Bayes-Verfahren in klinischen Studien€¦ · J. Gerß: Bayesian Methods in Clinical Trials 8 1....

Bayes-Verfahren in klinischen Studien

Dr. rer. nat. Joachim Gerß, Dipl.-Stat.

[email protected] of Biostatistics and Clinical Research

J. Gerß: Bayesian Methods in Clinical Trials 2

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 stoppingin interim analyses

• Prediction of the study result using predictiveprobabilities

• Borrowing of information across relatedpopulations

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


Contents






1. Combination of prior ‘beliefs’ with data from a studyExample: Two groups, survival data

HR=2.227 95% CI 0.947-5.238p=0.0990

Classical „frequentist“ statistical analysis

Bayesian analysis

Survival after (years)1614121086420

Sur

viva

l rat

e

1,0

0,8

0,6

0,4

0,2

0,0

Group 2

Group 1

Normal-Normal Model

Let θ:=ln(Hazard Ratio)

Data Model: | ~ ,

with , total no. observed eventsPrior distribution: ~ μ ,

Posterior distribution:

| = ,

∝ ,

= | ∙

=> | ~∙ ∙

,


1. Combination of prior ‘beliefs’ with data from a study

HR=2.227 95% CI 0.947-5.238p=0.0990


Sur

viva

l rat

e

1,0

0,8

0,6

0,4

0,2

0,0

Group 2

Group 1

1 2 3 4 5 6 87 9

95% Confidence interval: (0.947,5.238)Hazardratio

Data

1 2 3 4 5 6 87 9Hazardratio

Prior

1 2 3 4 5 6 87 9


Prior+ Data= Posterior

95% Credible Interval: (1.074,4.285)

Example 1Bayesian analysisClassical „frequentist“

statistical analysis


1. Combination of prior ‘beliefs’ with data from a study

1 2 3 4 5 6 87 9




1 2 3 4 5 6 87 9




1 2 3 4 5 6 87 9




Example 1 Example 2 Example 3„Noninformative“ prior


1. Combination of prior ‘beliefs’ with data from a studyFrequentist and Bayesian analysis

HR=2.227 95% CI 0.947-5.238p=0.0990

Classical „frequentist“ statistical analysis

Bayesian analysis


Sur

viva

l rat

e

1,0

0,8

0,6

0,4

0,2

0,0

Group 2

Group 1

1 2 3 4 5 6 87 9


Data

1 2 3 4 5 6 87 9Hazardratio

Prior

If p≤0.05 (<=> 1 Confidence Interval) => „significant“

Prob(HR>1|Data) ≥ 97.5% => „significant“

1 2 3 4 5 6 87 9





1. Combination of prior ‘beliefs’ with data from a studyType I error and power

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

0.0

0.2

0.4

0.6

0.8

1.0

prior

hazard ratio >> favours experimental therapyfavours standard << |

n=100 events Bayesian power

Classical power


Contents






2. Response-adaptive randomizationExample: Randomized 3-arm trial• Untreated patients aged ≥50 years with Adverse Karyotype Acute Myeloid Leukemia

• Random trt allocation: - Standard arm: Idarubicin + Ara-C (IA), with prob. π0

- 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

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-q2

2

5. If q2<0.15 or r>0.85, drop arm 2 (TI), set randomization probabilities π1=q12, π0=1-q1

2

6. Otherwise assign investigational treatments with probabilities π1 q12 , π2 q2

2any

time

durin

gth

etri

al


2. Response-adaptive randomizationExample: Randomized 3-arm trial• Untreated patients aged ≥50 years with Adverse Karyotype Acute Myeloid Leukemia

• Random trt allocation: - Standard arm: Idarubicin + Ara-C (IA), with prob. π0

- 1st investigational arm: Troxacitabine + Ara-C (TA),with prob. π1

- 2nd investigational arm: Troxacitabine + Idarubicin (TI), with prob. π2

1 5 10 15 20 25 30 34

0.0

0.2

0.4

0.6

0.8

1.0

Pat.-No.

IA

TA

TI

Pro

b (T

reat

men

t ass

ignm

ent)

CR No CR Total

IA 10 (56%) 8 18

TA 3 (27%) 8 11

TI 0 (0%) 5 5

Fisher‘s exact test: p = 0.057


Contents






3. Borrowing of information across related populationsBiomarkers in JIA

Gerß et al.Ann Rheum Dis 2012;71:1991–1997.

No. Flares / Patients (%)OR Fisher‘s Exact

TestMRP8/14 ≥690 ng/ml

MRP8/14 <690 ng/ml

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


3. Borrowing of information across related populationsHierarchical model

Let := Observed ln(Odds Ratio) in subgroup i

Observed lnOR‘s: | ~ , , 1,2,3 with assumed known

Parameter model: ~ ,

Prior: f ∝ 1 (noninformative)

f ln ∝ 1 (noninformative)

MCMC Sampling (Gibbs sampler, Metropolis algorithm)

• Burn-in: n=5000

• No. samples: n=100000

• | , , , 1,2,3

• | , ,


3. Borrowing of information across related populationsBiomarkers in JIA: Results

Observed Odds Ratio

Fully Bayesian Estimator

0.25 0.5 1 2 5 10

SubgroupOligoarthritis(n=86)

SubgroupPolyarthritis(n=74)

SubgroupOther(n=28)

Pooled OR


3. Borrowing of information across related populationsBiomarkers in JIA: Results

Observed Odds Ratio

Empirical Bayes Estimator

Fully Bayesian Estimator

0.25 0.5 1 2 5 10

SubgroupOligoarthritis(n=86)

SubgroupPolyarthritis(n=74)

SubgroupOther(n=28)

Pooled OR


Contents






4. Summary and ConclusionBayesian methods: Operating characteristics


Fully Bayesian final analysis usingposterior 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.


4. Summary and ConclusionBayesian supplements



# Bayesian interim analysis

Fully Bayesian final analysis usingposterior distribution



Bayesian „supplement“Final frequentist statistical analysis

UsesupplementaryBayesianinterimanalysisto determine the time to stop recruitmentFinal frequentist statistical analysis


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

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


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 yearsor 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.

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Bayes-Verfahren in klinischen Studien€¦ · J. Gerß: Bayesian Methods in Clinical Trials 8 1....

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