Hierarchical expectation propagation for Bayesian aggregation of average data

Transcription

1 Hierarchical expectation propagation for Bayesian aggregation of average data Andrew Gelman, Columbia University Sebastian Weber, Novartis also Bob Carpenter, Daniel Lee, Frédéric Bois, Aki Vehtari, and Amy Racine 23 Sept 2015

2 Motivation Trial design with hierarchical models using patient data and summaries of external data More generally: incorporating aggregate data into statistical analysis

3 Scenario Non-inferiority and bio-similarity trials Test of a candidate substance against an active control Candidate substance developed in-house: lots of raw data, individual patient level longitudinal data Active control developed externally: we only have data summaries from publications or submission documents New trial will be similar to earlier trials of the active control drug, but still to some extent different Also the conditions differ in the two experiments

4 Meta-analysis Partial pooling of information from two studies Statistical challenges: Only summary data are available on one study Nonlinear model, so no closed-form model for average data Only 2 groups so hierarchical modeling is not so easy, requires strong prior information

5 Drug disease modeling of drug responses Key application is clinical trial simulations for study design Simulation of drug responses of patients over time New designs can be considered Different endpoints can be explored including time to event Hierarchical (population) based models require patient-level data Problem: Same disease progression, same patient population and likely similar mechanism of action, but population model describes only in-house drug How to learn from published summaries of longitudinal data in the context of nonlinear hierarchical models?

6 Semi-Mechanistic Turn-Over Models Linking Pharmacokinetics (PK) with Pharmacodynamics (PD) PD response «R» can be safety or efficacy related driven by PK effect on «bio-compartement» Zero order «production» / first order «elimination» of response R 4 variants: zero / first order inhibition / stimulation due to PK Drug response with respect to reference state (placebo) Some regimens may lead to PK causing oscillations and hence oscillations in response Source: Peletie LA et al.; J Pharmacokinet Pharmacodyn Bayes Pharma S. Weber et al. 20. May 15 Bayesian aggregation of summary data Public

7 General formulation Direct data y = (y jt ; j = 1,..., J; t = 1,..., T ) with model p(y j α j, φ) Hierarchical model p(α, φ) = p(φ) J j=1 p(α j φ) Use Stan to draw posterior simulations from p(α, φ y) p(φ) J j=1 p(α j φ) J j=1 p(y j α j, φ) External dataset y = (y jt ; j = 1,..., J ; t = 1,..., T ) Observe time series of averages: ȳ = (ȳ 1,..., ȳ T ) External dataset has parameters φ = φ + δ Informative prior on δ

8 Simulated Example Data Set 50 Patients per Treatment Arm Placebo, Treatment 1 & 2 Solid line is true population mean No population differences No between-trial variation 8 Bayes Pharma S. Weber et al. 20. May 15 Bayesian aggregation of summary data Public

9 Integrating external summary data Simplest case: 2 datasets follow the same model Populations must be comparable Parameters must be identical Natural disease progression must be the same Relax these assumptions: φ = φ + δ Extreme cases δ 0: complete pooling p(δ) 1: no pooling

10 Natural Bayesian approach doesn t work Consider external data as latent variables y not observed; all we see is ȳ Computationally expensive For example, 300 patients and 15 measurements per patient Instead, we ll model ȳ directly Take advantage of central limit theorem

11 Multivariate normal approx to p(y 0 φ0 ) I I I I Given φ0 simulate data from 1000 hypothetical patients Compute mean M and T T covariance matrix S Approx p(y 0 φ0 ) by N(y 0 M, S/J 0 ) J 0 = number of patients in external data, not the same as the 1000

12 Importance sampling algorithm Fit model to direct data; get draws from p(α, φ y) For each draw of α, φ: Draw δ from prior p(δ) Compute φ = φ + δ Simulate data from 1000 hypothetical patients Approx p(ȳ φ ) by N(ȳ M, S/J ) Compute importance ratio N(ȳ M, S/J )

13 Hierarchical expectation propagation (EP) algorithm Simple importance sampling won t work if p(δ) is broad Need iterative algorithm At each step, sample from pseudo-prior g(φ, δ) Multiply importance ratios by p(φ,δ) g(φ,δ) EP: match moments to get update for g Use smoothed importance weights and stable moment matching

14 Examples Hierarchical linear model: Local data: yjt N(α j1 + α j2 x t + βx 2 t, σ 2 y ) αj N(µ α, Σ α ) External data: y jt N(α j1 + α j2 x t + βx 2 t, σ 2 y ) α j N(µ α, Σ α ). δ = µ α µ α. Hierarchical logistic Hierarchical PKPD

15 Hierarchical linear model example: Posterior mean +/ sd from EP algorithm from 3 starting points (Red lines show estimate from local data, blue includes aggregate data, green uses complete data) Updating of mu_a[1] Updating of mu_a[2] Updating of b Updating of sigma_a[1] Updating of sigma_a[2] Updating of delta[1] Step of algorithm Updating of delta[2] Pareto shape k^ of importance weights Efficiency of importance samples

16 Hierarchical logistic example: Posterior mean +/ sd from EP algorithm from 3 starting points (Red lines show estimate from local data, green uses complete data) Updating of mu_a[1] Updating of mu_a[2] Updating of b Updating of sigma_a[1] Updating of sigma_a[2] Updating of delta[1] Updating of delta[2] Pareto shape k^ of importance weights Efficiency of importance samples

17 Hierarchical PK/PD example: Posterior mean +/ sd from EP algorithm from 3 starting points (Red lines show estimate from local data, green uses complete data) Updating of lalpha_0 Updating of lalpha_s Updating of lkappa Updating of lemax_s Updating of sigma_lalpha_0 Updating of sigma_lalpha_s Updating of delta[1] Pareto shape k^ of importance weights Efficiency of importance samples

18 Discussion Use fake-data simulation to build trust in results Hope to improve efficiency of EP by approximating p(φ, δ) (currently approximating p(φ) and p(δ)) Shift parameter δ instead of cut New way to think about meta-analysis