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. Isaac Gravestock (UZH) Adaptive prior weighting Leuven 19 May 2016 2 / 21

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] Posterior 0.6 0.5 Prior Likelihood 0.4 density 0.3 0.2 0.1 0.0 -2 0 2 4 6 8 Isaac Gravestock (UZH) Adaptive prior weighting Leuven 19 May 2016 3 / 21

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] Adapted Posterior 0.6 0.5 Conflict Prior Likelihood 0.4 density 0.3 0.2 0.1 0.0 -2 0 2 4 6 8 Isaac Gravestock (UZH) Adaptive prior weighting Leuven 19 May 2016 4 / 21

Incorporating Historical Information: The Power Prior Prior Distribution Historical Data Power Prior Current Data Posterior Distribution Isaac Gravestock (UZH) Adaptive prior weighting Leuven 19 May 2016 5 / 21

The Power Prior 1 Start with a prior p 0 ( θ ) on the parameter θ , e.g. uniform 2 Update the prior based on the likelihood L ( θ ; x 0 ) of historical data x 0 raised to a power δ (“prior weight”) between 0 and 1 p ( θ | δ, x 0 ) ∝ L ( θ ; x 0 ) δ p 0 ( θ ) Ibrahim and Chen [2000] → If the historical study has n 0 patients, then the prior sample size is δ n 0 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 ⋆ , x 0 , δ ) ∝ L ( θ ; x ⋆ ) · p ( θ | δ, x 0 ) , → Total sample size is n ⋆ + δ n 0 Isaac Gravestock (UZH) Adaptive prior weighting Leuven 19 May 2016 6 / 21

Unknown Power Parameter ◮ Treat δ as unknown and include prior p 0 ( δ ) ◮ Requires normalisation of power prior: p ( θ, δ | x 0 ) = p ( θ | δ, x o ) p 0 ( δ ) L ( θ ; x 0 ) δ p 0 ( θ ) = L ( θ ; x 0 ) δ p 0 ( θ ) d θ p 0 ( δ ) � Duan et al. [2006] ◮ Joint and marginal posterior: Neuenschwander et al. [2009] p ( θ, δ | x ⋆ , x 0 ) ∝ L ( θ ; x ⋆ ) · p ( θ, δ | x 0 ) � p ( θ | x ⋆ , x 0 ) = p ( θ, δ | x ⋆ , x 0 ) d δ Isaac Gravestock (UZH) Adaptive prior weighting Leuven 19 May 2016 7 / 21

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 δ EB = arg max L ( δ ; x 0 , x ⋆ ) δ ∈ [0 , 1] � L ( θ ; x ⋆ ) L ( θ ; x 0 ) δ p 0 ( θ ) d θ = arg max . � L ( θ ; x 0 ) δ p 0 ( θ ) d θ δ ∈ [0 , 1] → ˆ δ EB can also be viewed as posterior mode under uniform prior on δ Isaac Gravestock (UZH) Adaptive prior weighting Leuven 19 May 2016 8 / 21

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) ˆ ˆ Prior weight δ = 1 δ = 0 . 44 ˆ ˆ Prior sample size δ n 0 = 91 δ n 0 = 86 Isaac Gravestock (UZH) Adaptive prior weighting Leuven 19 May 2016 9 / 21

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: x 0 / n 0 = 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] Isaac Gravestock (UZH) Adaptive prior weighting Leuven 19 May 2016 10 / 21

Expected Prior Sample Size against Fixed Borrowing 100 Number of Patients Borrowed from Historical Trial 80 60 δ = 0 40 δ = 0 . 2 δ = 0 . 4 δ = 0 . 6 δ = 0 . 8 20 δ = 1 EB FB B(1,1) 0 FB B(.5,.5) 0.50 0.55 0.60 0.65 0.70 0.75 0.80 True π ⋆ Isaac Gravestock (UZH) Adaptive prior weighting Leuven 19 May 2016 11 / 21

Mean Square Error against Fixed Borrowing 0.0030 δ = 0 δ = 0 . 2 δ = 0 . 4 0.0025 δ = 0 . 6 δ = 0 . 8 δ = 1 0.0020 EB Mean Square Error FB B(1,1) FB B(.5,.5) 0.0015 0.0010 0.0005 0.0000 0.50 0.55 0.60 0.65 0.70 0.75 0.80 True π ⋆ Isaac Gravestock (UZH) Adaptive prior weighting Leuven 19 May 2016 12 / 21

Bias against Fixed Borrowing 0.04 0.02 Bias 0.00 -0.02 -0.04 0.0 0.2 0.4 0.6 0.8 1.0 π Isaac Gravestock (UZH) Adaptive prior weighting Leuven 19 May 2016 13 / 21

Coverage against Fixed Borrowing 0.975 0.950 Coverage 0.925 0.900 0.875 0.2 0.4 0.6 0.8 π 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 π T − π ⋆ = 12% ◮ Type I Error which require inclusion of a treatment arm: ◮ Binomial setting: Fixed historical control arm data: x 0 / n 0 = 65 / 100 Current control arm data: X ⋆ ∼ Bin( n ⋆ = 200 , π ⋆ ) Current treatment arm data: X T ∼ Bin( n T = 200 , π T ) ◮ Bayesian “significance” if Pr( π T > π ⋆ | x 0 , x ⋆ , x T ) > 0 . 975 Viele et al. [2014] Isaac Gravestock (UZH) Adaptive prior weighting Leuven 19 May 2016 15 / 21

Power against Fixed Borrowing 1.0 0.9 0.8 0.7 Power 0.6 δ = 0 δ = 0 . 2 δ = 0 . 4 0.5 δ = 0 . 6 δ = 0 . 8 δ = 1 0.4 EB FB B(1,1) 0.3 FB B(.5,.5) 0.50 0.55 0.60 0.65 0.70 0.75 0.80 True π ⋆ Isaac Gravestock (UZH) Adaptive prior weighting Leuven 19 May 2016 16 / 21

Type I Error against Fixed Borrowing 0.20 δ = 0 δ = 0 . 2 δ = 0 . 4 δ = 0 . 6 δ = 0 . 8 0.15 δ = 1 EB FB B(1,1) Type I Error FB B(.5,.5) 0.10 0.05 0.00 0.50 0.55 0.60 0.65 0.70 0.75 0.80 True π ⋆ 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 Isaac Gravestock (UZH) Adaptive prior weighting Leuven 19 May 2016 18 / 21

Expected Total Sample Size in Adaptive Trial ◮ Use Rubinstein mortality as historical data: x 0 / n 0 = 49 / 193 = 25 . 4% ◮ Target sample size: 300 → include between 107 and 300 new patients. 300 250 Expected Total Number of Patients 200 150 100 50 π ⋆ = 0 . 254 π ⋆ = 0 . 2 π ⋆ = 0 . 3 π ⋆ = 0 . 35 π ⋆ = 0 . 4 0 0 50 100 150 200 250 300 New Patients Recruited Isaac Gravestock (UZH) Adaptive prior weighting Leuven 19 May 2016 19 / 21

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] Isaac Gravestock (UZH) Adaptive prior weighting Leuven 19 May 2016 20 / 21