Transcript No Slide Title

Sample Size Calculation for Comparing Strategies in Two-Stage
Randomizations with Censored Data
Zhiguo Li and Susan Murphy
Institute for Social Research and Departments of Statistics, University of Michigan, Ann Arbor
Introduction
Sample size calculation
Test statistics: comparing two strategies
Clinical trials with two-stage randomizations become
increasingly popular, especially in areas such as cancer
research, substance abuse, mental illness, etc.
Notation:
Strategy “11”: get A1 first, and if no response then get B1.
Strategy “22”: get A2 first, and if no response then get B2.
Patients are first randomized to a primary therapy. Then
non-responders (defined by some criterion) are further
randomized to a second stage treatment, and responders
are treated with a maintenance therapy (depending on the
area, sometimes responders are randomized instead of nonresponders).
 Interest is in comparing treatment strategies
(combinations of first stage treatment and second stage
treatment if eligible) and select the best strategy.
 Assuming proportional hazards
 Using asymptotic distribution of test statistics under local alternative
hypothesis
T: time to event, S: time to response, C: censoring time
X=I(A1), Z=I(B1), p=P(X=1), q=P(Z=1)
R: response indicator=I(S<T, S<C)
T11: time to event under strategy “11”, T22: time to event under strategy “22”
 : the time of the end of study
 Using test statistics based on estimation of S jj ( )
S jj ( ) : survival probability under policy jj
 Using weighted log-rank test
n
 Test statistics based on estimation of S jj ( )
 A question of interest is the determination of the
necessary sample size to achieve a certain power for testing
the equivalence of two strategies.
In particular, our interest is in cases where time to some
event may be censored.
X
RZ 

Q1  1  R 
p
q 
1 X 
R(1  Z ) 
1  R 

Q2 
1 p 
1 q 
 dN Q j (t ) 

• Weighted Kaplan-Meier estimator: Sˆ jj ( )   1 


Y
(
t
)
t  
Q1

Patients
with
cancer
Illustration of a twostage randomization
n
d
N
(
t
)
ˆ ( )


Q
jj
1
ˆ
ˆ
 jj ( )  
, S jj ( )  e
YQ1 (t )
0
R = randomization
R
E   Q1 [dN (t )  Y (t )d11 (t ) 
 0

Respond
Gn  
YQ2 (t )dN Q1 (t )
Y
(
t
)

Y
(
t
)
Q2
0 Q1


E   Q1[dN (t )  Y (t )d11 (t ) 
 0

1 

SC (t )dF11 (t )
Guess at the upper bound is relatively easy

p1q1 0

Hazard ratio Percent of
(sample size) subjects
Weighted
randomized Kaplanat the
Meier
second
stage

YQ1 (t )dN Q2 (t )
Y
(
t
)

Y
(
t
)
Q2
0 Q1
18.2
2 (301)
R
Tn | Gn | / SE
1.25 (3376)
0.902
Weighted
NelsonAalen
0.902
Weighted
sample
proportion
0.701
Treatment
B2
Maintenance
therapy
Treatment
B1
Treatment
B2
Maintenance
therapy
Hazard ratio
(sample size)
2 (301)
Percent of
subjects
randomized
at the
second
stage
Weighted
KaplanMeier
Weighted
NelsonAalen
Weighted
sample
proportion
19.4
0.925
0.910
0.713
51.7
0.847
0.845
0.613
65.4
0.812
0.805
0.560
20.8
0.926
0.926
0.762
52.1
0.827
0.827
0.734
53.4
0.850
0.847
0.675
68.2
0.803
0.798
0.590
19.3
0.903
0.903
0.793
50.6
0.832
0.832
0.760
67.1
0.811
0.921
0.675
66.8
0.804
0.802
0.608
19.8
0.918
0.918
0.812
22.1
0.912
0.912
0.784
52.8
0.855
0.855
0.764
52.6
0.847
0.841
0.716
68.5
0.823
0.823
0.700
69.4
0.815
0.815
0.660
Failure time and time to response are generated from a
Frank copula model with a negative association
parameter
Treatment
B1
Achieved power of different tests
Achieved power of different tests
1.5 (965)
R
2

Sample sizes calculated from the test based on the weighted Kaplan-Meier estimator and power of different tests under this sample size
 Weighted log-rank test
Not
respond
2
Using martingale property, this can be bounded by
Tn | Sˆ11 ( )  Sˆ22 ( ) | / SE
Respond
0
Simulation results
•Test statistic:
Not
respond

S C (t ) F11 (t )dt


Q
I
(
U


)
1
i
ji
i
Fˆ jj ( )  1  Sˆ jj ( )  
n i 1
SC (Ui )



2
 Difficulty: variance depends on quantities like the following, which involves time to
response and time to response is correlated with time to event
n
Treatment
A2
( Z1 / 2  Z1  ) 
2
 Most important issue: guess of variance based on prior knowledge before data
collection: usually get an upper bound—conservative sample size
•Weighted sample mean: Lunceford et al. (2002)
Treatment
A1
2
 : Significance level, 1   : power,  : log hazard ratio
2
 : asymptotic variance of (numerator of) test statistic
• Weighted Aalen-Nelson estimator: Guo and Tsiatis (2006):


2
2
Each subject is associated with a weight when estimating survival
probabilities: inverse of the probability that a subject is consistent with a
strategy
(Z1 / 2  Z1 ) 
2
1.5 (965)
1.25 (3376)
Failure time and time to response are generated from a
Clayton copula model with a positive association parameter