Priors, Utilities, Elicitation & Pharmaceutical R&D

1. Priors, Utilities, Elicitation & Pharmaceutical R&D Andy Grieve Statistical Research Centre Pfizer Global R&D

2. Outline • Use of Bayesian Methods in Pharmaceutical R&D • Three Prior Elicitation Examples • Acute toxicity – LD50 • Sample Sizing & Confidence Intervals • Counting Tablets in Dosing Dogs • Elicitation for Internal Company Decision Making – Portfolio Management

3. Are Bayesian Methods Acceptable in Drug Development? Not Forbidden by Regulation

4. = International Conference on Harmonisation Extracts from Drug Regulations E4 : Dose Response 1993 Agencies should be open to the use of various statistical and pharmacometric techniques such as Bayesian and population methods, modelling, and pharmacokinetic-pharmacodynamic approaches.

5. = European Medicines Evaluation Agency Extracts from Drug Regulations CPMP Biostatistical Guidelines 1994 .. the use of Bayesian or other approaches may be considered when the reasons for their use are clear and when the resulting conclusions are sufficiently robust to alternative assumptions

6. Extracts from Drug Regulations E9 : Statistical Principles in Clinical Trials1998 Essentially same as CPMP Guidelines .. the use of Bayesian or other approaches may be considered when the reasons for their use are clear and when the resulting conclusions are sufficiently robust (to alternative assumptions) LUKEWARM !!!

7. Why ? - Trust • Background • Hair-thinning • Researcher Bias “If I hadn’t believed it, I wouldn’t have seen it with my own eyes” Clinical_Trials List (1996)

8. Trust • Feeling that use of “subjective priors may allow unscrupulous companies and/or statisticians to attempt to pull the wool over the regulators eyes.” (Greg Campbell – FDA Centre for Devices & Radiological Health) • If it were that easy they are not very good and we probably need new regulators

9. Trust • Stephen Senn - “nowhere is the discipline of statistics conducted with greater discipline than in the pharmaceutical industry” • Nowhere will Bayesian statistics be conducted with more discipline than in the pharmaceutical industry • Document

10. Trust • Document • Where did the prior come from ? • Is it based on data? Is it subjective ? • “to present a Bayesian analysis in which the company’s own prior beliefs are used to augment the trial data will in general not be acceptable to a regulatory agency” (O’Hagan & Stevens, 2001) • “the frequentist approach is less assumption dependent and can provide the statistical strength of evidence required for a confirmatory trial that may be lacking in a more assumption dependent Bayesian approach” (Chi, Hung & O’Neill – Biopharmaceutical Report, Vol. 9, No.2, 2001) • SENSITIVITY ANALYSIS

11. Trust • We work in a Frequentist World • Remember Acceptance of Bayesian methods is Lukewarm • We will be asked about false positive rates • We will be asked about the impact of multiple looks at the data • We need to be calibrated

12. Assessing a Prior in Acute toxicity

13. Motivating Example Based on these data we wish to determine the LD50 to classify the drug according to the following classification scheme (Swiss Poison Regs.)

14. Model • Data triplets • { di , ni , ri } : i=1,..,k • Probabilities of response • pi : i=1,…,k • Logistic Model • log [ pi / (1-pi)] = a + blog(di) • Median Lethal dose (LD50) • log(LD50) = -a/b = m Probit Model : pi =F(a + blog(di))

15. Bayesian Solutions • Likelihood Function • Prior distribution - p(a,b) (b > 0 ) • Define : m=-a/b - log(LD50) • Inference

16. Likelihood Contours - Motivating Example

17. Likelihood Function - Hypothetical Example Normal Analytic Approximation

18. Motivating Example • Experienced toxicologists • will know that they need to span the LD50 with the doses they choose. • The choice of doses contains information concerning the toxicologist’s beliefs about the likely value of the LD50.

19. 1 p2 >p1 p2 0 p1 0 1 Choice of Prior 1) Tsutakawa (1975) : logit • Choose doses d1 & d2 s.t. P(d1<LD50<d2)=0.5 • Implies p1 and p2 uniform over the half square • p(a,b) : logit  n.c.p. probit  BN (truncated) (Grieve , 1988) • Implies knowledge of pi :  i ≠1,2

20. Specify modal responses probabilities and Choice of Prior 2) Tsutakawa (1975) - logit • Choose doses d1 & d2 • Assume n.c.p. for p1 and p2 • p(a,b) : logit  n.c.p. probit  BN (truncated) (Grieve, 1988) • Implies knowledge of pi :  i ≠1,2

21. Suppose p(a,b) is bivariate normal : Choice of Prior 3) Grieve (1988) - probit • Can the parameters be determined ? • Not uniquely !!! • The c.d.f. of –a/b depends only on : • Implying any 4 probabilities are sufficient to determine c1,c2,c3 and c4 • Any one of the 5 parameters is also needed • Modal slope ? How about median ? • Feedback

22. Determining a Prior for Sample Sizing

23. Sample Sizing CIs : Simon Day (Lancet, 1988) 2n patients : (1-a)% CI : Width : Acceptable Width =

24. Alternative Approach : Grieve - Lancet, 1989 Required : Solve by search

25. Simon Day’s Example Two Anti-Hypertensives Difference in Diastolic BP - 95% CI s = 10 mm Hg , w0=10 Grieve - Lancet , 1989 1-y 0.5 0.8 0.9 0.95 n 32 37 39 41 n=32

26. Never be absolutely certain of anything Bertrand Russell A Bayesian approach is an unconditional approach accounting for uncertainty in parameters

27. Beal - Biometrics , 1989 “ A prior estimate of s …… is needed. This clearly introduces some uncertainty regarding the required sample size “ Conditionality

28. Relation Between s and n s 7 8 9 10 11 12 13 14 n 21 27 33 39 46 54 62 71 Suppose we have some idea about the likely value of s through a probability distribution

29. Unconditional Approach

30. Where do we get p(s2) from ? • Previous studies • Expert opinion - subjective ? • Estimate of s2 : based on u0 d.f. • Inverse-Gamma prior

31. Conditional Formula Unconditional Formula – Grieve(1991)

32. Elicitation of Inverse-Gamma • Expert provides and s.t. • Not enough information – assume upper and lower limits are (1-p0)/2 percentiles • Solve directly or modify algorithm in Martz and Waller(1982 – Bayesian Reliability Analysis), Grieve (1987,1991)

33. Illustrative Example Probability ( 8 < s < 13 ) = 0.8 Implies u0=14.66 , =95.55

34. Relation Between P(w<w0) and n n 51 52 53 54 55 56 57 P(w<w0) 0.873 0.882 0.892 0.899 0.906 0.912 0.918 In this example accounting for uncertainty increases the sample size by 40 %

35. Elaborating a Prior for Tablet Counting

36. Checking the Dosing of Dogs • dogs dosed on mg/kg basis • adjusted weekly • Example • Unit Dose : 36 mg/kg • Weight : 19.2 kg • Required dose : 691 mg

37. Pre-Manufactured Tablet Strengths 4 - 5 0 -11 0 - 4 0 - 4 25 mg 300 mg 5 mg 0.5 mg 2 3 3 2 691 mg

38. Dog Dosing • Tablets placed in a gelatine capsule • Are the correct number of tablets in the capsule ?

39. Possible Approaches • Do Nothing • Hope - No : Inspection • Acceptance Sampling • too few samples - 308 capsules/wk • checking creates errors • Check Everything • checking creates errors • Weigh Capsules & Contents

40. Tablet /Capsule Weights

41. Statistical Model • T tablet sizes • tablet weights are : • capsule weights are : • Ni tablets of each size chosen • Total weight w is distributed as

42. Hypothetical Example 2 X 300 mg = 600 mg 3 X 25 mg = 75 mg 3 X 5 mg = 15 mg 2 x 0.5 mg = 1 mg 691 mg • Given a total weight of 3.397g (simulated) • What can we say about the likely numbers of tablets?

43. Dog Dosing - Solution (1) • Co-primal Weights • 3 : 7 : 13 : 23 instead of 1 : 2 : 4 : 8

44. Dog Dosing - Solution (2) • Co-primal Weights • 3 : 7 : 13 : 23 instead of 1 : 2 : 4 : 8 • Pre-Weighing of Capsules

45. Dog Dosing - Weights

46. Solution (3) • Co-primal Weights • 3 : 7 : 13 : 23 instead of 1 : 2 : 4 : 8 • Pre-Weighing of Capsules • Prior distribution belief in ability to count to 5 greater than belief in ability to count to 19

47. Elaborating a Prior Grieve et al (1994) • Suppose a technician tries to count to M tablets of a given strength • A model of the process could be : • The total of M tablets is “achieved” by M individual operations each attempting to count to 1 • An error can be made in either direction : xj=0,1 or 2 • The total count is : x1+x2+ …. + xM=N

48. Assuming independent counts the p.g.f. of N is : • Giving : Elaborating a Prior • Suppose the probability distribution of results from a single count is given by : with p.g.f. - P(t)=q+rt+pt2

49. Elaborating a Prior ADVANTAGES • A prior distribution need not be elicited for every M • Elaboration ensures consistency • If Mk=Mj+1 then P(Nk=Mk) < P(Nj=Mj) DISADVANTAGES • Need to elicit p,q (r=1-p-q) • Assumptions

50. Feeding Back