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Modification of Sample Size in Group Sequential Clinical Trials

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### Modification of Sample Size in Group Sequential Clinical Trials

MadanGopalKundu

PhD (Biostatistics) student, IUPUI

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.

Determine

N

Conduct of Clinical Trial

Final Analysis

Interim analysis 1

Interim analysis 2

Interim analysis 3

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

Introduction

A case study

Group sequential Z test

A sequential test procedure with sample size modification (based on Z test)

Generalization: Brownian motion

Introduction

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?

Motivation: A case study

Planning

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

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%)

Motivation: A case study

Concern

Finally…

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

Does it inflate overall type I error?

No valid testing procedure was available to account for such an outcome –dependent adjustment of sample size

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

Theoretical set-up

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

Group-sequential structure

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

Conditional Power

- Sample size may be modified based on conditional power.
- { } is the Rejection Region.

The conditional power evaluated at the Lth interim analysis

Sample Size Modification

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

Calculate and

Decide two positive constants ≤1 ≤

If or , N should be modified to

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

Sample size modification

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

Effect on Test Statistic because of sample size modification

Test statistic at (L+j)th interim analysis,

… (Eq. 1)

Where,

When sample size is NOT allowed to increase

Effect on Test Statistic because of sample size modification

… (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

A different group sequential test procedure (CHW)

… (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

Distribution of UL+j

Under H0 : µ1 - µ2 =0

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 = 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.

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 t=tL sample size increased to M
- w = N/M
- b =(w – tL)/(1-tL)

Test Statistic

U(t) = T(t) , if t≤tL

= , if t>tL

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: 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)

Reference

Cui L, Hung H J and Wang S J (1999). Modification of sample size in Group Sequential Clinical Trials. Biometrics55: 853-857.

Pocock S J (1977). Group sequential methods in the design and analysis of clinical trials. Biometrika64: 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. Biometrika70: 659-663.

Reference

Shih W J (2003). Group Sequential Methods. Encyclopedia of Biopharm. Statistics1:1 423-432.

Tsiatis A A (1982). Repeated significance testing for a general class of statistics used in censored survival analysis. JASA77: 855-861.

Sellke T and Siegmund D (1983). Sequential analysis of the proportional hazard model. Biometrika70: 315-326

Slud E V (1984). Sequential linear rank tests for two sample censored survival data. Annals of Statistics12: 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.

Therefore,

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