; X : time independent covariates;
e: random error
Suppose S0 : baseline survival function ( T | X  0)
( ' X )
S
(
t
)

S
{
te
} =S0 (u )

0
U ~ S0  Te  ' X  U
U : a subject would have lived if there's no exposure (X  0)

u  t exp(  ' X )  t  2
t  30 (years old) with the same survival probability as u  60
(year old), (aging twice faster)
For
time dependent covariates X(t), we consider
the AFT model in Cox and Oakes (1984):
T
U ~ S0 , where U   { X (T );  }   exp{ X ( s)}ds
S{t | X (t )}  S0[ {X (t;  )}]
 Biological
0
meaning: Allows the influence of entire
covariate history on subject specific risk.
For
an absolutely continuous S0, the hazard rate
function with covariate history:
t
{t | X (t )}  0 [  e ' X ( s ) ds]e ' X (t )  0 [ { X (t );  }] '{ X (t );  }
0
If
baseline hazard is unspecified, the expression
corresponds to a semi-parametric model.
 Robins
and Tsiatis (1992)– rank estimating equation
Lin and Ying (1995)– asym. consistency and Normality
Hsieh (2003)– over-identified estimating equation

Goal of the study: provide an effective estimators for
β with unspecified baseline hazard and the parameters
of longitudinal process

Different assumptions on baseline hazard:
-- Wulfsohn and Tsiatis (1997)
Discrete baseline hazard with jumps at event times
-- Our assumption:
The baseline hazard is a step function.
II. Joint AFT and Longitudinal model

Notations:
Ti : event time of subject i, i  1, ,n
Ci : censoring time
Vi : observed time  min (Ti , Ci )
 i : 1(Ti  Ci ), event time indicator
ti
: measurement schedule  (tij : tij  Vi ), j  1,...mi
Wi : response  (Wij : tij  Vi )
X i () : time dependent covariate
ei

: measurement error
Observed data for each i:
(Vi ,  i , Wi , t i ), independent across i.

Model for longitudinal data:
Wi  X i (t i )  ei
X i (t )  bi T  (t ) (linear mixed effect model)
 (t )  {1 (t ),...,  p (t )}T : vector of known functions of time t
bi T  (bi1 ,..., bip ) : p-dimensional random effects ~ N p (  , )  ei
ei ~ N (0,  e2 I )

Examples :
p  2, {0 (t ), 1 (t )}  (1, t )
p  k , {0 (t ),...,  p 1 (t )}  (1, t ,..., t k 1 )
p  2, {0 (t ), 1 (t )}  {log(t ), t  1}
Model
for survival:
 (t | X (t ))   (t |  , bi )  0 ( (t;  , bi ) ' (t;  , bi )
Where
t
 (t ;  , bi )   e
 X (s)
ds   e
0
 (t ;  , bi )  e
'
 Joint
t
 biT  ( s )
ds,
0
 X (t )
e
 biT  ( t )
likelihood:
Assumptions
-- noninformative censoring
-- noninformative measurement schedule tij ,
both are independent of future covariate
history and random effects bi

L( )  L(  ,  , ,  e2 , 0 )
  i 1[  { j 1 f (Wij | bi , t i ,  e2 )} f (Vi , i | bi , t i , 0 ,  ) f (bi | ,  ) dbi ]
n

mi
f (Wij | bi , t i ,  e2 ) ~ N{biT  ( s),  e2 }
f (bi | ,  ) ~ N (  , )
f (Vi ,  i | bi , t i , 0 ,  )  [0 ( (Vi ;  , bi ) (Vi ;  , bi )] exp{ 
'
i
 (Vi ;  ,bi )
0
0 (t )dt}
III. EM Algorithm
Complete
data likelihood:
L ( )  i 1[ j 1 f (Wij | bi , ti ,  e2 )} f (Vi , i | bi , ti , 0 ,  ) f (bi | ,  )]
*
n
mi
M-step:
Let E{ h(bi ) | Vi , i ,Wi , ti , }  Ei { h(bi ) }
2
e
be the conditional expectation based on the current estimate   (  ,  , ,  ,  0 ).
Dfferentiating Ei {log L* ( )}

n
   Ei (bi ) / n,
i 1
n
   Ei (bi   )(bi   )T / n,
i 1
n
mi
n
   Ei {Wij biT  (tij )}2 /  mi
2
e
i 1 j 1
i 1
 For  0 :
Let T1 ,..., Td denote d distinct uncensored event time
The corresponding baseline survival time are:
Tk
uk   exp{ bkT  ( s)}ds, k  1,..., d
0
Estimate uk by current estimate of  and the current empirical Bayes estimate of bi
u ( k ) denote these estimates in ascending order--- 0 =u (0)  u (1) 
 u(d )
Therefore, we have
d
0 (u )   Ck 1{u
k 1
Ck 
 For


n
i 1
n
i 1
( j 1)  u  u ( j ) }
Ei [ i 1{u ( k 1) u u ( k ) } ]
i
Ei [{u ( k )  u ( k 1) }1{u ( k 1) u u ( k ) } ]
i
:
Plug  0 in Ei {log L* ( )},
d
d


T
Ei  i log[ C j 1{u( j1) u u( j ) } ]   i  {bi  (Vi )}   C j {u ( k )  u ( k 1) }1{u( k 1) u u( k ) } 

i
i 1
j 1
j 1


n
n
n
mi
i 1
i 1
j 1
  Ei {log f (bi | ,  )}   Ei { log f (Wij |bi ,  e2 )}
no closed form expression for . We may
maximize the conditional likelihood by numerical
method.
There’s
E-step:
To compute Ei (.),we need knowledge of f (bi | Vi , i ,Wi , ti , )
which can be expressed as:
f (Vi , i | bi , ti ,  ) f (bi | Wi , ti ,  )
 f (Vi , i | bi , ti , ) f (bi | Wi , ti , )dbi
Let  *  { T (ti1 )  ,...,  T (timi ) }T , A  { (ti1 ),...,  (timi )}T .
  *   11 12  
 Wi 
Then   ~ N   , 
  , and therefore
     21  22  
 bi 
1
1
bi | Wi , ti ,  ~ N {   2111
(Wi  A ),  22   2111
12 }
To
derive Ei (.), we may generating M multivariate
normal sequences for bi | Wi , ti , , denoted by Ni  ( Ni1 ,...NiM )

E {h(b )} 
i
The
i
M
j 1
h( N ij ) f (Vi ,  i | N ij , ti ,  )

M
j 1
f (Vi ,  i | N ij , ti ,  )
, M is large.
T accuracy increases as M increases. In order to
have
h higher accuracy and less computing time, we
may follow the suggestion in Wei and Tanner (1990)
. That is, to use small value of M in the initial iterations
of the algorithm, and increase the values of M as the
algorithm moves closer to convergence.
We encounter two difficulties when estimating standard
error of  :
EM
algorithm involved missing information
-Remedies in Louis (1982) and McLachlan and
Krishnan (1997) are valid for finite dimensional
parameter space.
No
explicit profile likelihood
- Need projection onto all other parameters
- However, it’s very hard to derive due to λ0
Bootstrap
technique in Efron(1994):
1. Generating bootstrap sample 0* from original observed data 0 .
*
2. The EM algorithm is applied to the bootstrap sample  to derive the MLE  .
*
0
3. Repeat step 1 and 2 B times.
B
B
4. Compute Cov( )  1/( B  1) (   b )   b ) , where  b =  / B.
*
b 1
*
b
*
b
T
b 1
*
b
IV. Simulation Studies
Sample
size n=100 with 100 MC replications
-- preliminary scheduled measurement times: (0, 1, ... , 7)
--  (t )  (1, t )
--   (1,0.5)T
-- 0  1,   1,  e2  0.25
(i) No censoring with ( 11 ,  12 ,  22 )  (0.01, 0.001,0.01)
(ii) With censoring time ~ exponential distribution with mean 25.
(iii) With same setting except  22  0.3 and 35% negative values of bi are truncated
(i) Normal random effects without censoring
β
μ1
μ2
σ11
σ12
σ22
σe2
target
1
1
0.5
0.01
-0.001
0.001
0.25
mean
1.0075
0.9955
0.5013
0.0087
-0.0011
0.0009
0.2528
SD
0.0945
0.0163
0.0055
0.0015
0.0002
0.0002
0.0135
(ii) Normal random effects with censoring
β
μ1
μ2
σ11
σ12
σ22
σe2
target
1
1
0.5
0.01
-0.001
0.001
0.25
mean
0.9918
0.9944
0.5015
0.0083
-0.0011
0.0009
0.2516
SD
0.1272
0.0249
0.0056
0.0023
0.0004
0.0002
0.0198
(iii) Nonnormal random effects with censoring
β
μ1
μ2
σ11
σ12
σ22
σ2e
target
1
1
0.5
0.01
-0.001
0.001
0.25
empirical
target
1
0.9993
0.6758
0.0104
-0.0058
0.1358
0.2753
mean
0.9950
1.0007
0.6682
0.0099
-0.0006
0.1627
0.2500
SD
0.1091
0.0140
0.0535
0.0004
0.0036
0.0318
0.0223
V. Application on Medfly data

The medfly (Mediterranean fruit fly) data:
--From Carey, et al. (1998)
-- We focus on 251 female medflies which have the
most egg reproduction (>1150).
--Range of event time from 22 to 99
-- Range of total reproduction from 1151 to 2349
--No censoring and missing

Relationship between daily egg laying and mortality
--Violate the proportionality (By scaled
Schoenfeld residual test with p-value 0.00305)
Profiles of daily egg laying of first three flies
5
subject1
subject2
subject3
4.5
4
log(# of daily egg laying+1)
3.5
3
2.5
2
1.5
1
0.5
0
0
20
40
60
Time
80
100
120

Initial model:
W (t )  X (t )  e (t )
*
*
*
X (t )  t  exp[b1 (t )]
*

b0
Log transformed model:
W (t )  log[W * (t )  1]  X (t )  e(t )
X (t )  b0 log(t )  b1 (t  1)
The
parameter estimates derived from original data
and 100 bootstrap samples under the joint AFT
β
μ1
μ2
σ11
σ12
σ22
σe
fitted
values
-0.4340
2.1227
-0.1442
0.3701
-0.0482
0.0068
0.8944
bootstrap
mean
-0.4313
2.1112
-0.1429
0.3651
-0.0483
0.0066
0.8958
bootstrap
SD
0.0115
0.0375
0.0051
0.0353
0.0002
0.0005
0.0223
Fitting
incomplete medfly data:
--Randomly select 1-7 days as the corresponding schedule
times for each individual.
--Then, add the day of death as the last schedule time.
Therefore, each individual may have 2-8 repeated
measurements.
-- The sub data set is further censored by exponential
distribution with mean 500 (20% censoring rate)
The
parameter estimates derived from incomplete
data and 100 bootstrap samples under the joint AFT
β
μ1
μ2
σ11
σ12
σ22
σe
fitted
values
-0.3890
2.2011
-0.1665
0.2833
-0.0382
0.0051
0.9775
bootstrap
mean
-0.3526
2.1986
-0.1575
0.2862
-0.0398
0.0057
0.9712
bootstrap
SD
0.0323
0.0461
0.0074
0.0351
0.0046
0.0006
0.0570
The End