why bayes? innovations in clinical trial design & analysis
DESCRIPTION
WHY BAYES? INNOVATIONS IN CLINICAL TRIAL DESIGN & ANALYSIS. Donald A. Berry [email protected]. 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. - PowerPoint PPT PresentationTRANSCRIPT
WHY BAYES?INNOVATIONS IN CLINICAL TRIAL DESIGN & ANALYSIS
Donald A. [email protected]
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Conclusion These data add to the growing evidence thatsupports the regular use of aspirin and other NSAIDs … aseffective chemopreventive agents for breast cancer.
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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).
4
Bayesian analysis? Naïve Bayesian analysis of
“Results” is wrong Gives Bayesians a bad
name Any naïve frequentist
analysis is also wrong
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What is Bayesian analysis?
Bayes' theorem:'( | X ) () * f( X | ) Assess prior (subjective,
include available evidence) Construct model f for data
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Implication: The Likelihood PrincipleWhere X is observed data, the likelihood function
LX() = f( X | )contains all the information in an experiment relevant for inferences about
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Short version 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
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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!!)
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Silent multiplicities
Are the most difficult problems in statistical inference
Can render what we do irrelevant—and wrong!
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Which city is furthest north? Portland, OR Portland, ME Milan, Italy Vladivostok, Russia
11
Beating a dead horse . . . Piattelli-Palmarini (inevitable illusions)
asks: “I have just tossed a coin 7 times.” Which did I get?
1: THHTHTT2: 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!
Piattelli-Palmarini (inevitable illusions) asks: “I have just tossed a coin 7 times.” Which did I get?
1: THHTHTT2: 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!
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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]
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0.00.10.20.30.40.50.60.70.80.91.0
0 1 2 3 4 5 6 7Years
DFS
Std (96)
Hi (95)
Low (93)
0.00.10.20.30.40.50.60.70.80.91.0
0 1 2 3 4 5 6 7Years
DFS
Std (41)
Hi (38)
Low (36)
Marker/dose interactionMarker/dose interactionMarker negative Marker positiveMarker negative Marker positive
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Proportional hazards modelVariable Comp RelRisk P#PosNodes 10/1 2.7 <0.001MenoStatus pre/post 1.5 0.05TumorSize T2/T1 2.6 <0.001Dose –– –– NSMarker 50/0 4.0 <0.001MarkerxDose –– –– <0.001
This analysis is wrong!
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Data at face value? How identified? Why am I showing you these
results? What am I not showing you? What related studies show?
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Solutions? Short answer: I don’t know! A solution:
Supervise experiment 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
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A consequence
Statisticians come to believeNOTHING!!
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OUTLINE Silent multiplicities Bayes and predictive probabilities Bayes as a frequentist tool Adaptive designs:
Adaptive randomization Investigating many phase II drugs Seamless Phase II/III trial Adaptive dose-response Extraim analysis
Trial design as decision analysis
19
Bayes in pharma and FDA …
http://www.cfsan.fda.gov/~frf/bayesdl.htmlhttp://www.prous.com/bayesian2004/
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BAYES AND PREDICTIVE PROBABILITY
Critical component of experimental design
In monitoring trials
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Example calculationData: 13 A's and 4 B's
Likelihood p13 (1–p)4
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Posterior density of p for uniform prior: Beta(14,5)
0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1p
p (1–p)13 4
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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
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Updating w/next observation
0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1p
Beta(15, 5)
Beta(14, 6)
prob 5/19 prob 14/19
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Suppose 17 more observations
P(A wins x of 17 | data)= EP(A wins x | data, p)= E[ px(1–p)17–x | data, p]17
x( )
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Best fitting binomial vs. predictive probabilities
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Binomial, p=14/19Binomial, p=14/19
Predictive, p ~ beta(14,5)Predictive, p ~ beta(14,5)
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Comparison of predictive with posterior
.00
.05
.10
.15
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 170 .1 .2 .3 .4 .5 .6 .7 .8 .9 1p
p (1–p)13 4
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Example: Baxter’s DCLHb & predictive probabilities
Diaspirin Cross-Linked Hemoglobin Blood substitute; emergency trauma Randomized controlled trial (1996+)
Treatment: DCLHb Control: saline N = 850 (= 2x425) Endpoint: death
32
Waiver of informed consent Data Monitoring Committee First DMC meeting:
DCLHb SalineDead 21 (43%) 8 (20%)Alive 28 33Total 49 41
P-value? No formal interim analysis
33
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
34
OUTLINE Silent multiplicities Bayes and predictive probabilities Bayes as a frequentist tool Adaptive designs:
Adaptive randomization Investigating many phase II drugs Seamless Phase II/III trial Adaptive dose-response Extraim analysis
Trial design as decision analysis
35
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
36
OUTLINE Silent multiplicities Bayes and predictive probabilities Bayes as a frequentist tool Adaptive designs:
Adaptive randomization Investigating many phase II drugs Seamless Phase II/III trial Adaptive dose-response Extraim analysis
Trial design as decision analysis
37
ADAPTIVE DESIGN 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
38
OUTLINE Silent multiplicities Bayes and predictive probabilities Bayes as a frequentist tool Adaptive designs:
Adaptive randomization Investigating many phase II drugs Seamless Phase II/III trial Adaptive dose-response Extraim analysis
Trial design as decision analysis
39
Giles, et al JCO (2003) Troxacitabine (T) in acute myeloid
leukemia (AML) when combined with cytarabine (A) or idarubicin (I)
Adaptive randomization to:IA vs TA vs TI
Max n = 75 End point: CR (time to CR < 50 days)
40
Randomization Adaptive Assign 1/3 to IA (standard)
throughout (unless only 2 arms) Adaptive to TA and TI based on
current results Final results
41
Patient Prob IA Prob TA Prob TI Arm CR<501 0.33 0.33 0.33 TI not2 0.33 0.34 0.32 IA CR3 0.33 0.35 0.32 TI not4 0.33 0.37 0.30 IA not5 0.33 0.38 0.28 IA not6 0.33 0.39 0.28 IA CR7 0.33 0.39 0.27 IA not8 0.33 0.44 0.23 TI not9 0.33 0.47 0.20 TI not
10 0.33 0.43 0.24 TA CR11 0.33 0.50 0.17 TA not12 0.33 0.50 0.17 TA not13 0.33 0.47 0.20 TA not14 0.33 0.57 0.10 TI not15 0.33 0.57 0.10 TA CR16 0.33 0.56 0.11 IA not17 0.33 0.56 0.11 TA CR
42
Patient Prob IA Prob TA Prob TI Arm CR<5018 0.33 0.55 0.11 TA not19 0.33 0.54 0.13 TA not20 0.33 0.53 0.14 IA CR21 0.33 0.49 0.18 IA CR22 0.33 0.46 0.21 IA CR23 0.33 0.58 0.09 IA CR24 0.33 0.59 0.07 IA CR25 0.87 0.13 0 IA not26 0.87 0.13 0 TA not27 0.96 0.04 0 TA not28 0.96 0.04 0 IA CR29 0.96 0.04 0 IA not30 0.96 0.04 0 IA CR31 0.96 0.04 0 IA not32 0.96 0.04 0 TA not33 0.96 0.04 0 IA not34 0.96 0.04 0 IA CR
Compare n = 75
DropTI
43
Summary of results
CR rates: IA: 10/18 = 56% TA: 3/11 = 27% TI: 0/5 = 0%
Criticisms . . .
44
OUTLINE Silent multiplicities Bayes and predictive probabilities Bayes as a frequentist tool Adaptive designs:
Adaptive randomization Investigating many phase II drugs Seamless Phase II/III trial Adaptive dose-response Extraim analysis
Trial design as decision analysis
45
Example: Adaptive allocation of therapies
Design for phase II: Many drugsAdvanced breast cancer (MDA);
endpoint is tumor responseGoals:
Treat effectively Learn quickly
46
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!
47
Standard designsOne 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
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An adaptive allocation 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
49
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%
Suppose 10 drugs, 200 patients
50
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%
51
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
52
OUTLINE Silent multiplicities Bayes and predictive probabilities Bayes as a frequentist tool Adaptive designs:
Adaptive randomization Investigating many phase II drugs Seamless Phase II/III trial Adaptive dose-response Extraim analysis
Trial design as decision analysis
53
Example: Seamless phase II/III
Drug vs placebo, randomized Local control (or biomarker, etc):
early endpoint related to survival? May depend on treatment
*Inoue et al (2002 Biometrics)
54
LocalcontrolNo localcontrol
SurvivaladvantageNo survivaladvantage
Phase II Phase III
Conventional drug development
6 mos 9-12 mos > 2 yrs
Stop
Seamless phase II/III
< 2 yrs (usually)
Not
Market
55
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]
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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 futilityFor efficacy
To submit NDA
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Conventional Phase III designs: Conv4 & Conv18, max N = 900(samepower as adaptive design)
Comparisons
58
Expected N under H0
0
200
400
600
800
1000
431
855 884
Bayes Conv4 Conv18
59
Expected N under H1
0
200
400
600
800
1000
649
887 888
Bayes Conv4 Conv18
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Benefits Duration of drug development is
greatly shortened under adaptive 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
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Possibility of large N N seldom near 900 When it is, it’s necessary! This possibility gives Bayesian
design its edge[Other reason for edge is modeling local control/survival]
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OUTLINE Silent multiplicities Bayes and predictive probabilities Bayes as a frequentist tool Adaptive designs:
Adaptive randomization Investigating many phase II drugs Seamless Phase II/III trial Adaptive dose-response Extraim analysis
Trial design as decision analysis
*Berry, et al. Case Studies in Bayesian Statistics 2001
*
64
Example: Stroke and adaptive dose-response
Adaptive doses in Phase II setting: learn efficiently and rapidly about dose-response relationship
Pfizer trial of a neutrofil inhibitory factor; results recently announced
Endpoint: stroke scale at week 13 Early endpoints: weekly stroke scale
65Doses
Standard Parallel Group DesignStandard Parallel Group DesignEqual sample sizes at each of k doses.
66
Res
pons
e
Doses
True dose-response curve True dose-response curve (unknown)(unknown)
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Res
pons
e
Doses
Observe responses (with error) Observe responses (with error) at chosen dosesat chosen doses
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Res
pons
e
Doses
True EDTrue ED9595
Dose at which 95% max effectDose at which 95% max effect
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Res
pons
e
Dose
True EDTrue ED9595
Uncertainty about ED95Uncertainty about ED95
??
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Res
pons
e
Dose
Uncertainty about ED95Uncertainty about ED95
??
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Res
pons
e
Doses
Solution: Solution: Increase number of dosesIncrease number of doses
EDED9595
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Res
pons
e
Doses
But, enormous sample size, and . . . But, enormous sample size, and . . . wasted dose assignments—always!wasted dose assignments—always!
EDED9595
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Our adaptive approach Observe data continuously Select next dose to maximize
information about ED95, given available evidence
Stop dose-ranging trial when know ED95 & response at ED95 “sufficiently well”
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Our approach (cont’d)Info accrues gradually about each patient; prediction using longitudinal model
Longitudinal Model Longitudinal Model Copenhagen Stroke DatabaseCopenhagen Stroke Database
Difference from baseline in SSS week 3
-30
-20
-10
0
10
20
30
40
50
-40 -30 -20 -10 0 10 20 30 40
Diff
eren
ce fr
om b
asel
ine
in S
SS
wee
k 12
50
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Our approach (cont’d)
Model dose-response (borrow strength from neighboring doses)
Many doses (logistical issues)
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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)
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Dose-response trial Learn efficiently and rapidly about
dose-response; if + go to Phase III Assign dose to maximize info
about dose-response parameters given current info
Use predictive probabilities, based on early endpoints
Doses in continuum, or preset grid
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Dose-response trial (cont’d) Learn about SD on-line Halt dose-ranging when know
dose sufficiently well Seamless switch from dose-
ranging to confirmatory trial—2 trials in 1!
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Advantages over standard design
Fewer patients (generally); faster & more effective learning
Better at finding ED95Tends to treat patients in trial
more effectivelyDrops duds early —actual trial!
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Dose-assignment simulation
Assumes particular dose-response curve
Assumes SD = 12Shows weekly results, several
patients at a time (green circles)
81DOSE
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96DOSE
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97DOSE
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98DOSE
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EstimatedED95
Confirmatory
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0 10 20 30
WEEK
0.0
0.5
1.0
1.5
DOSE
Assigned Doses by Week - one simulationD
OS
E0.
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135Z
F
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Estimated functions
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ASSIGNED DOSES
Proportion
Doses assigned across all simulations
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Black: median; Red: upper & lower quartiles; Green: Nominal
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F
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Estimated functions
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(no dose effect)
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ASSIGNED DOSES
Proportion
Doses assigned across all simulations
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Black: median; Red: upper & lower quartiles; Green: Nominal
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Consequences of Using Bayesian Adaptive ApproachFundamental change in the
way we do medical researchMore rapid progressWe’ll get the dose right!Better treatment of patients . . . at less cost
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ReactionsFDA: Positive. “Makes coming to
work worthwhile.” “In five years all trials may be seamless.”
Pfizer management: EnthusiasticOther companies: Cautious
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OUTLINE Silent multiplicities Bayes and predictive probabilities Bayes as a frequentist tool Adaptive designs:
Adaptive randomization Investigating many phase II drugs Seamless Phase II/III trial Adaptive dose-response Extraim analysis
Trial design as decision analysis
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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
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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
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
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
vs 0.80
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OUTLINE Silent multiplicities Bayes and predictive probabilities Bayes as a frequentist tool Adaptive designs:
Adaptive randomization Investigating many phase II drugs Seamless Phase II/III trial Adaptive dose-response Extraim analysis
Trial design as decision analysis
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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
Decision-analytic approach
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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*Cheng et al (2003 Biometrika)
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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.”
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Maximize effective treatment overall
What is “overall”? All patients who will be treated
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?
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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 =
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Optimal allocations in a two-armed trial
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Knowledge about success rate r
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OUTLINE Silent multiplicities Bayes and predictive probabilities Bayes as a frequentist tool Adaptive designs:
Adaptive randomization Investigating many phase II drugs Seamless Phase II/III trial Adaptive dose-response Extraim analysis
Trial design as decision analysis