EÆcient Estimation of the Mean of a
Time-Lagged Response Subject to Right
Censoring
Anastasios A. Tsiatis
Department of Statistics
North Carolina State University
OUTLINE OF TOPICS
Examples of time-lagged response
Censoring
Inverse probability weighted complete case
http://www.stat.ncsu.edu/tsiatis/
estimator
Asymptotic properties of weighted estimator
EÆciency issues
Slide 1
Slide 2
General set-up
Data are collected on each individual from
clinical trial
Primary goal
{ Estimate R = E(R)
{ Model the relationship of R to covariates
For a variety of reasons, the response for the
ith individual, Ri , is not available
immediately but rather becomes known after
some time Ti which may vary by individual
the time they enter a study until time Ti
fZi(u); u Tig
Denote by ZiH (x) the history of data
collected on the ith individual up to time x
ZiH (x) = fZi (u); u xg
{ Such data may be necessary to dene the
response outcome Ri
{ Auxiliary information
The response is given by
Ri = g(Ti ; ZiH (Ti ))
where g() maps a realization of the data
(t; Z H (t)) to a real number
Slide 3
Slide 4
Statement of Problem
Let R denote the outcome of interest in a
Examples
Survival time
is survival time Ti
R = E (T )
R i = I ( Ti u )
R = P (T u) = S (u) is the survival
distribution at time u
Ri
Medical costs
Ri denotes cost of care for duration of disease
Ti duration of disease
Zi(u) data collected on individual i at u units
of time after entry into study
{ hospital bills
{ physician fees
{ Zi(0) may include patient's baseline
characteristics
Slide 5
Slide 6
Quality-Adjusted Life
Examples
K health states
(q1; : : : ; qK ) denotes utility (0-1) associated
with each health state
(T i; : : : ; TKi) denotes times in each health
state
Quality-adjusted life is given by
TWiST is dened as Time without symptoms
and toxicity
TWiST=T i , q = 0; q = 1;
2
1
2
q3 = 0
1
QALi =
Ti =
K
X
j =1
K
X
j =1
qj Tji
Tji
Ri = QALi: mean quality-adjusted life
Ri = I (QALi u): distribution function of
quality-adjusted life
Slide 7
Q-TWiST is dened as Quality adjusted time
without symptoms and toxicity
e.g. T i + T i + T i
1
2
1
2
1
3
3
Slide 8
other examples
recurrent events, gap times
Lagged Response
Ri denotes dichotomous response (0,1)
R = P (R = 1)
response is available after some lag time Ti
{ biological lag
{ administrative lag
Slide 9
(joint distribution of )
In some cases, the response Ri is not known
until time Ti with no intervening information:
marked point processes (Huang and
Louis)
In other cases Ri = Ri(Ti ) might be the
realization of an increasing stochastic process
Ri (u) at a stopping time Ti , where the
process Ri (u) is observed
(Strawderman, R.)
{ quality-adjusted life
{ medical costs
Slide 10
Censoring
In most clinical trials, the time Ti , which is
No censoring
Data: (Ri; Ti; ZiH (Ti )); i = 1; : : : ; n represent
a sample of data from some underlying
population
^R = n Pni Ri
1
=1
Slide 11
necessary to observe the response Ri , may
not be observed for all individuals due to
censoring
Censoring may result from
{ staggered entry and limited follow-up
(administrative censoring)
{ lost to follow-up or drop out
Let Ci denote the potential censoring time for
ith individual
Slide 12
In the presence of censoring, we observe for
individuals i = 1; : : : ; n
fXi = min(Ti; Ci ); i = I (Ti Ci );
ZiH (Xi ); i = 1; : : : ; ng
How do we estimate
R = E (R )
with a sample of censored data?
Assume that C is independent of (T; Z H (T ))
If i = 1 we observe Ri, otherwise, we only
have partial information
Delete censored observations
For increasing stochastic processes, where
Ri = Ri (Ti ) and (Ri (u); u Xi ) is observed,
we may want to treat Ri(Ci ) as a censored
observation for individual i, if i = 0, and
then use censored data methods such as
Kaplan-Meier estimate
This induces informative censoring
These naive methods can result in severely
biased estimators
Slide 13
Slide 14
Informative Censoring
Informative Censoring
If we were estimating distribution of TWiST
Naive methods
is independent of (T i; T i)
variable of interest T i
This is censored by (Ci T i)
but (Ci T i) is not necessarily indpendent of
T i which is needed for the Kaplan-Meier
estimator to be consistent
Ci
1
2
2
1
1
2
Slide 15
If we were estimating distribution of Medical
Costs
Suppose medical cost for each individual
increased proportionally over time
Ri = i Ti, where i varied by individual
Ci is independent of Ti and i
However, in a typical censored analysis, we
would censor Ri by the cost at censoring;
i Ci .
let us even assume that i is independent of
Ti
cov(i Ci ; i Ti) = var(i )E (Ti)E (Ci ) 6= 0
Thus the censoring variable i Ci is not
independent of the variable of interest i Ti,
hence Informative Censoring.
Slide 16
Inverse probability weighted complete case
estimator
Robins & Rotnitzky (1992)
Let K (u) = pr(C u)
Each person that is observed at time u is
representative of 1=K (u) people that might
have been observed if there was no censoring
This suggests using
n X
i R
^R = n
i
K
(
T
i)
i
(i = 1) corresponds to a complete
observation
i = I ( C i Ti )
1
=1
Unbiasedness
E f^R g
E f i Ri g
K (Ti )
i Ri jTi ; Z H (Ti)
E E
i
K (Ti )
Ri
E
E I (Ci Ti )jTi ; ZiH (Ti )
K (T i )
R
=
=
=
=
Slide 17
Slide 18
Since K (Ti) is not known, we suggest using
Asymptotic properties of our weighted
estimator
K-M estimator Kb (Ti) for the censoring
random variable. This can be done by using
the data
fXi = min(Ti; Ci ); 1 i ; i = 1; : : : ; ng
Our estimator is therefore
n X
i R
^R = n
i
i Kb (Ti )
Note that when Ri = I (Ti u) then the
above estimator is identical to the
(1-Kaplan-Meier estimator) for the
distribution function of T
1
=1
Slide 19
n 2 f^R R g
n
1 X i Ri
(
R )
n 2
i=1 Kb (Ti )
n R
1 X
n 2
( K (iT i) R)
i
i=1
(
)
n
1 X
1
1
2
n
i Ri b
K (T i ) K (T i )
i=1
1
=
=
+
=
n
n
1
2
n
X
i=1
n
1 X
2
i=1
( K(iTRi)
i
i Ri
(
R )
b (T i ) K (T i )
K
b (T i )K (T i )
K
Slide 20
)
Using the following equalities
i = 1 Z 1 dMic(u)
K (T i )
K ( u)
n Z T Kb (u )dM c(u)
X
b (T i ) K (T i )
K
j
=
K (Ti )
K ( u) Y ( u )
j
We can write
n f^R R g
n
X
fR i R g
= n
IMPROVING EFFICIENCY
0
i
=1
0
1
2
1
2
n
where
1
2
i=1
n Z L dM c(u)
X
i f Ri
i=1 0 K (u)
b (R; u) =
G
b (R; u)g
G
n R I (T u )
1 X
i i i
:
b ( Ti )
nSb(u ) i
K
The weighted estimator we presented so far
does not use the individual's data history
This naive estimator would be eÆcient if no
intervening information was available
regarding a patients history
Identical to the Huang-Louis estimator for
marked point processes
With additional information, the eÆciency of
the estimator can be improved greatly
=1
Slide 21
Slide 22
IMPROVING EFFICIENCY
IMPROVING EFFICIENCY
All estimators for R are asymptotically
equivalent to one of the estimators in the class
n R
X
i i
n
^
K
(
Ti )
i
1
=1
+n
n
X
1
(1
i)[efZiH (Xi)g e(Xi)]
i=1
where efZiH (u)g is any functional of the
history ZiH (u), and
X efZiH (u)gYi (u)
e(u) =
Y (u )
Knowledge of Ef KR(u) jZiH (u)g is needed
This can be very complicated
One simple way to improve eÆciency is to
i
take the weighted estimator, say ^R, and any
functional, say efZiH (u)g = Ri (u), or vector
of functionals, and compute
^R +
C
"
n
1
n
X
i=1
(1
i )[efZiH (Xi)g e(Xi)]
#
The value C, which minimizes the variance
nd the most eÆcient inuence function for
our problem using
R
efZiH (u)g = Ef i jZiH (u)g
K (u )
above, can be estimated straightforwardly, as
well as the asymptotic variance of the
resulting estimator
Approach taken by (Zhao and Tsiatis) and
(Bang and Tsiatis)
Slide 23
Slide 24
Using the Cauchy Schwartz inequality, we can
Dependent Censoring
If censoring is not independent of a patient's
SIMULATION STUDY
TOX uniform [0,72]
TR exp(120)
FU uniform[48,96]
Sample Size = 100
Simulation times = 2000
TWiST=TR-min(TR,TOX)
Estimate P (TWiST u)
Slide 25
Conclusions and Future Directions
A consistent estimator has been found, whose
asymptotic variance can be derived
Using semiparametric theory, we can also
derive the formula for the most eÆcient
estimator
Using weighted \complete-case estimating
equations", estimates for the parameters in
regression models can be obtained
i.e. E(RjX ) = (X; ), where X is a set of
covariates
Slide 27
history then the simple weighted estimator
may also be biased
By modeling the propensity of censoring as a
function of history, one can construct an
unbiased estimator by using
n X
i R
n
i
^
K
(
T
i
i)
i
where K^ i (Ti) is the estimated
P f(Ci Ti )jTi ; ZiH (Ti )g
Modeling the censoring distribution to health
history can be accomplished using a
proportional hazards model with
time-dependent covariates
Modeling the censoring distribution results in
improved eÆciency even if censoring is
independent of health history
1
=1
Slide 26