Bayesian Statistics for Personalized Medicine. David Yang

Transcription

1 Bayesian Statistics for Personalized Medicine David Yang

2 Outline Why Bayesian Statistics for Personalized Medicine? A Network-based Bayesian Strategy for Genomic Biomarker Discovery

3 Part One Why Bayesian Statistics for Personalized Medicine?

4 Data A 60- or 20-yr old woman?

5 Prior

6 Bayes Rule Bayes Rule Prior distribution + Data likelihood Posterior distribution Thomas Bayes ( )

7 Clinical Decision-Making Non-smoker

8 Personalized Medicine - My View Give right medicine to the right people at right time & location. It is a Decision-Making X: Genotype, smoking, climate,. A Statistical Model

9 Bayesian probability is personalized Small n for a rare disease: n 1 Flexible and adaptive: B.A.D.

10 Personalized Medicine Today Choose the right medicine to the right sub-population of patients sharing the same biomarker Pharmacogenetics Pharmacogenomics Pharmacoproteomics Pharmacometabonomics

11 Biomarkers

12 Biomarker and Drug Co-Development

13 Clinical Trials Simple 2-arm Random Trial design Based on Biomarkers Prospective-Retrospective Marker by Treatment Bayesian Adaptive Design Adaptively assign patients Interim Analyses Save time & Money

14 Part Two An Integrative Bayesian Framework for Biomarker Discovery

15 Model Selection A Fundamental Task Goodness of Fit + Parsimony IC γ = 2 log[ f (D θ γ )]+ d γ F The simpler the better! #Parameters (complexity) 27

16 Five Phases A Minimal Causal Network Model 5 Latent Variables TCM Example

17 Model/Variable Selection In Generalized Linear Models Indicator Variable # 1 if γ j = "! 0 Model Indicator γ = X j ( γ 1,..., γ p is selected Otherwise ) γ Γ

18 BVS and BMS Bayesian Model Selection Bayesian Variable Selection = Γ γ γ γ D γ γ D D γ ) ( ) ( ) ( ) ( ) ( p p p p p ) ( 1) ( ) 1 ( = = = Γ γ D γ D p I p j j γ γ

19 Models X1 X2 X3 BMS p( γ D) Illustration BVS p( γ j = 1 D) 62% 75% 35% 40% 30% 20% 5% 3% 2% 0% 0%

20 Challenges in Biomarker Discovery High Dimension (P>N) 20-25k genes Multiple Comparison issue Hard to Incorporate Knowledge Results vary from lab to lab Ignore structural, correlational, or functional relationships between genes

21 The Big Data Era

22 Gene-Pathway-Outcome Pathway 1 Pathway 2 Pathway K Clinical Outcome 1 Clinical Outcome 2

23 Identify both Pathways & Genes K Z =1α + ξ k T k(γ ) β k(γ ) +ε ε ~ N(0,σ 2 I ) k=1 " $ 1 if pathway k is selected ξ k = # %$ 0 otherwise ξ " $ γ j = # %$ 1 if gene j is selected 0 otherwise T X T k(γ ) = X k(γ ) U k 1 Z Y

24 Hierarchical Structure Genes (γ 1,...,γ p ) Pathways 1 2 K (ξ 1,...,ξ K ) Z

25 Prior Distributions Bernoulli Distributions for Pathway Indicators p(ξ ϕ) ~ K k=1 ϕ k ξ k (1 ϕ k ) 1 ξ k PLS (Partial Least Square) G-Prior b (j,c) Dc*N (0,c(T (j,c) T (j,c) ) z ),

26 Markov Random Field for Gene Selection Indicators ' p(γ µ,η) = exp(µ1 p γ +ηγ ' Rγ) p(c j Dc i,i[n j )! exp (c j (mzg X i[n j c i )),

27 Model Fitting via MCMC Marginalized Posterior p(z,a,b (j,c),j,cdy,x)! P n Hybrid Gibbs Sampler Sampling Z from i~1 p(z idy,x,a,b (j,c),j,c)p(a)p(b (j,c) DX,j,c)p(j,c) P p(zdy,x,j,c)!n (0,S (j,c) ) Xn M-H for simulating indicators p(j,cdy,x,z)! 1 DS (j,c) D 1 2 i~1 I(A i ): exp ({ Z S{1 (j,c) Z ) 2 P K j i~1 pj i i (1{p i ) 1{j i exp (m1 0 p czgc0 Rc):

28 Simulation Design K=19 Pathways P=315 Genes N=50 Samples X ~ MN according to the gene-networks in each pathway Z = X 4(γ ) β 4 + X 6(γ ) β 6 + X 11(γ ) β 11 + X 12(γ ) β 12 + X 15(γ ) β 15 +ε ε ~ N(0,σ 2 )

29 Posterior Selection Probabilities ε ~ N(0, 0.1) Genes Causal Pathway Causal Genes Effect Size 4 56, , , Pathways

30 Posterior Selection Probabilities ε ~ N(0, 0.5) Where to set the threshold? Pathway Genes Causal Pathway Causal Genes Effect Size 4 56, , ,

31 Where to cut? Test Error Idea of Cross-Validation fold -- Leave-one out Number of Selected Genes

32 Sensitivity and Specificity

33 0.8 Application to Acute Ischemic Stroke Gene Index ARG POLR2K IL1B IL23R Pathway Index Cytokme-Cytokim Receptor Interaction (POLR2K) Glioma (IL1B, IL23R) Cytosolic DNA Sensing Pathway (ARG-1)

34 Discussion Personalized Medicine is Bayesian BVS a powerful hierarchical strategy Sparse parameterization (e.g., RMF) Bayesian Model Averaging Deep Learning + Big Data Biologically-based interpretation Statistical & Clinical Significance Systems Biology

35 Acknowledgement Funded by NICHD: 1R01HD061404: Bayesian Variable Selection in Generalized Linear and Hierarchical Models with Missing Values Team Members Ben Peng - Bioinformatics James Zhu - Applied Mathematician Xiaoshuai Zhang - Biostat Brad Ander - Computational Biologist

36 All models are wrong, But some are useful! Thank You

37 Future Plan 37

38 Agilent Technology 38

39 Symptom Disease Bio-Networks 2014 Zhou, et al. Nature Communications 39

40 Gene-Environment Interactions Genetics365 EMA365

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42 ibvs for GWAS

43 ibvs for GWAS

44 ibvs for GWAS

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