adaptive enrichment designs for clinical trials...closed testing procedures for testing h 01 and h...
TRANSCRIPT
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Adaptive Enrichment Designs for Clinical Trials
Thomas BurnettUniversity of Bath
30/1/2017
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Overview
What is Adaptive Enrichment
Familywise Error Rate (FWER)
Hypothesis testing
Bayes optimal decisions
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What is Adaptive Enrichment
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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](https://reader033.vdocuments.net/reader033/viewer/2022053017/5f1b636ac35de14760676a30/html5/thumbnails/5.jpg)
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](https://reader033.vdocuments.net/reader033/viewer/2022053017/5f1b636ac35de14760676a30/html5/thumbnails/6.jpg)
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](https://reader033.vdocuments.net/reader033/viewer/2022053017/5f1b636ac35de14760676a30/html5/thumbnails/7.jpg)
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](https://reader033.vdocuments.net/reader033/viewer/2022053017/5f1b636ac35de14760676a30/html5/thumbnails/8.jpg)
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](https://reader033.vdocuments.net/reader033/viewer/2022053017/5f1b636ac35de14760676a30/html5/thumbnails/9.jpg)
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](https://reader033.vdocuments.net/reader033/viewer/2022053017/5f1b636ac35de14760676a30/html5/thumbnails/10.jpg)
Conducting Adaptive Enrichment
Pre-interim recruitment
Post-interim recruitment
H01 : θ1 ≤ 0
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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](https://reader033.vdocuments.net/reader033/viewer/2022053017/5f1b636ac35de14760676a30/html5/thumbnails/12.jpg)
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](https://reader033.vdocuments.net/reader033/viewer/2022053017/5f1b636ac35de14760676a30/html5/thumbnails/13.jpg)
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](https://reader033.vdocuments.net/reader033/viewer/2022053017/5f1b636ac35de14760676a30/html5/thumbnails/14.jpg)
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](https://reader033.vdocuments.net/reader033/viewer/2022053017/5f1b636ac35de14760676a30/html5/thumbnails/15.jpg)
Familywise Error Rate (FWER)
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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
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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](https://reader033.vdocuments.net/reader033/viewer/2022053017/5f1b636ac35de14760676a30/html5/thumbnails/18.jpg)
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](https://reader033.vdocuments.net/reader033/viewer/2022053017/5f1b636ac35de14760676a30/html5/thumbnails/19.jpg)
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](https://reader033.vdocuments.net/reader033/viewer/2022053017/5f1b636ac35de14760676a30/html5/thumbnails/20.jpg)
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](https://reader033.vdocuments.net/reader033/viewer/2022053017/5f1b636ac35de14760676a30/html5/thumbnails/21.jpg)
Hypothesis testing
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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](https://reader033.vdocuments.net/reader033/viewer/2022053017/5f1b636ac35de14760676a30/html5/thumbnails/23.jpg)
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](https://reader033.vdocuments.net/reader033/viewer/2022053017/5f1b636ac35de14760676a30/html5/thumbnails/24.jpg)
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](https://reader033.vdocuments.net/reader033/viewer/2022053017/5f1b636ac35de14760676a30/html5/thumbnails/25.jpg)
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](https://reader033.vdocuments.net/reader033/viewer/2022053017/5f1b636ac35de14760676a30/html5/thumbnails/26.jpg)
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
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Bayes optimal decisions
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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](https://reader033.vdocuments.net/reader033/viewer/2022053017/5f1b636ac35de14760676a30/html5/thumbnails/29.jpg)
Simple optimisation
Eπ(θ)(G (θ))
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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))
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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](https://reader033.vdocuments.net/reader033/viewer/2022053017/5f1b636ac35de14760676a30/html5/thumbnails/32.jpg)
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](https://reader033.vdocuments.net/reader033/viewer/2022053017/5f1b636ac35de14760676a30/html5/thumbnails/33.jpg)
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
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Any questions?