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ADAPTIVE BAYESIAN DESIGNS FOR DOSE-RANGING DRUG TRIALS

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					Bayesian Clinical Trials
         Scott M. Berry
   scott@berryconsultants.com



          BERRY
       CONSULTANTS
      STATISTICAL INNOVATION




                                1
          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
                                                   2
      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 '

                                        3
            Simple Example
•   Coin, P(HEADS) = 
•    = 0.25 or =0.75, equally likely.
•   DATA: Flip coin twice, both heads.
•   ???




                                           4
                      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)
  ----------------------------------- = 0.90
  (0.75)2 (0.5) + (0.25)2 (0.5)
                                                 Posterior Probabilities
 Likelihood           Prior Probabilities


                                                                              5
       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)




                                                                 6
         Bayesian Statistics
• A subjective probability axiomatic
  approach was developed with Bayes
  theorem as the ―mathematical crank‖--
  Savage, Lindley (1950’s)
• Very different than classical statistics: a
  collection of tools
• Before 1980-1990?: A philosophical
  niche, calculation very hard.
• Early 1990’s: Computers and methods
  made calculation possible…and more!
                                                7
       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
                                           8
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?

                             9
  Consequence of Bayes rule:
   The Likelihood Principle
The likelihood function
                LX(q) = f( X | q)
contains all the information in an experiment
relevant for inferences about q

             LX q  q 
          LX q  q dq
                                          10
• Short version of LP: Take data at
  face value
• But ―data‖ can be deceptive
• Caveats . . .
  – How identified?
  – Why are they showing me this?



                                      11
          Example
• Data: 13 A's and 4 B's
• Parameter =  = P(A wins)
• Likelihood   13 (1–)4
• Frequentist conclusion?
  Depends on design


                              12
    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"



                                                 13
  Design (1): 17 results
Binomial distribution
with n = 17,  = 0.5;
P-value = 0.049




                           14
Design (2): Stop when
 both 4 A’s and 4 B’s

           Two-sided negative
        binomial with r = 4,  = 0.5;
            P-value = 0.021




                                15
      Design (3): Interim analysis
      at n=17, possible total is 44


Analyses at
n = 17 & 44;
stop @ 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 16
   Design (4): Scientist’s
 stopping rule: Stop when
   you know the answer

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


                                   17
      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

                                           18
Likelihood function of 

                     13   4
                    p (1–p)




 0   .1   .2   .3   .4   .5   .6   .7   .8       .9   1
                                             p

                                                          19
         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




                                            20
    Posterior density of 
for uniform prior: Beta(14,5)

                        13   4
                       p (1–p)




    0   .1   .2   .3   .4   .5   .6   .7   .8       .9   1
                                                p
                                                             21
              Pr[ > 0.5 ]

     P(p > 0.5 | data)
           = 0.985




0   .1   .2   .3   .4   .5   .6   .7   .8       .9   1
                                            p            22
PREDICTIVE PROBABILITIES
 • Distribution of future data?


 • P(next is an A) = ?


 • Critical component of experimental design


 • In monitoring trials



                                          23
Laplace’s 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
                                     24
Updating w/next observation
                                                   Beta(15, 5)

                 Beta(14, 6)



          prob 5/19                                     prob 14/19




 0   .1     .2   .3   .4   .5   .6   .7   .8       .9    1
                                               p

                                                                     25
       Suppose 17 more
         observations
P(A wins x of 17 | data)
= EP(A wins x | data, )

     17 x                       
= E  x   1 -  
                     17- x
                          |data,  
                                

  Beta-Binomial Distribution          26
        Predictive distribution
Predictive      .15

distribution of
# of successes .10
in next
                                                               88% probability
17 tries:                                                      of statistical
                 .05                                           significance



                 .00
                       0   1   2   3   4   5   6   7   8   9 10 11 12 13 14 15 16 17



   Has more variability than any binomial 
                                                                               27
    Best fitting binomial vs.
    predictive probabilities
Binomial, p=14/19


                                             96% probability
                                              of statistical
                                              significance
0   1   2   3   4   5   6   7   8   9   10   11   12   13   14   15   16   17



Predictive, p ~ beta(14,5)
                                             88% probability
                                              of statistical
                                              significance

0   1   2   3   4   5   6   7   8   9   10   11   12   13   14   15   16   17
                                                                                28
        Possible Calculation
    17 x        17- x  14  5  13
   x  1 -                      1 -   d
                                               4

                          19 

• Simulate a  from the beta(14,5)

• Simulate an x from binomial(17, )

• Distribution of x’s is beta-binomial--the
  predictive distribution
                                                     29
           Posterior and
         Predictive…same?
• Clinical Trial, 100 subjects. HA:  > 0.25?
  FDA will approve if # success ≥ 33 [post >
  0.95, beta(1,1)]
• See 99 subjects, 32 successes

• Pr[  > 0.25 | data ] = 0.955
• Predictive prob trial success = 0.327


                                                30
   Predictive Probabilities
     for Medical Device
• Bayesian calculations  FDA:
  – Some patients have reached 2 years
  – Some patients have only 1-yr follow-
    up




                                       31
Continuous data; Patients w/both
       12 and 24 months

    2
    4
    -
    m
    o
    n
    t
    n




          12 - m o n t h O s w e s t r y   32
    Some patients with
    only 12-month data

2
4
-
m
o
n
t
n




                             33
         Missing 24 months
    Kernel density estimates

2
4

m
o
n
t                                         
h




         12 - m o n t h O s w e s t r y
                                              34
Small bandwidth (0.2)
  a




                         35
Larger bandwidth (0.3)
  b




                          36
Still larger bandwidth (0.4)
   c




                                37
Very large bandwidth (0.5)
 (nearly bivariate normal)
   d




                              38
Condition on 12-month value
    a




                          39
 Conditional distribution
of 24-month value (0.2)
  a




      24-month Oswestry     40
For largest bandwidth (0.5)
    d




        24-month Oswestry      41
Multiple imputation: simulate full
      set of 24-month data

    2
    4

    m
    o
    n
    t
    h




           12 - m o n t h O s w e s t r y   42
• 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‖


                                             43
        Monitoring example:
         Baxter’s DCLHb
• Diaspirin Cross-Linked Hemoglobin
• Blood substitute; emergency trauma
• Randomized controlled trial (1996+)
  – Treatment: DCLHb
  – Control: saline
  – N = 850 (= 425x2)
  – Endpoint: death


                                        44
• 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


                                      45
     Bayesian predictive
  probability of future results
        (no stopping)
• Probability of significant survival benefit
  for DCLHb after 850 patients: 0.00045
  (PP=0.0097)
• DMC paused trial: Covariates?
• DMC stopped the trial

                                            46
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 JCO—reactions …
                                       47
       Mixtures:
• Data: 13 A's and 4 B's


• Likelihood  p13 (1–p)4




                            48
                 Mixture Prior
       ~ 0 I[p=p0] + (1-0) Beta(a,b)
   0 I0 p013(1-p0)4 + (1-0) Kpa+13-1(1-p)b+4-1

    ~ 0 I0 + (1-0) Beta(a+13,b+4)
                    0 p13(1-p)4
0 = ---------------------------------------------------
                              (a)(b)(a+b+17)
   0 p 13(1-p)4 + (1- ) --------------------------
                          0
                            (a+13)(b+4)(a+b)
                                                   49
        Mixture Posterior 0=.5



Pr(p=0.5) = 0.246




                        P(p > 0.5) = 0.742

                                       50
   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?


                                       51
Numbers of heads




                   This is you




                                 52
     One-Sample Problem
[q]~ Beta(a,b
• [X] ~ Binomial(n,q)
• [q|X]~Beta(a+X,b+n-X)
• Mean = (a + X/a+b+n)




                           53
For uniform prior (a= b= 1)

                   Posterior:
                   q~ Beta(17, 5



         Prior:
         q~ Beta(1, 1


     0   .1   .2   .3   .4   .5   .6   .7   .8   .9   1
                             q         0.77
                                                          54
For a= b= 10
           Posterior:
           q~ Beta(26, 14

         Prior: q~
         Beta(10, 10




     0    .1   .2   .3   .4   .5   .6    .7    .8   .9   1
                              q         0.65
                                                             55
Remember the other coins . . .




                            This is you




                                          56
    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.


                                             57
HIERARCHICAL MODELING

         Population:




           Sample:
                       Inferential
                        problems
Sample from sample:
                                 58
         Selecting coins
Population of coins—population of q’s:




Select two coins and toss each coin
10 times: one 9 heads, other 4 heads.
•Estimate q1, q2.
•Estimate distribution of q’s in population.
                                           59
Generic example: Unit is lab or
 drug variation or lot or study
  Unit     s      n     s/n
    1      20     20     1.00   n = #observations
    2        4    10     0.40
    3      11     16     0.69
    4      10     19     0.53   s = #successes
    5        5    14     0.36
    6      36     46     0.78
    7        9    10     0.90   s/n = success
    8        7      9    0.78
    9        4      6    0.67         proportion
  Total   106    150     0.71


                                                 60
  If q1 = q2 = . . . = q9 = q
(all 150 units exchangeable)
                106            44
            p         (1-p)




  0   .1   .2    .3    .4     .5    .6   .7   .8   .9       1
                                                        p       61
     Assuming equal q’s,
   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?


                                          62
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

                                63
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




                                                  64
Beta(a,b) for a, b = 1, 2, 3, 4:




                               65
 Suppose uniform prior for
a & b on integers 1, . . ., 10




                                 66
Posterior probabilities
      for a & b
                 a
                    = 0.68
                a+b




                             67
   Calculating posterior
     distribution of G
• Direct in this example
• Can be more complicated, and
  require:
  – Gibbs sampling (BUGS)
  – Other Markov chain Monte Carlo




                                     68
     Posterior mean of G
(also predictive density for q)




  0   .1   .2   .3   .4   .5   .6   .7   .8   .9       1
                                                   p       69
Contrast with likelihood
assuming all p’s equal
     All p's equal


     Posterior
     mean of G




0   .1   .2   .3   .4   .5   .6   .7   .8   .9       1
                                                 p
                                                         70
       Bayesian questions:
• P(q > 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?



                                           71
            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)

                                   72
Bayes estimates are regressed
or shrunk toward overall mean
                            Bayes estimates


    0   .1   .2   .3   .4    .5   .6   .7   .8   .9       1
                                                      p

                  Unadjusted estimates

                                                              73
         Baseball Example
• 446 players in 2000 with > 100 at bats




                                 Jose Vidro




                                              74
How good was Jose Vidro?
      (200 hits in 606 at bats, 0.330)



       X ~ Binomial(606, qJV
     (hits)


             qJV ~Beta(a,b


                                         75
• Empirical Bayes:
        aEB =95.5bEB=258.9
mean = 0.269; var = 0.0362-.0272)

[q|X] ~ Beta(200+95.5, 406+258.9)
  (approx)

Posterior mean = 0.308
Posterior st. dev. = 0.015

                                           76
Science, Feb 6, 2004, pp 784-6




                            77
www.fda.gov/ ohrms/dockets/ac/02/slides/
3829s2_03_Bristol-Meyers-meta-analysis.ppt

Efficacy of Pravastatin + Aspirin:
         Meta-Analyses




[For statistical analysis, S.M. Berry et al.,
Journal of the American Statistical Association, 2004]
                                                   78
             Meta-Analysis of these
     Pravastatin Secondary Prevention Trials
                 Number of
  Trial          Subjects*           % on Aspirin               Primary Endpoint

 LIPID              9014                  82.7                    CHD mortality

 CARE               4159                  83.7              CHD death & non-fatal MI

                                                            Atherosclerotic progression
REGRESS              885                  54.4
                                                                    (& events)
                                                            Atherosclerotic progression
 PLAC I              408                  67.5
                                                                    (& events)
                                                            Atherosclerotic progression
PLAC II              151                  42.7
                                                                    (& events)

 Totals            14,617                 80.4

*99.7% of pravastatin-treated subjects received 40mg dose
                                                                                    79
               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




                                                    80
    Patient Group Comparisons


                    Randomized Groups

                Pravastatin       Placebo

                                                Randomized
Aspirin Users   Prava+ASA       Placebo+ASA     Comparison
  Aspirin
                Prava alone     Placebo alone
 Non-Users
                Observational
                 Comparison




                                                             81
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




                                                       82
Is the Combination More Effective
      than Pravastatin Alone?

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




                                           83
Event Rates for Primary Endpoints in LIPID
                and CARE
              Pravastatin-treated Subjects Only



             Trial:        LIPID              CARE
   Primary Endpoint:     CHD Death          CHD Death or
                                            Non-fatal MI

 Aspirin Users              5.8%                  9.3%

 Aspirin Non-Users          8.8%               14.8%


                        Observational      Observational
                         Comparison         Comparison

                                                           84
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

                                                   85
Meta-Analysis Endpoints Considered


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




                                      86
           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)


Baseline Hazards        Covariates    Study
                                                   Treatment Effects
constant                              effects
                                                               87
        Relative Risk Reduction
   Cox Proportional Hazards – All Trials
Fatal or Non-Fatal MI             Relative Risk (95% CI)                     RRR
                                             0.69
    Prava+ASA vs ASA alone                                                   31%
                                                    0.74
   Prava+ASA vs Prava alone                                                  26%
                          0.400      0.600            0.800          1.000

Ischemic Stroke
                                               0.71
    Prava+ASA vs ASA alone                                                   29%
                                             0.69
   Prava+ASA vs Prava alone                                                  31%

                          0.400      0.600            0.800          1.000

CHD Death, Non-Fatal MI, CABG, PTCA, or Ischemic Stroke
                                                    0.76
    Prava+ASA vs ASA alone                                                   24%
                                                              0.87
   Prava+ASA vs Prava alone                                                  13%
                          0.400      0.600            0.800          1.000
                                                                                     88
                                                     RRR = Relative Risk Reduction
           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)


Baseline Hazards     Covariates      Study
                                                 Treatment Effects
piecewise-constant                   effects                 89
                                  Hierarchical
        Model 2 – Hierarchical, Random Effects

                       Fatal or Non-Fatal MI
           0.100
                                                      Placebo
                                                      ASA alone
                                                      Prava alone
           0.075
Cumulative
Proportion                                            Prava+ASA
 of Events 0.050



           0.025



           0.000
                   0      1    2          3   4   5
                                   Year
                                                         90
        Model 2 – Hierarchical, Random Effects

                       Ischemic Stroke Only
           0.025
                                                      Placebo
                                                      Prava alone
                                                      ASA alone
           0.020


           0.015                                      Prava+ASA
Cumulative
Proportion
 of Events 0.010


           0.005


           0.000
                   0     1     2          3   4   5
                                   Year
                                                         91
        Model 2 – Hierarchical, Random Effects
              CHD Death, Non-Fatal MI, CABG,
              PTCA, or Ischemic Stroke
           0.25
                                                  Placebo
                                                  ASA alone
           0.20                                   Prava alone

                                                  Prava+ASA
           0.15
Cumulative
Proportion
 of Events 0.10


           0.05


           0.00
                  0   1    2          3   4   5
                               Year
                                                     92
      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


                                                         93
                                              Model 2:
                                       Fatal or Non-Fatal MI
            Cumulative Proportion of Events                                 Hazard
0.100                                                       0.025
                                              Placebo
                                              ASA alone
                                                                                               Placebo
                                              Prava alone   0.020
0.075                                                                                          ASA alone
                                                                                               Prava alone

                                              Prava+ASA     0.015
0.050                                                                                          Prava+ASA

                                                            0.010

0.025
                                                            0.005


0.000                                                       0.000
        0        1      2          3     4    5                     0   1   2          3   4        5
                            Year                                                Year
                                                                                               94
             Meta-Analysis Models
 •Model 3: Same as Model 2 except
   – Treatment hazard ratios vary over time



    H(t) =           l0(t)exp(Zb
                      T                + fS)

Baseline Hazards          Covariates      Study
piecewise-constant
                                         Effects
Within treatment
                                       Hierarchical

                                                      95
                                          Model 3:
                                   Fatal or Non-Fatal MI
            Cumulative Proportion of Events                                 Hazard
0.100                                                       0.030
                                              Placebo

                                              ASA alone
                                                            0.025
                                              Prava alone
0.075
                                                                                               Placebo
                                                            0.020
                                                                                               ASA alone
                                              Prava+ASA
0.050                                                       0.015                              Prava alone
                                                                                               Prava+ASA
                                                            0.010
0.025
                                                            0.005


0.000                                                       0.000
        0        1      2          3   4      5                     0   1   2          3   4      5
                            Year                                                Year
                                                                                          Year
                                                             5 Separate Analyses: One per 96
      Probability of synergy
  between pravastatin & aspirin

Endpoint         Model 2   Model 3
All events        0.983     0.985
Cardiac events    0.945     0.947
Any MI            0.911     0.923
Stroke            0.924     0.906
Death             0.997     0.997

                                     97
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



                                                                   98
   Hierarchical modeling
         in design
• Using historical information
• Combining results from multiple
  concurrent trials (or many
  centers)



                                    99
   Hierarchical modeling
     & dose-response
• Example: drug Z (rozuvastatin) vs drug
  A (atorvastatin)
  (Berry et al., 2002, American Heart
  Journal)




                                     100
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                101
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     0.71
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   102
        31   80*       –62.9    7.8   0.371
       Dose-response model
• Yij = exp{as + 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, 2)
• Priors don’t matter much, except . . .
                                      103
      Prior for as ~ N(0, t2)
t2 is important
t2 large means studies heterogeneous—
  little borrowing
t2 small means studies homogeneous—
  much borrowing
• Prior of t2 is IG(10, 10)
• Prior mean and sd are 0.10 & 0.017


                                    104
                    Likelihood

      ni                                  2 
 exp- 2  Y - expa s + at + bt log(d) 
               43
 -n

      2 i=1                                
               i



           Calculations of posterior &
       predictive distributions by
       MCMC
                                          105
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
            0.152    0.024
     t       0.087    0.011



                              106
      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




                                                   107
Model fit




            108
  Interval
 estimates
  for pop.
   mean:
model (line)
vs standard
   (box)



               109
 Study/dose-
   specific
   interval
  estimates:
model (line) vs
standard (box)



                  110
Posterior
 dist’n of
reduction
  (95%                Drug A
intervals)

             Drug Z

                               111
   Posterior
dist’n of mean
      diff,
     A–Z




                 112
         Really neat . . .

• Using predictive probabilities for
  designing future studies

• Contour plots


                                       113
                       90
                   %        Mean Z10 better than A10: 12.1%
                   Y        P(Z10 betters A10) = 99.4%
                       80
                   f
                   o 70
                   r

                   A 60
                   Z

 Observed %Y           50
for future study
                       40
 with nA=nZ=20                     Z10 = A10

    dA=dZ=10                  40       50     60      70      80    90
                                                     A
                                            %Y f o r Z

                   Height       20              40            60
                                80              100           120
                                140             160           180
                                200             220           240
                                260             280           300    114
                                320             340
                       90
                   %        Mean Z10 better than A10: 12.1%
                   Y        P(Z10 betters A10) = 99.99%
                       80
                   f
                   o 70
                   r

                   A 60
                   Z

Observed %Y            50
for future study
                       40
with nA=nZ=100                  Z10 = A10

 dA=dZ=10                     40      50      60      70      80    90
                                                    A
                                           %Y f o r Z

                   Height       30             60             90
                                120            150            180
                                210            240            270
                                300            330            360
                                390            420            450    115
                                480            510            540
                       90
                   %        Mean Z5 better than A10: 7.0%
                   Y 80     P(Z5 betters A10) = 92.5%

                   f
                   o   70
                   r
                   Z 60
Observed %Y
                   A


for future study       50

with nA=nZ=20          40
                                  Z5 = A10
 dA=10,
                             40      50     60      70      80    90
 dZ=5                                              A
                                          %Y f o r Z

                   Height      20             40            60
                               80             100           120
                               140            160           180
                               200            220           240
                               260            280           300    116
                               320            340
                       90
                   %        Mean Z5 better than A10: 7.0%
                   Y        P(Z5 betters A10) = 98.6%
                       80
                   f
                   o 70
                   r

                   A 60
                   Z
Observed %Y
for future study       50

with nA=nZ=100         40
                                   Z5 = A10
 dA=10,
                              40      50      60     70     80    90
 dZ=5                                               A
                                           %Y f o r Z

                   Height       30             60           90
                                120            150          180
                                210            240          270
                                300            330          360
                                390            420          450    117
                                480            510          540
                                                       STELLAR trial results (each n≈160)
                                                                      Dose (log scale)
                                                           10mg        20mg        40mg            80mg
                                                       0
    Mean percentage change in
                                LDL -C from baseline
                                        -10

                                        -20                   -20%
                                                                            -24%
                                                              -28%
                                        -30                                               -30%
                    Predicted                                               -35%
                                -37%
                     atorva -36%
                      -40
                                                                                          -39%
                                                                            -43%
                    Predicted -46%                                        -41%            -48%          -46%
                      -50
                     rosuva -50%
                              *                                             -52%         -46%           -51%

                                                                          *
                                                                          -54%
                                                                                          -55%        -52%
                                        -60                                              *-58%

                                                             Rosuvastatin n=473           Atorvastatin n=634
                                                             Simvastatin n=648            Pravastatin n=485

*p<0.002 rosuvastatin 10mg vs. atorvastatin 10mg, simvastatin 10, 20 & 40mg,
pravastatin 10, 20 & 40mg; rosuvastatin 20mg vs. atorvastatin 20mg and 40mg,
simvastatin 20, 40 & 80mg, pravastatin 20 & 40mg; rosuvastatin 40mg vs.                                        118
atorvastatin 40mg, simvastatin 40 & 80mg, pravastatin 40mg
Recall:

 Posterior
  dist’n of
 reduction
   (95%                Drug A
 intervals)

              Drug Z

                                119
Adaptive Phase II: Finding
    the ―Best‖ Dose
          Scott M. Berry
    scott@berryconsultants.com




            BERRY
         CONSULTANTS
        STATISTICAL INNOVATION


                                 120
Standard Parallel Group Design
Equal sample sizes at each of k doses.




                          Doses
                                         121
True dose-response curve
       (unknown)
 Response




            Doses
                           122
Observe responses (with error) at
         chosen doses
    Response




               Doses
                             123
Dose at which 95% max effect
   Response




                      True ED95




              Doses
                             124
Uncertainty about ED95
Response




                               ?
           Doses
                         125
        Solution:
Increase number of doses
 Response




                    True ED95




            Doses
                           126
But, enormous sample size, and . . .
wasted dose assignments—always!
     Response




                          True ED95




                Doses
                                 127
              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 don’t know where
  you are going, how do you know when
  you get there?
                                       128
          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
                                                129
Adaptive Approach
                                Decisions:
                                Continue
                               Study, Stop
   Subject Enters               Phase III

                     Utility


                                    Statistical
     Allocator                        Model




Wait Event
                     Results
                    Measured

                                                  130
         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

                                        131
           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

                                           132
          Efficacy Endpoints
• Let d be the dose
• Pj(d) probability of event j, dose d.
                  Pj d                        d 
  q j d  = log              = a j d  + b j  d + g 
                  1 - Pj d                   
                                                       j 
                                                          

               aj(d) ~ N(j, 2)                      G(1,1)


               N(–2,1)                        N(1,1)
                                  IG(2,2)
                                                         133
            Safety Endpoint
• Let di be the dose for subject i
• Pj(d) probability of safety j, dose d.
                    Pj d                    d 
    q j d  = log              = a j + bj           
                    1 - Pj d               d + g j 
                                                       


                       N(-2,1)                       G(1,1)
                                        N(1,1)


                                                        134
         Utility Function

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

                                       135
       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 d=0
                                        136
       Monetary Utility

             P1 - 0.1
U1 P1  =            for P1  0.1
               0.4


U2 P3  = 1- 2 P3  for P3  0.5
                   1.5




                                     137
138
139
U1 P1 U 2 P3 



                         P1

                    P3
                              140
          U3: FDA Success


U 3 P2 , P0  = Pr P2  P0 | 250/arm trial

     U 4 P4  = 0 if any Major safety
    ―DSMB?‖


                                         141
       Statistical + Utility Output
   •   E[U(d)]
   •   E[qj(d)], V[qj(d)]
   •   E[Pj(d)], V[Pj(d)]
   •   Pr[dj max U]
   •   Pr[P2(d) > P0]
   •   Pr[ P2 >> P0 | 250/per arm) each d

>> means statistical significance will be achieved
                                               142
                  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!


                                                 143
                   Allocator
• Find best dose?        
                    V P d -P d
                       1    1
                             *           **



• Learn about best dose? V  d - P 
                                     *
                            P    2            0


 d* is the max utility dose, d** second best

             + w V P d - P 
V* = w1V P d - P d
          1
               *
                1
                        **
                                 2       2
                                              *
                                                        0



 Find the V* for each dose ==> allocation probs
                                                  144
                     Allocator
                    V*(d≠0) =
V P d 
    1

 nd + 1 
                               
         w1 Pr d = d * + w2 Pr d = d **  +
                                           
                            V P2 d 
                              n +1 
                                                
                                         w2 Pr d = d * 
                                                         
                                 d


                               V P0 
             V*(d=0) =        n0 + 1
                                       w2

                                                    145
                     Allocator
                V d 
                     *
       rd =    k            for all d
               V * d 
              d =0



• ―Drop‖ any rd<0.05

• Renormalize

                                        146
               Decisions
• Find best dose? If found, stop: Pr(d = d*) > C1
• Learn about best dose?
                If found, stop: Pr(P2(d*) >> P0)>C2
• Shut down allocator wj if stop!!!!
• Stop trial when both wj = 0

• If Pr(P2(d*) >> P0) < 0.10 stop for
  futility
                                              147
           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.
                                           148
               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


                                             149
            Scenario #1
Dose   P1      P2    P3     P4    UTIL
 0     0.06   0.05   0.05    0      0
0.25   0.10   0.05   0.06    0      0
0.5    0.13   0.08   0.07    0    0.063
 1     0.17   0.12   0.08    0    0.323
2.5    0.20   0.15   0.09    0    0.457
 5     0.23   0.18   0.10    0    0.532
                                                MAX
 10    0.30   0.25   0.11    0    0.656

 Stopping Rules: C1 = 0.80, C2 = 0.90     150
18 20 18   15   19   17   18
 1 2 2      5    5    4    5
 1 0 2      0    0    4    3
 0 1 0      3    3    2    2
 2 0 2      5    1    3    2




                               151
           Dose Probabilities
           0   .25 .5   1   2.5   5    10
P(>>Pbo)       .18 .33 .27 .29 .67 .67
P(max)         .01 .04 .06 .04 .33 .52
P(2nd)         .03 .06 .10 .13 .35 .32
 Alloc     .06 .01 .02 .04 .06 .35 .46

                                      152
20 20 18   19   19   25   24
 1 2 2      5    5    7    7
 1 0 2      1    0    8    5
 0 1 0      4    3    2    2
 3 0 2      1    3    7    7




                               153
           Dose Probabilities
           0   .25 .5   1   2.5   5    10
P(>>Pbo)       .12 .38 .36 .38 .92 .91
P(max)         .00 .00 .02 .04 .41 .53
P(2nd)         .00 .03 .06 .07 .47 .37
 Alloc     .00 .00 .02 .04 .09 .34 .51

                                      154
21 20 19   20   21   29   31
 2 2 3      5    5    7   11
 1 0 2      1    0    9    6
 0 1 0      4    3    2    3
 2 0 1      0    4   11   17




                               155
           Dose Probabilities
           0   .25 .5   1   2.5   5    10
P(>>Pbo)       .13 .39 .38 .26 .97 .85
P(max)         .00 .02 .03 .01 .39 .55
P(2nd)         .00 .03 .10 .05 .46 .35
 Alloc     .11 .00 .03 .10 .05 .46 .35

                                      156
23 20 20   21   25   36   45
 2 2 4      5    5    7   12
 1 0 2      1    1   10   10
 0 1 0      4    4    3    3
 4 0 0      4    0   10   16




                               157
           Dose Probabilities
           0   .25 .5   1   2.5   5    10
P(>>Pbo)       .16 .41 .38 .48 .93 .93
P(max)         .00 .02 .03 .04 .26 .65
P(2nd)         .00 .05 .07 .10 .49 .29
 Alloc     .00 .00 .08 .11 .18 .35 .28

                                      158
26 20 20   25   26   44   52
 2 2 4      5    6    7   13
 1 0 2      1    2   13   10
 0 1 0      4    4    3    4
 1 0 0      6    5   12   15




                               159
           Dose Probabilities
            0   .25 .5   1   2.5   5     10
P(>>Pbo)        .16 .40 .31 .41 .98 .89
P(max)          .00 .02 .03 .06 .27 .63
P(2nd)          .00 .06 .06 .12 .48 .28
 Alloc     .16 .00 .10 .04 .13 .26 .30

                                       160
26 20 21   26   33   52   61
 2 2 4      6    7    8   18
 1 0 2      1    3   13   15
 0 1 0      4    4    4    4
 6 0 3      5    5   10   12




                               161
           Dose Probabilities
           0   .25 .5   1   2.5   5    10
P(>>Pbo)       .13 .36 .32 .65 .96 .96
P(max)         .00 .01 .01 .09 .08 .81
P(2nd)         .00 .05 .05 .23 .52 .15
 Alloc

                                      162
             Trial Ends
• P(10-Dose max Util dose) = 0.907
• P(10-Dose >> Pbo 250/arm) = 0.949
• 280 subjects:
   32, 20, 24, 31, 38, 62, 73 per arm




                                    163
 Operating Characteristics




       Pbo    0.25   0.5      1    2.5      5     10
 SS     39     21     25     37     63     89    110
Pmax    ---   0.00   0.00   0.00   0.00   0.04   0.96
 SS     66     66     66     66     66     66     66
Pmax    ---   0.00   0.00   0.00   0.01   0.06   0.93
                                                   164
Operating Characteristics
               Adaptive   Constant
P(Success)      0.936      0.810
  P(Cap)        0.064      0.190
 P(Futility)    0.000      0.000
 Mean SS         384        459
  SD SS          186        224
Mean TDose      1754       1263
Max TDose       4818       2370


                                     165
            Scenario #2
Dose   P1      P2    P3     P4    UTIL
 0     0.06   0.05   0.05    0      0
0.25   0.10   0.05   0.06    0      0
0.5    0.13   0.08   0.07    0    0.063
 1     0.17   0.12   0.08    0    0.323
2.5    0.20   0.15   0.10    0    0.452
 5     0.23   0.18   0.15    0    0.502
 10    0.25   0.20   0.40    0    0.302

 Stopping Rules: C1 = 0.80, C2 = 0.90     166
  Operating Characteristics




       Pbo    0.25   0.5     1     2.5     5     10
 SS    71      27     41     81    137    172    164
Pmax    ---   0.00   0.00   0.03   0.22   0.60   0.16
 SS    100    100    100    100    100    100    100
Pmax    ---   0.00   0.00   0.03   0.20   0.44   0.33
                                                  167
Operating Characteristics
               Adaptive   Constant
P(Success)      0.314      0.266
  P(Cap)        0.686      0.734
 P(Futility)    0.000      0.000
 Mean SS         694        702
  SD SS          193        190
Mean TDose      2954       1937
Max TDose       4489      2455.25


                                     168
            Simulation #3
Dose   P1      P2    P3     P4    UTIL
 0     0.06   0.05   0.05    0      0
0.1    0.10   0.05   0.06    0      0
0.5    0.13   0.08   0.07    0    0.063
 1     0.30   0.25   0.11    0    0.656
2.5    0.17   0.12   0.08    0    0.323
 5     0.20   0.15   0.09    0    0.457
 10    0.23   0.18   0.10    0    0.532

 Stopping Rules: C1 = 0.80, C2 = 0.90     169
  Operating Characteristics




       Pbo    0.25   0.5      1    2.5      5     10
 SS     53     23     28    119     52     76    102
Pmax    ---   0.00   0.00   0.92   0.00   0.01   0.07
 SS     87     87     87     87     87     87     87
Pmax    ---   0.00   0.00   0.83   0.00   0.02   0.15
                                                   170
Operating Characteristics
               Adaptive   Constant
P(Success)      0.906      0.596
  P(Cap)        0.092      0.404
 P(Futility)    0.002      0.000
 Mean SS         453        606
  SD SS          187        205
Mean TDose      1663       1662
Max TDose       3771      2384.25


                                     171
            Scenario #4
Dose   P1      P2    P3     P4    UTIL
 0     0.06   0.05   0.05    0      0
0.25   0.06   0.05   0.05    0      0
0.5    0.06   0.05   0.05    0      0
 1     0.06   0.05   0.05    0      0
2.5    0.25   0.20   0.10    0    0.573
 5     0.25   0.20   0.10    0    0.573
 10    0.25   0.20   0.10    0    0.573

 Stopping Rules: C1 = 0.80, C2 = 0.90     172
  Operating Characteristics




       Pbo   0.25   0.5     1     2.5     5     10
 SS    53     21     22     23    150    160    163
Pmax   ---   0.00   0.00   0.00   0.27   0.32   0.40
 SS    92     92     92     92     92     92     92
Pmax   ---   0.00   0.00   0.00   0.28   0.33   0.40
                                                 173
Operating Characteristics
               Adaptive   Constant
P(Success)      0.514      0.408
  P(Cap)        0.486      0.592
 P(Futility)    0.000      0.000
 Mean SS         591        647
  SD SS          239        220
Mean TDose      2840       1780
Max TDose       4815      2448.25


                                     174
            Scenario #5
Dose   P1      P2    P3     P4    UTIL
 0     0.06   0.05   0.05    0      0
0.1    0.07   0.06   0.06    0      0
0.5    0.08   0.07   0.07    0      0
 1     0.09   0.08   0.08    0      0
2.5    0.09   0.08   0.09    0      0
 5     0.09   0.08   0.10    0      0
 10    0.09   0.08   0.11    0      0

 Stopping Rules: C1 = 0.80, C2 = 0.90    175
  Operating Characteristics




       Pbo   0.25   0.5     1     2.5     5      10
 SS    92     91     75     66     76     83     90
Pmax   ---   0.45   0.04   0.07   0.10   0.13   0.21
 SS    84     84     84     84     84     84     84
Pmax   ---   0.44   0.04   0.08   0.12   0.15   0.17
                                                  176
Operating Characteristics
               Adaptive   Constant
P(Success)      0.004      0.006
  P(Cap)        0.484      0.544
 P(Futility)    0.512      0.450
 Mean SS         574        589
  SD SS          250        258
Mean TDose      1637       1615
Max TDose       3223.5    2523.75


                                     177
            Scenario #6
Dose   P1      P2    P3     P4    UTIL
 0     0.06   0.05   0.05    0      0
0.1    0.06   0.05   0.05    0      0
0.5    0.06   0.05   0.05    0      0
 1     0.06   0.05   0.05    0      0
2.5    0.06   0.05   0.05    0      0
 5     0.06   0.05   0.05    0      0


 Stopping Rules: C1 = 0.80, C2 = 0.90
                                         178
  Operating Characteristics




       Pbo   0.25   0.5     1     2.5     5     10
 SS    66     77     51     34     38     41     43
Pmax   ---   0.90   0.01   0.02   0.02   0.02   0.03
 SS    56     56     56     56     56     56     56
Pmax   ---   0.86   0.01   0.02   0.03   0.03   0.05
                                                 179
Operating Characteristics
               Adaptive   Constant
P(Success)      0.000      0.000
  P(Cap)        0.122      0.190
 P(Futility)    0.878      0.810
 Mean SS         350        395
  SD SS          215        241
Mean TDose       811       1086
Max TDose       2404      2428.75


                                     180
            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!!!
                                            181
                 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!
                                               182

				
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