48x36 Poster Template

1
Weighted Kaplan-Meier Estimator for Adaptive
Treatment Strategies in Two-Stage Randomization
Designs Sachiko Miyahara, Abdus S. Wahed, Ph.D
Department of Biostatistics, Graduate School of
Public Health, University of Pittsburgh Summer
2007 Program on Challenges in Dynamic Treatment
Regimes and Multistage Decision-Making , SAMSI,
Research Triangle Park, NC
Leukemia Data Analysis
Simulation Results
Abstract
Two Existing Methods
Statistical inference for adaptive
treatment strategies from two-stage
randomization designs is mainly done through
marginal mean models using inverse-probability-we
ighting (Murphy et al., 2001, JASA). In survival
settings, Lunceford et al. (2002, Biometrics)
proposed an estimating equation based
semi-parametric approach for estimating the
survival distribution for treatment strategies.
A weighted risk set estimator (weighted
Nelson-Aalen estimator) has recently been
proposed in Guo and Tsiatis (2005, International
Journal of Biostatistics). Although these methods
have provided methodologies for consistently
estimating parameters from two-stage
randomization designs, their implementation is
not as straightforward. In this study we propose
a weighted Kaplan-Meier estimator and compare its
properties to that of the estimating equations
based estimator and the weighted risk set
estimator through simulation. We analyzed a
Leukemia data set to demonstrate the application
of the weighted Kaplan-Meier estimator.

The two stage double-blind placebo-controlled
randomized clinical trial, Protocol 8923, was
conducted by the Cancer and Leukemia Group
(CALGB) . The data set contains 388 elderly
patients with acute myelogenous leukemia. At
the first stage, patients were randomized into
one of the initial treatments (GM-CSF or Placebo
following standard chemotherapy). At the second
stage, the responders were randomized into one of
two intensification treatments. Data
Modification Since there was no second stage
treatment for the non-responders, we have
simulated the second treatment B2 using
Bernoulli(0.5) distribution.
Monte Carlo mean and standard deviations are
shown in Table 1 below.
1. Estimating Equation Based Approach (Lunceford
et al., 2002)
N T P(Resp) S(t) WKM(t) (MCSE(t)) EEB(t) WRSE(t) (MCSE(t)) (MCSE(t)) EEB(t) WRSE(t) (MCSE(t)) (MCSE(t))
200 0.5 0.4 0.792 0.804 0.806 0.800
200 0.5 0.4 (0.051) (0.051) (0.045)
200 0.5 0.7 0.846 0.835 0.838 0.832
200 0.5 0.7 (0.051) (0.051) (0.044)
200 1.0 0.4 0.461 0.440 0.484 0.489
200 1.0 0.4 (0.114) (0.067) (0.056)
200 1.0 0.7 0.574 0.552 0.563 0.567
200 1.0 0.7 (0.097) (0.069) (0.057)
500 0.5 0.4 0.792 0.807 0.800 0.808
500 0.5 0.4 (0.032) (0.028) (0.032)
500 0.5 0.7 0.846 0.838 0.839 0.831
500 0.5 0.7 (0.032) (0.033) (0.028)
500 1.0 0.4 0.461 0.489 0.486 0.488
500 1.0 0.4 (0.042) (0.036) (0.043)
500 1.0 0.7 0.574 0.568 0.565 0.563
500 1.0 0.7 (0.042) (0.044) (0.036)
.
where
Leukemia Data Analysis Results
Applying the three methods to the modified CALGB
8923 data, the survival distributions of the
treatment policy A1B1B2 were estimated.
The results are shown in Figure 1 blow.
Two-Stage Randomized Trials
is a consistent estimate of p from the sample
data.
The two-stage adaptive treatment regimes design
is as follows
Table 1. Simulation Results True Survival Rate
and Three Survival Estimators with n200 and 500,
t0.5 and 1.0, P(Response)0.4 and 0.7
2. Weighted Risk Set Based Approach (Guo and
Tsiatis, 2005)

• All three estimators were approximately
  unbiased.
• Absolute relative biases were very small and
  ranged from 0.038 to 0.061.
• The Monte Carlo standard deviations for the
  estimators were comparable.

where
Conclusions
These two estimators are consistent however, the
implementation is not as user-friendly.

• We proposed a weighted Kaplan Meier estimator to
  estimate the survival distribution of treatment
  strategies in two stage randomization designs.
• The estimator performs well in terms of bias and
  variances compared to the existing estimators.
• It is easy to implement using standard software
  packages.

Proposed Weighted Kaplan-Meier
Figure 1. Three Estimators with Modified Leukemia
Data.
At the first stage, patients are randomized into
one of the initial treatments (A1 or A2). At the
second stage, depending on how they respond to
the initial treatment, they are randomized again
for the second treatments (B1 or B1, and B2 or
B2). With this design, there are eight treatment
policies (A1B1B2, A1B1B2, etc.), and the goal is
to consistently estimate the survival function
for each treatment policy.
If everyone in the sample were treated with the
policy A1B1B2, their survival rate at time t
could be estimated using the Kaplan-Meier

• For this data set, the weighted risk set
  estimator (WRSE) estimates
• were always larger than the estimating
  equation based (EEB) and
• the weighted Kaplan-Meier (WKM) estimates.
• The EEB and the WKM estimators followed each
  other very closely.

estimator as follows
1 If t lt t1
Acknowledgement
If t t1
Notation
Simulation Study
We thank SAMSI, and Professor Steve Wisniewski of
University of Pittsburgh for funding, and the
CALGB group and Professor Butch Tsiatis of North
Carolina State University for kind permission to
use the leukemia dataset.
The following notations were used throughout this
poster R Response status
TR Time to second treatment C
Time to Censoring T Time to death
V Minimum of T or C X1 A1
treatment indicator Z1 Indicator
variable for the second treatment B1
(If Z11, then B1) Z2
Indicator variable for the second treatment B2
(If Z21, then B2) ?
Indicator variable for death pA1B1B2
Probability that a subject will follow the policy
A1B1B2 pA1 Proportion of subject s
with A1 pRA1 Response rate for
A1group pZ1 Proportion of
subjects receiving B1 within responders
pZ2 Proportion of subjects with B2 within
non-responders
True population
where

• R Bernoulli(0.4) and Bernoulli(0.7)
• Z1 Bernoulli(0.5) for R1 only
• Z2 Bernoulli(0.5) for R0 only
• TR Exp(1)
• C Uniform(0, 2.5)
• TA1B1 TR Exp(1) --Time to death for
  patients receiving A1B1
• TA1B2 TR Exp(0.5) --Time to death for
  patients receiving A1B2
• T (1-R)TA1B2 RTA1B1

tj , j 1, 2, .., are distinct ordered failure
times.
However, in the two-stage randomized trial
settings, subjects receive a subsequent treatment
based on the result of the first treatment.
Therefore some patients will potentially receive
treatment inconsistent with the policy (e.g.,
responders who receive B1). Therefore, we need
to adjust for the loss of these patients. We
propose to use the weighted
References
Murphy S.A., van der Laan M.J., Robins J.M.
CPPRG (2001) Marginal Mean Models for Dynamic
Regimes. JASA 96 1410-1423 Lunceford, J.K.,
Davidian, M., and Tsiatis, A.A. (2002) Estimation
of Survival Distribution of Treatment Policies in
Two-Stage Randomization Designs in Clinical
Trials. Biometrics, 58, 48-57 Xiang Guo
Anastasios Tsiatis, 2005. "A Weighted Risk Set
Estimator for Survival Distributions in Two-Stage
Randomization Designs with Censored Survival
Data," International Journal of Biostatistics,
Berkeley Electronic Press, vol. 1(1), pages
1000-1000.
Kaplan Meier estimator.
1 If t lt t1
Sample

• 5000 datasets
• 200 and 500 observations
• R, Z1, Z2, TR, C, TA1B1, TA1B2 are the same as
  above
• TA1B1, TA1B2 are generated from exponential
  distributions
• T (1-R)Z2 TA1B2(1- Z2) TA1B2 RZ1
  TA1B1(1- Z1) TA1B1
• V min(T, C)

If t t1
where
QA1B1B2 is defined at the estimator 1