Group Sequential Clinical Trial Designs for Normally ... · Group Sequential Clinical Trial Designs...

Introduction Group Sequential Design Theory Commands Discussion

Group Sequential Clinical Trial Designs forNormally Distributed Outcome Variables

Michael Grayling James Wason Adrian Mander

Hub for Trials Methodology ResearchMRC Biostatistics Unit

Cambridge, UK

Stata UK Users Group Meeting, September 2017

M. J. Grayling ([email protected]) Group sequential trial designs MRC Biostatistics Unit 1 / 20

Outline

1 Introduction

2 Group Sequential Design Theory

3 Commands

4 Discussion

Outline

1 Introduction

3 Commands

4 Discussion

Randomised Controlled Trial Design

Choose a sample size that provides some level of statisticalpower for a target treatment effect.

Recruit the number of patients required.

Perform an analysis after all patients have been assessed.

Design, analysis, and reporting of such trials well characterised.

Incredibly effective way to assess the efficacy of a treatment.

But is this the best we can do?

Adaptive Trial Design

Trials gather a lot of data during their progress!

What if we are unsure about the sample size to use?

What if the new treatment is harmful?

What if the new treatment works only in a subset of patients?

This is where adaptive trial design comes in.

Here discuss group sequential trials.

Group Sequential Trials

Can bring substantial administrative, ethical, monetary advan-tages.

Origins in industrial sampling and Wald’s SPRT.

First proposals were fully sequential, but this proved to be im-practical.

Idea therefore is to conduct analyses after particular landmarknumbers of patients recruited.

Trial may be stopped early to accept or reject null hypotheses.

Expected sample size typically reduced.

Outline

1 Introduction

3 Commands

4 Discussion

Overview

Focus on the design of a two-arm group sequential trial testingfor superiority, with normally distributed outcomes.

Assume a maximum of L analysis planned, and that analysisl = 1, . . . , L takes place after n0l = ln and n1l = rln patientsevaluated in arms 0 and 1 respectively.

Suppose that Ydli ∼ N(µd, σ2d) for d = 0, 1.

Defining τ = µ1 − µ0, interest is in testing

H0 : τ ≤ 0, H1 : τ > 0.

Want overall type-I error-rate when τ = 0 of α, and power of1− β when τ = δ > 0.

Overview

H0 : τ ≤ 0, H1 : τ > 0.

Overview

H0 : τ ≤ 0, H1 : τ > 0.

Overview

H0 : τ ≤ 0, H1 : τ > 0.

Overview

H0 : τ ≤ 0, H1 : τ > 0.

Analysis

To test H0, the following test statistic is used after analysisl = 1, . . . , L

l∑j=1

rn∑i=1

Y1jl −1

l∑j=1

n∑i=1

I1/2l ,

n0l+σ2

Importantly (Z1, . . . , ZL) is multivariate normal with

E(Zl) = τI1/2l , l = 1, . . . , L,

Cov(Zl, Zk) = (Il/Ik)1/2, 1 ≤ l ≤ k ≤ L.

Means operating characteristics can be calculated analytically...

Analysis

l∑j=1

rn∑i=1

Y1jl −1

l∑j=1

n∑i=1

I1/2l ,

n0l+σ2

E(Zl) = τI1/2l , l = 1, . . . , L,

Cov(Zl, Zk) = (Il/Ik)1/2, 1 ≤ l ≤ k ≤ L.

Analysis

l∑j=1

rn∑i=1

Y1jl −1

l∑j=1

n∑i=1

I1/2l ,

n0l+σ2

E(Zl) = τI1/2l , l = 1, . . . , L,

Cov(Zl, Zk) = (Il/Ik)1/2, 1 ≤ l ≤ k ≤ L.

Stopping Rules

...given choices for f1, . . . , fL and e1, . . . , eL. Use these in thefollowing stopping rules at analysis l = 1, . . . , L

If Zl ≥ el stop and reject H0.If Zl < fl stop and accept H0.otherwise continue to stage l + 1.

P(Reject H0 | τ) =L∑

P(Reject H0 at stage l | τ),

= P(Z1 ≥ e1 | τ)

P(f1 ≤ Z1 < e1, . . . , fl−1 ≤ Zl−1 < el−1, Zl ≥ el | τ).

Similar formulae for E(N | τ).Evaluate these formulae using mvnormal mata().

Stopping Rules

= P(Z1 ≥ e1 | τ)

Stopping Rules

= P(Z1 ≥ e1 | τ)

Stopping Rules

= P(Z1 ≥ e1 | τ)

Stopping Rules

= P(Z1 ≥ e1 | τ)

Stopping Rules

= P(Z1 ≥ e1 | τ)

Stopping Rules

= P(Z1 ≥ e1 | τ)

Boundaries

Functional form assumed, then search to find group size andexact values for correct operating characteristics.

For example

el = Ce(l/L)Ω−1/2,

fl = δI1/2l − Cf (l/L)Ω−1/2.

Then take I1/2L = (Ce + Cf )/δ, to ensure eL = fL.

Search over Ce and Cf using optimize().

Boundaries

For example

el = Ce(l/L)Ω−1/2,

fl = δI1/2l − Cf (l/L)Ω−1/2.

Boundaries

For example

el = Ce(l/L)Ω−1/2,

fl = δI1/2l − Cf (l/L)Ω−1/2.

Boundaries

For example

el = Ce(l/L)Ω−1/2,

fl = δI1/2l − Cf (l/L)Ω−1/2.

Outline

1 Introduction

3 Commands

4 Discussion

Commands

Six commands in total. Four for two-sided tests, and two forone-sided tests as discussed here.

One-sided tests as follows

powerFamily, [l(integer 3) delta(real 0.2)

alpha(real 0.05) beta(real 0.2)

sigma(numlist) ratio(real 1) Omega(real 0.5)

performance *]

triangular, [l(integer 3) delta(real 0.2) alpha(real 0.05)

beta(real 0.2) sigma(numlist) ratio(real 1)

performance *]

Commands

Six commands in total. Four for two-sided tests, and two forone-sided tests as discussed here.

One-sided tests as follows

powerFamily, [l(integer 3) delta(real 0.2)

alpha(real 0.05) beta(real 0.2)

sigma(numlist) ratio(real 1) Omega(real 0.5)

performance *]

triangular, [l(integer 3) delta(real 0.2) alpha(real 0.05)

beta(real 0.2) sigma(numlist) ratio(real 1)

performance *]

Example Output

Example: Comparison

t H0|τ

-1 -.5 0 .5 1τ

P(Reject H0|τ) E(N|τ)

Power family with Ω = -0.25

t H0|τ

-1 -.5 0 .5 1τ

Power family with Ω = 0

t H0|τ

-1 -.5 0 .5 1τ

Power family with Ω = 0.25

t H0|τ

-1 -.5 0 .5 1τ

Triangular test

Outline

1 Introduction

3 Commands

4 Discussion

Discussion

Group sequential designs provide gains in efficiency, easy to find(at least in this case).

Key commands working.

Only considered design so far.

Only considered two-arm; multi-arm multi-stage designs of in-creasing interest.

An option to use simulation instead of integration would alsobe a good step.

Discussion

References

1 Jennison C, Turnbull BW (2000) Group Sequential Methods with Applica-tions to Clinical Trials. Boca Raton, FL: Chapman & Hall/CRC.

2 Pampallona S, Tsiatis AA (1994) Group sequential designs for one-sidedand two-sided hypothesis testing with provision for early stopping in favorof the null hypothesis. Journal of Statistical Planning and Inference 42(1-2):19-35.

3 Whitehead J (1997) The Design and Analysis of Sequential Clinical Trials.Revised 2nd ed. Chichester: John Wiley & Sons.

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