Adaptive Enrichment Designs for Clinical Trials...Closed testing procedures For testing H 01 and H...

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Adaptive Enrichment Designs for Clinical Trials Thomas Burnett University of Bath 30/1/2017

Transcript of Adaptive Enrichment Designs for Clinical Trials...Closed testing procedures For testing H 01 and H...

Page 1: Adaptive Enrichment Designs for Clinical Trials...Closed testing procedures For testing H 01 and H 02 we construct level tests for I H 01: 1 0, I H 02: 2 0, I H 01 \H 02: 1 0 and 2

Adaptive Enrichment Designs for Clinical Trials

Thomas BurnettUniversity of Bath

30/1/2017

Page 2: Adaptive Enrichment Designs for Clinical Trials...Closed testing procedures For testing H 01 and H 02 we construct level tests for I H 01: 1 0, I H 02: 2 0, I H 01 \H 02: 1 0 and 2

Overview

What is Adaptive Enrichment

Familywise Error Rate (FWER)

Hypothesis testing

Bayes optimal decisions

Page 3: Adaptive Enrichment Designs for Clinical Trials...Closed testing procedures For testing H 01 and H 02 we construct level tests for I H 01: 1 0, I H 02: 2 0, I H 01 \H 02: 1 0 and 2

What is Adaptive Enrichment

Page 4: Adaptive Enrichment Designs for Clinical Trials...Closed testing procedures For testing H 01 and H 02 we construct level tests for I H 01: 1 0, I H 02: 2 0, I H 01 \H 02: 1 0 and 2

Pre-identified sub-populations

H01 : θ1 ≤ 0 H02 : θ2 ≤ 0

Page 5: Adaptive Enrichment Designs for Clinical Trials...Closed testing procedures For testing H 01 and H 02 we construct level tests for I H 01: 1 0, I H 02: 2 0, I H 01 \H 02: 1 0 and 2

Pre-identified sub-populations

H01 : θ1 ≤ 0 H02 : θ2 ≤ 0

Page 6: Adaptive Enrichment Designs for Clinical Trials...Closed testing procedures For testing H 01 and H 02 we construct level tests for I H 01: 1 0, I H 02: 2 0, I H 01 \H 02: 1 0 and 2

Pre-identified sub-populations

H01 : θ1 ≤ 0 H02 : θ2 ≤ 0

Page 7: Adaptive Enrichment Designs for Clinical Trials...Closed testing procedures For testing H 01 and H 02 we construct level tests for I H 01: 1 0, I H 02: 2 0, I H 01 \H 02: 1 0 and 2

Pre-identified sub-populations

H01 : θ1 ≤ 0 H02 : θ2 ≤ 0

Page 8: Adaptive Enrichment Designs for Clinical Trials...Closed testing procedures For testing H 01 and H 02 we construct level tests for I H 01: 1 0, I H 02: 2 0, I H 01 \H 02: 1 0 and 2

Pre-identified sub-populations

H01 : θ1 ≤ 0 H02 : θ2 ≤ 0

H03 : θ3 ≤ 0 where θ3 = λθ1 + (1− λ)θ2

Page 9: Adaptive Enrichment Designs for Clinical Trials...Closed testing procedures For testing H 01 and H 02 we construct level tests for I H 01: 1 0, I H 02: 2 0, I H 01 \H 02: 1 0 and 2

Pre-identified sub-populations

H01 : θ1 ≤ 0 H02 : θ2 ≤ 0

H03 : θ3 ≤ 0 where θ3 = λθ1 + (1− λ)θ2

Page 10: Adaptive Enrichment Designs for Clinical Trials...Closed testing procedures For testing H 01 and H 02 we construct level tests for I H 01: 1 0, I H 02: 2 0, I H 01 \H 02: 1 0 and 2

Conducting Adaptive Enrichment

Pre-interim recruitment

Post-interim recruitment

H01 : θ1 ≤ 0

Page 11: Adaptive Enrichment Designs for Clinical Trials...Closed testing procedures For testing H 01 and H 02 we construct level tests for I H 01: 1 0, I H 02: 2 0, I H 01 \H 02: 1 0 and 2

Conducting Adaptive Enrichment

Pre-interim recruitment

Post-interim recruitment

H01 : θ1 ≤ 0

Page 12: Adaptive Enrichment Designs for Clinical Trials...Closed testing procedures For testing H 01 and H 02 we construct level tests for I H 01: 1 0, I H 02: 2 0, I H 01 \H 02: 1 0 and 2

Conducting Adaptive Enrichment

Pre-interim recruitment

Post-interim recruitment

H01 : θ1 ≤ 0 H02 : θ2 ≤ 0

Page 13: Adaptive Enrichment Designs for Clinical Trials...Closed testing procedures For testing H 01 and H 02 we construct level tests for I H 01: 1 0, I H 02: 2 0, I H 01 \H 02: 1 0 and 2

Conducting Adaptive Enrichment

Pre-interim recruitment

Post-interim recruitment

H01 : θ1 ≤ 0

Page 14: Adaptive Enrichment Designs for Clinical Trials...Closed testing procedures For testing H 01 and H 02 we construct level tests for I H 01: 1 0, I H 02: 2 0, I H 01 \H 02: 1 0 and 2

Conducting Adaptive Enrichment

Pre-interim recruitment

Post-interim recruitment

H02 : θ2 ≤ 0

Page 15: Adaptive Enrichment Designs for Clinical Trials...Closed testing procedures For testing H 01 and H 02 we construct level tests for I H 01: 1 0, I H 02: 2 0, I H 01 \H 02: 1 0 and 2

Familywise Error Rate (FWER)

Page 16: Adaptive Enrichment Designs for Clinical Trials...Closed testing procedures For testing H 01 and H 02 we construct level tests for I H 01: 1 0, I H 02: 2 0, I H 01 \H 02: 1 0 and 2

Defining FWER

Pθ(Reject any combination of true null hypotheses) for all θ

True null hypotheses Familywise error to reject

H01 H01

H02 H02

H01 and H02 H01, H02 or both

Page 17: Adaptive Enrichment Designs for Clinical Trials...Closed testing procedures For testing H 01 and H 02 we construct level tests for I H 01: 1 0, I H 02: 2 0, I H 01 \H 02: 1 0 and 2

Defining FWER

Pθ(Reject any combination of true null hypotheses) for all θ

True null hypotheses Familywise error to reject

H01 H01

H02 H02

H01 and H02 H01, H02 or both

Page 18: Adaptive Enrichment Designs for Clinical Trials...Closed testing procedures For testing H 01 and H 02 we construct level tests for I H 01: 1 0, I H 02: 2 0, I H 01 \H 02: 1 0 and 2

Achieving strong control of FWER

Pθ(Reject any combination of true null hypotheses) ≤ α for all θ

To achieve this we require:

I If H01 true, P(Reject H01) ≤ αI If H02 true, P(Reject H02) ≤ αI If H01 and H02 true, P(Reject both H01 and H02) ≤ α

The Bonferroni correction where both H01 and H02 are tested atα/2 is the simplest procedure that ensures strong control of theFWER.

Page 19: Adaptive Enrichment Designs for Clinical Trials...Closed testing procedures For testing H 01 and H 02 we construct level tests for I H 01: 1 0, I H 02: 2 0, I H 01 \H 02: 1 0 and 2

Achieving strong control of FWER

Pθ(Reject any combination of true null hypotheses) ≤ α for all θ

To achieve this we require:

I If H01 true, P(Reject H01) ≤ αI If H02 true, P(Reject H02) ≤ αI If H01 and H02 true, P(Reject both H01 and H02) ≤ α

The Bonferroni correction where both H01 and H02 are tested atα/2 is the simplest procedure that ensures strong control of theFWER.

Page 20: Adaptive Enrichment Designs for Clinical Trials...Closed testing procedures For testing H 01 and H 02 we construct level tests for I H 01: 1 0, I H 02: 2 0, I H 01 \H 02: 1 0 and 2

Achieving strong control of FWER

Pθ(Reject any combination of true null hypotheses) ≤ α for all θ

To achieve this we require:

I If H01 true, P(Reject H01) ≤ αI If H02 true, P(Reject H02) ≤ αI If H01 and H02 true, P(Reject both H01 and H02) ≤ α

The Bonferroni correction where both H01 and H02 are tested atα/2 is the simplest procedure that ensures strong control of theFWER.

Page 21: Adaptive Enrichment Designs for Clinical Trials...Closed testing procedures For testing H 01 and H 02 we construct level tests for I H 01: 1 0, I H 02: 2 0, I H 01 \H 02: 1 0 and 2

Hypothesis testing

Page 22: Adaptive Enrichment Designs for Clinical Trials...Closed testing procedures For testing H 01 and H 02 we construct level tests for I H 01: 1 0, I H 02: 2 0, I H 01 \H 02: 1 0 and 2

Closed testing procedures

For testing H01 and H02 we construct level α tests for

I H01 : θ1 ≤ 0,

I H02 : θ2 ≤ 0,

I H01 ∩ H02 : θ1 ≤ 0 and θ2 ≤ 0.

Then:

I Reject H01 globally if H01 and H01 ∩ H02 are both rejected.

I Reject H01 globally if H01 and H01 ∩ H02 are both rejected.

Simes rule and the method proposed by Dunnett provide twomethods for testing the intersection hypothesis that are suitable forAdaptive Enrichment designs.

Page 23: Adaptive Enrichment Designs for Clinical Trials...Closed testing procedures For testing H 01 and H 02 we construct level tests for I H 01: 1 0, I H 02: 2 0, I H 01 \H 02: 1 0 and 2

Closed testing procedures

For testing H01 and H02 we construct level α tests for

I H01 : θ1 ≤ 0,

I H02 : θ2 ≤ 0,

I H01 ∩ H02 : θ1 ≤ 0 and θ2 ≤ 0.

Then:

I Reject H01 globally if H01 and H01 ∩ H02 are both rejected.

I Reject H01 globally if H01 and H01 ∩ H02 are both rejected.

Simes rule and the method proposed by Dunnett provide twomethods for testing the intersection hypothesis that are suitable forAdaptive Enrichment designs.

Page 24: Adaptive Enrichment Designs for Clinical Trials...Closed testing procedures For testing H 01 and H 02 we construct level tests for I H 01: 1 0, I H 02: 2 0, I H 01 \H 02: 1 0 and 2

Combination tests

Suppose P(1) and P(2) are the p-values from the first and secondstages of the trial. Using the weighted inverse Normal we find P(c)

the p-value for the whole trial.

Z (1) = φ−1(1− P(1)) and Z (2) = φ−1(1− P(2))

With w1 and w2 such that w21 + w2

2 = 1

Z (c) = w1Z(1) + w2Z

(2)

P(c) = 1− φ(Z (c))

Choosing w1 and w2 in proportion to the sample size from eachstage gives the optimal procedure.

Page 25: Adaptive Enrichment Designs for Clinical Trials...Closed testing procedures For testing H 01 and H 02 we construct level tests for I H 01: 1 0, I H 02: 2 0, I H 01 \H 02: 1 0 and 2

Combination tests

Suppose P(1) and P(2) are the p-values from the first and secondstages of the trial. Using the weighted inverse Normal we find P(c)

the p-value for the whole trial.

Z (1) = φ−1(1− P(1)) and Z (2) = φ−1(1− P(2))

With w1 and w2 such that w21 + w2

2 = 1

Z (c) = w1Z(1) + w2Z

(2)

P(c) = 1− φ(Z (c))

Choosing w1 and w2 in proportion to the sample size from eachstage gives the optimal procedure.

Page 26: Adaptive Enrichment Designs for Clinical Trials...Closed testing procedures For testing H 01 and H 02 we construct level tests for I H 01: 1 0, I H 02: 2 0, I H 01 \H 02: 1 0 and 2

Overall testing procedure

H01 H02 H01 ∩ H02

Pre-interim recruitment cohort p(1)1 p

(1)2 p

(1)12

Post-interim recruitment cohort p(2)1 p

(2)2 p

(2)12

Overall p(c)1 p

(c)2 p

(c)12

Page 27: Adaptive Enrichment Designs for Clinical Trials...Closed testing procedures For testing H 01 and H 02 we construct level tests for I H 01: 1 0, I H 02: 2 0, I H 01 \H 02: 1 0 and 2

Bayes optimal decisions

Page 28: Adaptive Enrichment Designs for Clinical Trials...Closed testing procedures For testing H 01 and H 02 we construct level tests for I H 01: 1 0, I H 02: 2 0, I H 01 \H 02: 1 0 and 2

Optimisation components

Prior distribution:π(θ)

Utility function (gain):

G (θ) = γ1(θ1)I(Reject H01) + γ2(θ2)I(Reject H02)

Page 29: Adaptive Enrichment Designs for Clinical Trials...Closed testing procedures For testing H 01 and H 02 we construct level tests for I H 01: 1 0, I H 02: 2 0, I H 01 \H 02: 1 0 and 2

Simple optimisation

Eπ(θ)(G (θ))

Page 30: Adaptive Enrichment Designs for Clinical Trials...Closed testing procedures For testing H 01 and H 02 we construct level tests for I H 01: 1 0, I H 02: 2 0, I H 01 \H 02: 1 0 and 2

Bayes optimal decision

Data available at the interim analysis:

X

Bayes expected gain:Eπ(θ),X (G (θ))

Choose sub-population that maximises Eπ(θ),X (G (θ))

Page 31: Adaptive Enrichment Designs for Clinical Trials...Closed testing procedures For testing H 01 and H 02 we construct level tests for I H 01: 1 0, I H 02: 2 0, I H 01 \H 02: 1 0 and 2

Bayes optimal decision

Data available at the interim analysis:

X

Bayes expected gain:Eπ(θ),X (G (θ))

Choose sub-population that maximises Eπ(θ),X (G (θ))

Page 32: Adaptive Enrichment Designs for Clinical Trials...Closed testing procedures For testing H 01 and H 02 we construct level tests for I H 01: 1 0, I H 02: 2 0, I H 01 \H 02: 1 0 and 2

Bayes optimal rule

Eπ(θ),X (G (θ))

Page 33: Adaptive Enrichment Designs for Clinical Trials...Closed testing procedures For testing H 01 and H 02 we construct level tests for I H 01: 1 0, I H 02: 2 0, I H 01 \H 02: 1 0 and 2

Key points

I Strong control of FWER

I Prior distribution

I Gain function

I Eπ(θ),X (G (θ)) for optimisation

I Eπ(θ)(G (θ)) for overall behaviour

Page 34: Adaptive Enrichment Designs for Clinical Trials...Closed testing procedures For testing H 01 and H 02 we construct level tests for I H 01: 1 0, I H 02: 2 0, I H 01 \H 02: 1 0 and 2

Any questions?