Bayesian Methods in Regulatory Science

Transcription

1 Bayesian Methods in Regulatory Science Gary L. Rosner, Sc.D. Regulatory-Industry Statistics Workshop Washington, D.C. 13 September 2018

2 What is Regulatory Science? US FDA Regulatory Science is the science of developing new tools, standards, and approaches to assess the safety, efficacy, quality, and performance of all FDAregulated products. Accessed 5 June

3 What Do I Mean by Bayesian Approaches? Truly Bayes: Choose design points to maximize expected utility - E.g., sample size for certain precision at fixed cost Stylized Bayes (or calibrated Bayes, etc.) Bayesian formulation of study data Choose design points to achieve desirable frequentist operating characteristics - E.g., choose prior parameters to attain monitoring rules good sensitivity, specificity, & expected N 3

4 Acceptance When Want to Borrow Information Change in device from previous version Historical information reasonable if minor change Population pharmacokinetics & pharmacodynamics Population modeling & hierarchical models Pediatric drug development Borrow information Rare diseases or adverse events Early phase studies (e.g., phases 1 & 2) 4

5 When Are Bayesian Statistical Methods Not Accepted When want to be objective Concern about influence of prior distribution - But frequentist methods are subjective, too! Model, Null & alternative hypotheses, Incorporating current trial in context of others Confirmatory drug studies 5

6 Sheiner: Two Major Learn-Confirm Cycles in Clinical Drug Development 1 st cycle: Clin Pharmacol Ther 61:275-91, 1997 Phase 1: Learn what dose is tolerated Phase 2: Confirm dose has promise of efficacy - Make decision based on this learn-confirm cycle 2 nd cycle: Phase 2B: Learn how to use the drug in patients Phase 3: Confirm in large representative pt pop n that therapy achieves acceptable benefit:risk ratio - If acceptable, approval is granted 6

7 Sheiner (cont d) Learning & confirming are distinct Different goals, designs, methods of analysis - Analysis choice: Hypothesis testing or estimation? - Learning involves estimation The [B]ayesian view is well suited to this task because it provides a theoretical basis for learning from experience; that is, for updating prior beliefs in the light of new evidence. Clin Pharmacol Ther 61:275-91, Confirming involves hypothesis testing 7

8 Neyman & Pearson Need more than a test based on probability to establish truth of a particular hypothesis Neyman, J., and Pearson, E. S. (1933). On the Problem of the Most Efficient Tests of Statistical Hypotheses. Philosophical Transactions of the Royal Society of London. Series A, 231,

9 R. A. Fisher Discusses forming summary statistics & evaluating deviation relative to a dist n (for tests of significance ). Fisher, R. A. (1934). Statistical Methods for Research Workers, 5th ed. Edinburgh: Oliver and Boyd. 9

10 SN Goodman: Toward Evidence- Based Medical Statistics 1 & 2 A p value seems to provide a measure of evidence against H0 from this study and a means to control Type 1 error regarding rejecting H0 in the long run. But, no single number can do both jobs! Describe long-run behavior & meaning in this study. Treat this study as one of infinitely many & provide a measure of evidence for results in this study. Goodman, S. N. (1999). Toward Evidence-Based Medical Statistics 1 & 2 Ann Intern Med 130, &

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sha1_base64="qr1zwnehv3agya+xuykh2l9xjjo=">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</latexit> Hypothesis Testing Answer a question Do observations agree with predictions based on one hypothesis more than the other? N-P testing Decision rule to limit long-run risks of errors Bayesian testing Pr(y H 0 )vs.pr(y H A ) Compare Pr(H A y) topr(h <latexit sha1_base64="jufvd58b/uj+mi6g5pl0ra18ffm=">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</latexit> 0 y) Condition on data - Could be with Bayes factor, and Jeffreys s criteria Pr(H A y) Pr(H A ) Pr(H 0 y) Pr(H 0 ) 11

12 Which Do We Want? Evidence one trt is superior to the other? Estimate with precision? Decision rule regarding hypothesis? Yes reject? No do not reject? Decision rule regarding next step? Continue to next phase of study? Approve the treatment for indication? 12

13 Decision Theory & Clinical Trials Why decision theory? Clinical trials: Purpose is to lead to decisions - What dose(s) to use? - How best to apply the therapy? - What is the next step for evaluating therapy? - Should patients receive this therapy from now on? - Which patients receive the most benefit? Why not make decisions explicit and coherent? Put results in context via formal decision analysis 13

14 Decision Theory & Clinical Trials Clinical trial design involves decisions, too Sample size, Duration of follow-up, Stopping rules, Whether to run the study in the 1 st place Why not make these decisions explicit and coherent? 14

15 Value model Weight each value (utility for each outcome) Develop scenarios (simulate the trial) Multiply each grade by the weight of each value and add up the numbers for each scenario. The scenario with the highest score wins.

16 Decision Theory Consider Set of possible actions: A - E.g., stop the study; move to phase 3 Set of possible outcomes:y Parameters characterizing stochastic nature: - E.g., treatment effect, model parameters Utility function: u(a) 16

17 Bayesian Optimal Design Bayes action maximizes expected utility Expectation to account for sources of uncertainty - Uncertainty in parametersp( ) - Variation in data resulting from action p a (y ) R R U(a) = Y u(y)p a(y )p( )d dy - Choose: a = arg max U(a) 17

18 Application Cancer & Leukemia Group B wanted to study Taxol Large population of women 3-hour infusion Many participating hospitals Outpatient 18

19 Problem Cannot carry out extensive sampling Large study Many institutions Devise limited-sampling scheme Optimal sampling times Stroud JR, Müller P, Rosner GL. Optimal sampling times in population pharmacokinetic studies. Applied Statistics. 2001;50(3):

20 Objective: Maximize Precision Toxicity risk = f(auc) or g(tcv)!20

21 Paclitaxel PK Sampling Times Log Concentration log(µm/l) Drug Infusion 100 mg/m2 75 mg/m2 50 mg/m2 k k 13 infusion ov m ok m V m / k 21 K m Time (Hours)!21

22 Optimal Sampling Times for AUC u(t, y) = applez { ( ) E [ ( ) y, Y, t]} 2 p( y, Y, t)d 1 k px i=1 t 2 i I {ti >8} n=1 n=2 Cost Coeff k t * U * t * U * (3) 0.53 (3,25) (3) 0.53 (3,10) (3) 0.53 (3,7) 0.70!22

23 Dose Optimization AUC too low! TARGET AUC too high! 23

24 Asymmetric Loss Function Want AUC in optimal range AUC ll apple AUC apple AUC ul Loss function Loss L(auc) = 8 >< L (auc, AUC ll ) if auc < AUC ll AUC 0 if AUC >: ll apple auc apple AUC ul L + (auc, AUC ul ) if auc > AUC ul 24

25 Posterior for Pt s PK Parameters 1 st pt in 3 rd study, accounting for studies 1 & 2 25

26 Optimal Dose w.r.t. Posterior Z E[u(y, d, )] = L[AUC(y)] p(y d, )p( Data)dyd Loss Function Expected Loss Loss AUC 26

27 Learn About Benefit:Risk Weigh chance of benefit against risk of adverse event Dose finding based on trade-offs between probs of efficacy and toxicity Thall PF, Cook JD (2004). Dose-Finding Based on Efficacy-Toxicity Trade-Offs. Biometrics 60,

28 Trippa: Response-Adaptive Designs for Basket Trials Strategies for choosing among baskets Objective function is Bayesian decision rule (Exp. utility) Decisions during study work toward maximum information at the end of the study Fully sequential hard! Bayesian Uncertainty Directed (BUD) designs - Approximate optimal decision rules (myopic) - Goal: Design rule that approximates Bayesian decision and that one can compute 28

29 <latexit sha1_base64="nzujt6duni8+vjs/3ykhn+zzpxg=">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</latexit> <latexit sha1_base64="kpo7qn/jciwexfpxmkod83rv/te=">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</latexit> <latexit sha1_base64="kpo7qn/jciwexfpxmkod83rv/te=">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</latexit> <latexit sha1_base64="56nzo5i4xdbwswrm+e2dkpsx9y4=">aaaesxichzndb9mwfia9lcaoh+vgkhuldmlimcvwaqihtqwkxq6jbkvnvhzntlvqx8f2wnsop4zfwy1c8m9wpjqtargwyr32857y59goe860cd0/w9s7zo2bt3zvt+7cvxd/r73/4ezlvfhoucml6odea2cx9awzhpqjaijcdufh9h3bz7+d0kzgn808gucqccxgjbjjp4btt33sayzwcth/+gz731is4znbmqrwo/z848dnclxs7aswyd/azgqficf5mgx33co3bhhdelxoolqddvd3lvxi0lrabcgnwg88nzfbrprhlepe8lmncaftmoablteroioszdlht+xmheds2s82ujy9hperofvchnypijnojismn7fbakavg4zfswogptvco5rji3frmhwxbdtwurwekmb3iumekeknlwzlj+gssifihgu+jdove8zlbpnrcj3a9ysobpfwwzqb52uvmis05gkxll6yckzyrdqwipq/huif2hcvv/lclwerngj4lanug12zyjh7nwylrcsvo2fzvpgljffimyrhpijmnio7q7zb5pz0iiqkum6yujflipkbf9qgp5q8kq4cpsad72dnw+crnsfsmwatzgswbwkr1qq4me1vx+jy0htwkfe7lq97pidkcz3bseotswoalcdm8mag2hfpnd/fujh7ces5r94nr3pcrd/mlnqehqnd5kfx6bidofpuqxt9qd/rl/tbeel8cb46ywxd3qpjhqkv5kz/asz+pti=</latexit> Utility Function: Information Measure Information measure consistent with goal of study E.g., entropy of estimate, posterior variance, entropy of posterior distribution of an indicator - Entropy of posterior distribution X p(x), H[p(X)] = E[log p(x) Data] see also Piantadosi (2005) Clinical Trials 2:

30 Can Incorporate Frequentist Criteria in Utility Function Bayesian design optimizes expected utility Utility function can include different considerations - Sample size or cost - Precision - Number of patients who benefit - Prediction of future study outcomes E.g., Anscombe ( 63), Berry & Ho ( 88), Lewis & Berry ( 94), Carlin, Kadane, & Gelfand ( 98), Stallard, Thall, & Whitehead ( 99), Lewis, Lipsky, & Berry ( 07), Trippa, Rosner, & Müller ( 12), Ventz & Trippa ( 15) 30

31 Platform or Master Protocols At any one time, multiple phase II studies Rossell, Müller, Rosner (2007) Biostatistics Ding, Rosner, Müller (2008) Biometrics 31

32 Decisions At each analysis-decision time t Make decision dti for current study or trt d ti = 1: - Abandon current study trt d ti = 2: - Stop study and move to phase 3 d ti = 3: - Continue current study Could have multiple trt-studies at t Then have d t =(d t1,d t2,...,d tk ) Only considering one at a time now d t = d t1 32

33 Utility Function Utility at decision-time t (for current trt) c 1 n 1 t if stop & discard u t (d t,θ, Y t,y III ) = Phase 3 sample size & cost {c 1 n 1 t + c 2 n 2 } + b {θ new θ old }I [z>z1 α ] Gain if significant phase 3; gain proportional to effect Predict phase 3 outcome 33

34 Stopping Boundaries 5 per cohort c1 = 0.14 c2 = 0.7 b = 290 n2 = 96 0 =0.2 A =0.5 old Beta(20, 80) 2000 sims 34

35 Our Colleagues in Other Disciplines Want Our Help Work with colleagues in other fields Clin Chem 62:1285-6, 2016 They want our help Operskalski JT, Barbey AK (2016). Risk literacy in medical decisionmaking. Science 352:

36 Opportunities Work with colleagues in other fields 36

37 Summary: Why Bayes? Easier to combine or incorporate information Bayesian paradigm corresponds to learning - External information feeds priors Frequentist methods seem impractical in some cases Evidence of treatment benefit for rare diseases Interest in complex designs & decision making Outcome adaptive randomization Matching treatments to subgroups 37

38 Conclusions Clinical drug development involves learning & confirming Bayesian inference has a place in regulatory science Hypothesis testing in and of itself is not evil Need better measures of weight of evidence than p Clinical research involves decisions Incorporate statistical decision theory Include predictive dist n for frequentist tests 38

39 Thank You!