Modification of Sample Size in Group Sequential Clinical Trials. Madan Gopal Kundu PhD (Biostatistics) student, IUPUI. Sample Size. Less expensive. More expensive. LESS SAMPLE SIZE MORE. Less Statistical Power. More Statistical Power.
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Modification of Sample Size in Group Sequential Clinical Trials
MadanGopalKundu
PhD (Biostatistics) student, IUPUI
Sample Size
Less expensive
More expensive
LESS SAMPLE SIZE MORE
Less Statistical Power
More Statistical Power
Sample size (N) = No. of patients
Estimation of Sample size depends on - Type I error, Power and Expected effect size
Reasonable power at reasonable cost (Best deal!!)
Sample size is determined at the beginning of trial.
Group Sequential Design
Determine
N
Conduct of Clinical Trial
Final Analysis
Interim analysis 1
Interim analysis 2
Interim analysis 3
Outline
Introduction
A case study
Group sequential Z test
A sequential test procedure with sample size modification (based on Z test)
Generalization: Brownian motion
From Interim analyses
we have….
Sample size is planned based on….
Less than
Expected Effect Size
Observed Effect Size
Planned sample size is NOT sufficient
There is scope to modify sample size when trial is ongoing
Q: Does it increase overall type I error?
Q: Is there any testing strategy to modify sample size without increasing overall type I error?
Planning
Expected
Effect size = 0.30
Interim Analysis
Observed Effect size = 0.14
Phase III, comparative, placebo-controlled trial for prevention of myocardial infarction.
Assumption:
Incidence rate in Placebo: 22%
Incidence rate in New Drug: 11%
Planned sample size = 600 (power>95%)
Concern
Finally…
Does it inflate overall type I error?
No valid testing procedure was available to account for such an outcome –dependent adjustment of sample size
To have accurate estimate of treatment effect size at the beginning of trial
- Less likely!!
Implementation of valid inferential procedure that allows adjustment of sample size in the mid-course of trial
- Cui, Hung and Wang method
Population I
N (µ1, σ2=1)
Population II
N (µ2, σ2=1)
x1, x2, …. , xN
y1, y2, …. , yN
Effect size, ∆ = µ1 - µ2
Our interest is to test (using two sample Z-test)
Ho : ∆ = 0 vs Ha : ∆ > 0
Assuming ∆ = δ, total sample size (N) per population
Trial Initiation
(K-1) Interim Analyses
Final Analysis
0
1
2
L-1
L
K-1
K
Additional Subjects
n1
n2
nL-1
nL
nK-1
nK
Cumulative Subjects
N1
N2
NL-1
NL
NK-1
NK=N
Information Time
Observed Effect size
∆ 1
∆ 2
∆ L-1
∆ L
∆ K-1
∆ K
2-sample Z Test Statistic
T1
T2
TL-1
TL
TK-1
TK
Critical values
C1
C2
CL-1
CL
CK-1
CK
Reject Ho & Stop trial if:
T1>C1
T2>C2
TL-1>CL-1
TK-1>CK-1
TK>CK
TL>CL
The conditional power evaluated at the Lth interim analysis
Calculate and
Decide two positive constants ≤1 ≤
If or , N should be modified to
Simulation studies
Increase in sample size
Substantial inflation in Type I error rate
Decrease in sample size
Mild effect on Type I error rate and power
Trial Initiation
(K-1) Interim Analyses
Final Analysis
0
1
2
L
L+1
L+j
K-1
K
Cumulative sample size
N1
N2
NL
NL+1
NL+j
NK-1
N
Cumulative sample size
(with adjust)
N1
N2
NL
ML+1
ML+j
M
MK-1
Modify sample size in Lth interim analysis: N→M
Test statistic at (L+j)th interim analysis,
… (Eq. 1)
Where,
When sample size is NOT allowed to increase
… (Eq. 2)
Where,
(Eq. 1) versus (Eq. 2)
Note: Here is replaced by
Note: Weights are also changed and become random as is a function of
When sample size is allowed to increase
… (Eq. 3)
Note: UL+jreduces to TL+j, when ML+j = NL+j
Trial Initiation
Final Analysis
Test procedure
(K-1) Interim Analyses
0
1
2
L
L+1
L+j
K-1
K
Test Statistic
T1
T2
TL
UL+1
UL+j
UK-1
UK
Critical values
C1
C2
CL
CL+1
CL+j
CK-1
CK
Reject Ho & Stop trial if:
T1>C1
T2>C2
TL>CL
UL+1>CL+1
UL+j>CL+j
UK-1>CK-1
UK>CK
Here weights are kept fixed but are replaced by
Under H0 : µ1 - µ2 =0
D
So,
D
Overall Type I error of the new test Procedure
=
= Overall Type I error of the Original test Procedure = α
Monte Carlo Simulation:
New test has its type I error rate maintained at α.
Conclusion:
New test procedure allows to modify sample size without increase in overall type I error.
Scope
Repeated significance test with Brownian motion process and Independent increment
Steps
Test Statistic
U(t) = T(t) , if t≤tL
= , if t>tL
Brownian Motion
B(t|tЄT) is known as Brownian motion process if
Multivariate Normal Distribution
Mean = 0
Var {B(t)}= t
Var {B(t2) – B(t1)} = t2 – t1
Cov {B(t2), B(t1)} = min(t2, t1)
Brownian Motion: Z test statistic
Brownian Motion: Other Test statistic
T-test
Approx. Brownian motion (see Pocock 1977)
Log-rank test
Wilcoxon test
Approx. Brownian motion (see Slud & Wei 1982)
Approx. Brownian motion (see Tsiatis 1982, Sellke & Siegmund 1983, Slud 1984)
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Thank You…!
Now,
Therefore,