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1 Bayesian Clinical Trials Scott M. Berry Scott M. Berry

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2 Bayesian Statistics Reverend Thomas Bayes ( ) Essay towards solving a problem in the doctrine of chances (1764) This paper, on inverse probability, led to Bayes theorem, which led to Bayesian Statistics

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3 Bayes Theorem Bayesian inferences follow from Bayes theorem: '( | X) ( )*f (X | ) Assess prior ; subjective, include available evidence Construct model f for data Find posterior ' Bayesian inferences follow from Bayes theorem: '( | X) ( )*f (X | ) Assess prior ; subjective, include available evidence Construct model f for data Find posterior '

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4 Simple Example Coin, P(HEADS) = = 0.25 or =0.75, equally likely. DATA: Flip coin twice, both heads. ???

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5 Bayes Theorem Pr[ = 0.75 | DATA] = Pr[DATA | p=0.75] Pr[ =0.75] Pr[DATA | =0.75] Pr[ =0.75]+ Pr[DATA | =0.25] Pr[ =0.25] (0.75) 2 (0.5) (0.75) 2 (0.5) + (0.25) 2 (0.5) = 0.90 Likelihood Prior Probabilities Posterior Probabilities

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6 Rare Disease Example Suppose 1 in 1000 people have a rare disease, X, for which there is a diagnostic test which is 99% effective. A random subject takes the test, which says POSITIVE. What is the probability they have X? (0.99) (0.001) (0.99) (0.001) + (0.01) (0.999) = !!!

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7 Bayesian Statistics A subjective probability axiomatic approach was developed with Bayes theorem as the mathematical crank-- Savage, Lindley (1950s) Very different than classical statistics: a collection of tools Before ?: A philosophical niche, calculation very hard. Early 1990s: Computers and methods made calculation possible…and more!

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8 Bayesian Approach Probabilities of unknowns: hypotheses, parameters, future data Hypothesis test: Probability of no treatment effect given data Interval estimation: Probability that parameter is in the interval Synthesis of evidence Tailored to decision making: Evaluate decisions (or designs), weigh outcomes by predictive probabilities Probabilities of unknowns: hypotheses, parameters, future data Hypothesis test: Probability of no treatment effect given data Interval estimation: Probability that parameter is in the interval Synthesis of evidence Tailored to decision making: Evaluate decisions (or designs), weigh outcomes by predictive probabilities

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9 Frequentist vs. Bayesian Seven comparisons 1. Evidence used? 2. Probability, of what? 3. Condition on results? 4. Dependence on design? 5. Flexibility? 6. Predictive probability? 7. Decision making? 1. Evidence used? 2. Probability, of what? 3. Condition on results? 4. Dependence on design? 5. Flexibility? 6. Predictive probability? 7. Decision making?

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10 Consequence of Bayes rule: The Likelihood Principle The likelihood function L X ( ) = f( X | ) contains all the information in an experiment relevant for inferences about The likelihood function L X ( ) = f( X | ) contains all the information in an experiment relevant for inferences about

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11 Short version of LP: Take data at face value But data can be deceptive Caveats... –How identified? –Why are they showing me this? Short version of LP: Take data at face value But data can be deceptive Caveats... –How identified? –Why are they showing me this?

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12 Example Data: 13 A's and 4 B's Parameter = = P(A wins) Likelihood 13 (1– ) 4 Frequentist conclusion? Depends on design Data: 13 A's and 4 B's Parameter = = P(A wins) Likelihood 13 (1– ) 4 Frequentist conclusion? Depends on design

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13 Frequentist hypothesis testing P-value = Probability of observing data as or more extreme than results, assuming H 0. P-V = P(tail of dist. | H 0 ) Four designs: (1) Observe 17 results (2) Stop trial once both 4 A's and 4 B's (3) Interim analysis at 17, stop if or A's, else continue to n = 44 (4) Stop when "enough information" P-value = Probability of observing data as or more extreme than results, assuming H 0. P-V = P(tail of dist. | H 0 ) Four designs: (1) Observe 17 results (2) Stop trial once both 4 A's and 4 B's (3) Interim analysis at 17, stop if or A's, else continue to n = 44 (4) Stop when "enough information"

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14 Design (1): 17 results Binomial distribution with n = 17, = 0.5; P-value = Binomial distribution with n = 17, = 0.5; P-value = 0.049

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15 Design (2): Stop when both 4 As and 4 Bs Two-sided negative binomial with r = 4, = 0.5; P-value = Two-sided negative binomial with r = 4, = 0.5; P-value = 0.021

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16 Design (3): Interim analysis at n=17, possible total is 44 Analyses at n = 17 & 44; 17 if 0-4 or 13-17; P = Analyses at n = 17 & 44; 17 if 0-4 or 13-17; P = Both shaded regions = P(both) = 0.013; net = 2(0.049) – = P(both) = 0.013; net = 2(0.049) – = 0.085

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17 Design (4): Scientists stopping rule: Stop when you know the answer Cannot calculate P-value Strictly speaking, frequentist inferences are impossible Cannot calculate P-value Strictly speaking, frequentist inferences are impossible

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18 Bayesian Calculations Data: 13 A's and 4 B's Parameter = = P(A wins) For ANY design with these results, the likelihood function is P(data | p) 13 (1– ) 4 Posterior probabilities & Bayesian conclusion same for any design Data: 13 A's and 4 B's Parameter = = P(A wins) For ANY design with these results, the likelihood function is P(data | p) 13 (1– ) 4 Posterior probabilities & Bayesian conclusion same for any design

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19 Likelihood function of

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20 Posterior Distribution Prior: 1 0 < < 1 Posterior 1 * 13 (1– ) 4 = 1 * 13 (1– ) 4 / 1 * 13 (1– ) 4 d = {13!4!/18!} 13 (1– ) 4 Prior: 1 0 < < 1 Posterior 1 * 13 (1– ) 4 = 1 * 13 (1– ) 4 / 1 * 13 (1– ) 4 d = {13!4!/18!} 13 (1– ) 4

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21 Posterior density of for uniform prior: Beta(14,5)

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22 Pr[ > 0.5 ]

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23 PREDICTIVE PROBABILITIES Distribution of future data? P(next is an A) = ? Critical component of experimental design In monitoring trials

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24 Laplaces rule of succession P(A wins next pair | data) = EP(A wins next pair | data, ) = E( | data) = mean of Beta(14, 5) = 14/19 Laplace uses Beta(1,1) prior

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25 Updating w/next observation

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26 Suppose 17 more observations P(A wins x of 17 | data) = EP(A wins x | data, ) = Beta-Binomial Distribution

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27 Predictive distribution Predictive distribution of # of successes in next 17 tries: Has more variability than any binomial Has more variability than any binomial 88% probability of statistical significance

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28 Best fitting binomial vs. predictive probabilities Binomial, p=14/19 Predictive, p ~ beta(14,5) 88% probability of statistical significance 96% probability of statistical significance

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29 Possible Calculation Simulate a from the beta(14,5) Simulate an x from binomial(17, ) Distribution of xs is beta-binomial--the predictive distribution

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30 Posterior and Predictive…same? Clinical Trial, 100 subjects. H A : > 0.25? FDA will approve if # success 33 [post > 0.95, beta(1,1)] See 99 subjects, 32 successes Pr[ > 0.25 | data ] = Predictive prob trial success = 0.327

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31 Predictive Probabilities for Medical Device Bayesian calculations FDA: –Some patients have reached 2 years –Some patients have only 1-yr follow- up

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32 Continuous data; Patients w/both 12 and 24 months

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33 Some patients with only 12-month data

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34 Kernel density estimates

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35 Small bandwidth (0.2 )

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36 Larger bandwidth (0.3 )

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37 Still larger bandwidth (0.4 )

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38 Very large bandwidth (0.5 ) (nearly bivariate normal)

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39 Condition on 12-month value

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40 Conditional distribution of 24-month value (0.2 )

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41 For largest bandwidth (0.5 )

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42 Multiple imputation: simulate full set of 24-month data

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43 Simulate experimental patients and controls in this way multiple imputation Make inferences with full data (for example, equivalent improvement) Repeat simulations (10,000 times) Gives probability of future results– for example, of equivalence Simulate experimental patients and controls in this way multiple imputation Make inferences with full data (for example, equivalent improvement) Repeat simulations (10,000 times) Gives probability of future results– for example, of equivalence

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44 Monitoring example: Baxters DCLHb Diaspirin Cross-Linked Hemoglobin Blood substitute; emergency trauma Randomized controlled trial (1996+) –Treatment: DCLHb –Control: saline –N = 850 (= 425x2) –Endpoint: death

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45 Waiver of informed consent Data Monitoring Committee First DMC meeting: DCLHbSaline Dead 21 (43%) 8 (20%) Alive2833 Total No formal interim analysis

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46 Bayesian predictive probability of future results (no stopping) Probability of significant survival benefit for DCLHb after 850 patients: (PP=0.0097) DMC paused trial: Covariates? DMC stopped the trial

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47 Herceptin in Neoadjuvant BC Endpoint: tumor response Balanced randomized, A & B Sample size planned: 164 Interim results after n = 34: –Control: 4/16 = 25% (pCR) –Herceptin: 12/18 = 67% (pCR) Not unexpected (prior?) Predictive prob of stat sig: 95% DMC stopped the trial ASCO and JCOreactions …

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48 Mixtures: Data: 13 A's and 4 B's Likelihood p 13 (1–p) 4

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49 Mixture Prior ~ 0 I[p=p 0 ] + (1 0 ) Beta(, ) 0 I 0 p 0 13 (1 p 0 ) 4 + (1 0 ) Kp (1 p) +4 1 ~ 0 I 0 + (1 0 ) Beta(, ) 0 p 13 (1 p) 4 0 = p 13 (1 p) 4 + (1 0 ) ( ) ( ) ( + +17) ( ) ( ) ( + )

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50 Pr(p=0.5) = P(p > 0.5) = Mixture Posterior 0 =.5

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51 Crooked-Penny Example Flip the coin 20 times. What is for your coin? Everyone reports p for their coin. ^ A new estimate for ? Are others relevant for you? A new estimate for ? Are others relevant for you?

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52 Numbers of heads This is you

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53 One-Sample Problem ~ Beta( [X] ~ Binomial(n, ) [ X]~Beta( +X, +n-X) Mean = ( + X n)

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54 Prior: ~ Beta(1, 1 Posterior: ~ Beta(17, For uniform prior ( = = 1)

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55 Prior: ~ Beta(10, 10 Posterior: ~ Beta(26, For = = 10 Prior: ~ Beta(10, 10

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56 Remember the other coins... This is you

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57 Learning about the prior In your setting the other coins give you information about the prior…which helps!!!! The coins do not have to be the same or close, you learn the appropriate amount of borrowing.

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58 HIERARCHICAL MODELING Population: Sample: Sample from sample: Inferential problems problemsInferential

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59 Selecting coins Population of coinspopulation of s: Select two coins and toss each coin 10 times: one 9 heads, other 4 heads. Estimate 1, 2. Estimate 1, 2. Estimate distribution of s in population. Estimate distribution of s in population.

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60 Generic example: Unit is lab or drug variation or lot or study Unit s n s/n Total n = #observations s = #successes s/n = success proportion proportion

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61 If 1 = 2 =... = 9 = (all 150 units exchangeable)

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62 Assuming equal s, 95% CI for : (0.63, 0.77) But 7 of 9 estimates lie outside this interval. Combined analysis unsatisfactory. Nine different analyses even worse: nine individual CIs?

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63 Suppose n i independent observations on unit i Suppose each unit has its own, with 1,..., 9 having distribution G. Observe x's, not 's. X i ~ binomial(n i, i ). Likelihood is product of likelihoods of i

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64 Bayesian view: G unknown = G has probability distribution Prior distribution reflects heterogeneity vs homogeneity. Assume G is Beta(a,b), a > 0, b > 0 with a and b unknown. Study heterogeneity: –little if a+b is large –lots if a+b is small

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65 Beta(a,b) for a, b = 1, 2, 3, 4:

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66 Suppose uniform prior for a & b on integers 1,..., 10

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67 Posterior probabilities for a & b

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68 Calculating posterior distribution of G Direct in this example Can be more complicated, and require: –Gibbs sampling (BUGS) –Other Markov chain Monte Carlo

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69 Posterior mean of G (also predictive density for )

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70 Contrast with likelihood assuming all ps equal

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71 Bayesian questions: P( > 1/2) = ???? P(next unit in study i is success) = ? –How to weigh results in unit i? –How to weigh results in unit j? P(unit in 10th study is success) = ? –How to weigh results in study i?

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72 Bayes estimates Unit x n x/n Bayes Total (0.71)

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73 Bayes estimates are regressed or shrunk toward overall mean Bayes estimates Unadjusted estimates

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74 Baseball Example 446 players in 2000 with > 100 at bats Jose Vidro

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75 How good was Jose Vidro? (200 hits in 606 at bats, 0.330) X ~ Binomial(606, JV X ~ Binomial(606, JV (hits) JV Beta( JV Beta(

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76 Empirical Bayes: EB EB mean = 0.269; var = ) |X] ~ Beta( , ) (approx) Posterior mean = Posterior st. dev. = 0.015

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77 Science, Feb 6, 2004, pp 784-6

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78 Efficacy of Pravastatin + Aspirin: Meta-Analyses [For statistical analysis, S.M. Berry et al., Journal of the American Statistical Association, 2004] ohrms/dockets/ac/02/slides/ 3829s2_03_Bristol-Meyers-meta-analysis.ppt ohrms/dockets/ac/02/slides/ 3829s2_03_Bristol-Meyers-meta-analysis.ppt

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79 Trial LIPID CARE REGRESS PLAC I PLAC II Totals Number of Subjects*% on Aspirin Primary Endpoint CHD mortality CHD death & non-fatal MI Atherosclerotic progression (& events) ,617 Atherosclerotic progression (& events) *99.7% of pravastatin-treated subjects received 40mg dose Meta-Analysis of these Pravastatin Secondary Prevention Trials

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80 Trial Commonalities –Similar entry criteria –Patient populations with clinically evident CHD –Same dose of pravastatin (40mg) –Randomized comparison against placebo –All trials with durations of 2 years –Pre-specified endpoints –Covariates recorded –Common meta-analysis data management

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81 Patient Group Comparisons PlaceboPravastatin Aspirin Users Aspirin Non-Users Prava+ASA Prava alone Placebo+ASA Placebo alone Randomized Groups Randomized Comparison Observational Comparison

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82 Is Pravastatin+Aspirin More Effective than Pravastatin Alone? –Aspirin studies were conducted before statins were widely used –Placebo-controlled trial with aspirin is not feasible –Investigation of pravastatin database to explore this question

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83 Is the Combination More Effective than Pravastatin Alone? –Unadjusted event rates in LIPID and CARE suggest pravastatin + aspirin is more effective than pravastatin alone

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84 Event Rates for Primary Endpoints in LIPID and CARE Aspirin Users Aspirin Non-Users 5.8% 8.8%14.8% 9.3% LIPID CHD Death CARE CHD Death or Non-fatal MI Pravastatin-treated Subjects Only Trial: Primary Endpoint: Observational Comparison

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85 Accounting for Baseline Risk Factors –Age –Gender –Previous MI –Smoking status –Baseline LDL-C, HDL-C, TG –Baseline DBP & SBP Additional analyses also included revascularization, diabetes and obesity

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86 Meta-Analysis Endpoints Considered –Fatal or non-fatal MI –Ischemic stroke –Composite: CHD death, non-fatal MI, CABG, PTCA or ischemic stroke

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87 Model 1: –Multivariate Cox proportional hazards model –Patients combined across trials; trial effect is a fixed covariate Meta-Analysis Models H(t) = 0 (t)exp(Z + S + T ) Baseline Hazards constant Covariates Study effects Treatment Effects

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88 RRR = Relative Risk Reduction Relative Risk (95% CI) RRR Relative Risk Reduction Cox Proportional Hazards – All Trials Prava+ASA vs ASA alone Prava+ASA vs Prava alone Fatal or Non-Fatal MI CHD Death, Non-Fatal MI, CABG, PTCA, or Ischemic Stroke Prava+ASA vs ASA alone Prava+ASA vs Prava alone 24% % % % 0.74 Prava+ASA vs ASA alone Prava+ASA vs Prava alone 29% % 0.69 Ischemic Stroke

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89 Model 2: Same as Model 1 except –Allows trial heterogeneity: Bayesian hierarchical (random effects) model of trial effect Meta-Analysis Models H(t) = 0 (t)exp(Z + S + T ) Baseline Hazards piecewise-constant Covariates Study effects Hierarchical Treatment Effects

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Year Model 2 – Hierarchical, Random Effects Fatal or Non-Fatal MI Placebo Prava alone ASA alone Prava+ASA Cumulative Proportion of Events

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Model 2 – Hierarchical, Random Effects Ischemic Stroke Only ASA alone Prava+ASA Year Cumulative Proportion of Events Prava alone Placebo

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Year Model 2 – Hierarchical, Random Effects CHD Death, Non-Fatal MI, CABG, PTCA, or Ischemic Stroke Prava+ASA ASA alone Prava alone Placebo Cumulative Proportion of Events

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93 Combination is More Effective than Either Agent Alone –Pravastatin + aspirin provides benefit for all three endpoints: 24% - 34% RRR compared with aspirin 13% - 31% RRR compared with pravastatin This benefit was similar in Models 1 and 2 This benefit was consistent in both LIPID and CARE trials This benefit was similar in Models 1 and 2 This benefit was consistent in both LIPID and CARE trials

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94 Model 2: Fatal or Non-Fatal MI Cumulative Proportion of Events Year Prava+ASA ASA alone Prava alone Placebo Year HazardPrava+ASA ASA alone Prava alone Placebo

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95 Model 3: Same as Model 2 except –Treatment hazard ratios vary over time Meta-Analysis Models H(t) = 0 (t)exp(Z + S ) Baseline Hazards piecewise-constant Within treatment Covariates Study Effects Hierarchical

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96 Model 3: Fatal or Non-Fatal MI Cumulative Proportion of Events Year Prava+ASA ASA alone Prava alone Placebo Year 5 Separate Analyses: One per Year Hazard Prava+ASA ASA alone Prava alone Placebo

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97 Probability of synergy between pravastatin & aspirin EndpointModel 2Model 3 All events Cardiac events Any MI Stroke Death0.997

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98 Conclusion of Hazard Analysis over Time –Benefit of pravastatin+aspirin over aspirin was present in each year of the 5-year duration of the trials Benefit of pravastatin+aspirin over pravastatin was present in each year of the 5-year duration of the trials Benefits estimated from Model 1 (and confidence intervals) confirmed by more general models and fewer assumptions Benefit of pravastatin+aspirin over pravastatin was present in each year of the 5-year duration of the trials Benefits estimated from Model 1 (and confidence intervals) confirmed by more general models and fewer assumptions

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99 Hierarchical modeling in design Using historical information Combining results from multiple concurrent trials (or many centers)

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100 Hierarchical modeling & dose-response Example: drug Z (rozuvastatin) vs drug A (atorvastatin) (Berry et al., 2002, American Heart Journal)

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101 Studies involving drugs A and Z*, with %change from baseline. %Change Study n DoseMeanSD Y – – – Placebo – – – – – – – – Placebo – Placebo – – – – – –

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102 Study n DoseMeanSD Y – –37.6NA Placebo – – – – – – – –46NA Placebo – – *– *– *– *– *– *– Placebo *– *–

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103 Dose-response model Y ij = exp{ s + a t + b t log(d)} + ij s for study t for drug d for dose i for observation (1,..., 43) j for patient within study/dose ij is N(0, 2 ) Priors dont matter much, except...

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104 Prior for s ~ N(0, 2 ) 2 is important 2 large means studies heterogeneous little borrowing 2 small means studies homogeneous much borrowing Prior of 2 is IG(10, 10) Prior mean and sd are 0.10 & 0.017

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105 Likelihood Calculations of posterior & predictive distributions by MCMC

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106 Posterior means and SDs Parameter MeanStDev a P – a A – a Z – b A – b Z –

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107 Posterior means and SDs Par. MeanStDevPar. MeanStDev – – – – – – – – – –

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108 Model fit

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109 Interval estimates for pop. mean: model (line) vs standard (box)

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110 Study/dose- specific interval estimates: model (line) vs standard (box)

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111 Posterior distn of reduction (95% intervals) Drug A Drug Z

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112 Posterior distn of mean diff, A – Z

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113 Really neat... Using predictive probabilities for designing future studies Contour plots

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114 Observed %Y for future study with n A =n Z =20 d A =d Z =10 Z A

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115 Observed %Y for future study with n A =n Z =100 d A =d Z =10 Z A

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116 Observed %Y for future study with n A =n Z =20 d A =10, d Z =5 Z A

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117 Observed %Y for future study with n A =n Z =100 d A =10, d Z =5 Z A

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118 STELLAR trial results (each n160) -50% -54% -58% -36% -41% -46% -52% Predictedatorva Predictedrosuva

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119 Posterior distn of reduction (95% intervals) Drug A Drug Z Recall:

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120 Adaptive Phase II: Finding the Best Dose Scott M. Berry Scott M. Berry

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121 Doses Standard Parallel Group Design Equal sample sizes at each of k doses.

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122 Response Doses True dose-response curve (unknown)

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123 Response Doses Observe responses (with error) at chosen doses

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124 Response Doses True ED 95 Dose at which 95% max effect

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125 Response Doses Uncertainty about ED95 ?

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126 Response Doses True ED 95 Solution: Increase number of doses Solution:

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127 Response Doses True ED 95 But, enormous sample size, and... wasted dose assignmentsalways!

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128 Solutions Lots of doses (continuum?) Adaptive Allocation Model dose response Define what you are looking for Stop when you find what you are looking for… Yogi Berra-ism: If you dont know where you are going, how do you know when you get there?

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129 Dose Finding Trial Real example (all details hidden, but flavor is the same) Delayed Dichotomous Response (random waiting time) Combine multiple efficacy + safety in the dose finding decision Use utility approach for combining various goals Multiple statistical goals Adaptive stopping rules

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130 Adaptive Approach

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131 Statistical Model The statistical model captures all the uncertainty in the process. Capture data, quantities of interest, and forecast future data Be flexible, (non-monotone?) but capture prior information on model behavior. Invisible in the process

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132 Empirical Data Observe Y ij for subject i, outcome j Y ij = 1 if event, 0 otherwise j = 1 is type #1 efficacy response j = 2 is type #2 efficacy response j = 3 is minor safety event j = 4 is major safety event

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133 Efficacy Endpoints Let d be the dose P j (d) probability of event j, dose d. j (d) ~ N( j, 2 ) IG(2,2) N(–2,1) N(1,1) G(1,1)

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134 Safety Endpoint Let d i be the dose for subject i P j (d) probability of safety j, dose d. N(-2,1) N(1,1) G(1,1)

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135 Utility Function Multiple Factors: Monetary Profile (value on market) FDA Success Safety Factors Utility is critical: Defines ED ?

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136 Utility Function U(d)=U 1 (P 1 )U 2 (P 3 )*U 3 (P 0,P 2 )*U 4 (P 4 ) Monetary FDA Approval Extra Safety P 0 is prob efficacy 2 success for d=0

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137 Monetary Utility

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141 U 3 : FDA Success DSMB?

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142 Statistical + Utility Output E[U(d)] E[ j (d)], V[ j (d)] E[P j (d)], V[P j (d)] Pr[d j max U] Pr[P 2 (d) > P 0 ] Pr[ P 2 >> P 0 | 250/per arm) each d >> means statistical significance will be achieved

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143 Allocator Goals of Phase II study? Find best dose? Learn about best dose? Learn about whole curve? Learn the minimum effective dose? Allocator and decisions need to reflect this (if not through the utility function) Calculation can be an important issue!

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144 Allocator Find best dose? Learn about best dose? Find the V* for each dose ==> allocation probs d* is the max utility dose, d** second best

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145 Allocator V*(d0) = V*(d=0) =

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146 Allocator Drop any r d <0.05 Renormalize

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147 Decisions Find best dose? Learn about best dose? Shut down allocator w j if stop!!!! Stop trial when both w j = 0 If Pr(P 2 (d*) >> P 0 ) < 0.10 stop for futility If found, stop: Pr(d = d*) > C 1 Pr(P 2 (d*) >> P 0 )>C 2

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148 More Decisions? Ultimate: EU(dosing) > EU(stopping)? Wait until significance? Goal of this study? Roll in to phase III: set up to do this Utility and why? are critical and should be done--easy to ignore and say it is too hard.

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149 Simulations Subject level simulation Simulate 2/day first 70 days, then 4/day Delayed observation –exponential with mean 10 days Allocate + Decision every week First 140 subjects 20/arm

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150 Scenario #1 DoseP1P1 P2P2 P3P3 P4P4 UTIL Stopping Rules: C 1 = 0.80, C 2 = 0.90 MAX

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152 Dose Probabilities P(>>Pbo) P(max) P(2nd) Alloc

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154 Dose Probabilities P(>>Pbo) P(max) P(2nd) Alloc

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156 Dose Probabilities P(>>Pbo) P(max) P(2nd) Alloc

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158 Dose Probabilities P(>>Pbo) P(max) P(2nd) Alloc

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160 Dose Probabilities P(>>Pbo) P(max) P(2nd) Alloc

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162 Dose Probabilities P(>>Pbo) P(max) P(2nd) Alloc

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163 Trial Ends P(10-Dose max Util dose) = P(10-Dose >> Pbo 250/arm) = subjects: 32, 20, 24, 31, 38, 62, 73 per arm

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164 Operating Characteristics Pbo SS Pmax SS66 Pmax

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165 Operating Characteristics AdaptiveConstant P(Success) P(Cap) P(Futility)0.000 Mean SS SD SS Mean TDose Max TDose

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166 Scenario #2 DoseP1P1 P2P2 P3P3 P4P4 UTIL Stopping Rules: C 1 = 0.80, C 2 = 0.90

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167 Operating Characteristics Pbo SS Pmax SS100 Pmax

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168 Operating Characteristics AdaptiveConstant P(Success) P(Cap) P(Futility)0.000 Mean SS SD SS Mean TDose Max TDose

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169 Simulation #3 DoseP1P1 P2P2 P3P3 P4P4 UTIL Stopping Rules: C 1 = 0.80, C 2 = 0.90

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170 Operating Characteristics Pbo SS Pmax SS87 Pmax

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171 Operating Characteristics AdaptiveConstant P(Success) P(Cap) P(Futility) Mean SS SD SS Mean TDose Max TDose

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172 Scenario #4 DoseP1P1 P2P2 P3P3 P4P4 UTIL Stopping Rules: C 1 = 0.80, C 2 = 0.90

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173 Operating Characteristics Pbo SS Pmax SS92 Pmax

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174 Operating Characteristics AdaptiveConstant P(Success) P(Cap) P(Futility)0.000 Mean SS SD SS Mean TDose Max TDose

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175 Scenario #5 DoseP1P1 P2P2 P3P3 P4P4 UTIL Stopping Rules: C 1 = 0.80, C 2 = 0.90

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176 Operating Characteristics Pbo SS Pmax SS84 Pmax

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177 Operating Characteristics AdaptiveConstant P(Success) P(Cap) P(Futility) Mean SS SD SS Mean TDose Max TDose

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178 Scenario #6 DoseP1P1 P2P2 P3P3 P4P4 UTIL Stopping Rules: C 1 = 0.80, C 2 = 0.90

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179 Operating Characteristics Pbo SS Pmax SS56 Pmax

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180 Operating Characteristics AdaptiveConstant P(Success)0.000 P(Cap) P(Futility) Mean SS SD SS Mean TDose Max TDose

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181 Bells & Whistles Interest in Quantiles Minimum Effective Dose Significance, control type I error Seamless phase II --> III Partial Interim Information Biomarkers of endpoint Continuous (& Poisson) Continuum of doses (IV)--little additional n!!!

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182 Conclusions Approach, not answers or details! Shorter, smaller, stronger! Better for company, FDA, Science, PATIENTS Why study?--adaptive can help multiple needs. Adaptive Stopping Bid Step!

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