A SEMIPARAMETRIC ESTIMATOR
FOR THE PROPORTIONAL HAZARDS
MODEL WITH LONGITUDINAL
COVARIATES MEASURED WITH
ERROR
Marie Davidian and Anastasios A. Tsiatis
North Carolina State University
http://www.stat.ncsu.edu/davidian/
http://www.stat.ncsu.edu/tsiatis/
OUTLINE
1. Introduction and example
2. Conditional score estimator
3. Large-sample properties
4. Simulation evidence
5. Example, revisited
6. Discussion
Slide 1
Slide 2
1. INTRODUCTION AND EXAMPLE
Classic example: Relationship of immunologic/virologic status to
Longitudinal studies in medical research:
Time-to-event, e.g. death, progression to AIDS, etc. (possibly
censored)
Intermittent measurements on a covariate (time-dependent)
thought to be related to prognosis
Other covariates, e.g. treatment group, age at baseline
A question of interest: Relationship between time-to-event and
covariate process
Slide 3
survival in HIV-infected subjects
\Surrogate marker?"
AIDS Clinical Trials Group (ACTG) 016: Ecacy of ZDV in
mildly symptomatic HIV-infected patients
Initiated 1987, follow-up data to 1991
200 mg ZDV / 4 hours vs. placebo
\Survival" { time to death or progression to AIDS (develop OI)
Slide 4
log CD4 trajectories and times-to-event, placebo group
(first 50 subjects with >3 obs)
8
CD4 lymphocyte count: Immunologic status
0
7
Measured at baseline and intermittently every 4 weeks until week
00
0
0
0
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0
0
0
0 0
0 0
0
0
1
1
0
0
0
0
0 0
0
0
0
0
0
5
log CD4
Low CD4 is BAD
Well-known to be subject to substantial measurement error,
1
00
0
6
16 and then every 8{12 weeks
1
1
0
0
0
0
0
0
00
0
1
0
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4
biological variation
3
Most subjects show rough linear trend over time in log CD4
0
1
censored
event
2
(many decline)
0
20
40
60
80
100
weeks
Slide 5
Proportional hazards model:
Xi (u) = covariate value at time u (e.g. CD4) and history
XiH (t) = fXi(u); u tg
Zi other time-independent covariates
Ti time-to-event
i(u) = dulim
du,1prfu Ti < u + dujTi u; XiH (u); Zig
!0
T
= 0 (u) expf |X{z
i (u) + Zi g
}
Interest in estimating , ; (q 1):
Slide 7
Slide 6
Censored data: For subject i = 1; : : : ; n ( k )
Underlying censoring variable Ci
(Ti ; Ci ) = (survival,censoring) time for i
Observe Vi = min(Ti ; Ci ), i = I (Ti Ci )
Zi time-independent covariates (e.g. treatment)
Cox (1972, 1975) partial likelihood: maximize
"
expfXi (Vi ) + T Zi g
Pn
exp
fXj (Vi) + T Zj gI (Vj Vi)
j =1
i=1
n
Y
#i
Presumes Xi (u); i = 1; : : : ; n is available at all failure times
Slide 8
Popular approach: Joint longitudinal covariate-survival model
In actuality:
Xi(u) = 0i + 1iu; i = (0i; 1i)T (generalized to more
Xi (u) only available intermittently
Subject to error
Naive approach: Extrapolate Xi(u) at failure times u using
available longitudinal data, substitute in partial likelihood
E.g. use most recent observed, error-prone covariate value (in
place of unobserved, \true" covariate)
\Last Value Carried Forward" (LVCF)
Such substitution ) biased estimation (Prentice, 1982)
complex models)
Observe at times tij , j = 1; : : : ; mi
Wi(tij ) = Xi(tij ) + eij ; eij iid
N (0; 2) k (Ti; Ci; i; Zi; tij )
ti(u) = (tij < u) = measurement times prior to u
i usually assumed normal
Proportional hazards model
i (u) = dulim
du,1prfu Ti < u + dujTi u; i; Zi; Ci ; eij ; ti(u)g
!0
T
= 0 (u)expf X
i (u) + Zi g
}
| {z
Slide 9
Implementation:
Two-stage (\Regression calibration"): (i) EBLUPs from mixed
model t at each survival time, (ii) insert in Cox partial
likelihood ) reduces but doesn't eliminate bias (Pawitan & Self,
1993; Tsiatis et al., 1995; Dafni & Tsiatis, 1998)
Likelihood: NPML via EM (Wulfsohn & Tsiatis, 1997;
Henderson et al., 2000)
Fully parametric Bayesian: Faucett & Thomas (1996)
Related: DeGruttola & Tu (1994), Henderson et al. (2000)
Drawbacks:
Heavy reliance on normality of underlying random eects {
consequences if violated?
Computational issues : : :
Slide 11
Slide 10
Our objective: For mixed eects/proportional hazards model
Simple method (computationally and conceptually
straightforward) that yields consistent, asymptotically normal
estimator for Cox parameters
: : : And that furthermore makes no distributional assumption
about the underlying random eects
Our approach: Exploit the conditional score idea of Stefanski &
Carroll (1987) for GLIMs
\Condition away" the random eects i
Slide 12
Notation:
2. CONDITIONAL SCORE ESTIMATOR
Assume:
Timing of measurements is not prognostic
Noninformative censoring
Nothing about i (usually assumed normal)
Assume 2 known for now
X^i(u) OLS estimator for Xi(u) based on ti(u) (requires 2
measurements prior to u)
X^i(u)ji; ti(u) NfXi(u); 2i (u)g, Xi(u) = 0i + 1iu
2i(u) (prediction variance) depends on ti(u)
dNi(u) = I (u Vi < u + du; i = 1; ti2 u) puts point mass at
u for observed death time if after 2nd measurement
Yi(u) = I (Vi u; ti2 u) = at risk with 2 measurements
at u
Slide 13
Slide 14
At any time u: Conditional on being at risk at u
prfdNi(u) = r; X^i (u) = xjYi (u) = 1; i ; Zi ; ti(u)g
= prfdNi(u) = rjYi (u) = 1; X^ i (u) = x; i ; Zi ; ti (u)g
prfX^i(u) = xjYi(u) = 1; i; Zi; ti(u)g
1st piece: Bernoulli[0(u)du expfXi(u) + T Zi g]
2nd piece: NfXi(u); 2i(u)g
At time u, likelihood fdNi(u); X^i(u)gjYi(u) = 1; i; Zi; ti(u): To
order du
"
)#
(
2
^i (u)
i (u) + X
= exp Xi (u) i (u)dN
2 i (u)
(
)
T Zi)dugdN (u) exp , X^i2(u) + Xi2 (u)
f0(u)f2exp(
22i (u)
2i (u)g1=2
Thus: Sucient statistic for i
Si (u; ; 2) = 2i(u)dNi(u) + X^i (u)
(conditional on Yi (u) = 1)
Suggestion: Conditioning on Si(u; ; 2) would remove the
dependence of the conditional distribution on (the nuisance
parameter) i
Form estimating equations in same spirit as the partial likelihood
score equations
i
Slide 15
Slide 16
Heuristically: Compare observed number of failures at u (0 or 1) to
expected number conditional on being at risk at u and covariates
0 0 0 dNi(u)
0 1 0 Usual partial likelihood score equations: Xi(u) known
limdu!0 du,1prfdNi(u) = 1jXi (u); Zi; Yi (u)g =
E0(u; ; ) = Pnj=1 E0j (u; ; ), dN (u) = Pnj=1 dNj (u)
^0 (u)du = dN (u)=E0(u; ; )
Substitute ^0(u)du, rearrange { solve in (; )
n Z X
0 (u) expfXi(u) + T ZigYi(u) = 0(u)E0i(u; ; )
Estimating equations
n Z
X
=1
i
fXi(u); Z g
n
X
=1
i
T T
i
fdNi(u) , E0i(u; ; ; 2)0(u)dug = 0
=1
i
= 0 (u)E0i(u; ; ; 2 )
fSi(u; ; 2); ZiT gT fdNi(u) , E0i(u; ; ; 2)0(u)dug = 0
n
X
fdNi(u) , E0i(u; ; ; 2)0 (u)dug = 0
=1
P
P
2
E0 (u; ; ; ) = nj=1 E0j (u; ; ; 2), dN (u) = nj=1 dNj (u)
^0(u)du = dN (u)=E0(u; ; ; 2)
i
j
=1
E1j (u; ; )
Slide 18
Substitute ^0(u)du, rearrange { solve in (; )
n Z X
=1
i
Slide 19
2 ; ; )
fSi(u; ; 2); ZiT gT , EE1 ((u;
u; ; ; 2) dNi(u) = 0
0
E1j (u; ; ; 2) = fSj (u; ; 2); ZjT gT E0j (u; ; ; 2);
Estimating equations
=1
n
X
fdNi(u) , E0i(u; ; ; 2)0 (u)dug = 0
Analogy:
Conditional intensity { can show
lim du,1 prfdNi(u) = 1jSi (u; ; 2); Zi ; ti (u); Yi(u)g
du!0
= 0 (u) expfSi (u; ; 2) , 22 i (u)=2 + T Zi gYi(u)
i
0
E1j (u; ; ) = fXi(u); ZiT gT E0j (u; ; ); E1 (u; ; ) =
Slide 17
n Z
X
; )
fXi(u); ZiT gT , EE1((u;
u; ; ) dNi(u) = 0
E1 (u; ; ; 2) =
n
X
E1j (u; ; ; 2)
j
=1
Remarks:
Reduces to usual equations when 2 = 0 [so X^i(u) = Xi(u)]
Easy to compute
Slide 20
At true values (0; 0):
3. LARGE-SAMPLE PROPERTIES
(2) may be shown !p 0
(1) has expectation 0: expectation of ith summand is
Heuristically: Show estimating equations are unbiased
S(u; ; ; 2) = E1 (u; ; ; 2)=E0(u; ; ; 2) !p (u; ; ; 2)
Can write estimating equations as
0 = n,1=2
n Z
X
=1
i
n Z
X
=1
i
(1)
Integrand becomes (iterated conditional expectation)
n
E [fSi(u; 0; 2); ZiT gT , (u; 0; 0; 2)]
E [dNi(u) j fSi(u; 0; 2); ZiT gT ; Zi; ti(u); Yi(u)]
o
,E0i(u; 0; 0; 2)0(u)du] = 0
f(u; ; ; 2) , S(u; ; ; 2)g
fdNi(u) , E0i(u; ; ; 2)0(u)dug
E [fSi(u; 0; 2); ZiT gT , (u; 0; 0; 2)]
fdNi(u) , E0i(u; 0; 0; 2)0(u)dug
[fSi (u; ; 2); ZiT gT , (u; ; ; 2)]
fdNi(u) , E0i(u; ; ; 2)0(u)dug
+n,1=2
Z
(2)
Slide 21
Slide 22
Unbiased estimating equation: Consistency, asymptotic
normality
2 unknown: Replace by pooled estimator
Pn
> 2)Ri
^2 = Pn iI=1(mI (m>i 2)(
mi , 2)
i
i=1
Ri = RSS from all mi measurements on i
n1=2(^2 , 2) =
n,1=2
n
X
=1
i
fE (mi) , 2g,1fRi , (mi , 2)2g + op(1)
Consistency, asymptotic normality preserved by standard
arguments
Standard errors for (^; ^): Usual sandwich technique
Slide 23
4. SIMULATION STUDIES
Simple situation: = 0
E (i) = (4:173; ,0:0103)T , = ,1:0, 0 (u) = 1,
Ci exp(1=110), additional censoring at 80 weeks
Nominal tij = (0; 2; 4; 8; 16; 24; 32; 40; 48; 56; 64; 72; 80), 10%
missing
Hazard u 16, 0 otherwise
i mean E (i) (a) normal, (b) 50-50 mixture of normals, (c)
bivariate skew-normal (Azzalini & Dalla Valle, 1996) chosen to
have similar covariance matrix
Slide 24
Methods: All \simple" to compute
i, Ideal, Xi(u) known for all u
cs, Conditional score, 2 estimated
rc, Regression calibration at quartiles
nr, Naive regression (all mi measurements)
lv, LVCF
Next slide:
500 Monte Carlo replications, n = 200, 2 = 0:30
MC SD, Average estimated SE, Coverage of 95% Wald CI
i
cs
rc
nr
lv
Mean
,1:01
,1:01
,0:93
,0:88
,0:87
i
cs
rc
nr
lv
,1:01
,1:01
,0:92
,0:88
,0:86
Normal
sd
se
0:08 0:09
0:11 0:12
0:08 0:09
0:07 0:08
0:07 0:08
Skewed
0:09 0:08
0:12 0:12
0:07 0:09
0:07 0:08
0:07 0:08
Cov
0:96
0:95
0:87
0:65
0:67
Mean
,1:02
,1:03
,0:88
,0:83
,0:87
Mixture
sd
0:10
0:24
0:07
0:06
0:06
Slide 26
5. EXAMPLE, REVISITED
6. DISCUSSION
Subset of 557 patients (294 placebo, 263 ZDV)
61 events (death or AIDS)
Maximum time on study 180 weeks
Assume: i(u) = 0 (u)expfXi(u)g, ( = 0) Xi(u) = 0i + 1iu
0:12
0:25
0:09
0:08
0:08
Cov
0:95
0:95
0:75
0:44
0:65
0:97
0:95
0:85
0:68
0:61
Slide 25
Subset: Individuals failing within 24 weeks of last CD4
se
Caveats:
Long lags between nal longitudinal measurement and failure or
censoring causes instability
If very few subjects have measurements out to failure or
censoring, probably shouldn't be doing this kind of modeling
anyway, very little info on hazard
,1:611 ,1:385 ,0:782 ,1:307
Not suitable when most subjects have very few measurements
Multiple solutions (hasn't been a problem)
Slide 27
Slide 28
cs
rc
nr
lv
(0.299) (0.195) (0.090) (0.149)
Advantages and extensions:
Works well otherwise
Easy to compute method for a popular model
Unbiased estimation in nite samples
Normal measurement error may be relaxed
Extension to nonlinear models
Improve eciency?
Slide 29