Statistical Analysis for Two-stage Seamless Design with Different Study Endpoints

Shein-Chung Chow, Duke U, Durham, NC, USAQingshu Lu, U of Science and Technology of ChinaSiu-Keung Tse, City U of Hong Kong, Hong Kong

Presented at

ICSA 2007 Applied Symposium – JP Hsu Memorial Session

Raleigh, North CarolinaJune 4, 2007

Outline

Adaptive seamless design Practical issues Statistical methods Sample size calculation Concluding remarks

Definition

There is no universal definition. Adaptive randomization, group sequential,

and sample size re-estimation, etc. Chow, Chang, and Pong (2005) PhRMA (2006)

Adaptive design is also known as Flexible design (EMEA, 2002, 2006) Attractive design (Uchida, 2006)

PhRMA’s definition

PhRMA (2006), J. Biopharm. Stat., 16 (3), 275-283.

An adaptive design is referred to as a clinical trial design that uses accumulating data to decide on how to modify aspects of the study as it continues, without undermining the validity and integrity of the trial.

PhRMA’s definition

Characteristics Adaptation is a design feature. Changes are made “by design” not

on an “ad hoc” basis. Comments

It does not reflect real practice. It may not be flexible as it means

to be.

Types of adaptation Prospective adaptation

Adaptive randomization Interim analysis Stopping trial early due to safety, futility, or

efficacy Sample size re-estimation

etc. Concurrent adaptation

Trial procedures Retrospective adaptation

Statistical procedures

Adaptive designs Adaptive randomization design Adaptive group sequential design N-adjustable design Drop-the-loser design Adaptive dose-escalation design Biomarker-adaptive design Adaptive treatment-switching design Adaptive-hypotheses design Adaptive seamless phase II/III trial design Any combinations of the above (multiple

adaptive design)

Seamless design

A seamless trial design is referred to a program that addresses within a single trial objectives that are normally achieved through separate trials of clinical development

Adaptive seamless design

An adaptive seamless design is a seamless trial design that would use data from patients enrolled before and after the adaptation in the final analysis.

Adaptive seamless trial design

Characteristics Combine two separate trials into

a single trial The single trial consists of two

phases Learning phase Confirmatory phase

Opportunity for adaptation based on accrued data at the end of learning phase

Advantages of adaptive seamless design

Opportunities for saving Stopping early for futility Stopping early for efficacy

Efficiency Can reduce lead time between the

learning and confirmatory phases Combined analysis

Data collected at the learning phase are combined with those data obtained at the confirmatory phase for final analysis

Seamless phase II/III design

A seamless phase II/III trial design is referred to a program that addresses within a single trial objectives that are normally achieved through separate trials in phase IIb and phase III of clinical development

Adaptive seamless phase II/III design

An adaptive seamless phase II/III design is a seamless phase II/III trial design that would use data from patients enrolled before and after the adaptation in the final analysis.

Comparison of type I errors

Let and be the type I error for phase II and phase III studies, respectively. Then the alpha for the traditional approach is given by

if one phase III study is required if two phase III studies are

required In an adaptive seamless phase II/III

design, the actual alpha is The alpha for a seamless design is actually

times larger than the traditional design

II III

IIIII IIIIIIII

III II/1

Comparison of powers Let and be the power for phase

II and phase III studies, respectively. Then the power for the traditional approach is given by

if one phase III study is required if two phase III studies are required

In an adaptive seamless phase II/III design, the power is

The power for a seamless design is actually times larger than the traditional design

IIPower IIIPower

IIIII PowerPowerPower *

IIIIIIII PowerPowerPowerPower **

IIIPowerPower IIPower/1

Comparison

Traditional Approach

Seamless Design

Significance level

1/20 * 1/20 1/20

Power 0.8 * 0.8 0.8

Lead time 6 m – 1 yr Reduce lead time

Sample size

n1+n2 n3

Multiple-stage design

An adaptive seamless trial design is a multiple-stage design

Adaptations Stop the trial early for

futility/efficacy Drop the losers Sample size re-estimation

etc

Multiple-stage design

Statistical approaches Hypotheses testing Stopping rules Decision rules

Hypotheses testing

Null hypothesis

where is the null hypothesis at the kth stage

KHHHH 002010 ...:

kH 0

Stopping rules

Let be the test statistic associated with the null hypothesis

Stop for efficacy ifStop for futility ifContinue with adaptations if Where and

kT

,kkT ,kkT

,kkk T

)1,..,1( Kkkk KK

Test based on individual p-values

This method is referred to as method of individual p-values (MIP)

Test statistics

For a two-stage design, we have

KkpT kk ,...,1,

)( 1121

Stopping boundaries based on MIP

Test based on sum of p-values

This method is referred to as the method of sum of p-values (MSP)

Test statistic

For a two-stage design, we have

KkpTk

iik ,...,1,

1

,)(2

1

),(2

1)(

2121

21

211121

for

for

,

,

21

21

Stopping boundaries based on MSP

Test based on product of p-values

This method is known as the method of products of p-values (MPP)

Test statistic

For a two-stage design, we have

KkpT ikik ,...,1,1

),(ln

,ln

211

121

1

121

for

for

,

,

21

21

Stopping boundaries based on MPP

Practical issues Similar but different study objectives

Learning phase is to select optimal dose for confirmatory phase

Confirmatory phase is to evaluate efficacy of the treatment

Different study endpoints Same study endpoints with different duration Different study endpoints, e.g.,

biomarker (surrogate) versus clinical Moving target patient population

Protocol amendments

Statistical method

Let be the data observed from stage 1 (learning phase) and stage 2 (confirmatory phase), respectively.

Assume that there is a relationship between and , i.e., .

The idea is to use the predicted values of at the first stage for the final combined analysis.

),( yx

x y )(xfy

y

Assumptions

and and

and can be related by

where is an error term with

zero mean and variance

)(xE 2)( xVar

)(yE 2)( yVar

xy 10

2

x y

Weighted-mean approach

Graybill-Deal estimator

where

21 )ˆ1(ˆˆˆ yyGD

22

21

21

//

/ˆ

smsn

sn

.1

1

1

1)ˆ1(ˆ41

//

1)ˆ(

22

21

mnsmsnVar GD

Sample size For simplicity, let .

Then the total sample size For testing the hypothesis of

equality, it can be verified that an approximate formula for n is given as

where and

nm nN )1(

0

10

)1(

)1(811

)(2 Nrr

Nn

22

2/2

0 /)( zzN 2221 /r

Concluding remarks The usual sample size calculation for an

adaptive two-stage design with different study endpoints needs adjustment.

Key assumptions in the above derivation are (i) there is a well-established relationship between the endpoints and (ii) the responses are continuous.

When there is a shift in patient population (e.g., as the result of protocol amendments), the above method needs to be modified.