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ADAPTIVE BAYESIAN DESIGNS FOR DOSE RANGING DRUG TRIALS

VIEWS: 14 PAGES: 44

									                         Today’s Schedule
                8:00 Introduction
                8:15 Intro to Bayesian approach
               10:00 Break
               10:15 Hierarchical Distributions
               11:30 Adaptive Trial designs, part 1
               12:00 Lunch
                1:00 Adaptive Trial designs, part 1
                2:00 Adaptive Trial designs, part 2
                3:00 Break
                3:15 Multiplicities
                4:00 Decision analysis
                5:00 Adjourn

                                                      1
   BERRY
CONSULTANTS
STATISTICAL INNOVATION
                         Using Historical Data
               RCT:         X ~ f(x|qx)
               Historical: Y ~ f(y|qy)
               Examples include non-
                randomized data, different study,
                different population, . . .
               Hard Problem: No right answers
                (negotiate with FDA)
                                                    2
   BERRY
CONSULTANTS
STATISTICAL INNOVATION
                         Bayesian Approaches
                          Use historical data to
                           choose prior for q
                          Weighted likelihood
                          Functional dependence
                          Hierarchical models



                                                    3
   BERRY
CONSULTANTS
STATISTICAL INNOVATION
                             Example
                  25   successes in 100 trials in
                   historical, open-label setting
                  RCT: X ~ Binomial(100, q)
                  Priors:
                   q ~ Beta(2.5, 7.5), q ~ Beta(5, 15),
                   . . . , q ~ Beta(25, 75)

                                                          4
   BERRY
CONSULTANTS
STATISTICAL INNOVATION
                         Weighted Likelihood

          L(q)=  qx(1-q)n-x [q25(1-q)75]q
          [q|X] = Beta(x + 25q, n – x + 75q)
          Same as historical prior method
          Both are static borrowing



                                               5
   BERRY
CONSULTANTS
STATISTICAL INNOVATION
                         Functional Modeling
                              (In Prior)
         Incorporate   a variable amount of
          borrowing—through modeling
         Many examples. One, mixtures:
          qx ~ p Beta(2.5,7.5)
                     + (1-p)I[qx=qy]
                                               6
   BERRY
CONSULTANTS
STATISTICAL INNOVATION
                            Actual example
                  Historical data on cases:
                  Control: XH ~ Bin(81, qH) (XH=14)
                  TRT: YH ~ Bin(71, pH) (YH=5)

                           qH = log(qH/(1–qH))
                         dH + qH = log(pH/(1–pH))
                            qH ~ N(–1.8, 1)
                             dH ~ N(0, 1)
                                                      7
   BERRY
CONSULTANTS
STATISTICAL INNOVATION
                                 RCT Data
                         Control: X ~ Bin(153, q)
                         TRT: Y ~ Bin(153, p)

                               q = log(q/(1-q))
                             q + d = log(p/(1-p))
                                 q ~ N(qH, 1)
                         d ~ (.5)N(0, 1) + (.5)I[q=qH]
                                                         8
   BERRY
CONSULTANTS
STATISTICAL INNOVATION
                                 Results
                         #1: X=25, Y=10 (z=2.69)
                              P(d=dH|X,Y) = 0.76
                              P(d<0|X,Y) = 0.998
                         #2: X=25, Y=16 (z=1.51)
                              P(d=dH|X,Y) = 0.70
                               P(d<0|X,Y) = 0.98
                                                   9
   BERRY
CONSULTANTS
STATISTICAL INNOVATION
―Heat‖
Image




                         10
   BERRY
CONSULTANTS
STATISTICAL INNOVATION
                                   OUTLINE
                    Usinghistorical data
                    Bayesian multiple comparisons
                          Attitudes
                          Methodology

                    Application    of hierarchical
                         modeling to ―Tier 2‖ safety data

                                                            11
   BERRY
CONSULTANTS
STATISTICAL INNOVATION
                             Multiplicities

                         Two types of Bayesian
                          contributions:
                          Attitude/philosophy
                          Methodology



                                                 12
   BERRY
CONSULTANTS
STATISTICAL INNOVATION
                     Multiplicities problematic
                           for Bayesians
                   Multiple comparisons
                    (relation to frequentist solution)
                   Variable selection in regression
                   Transformations in regression
                   Subgroup analyses
                   Data dredging
                   Silent multiplicities
                                                         13
   BERRY
CONSULTANTS
STATISTICAL INNOVATION
                      Multiplicities not
                  problematic for Bayesians
                      Metaanalysis
                      Multicenter trials
                      Multiple test statistics
                      Repeated significance testing
                       (interim analysis)

                                                       14
   BERRY
CONSULTANTS
STATISTICAL INNOVATION
Need for subjectivity in handling
multiplicities: Subgroup analyses
                  Examining    outcomes in subsets
                   of patients is
                    Natural
                    Compelling
                    Inevitable
                    Dangerous!
                  If there are subgroup
                   differences, you can’t find them
                   if you don’t look!
                  But Type I error rates increase    15
   BERRY
CONSULTANTS
STATISTICAL INNOVATION
                         Type I vs Type II
                  8-team single elimination tournament
                   Results:     4 teams: 0-1
                                 2 teams: 1-1
                                 1 team: 2-1
                                 1 team: 3-0
                   Type I: Some team had to be 3-0;
                    nothing special about the winner
                   Type II: Take records at face value;
                    3-0 team most likely best
                                                           16
   BERRY
CONSULTANTS
STATISTICAL INNOVATION
      Bayesian choice: Type I or Type II?
                     In   tournament:
                           What’s the game?
                           Who’s playing?
                     In   science or medicine:
                           What’s the context?
                           What’s the other evidence?
                           What’s the biology?
                                                        17
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CONSULTANTS
STATISTICAL INNOVATION
                         ―Silent multiplicities‖

                         Which is northernmost city:
                          Portland, OR?
                          Portland, ME?
                          Milan, Italy?
                          Vladivostok, Russia?



                                                       18
   BERRY
CONSULTANTS
STATISTICAL INNOVATION
                         ―Silent multiplicities‖

                  Numerical order of northernness:
                   2. Portland, OR
                   3. Portland, ME
                   1. Milan, Italy
                   4. Vladivostok, Russia


                                                     19
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CONSULTANTS
STATISTICAL INNOVATION
                         inevitable illusions ?
          Piattelli-Palmarini:―I have just
           tossed a coin 7 times.‖ Which did I
           get?        1: THHTHTT
                       2: TTTTTTT
          Most people say 1. But ―the
           probabilities are totally even‖
          Most people are right. He’s totally
           wrong!
                                                  20
   BERRY
CONSULTANTS
STATISTICAL INNOVATION
                         THHTHTT or TTTTTTT?
            LR = Bayes factor of 1 over 2 =
                     P(wrote 1&2 | Obs 1)
                     P(wrote 1&2 | Obs 2)
            LR = (1/2)/(1/128) = 64?
             In any case, much greater than 1
            LR = 64  P(Obs 2 | wrote 1&2) = 64/65
            [Probs equal if he used the coin to
             generate the alternative sequence]
                                                      21
   BERRY
CONSULTANTS
STATISTICAL INNOVATION
            Bayesian multiple comparisons
              using hierarchical models
                         Apply to (units in parens):
                          Metaanalysis (studies)
                          Multicenter trials (centers)
                          Pharmacokinetics (patients)
                          Multiple comparisons (treatments)
                          Variable selection in regression
                           (coefficients)
                          Subgroup analyses (subsets)
                          Data dredging (subsets, interactions, & ?)

                                                                    22
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CONSULTANTS
STATISTICAL INNOVATION
                                Application:
                            ―Tier 2‖ safety data

                          Example   1 (Mehrotra & Heyse)
                          Vaccine

                          40   AEs within 8 body systems
                          Which   to flag?

                                                            23
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CONSULTANTS
STATISTICAL INNOVATION
              George Chi, H.M. James Hung,
               Robert O’Neill (FDA CDER)
                   ―Safety assessment is one area
                   where frequentist strategies have
                   been less applicable. Perhaps
                   Bayesian approaches in this area
                   have more promise.‖

                   (Pharmaceutical Report, 2002)
                                                       24
   BERRY
CONSULTANTS
STATISTICAL INNOVATION
                                    Example 1
                     #   BS   AE                       y (/148) x (/132)   P-value
                     1    1   Asthenia/Fatigue          57       40        0.17
                     2    1   Fever                     34       26        0.56
                     3    1   Infection, fungal           2       0        0.50
                     4    1   Infection, viral            3       1        0.62
                     5    1   Malaise                   27       20        0.52
                     6    3   Anorexia                    7       2        0.18
                     7    3   Candidiasis, oral           2       0        0.50
                     8    3   Constipation                2       0        0.50
                     9    3   Diarrhea                  24       10        0.029*
                    10    3   Gastroenteritis, infect.    3       1        0.62
                    11    3   Nausea                      2       7        0.090
                    12    3   Vomiting                  19       19        0.73
                    13    5   Lymphadenopathy             3       2        1
                    14    6   Dehydration                 0       2        0.22
                    15    8   Crying                      2       0        0.50
                    16    8   Insomnia                    2       2        1
                    17    8   Irritability              75       43        0.0025***
                    18    9   Bronchitis                  4       1        0.37
                    19    9   Congestion, nasal           4       1        0.37
                    20    9   Congestion, respiratory 1           2        0.60
                                                                                       25
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CONSULTANTS
STATISTICAL INNOVATION
                                   Example 1 (cont’d)
                    #      BS      AE                       y (/148) x (/132)   P-value
                    21      9      Cough                      13       8        0.50
                    22      9      Infect., respiratory,upper 28      20        0.43
                    23      9      Laryngotracheobronch        2       1        1
                    24      9      Pharyngitis                13       8        0.50
                    25      9      Rhinorrhea                 15      14        1
                    26      9      Sinusitis                   3       1        0.62
                    27      9      Tonsillitis                 2       1        1
                    28      9      Wheezing                    3       1        0.62
                    29     10      Bite/sting, non-venom.      4       0        0.12
                    30     10      Eczema                      2       0        0.50
                    31     10      Pruritus                    2       1        1
                    32     10      Rash                       13       3        0.021*
                    33     10      Rash, diaper                6       2        0.29
                    34     10      Rash, meas./rubella-like 8          1        0.039*
                    35     10      Rash, varicella-like        4       2        0.69
                    36     10      Urticaria                   0       2        0.22
                    37     10      Viral exanthema             1       2        0.50
                    38     11      Conjunctivitis              0       2        0.22
                    39     11      Otitis media               18      14        0.71
                    40     11      Otorrhea                    2       1        1
                                                                                          26
   BERRY
CONSULTANTS              * DFDR = 0.15 *** DFDR = 0.05 (or 0.10) of Mehrotra&Heyse
STATISTICAL INNOVATION
                           Observed proportions
                                 0.6
                                                     Irritability
                                 0.5

                                 0.4
                         y/148                                      No difference
                                 0.3

                                 0.2

                                 0.1

                                   0
                                       0   .1   .2      .3     .4   .5    .6
                                                                                    27
   BERRY
CONSULTANTS
                                                       x/132
STATISTICAL INNOVATION
                           Observed proportions
                                 0.6
                                                             Log odds contour
                                 0.5

                                 0.4
                         y/148                                    No difference
                                 0.3

                                 0.2

                                 0.1

                                   0
                                       0   .1   .2    .3     .4   .5    .6
                                                                                  28
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CONSULTANTS
                                                     x/132
STATISTICAL INNOVATION
             Levels of experimental units

                  Body      systems
                  Adverse         effects within body
                         system
                  Patient        (depending on treatment)
                  3-way      borrowing

                                                             29
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CONSULTANTS
STATISTICAL INNOVATION
                                        Model
                 B body systems
                 ki adverse effects within body system i
                 Data: For AEij, i = 1, . . ., B; j=1, . . .,ki
                         Control: xij events in nC patients
                         Treatment: yij events in nT patients
                 H0: cij = tij, where cij & tij are event rates
                 logit(cij) = gij
                 logit(tij) = gij + qij

                                                                  30
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CONSULTANTS
STATISTICAL INNOVATION
                                     Model 1
                         gij ~ N(mgi, sgi2)
                                mgi ~ N(mg0, tg02)
                                         mg0 ~ N(mg00, tg002)
                                       tg02 ~ IG(atg,btg)
                                sgi2 ~ IG(asg,bsg)


                                                                31
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CONSULTANTS
STATISTICAL INNOVATION
                                     Model 1
                         qij ~ N(mqi, sqi2)
                                mqi ~ N(mq0, tq02)
                                        mq0 ~ N(mq00, tq002)
                                      tq02 ~ IG(atq,btq)
                                sqi2 ~ IG(asq,bsq)


                                                               32
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CONSULTANTS
STATISTICAL INNOVATION
                             Model 2: Mixtures

                     Associated     with body system i
                         is a probability pi that drug has
                         no effect on AEs in that system
                     In   previous model each pi = 0
                     Assume      pi ~ Beta(1, 3)

                                                             33
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CONSULTANTS
STATISTICAL INNOVATION
                                    Model 2

                         qij ~ pi I{0} + (1–pi)N(mqi, sqi2)
                                 mqi ~ N(mq0, tq02)
                                        mq0 ~ N(mq00, tq002)
                                       tq02~IG(atq,btq)
                                 sqi2~IG(asq,bsq)
                                pi ~ Beta(a, b)

                                                               34
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CONSULTANTS
STATISTICAL INNOVATION
                         Parameter values assumed
                             gij ~ N(mgi, sgi2)
                                    mgi ~ N(mg0, tg02)
                                            mg0 ~ N(0, 102)
                                           tg02 ~IG(3, 1)
                                    sgi2~IG(3, 1)
                             qij ~ N(mqi, sqi2)
                                    mqi ~ N(mq0, tq02)
                                            mq0 ~ N(0, 102)
                                           tq02 ~IG(3, 1)
                                    sqi2~IG(3, 1)
                                                              35
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CONSULTANTS
STATISTICAL INNOVATION
            #       BS   AE                       y (of 148) x (of 132) P1(q>0) P2(q=0) P2(q>0)
            1       1    Asthenia/Fatigue               57       40      0.956    0.56     0.44
            2       1    Fever                          34       26      0.89     0.77     0.21
            3       1    Infection, fungal               2        0      0.83     0.84     0.14
            4       1    Infection, viral                3        1      0.84     0.83     0.15
            5       1    Malaise                        27       20      0.89     0.75     0.23
            6       3    Anorexia                        7        2      0.90     0.88     0.11
            7       3    Candidiasis, oral               2        0      0.79     0.95     0.04
            8       3    Constipation                    2        0      0.78     0.96     0.03
            9       3    Diarrhea                       24       10      0.987    0.48     0.52
           10       3    Gastroenteritis, infectious     3        1      0.79     0.94     0.04
           11       3    Nausea                          2        7      0.46     0.94     0.01
           12       3    Vomiting                       19       19      0.70     0.94     0.04
           13       5    Lymphadenopathy                 3        2      0.77     0.59     0.24
           14       6    Dehydration                     0        2      0.51     0.56     0.11
           15       8    Crying                          2        0      0.86     0.58     0.34
           16       8    Insomnia                        2        2      0.80     0.63     0.24
           17       8    Irritability                   75       43      0.999    0.02     0.981
           18       9    Bronchitis                      4        1      0.86     0.99     0.01
           19       9    Congestion, nasal               4        2      0.86     0.99     0.00
           20       9    Congestion, respiratory         1        2      0.73     0.99     0.00

                                                                                                   36
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CONSULTANTS
STATISTICAL INNOVATION
           #        BS   AE                      y (of 148) x (of 132) P1(q>0) P2(q=0) P2(q>0)
           21       9    Cough                         13        8      0.92     0.97     0.03
           22       9    Infection,respiratory,upper 28         20      0.943    0.97     0.03
           23       9    Laryngotracheobronchitis       2        1      0.80     0.99     0.00
           24       9    Pharyngitis                   13        8      0.91     0.98     0.02
           25       9    Rhinorrhea                    15       14      0.83     0.99     0.01
           26       9    Sinusitis                      3        1      0.84     0.99     0.00
           27       9    Tonsillitis                    2        1      0.81     0.99     0.00
           28       9    Wheezing                       3        1      0.84     0.99     0.00
           29      10    Bite/sting, non-venomous       4        0      0.93     0.90     0.10
           30      10    Eczema                         2        0      0.84     0.96     0.04
           31      10    Pruritus                       2        1      0.82     0.97     0.03
           32      10    Rash                          13        3      0.997    0.42     0.58
           33      10    Rash, diaper                   6        2      0.946    0.88     0.12
           34      10    Rash, measles/rubella-like     8        1      0.976    0.67     0.33
           35      10    Rash, varicella-like           4        2      0.87     0.93     0.07
           36      10    Urticaria                      0        2      0.62     0.97     0.01
           37      10    Viral exanthema                1        2      0.71     0.97     0.02
           38      11    Conjunctivitis                 0        2      0.50     0.78     0.05
           39      11    Otitis media                  18       14      0.82     0.73     0.23
           40      11    Otorrhea                       2        1      0.71     0.80     0.10

                                                                                                 37
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CONSULTANTS
STATISTICAL INNOVATION
                                      Body Systems
                                 p                   mq              sq
           Body          MEAN        STDEV   MEAN    STDEV   MEAN    STDEV
              1          0.528       0.225   0.182   0.159   0.323    0.088
              3          0.642       0.176   0.159   0.168   0.355    0.102
              5          0.316       0.209   0.146   0.207   0.375    0.127
              6          0.312       0.210   0.104   0.216   0.398    0.151
              8          0.320       0.201   0.242   0.189   0.387    0.126
              9          0.794       0.105   0.131   0.152   0.305    0.075
              10         0.665       0.176   0.240   0.183   0.387    0.122
              11         0.470       0.221   0.102   0.188   0.377    0.131



                                                                              38
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CONSULTANTS
STATISTICAL INNOVATION
                                    Example 2
                         AE   BS x (/340) y (/340) Z P1(q>0) P2(q>0)
                          1    1      1      5    1.64 0.72   0.00
                          2    1      4      6    0.64 0.71   0.00
                          3    1      5      6    0.30 0.66   0.00
                          4    1     21    30     1.31 0.90   0.02
                          5    1     27    21 –0.90 0.41      0.00
                          6    1     82    94     1.05 0.90   0.03
                          7    1   136 137        0.08 0.65   0.00
                          8    1      7      9    0.51 0.71   0.00
                          9    1      9      8 –0.25 0.57     0.00
                         10    1     18    17 –0.17 0.61      0.00
                         11    1      6      7    0.28 0.67   0.00
                         12    1     36    47     1.29 0.91   0.03
                         13    1     52    77     2.45 0.993 0.30
                         14    1     14    10 –0.83 0.46      0.00
                         15    1      5      5    0    0.59   0.00
                         16    1      8    11     0.70 0.74   0.00
                         17    2      5      4 –0.34 0.45     0.00
                         18    2      5      1 –1.64 0.29     0.00
                         19    2      3      6    1.01 0.69   0.00
   BERRY                 20    2      4      6    0.64 0.65   0.00     39
CONSULTANTS
STATISTICAL INNOVATION
                                    Example 2
                         AE   BS x (/340)   y (/340) Z P1(q>0) P2(q>0)
                         21    2      5       0    –2.24 0.20   0.00
                         22    2      3       6     1.01 0.73   0.00
                         23    2      5       4    –0.34 0.49   0.00
                         24    2      6      11     1.23 0.82   0.01
                         25    2    40       64     2.56 0.994 0.59
                         26    2    19       23     0.64 0.81   0.02
                         27    2    13       12    –0.20 0.58   0.00
                         28    2      7      14     1.55 0.88   0.02
                         29    2      4       5     0.34 0.61   0.00
                         30    2      5       5     0    0.54   0.00
                         31    2      5       1    –1.64 0.29   0.00
                         32    3      1       5     1.64 0.76   0.02
                         33    3      4       5     0.34 0.64   0.01
                         34    3    258      252 –0.53 0.47     0.01
                         35    3      3       5     0.71 0.66   0.01
                         36    3    39       35    –0.49 0.45   0.01
                         37    3      5       9     1.08 0.77   0.02
                         38    3      5       5     0    0.53   0.01
                         39    3      5       5     0    0.58   0.00
   BERRY                 40    4    37       30    –0.90 0.28   0.00     40
CONSULTANTS
STATISTICAL INNOVATION
                                    Example 2
                         AE   BS x (/340)   y (/340) Z P1(q>0) P2(q>0)
                         41    4      5       4    –0.34 0.39   0.00
                         42    4    76       86     0.90 0.83   0.06
                         43    4      5       4    –0.34 0.43   0.00
                         44    4    11        9    –0.45 0.45   0.00
                         45    4      6       7     0.28 0.50   0.01
                         46    4    58       60     0.20 0.64   0.03
                         47    5    17       16    –0.18 0.49   0.04
                         48    5      2       5     1.14 0.61   0.06
                         49    5    23       21    –0.31 0.48   0.04
                         50    5    97      109     1.00 0.88   0.21
                         51    5    29       25    –0.57 0.43   0.03
                         52    6    16       12    –0.77 0.41   0.05
                         53    6    22       29     1.02 0.83   0.21
                         54    6    118      121    0.24 0.70   0.13
                         55    6    10       12     0.43 0.67   0.11
                         56    7      6       3    –1.01 0.31   0.10
                         57    8      5       3    –0.71 0.34   0.12
                         58    9      6       2    –1.42 0.25   0.08

                                                                         41
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CONSULTANTS
STATISTICAL INNOVATION
                                           Example 2
                               0.8
                               0.7
                               0.6

                               0.5
                         y/340 0.4

                               0.3
                               0.2

                               0.1

                               0.0
                                     .0   .1   .2   .3     .4 .5   .6   .7   .8
                                                         x/340
                                                                                  42
   BERRY
CONSULTANTS
STATISTICAL INNOVATION
                     AE25 in Example 2, Model 2

            AE           BS   x    y     Z   P2(q>0) P2(q>0)
            25           2    40   64   2.56  0.59    0.96
            251          2    40   69   3.03  0.86    0.98
            252          2    40   74   3.49  0.95    0.996
            253          2    40   79   3.94  0.99    0.998

            In body system with 14 neutral AEs
            In body system by itself!
                                                          43
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CONSULTANTS
STATISTICAL INNOVATION
                         Today’s Schedule
                8:00 Introduction
                8:15 Intro to Bayesian approach
               10:00 Break
               10:15 Hierarchical Distributions
               11:30 Adaptive Trial designs, part 1
               12:00 Lunch
                1:00 Adaptive Trial designs, part 1
                2:00 Adaptive Trial designs, part 2
                3:00 Break
                3:15 Multiplicities
                4:00 Decision analysis
                5:00 Adjourn

                                                      44
   BERRY
CONSULTANTS
STATISTICAL INNOVATION

								
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