Bayes-Verfahren in klinischen Studien
Dr. rer. nat. Joachim Gerß, Dipl.-Stat.
[email protected]
Institute of Biostatistics and Clinical Research
Popular Bayesian Methods in Clinical Trials
•
Combination of knowledge from previous data
or prior ‘beliefs’ with data from a current study
•
Dose finding: Continual reassessment method
•
Response-adaptive randomization
•
Bayesian data monitoring / sequential stopping
in interim analyses
•
Prediction of the study result using predictive
probabilities
•
Borrowing of information across related
populations
J. Gerß: Bayesian Methods in Clinical Trials
2
Thomas Bayes
(1702-1761)
Contents
1. Combination of prior beliefs with data from a study
2. Response-adaptive randomization
3. Borrowing of information across related populations
4. Summary and Conclusion
J. Gerß: Bayesian Methods in Clinical Trials
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Contents
1. Combination of prior beliefs with data from a study
2. Response-adaptive randomization
3. Borrowing of information across related populations
4. Summary and Conclusion
J. Gerß: Bayesian Methods in Clinical Trials
4
1. Combination of prior ‘beliefs’ with data from a study
Example: Two groups, survival data
Classical „frequentist“
statistical analysis
Bayesian analysis
Normal-Normal Model
Let θ:=ln(Hazard Ratio)
Data Model: | ~
,
with
, total no. observed events
Prior distribution: ~
1,0
μ ,
Survival rate
0,8
0,6
Group 1
0,4
Group 2
Posterior distribution:
|
=
,
∝
,
=
|
0,2
0,0
0
2
4
6
8
10
12
Survival after (years)
14
16
HR=2.227
95% CI 0.947-5.238
p=0.0990
J. Gerß: Bayesian Methods in Clinical Trials
=>
5
| ~
∙
∙
∙
,
1. Combination of prior ‘beliefs’ with data from a study
Classical „frequentist“
statistical
analysis
Example
1
Bayesian analysis
Prior
Prior
+ Data
= Posterior
1
1,0
2
3
4
Survival rate
0,8
0,6
Group 1
0,4
Group 2
5 6 7 8 9
Hazard
ratio
Data
1
5 6 7 8 9
Hazard
95% Confidence interval: (0.947,5.238) ratio
0,2
2
3
4
0,0
0
2
4
6
8
10
12
Survival after (years)
14
16
HR=2.227
95% CI 0.947-5.238
p=0.0990
J. Gerß: Bayesian Methods in Clinical Trials
95% Credible Interval: (1.074,4.285)
1
2
3
4
5 6 7 8 9
Hazard
95% Confidence interval: (0.947,5.238) ratio
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1. Combination of prior ‘beliefs’ with data from a study
Example 1
Example 2
Prior
+ Data
= Posterior
1
2
3
4
5 6 7 8 9
Hazard
95% Confidence interval: (0.947,5.238) ratio
95% Credible Interval: (1.074,4.285)
J. Gerß: Bayesian Methods in Clinical Trials
Example 3
„Noninformative“ prior
Prior
+ Data
= Posterior
1
2
3
4
5 6 7 8 9
Hazard
95% Confidence interval: (0.947,5.238) ratio
95% Credible Interval: (1.822,4.264)
7
Prior
+ Data
= Posterior
1
2
3
4
5 6 7 8 9
Hazard
95% Confidence interval: (0.947,5.238) ratio
95% Credible Interval: (0.947,5.238)
1. Combination of prior ‘beliefs’ with data from a study
Frequentist and Bayesian analysis
Classical „frequentist“
statistical analysis
Bayesian analysis
Prior
Prior
+ Data
= Posterior
1
1,0
2
3
4
Survival rate
0,8
0,6
Group 1
0,4
Group 2
5 6 7 8 9
Hazard
ratio
Data
1
5 6 7 8 9
Hazard
95% Confidence interval: (0.947,5.238) ratio
0,2
2
3
4
0,0
0
2
4
6
8
10
12
Survival after (years)
14
95% Credible Interval: (1.074,4.285)
16
HR=2.227
95% CI 0.947-5.238
p=0.0990
If p≤0.05 (<=> 1 Confidence Interval)
=>
„significant“
J. Gerß:
Bayesian Methods in Clinical Trials
1
2
3
4
5 6 7 8 9
Hazard
95% Confidence interval: (0.947,5.238) ratio

8
Prob(HR>1|Data) ≥ 97.5%
=> „significant“
1. Combination of prior ‘beliefs’ with data from a study
Type I error and power
Bayesian power
n=100 events
1.0
Classical power
0.8
0.6
0.4
prior
0.2
0.0
0.5
1.0
1.5
2.0
hazard ratio
favours standard << | >> favours experimental therapy
J. Gerß: Bayesian Methods in Clinical Trials
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2.5
3.0
3.5
4.0
Contents
1. Combination of prior beliefs with data from a study
2. Response-adaptive randomization
3. Borrowing of information across related populations
4. Summary and Conclusion
J. Gerß: Bayesian Methods in Clinical Trials
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2. Response-adaptive randomization
Example: Randomized 3-arm trial
•
Untreated patients aged ≥50 years with Adverse Karyotype Acute Myeloid Leukemia
•
(IA), with prob. π0
Random trt allocation: - Standard arm:
Idarubicin + Ara-C
- 1st investigational arm: Troxacitabine + Ara-C
(TA),with prob. π1
- 2nd investigational arm: Troxacitabine + Idarubicin (TI), with prob. π2
•
Response: Time to Complete Remission (CR) within 49 days of starting treatment
•
Denote mk := Current posterior median time to CR in arm k{0,1,2}
qk := Current posterior Prob(mk<m0|data), k{1,2}
r := Current posterior Pr(m1<m2|data)
•
Algorithm
any time
during the trial
1. Initially, balanced randomization, with a probabilities π0 = π1 = π2 =1/3
2. Standard arm: Fixed probability π0 = 1/3, as long as all three arms remain in the trial.
3. If q1≥0.85 or q2≥0.85, drop standard arm, set randomization probabilities π1=r2, π2=1-r2
4. If q1<0.15 or r<0.15, drop arm 1 (TA), set randomization probabilities π2=q22, π0=1-q22
5. If q2<0.15 or r>0.85, drop arm 2 (TI), set randomization probabilities π1=q12, π0=1-q12
6. Otherwise assign investigational treatments with probabilities π1  q12 , π2  q22
J. Gerß: Bayesian Methods in Clinical Trials
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2. Response-adaptive randomization
Example: Randomized 3-arm trial
Untreated patients aged ≥50 years with Adverse Karyotype Acute Myeloid Leukemia
•
(IA), with prob. π0
Random trt allocation: - Standard arm:
Idarubicin + Ara-C
- 1st investigational arm: Troxacitabine + Ara-C
(TA),with prob. π1
- 2nd investigational arm: Troxacitabine + Idarubicin (TI), with prob. π2
Prob (Treatment assignment)
•
1.0
0.8
TI
0.6
CR
No CR
Total
IA
10 (56%)
8
18
TA
3 (27%)
8
11
TI
0 (0%)
5
5
TA
0.4
0.2
IA
0.0
1
5
10
15
Pat.-No.
20
J. Gerß: Bayesian Methods in Clinical Trials
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30
Fisher‘s exact test: p = 0.057
34
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Contents
1. Combination of prior beliefs with data from a study
2. Response-adaptive randomization
3. Borrowing of information across related populations
4. Summary and Conclusion
J. Gerß: Bayesian Methods in Clinical Trials
13
3. Borrowing of information across related populations
Biomarkers in JIA
No. Flares / Patients (%)
Fisher‘s Exact
Test
MRP8/14
≥690 ng/ml
MRP8/14
<690 ng/ml
OR
All patients (n=188)
22 / 75 (29%)
13/ 113 (12%)
3.2
p=0.0036
Subgroup Oligoarthritis (n=86)
9 / 34 (26%)
8 / 52 (15%)
2.0
p=0.2700
Subgroup Polyarthritis (n=74)
11 / 25 (44%)
5 / 49 (10%)
6.9
p=0.0019
Subgroup Other (n=28)
2 / 16 (13%)
0 / 12 (0%)
4.3
p=0.4921
Gerß et al.
Ann Rheum Dis 2012;71:1991–1997.
J. Gerß: Bayesian Methods in Clinical Trials
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3. Borrowing of information across related populations
Hierarchical model
Let
:= Observed ln(Odds Ratio) in subgroup i
Observed lnOR‘s:
| ~
Parameter model:
~
f
Prior:
f ln
,
,
1,2,3 with
,
∝ 1 (noninformative)
∝ 1 (noninformative)
MCMC Sampling (Gibbs sampler, Metropolis algorithm)
• Burn-in: n=5000
• No. samples: n=100000
•
| ,
,
•
| ,
,
,
1,2,3
J. Gerß: Bayesian Methods in Clinical Trials
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assumed known
3. Borrowing of information across related populations
Biomarkers in JIA: Results
Observed Odds Ratio
Subgroup
Oligoarthritis
(n=86)
Fully Bayesian Estimator
Subgroup
Polyarthritis
(n=74)
Subgroup
Other
(n=28)
Pooled OR
0.25
0.5
J. Gerß: Bayesian Methods in Clinical Trials
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2
5
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10
3. Borrowing of information across related populations
Biomarkers in JIA: Results
Observed Odds Ratio
Subgroup
Oligoarthritis
(n=86)
Empirical Bayes Estimator
Fully Bayesian Estimator
Subgroup
Polyarthritis
(n=74)
Subgroup
Other
(n=28)
Pooled OR
0.25
0.5
J. Gerß: Bayesian Methods in Clinical Trials
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2
5
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Contents
1. Combination of prior beliefs with data from a study
2. Response-adaptive randomization
3. Borrowing of information across related populations
4. Summary and Conclusion
J. Gerß: Bayesian Methods in Clinical Trials
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4. Summary and Conclusion
Bayesian methods: Operating characteristics
1.
Combination of prior beliefs with data from a study
Fully Bayesian final analysis using
posterior distribution
•
increased power
•
… but also increased type I error
•
Bayesian methods in a strict corset of
frequentist quality criteria are usually
not much more powerful than classical
frequentist methods.
J. Gerß: Bayesian Methods in Clinical Trials
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4. Summary and Conclusion
Bayesian supplements
1.
Combination of prior beliefs with data from a study
Fully Bayesian final analysis using
posterior distribution
2.
Response-adaptive randomization
Fully Bayesian final analysis using
posterior distribution
#
Bayesian „supplement“
Final frequentist statistical analysis
Bayesian interim analysis
Fully Bayesian final analysis using
posterior distribution
UsesupplementaryBayesianinterimanalysis
to determine the time to stop recruitment
Final frequentist statistical analysis
J. Gerß: Bayesian Methods in Clinical Trials
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Bayesian Methods in Clinical Trials
• Early phase clinical trials („in-house
studies“ w/o strict regulatory control)
• Trials in small populations
• Medical device trials
• Exploratory studies
• Large scale confirmatory trials with
strict type I error control
J. Gerß: Bayesian Methods in Clinical Trials
Use fully Bayesian approach, paying
attention to
• choose the appropriate model
carefully,
• choose the inputted (prior)
information carefully and
• check (classical) operating
characteristics (type I error, power)
Use of Bayesian supplements
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Literature
•
Berry SM, Carlin BP, Lee JJ , Müller P (2010): Bayesian Adaptive Methods for Clinical Trials.
Chapman & Hall/CRC Biostatistics.
•
Spiegelhalter DJ, Abrams KR, Myles JP (2004): Bayesian Approaches to Clinical Trials and
Health-Care Evaluation. Wiley Series in Statistics in Practice.
•
Tan SB, Dear KBG, Bruzzi P, Machin D (2003): Strategy for randomised clinical trials in rare
cancers. British Medical Journal 327;47-49.
•
Giles FJ et al. (2003): Adaptive randomized study of Idarubicin and Cytarabine versus
Troxacitabine and Cytarabine versus Troxacitabine and Idarubicin in untreated patients 50 years
or older with adverse karyotype Acute Myeloid Leukemia. Journal of Clinical Oncology
21(9);1722-1727
•
Gerss J et al. (2012): Phagocyte-specific S100 proteins and high-sensitivity C reactive protein as
biomarkers for a risk-adapted treatment to maintain remission in juvenile idiopathic arthritis: a
comparative study. Annals of the rheumatic diseases 71(12);1991-1997.
J. Gerß: Bayesian Methods in Clinical Trials
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