A Flexible Two Stage Design in Active Control Non-inferiority Trials

Gang Chen, Yong-Cheng Wang, and George Chi†

Division of Biometrics I, CDER, FDA

Qing Liu*

J & J PRD

MCP 2002, Bethesda MD

August 5-7, 2002

†: The views expressed in this presentation do not necessarily represent those of the U.S. Food and Drug Administration.

*: This is a continuing research based on the work initiated while Dr. Qing Liu was affiliated with FDA.

Outline

• An Example

• Non-inferiority trial:

– objectives, hypotheses, tests, type I error, sample size

• Two stage adaptive designs for sample size re-estimation

• A flexible design for sample size calculation

– superiority trials

– non-inferiority trials

• Summary and issues

An Example

• An example of a flexible design for sample size calculation in a non-inferiority trial:

T is an approved dose for a treatment. Investigator (sponsor) wants to lower the dose to improve the toxicity profile without compromising much loss in treatment effect.

An Example

Trial design:

• A randomized, active-control trial

• Primary efficacy endpoint: response rate

• Two arms: – T: approved dose

– C: a low dose never studied

• Non-inferiority hypothesis: low treatment dose can preserve at least 75% effect of the approved treatment dose

An Example

• Since there is no information on the efficacy for this low dose treatment, a two stage non-inferiority design is proposed

– stage 1: recruit 100 patient/arm to evaluate the treatment effect of low dose and calculate sample size

– stage 2: recruit n patients (calculated based on stage 1 data) for the non-inferiority trial.

• Sponsor’s Question: Can we include stage 1 data in the final analysis?

Non-inferiority trial - objectives

Brief introduction of objectives, hypotheses, tests, type I error

control, sample size determination in the design of

active control non-inferiority trials:

• Objectives:– To establish efficacy through testing a fraction retention of control

effect

– To establish non-inferiority or equivalence

Non-inferiority trial - hypotheses

Some notations:

• T, C and P denote the treatment, control and placebo respectively.

• µtp=T-P: treatment effect relative to the placebo P

µcp=C-P: control effect relative to the placebo P

µtc=T-C: the treatment effect relative to C.

• The proportion of the active control effect: =µtp/µcp.

Non-inferiority trial - hypotheses

• Non-inferiority hypotheses: – hypotheses with a pre-selected fixed margin – hypotheses with a fixed fraction retention

The detailed discussion on those hypotheses is given in [1].

• When testing whether the treatment maintains a proportion 0 (<= 1) of active control effect, hypotheses are:

H0: µtp< 0 µcp vs. H1: µtp> 0 µcp or

(under constancy assumption for the control effect)

H0: µtc< -(1-0)µcp vs. H1: µtc> -(1-0)µcp

[1]: Chen et al (2001), Active control trials - hypotheses and issues. ASA Proceedings

Non-inferiority trial - test statistic

• The test statistic for the above hypotheses:

Non-inferiority trial - type I error

• Asymptotic alpha of the test [2]:

[2]: Rothmann et al (2001), Non-inferiority methods for mortality trials. ASA Proceedings.

Non-inferiority trial - sample size

• The sample size n (under Ha: T=C) for a binary endpoint:

Non-inferiority trial - sample size

• The sample size determination in a non-inferiority trial depends on the following factors

– control effect size and a proportion retention – standard errors from current and historical trials– alpha and power

Non-inferiority trial - sample size

• At the design stage: – Known: alpha, beta, fraction retention and control effect size

(estimate) and its associated variation,

– Unknown: treatment effect and its associated variation (relative to control).

• A two stage flexible design can be used for sample size determination. The purpose for the 1st stage is only for evaluation of treatment effect and its associated variation – Question: Can we include stage 1 data in the final analysis?

Non-inferiority trial - sample size

Answer: Yes, but

• the overall type I error should be controlled at the desired level.

Two stage adaptive designs for sample size re-estimation

• Existing methods for two stage adaptive designs, example: those methods proposed by Bauer & Kieser, Proschan and Hunsbarger, Liu & Chi, Cui et al: – choosing a conditional error function to control type I error

– down-weight data collected from second stage to preserve the overall type I error.

– base sample size on conditional power

– other

• In non-inferiority trials, those methods above can not apply directly and need to be modified.

A flexible two stage design

To calculate sample size:

• Stage 1 sample size n1 is selected arbitrarily, and stage 1 data provides information on treatment effect size, variation and futility.

• Total sample size can be estimated based on stage 1 data.

• Final analysis includes the pooled data of both stages.

A flexible two stage design

Major issues:

• Two sources for the type I error inflation: – arbitrary selection of the size of stage 1 (n1)

– total sample size (n) calculation based on the first stage data

• Distribution of the final test statistic: – test: T=T(n, control information, stage 1 & 2 data)

– sample size: n = n(stage 1data, control information)

Type I error inflation due to arbitrary selection of stage 1 size superiority test, alpha=.05, power=.8

Type I error inflation due to arbitrary selection of stage 1 size non-inferiority test, alpha=.05, power=.8, Nc = 300

A flexible two stage superiority design

The procedure :

• At stage 1– Effect size n1 and its associated variation are estimated

– Overall sample size n can be calculated based on n1..

• At the final – Let Tn( n1) be a test statistic

– Conditional type I error to reject the null is

( n1) = Pr (Tn( n1) >C /2), where C /2 is the critical value

– Overall type I error becomes: =E ( n1).

A flexible two stage superiority design

• Simulation results for type I error inflation *=E ( n1) due to the inclusion of stage 1 trial data.

Table 1. Simulation Study for Superiority Two-Stage Design

( simulation runs = 5000, = 0.05, = 0.20, r0 = 0.005 )

N1

T = C

25 0.30 0.07725 0.75 0.07750 0.30 0.07550 0.75 0.07775 0.30 0.07475 0.75 0.078100 0.30 0.077100 0.75 0.073

A flexible two stage non-inferiority design

Similarly, the procedure :

• At stage 1– Treatment effect size TC n1 and its associated variation SETC n1 are

estimated based on stage 1 data and control information

– Overall sample size n is calculated

• At the final– Let Tn(TC n1, control info) be a test statistic

– Conditional type I error to reject the null is

(TC n1) = Pr (Tn(TC n1) >C /2), where C /2 is the critical value

– Overall type I error becomes: =E ( TC n1).

A flexible two stage non-inferiority design

• Simulation results * = E(TC n1) due to the inclusion of stage 1 trial data.

Table Simulation Study for Non-inferiority Two-Stage Design

( simulation times = 5000, = 0.05, = 0.20, r0 = 0.005NC = 300)

N1

0

C1 = C2

P

T

100 0.50 0.30 0.05 0.175 0.063100 0.50 0.75 0.05 0.4 0.078100 0.75 0.30 0.05 0.2375 0.082100 0.75 0.75 0.05 0.575 0.088200 0.50 0.30 0.05 0.175 0.056200 0.50 0.75 0.05 0.4 0.071200 0.75 0.30 0.05 0.2375 0.078200 0.75 0.75 0.05 0.575 0.086400 0.50 0.30 0.05 0.175 0.039400 0.50 0.75 0.05 0.4 0.057400 0.75 0.30 0.05 0.2375 0.070400 0.75 0.75 0.05 0.575 0.079

Summary and issues

Question: Can we include stage 1 data in the final analysis?

Response: Yes, but the inclusion of stage 1 data needs

• to control both sources for overall Type I error inflation (under research)– due to arbitrary selection of the sample size of stage 1

– due to the inclusion of stage 1 data in the final test

• to assess the distribution of of final test statistic:

(under research)– test: T=T(n, control information, stage 1 & 2 data)

– sample size: n = n(stage 1data, control information)