Download 2D - Hee - Society for Clinical Trials

Designing a Series of Hybrid Trials
with Continuous Outcomes
Siew Wan Hee
SCT 34th Annual Meeting (2013)
Motivation: Phase II
Proposal
…………
X1  N(q1, 2/n)
X2  N(q2, 2/n)
Xk  N(qk, 2/n)
Random parameter
qi  N(m, t2),
i = 1, 2, …, k
Sample Size Determination
H0: qi = q0
vs.
H1: qi ≠ q0
• q0 mean of historical control
• Controlling type I and II errors
Pr( X i  c | qi  q0 )   / 2
Pr( X i  c | q i  q A )  1  
Sample Size Determination
 ( z1 / 2  z  )
n
2
(q A  q 0 )
2
2
Assurance (Average Power)
A(n)   ( Power ) f (q i ) dq i
2
z


n
/

(
m

q
)
1 / 2
0 

 1 
2
2


1  nt / 


Average probability of rejecting H0 over all
possible values of the parameter of
interest based on prior density
Limits of Assurance
lim A(n)  1  z
n 0
1 / 2

2
2
z


n
/

(
m

q
)
0 
 1 / 2
A
(
n
)

1


lim
lim
2
2


n 
n 
n
t
/



 m  q0 
 

 t 
Maximization of No. of Successful
Trials
Population of N
Sample size of n
K = N/n
𝐾 𝑛 = no. of trials reject 𝐻0
𝐸 𝐾 𝑛
max 𝐸 𝐾 𝑛 ∗
𝑛∗
= 𝐾𝐴 𝑛
= 𝐾𝐴(𝑛∗ )
Limits of 𝑬(𝑲 𝒏 )


N
~
E K (n)  KA(n)  A(n)
n
n0
~
E
K
(
n
)


lim
n 0


Limits of 𝑬(𝑲 𝒏 )


N
~
E K (n)  KA(n)  A(n)
n
n


~
 m  q0 
E K (n)  lim A(n)  

lim
 t 
n 
n 
Summary
Optimal sample size, n* = 0
Maximum of expected number of successful
trials is attainable by not recruiting any
patient to the trial
Include Start-up Cost
1 successful trial = 1 unit of gain
𝐸 𝐾 𝑛
successful trials = 𝐸 𝐾 𝑛
units
of gain
l2 = fixed start-up cost of a trial; relative to 1
unit of gain
l2K = total cost for all trials
𝐺 𝑛 =𝐸 𝐾 𝑛
− 𝑙2 𝐾
Maximization of Expected Utility
lim 𝐺 𝑛 = 𝐸 𝐾 𝑛
𝑙2 →0
lim 𝐺 𝑛 < 0
𝑙2 →1
Example: N = 1000

m
t
1.25
1
1
0.93
132.642
1.25
1
2
0.40
258.465
1.25
1
5
0.09
949.198
2
1
1
2.37
51.813
2
1
2
1.02
100.963
2
1
5
0.24
370.974
5
1
1
14.83
8.290
5
1
2
6.38
16.154
5
1
5
1.51
59.356
n*
G(n)
Summary
Negligible cost; optimal sample size, n* = 0.
High cost; not worth starting any trial at all.
Variance of prior increases; expected utility
increases, sample size decreases.
Variance of likelihood increases; expected
utility decreases, sample size increases.
Conclusion
Combination of classical frequentist and
Bayesian.
Design each trial as part of a series of trials
so that the long-term gain will be the
greatest.
Flexibility to run either sequentially or
concurrently.
References
O'Hagan, A. and Stevens, J. W. (2001). Bayesian assessment of
sample size for clinical trials of cost-eectiveness. Medical Decision
Making, 21(3):219-230.
Whitehead, J. (1985). Designing Phase II Studies in the Context of a
Programme of Clinical Research. Biometrics 41, 373-383.
Whitehead, J. (1986). Sample sizes for phase II and phase III clinical
trials: An integrated approach. Statistics in Medicine 5, 459-464.
Acknowledgement
This work forms part of my PhD which was
supervised by Nigel Stallard.
Thanks to Roche Products Limited and
Warwick Medical School for the financial
support for this PhD project.