Bayesian ANalysis of Variance for Microarray Analysis

Transcription

1 Bayesian ANalysis of Variance for Microarray Analysis c These notes are copyrighted by the authors. Unauthorized use is not permitted. Bayesian ANalysis of Variance p.1/19

2 Normalization Nuisance effects, e.g. technological variation between arrays and within arrays, can obscure the biology. The heuristic approach to removing technological variation relies heavily on the assumption that a step-by-step analysis, i.e. cleaning and analyzing the data in stages, is appropriate. This is not always a valid assumption. Bayesian ANalysis of Variance p.2/19

3 Normalization Fig pairwise M versus A plots using liver (at concentration 10) dilution series data for unadjusted data. Bolstad et al. (2002) Bayesian ANalysis of Variance p.3/19

4 Heuristic Approach 1. Remove the trends, e.g., i) by a non-parametric smoother, or ii) lining up the distributions between arrays. 2. Estimate the gene-wise effects. This assumes implicitely that the distribution of intensity values for genes that change are relatively symmetric (up-regulated versus down-regulated) and uniform over the range of possible intensity values. Bayesian ANalysis of Variance p.4/19

5 Heuristic Approach Shortcommings of the Heuristic approach Technology can obscure biology Underestimates uncertainty due to technology Loss of Power Lewin et al. (2006) devised a Bayesian approach to simultaneously model gene and array effects. Bayesian ANalysis of Variance p.5/19

6 Lewin et al. (2006) Bayesian Modeling of Differential Gene Expression Log-gene expression, y gsr, for gene g, condition (k) s = 1, 2 and replicate r, is modeled as y g1r N y g2r N ( α g 1 ) 2 δ g + β g1r,σg1 2 ( α g + 1 ) 2 δ g + β g2r,σg2 2 σ 2 g2 lognorm(µ s,η 2 s) where α g is the overall log-intensity for gene g, δ g accounts for differential expression, and β gsr accounts for the array-effect (normalization). Bayesian ANalysis of Variance p.6/19

7 Lewin et al. (2006) The array-effects are modeled with quadratic splines β gsr = b (0) sr0 + b (1) sr0(α g α 0 ) + b (2) sr0(α g α 0 ) 2 K + b (2) srk (α g α srk ) 2 I [α g α srk ] k=1 with unknown coefficients b (p) srk and knots α srk. Noninformative priors are specified for all unknown variables. In order to identify the model, the authors require R r=1 β gsr = 0. Bayesian ANalysis of Variance p.7/19

8 Lewin et al. (2006) Notice that the model allows for normalization within condition s = 1, 2, rather than across replicates and conditions. This is possble by imposing the constraint R r=1 2 s=1 β gsr = 0. If control genes are placed on the array, that always fluoresce (positive controls), or never light up (negative controls), then we may require δ g = 0, to improve our estimates of β gsr. Note: Typically, controls do show variation, so beware! Bayesian ANalysis of Variance p.8/19

9 Lewin et al. (2006) Gene g is selected as differentially expressed if: p g = P( δ g > δ cut and α g > α cut data) p g p cut. Consider again the heuristic approach where 1. data is normalized, ŷ g = y g ˆf( ˆα g ) 2. individual gene variances are estimated, ˆσ g 2 3. tests are performed for gene-wise differential expression t = ŷg ˆδ g ˆσ g Bayesian ANalysis of Variance p.9/19

10 Gottardo et al. (2006) Bayesian Robust Inference for Differential Gene Expression in Microarrays with Multiple Samples Due to all of the steps in microarray experiments, technical variation is comming in to play from many sources. The potential for outliers is high. The use of robotics has helped reduce variation, but outliers still pose a problem, from say, imperfections in the slides, or experimental efficiency which can vary. Bayesian ANalysis of Variance p.10/19

11 Gottardo et al. (2006) The following assumes that the arrays are two-color technology, i.e., two channels each labeled with a different dye (Cy3 and Cy5). The log transformed signal, y isr, is modeled as y isr = γ is + ǫ isr ωir for i = 1,...,I genes; s = 1, 2 samples (separate for each channel/dye); and r = 1,...,R replicates. γ is is the effect of interest, modeled as a mixture below. (ǫ i1r,ǫ i2r ) and ω ir are assumed independent (ǫ i1r,ǫ i2r ) V i N 2 (0,V i ) ω ir ν r Ga(ν r /2,ν r /2). Bayesian ANalysis of Variance p.11/19

12 Gottardo et al. (2006) such that residuals have a bivariate T distribution ( ) ǫi1r ǫ i2r, T(ν r, 0,V i ) ωir ωir with ν r d.f. and covariance V i. This allows the measurements at the same spot between the channels to be correlated. The precision is given by: V 1 1 i = (1 ρ 2 ) λ ǫi1 λ ǫi1 λ ǫi2 ρ λ ǫi1 λ ǫi2 ρ λ ǫi1. Bayesian ANalysis of Variance p.12/19

13 Gottardo et al. (2006) One reason that there may be correlation between the dye measurements, at each spot, is that the amount of material that binds at a spot is a random variable. This can lead to true correlation between the channel-wise measurements. In the case of a one channel experiment, the model is modified y isr = γ is + ǫ isr ωir ǫ isr λ ǫis N 2 (0,λ 1 ǫ is ) ω ir ν r Ga(ν r /2,ν r /2). that is, ω ir and ǫ isr are modeled as a priori independent. Bayesian ANalysis of Variance p.13/19

14 Gottardo et al. (2006) Returning to γ is, the effect of interest, accounting for differential expression between the samples in the different channels, there are two cases, γ i1 = γ i2 andγ i1 γ i2. The mixture model is and (γ i λ γ,p) (1 p)n(γ i1 ; 0,λ 1 γ 12 )1 [γi1 =γ i2 ] +pn(γ i1 ; 0,λ 1 γ 1 )N(γ i2 ; 0,λ 1 γ 2 )1 [γi1 γ i2 ] λ ǫis Ga(a 2 ǫ is /b ǫis,a ǫis /b ǫis ). Bayesian ANalysis of Variance p.14/19

15 Gottardo et al. (2006) Differential gene expression is identified by the marginal posterior pobability B Pr(γ i1 γ i2 y) 1 B k=1 1 [γ (k) i1 γ(k) i2 ] Their code, BRIDGE, is freely available at: Bayesian ANalysis of Variance p.15/19

16 Gottardo et al. (2006) HIV24 Posterior probability posterior probability t distribution Gaussian distribution posterior mean (gamma1 gamma2) Figure 1. Posterior probabilities from the BRIDGE method with both Gaussian and t-errors plotted against the posterior differences between γ 1 and γ 2 (estimated log-ratios) from the model with t-distribution for the HIV24 data. Most of the log-ratios are shrunk close to zero and have very low posterior probabilities of differential expression. The use of the t-distribution increases the posterior probabilities of expression for several of the genes. Bayesian ANalysis of Variance p.16/19

17 Gottardo et al. (2006) Table 2 Agreement and disagreement about the differential expression of genes in the HIV data when the four replicates are divided into two sets of two. For each method, Agreement denotes the number of genes declared to be differentially expressed based on both sets of two replicates, while Disagreement refers to the number of genes that were declared to be differentially expressed based on one set of two replicates, and not to be differentially expressed based on the other set of two replicates. Using BRIDGE and EBarrays, we report three numbers, the first two corresponding to posterior probability thresholds of 0.5 and 0.9, while the third controls the FDR at 10%. Rep. 1&3 vs. 2&4 Rep. 1&4 vs. 2&3 Agreement Disagreement Agreement Disagreement BRIDGE PP > PP > FDR < SAM FDR < EBarrays GG PP > PP > FDR < EBarrays LNN PP > PP > FDR < Efron s method Local FDR < t-test Raw p < Adj. p < Bayesian ANalysis of Variance p.17/19

18 Gottardo et al. (2006) With 3 factors the prior for γ is modified as (γ λ, p) p 1 N(γ 1 ; 0,λ 1 γ 1 )1 [γ1 =γ 2 =γ 2 ] +p 2 N(γ 1 ; 0,λ 1 γ 1 )N(γ 2 ; 0,λ 1 γ 23 )1 [γ1 γ 2 =γ 3 ] +p 3 N(γ 2 ; 0,λ 1 γ 2 )N(γ 1 ; 0,λ 1 γ 13 )1 [γ1 =γ 3 γ 2 ] +p 4 N(γ 3 ; 0,λ 1 γ 3 )N(γ 1 ; 0,λ 1 γ 12 )1 [γ1 =γ 2 γ 3 ] +p 5 N(γ 1 ; 0,λ 1 γ 1 )N(γ 2 ; 0,λ 1 γ 2 )N(γ 3 ; 0,λ 1 γ 3 )1 [γ1 γ 2 γ 3 ] Bayesian ANalysis of Variance p.18/19

19 Gottardo et al. (2006) Table 3 Estimates of the mixing probabilities for the five patterns of expression, and the numbers of genes classified into each pattern on the BRCA data from model (2) with three samples, and from EBarrays. A gene was classified into a pattern if the corresponding posterior probability of its conforming to that pattern was greater than 0.5. The null pattern has the highest average posterior probability according to both methods. P 1 P 2 P 3 P 4 P 5 BRIDGE Mix. prob No. of genes EBarrays Mix. prob (LNN) No. of genes Bayesian ANalysis of Variance p.19/19

20 Project : Due March 22nd. (i) Ishwaran and Rao: JASA, 2003, P:438: Soma, Hwang. (ii) Lewin, Richardson et al.: Biometrics, 2006 P:1-9: Rajesh, Litton. (iii) Gottardo, Raftery et al., Biometrics, 2006, P10-18: Souparno, Beverly. p.1/1