Bayesian Analysis of Differential Gene Expression

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1 Bayesian Analysis of Differential Gene Expression Biostat Journal Club Chuan Zhou Department of Biostatistics Vanderbilt University Bayesian Modeling p. 1/1

2 Lewin et al., 2006 Goals and Data Model specification Model checking Integrated vs. non-integrated Application to mouse data Discussion Bayesian Modeling p. 2/1

3 Goals Difficulties with differential gene expression analysis Fold change not comparable between genes Small numbers of replicates Multiple sources of variability Proposed approach: model biological variability, systematic array effects and differential expression simultaneously A fully Bayesian model, take into account uncertainty in parameter estimates Decision rules based on posterior distributions Use Bayesian FDR estimate to select cutoff point Bayesian Modeling p. 3/1

4 Data Three wild-type mice and three mice with Cd36 gene removed Hybridized to Affymetrix U74A, U74B and U74C chips, total 18 microarrays U74A chip data: three repeated measurements of two conditions for each of the 12,488 genes Data pre-processed by Affymetrix MAS 5.0 software Skewed data use log transformation Bayesian Modeling p. 4/1

5 Bayesian hierarchical model Intuition Observed = Gene effect + Differential effect + Array effect Notations: y gsr = log-expression of gene g, condition s = 1, 2, and replicate r. An ANOVA model y g1r N(α g 1 2 δ g + β g1r, σ 2 g1) y g2r N(α g δ g + β g2r, σ 2 g2) Identifiability constraint: β gs = 0, g, s Bayesian Modeling p. 5/1

6 The Model Model array effect as a function of the expression level β gsr = f sr (α g ). For example, a quadratic spline β gsr =b (0) sr0 + b(1) sr0 (α g a 0 ) + b (2) sr0 (α g a 0 ) 2 K + b (2) sr0 (α g a 0 ) 2 I[α g a srk ], k=1 Assume variances are exchangeable within condition σ 2 gs lognorm(µ s, η 2 s). Gene effects α g and knots a srk are uniform between (a 0, a K+1 ) which are pre-specified bounds. Bayesian Modeling p. 6/1

7 The Model Confounding: normalizing across replicates and conditions in a preprocessing step implicitly assume there is no differential effects Implementation: MCMC using WinBUGS, code available online, 74,922 data points, 1000 iterations took approximately 3 hours on a dual processor 2.4 GHz machine Rules for selecting genes p g P( δ g > δ cut and α g > α cut data) Genes are selected if p g p cut. The choice of p cut is determined by the evaluation of FDR. Bayesian Modeling p. 7/1

8 Model checking Use biological replicate data Exploratory analysis of array effects Divide genes into J groups with similar expression levels y g1r N(α g + β j1r, σ 2 j1 ) Try various functional forms f( ) for array effects determined by DIC Clearly see non-linear relationship between gene effects and array effects Bayesian Modeling p. 8/1

9 Non-linear array effects Bayesian Modeling p. 9/1

10 Predictive checks on prior for variance Four possibilities: equal variance model σgs 2 σs, 2 σs 2 lognorm(0, 10 4 ); exchangeable log-normal variance model; exchangeable with 1-parameter gamma σgs 2 Gam(2, βs prior ), βs prior Gam(10 2, 10 2 ); exchangeable model with 2-parameter gamma σgs 2 Gam(αs prior, βs prior ), αs prior Gam(10 2, 10 2 ) and βs prior Gam(10 2, 10 2 ). Use predictive p-values under the model Simulate y (pred) gs p gs P(S 2(pred) gs N(α g + f(α g ), σ 2 gs) > S 2(obs) gs ) Under the null hypothesis of the model being true, the distribution of p-values is almost uniform. Bayesian Modeling p. 10/1

11 Non-linear array effects Bayesian Modeling p. 11/1

12 Integrated vs. non-integrated analysis Expect to obtain biased estimates of the array effects if they are estimated in a preprocessing step, similar to a measurement error problem A simulation study compare to pre-normalization using loess smoothing The ratios of MSE of array effects (loess vs. full model) are 1.5, 1.3, 1.2, 1.2, 1.4, 1.3 (averaged for 5 simulation each chip). Lower estimated FDR with full model, but no difference in FNR The larger the magnitude of array effects, the larger the difference between the pre-normalized and integrated models Bayesian Modeling p. 12/1

13 Non-linear array effects Bayesian Modeling p. 13/1

14 Application to mouse data Choose α cut = 4, δ cut = log(2) a priori FDR = [1/ S(p cut ] g S(p cut ) (1 p g), where S(p cut ) is the group of genes with p g > p cut and S(p cut ) is its cardinality. standardized log-fold difference t g E(δ g /[σ 2 g1 + σ 2 g2/3] 1/2 data) Bayesian Modeling p. 14/1

15 Non-linear array effects Bayesian Modeling p. 15/1

16 Non-linear array effects Bayesian Modeling p. 16/1

17 Discussion Unified Bayesian hierarchical model Justify functional choice by exploratory analysis Joint estimation of differential effects and array effects Richer output Expert opinions expressed in pre-selected cutoffs Bayesian Modeling p. 17/1