Practical Priors with Empirical Bayes Isaac Gravestock and Leonhard - - PowerPoint PPT Presentation

Practical Priors with Empirical Bayes

Isaac Gravestock and Leonhard Held Leuven 19 May 2016

COMBACTE STAT-Net

◮ STAT-Net is a network of academic and EFPIA partners with specific

expertise in PK/PD, modelling, biostatistics, infectious diseases, antimicrobial agents, microbiology, epidemiology and clinical development.

◮ The objective of STAT-Net is to support the clinical development of

new antibacterial drugs by investigating approaches to improve the data-driven design of Phase 2 and 3 clinical trials.

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Prior-Data Conflict

“Bayesian: One who, vaguely expecting a horse and catching a glimpse of a donkey, strongly concludes that he has seen a mule.” Senn [2007]

• 2

2 4 6 8 0.0 0.1 0.2 0.3 0.4 0.5 0.6 density

Prior Likelihood Posterior

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Adaptive Prior Weighting

“Bayesian: One who, vaguely expecting a horse and catching a glimpse of a donkey, concludes that he has seen . . . most likely a donkey!” not Senn [2007]

• 2

2 4 6 8 0.0 0.1 0.2 0.3 0.4 0.5 0.6 density

Conflict Prior Likelihood Adapted Posterior

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Incorporating Historical Information: The Power Prior

Prior Distribution Historical Data Power Prior Current Data Posterior Distribution

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The Power Prior

1 Start with a prior p0(θ) on the parameter θ, e.g. uniform 2 Update the prior based on the likelihood L(θ; x0) of historical data x0

raised to a power δ (“prior weight”) between 0 and 1 p(θ | δ, x0) ∝ L(θ; x0)δp0(θ)

Ibrahim and Chen [2000]

→ If the historical study has n0 patients, then the prior sample size is δn0

3 Data x⋆ from current study arrives, from n⋆ patients, say. 4 Current study is combined with power prior to (final) posterior:

p(θ | x⋆, x0, δ) ∝ L(θ; x⋆) · p(θ | δ, x0), → Total sample size is n⋆ + δn0

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Unknown Power Parameter

◮ Treat δ as unknown and include prior p0(δ) ◮ Requires normalisation of power prior:

p(θ, δ | x0) = p(θ | δ, xo) p0(δ) = L(θ; x0)δp0(θ)

• L(θ; x0)δp0(θ)dθ p0(δ)

Duan et al. [2006] Neuenschwander et al. [2009] ◮ Joint and marginal posterior:

p(θ, δ | x⋆, x0) ∝ L(θ; x⋆) · p(θ, δ | x0) p(θ | x⋆, x0) =

• p(θ, δ | x⋆, x0) dδ

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Choosing δ

Possible approaches for choosing δ from the literature:

1 Pick some fixed values and do a sensitivity analysis afterwards 2 Integrate it out of joint posterior and use a fully Bayesian approach 3 We have developed an empirical Bayes (EB) method: ◮ Maximise the marginal likelihood to choose δ:

ˆ δEB = arg max

δ∈[0,1]

L(δ; x0, x⋆) = arg max

δ∈[0,1]

• L(θ; x⋆)L(θ; x0)δp0(θ)dθ
• L(θ; x0)δp0(θ)dθ

.

→ ˆ δEB can also be viewed as posterior mode under uniform prior on δ

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Antibiotics Trials

◮ Treating nosocomial (caught in hospital) pneumonia ◮ 2 studies comparing Linezolid and Vancomycin ◮ Use Rubinstein (2001) study to improve estimate of Vancomycin arm

in Wunderink (2003)

◮ 2 binary outcomes: Clinical Cure Mortality Rubinstein (2001) 62/91 (68%) 49/193 (25%) Wunderink (2003) 111/171 (65%) 61/302 (20%) Difference (95%-CI, p) 3% (−9 to 15%, p=0.68) 5% (−2 to 13%, p=0.18) Prior weight ˆ δ = 1 ˆ δ = 0.44 Prior sample size ˆ δn0 = 91 ˆ δn0 = 86

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Operating Characteristics

◮ We looked at performance of various borrowing methods for the

control arm of a randomized controlled clinical trial.

◮ Expected Prior Sample Size ◮ Mean Square Error ◮ Bias ◮ Coverage

◮ Binomial setting:

Fixed historical control arm data: x0/n0 = 65/100 Random current control arm data: X⋆ ∼ Bin(n⋆ = 200, π⋆)

◮ Comparison with

◮ fixed borrowing (δ = 0, 0.2, . . . , 1) ◮ (test-than-pool)

Viele et al. [2014]

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Expected Prior Sample Size against Fixed Borrowing

0.50 0.55 0.60 0.65 0.70 0.75 0.80 20 40 60 80 100 True π⋆ Number of Patients Borrowed from Historical Trial δ = 0 δ = 0.2 δ = 0.4 δ = 0.6 δ = 0.8 δ = 1 EB FB B(1,1) FB B(.5,.5) Isaac Gravestock (UZH) Adaptive prior weighting Leuven 19 May 2016 11 / 21

Mean Square Error against Fixed Borrowing

0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 True π⋆ Mean Square Error δ = 0 δ = 0.2 δ = 0.4 δ = 0.6 δ = 0.8 δ = 1 EB FB B(1,1) FB B(.5,.5) Isaac Gravestock (UZH) Adaptive prior weighting Leuven 19 May 2016 12 / 21

Bias against Fixed Borrowing

0.0 0.2 0.4 0.6 0.8 1.0

• 0.04
• 0.02

0.00 0.02 0.04 π Bias Isaac Gravestock (UZH) Adaptive prior weighting Leuven 19 May 2016 13 / 21

Coverage against Fixed Borrowing

0.2 0.4 0.6 0.8 0.875 0.900 0.925 0.950 0.975 π Coverage Isaac Gravestock (UZH) Adaptive prior weighting Leuven 19 May 2016 14 / 21

Operating Characteristics cont.

◮ We look at additional performance criteria:

◮ Power to detect difference πT − π⋆ = 12% ◮ Type I Error

which require inclusion of a treatment arm:

◮ Binomial setting:

Fixed historical control arm data: x0/n0 = 65/100 Current control arm data: X⋆ ∼ Bin(n⋆ = 200, π⋆) Current treatment arm data: XT ∼ Bin(nT = 200, πT)

◮ Bayesian “significance” if Pr(πT > π⋆ | x0, x⋆, xT) > 0.975 Viele et al. [2014]

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Power against Fixed Borrowing

0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 True π⋆ Power δ = 0 δ = 0.2 δ = 0.4 δ = 0.6 δ = 0.8 δ = 1 EB FB B(1,1) FB B(.5,.5) Isaac Gravestock (UZH) Adaptive prior weighting Leuven 19 May 2016 16 / 21

Type I Error against Fixed Borrowing

0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.00 0.05 0.10 0.15 0.20 True π⋆ Type I Error δ = 0 δ = 0.2 δ = 0.4 δ = 0.6 δ = 0.8 δ = 1 EB FB B(1,1) FB B(.5,.5) Isaac Gravestock (UZH) Adaptive prior weighting Leuven 19 May 2016 17 / 21

Application in Adaptive Trial Design

Idea:

◮ Sequentially assess the compatibility between historical and current

data and re-estimate δ

◮ Goal: Incorporate historical data and use less patients in current

control arm if there is no conflict between historical and current data, e.g. by adapting the randomisation ratio

◮ Otherwise downweight historical data appropriately

Operating characteristic:

◮ Expected total sample size ◮ as a function of the number of new patients recruited

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Expected Total Sample Size in Adaptive Trial

◮ Use Rubinstein mortality as historical data: x0/n0 = 49/193 = 25.4% ◮ Target sample size: 300 → include between 107 and 300 new patients.

50 100 150 200 250 300 50 100 150 200 250 300 New Patients Recruited Expected Total Number of Patients π⋆ = 0.254 π⋆ = 0.2 π⋆ = 0.3 π⋆ = 0.35 π⋆ = 0.4

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Summary and Outlook

◮ Empirical Bayes is useful to downweight historical data using the

power prior.

◮ Empirical Bayes procedure has good frequentist properties! Bayarri and Berger [2004] ◮ If more that one historical study is to be incorporated, the robust

meta-analytic-predictive-prior is a valuable alternative to incorporate between-study variation.

Schmidli et al. [2014]

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Literature I

• M. J. Bayarri and J. O. Berger. The interplay of Bayesian and frequentist analysis. Statistical

Science, 19(1):58–80, 2004. doi: 10.1214/088342304000000116. Yuyan Duan, Eric P. Smith, and Keying Ye. Using Power Priors to Improve the Binomial Test of Water Quality. Journal of Agricultural, Biological, and Environmental Statistics, 11(2): 151–168, 2006. Joseph G. Ibrahim and Ming-Hui Chen. Power prior distributions for regression models. Statistical Science, 15(1):46–60, 2000. ISSN 08834237. Beat Neuenschwander, Michael Branson, and David J. Spiegelhalter. A note on the power prior. Statistics in Medicine, 28(28):3562–3566, 2009. ISSN 1097-0258. Heinz Schmidli, Sandro Gsteiger, Satrajit Roychoudhury, Anthony O’Hagan, David Spiegelhalter, and Beat Neuenschwander. Robust Meta-Analytic-Predictive Priors in Clinical Trials with Historical Control Information. Biometrics, 70(4):1023–1032, 2014. Steven Senn. Statistical Issues in Drug Development. Wiley, 2nd edition, 2007. Kert Viele, Scott Berry, Beat Neuenschwander, Billy Amzal, Fang Chen, Nathan Enas, Brian Hobbs, Joseph G Ibrahim, Nelson Kinnersley, Stacy Lindborg, et al. Use of historical control data for assessing treatment effects in clinical trials. Pharmaceutical Statistics, 13(1):41–54, 2014.

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