ADAPTIVE BAYESIAN DESIGNS FOR DOSE RANGING DRUG

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```					      WHY BAYES?
INNOVATIONS IN CLINICAL
TRIAL DESIGN & ANALYSIS
Donald A. Berry
dberry@mdanderson.org
Conclusion These data add to the growing evidence that
supports the regular use of aspirin and other NSAIDs … as
effective chemopreventive agents for breast cancer.

2
Results Ever use of aspirin or other NSAIDs … was
reported in 301 cases (20.9%) and 345 controls (24.3%)
(odds ratio 0.80, 95% CI 0.66-0.97).

3
Bayesian analysis?
Bayesian analysis of
 Naïve
―Results‖ is wrong
name
 Anynaïve frequentist
analysis is also wrong
4
What is Bayesian analysis?
Bayes' theorem:
'( q| X )  (q) * f( X | q )
 Assess prior  (subjective,
include available evidence)
 Construct model f for data

5
Implication:
The Likelihood Principle
Where X is observed data,
the likelihood function
LX(q) = f( X | q )
contains all the information
in an experiment relevant for
6
 Shortversion of LP:
Take data at face value
 Data:
 Among cases:       301/1442
 Among controls:    345/1420
 But   ―Data‖ is deceptive
 These   are not the full data
7
The data
 Methods:
 ―Population-based  case-control
study of breast cancer‖
 ―Study design published previously‖

 Aspirin/NSAIDs?  (2.25-hr ?naire)
 Includes superficial data:
 Among cases:        301/1442
 Among controls:     345/1420
 Other   studies (& fact published!!)
8
Silent multiplicities

 Are the most difficult problems in
statistical inference
render what we do irrelevant
 Can
—and wrong!

9

Which city is furthest north?
 Portland,   OR
 Portland,   ME
 Milan,   Italy

10
Beating a dead horse . . .
 Piattelli-Palmarini (inevitable illusions)
asks: ―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!
 Data: He presented us with 1 & 2!
11
THHTHTT or TTTTTTT?
   LR = Bayes factor of 1 over 2 =
P(Wrote 1&2 | Got 1)
P(Wrote 1&2 | Got 2)
 LR > 1  P(Got 1 | Wrote 1&2) > 1/2
 Eg: LR = (1/2)/(1/42) = 21 
P(Got 1 | Wrote 1&2) = 21/22 = 95%
 [Probs ―totally even‖ if a coin was used
to generate the alternative sequence]
12
Marker/dose interaction
Marker negative Marker positive
1.0                                          1.0
0.9                                          0.9
0.8                       Std (96)           0.8                             Hi (38)
0.7                                          0.7
D                                            D
F 0.6               Low (93)                 F 0.6                           Std (41)
S 0.5                          Hi (95)       S 0.5
0.4                                          0.4
Low (36)
0.3                                          0.3
0.2                                          0.2
0.1                                          0.1
0.0                                          0.0
0   1   2   3 4        5    6    7           0   1   2   3 4     5      6        7
Years                                        Years
13
Proportional hazards model
Variable      Comp       RelRisk     P
#PosNodes     10/1        2.7      <0.001
MenoStatus    pre/post    1.5       0.05
TumorSize     T2/T1       2.6      <0.001
Dose          ––          ––        NS
Marker        50/0        4.0      <0.001
MarkerxDose   ––          ––       <0.001
This analysis is wrong!
14
Data at face value?
 How   identified?
 Why am I showing you these
results?
 What   am I not showing you?
 What   related studies show?

15
Solutions?
 Short answer: I don’t know!
 A solution:
 Superviseexperiment yourself
 Become an expert on substance
 Partial   solution:
 Supervise supervisors
 Learn as much substance as you can
 Danger: You risk projecting
yourself as uniquely scientific
16
A consequence

 Statisticians   come to believe
NOTHING!!

17
OUTLINE
 Silentmultiplicities
 Bayes and predictive probabilities
 Bayes as a frequentist tool
 Investigating many phase      II drugs
 Seamless Phase II/III trial
 Extraim analysis
 Trial   design as decision analysis        18
Bayes in pharma
and FDA …

19
http://www.cfsan.fda.gov/~frf/bayesdl.html
http://www.prous.com/bayesian2004/
BAYES AND
PREDICTIVE PROBABILITY

component of
 Critical
experimental design

 In   monitoring trials

23
Example calculation
 Data:   13 A's and 4 B's

 Likelihood    p13 (1–p)4

24
Posterior density of p
for uniform prior: Beta(14,5)

13   4
p (1–p)

0   .1   .2   .3   .4   .5   .6   .7   .8       .9   1
p            25
Laplace’s rule of
succession
P(A wins next pair | data)
= EP(A wins next pair | data, p)
= E(p | data)
= mean of Beta(14, 5)
= 14/19
26
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
27
Suppose 17 more
observations
P(A wins x of 17 | data)
= EP(A wins x | data, p)
=E   [( )
17
x    p x(1–p)17–x   | data, p]
         28
Best fitting binomial vs.
predictive probabilities
Binomial, p=14/19

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

Predictive, p ~ beta(14,5)

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

29
Comparison of predictive
with posterior
13   4
.15                   p (1–p)

.10

.05

.00
1    3
0 0 .1 2 .2 4 .35   6.47   8 9 10 11 .7 13 .8 15 .9 17 1
12     14    16
.5 .6
p            30
Example: Baxter’s DCLHb &
predictive probabilities
 Diaspirin
 Blood substitute; emergency trauma
 Randomized controlled trial (1996+)
 Treatment:  DCLHb
 Control: saline
 N = 850 (= 2x425)
 Endpoint: death

31
 Waiver of informed consent
 Data Monitoring Committee
 First DMC meeting:
DCLHb      Saline
Alive      28         33
Total      49         41
 P-value?   No formal interim analysis
32
Predictive probability of
future results (after n = 850)
 Probability of significant
survival benefit for DCLHb
after 850 patients: 0.00045
 DMC   paused trial: Covariates?
 No   imbalance
 DMC   stopped trial
33
OUTLINE
 Silentmultiplicities
 Bayes and predictive probabilities
 Bayes as a frequentist tool
 Investigating many phase      II drugs
 Seamless Phase II/III trial
 Extraim analysis
 Trial   design as decision analysis        34
BAYES AS A
FREQUENTIST TOOL
 Design  a Bayesian trial
 Check operating characteristics
 Adjust design to get  = 0.05
  frequentist design
 That’s fine!
 We have 50+ such trials at MDACC

35
OUTLINE
 Silentmultiplicities
 Bayes and predictive probabilities
 Bayes as a frequentist tool
 Investigating many phase      II drugs
 Seamless Phase II/III trial
 Extraim analysis
 Trial   design as decision analysis        36
 Look at accumulating data …
without blushing
 Update probabilities
 Find predictive probabilities
 Modify future course of trial
 Give details in protocol
 Simulate to find operating
characteristics
37
OUTLINE
 Silentmultiplicities
 Bayes and predictive probabilities
 Bayes as a frequentist tool
 Investigating many phase      II drugs
 Seamless Phase II/III trial
 Extraim analysis
 Trial   design as decision analysis        38
Giles, et al JCO (2003)
 Troxacitabine(T) in acute myeloid
leukemia (AML) when combined with
cytarabine (A) or idarubicin (I)
IA vs TA vs TI
 Max   n = 75
 End   point: CR (time to CR < 50 days)
39
Randomization

 Assign1/3 to IA (standard)
throughout (unless only 2 arms)
 Adaptive to TA and TI based on
current results
 Final   results   
40
Patient   Prob IA   Prob TA   Prob TI   Arm   CR<50
1       0.33      0.33      0.33     TI     not
2       0.33      0.34      0.32     IA     CR
3       0.33      0.35      0.32     TI     not
4       0.33      0.37      0.30     IA     not
5       0.33      0.38      0.28     IA     not
6       0.33      0.39      0.28     IA     CR
7       0.33      0.39      0.27     IA     not
8       0.33      0.44      0.23     TI     not
9       0.33      0.47      0.20     TI     not
10       0.33      0.43      0.24     TA     CR
11       0.33      0.50      0.17     TA     not
12       0.33      0.50      0.17     TA     not
13       0.33      0.47      0.20     TA     not
14       0.33      0.57      0.10     TI     not
15       0.33      0.57      0.10     TA     CR
16       0.33      0.56      0.11     IA     not
17       0.33      0.56      0.11     TA     CR
41
Patient    Prob IA   Prob TA   Prob TI   Arm   CR<50
18        0.33      0.55      0.11     TA     not
19        0.33      0.54      0.13     TA     not
20        0.33      0.53      0.14     IA     CR
21        0.33      0.49      0.18     IA     CR
22        0.33      0.46      0.21     IA     CR
IA
Drop 23        0.33      0.58      0.09            CR
24        0.33      0.59      0.07     IA     CR
TI 25         0.87      0.13        0      IA     not
26        0.87      0.13        0      TA     not
27        0.96      0.04        0      TA     not
28        0.96      0.04        0      IA     CR
29        0.96      0.04        0      IA     not
30        0.96      0.04        0      IA     CR
31        0.96      0.04        0      IA     not
32        0.96      0.04        0      TA     not
33        0.96      0.04        0      IA     not
34        0.96      0.04        0      IA     CR
42
Compare n = 75
Summary of results

CR rates:
 IA: 10/18 = 56%
 TA: 3/11 = 27%
 TI:   0/5 = 0%
Criticisms . . .
43
OUTLINE
 Silentmultiplicities
 Bayes and predictive probabilities
 Bayes as a frequentist tool
 Investigating many phase      II drugs
 Seamless Phase II/III trial
 Extraim analysis
 Trial   design as decision analysis        44
allocation of therapies
 Design for phase II: Many drugs
endpoint is tumor response
 Goals:
   Treat effectively
   Learn quickly
45
Comparison:
Standard designs
 One drug (or dose) at a time;
no drug/dose comparisons
 Typical comparison by null
hypothesis:
response rate = 20%
 Progress is slow!

46
Standard designs
 One   stage, 14 patients:
 If 0 responses then stop
 If ≥ 1 response then phase III

 Two stages, first stage 20
patients:
 If≤ 4 or ≥ 9 responses then stop
 Else second set of 20 patients
47
 When   assigning next patient, find
r = P(rate ≥ 20%|data) for each drug
[Or, r = P(drug is best|data)]
 Assign drugs in proportion to r
 Add drugs as become available
 Drop drugs that have small r
 Drugs with large r  phase III

48
Suppose 10 drugs, 200 patients
  9 drugs have mix of response rates
20% & 40%, 1 (―nugget‖) has 60%
 Standard 2-stage design finds nugget
with probability < 70% (After 110
patients on average)
 Adaptive design finds nugget with
probability > 99% (After about 50
patients on average)
 Adaptive also better at finding 40%

49
Suppose 100 drugs, 2000 patients
 99 drugs have mix of response rates
20% & 40%, 1 (―nugget‖) has 60%
 Standard 2-stage design finds nugget
with probability < 70% (After 1100
patients on average)
 Adaptive design finds nugget with
probability > 99% (After about 500
patients on average)
 Adaptive also better at finding 40%

50
Consequences
 Recall   goals:
(1) Treat effectively
(2) Learn quickly
 Attractive   to patients, in and out
of the trial
 Better drugs identified faster;
move through faster
51
OUTLINE
 Silentmultiplicities
 Bayes and predictive probabilities
 Bayes as a frequentist tool
 Investigating many phase      II drugs
 Seamless Phase II/III trial
 Extraim analysis
 Trial   design as decision analysis        52
Example: Seamless phase II/III

 Drug   vs placebo, randomized
 Local  control (or biomarker, etc):
early endpoint related to survival?
 May   depend on treatment

53
*Inoue et al (2002 Biometrics)
Conventional drug development
Survival
control           No survival
No local
control Stop
Phase II               Phase III

6 mos   9-12 mos       > 2 yrs

Seamless phase II/III
< 2 yrs (usually)                       54
Seamless phases
 Phase II: Two centers; 10 pts/mo.
drug vs placebo. If predictive
probabilities look good, expand to
 Phase III: Many centers; 40+ pts/mo.
(Initial centers accrue during set-up)
 Max sample size: 900

[Single trial: survival data from both
phases combined in final analysis]
55
Early stopping
 Use predictive probs of stat. signif.
 Frequent analyses (total of 18)
using predictive probabilities:
 To switch to Phase III
 To stop accrual
 For futility
 For efficacy

 To   submit NDA
56
Comparisons

Conventional Phase III designs:
Conv4 & Conv18, max N = 900

57
Expected N under H0
1000
884
855
800

600

431
400

200

0
Bayes   Conv4   Conv18

58
Expected N under H1
1000
887      888

800

649
600

400

200

0
Bayes   Conv4   Conv18
59
Benefits
 Durationof drug development is
design:
 Fewer  patients in trial
 No hiatus for setting up phase III
 Use all patients to assess phase III
endpoint and relationship between
local control and survival
60
Possibility of large N
N   seldom near 900
 When   it is, it’s necessary!
 Thispossibility gives Bayesian
design its edge
[Other reason for edge is
modeling local control/survival]

61
OUTLINE
 Silentmultiplicities
 Bayes and predictive probabilities
 Bayes as a frequentist tool
 Investigating many phase      II drugs
 Seamless Phase II/III trial
 Extraim analysis
 Trial   design as decision analysis        62
*

*Berry, et al. Case Studies in Bayesian Statistics 2001
Example: Stroke and
 Adaptive   doses in Phase II setting:
dose-response relationship
 Pfizer trial of a neutrofil inhibitory
factor; results recently announced
 Endpoint: stroke scale at week 13
 Early endpoints: weekly stroke scale
64
Standard Parallel Group Design
Equal sample sizes at each of k
doses.

Doses             65
True dose-response curve
(unknown)
Response

Doses          66
Observe responses (with error)
at chosen doses
Response

Doses          67
Dose at which 95% max effect
Response

True ED95

Doses               68
Response

True ED95

Dose
?    69
Response

Dose
?   70
Solution:
Increase number of doses
Response

ED95

71
Doses
But, enormous sample size, and . . .
wasted dose assignments—always!
Response

ED95

72

Doses
 Observe  data continuously
 Select next dose to maximize
available evidence
 Stop dose-ranging trial when
know ED95 & response at ED95
―sufficiently well‖

73
Our approach (cont’d)
Info accrues         Longitudinal Model
Copenhagen Stroke Database

40
patient;        20

prediction      10

using            0

-10
longitudinal    -20

model           -30
-40 -30 -20 -10 0 10 20 30 40 50
Difference from baseline in SSS week 3

74
Our approach (cont’d)

 Modeldose-response
(borrow strength from
neighboring doses)
 Many   doses (logistical issues)

75
Possible decisions each day:
 Stop trial and drug’s development
 Stop and set up confirmatory trial
 Continue dose-finding (what dose?)

Size of confirmatory trial based on
info from dose-ranging phase
Choices by decision analysis (Human
safeguard: DSMB)
76
Dose-response trial
 Learn efficiently and rapidly about
dose-response; if + go to Phase III
 Assign dose to maximize info
given current info
 Use predictive probabilities, based
on early endpoints
 Doses in continuum, or preset grid
77
Dose-response trial (cont’d)
dose-ranging when know
 Halt
dose sufficiently well
 Seamless   switch from dose-
ranging to confirmatory trial—
2 trials in 1!

78
standard design
 Fewer  patients (generally);
faster & more effective learning
 Better at finding ED95
 Tends to treat patients in trial
more effectively
 Drops duds early —actual trial!

79
Dose-assignment
simulation
 Assumes  particular dose-
response curve
 Assumes SD = 12

 Shows weekly results, several
patients at a time (green circles)

80
Prior
30
25
20
E(f)

15
10
5
0

0.0   0.5           1.0   1.5
81
DOSE
DATA
30
25
20
15
Y

10
5
0

0.0       0.5                        1.0       1.5
82
DOSE
green=obs, blue=imputed, black=true mn
DATA
30
25
20
15
Y

10
5
0

0.0       0.5                        1.0       1.5
83
DOSE
green=obs, blue=imputed, black=true mn
DATA
30
25
20
15
Y

10
5
0

0.0       0.5                        1.0       1.5
84
DOSE
green=obs, blue=imputed, black=true mn
DATA
30
25
20
15
Y

10
5
0

0.0       0.5                        1.0       1.5
85
DOSE
green=obs, blue=imputed, black=true mn
DATA
30
25
20
15
Y

10
5
0

0.0       0.5                        1.0       1.5
86
DOSE
green=obs, blue=imputed, black=true mn
DATA
30
25
20
15
Y

10
5
0

0.0       0.5                        1.0       1.5
87
DOSE
green=obs, blue=imputed, black=true mn
DATA
30
25
20
15
Y

10
5
0

0.0       0.5                        1.0       1.5
88
DOSE
green=obs, blue=imputed, black=true mn
DATA
30
25
20
15
Y

10
5
0

0.0       0.5                        1.0       1.5
89
DOSE
green=obs, blue=imputed, black=true mn
DATA
30
25
20
15
Y

10
5
0

0.0       0.5                        1.0       1.5
90
DOSE
green=obs, blue=imputed, black=true mn
DATA
30
25
20
15
Y

10
5
0

0.0       0.5                        1.0       1.5
91
DOSE
green=obs, blue=imputed, black=true mn
DATA
30
25
20
15
Y

10
5
0

0.0       0.5                        1.0       1.5
92
DOSE
green=obs, blue=imputed, black=true mn
DATA
30
25
20
15
Y

10
5
0

0.0       0.5                        1.0       1.5
93
DOSE
green=obs, blue=imputed, black=true mn
DATA
30
25
20
15
Y

10
5
0

0.0       0.5                        1.0       1.5
94
DOSE
green=obs, blue=imputed, black=true mn
DATA
30
25
20
15
Y

10
5
0

0.0       0.5                        1.0       1.5
95
DOSE
green=obs, blue=imputed, black=true mn
DATA
30
25
20
15
Y

10
5
0

0.0       0.5                        1.0       1.5
96
DOSE
green=obs, blue=imputed, black=true mn
DATA
30
25
20
15
Y

10
5
0

0.0       0.5                        1.0       1.5
97
DOSE
green=obs, blue=imputed, black=true mn
DATA
30
25
20
15
Y

10
5
0

0.0       0.5                        1.0       1.5
98
DOSE
green=obs, blue=imputed, black=true mn
DATA
30
25
20
15
Y

10
5
0

0.0       0.5                        1.0       1.5
99
DOSE
green=obs, blue=imputed, black=true mn
DATA
30
25
20
15
Y

10
5
0

0.0       0.5                        1.0       1.5
100
DOSE
green=obs, blue=imputed, black=true mn
DATA
30
25
20
15
Y

10
5
0

0.0       0.5                        1.0       1.5
101
DOSE
green=obs, blue=imputed, black=true mn
DATA
30
25
20
15
Y

10
5
0

0.0       0.5                        1.0       1.5
102
DOSE
green=obs, blue=imputed, black=true mn
DATA
30
25
20
15
Y

10
5
0

0.0       0.5                        1.0       1.5
103
DOSE
green=obs, blue=imputed, black=true mn
DATA
30
25
20
15
Y

10
5
0

0.0       0.5                        1.0       1.5
104
DOSE
green=obs, blue=imputed, black=true mn
DATA
30
25
20
15
Y

10
5
0

0.0       0.5                        1.0       1.5
105
DOSE
green=obs, blue=imputed, black=true mn
DATA
30
25
20
15
Y

10
5
0

0.0       0.5                        1.0       1.5
106
DOSE
green=obs, blue=imputed, black=true mn
DATA
30
25
20
15
Y

10
5
0

0.0       0.5                        1.0       1.5
107
DOSE
green=obs, blue=imputed, black=true mn
DATA
30
25
20
15
Y

10
5
0

0.0       0.5                        1.0       1.5
108
DOSE
green=obs, blue=imputed, black=true mn
DATA
30
25
20
15
Y

10
5
0

0.0       0.5                        1.0       1.5
109
DOSE
green=obs, blue=imputed, black=true mn
DATA
30
25
20
15
Y

10
5
0

0.0       0.5                        1.0       1.5
110
DOSE
green=obs, blue=imputed, black=true mn
DATA
30
25
20
15
Y

10
5
0

0.0       0.5                        1.0       1.5
111
DOSE
green=obs, blue=imputed, black=true mn
DATA
30
25
20
15
Y

10
5
0

0.0       0.5                        1.0       1.5
112
DOSE
green=obs, blue=imputed, black=true mn
DATA
30
25
20
15
Y

10
5
0

0.0       0.5                        1.0       1.5
113
DOSE
green=obs, blue=imputed, black=true mn
DATA
30
25
20
15
Y

10
5
0

0.0       0.5                        1.0       1.5
114
DOSE
green=obs, blue=imputed, black=true mn
DATA
30
25
20
15
Y

10
5
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DATA
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DOSE
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DOSE
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DOSE
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DOSE
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DOSE
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DOSE
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Estimated
ED95
0

0.0       0.5                        1.0              1.5
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Confirmatory
Ao e s n
sD e il
s s km
i d y-
ge
ns
e b oa
nt
Wu o
ei
1.5    1.5
1.0
0.5
0.DOSE 1.0
0.5
0.0

0    0
1   0
2   0
3
134
E
E
K
W
Estimated f unctions
20
15

d:/data/build13/run11/
10
F

5
0

0.0   0.5               1.0   1.5
135
Z
Doses assigned across all simulations
0.14

20
0.12

15
0.10
0.08
Proportion

10
0.06
0.04

5
0.02
0.0

0
0   0.1   0.2   0.3   0.4   0.5   0.6   0.7   0.8   0.9   1   1.1   1.2   1.3   1.4   1.5
136
ASSIGNED DOSES
Black: median; Red: upper & lower quartiles; Green: Nominal
Estimated f unctions (no dose effect)
20
15

d:/data/build13/run12/
10
F

5
0

0.0   0.5               1.0                1.5
137
Z
Doses assigned across all simulations

20
0.25
0.20

15
0.15
Proportion

10
0.10

5
0.05
0.0

0
0   0.1   0.2   0.3   0.4   0.5   0.6   0.7   0.8   0.9   1   1.1   1.2   1.3   1.4   1.5
138
ASSIGNED DOSES
Black: median; Red: upper & lower quartiles; Green: Nominal
Consequences of Using
 Fundamental     change in the
way we do medical research
 More rapid progress
 We’ll get the dose right!
 Better treatment of patients
 . . . at less cost
139
Reactions
 FDA:  Positive. ―Makes coming to
work worthwhile.‖ ―In five years
all trials may be seamless.‖
 Pfizer   management: Enthusiastic
 Other    companies: Cautious

140
OUTLINE
 Silentmultiplicities
 Bayes and predictive probabilities
 Bayes as a frequentist tool
 Investigating many phase      II drugs
 Seamless Phase II/III trial
 Extraim analysis
 Trial   design as decision analysis        141
Example: Extraim analysis
 Endpoint: CR (detect 0.42 vs 0.32)
 80% power: N = 800
 Two extraim analyses, one at 800
 Another after up to 300 added pts
 Maximum n = 1400 (only rarely)
 Accrual: 70/month
 Delay in assessing response
142
 After 800 patients, have response
info on 450
 Find predictive probability of stat
significance when full info on 800
 Also when full info on 1400
 Continue if . . .
 Stop if . . .
 If continue, n via predictive power
 Repeat at second extraim analysis
143
Table 1: p0=0.42
p1     P(succ)   meanSS    sdSS    P(800)   P(1400)   P(succ1)   P(succ2)
0.37 0.0001        844.6    122.0   0.8707   0.0194    0.0001     0.0001
0.42 0.0243       1011.2    247.6   0.5324   0.2360    0.0084     0.0059
0.47 0.4467       1188.5    254.5   0.2568   0.5484    0.1052     0.0914
0.52 0.9389       1049.9    248.7   0.4435   0.2693    0.4217     0.2590
0.57 0.9989        874.2    149.1   0.7849   0.0268    0.7841     0.1729

Table 2: p0=0.32
p1     P(succ)   meanSS    sdSS    P(800)   P(1400)   P(succ1)   P(succ2)
0.27 0.0001        836.5    111.1   0.8937   0.0152    0.0005     0.0000
0.32 0.0284       1013.1    246.3   0.5238   0.2338    0.0094     0.0083
0.37 0.4757       1186.6    252.0   0.2513   0.5339    0.1083     0.1044
0.42 0.9545       1045.5    245.9   0.4485   0.2449    0.4316     0.2505
0.47 0.9989        922.7    181.0   0.6632   0.0258    0.6632     0.2111
vs 0.80
Table 3: p0=0.22
p1     P(succ)   meanSS    sdSS    P(800)   P(1400)   P(succ1)   P(succ2)
0.17 0.0000        827.7     95.3   0.9163   0.0086    0.0000     0.0000
0.22 0.0288       1013.3    246.6   0.5242   0.2340    0.0090     0.0062
0.27 0.5484       1199.0    246.3   0.2313   0.5392    0.1089     0.1063
0.32 0.9749       1074.4    234.8   0.3702   0.2030    0.3577     0.2065
0.37 0.9995       1024.7    205.4   0.4121   0.0508    0.3977     0.1685
OUTLINE
 Silentmultiplicities
 Bayes and predictive probabilities
 Bayes as a frequentist tool
 Investigating many phase      II drugs
 Seamless Phase II/III trial
 Extraim analysis
 Trial   design as decision analysis        145
Decision-analytic approach
 For each trial design …
 List possible results
 Calculate their predictive probabilities
 Evaluate their utilities

 Average    utilities by probabilities to
give utility of trial with that design
 Compare utilities of various designs
 Choose design with high utility
146
Choosing sample size*
 Special case of above
 One utility: Effective overall
treatment of patients, both those
 after the trial
 in the trial

 Example,dichotomous endpoint:
Maximize expected number of
successes over all patients
147
*Cheng et al (2003 Biometrika)
Compare Joffe/Weeks
JNCI Dec 18, 2002
―Many respondents viewed the main
societal purpose of clinical trials as
benefiting the participants rather than as
creating generalizable knowledge to
advance future therapy. This view, which
was more prevalent among specialists
such as pediatric oncologists that
enrolled greater proportions of patients
in trials, conflicts with established
principles of research ethics.‖
148
Maximize effective
treatment overall
 What    is ―overall‖?
patients who will be treated
 All
with therapies assessed in trial
 Call   it N, ―patient horizon‖
 Enough     to know mean of N
 Enough to know magnitude of N:
100? 1000? 1,000,000?
149
   Goal: maximize expected number of successes in N
   Either one- or two-armed trial
   Suppose n = 1000 is right for N = 1,000,000
   Then for other N’s use n =

150
Optimal allocations
in a two-armed trial

151
152
OUTLINE
 Silentmultiplicities
 Bayes and predictive probabilities
 Bayes as a frequentist tool
 Investigating many phase      II drugs
 Seamless Phase II/III trial