Linear Models and Empirical Bayes Methods for. Assessing Differential Expression in Microarray Experiments

Transcription

1 Linear Models and Empirical Bayes Methods for Assessing Differential Expression in Microarray Experiments by Gordon K. Smyth (as interpreted by Aaron J. Baraff) STAT 572 Intro Talk April 10, 2014

2 Microarray Data G1 G2 G3 G4 G5 G6 G7 G8 G9 G17 G10 G18 G11 G19 G12 G20 G13 G21 G14 G22 G15 G23 G16 G24 Measure expression level across large numbers of genes simultaneously G25 G33 G26 G34 G27 G35 G28 G36 G29 G37 G30 G38 G31 G39 G32 G40 Genes express by producing mrna translated into proteins G41 G49 G42 G50 G43 G51 G44 G52 G45 G53 G46 G54 G47 G55 G48 G56 20,000 protein-coding genes in humans G57 G58 G59 G60 G61 G62 G63 G64 Microarray chip contains cdna for a different gene at each spot Sample cdna hybridizes with cdna on chip

3 Microarray Data Two-color: cdna from two samples dyed red and green Response is log-ratio of intensity y g = log 2 R g G g Relative expressions only (fold changes) Single-channel: cdna from a single dyed sample Absolute expressions

4 Microarray Data Issues: Expensive! - sample sizes are low, number of genes is high n << p Multiple comparisons control for false discovery rate (FDR), e.g. Benjamini and Hochberg (1995, 2000) often assumes independence between genes For two-color microarrays, experimental design is more complicated

5 Microarray Data Design: A B A B A B A B C C Matrix: X = ( 1 ) ( 1 X = 1 ) X = X = Coefficients: α = ( B A ) α = ( B A ) α = ( A C B A ) ( B A α = C B )

6 Assumptions Sample of n microarrays: Response vector y g = (y g1,..., y gn ) T for each gene g Assume E(y g ) = X α g, and Var(y g ) = W g σ 2 g for known design matrix X and weight matrix W g Usually interested in contrasts of coefficients for known contrast matrix C β g = C T α g

7 Assumptions Fitting the model gives: Coefficient estimators ˆα g for α g Contrast estimators ˆβ g = C T ˆα g for β g Variance estimators sg 2 for σg 2 (Note: no assumption that y g is normal or model is fit by OLS)

8 Assumptions Assume: covariance matrices distributions Var( ˆα g ) = V g σ 2 g and Var(ˆβ g ) = C T V g Cσ 2 g ˆβ gi β gj, σ 2 g N(β gj, v gj σ 2 g ) and s 2 g σ 2 g σ2 g d g χ 2 d g all independent, where v gj is the jth diagonal element of C T V g C

9 Differential Expression The problem: Would like to test H 0 : β gj = 0 vs H 1 : β gj 0 Too many genes! - multiple comparison methods assume independence across genes Instead, think of p-values as statistics used to rank genes

10 Differential Expression Previous methods for ranking genes: Fold changes - use ˆβ gj directly t-statistics t gj = ˆβ gj s g vgj Problem: s g small t gj large Offset t-statistics - inflate s g Tusher et al (2001) - minimize coefficient of variation Efron et al (2001) - percentile of sample variances Odds ratios - Lönnstedt and Speed (2002) - empirical Bayes methods to estimate odds of differential expression - replicated experiments only Other methods dependent on specific designs

11 Differential Expression Goal: Extend empirical Bayes method from Lönnstedt and Speed (2002) to more general experiments Priors: Variance Differential expression 1 σ 2 g 1 d 0 s 2 0 χ 2 d 0 P(β gj 0) = p j Fold change β gj σ 2 g, β gj 0 N(0, v 0j σ 2 g )

12 Differential Expression After a bunch of calculgebra that I haven t done yet, we get posterior mean s 2 g = and moderated t-statistic t gj = 1 E(σ 2 g s 2 g ) = d 0s d g s 2 g d 0 + d g, ˆβ gj s g vgj t d0 +d g, under H 0. Since this is an empirical Bayes method, estimate hyperparameters s 2 0 and d 0 from the data.

13 Results Simulation study: All parameters and hyperparameters held constant except d 0 = 1, 10, 1000 Moderated t has fewer false positives than other methods Rigging the game? Swirl data: Mutation in known gene in zebrafish Degrees of freedom for t increases from 4 to 7.17 Ranking more sensible than other methods

14 Discussion Modern relevance: Method included in R package limma as part of Bioconductor Later papers extended the idea of sharing variance - Cui et al (2005) uses a James-Stein-type shrinkage estimator Applications to other -omics data with similar high-dimensional problems Digital gene expression (DGE) starting to overtake microarrays observed as count data modeled with overdispersed Poisson empirical Bayes used to share data about overdispersion parameter across genes

15 Discussion What next? Do the mathy stuff calculation of posterior and marginal distributions estimation of hyperparameters Data normalization Perform simulations Develop critique paper makes a lot of unrealistic assumptions about distribution and independence of σ 2 g imperfect method for imperfect data?

16 The End Any questions?