Bayes-Verfahren in klinischen Studien

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1 Bayes-Verfahren in klinischen Studien Dr. rer. nat. Joachim Gerß, Dipl.-Stat. Institute of Biostatistics and Clinical Research

2 J. Gerß: Bayesian Methods in Clinical Trials 2

3 Contents 1. Prior and Posterior distribution 2. A Bayesian significance test? 3. Response-adaptive randomization Thomas Bayes ( ) 4. Bayesian decision making in interim analyses 5. Borrowing of information across related populations 6. Conclusion J. Gerß: Bayesian Methods in Clinical Trials 3

4 Contents 1. Prior and Posterior distribution 2. A Bayesian significance test? 3. Response-adaptive randomization Thomas Bayes ( ) 4. Bayesian decision making in interim analyses 5. Borrowing of information across related populations 6. Conclusion J. Gerß: Bayesian Methods in Clinical Trials 4

5 1. Prior and Posterior distribution Combination of prior beliefs with data from a study Classical frequentist statistical analysis Example 1 Bayesian analysis Prior Prior + Data = Posterior 1,0 0, Hazard ratio Survival rate 0,6 0,4 0,2 0,0 Group 1 Group 2 Data Hazard 95% Confidence interval: (0.947,5.238) ratio Survival after (years) % Credible Interval: (1.074,4.285) HR= % CI p= Hazard 95% Confidence interval: (0.947,5.238) ratio J. Gerß: Bayesian Methods in Clinical Trials 5

6 1. Prior and Posterior distribution Combination of prior beliefs with data from a study Example 1 Example 2 Example 3 Non-informative prior Prior + Data = Posterior Prior + Data = Posterior Prior + Data = Posterior Hazard 95% Confidence interval: (0.947,5.238) ratio Hazard 95% Confidence interval: (0.947,5.238) ratio Hazard 95% Confidence interval: (0.947,5.238) ratio 95% Credible Interval: (1.074,4.285) 95% Credible Interval: (1.822,4.264) 95% Credible Interval: (0.947,5.238) J. Gerß: Bayesian Methods in Clinical Trials 6

7 1. Prior and Posterior distribution Example: Two groups, normal data, known variance Classical frequentist statistical analysis Gauss Test X 1,,X n ~ N(µ 1,σ 2 ) Y 1,,Y m ~ N(µ 2,σ 2 ), σ 2 known H 0 : µ 1 -µ 2 0 versus H 1 : µ>0 <=> H 0 : µ 0 versus H 1 : µ>0 with μ Test statistic If µ=0 (H 0 ): Z~N(0,1) ~ μ, 1 p = Prob { Z z H 0 } = 1 Φ(z) Bayesian analysis Normal-Normal Model Data Model: μ ~ μ, 1 Prior distribution: f(µ) 1 (non-informative) or µ ~ N(µ 0,τ 02 ) Posterior distribution: μ = μ, μ, = μ μ => μ ~, 1 (non-informative prior) or μ ~, J. Gerß: Bayesian Methods in Clinical Trials 7

8 Contents 1. Prior and Posterior distribution 2. A Bayesian significance test? 3. Response-adaptive randomization 4. Bayesian decision making in interim analyses 5. Borrowing of information across related populations 6. Conclusion J. Gerß: Bayesian Methods in Clinical Trials 8

9 2. A Bayesian significance test? Example: Two groups, normal data, known variance H 0 : µ 0 versus H 1 : µ>0 Classical frequentist statistical analysis Gauss Test p = Prob { Z z H 0 } = 1 Φ(z) Reject H 0, if p 0.05 => Prob { Reject H 0 µ=0 } = 0.05 Bayesian analysis, (1) non-informative prior μ ~, 1 Reject H 0, if Prob { µ 0 z } α Bayes Now Prob { µ 0 z } = Prob { µ-z -z z } 95% µ = Φ(-z) = 1 Φ(z) = p, and with α Bayes = 0.05: Prob { Reject H 0 µ=0 } = Prob { p α Bayes µ=0} = α Bayes = 0.05 => Reject H 0, if Prob { µ 0 z } = p α Bayes = 0.05 J. Gerß: Bayesian Methods in Clinical Trials 9

10 2. A Bayesian significance test? Example: Two groups, normal data, known variance H 0 : µ 0 versus H 1 : µ>0 Classical frequentist statistical analysis Bayesian analysis, (2) informative prior Gauss Test p = Prob { Z z H 0 } = 1 Φ(z) Reject H 0, if p 0.05 => Prob { Reject H 0 µ=0 } = Bayesian analysis, informative prior (2a) μ ~ Bayesian analysis, informative prior (2b) J. Gerß: Bayesian Methods in Clinical µ Trials 10 1 μ 1, Reject H 0, if Prob { µ 0 z } α Bayes (a) α Bayes = 0.05 (b) Determine α Bayes, so that Prob { Reject H 0 µ=0 } = 0.05 Frequentist statistical analysis = Bayesian analysis, non-informative prior Prior distribution

11 2. A Bayesian significance test? with controlled type I error, but (1) Assume µ=0 (H 0 ) (Informative) prior (2) Prob(Type I error) is a long-run frequentist probability! Bayesian decision rule: 95% certainty is directly assured, according to the posterior distribution 95% µ -> Bayesian methods in a strict corset of frequentist quality criteria are usually not much more powerful than classical frequentist methods. J. Gerß: Bayesian Methods in Clinical Trials 11

12 2. A Bayesian significance test? with controlled type I error, but (1) Assume µ=0 (H 0 ) (Informative) prior (2) Prob(Type I error) is a long-run frequentist probability! Bayesian decision rule: 95% certainty is directly assured, according to the posterior distribution Application of Bayesian methods 95% (Interval) estimation Interpretation of results: E.g., Compute Prob { H 0 data } Model complex relationships Account for different levels of uncertainty (see below) -> Clinical trial: Basic frequentist design (classical significance test), with Bayesian supplements (see below) µ J. Gerß: Bayesian Methods in Clinical Trials 12

13 Contents 1. Prior and Posterior distribution 2. A Bayesian significance test? 3. Response-adaptive randomization 4. Bayesian decision making in interim analyses 5. Borrowing of information across related populations 6. Conclusion J. Gerß: Bayesian Methods in Clinical Trials 13

14 3. Response-adaptive randomization Consider a randomized two- or multi-arm clinical trial Response-adaptive randomization: Randomized Treatment Assignment not with equal and fixed probabilities, but increased assignment of patients to more promising treatments Prob (Treatment assignment) TI 0.6 TA IA Pat.-No. J. Gerß: Bayesian Methods in Clinical Trials 14

15 3. Response-adaptive randomization Example: Phase IIB design with binary response Two treatment arms k=1,2 Denote θ k the response probability in arm k {1,2} Goal: Compare response probabilities θ 1, θ 2 Algorithm 1. Randomize the first 14 patients to treatment arm 1 and 2 with equal probability 1/2. 2. After each observed outcome, compute the (posterior) probability of each arm k to be the best arm, using all currently available data ( Prob(arm k is best) ) 3. Assign patients to treatment groups with probability proportional to Prob(arm k is best) c (with tuning parameter c=1), but never lower than Final analysis with n=60 patients: Fisher s exact test J. Gerß: Bayesian Methods in Clinical Trials 15

16 3. Response-adaptive randomization Example: Phase IIB design with binary response Simulation ( runs) Response prob. θ 1 / θ 2 Mean number of patients in groups 1 / 2 Fixed alloc. rate Bayesian adaptive random. Mean total number of responses Fixed alloc. rate Bayesian adaptive random. Type 1 error / Power Fixed alloc. rate Bayesian adaptive random. 0.6 / / / / / / / / / / / / J. Gerß: Bayesian Methods in Clinical Trials 16

17 Contents 1. Prior and Posterior distribution 2. A Bayesian significance test? 3. Response-adaptive randomization 4. Bayesian decision making in interim analyses 5. Borrowing of information across related populations 6. Conclusion J. Gerß: Bayesian Methods in Clinical Trials 17

18 4. Bayesian decision making in interim analyses Klinische Studie mit 2 Behandlungsgruppen Normalverteilte Zielgröße Gruppe 1: ~ μ, Gruppe 2: ~ μ,, mit bekannter Varianz μ μ μ H 0 : µ 0 versus H 1 : µ>0, α=0.025 Teststatistik: p = 1 - Φ(z) ~ 0,1 unter H 0 J. Gerß: Bayesian Methods in Clinical Trials 18

19 4. Bayesian decision making in interim analyses Pocock-Design, Inverse-Normal-Methode n 1 =35 Fälle pro Gruppe Information rate 0.5 Teststatistik Z 1 p 1 = 1 - Φ(z 1 ) p 1 1 α 0 =0,5 Futility Stop Fortsetzung mit n 2 Fällen pro Gruppe Teststatistik Z 2, p 2 = 1 - Φ(z 2 ) H 0 ablehnen, falls, α c = α 1 = 0, : α 1 =0, H 0 ablehnen mit 1 2 J. Gerß: Bayesian Methods in Clinical Trials 19

20 4. Bayesian decision making in interim analyses Fallzahl-Rekalkulation Sei α 1 < p 1 α Conditional Power 1-β c 1,μ 1 1 Ziel: 1-β c =0.8 bei wahrem Effekt µ (!) Setze μ Conditional Power = 1 = z 1 = p 1 = (beobachtete Mittelwertdifferenz in 1. Stufe) n n2 J. Gerß: Bayesian Methods in Clinical Trials 20

21 4. Bayesian decision making in interim analyses Pocock-Design, Inverse-Normal-Methode n 1 =35 Fälle pro Gruppe Information rate 0.5 Teststatistik Z 1 p 1 = 1 - Φ(z 1 ) p 1 1 α 0 =0,5 Futility Stop Fortsetzung mit n 2 Fällen pro Gruppe so dass 1-β c =0.8 (35 n 2 100) Teststatistik Z 2, p 2 = 1 - Φ(z 2 ) H 0 ablehnen, falls, α c = α 1 = 0, : α 1 =0, H 0 ablehnen mit 1 2 J. Gerß: Bayesian Methods in Clinical Trials 21

22 4. Bayesian decision making in interim analyses Bayesian Predictive Power Sei α 1 < p 1 α 0 Conditional Power 1-β c 1,μ 1 1 Ziel: 1-β c =0.8 bei wahrem Effekt µ (!) Setze μ (beobachtete Mittelwertdifferenz in 1. Stufe) Bayesian Predictive Power (BPP),μ, μ μ mit μ, ~, 1 J. Gerß: Bayesian Methods in Clinical Trials 22

23 4. Bayesian decision making in interim analyses Bayesian Predictive Power Sei α 1 < p 1 α 0 Conditional Power 1-β c 1,μ 1 1 Ziel: 1-β c =0.8 bei wahrem Effekt µ (!) Setze μ (beobachtete Mittelwertdifferenz in 1. Stufe) Bayesian Predictive Power (BPP) 1,μ, μ μ mit μ, ~, 1 J. Gerß: Bayesian Methods in Clinical Trials 23

24 4. Bayesian decision making in interim analyses Bayesian Predictive Power : lim Sei α 1 < p 1 α 0 Conditional Power 1-β c 1,μ 1 1 Ziel: 1-β c =0.8 bei wahrem Effekt µ (!) Setze μ (beobachtete Mittelwertdifferenz in 1. Stufe) Bayesian Predictive Power (BPP) Bayesian Predictive Power = 1 = z 1 = p 1 = n2,μ, μ μ 1 mit μ, ~, J. Gerß: Bayesian Methods in Clinical Trials 24

25 4. Bayesian decision making in interim analyses Gruppensequentiell-adaptives Studiendesign Pocock-Design, Inverse-Normal-Methode n 1 =35 Fälle pro Gruppe Information rate 0.5 Teststatistik Z 1 p 1 = 1 - Φ(z 1 ) p 1 1 α 0 =0,5 Futility Stop Fortsetzung mit n 2 Fällen pro Gruppe so dass 1-β c BPP=0.8 (35 n 2 100) Teststatistik Z 2, p 2 = 1 - Φ(z 2 ) H 0 ablehnen, falls, α c = α 1 = 0, : α 1 =0, H 0 ablehnen mit 1 2 J. Gerß: Bayesian Methods in Clinical Trials 25

26 4. Bayesian decision making in interim analyses Gruppensequentiell-adaptives Studiendesign Pocock-Design, Inverse-Normal-Methode n 1 =35 Fälle pro Gruppe Information rate 0.5 Teststatistik Z 1 p 1 = 1 - Φ(z 1 ) p 1 1 α 0 =0,5 Futility Stop Fortsetzung mit n 2 Fällen pro Gruppe so dass 1-β c BPP=0.8 (35 n 2 100) Teststatistik Z 2, p 2 = 1 - Φ(z 2 ) H 0 ablehnen, falls, α c = α 1 = 0, : α 1 =0, H 0 ablehnen mit 1 2 J. Gerß: Bayesian Methods in Clinical Trials 26

27 4. Bayesian decision making in interim analyses Futility Stop Sei p 1 > α 1 Bayesian Predictive Power (BPP),μ, μ μ mit μ, ~, 1 ( 35 n ) Bei n 2 =100: BPP 100 =? Falls BPP 100 <0.2 => Futility Stop J. Gerß: Bayesian Methods in Clinical Trials 27

28 4. Bayesian decision making in interim analyses Futility Stop p n 1 =35 Fälle pro Gruppe n 2 =100 Fälle pro Gruppe =1 0.2 BPP Observed Mean Difference J. Gerß: Bayesian Methods in Clinical Trials 28

29 4. Bayesian decision making in interim analyses Gruppensequentiell-adaptives Studiendesign Pocock-Design, Inverse-Normal-Methode n 1 =35 Fälle pro Gruppe Information rate 0.5 Teststatistik Z 1 p 1 = 1 - Φ(z 1 ) BPP ,2 BPP 100 Futility Stop Fortsetzung mit n 2 Fällen pro Gruppe so dass 1-β c BPP=0.8 (35 n 2 100) Teststatistik Z 2, p 2 = 1 - Φ(z 2 ) H 0 ablehnen, falls, α c = α 1 = 0,0148 0,0148 p 1 α 1 =0,0148 0, H 0 ablehnen 1 : mit 1 2 J. Gerß: Bayesian Methods in Clinical Trials 29

30 4. Bayesian decision making in interim analyses Futility Stop 0.8 n 1 =35 Fälle pro Gruppe n 2 =100 Fälle pro Gruppe 0.6 BPP p1 J. Gerß: Bayesian Methods in Clinical Trials 30

31 4. Bayesian decision making in interim analyses Simulation Fehler 1. Art / Power Fallzahl (pro Gruppe) Futility Stops Klassisch Bayes Klassisch Bayes Klassisch Bayes δ/σ = 0 0,025 0, ,4 132,6 δ/σ = 0,2 0,269 0, ,6 122,6 δ/σ = 0,4 0,803 0,827 91,6 97,9 δ/σ = 0,5 0,940 0,955 75,7 81,7 δ/σ = 0,6 0,985 0,992 61,1 65,8 δ/σ = 0,8 0,999 1,000 42,4 44,0 δ/σ = 1 1,000 1,000 36,3 36,5 J. Gerß: Bayesian Methods in Clinical Trials 31

32 4. Bayesian decision making in interim analyses Simulation Fehler 1. Art / Power Fallzahl (pro Gruppe) Futility Stops Klassisch Bayes Klassisch Bayes Klassisch Bayes δ/σ = 0 0,025 0,025 81,4 62,6 0,499 0,700 δ/σ = 0,2 0,266 0,268 98,6 84,8 0,200 0,377 δ/σ = 0,4 0,791 0,776 86,9 85,2 0,046 0,125 δ/σ = 0,5 0,929 0,914 73,6 75,5 0,018 0,058 δ/σ = 0,6 0,980 0,970 60,7 63,3 0,006 0,024 δ/σ = 0,8 0,999 0,997 42,3 43,9 0,000 0,002 δ/σ = 1 1,000 1,000 36,2 36,5 0,000 0,000 J. Gerß: Bayesian Methods in Clinical Trials 32

33 Contents 1. Prior and Posterior distribution 2. A Bayesian significance test? 3. Response-adaptive randomization 4. Bayesian decision making in interim analyses 5. Borrowing of information across related populations 6. Conclusion J. Gerß: Bayesian Methods in Clinical Trials 33

34 5. Borrowing of information across related populations Biomarkers in JIA No. Flares / Patients (%) MRP8/ ng/ml MRP8/14 <690 ng/ml OR Fisher s Exact Test All patients (n=188) 22 / 75 (29%) 13/ 113 (12%) 3.2 p= Subgroup Oligoarthritis (n=86) 9 / 34 (26%) 8 / 52 (15%) 2.0 p= Subgroup Polyarthritis (n=74) 11 / 25 (44%) 5 / 49 (10%) 6.9 p= Subgroup Other (n=28) 2 / 16 (13%) 0 / 12 (0%) 4.3 p= Gerß et al. Ann Rheum Dis 2012;71: J. Gerß: Bayesian Methods in Clinical Trials 34

35 5. Borrowing of information across related populations Hierarchical model Let := Observed ln(odds Ratio) in subgroup i Observed lnor s: ~,, 1,2,3 with assumed known Parameter model: Prior: μ, ~, f, f f 1 (non-informative) 1 1 Bayesian approach: Sampling from f,,,, f,,,,,, f,,,,,,,,, Frequentist approach ( Empirical Bayes ):, ~ Bivariate Normal =>, plug in REML estimators of, J. Gerß: Bayesian Methods in Clinical Trials 35

36 5. Borrowing of information across related populations Biomarkers in JIA: Results Observed Odds Ratio Subgroup Oligoarthritis (n=86) Fully Bayesian Estimator Empirical Bayes Estimator Subgroup Polyarthritis (n=74) Subgroup Other (n=28) Pooled OR J. Gerß: Bayesian Methods in Clinical Trials 36

37 Contents 1. Prior and Posterior distribution 2. A Bayesian significance test? 3. Response-adaptive randomization 4. Bayesian decision making in interim analyses 5. Borrowing of information across related populations 6. Conclusion J. Gerß: Bayesian Methods in Clinical Trials 37

38 6. Conclusion Bayesian Methods in Clinical Trials Early phase clinical trials ( in-house studies w/o strict regulatory control) Trials in small populations Medical device trials Exploratory studies Use fully Bayesian approach, paying attention to choose the appropriate model carefully, choose the inputted (prior) information carefully and check (classical) operating characteristics (type I error, power) Large scale confirmatory trials with strict type I error control Use of Bayesian supplements Response-adaptive randomization Bayesian decision making in interim analyses (Bayesian data monitoring / sequential stopping, predictive probabilities) J. Gerß: Bayesian Methods in Clinical Trials 38

39 Literature Berry SM, Carlin BP, Lee JJ, Müller P (2010): Bayesian Adaptive Methods for Clinical Trials. Chapman & Hall/CRC Biostatistics. Spiegelhalter DJ, Abrams KR, Myles JP (2004): Bayesian Approaches to Clinical Trials and Health-Care Evaluation. Wiley Series in Statistics in Practice. U. S. Food and Drug Administration, Center for Devices and Radiological Health (2010): Guidance for the Use of Bayesian Statistics in Medical Device Clinical Trials. 2.htm Giles FJ et al. (2003): Adaptive randomized study of Idarubicin and Cytarabine versus Troxacitabine and Cytarabine versus Troxacitabine and Idarubicin in untreated patients 50 years or older with adverse karyotype Acute Myeloid Leukemia. Journal of Clinical Oncology 21(9); Gerss J et al. (2012): Phagocyte-specific S100 proteins and high-sensitivity C reactive protein as biomarkers for a risk-adapted treatment to maintain remission in juvenile idiopathic arthritis: a comparative study. Annals of the rheumatic diseases 71(12); J. Gerß: Bayesian Methods in Clinical Trials 39

Effective Implementation of Bayesian Adaptive Randomization in Early Phase Clinical Development. Pantelis Vlachos.

Effective Implementation of Bayesian Adaptive Randomization in Early Phase Clinical Development Pantelis Vlachos Cytel Inc, Geneva Acknowledgement Joint work with Giacomo Mordenti, Grünenthal Virginie