Bayesian Clinical Trials

1
Bayesian Clinical Trials

• Scott M. Berry
• scott_at_berryconsultants.com

2
Bayesian Statistics

• Reverend Thomas Bayes (1702-1761)
• 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
3
Bayes Theorem

• Bayesian inferences follow from Bayes theorem
• ?'(q? X) ? ?(q)f (X q)
• Assess prior ? subjective, include available
  evidence
• Construct model f for data
• Find posterior ?'

4
Simple Example

• Coin, P(HEADS) p
• p 0.25 or p 0.75, equally likely.
• DATA Flip coin twice, both heads.
• p ???

5
Bayes Theorem
Pr p 0.75 DATA
PrDATA p0.75 Prp0.75
--------------------------------------------------
---------------------------------
PrDATA p0.75 Prp0.75
PrDATA p0.25 Prp0.25
(0.75)2 (0.5)
-----------------------------------
0.90
(0.75)2 (0.5) (0.25)2 (0.5)
Posterior Probabilities
Likelihood
Prior Probabilities
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)
0.0902 !!!
---------------------------------------
(0.99) (0.001) (0.01) (0.999)
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 1980-1990? A philosophical niche,
  calculation very hard.
• Early 1990s Computers and methods made
  calculation possibleand more!

8
Bayesian Approach

• Probabilities of unknownshypotheses,
  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

9
Frequentist vs. BayesianSeven comparisons

• 1. Evidence used?
• 2. Probability, of what?
• 3. Condition on results?
• 4. Dependence on design?
• 5. Flexibility?
• 6. Predictive probability?
• 7. Decision making?

10
Consequence of Bayes ruleThe Likelihood
Principle

• The likelihood function
• LX(?) f( X ?)
• contains all the information in an experiment
  relevant for inferences about ?

11

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

12
Example

• Data 13 A's and 4 B's
• Parameter ? P(A wins)
• Likelihood ? ? 13 (1?)4
• Frequentist conclusion? Depends on design

13
Frequentist hypothesis testing

• P-value Probability of observing data as or
  more extreme than results, assuming H0. P-V
  P(tail of dist. H0)
• 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 0 - 4 or
• 13 - 17 A's, else continue to n 44
• (4) Stop when "enough information"

14
Design (1) 17 results
Binomial distribution with n 17, ?
0.5 P-value 0.049
15
Design (2) Stop when both 4 As and 4 Bs
Two-sided negative binomial with r 4, ?
0.5 P-value 0.021
16
Design (3) Interim analysis at n17, possible
total is 44
Analyses at n 17 44 stop _at_ 17 if 0-4 or
13-17 P 0.085
Both shaded regions 0.049
P(both) 0.013 net 2(0.049) 0.013
0.085
17
Design (4) Scientists stopping rule Stop when
you know the answer

• Cannot calculate P-value
• Strictly speaking, frequentist inferences are
  impossible

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

19
Likelihood function of ?
20
Posterior Distribution
Prior 1 0 lt ? lt 1 Posterior ? 1 ? 13
(1?)4 1 ? 13 (1?)4 / ? 1 ? 13 (1?)4
d ? 13!4!/18! ? 13 (1?)4
21
Posterior density of ? for uniform prior
Beta(14,5)
22
Pr? gt 0.5
23
PREDICTIVE PROBABILITIES

• Distribution of future data?
• P(next is an A) ?
• Critical component of experimental design
• In monitoring trials

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
25
Updating w/next observation
26
Suppose 17 more observations

• P(A wins x of 17 data)
• EP(A wins x data, ?)

?
Beta-Binomial Distribution
27
Predictive distribution

• Predictive distribution of of successes in
  next
• 17 tries

88 probability of statistical significance
Has more variability than any binomial ?
28
Best fitting binomial vs. predictive probabilities
Binomial, p14/19
96 probability of statistical significance
Predictive, p beta(14,5)
88 probability of statistical significance
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

30
Posterior and Predictivesame?

• Clinical Trial, 100 subjects. HA ? gt 0.25? FDA
  will approve if success 33 post gt 0.95,
  beta(1,1)
• See 99 subjects, 32 successes
• Pr ? gt 0.25 data 0.955
• Predictive prob trial success 0.327

31
Predictive Probabilities for Medical Device

• Bayesian calculations ? FDA
• Some patients have reached 2 years
• Some patients have only 1-yr follow-up

32
Continuous data Patients w/both 12 and 24 months
33
Some patients with only 12-month data
34
Kernel density estimates
35
Small bandwidth (0.2?)
36
Larger bandwidth (0.3?)
37
Still larger bandwidth (0.4?)
38
Very large bandwidth (0.5?)(nearly bivariate
normal)
39
Condition on 12-month value
40
Conditional distribution of 24-month value (0.2?)
41
For largest bandwidth (0.5?)
42
Multiple imputation simulate full set of
24-month data
43

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

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

45

• Waiver of informed consent
• Data Monitoring Committee
• First DMC meeting
• DCLHb Saline
• Dead 21 (43) 8 (20)
• Alive 28 33
• Total 49 41
• No formal interim analysis

46
Bayesian predictive probability of future results
(no stopping)

• Probability of significant survival benefit for
  DCLHb after 850 patients 0.00045 (PP0.0097)
• DMC paused trial Covariates?
• DMC stopped the trial

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

48
Mixtures

• Data 13 A's and 4 B's
• Likelihood ? p13 (1p)4

49
Mixture Prior
p p0 Ipp0 (1-p0) Beta(a,b)
p? ? p0 I0 p013(1-p0)4 (1-p0) Kpa13-1(1-p)b4-1
p? p?0 I0 (1-p?0) Beta(a13,b4 )
p0 p13(1-p)4
p?0 -------------------------------------------
--------
G(a)G(b)G(ab17)
--------------------------
p0 p13(1-p)4 (1-p0)
G(a13)G(b4)G(ab)
50
Mixture Posterior p0.5
Pr(p0.5) 0.246
P(p gt 0.5) 0.742
51
Crooked-Penny Example

• Flip the coin 20 times.
• What is q for your coin?
• Everyone reports p for their coin.

A new estimate for q? Are others relevant for
you?
52
Numbers of heads
This is you
53
One-Sample Problem

• q Beta(a,b)
• X Binomial(n,q)
• qXBeta(aX,bn-X)
• Mean (a X)/(abn)

54
For uniform prior (a b 1)
Posterior q Beta(17, 5)
Prior q Beta(1, 1)
0.77
55
For a b 10
Posterior q Beta(26, 14)
Prior q Beta(10, 10)
Prior q Beta(10, 10)
0.65
56
Remember the other coins . . .
This is you
57
Learning about the prior

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

58
HIERARCHICAL MODELING
Population
Sample
Inferential problems
Sample from sample
59
Selecting coins
Population of coinspopulation of qs

• Select two coins and toss each coin 10 times
  one 9 heads, other 4 heads.
• Estimate q1, q2.
• Estimate distribution of qs in population.

60
Generic example Unit is lab or drug variation
or lot or study

• Unit s n s/n
• 1 20 20 1.00
• 2 4 10 0.40
• 3 11 16 0.69
• 4 10 19 0.53
• 5 5 14 0.36
• 6 36 46 0.78
• 7 9 10 0.90
• 8 7 9 0.78
• 9 4 6 0.67
• Total 106 150 0.71

n observations s successes s/n success
proportion
61
If q1 q2 . . . q9 q(all 150 units
exchangeable)
62
Assuming equal qs,95 CI for q (0.63, 0.77)

• But 7 of 9 estimates lie outside this interval.
• Combined analysis unsatisfactory.
• Nine different analyses even worse nine
  individual CIs?

63
Suppose ni independent observations on unit i

• Suppose each unit has its own q, with q1, . .
  . , q9 having distribution G.
• Observe x's, not q's.
• Xi binomial(ni, qi).
• Likelihood is product of likelihoods of qi

64
Bayesian view G unknown G has probability
distribution

• Prior distribution reflects heterogeneity vs
  homogeneity.
• Assume G is Beta(a,b),a gt 0, b gt 0 with a and b
  unknown.
• Study heterogeneity
• little if ab is large
• lots if ab is small

65
Beta(a,b) for a, b 1, 2, 3, 4
66
Suppose uniform prior for a b on integers 1, .
. ., 10
67
Posterior probabilities for a b
68
Calculating posterior distribution of G

• Direct in this example
• Can be more complicated, andrequire
• Gibbs sampling (BUGS)
• Other Markov chain Monte Carlo

69
Posterior mean of G(also predictive density for
q)
70
Contrast with likelihood assuming all ps equal
71
Bayesian questions

• P(q gt 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?

72
Bayes estimates

• Unit x n x/n Bayes
• 1 20 20 1.00 0.90
• 2 4 10 0.40 0.53
• 3 11 16 0.69 0.69
• 4 10 19 0.53 0.57
• 5 5 14 0.36 0.48
• 6 36 46 0.78 0.77
• 7 9 10 0.90 0.80
• 8 7 9 0.78 0.73
• 9 4 6 0.67 0.68
• Total 106 150 0.68 0.68
• (0.71)

73
Bayes estimates are regressed or shrunk toward
overall mean
Bayes estimates
Unadjusted estimates
74
Baseball Example

• 446 players in 2000 with gt 100 at bats

Jose Vidro
75

• How good was Jose Vidro?
• (200 hits in 606 at bats, 0.330)

• X Binomial(606, qJV)
• (hits)

qJV Beta(a,b)
76

• Empirical Bayes
• aEB 95.5 bEB258.9
• (mean 0.269 var 0.0362-.0272)
• qX Beta(20095.5, 406258.9) (approx)
• Posterior mean 0.308
• Posterior st. dev. 0.015

77
Science, Feb 6, 2004, pp 784-6
78
Efficacy of Pravastatin Aspirin Meta-Analyses
www.fda.gov/ ohrms/dockets/ac/02/slides/ 3829s2_03
_Bristol-Meyers-meta-analysis.ppt
For statistical analysis, S.M. Berry et
al., Journal of the American Statistical
Association, 2004
79
Meta-Analysis of these Pravastatin Secondary
Prevention Trials
Trial
Number of Subjects
on Aspirin
Primary Endpoint
LIPID
82.7
CHD mortality
9014
CARE
83.7
CHD death non-fatal MI
4159
REGRESS
54.4
Atherosclerotic progression ( events)
885
PLAC I
67.5
408
Atherosclerotic progression ( events)
PLAC II
42.7
151
Atherosclerotic progression ( events)
Totals
80.4
14,617
99.7 of pravastatin-treated subjects received
40mg dose
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

81
Patient Group Comparisons
Randomized Groups
Placebo
Pravastatin
Aspirin Users
Randomized Comparison
Aspirin Non-Users
Observational Comparison
82
Is PravastatinAspirin More Effectivethan
Pravastatin Alone?

• Aspirin studies were conducted before statins
  were widely used
• Placebo-controlled trial with aspirinis not
  feasible
• Investigation of pravastatin databaseto explore
  this question

83
Is the Combination More Effectivethan
Pravastatin Alone?

• Unadjusted event rates in LIPID and CARE suggest
  pravastatin aspirin is more effective than
  pravastatin alone

84
Event Rates for Primary Endpoints in LIPID and
CARE
Pravastatin-treated Subjects Only
LIPID CHD Death
CARE CHD Death or Non-fatal MI
Trial Primary Endpoint
Aspirin Users
Aspirin Non-Users
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
86
Meta-Analysis Endpoints Considered

• Fatal or non-fatal MI
• Ischemic stroke
• Composite CHD death, non-fatal MI, CABG, PTCA or
  ischemic stroke

87
Meta-Analysis Models

• Model 1
• Multivariate Cox proportional hazards model
• Patients combined across trials trial effect is
  a fixed covariate

H(t) l0(t)exp(Zb fS gT)
Covariates
Study effects
Baseline Hazards constant
Treatment Effects
88
Relative Risk ReductionCox Proportional Hazards
All Trials
Relative Risk (95 CI)
RRR
RRR Relative Risk Reduction
89
Meta-Analysis Models

• Model 2 Same as Model 1 except
• Allows trial heterogeneity Bayesian
  hierarchical (random effects) model of trial
  effect

H(t) l0(t)exp(Zb fS gT)
Covariates
Study effects Hierarchical
Baseline Hazards piecewise-constant
Treatment Effects
90
Model 2 Hierarchical, Random Effects
Fatal or Non-Fatal MI
Cumulative Proportion of Events
Year
91
Model 2 Hierarchical, Random Effects
Ischemic Stroke Only
Cumulative Proportion of Events
Year
92
Model 2 Hierarchical, Random Effects
CHD Death, Non-Fatal MI, CABG,PTCA, or Ischemic
Stroke
Cumulative Proportion of Events
93
Combination is More Effectivethan 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
94
Model 2 Fatal or Non-Fatal MI
95
Meta-Analysis Models

• Model 3 Same as Model 2 except
• Treatment hazard ratios vary over time

H(t) lT0(t)exp(Zb fS)
Baseline Hazards piecewise-constant Within
treatment
Covariates
Study Effects Hierarchical
96
Model 3 Fatal or Non-Fatal MI
97
Probability of synergy between pravastatin
aspirin
98
Conclusion of Hazard Analysis over Time

• Benefit of pravastatinaspirin over aspirin was
  present in each year of the 5-year duration of
  the trials

Benefit of pravastatinaspirin 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
99
Hierarchical modeling in design

• Using historical information
• Combining results from multiple concurrent trials
  (or many centers)

100
Hierarchical modeling dose-response

• Example drug Z (rozuvastatin) vs drug A
  (atorvastatin) (Berry et al., 2002, American
  Heart Journal)

101
Studies involving drugs A and Z, with change
from baseline. Change Study n
Dose Mean SD Y 1. 46 10 27 10 0.73
45 20 34 10 0.66 2. 45 10 35.3 8 0.647
3. 14 Placebo 1.4 18 0.986 13
5 16.7 17 0.833 16 20 33.2 18 0.668
12 80 41.4 18 0.586 4. 222 10 35 14 0.65
5. 210 20 45.0 10 0.55 215 40 51.1 12 0.489
6. 132 10 37 13 0.63 7. 133 Placebo
1 12 1.01 707 10 36 13 0.64 8. 17 Placebo
0 8 1.00 18 10 35 8 0.65 9.
41 10 35 13 0.65 10. 73 10 38 10 0.62
51 20 46 8 0.54 61 40 51 10 0.49
10 80 54 9 0.46
102
Study n Dose Mean SD Y 11. 54 10 30 18 0.70
12. 1897 10 37.6 NA 0.624 13. 12 Placebo
7.6 9 1.076 11 2.5 25.0 9 0.75 13
5 29.0 9 0.71 11 10 41.0
9 0.59 10 20 44.3 9 0.557 11 40 49.7
9 0.503 11 80 61.0 9 0.39 14. 40 10 29 12 0.7
1 15. 164 80 46 NA 0.54 16. 12 Placebo
5.1 8.1 1.051 15 10 43.9
7.8 0.561 13 80 56.9 8.3 0.431 14
1 35.9 7.7 0.641 15 2.5 40.6
9.9 0.594 16 5 44.1 8.3 0.559 17 10 51.7
8.7 0.483 17 20 55.5 12.8 0.445 18 40 63.
2 8.7 0.368 17. 17 Placebo
0.8 10.6 1.008 15 40 61.9 7.2 0.381 31 80
62.9 7.8 0.371
103
Dose-response model

• Yij expas at bt log(d) eij
• s for study
• t for drug
• d for dose
• i for observation (1, . . . , 43)
• j for patient within study/dose
• eij is N(0, s2)
• Priors dont matter much, except . . .

104
Prior for as N(0, t2)

• t2 is important
• t2 large means studies heterogeneouslittle
  borrowing
• t2 small means studies homogeneousmuch borrowing
• Prior of t2 is IG(10, 10)
• Prior mean and sd are 0.10 0.017

105
Likelihood
Calculations of posterior predictive
distributions by MCMC
106
Posterior means and SDs

• Parameter Mean StDev
• aP 0.0016 0.027
• aA 0.073 0.055
• aZ 0.34 0.059
• bA 0.149 0.021
• bZ 0.146 0.019
• s 0.152 0.024
• t 0.087 0.011

107
Posterior means and SDs

• Par. Mean StDev Par. Mean StDev
• a1 0.102 0.023 a10 0.072 0.022
• a2 0.017 0.032 a11 0.052 0.027
• a3 0.062 0.025 a12 0.054 0.013
• a4 0.014 0.018 a13 0.028 0.024
• a5 0.072 0.035 a14 0.063 0.031
• a6 0.043 0.022 a15 0.104 0.042
• a7 0.015 0.013 a16 0.070 0.031
• a8 0.002 0.029 a17 0.017 0.033
• a9 0.013 0.033

108
Model fit
109
Interval estimates for pop. mean model (line) vs
standard (box)
110
Study/dose-specific interval estimates model
(line) vs standard (box)
111
Posterior distn of reduction (95 intervals)
Drug A
Drug Z
112
Posterior distn of mean diff, A Z
113
Really neat . . .

• Using predictive probabilities for designing
  future studies
• Contour plots

114
Observed Y for future study with nAnZ20
dAdZ10
Z
A
115
Observed Y for future study with nAnZ100
dAdZ10
Z
A
116
Observed Y for future study with nAnZ20
dA10, dZ5
Z
A
117
Observed Y for future study with nAnZ100
dA10, dZ5
Z
A
118
STELLAR trial results (each n160)
Predicted atorva
-36
Predicted rosuva
-41
-46
-50
-52
-54
-58
119
Posterior distn of reduction (95 intervals)
Recall
Drug A
Drug Z
120
Adaptive Phase II Finding the Best Dose
Scott M. Berry scott_at_berryconsultants.com
121
Standard Parallel Group Design
Equal sample sizes at each of k doses.
Doses
122
True dose-response curve (unknown)
Response
Doses
123
Observe responses (with error) at chosen doses
Response
Doses
124
Dose at which 95 max effect
Response
True ED95
Doses
125
Uncertainty about ED95
Response
?
Doses
126
Solution Increase number of doses
Response
True ED95
Doses
127
But, enormous sample size, and . . . wasted dose
assignmentsalways!
Response
True ED95
Doses
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?

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

130
Adaptive Approach
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

132
Empirical Data

• Observe Yij for subject i, outcome j
• Yij 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

133
Efficacy Endpoints

• Let d be the dose
• Pj(d) probability of event j, dose d.

?j(d) N(?j, ??2)
G(1,1)
N(1,1)
N(2,1)
IG(2,2)
134
Safety Endpoint

• Let di be the dose for subject i
• Pj(d) probability of safety j, dose d.

N(-2,1)
G(1,1)
N(1,1)
135
Utility Function

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

136
Utility Function
U(d)U1(P1)U2(P3)U3(P0,P2)U4(P4)
Monetary
FDA Approval
Extra Safety
P0 is prob efficacy 2 success for d0
137
Monetary Utility
138
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139
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140
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141
U3 FDA Success
DSMB?
142
Statistical Utility Output

• EU(d)
• E?j(d), V?j(d)
• EPj(d), VPj(d)
• Prdj max U
• PrP2(d) gt P0
• Pr P2 gtgt P0 250/per arm) each d

gtgt means statistical significance will be achieved
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!

144
Allocator

• Find best dose?
• Learn about best dose?

d is the max utility dose, d second best
Find the ?V for each dose gt allocation probs
145
Allocator
?V(d?0)
?V(d0)
146
Allocator

• Drop any rdlt0.05
• Renormalize

147
Decisions
Pr(d d) gt C1

• Find best dose?
• Learn about best dose?
• Shut down allocator wj if stop!!!!
• Stop trial when both wj 0
• If Pr(P2(d) gtgt P0) lt 0.10 stop for futility

If found, stop
Pr(P2(d) gtgt P0)gtC2
If found, stop
148
More Decisions?

• Ultimate EU(dosing) gt 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.

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

150
Scenario 1
MAX
Stopping Rules C1 0.80, C2 0.90
151
18 2 2 0 2
15 5 0 3 5
18 1 1 0 2
20 2 0 1 0
19 5 0 3 1
17 4 4 2 3
18 5 3 2 2
152
Dose Probabilities
153
18 2 2 0 2
19 5 1 4 1
20 1 1 0 3
20 2 0 1 0
19 5 0 3 3
25 7 8 2 7
24 7 5 2 7
154
Dose Probabilities
155
19 3 2 0 1
20 5 1 4 0
21 2 1 0 2
20 2 0 1 0
21 5 0 3 4
29 7 9 2 11
31 11 6 3 17
156
Dose Probabilities
157
20 4 2 0 0
21 5 1 4 4
23 2 1 0 4
20 2 0 1 0
25 5 1 4 0
36 7 10 3 10
45 12 10 3 16
158
Dose Probabilities
159
20 4 2 0 0
25 5 1 4 6
26 2 1 0 1
20 2 0 1 0
26 6 2 4 5
44 7 13 3 12
52 13 10 4 15
160
Dose Probabilities
161
21 4 2 0 3
26 6 1 4 5
26 2 1 0 6
20 2 0 1 0
33 7 3 4 5
52 8 13 4 10
61 18 15 4 12
162
Dose Probabilities
163
Trial Ends

• P(10-Dose max Util dose) 0.907
• P(10-Dose gtgt Pbo 250/arm) 0.949
• 280 subjects
• 32, 20, 24, 31, 38, 62, 73 per arm

164
Operating Characteristics
165
Operating Characteristics
166
Scenario 2
Stopping Rules C1 0.80, C2 0.90
167
Operating Characteristics
168
Operating Characteristics
169
Simulation 3
Stopping Rules C1 0.80, C2 0.90
170
Operating Characteristics
171
Operating Characteristics
172
Scenario 4
Stopping Rules C1 0.80, C2 0.90
173
Operating Characteristics
174
Operating Characteristics
175
Scenario 5
Stopping Rules C1 0.80, C2 0.90
176
Operating Characteristics
177
Operating Characteristics
178
Scenario 6
Stopping Rules C1 0.80, C2 0.90
179
Operating Characteristics
180
Operating Characteristics
181
Bells Whistles

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

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!