Modification of Sample Size in Group Sequential Clinical Trials

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Modification of Sample Size in Group Sequential
Clinical Trials

• Madan Gopal Kundu
• PhD (Biostatistics) student, IUPUI

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Sample Size

• 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.

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Group Sequential Design
Determine N
Final Analysis
Conduct of Clinical Trial

• Scope of Early termination of trial
• Overwhelming efficacy
• futility of the drug

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Outline

• Introduction
• A case study
• Group sequential Z test
• A sequential test procedure with sample size
  modification (based on Z test)
• Generalization Brownian motion

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Introduction
Less than
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?
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Motivation A case study
Planning

• 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 (powergt95)

Expected Effect size 0.30
Interim Analysis

• After evaluation of 300 patients
• Incidence rate in Placebo 22
• Incidence rate in New Drug 16.5
• Need to increase the sample size

Observed Effect size 0.14
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Motivation A case study
Concern

• Does it inflate overall type I error?
• No valid testing procedure was available to
  account for such an outcome dependent adjustment
  of sample size

Finally

• It was decided not to increase the sample size
• Trial eventually failed to show a statistically
  significant effect

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Solution to this Dilemma

• 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

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Theoretical set-up
Population I N (µ1, s21)
Population II N (µ2, s21)
x1, x2, . , xN
y1, y2, . , yN
Effect size, ? µ1 - µ2
Our interest is to test (using two sample
Z-test) Ho ? 0 vs Ha ? gt 0
Assuming ? d, total sample size (N) per
population
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Group-sequential structure
Additional Subjects
n1
n2
nL-1
nL
nK-1
nK
Cumulative Subjects
N1
N2
NL-1
NL
NK-1
NKN
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
C1
C2
CL-1
CL
CK-1
CK
Critical values
Reject Ho Stop trial if
T1gtC1
T2gtC2
TL-1gtCL-1
TLgtCL
TK-1gtCK-1
TKgtCK
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Conditional Power

• The conditional power evaluated at the Lth
  interim analysis

• Sample size may be modified based on conditional
  power.
• is the Rejection Region.

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Sample Size Modification

• Calculate and
• Decide two positive constants 1
• If or
  , N should be
  modified to

• This adjustment of sample size preserves the
  unconditional power at 1-ß when
• If is smaller than d then it gives large M.

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Does Sample Size adjustment inflates Type I
error??

• 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

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Sample size modification

• Modify sample size in Lth interim analysis N?M

Cumulative sample size (with adjust)
N1
N2
NL
M
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Effect on Test Statistic because of sample size
modification

• When sample size is NOT allowed to increase

Test statistic at (Lj)th interim analysis,
(Eq. 1)
Where,
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Effect on Test Statistic because of sample size
modification

• When sample size is 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
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A different group sequential test procedure (CHW)

• Here weights are kept fixed but are
  replaced by

(Eq. 3)
Note ULj reduces to TLj, when MLj NLj
Test procedure
(K-1) Interim Analyses
Test Statistic
T1
T2
TL
ULj
UK-1
UK
UL1
Critical values
C1
C2
CL
CLj
CK-1
CK
CL1
Reject Ho Stop trial if
T1gtC1
T2gtC2
TLgtCL
UL1gtCL1
ULjgtCLj
UK-1gtCK-1
UKgtCK
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Distribution of ULj

Under H0 µ1 - µ2 0

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Distribution of ULj


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Impact on Type-I error

D

So,
D
Overall Type I error of the new test Procedure

Overall Type I error of the Original test
Procedure a Monte Carlo Simulation
New test has its type I error rate
maintained at a. Conclusion New test procedure
allows to modify sample size without increase in
overall type I error.
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Generalization
Scope
Repeated significance test with Brownian motion
process and Independent increment
Steps

• B(t) be such repeated significance test at
  information time t.
• T(t) B(t)/t1/2
• Let at ttL sample size increased to M
• w N/M
• b (w tL)/(1-tL)

Test Statistic
U(t) T(t) , if ttL

, if tgttL
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Brownian Motion

• B(tt?T) 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)

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Brownian Motion Z test statistic
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Brownian Motion Other Test statistic
T-test
Approx. Brownian motion (see Pocock 1977)
Log-rank test

• Approx. Brownian motion (see Tsiatis 1982, Sellke
  Siegmund 1983, Slud 1984)

Wilcoxon test
Approx. Brownian motion (see Slud Wei 1982)
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Reference

• Cui L, Hung H J and Wang S J (1999). Modification
  of sample size in Group Sequential Clinical
  Trials. Biometrics 55 853-857.
• Pocock S J (1977). Group sequential methods in
  the design and analysis of clinical trials.
  Biometrika 64 191-199.
• Lan K K G and Wittes J (1988). The B-value A
  tool for monitoring data. Biometrics 44 579-585.
• Lan K K G and Zucker D M (1993). Sequential
  monitoring of clinical trials the role of
  information and brownian motion. Statistics in
  Medicine 12 753-765.
• Reboussin D M, DeMets D L, Kim K M and Lan K K G
  (2000). Computation for group sequential
  boundaries using the Lan-DeMets spending function
  method. Controlled Clinical Trials 21 190-207.
• Lan K K G and DeMets D L (1983). Discrete
  sequential boundaries for clinical trials.
  Biometrika 70 659-663.

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Reference

• Shih W J (2003). Group Sequential Methods.
  Encyclopedia of Biopharm. Statistics 11 423-432.
• Tsiatis A A (1982). Repeated significance testing
  for a general class of statistics used in
  censored survival analysis. JASA 77 855-861.
• Sellke T and Siegmund D (1983). Sequential
  analysis of the proportional hazard model.
  Biometrika 70 315-326
• Slud E V (1984). Sequential linear rank tests for
  two sample censored survival data. Annals of
  Statistics 12 551-571.
• Slud E and Wei L J (1982). Two-sample repeated
  significance tests based on the modified Wilcoxon
  test statistic . JASA 77(380) 862-867.

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Thank You!
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Now,
Therefore,