Bayesian Statistics Applied to Clinical Trials

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Bayesian Statistics Applied to Clinical Trials

• Scott M. Berry, Ph.D.

Berry Consultants, LLC
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Outline

• Intro Berry Consultants
• Bayesian Statistics
• Comparison to Classical
• Example Trial
• TFAS Trial

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Berry Consultants

• www.berryconsultants.com
• Me and Donald Berry, PhD
• Since 2000, mostly biostatistics Design (80)
  and Analysis (20)
• 95 Bayesian (Rare!)
• 25 trials, 5 spinal studies

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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 the
  name Bayesian Statistics

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Simple Example

• Coin, P(HEADS) p
• p 0.25 or p 0.75, equally likely.
• DATA Flip coin twice, both heads.
• p ???

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Bayes Theorem
Pr p 0.75 DATA
PrDATA p0.75 Prp0.75
--------------------------------------------------
---------------------------------
PrDATA p0.75 Prp0.75
PrDATA p0.75 Prp0.75
(0.75)2 (0.5)
-----------------------------------
0.90
(0.75)2 (0.5) (0.25)2 (0.5)
Posterior Probabilities
Likelihood
Prior Probabilities
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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 POSTIVE. What is the probability they
have X?
(0.99) (0.001)
0.0902 !!!
---------------------------------------
(0.99) (0.001) (0.01) (0.999)
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Bayesian Statistics

• A subjective probability axiomatic approach was
  developed with Bayes theorem as the mathematical
  crank--Savage, Lindley, Jeffreys (1950s)
• A philosophical niche, calculation very hard.
• Early 1990s Computers made calculation
  possibleand more!

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Bayesian Approach

• Probabilities of unknownshypotheses,
  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

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Classical Statistics

• Ad-hoc collection of techniques, many holes
• Address P(DATA q )
• BAYESIAN P( q DATA)
• VERY DIFFERENT!!!
• No Prior probabilities (good/bad)??

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Frequentist vs. BayesianSeven comparisons

• 1. Evidence used?
• 2. Probability, of what?
• 3. Condition on results?
• 4. Dependence on design?
• 5. Flexibility?
• 6. Predictive probability?
• 7. Decision making?

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Single-Arm Device Study

• A Revised Medical Device
• Looking for FDA approval
• A single-arm non-inferiority trial
• Sequential design
• Patients , accrual slow, get to market quickly

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• Endpoint 12-month failure-free is industry
  standardour ENTIRE focus
• PD and PH are probabilities of 12-month survival
  for device and historical
• We have historical RCT data on failure rates for
  FDA-okayed devices same PI and centers

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What is trial success?
d
From the historical data we calculate the
posterior distribution of PH
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Modeling Survival

• Not constant hazardsfailures tend to be early.
• Assume piecewise exponential rate with

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Notation

• Probability of 12-month survival
  exp(-l1-5l2-6l3)
• Trial parameter
• q exp(-l1-5l2-6l3)-PH
• For completed trial
• Pr(q gt -0.10) 0.95 ?

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Prior Distributions

• Flat priors for this final calculation.
• Informative (historical data) based priors for
  predictive analysis
• No FDA concerns/remarks

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Interim Look

• 3 types of subjects complete, partial, not
  accrued
• For partial information subjects we calculate the
  predictive distribution of their time to failure
  ( not accrued)
• Combining individual pred. distrn creates a
  pred. distrn for the end of trial results.

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Example Interim Analysis
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13/40
9
14/45
6
18/60
3
0
Max Sample Size60
Time subject enters trial
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• Predictive probability for the current trialto
  completion

QPrPr(qgt-0.10 Xn) gt 0.95

• With 10 more accrued patients

Q10Prn10Pr(qgt-0.10 Xn10) gt 0.95

• Taken to the cap

QcapPrcapPr(q gt-0.10 Xcap) gt 0.95
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Sequential Decisions

• Q gt 0.90 gt Stop accrual
• Q gt 0.99 gt Stop accrual file early to FDA
• Q, Q10, Qcap lt 0.05 then futility
• Otherwise continue(to cap)

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Simulation Results
200
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TFAS Trial

• 21 Randomization (?)
• N Subjects
• Subject Success when a success on all 4
  categories
• Parameters
• PT P(S _at_ 24 w/ TFAS)
• PC P(S _at_ 24 w/ Control)

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TFAS Trial

• Trial Success
• Pr(PT gt PC - d DATA) gt 0.96

1
PT PC
PT PC - d
PT
0
0
PC
1
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Trial Time Line
Interim analyses
N pts accrued
n2
n1
Pts in trial
with 24-mo FU
0
12
24
36
48
Months
End oftrial
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Early Success

• Two interim looks for early success.
• PP Predictive probability of trial success at
  interim analysis
• If PP gt 0.99 immediate success!

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Predictive Probability

• Subjects at 3, 6, 12, and 24 months.
• We predict the results of each subject based
  on completed subject data.

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EXAMPLE 1st Interim Data
Predict
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All 24-Month Completers
Beta-Binomial Statistical Model
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P(Successful Trial) 0.970
TRIAL FAILS HERE
55
50
15
20
Treatment Failures
45
25
Control Failures
30
35
40
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EXAMPLE 2nd Interim Data
Predict
P(Success) 0.999
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Imputation for LTFU

• A subject with missing data has information
  (12-M Suc, even 3M Fail)
• Impute (predict) those that are LTFU.
• ITT Analysis, better

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OCs for 350 (175,250)
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OCs for 400 (200,300)
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Summary

• Can ignore philosophy (but Bayesian better!)
• Better Statistics--better trials
• More Ethical, Scientific
• Better for Sponsor
• Better for FDA
• Decision Analysis Easier