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(d0) 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
BERRY
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
BERRY
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
BERRY
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
BERRY
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
BERRY
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