Dr. Rene Schmidt - uni-muenster.de · Dr. Rene Schmidt, Prof. Dr. Andreas Faldum 4 • Ethical...

Dr. Rene Schmidt

Institute of Biostatistics and Clinical Research University of Münster

[email protected]

http://ibkf.uni-muenster.de

Biometrie-Symposium: 40 Jahre Biometrie Münster 03. Juli 2013

Münster

Adaptive Designs für multiple Überlebenszeit-Endpunkte

Adaptive Designs for Multiple Time-To-Event Endpoints

Dr. Rene Schmidt, Prof. Dr. Andreas Faldum

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• An adaptive design is a sequential design, which:

− Forsees (usually planned) interim analyses.

− Allows to redesign the trial after each interim analysis.

− Uses (certain) information from the running trial.

− Without compromizing integrity of the trial.

What Are Adaptive Designs?



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• Ethical reasons:

− Evident superiority of therapy (ICH guidelines)

• Economic reasons:

− No therapy effect verifiable New therapy strategy.

• Information in planning phase might be unsure or biased.

• Changes of medical setting (applies to long-term trails).

• Here: Adaptive designs (two-stage).

Why Do We Need Interim Analyses/Adaptive Designs?



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• Example from pediatric oncology (brain tumors):

− SIOP-LGG 201X trial (successor of SIOP-LGG 2004).

− Objective: Reduce adverse long-term effects of treatment on neurologic functions without compromising efficacy.

− In short: Reduce toxicities without compromising efficacy.

− Proving non-inferior efficacy and less toxicity.

− Consider both as primary: Event free survival (EFS) and toxicity free survival (TFS).

Why Do We Need Multiple (Time-to-Event) Endpoints?



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• Treat several time-to-event outcomes as key in an adaptive setting: − E.g.: Event free survival (EFS) and toxicity free survival (TFS)

• This implies performance of several significance tests: − Multiplicity due to interim analyses (from adaptive design) − Multiplicity due to multiple endpoints.

• Doing several significance tests increases the overall chance of a type I error.

• Adjustment for multiple testing: Adaptive closed testing procedure (Hommel, 2001).

Trial Structure with Dependent Stages



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• Comparison of two treatment arms w.r.t. event free survival (EFS) and toxicity free survival (TFS).

• Notation: − T1(t): Brownian motion

(corresponding to Log-Rank statistic w.r.t. EFS) − T2(t): Brownian motion

(corresponding to Log-Rank statistic w.r.t. TFS) • Perform two-stage adaptive design.

Notation



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• Properties of the above p-values: − 𝑃𝑖,𝑗~𝑈(0,1) − 𝑃1,1 and 𝑃2,2: Dependent with unknown dependence structure.

• P-clud condition 𝑃𝐻0 𝑃2,2 ≤ 𝛼 𝑃1,1 ≤ 𝛼 might not be fulfilled!

Example Hierarchical Testing with Change of Order of

Hypotheses in Any Case Stage 1 Stage 2

𝐻1 𝑝1,1 ↔ 𝑇1(𝑡1,1) 𝑝1,2 ↔ 𝑇1 𝑡1,2 − 𝑇1(𝑡1,1)

𝐻2 𝑝2,1 ↔ 𝑇2(𝑡2,1) 𝑝2,2 ↔ 𝑇2 𝑡2,2 − 𝑇2(𝑡2,1)

𝐻1 ∩ 𝐻2 𝑝1∩2,1 = 𝑝1,1 𝑝1∩2,2 = 𝑝1,2

Stage 1 Stage 2 𝐻1 𝑝1,1 ↔ 𝑇1(𝑡1,1) 𝑝1,2 ↔ 𝑇1 𝑡1,2 − 𝑇1(𝑡1,1)

𝐻2 𝑝2,1 ↔ 𝑇2(𝑡2,1) 𝑝2,2 ↔ 𝑇2 𝑡2,2 − 𝑇2(𝑡2,1)

𝐻1 ∩ 𝐻2 𝑝1∩2,1 = 𝑝1,1 𝑝1∩2,2 = 𝑝1,2


𝐻2 𝑝2,1 ↔ 𝑇2(𝑡2,1) 𝑝2,2 ↔ 𝑇2 𝑡2,2 − 𝑇2(𝑡2,1)

𝐻1 ∩ 𝐻2 𝑝1∩2,1 = 𝑝1,1


𝐻2 𝑝2,1 ↔ 𝑇2(𝑡2,1) 𝑝2,2 ↔ 𝑇2 𝑡2,2 − 𝑇2(𝑡2,1)

𝐻1 ∩ 𝐻2 𝑝1∩2,1 = 𝑝1,1 𝑝1∩2,2 = 𝑝2,2



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• Properties of p-values: − 𝑝1,1 and 𝑝2,2: Dependent with unknown dependence structure. − min 𝑝1,1,𝑝2,1 and min 𝑝1,2,𝑝2,2 dependent.

• P-clud condition might not be fulfilled for design for 𝐻1 ∩ 𝐻2!

Example Bonferroni Adjustment


𝐻2 𝑝2,1 ↔ 𝑇2(𝑡2,1) 𝑝2,2 ↔ 𝑇2 𝑡2,2 − 𝑇2(𝑡2,1)

𝐻1 ∩ 𝐻2 min{2 min 𝑝1,1,𝑝2,1 , 1} min{2 min 𝑝1,2,𝑝2,2 , 1}



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• In order to guarantee control of type I error rate:

− Need adaptive designs with arbitrary dependent p-values.

Rationale



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• Two-stage adaptive design with arbitrary dependence structures:

− Arbitrary in the class of all dependence structures.

− Not restricted to the class of bivariate normal dependence structures.

• General approach via copulas (encode dependence structure).

• Example:

− Application to inverse normal designs

Contents



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• First stage: – p1 > α0 Stopping with non-rejection of H0 – p1 ≤ α1 Stopping with rejection of H0 – α1 < p1 ≤ α0 Continue with second stage

• Second stage: – Rejection of H0, if p2 ≤ α(p1).

Two-Stage Adaptive Designs



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• Determination of α1 or of CEF α(p1) such that:

• Depends on:

– α0, α1 and CEF α(p1)

– Joint distribution of P1, P2

Two-Stage Adaptive Designs

PH0(P1 ≤ α1 ) + PH0

(P2 ≤ α(P1 ), α1 < P1 ≤ α0 ) = α



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• Notation:

– 𝑇1,𝑇2 continuous statistics of stages with distribution 𝐹1,𝐹2.

– And with (unknown) copula C.

– 𝛼 𝑝1 = � 1 , 𝑖𝑖 𝑝1 ≤ 𝛼1𝑖(𝛼1) , 𝑖𝑖 𝑝1 > 𝛼1

– 𝑖: 0,1 → [0,1] strictly decreasing, absolutely continuous.

Adaptive Designs with Arbitrary Dependence Structure



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𝑃𝐻0 𝑅𝑅𝑅𝑅𝑅𝑡 𝐻0 = 1 − 𝑃𝐻0𝐶 (𝑋 + 𝑌 > 0,𝑋 > 𝛾)

• 𝑌 ≔ 𝑃2, 𝑋 ≔ −𝑖(𝑃1), 𝛾 = −𝑖(𝛼1).

• Control of type I error rate 𝛼 is guaranteed if:

inf𝐶𝑃𝐻0𝐶 (𝑋 + 𝑌 > 0,𝑋 > 𝛾) ≥ 1 − 𝛼

• Left hand side (type I error in the worst case) may be computed in analytically closed form.

Adaptive Designs with Arbitrary Dependence Structure



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𝛼 𝑝1 = �1 ,𝑝1 ≤ 𝛼1

1 −Φ( 2𝑅 − Φ−1(1 − 𝑝1)) ,𝑝1 > 𝛼1

• Let 𝑃𝑖 ∼ 𝑈(0,1) under H0. • Compute worst case type I error rate

(i.e. arbitrary dependence structure between 𝑃1 and 𝑃2).

Example: Worst Case Inverse Normal Designs



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• Worst case type I error rate 𝛼 (may be computed in analytically closed form):

𝛼 = 1 − inf𝐶𝑃𝐻0𝐶 𝑇1 + 𝑇2 < 2𝑅,𝑇1 < 𝛾

𝛾 = Φ−1(1 − 𝛼1) • “Reference” level for independent tests:

𝛼𝑖𝑖𝑖 = 1 − � Φ 2𝑅 − 𝑧1 𝜙 𝑧1 𝑑𝑧1

𝛾

−∞

• Design of Pocock type: 𝛾 = 𝑅 (Wassmer, 2001)

Example: Worst Case Inverse Normal Designs



21 Pocock design (no futility stop): Inflation of Type I error

18.5%

0.5%



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Summary

• General concept for adaptive designs with arbitrary dependence structure.

• Application to continuous test statistics T1, T2. • E.g.: Adaptive Change of Hypotheses in trial with multiple

time-to-event endpoints (e.g. in pediatric oncology) • Examples:

– Worst case inverse normal designs. • Further research:

– Designs with specific sort of dependence of T1, T2. – Discrete test statistics. – Study the worst case for other popular adaptive designs.



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Bibliography

• Brannath W, Posch M, Bauer P. Recursive combination tests. Journal of American Statistical Association 2002; 97: 236-244.

• Götte H, Hommel G, Faldum A. Adaptive designs with correlated test statistics. Statist. Med. 2009; 28: 1429-1444.

• Hommel G. Adaptive modifications of hypotheses after an interim analysis. Biometrical Journal 2001; 43, 581-589.

• Hommel G, Lindig V, Faldum A. Two-stage adaptive designs with correlated test statistics. Journal of Biopharmaceutical Statistics 2005; 15: 613-623.

• Wassmer G. 2001. Statistische Testverfahren für gruppensequentielle und adaptive Pläne in klinischen Studien (2. Auflage). Verlag Alexander Mönch: München.



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IBKF Institut für Biometrie und

Klinische Forschung

Dr. Rene Schmidt

Institute of Biostatistics and Clinical Research University of Münster

[email protected]

http://ibkf.uni-muenster.de

Dr. Rene Schmidt - uni-muenster.de · Dr. Rene Schmidt, Prof. Dr. Andreas Faldum 4 • Ethical...

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