Dispersion modeling for RNAseq differential analysis
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1 Dispersion modeling for RNAseq differential analysis E. Bonafede 1, F. Picard 2, S. Robin 3, C. Viroli 1 ( 1 ) univ. Bologna, ( 3 ) CNRS/univ. Lyon I, ( 3 ) INRA/AgroParisTech, Paris IBC, Victoria, July 2016 S. Robin (AgroParisTech / INRA) Dispersion modeling for RNAseq differential analysis IBC, Victoria 1 / 20
2 Modeling dispersion in RNAseq experiments 1 Modeling dispersion in RNAseq experiments 2 Statistical inference & Test statistics 3 Simulations & Illustration 4 Conclusions and future work S. Robin (AgroParisTech / INRA) Dispersion modeling for RNAseq differential analysis IBC, Victoria 2 / 20
3 Modeling dispersion in RNAseq experiments General problem RNAseq is a sequencing based technology that gives access to a measure of the expression level of all the genes from a given species in a given sample (condition) Differential analysis: p genes, d conditions (possibly with replicates). Find the genes, the expression of which vary across conditions. Data. Y ijr = RNAseq read count for gene i in replicate r of condition j S. Robin (AgroParisTech / INRA) Dispersion modeling for RNAseq differential analysis IBC, Victoria 3 / 20
4 Modeling dispersion in RNAseq experiments Negative binomial model RNAseq count = number of reads mapped onto a gene s sequence. Observed variability often exceeds this expected according to Poisson. S. Robin (AgroParisTech / INRA) Dispersion modeling for RNAseq differential analysis IBC, Victoria 4 / 20
5 Modeling dispersion in RNAseq experiments Negative binomial model RNAseq count = number of reads mapped onto a gene s sequence. Observed variability often exceeds this expected according to Poisson. Popular model. [14] Y ijr N B(λ ij, α i ) where 1/α = over-dispersion parameter: V(Y ijr ) = λ ij (1 + λ ij /α i ). S. Robin (AgroParisTech / INRA) Dispersion modeling for RNAseq differential analysis IBC, Victoria 4 / 20
6 Modeling dispersion in RNAseq experiments Negative binomial model RNAseq count = number of reads mapped onto a gene s sequence. Observed variability often exceeds this expected according to Poisson. Popular model. [14] Y ijr N B(λ ij, α i ) where 1/α = over-dispersion parameter: V(Y ijr ) = λ ij (1 + λ ij /α i ). Differential analysis. For each gene i, test H 0 = {λ i1 = λ i2 } vs H 1 = {λ i1 λ i2 }. S. Robin (AgroParisTech / INRA) Dispersion modeling for RNAseq differential analysis IBC, Victoria 4 / 20
7 Modeling dispersion in RNAseq experiments Modeling over-dispersion Assumption on α: Same α for all genes: unrealistic; Gene-specific α i : hard to estimate, especially with few replicates. S. Robin (AgroParisTech / INRA) Dispersion modeling for RNAseq differential analysis IBC, Victoria 5 / 20
8 Modeling dispersion in RNAseq experiments Modeling over-dispersion Assumption on α: Same α for all genes: unrealistic; Gene-specific α i : hard to estimate, especially with few replicates. Several approaches: Shrinkage: edger [15,13], DSS [18]; Function of mean expression: [8], NBPseq [5], DEseq [1,11]; Bayesian estimation: bayseq [6]; Mixture model: DEXUS [7] (components = conditions) + many others: NOISeq [17], PoissonSeq [10], QuasiSeq [12], TSPM [2] S. Robin (AgroParisTech / INRA) Dispersion modeling for RNAseq differential analysis IBC, Victoria 5 / 20
9 Modeling dispersion in RNAseq experiments Mixture model for the dispersion Poisson-Gamma representation. U Gam(α, α), Y U P(λU) Y N B(λ, α) Over-dispersion generated by the latent variable U. S. Robin (AgroParisTech / INRA) Dispersion modeling for RNAseq differential analysis IBC, Victoria 6 / 20
10 Modeling dispersion in RNAseq experiments Mixture model for the dispersion Poisson-Gamma representation. U Gam(α, α), Y U P(λU) Y N B(λ, α) Over-dispersion generated by the latent variable U. Mixture model for the dispersion. Genes belong to K different dispersion groups: S. Robin (AgroParisTech / INRA) Dispersion modeling for RNAseq differential analysis IBC, Victoria 6 / 20
11 Modeling dispersion in RNAseq experiments Mixture model for the dispersion Poisson-Gamma representation. U Gam(α, α), Y U P(λU) Y N B(λ, α) Over-dispersion generated by the latent variable U. Mixture model for the dispersion. Genes belong to K different dispersion groups: Dispersion group of gene i: Z i M (1; (ω k )) ; S. Robin (AgroParisTech / INRA) Dispersion modeling for RNAseq differential analysis IBC, Victoria 6 / 20
12 Modeling dispersion in RNAseq experiments Mixture model for the dispersion Poisson-Gamma representation. U Gam(α, α), Y U P(λU) Y N B(λ, α) Over-dispersion generated by the latent variable U. Mixture model for the dispersion. Genes belong to K different dispersion groups: Dispersion group of gene i: Z i M (1; (ω k )) ; Latent dispersion variable: U ijr Z i = k Gam(α k, α k ); S. Robin (AgroParisTech / INRA) Dispersion modeling for RNAseq differential analysis IBC, Victoria 6 / 20
13 Modeling dispersion in RNAseq experiments Mixture model for the dispersion Poisson-Gamma representation. U Gam(α, α), Y U P(λU) Y N B(λ, α) Over-dispersion generated by the latent variable U. Mixture model for the dispersion. Genes belong to K different dispersion groups: Dispersion group of gene i: Z i M (1; (ω k )) ; Latent dispersion variable: U ijr Z i = k Gam(α k, α k ); Observed count: Y ijr U ijr P(λ ij U ijr ). S. Robin (AgroParisTech / INRA) Dispersion modeling for RNAseq differential analysis IBC, Victoria 6 / 20
14 Modeling dispersion in RNAseq experiments Distributions U ijr Gam(α k, α k ) Y ijr N B(λ, α k ) S. Robin (AgroParisTech / INRA) Dispersion modeling for RNAseq differential analysis IBC, Victoria 7 / 20
15 Modeling dispersion in RNAseq experiments Graphical representation For gene i: Marginal distribution: Y ijr Z i = k N B(λ ij, α k ), Y ijr = k ω k N B(λ ij, α k ) S. Robin (AgroParisTech / INRA) Dispersion modeling for RNAseq differential analysis IBC, Victoria 8 / 20
16 Statistical inference & Test statistics 1 Modeling dispersion in RNAseq experiments 2 Statistical inference & Test statistics 3 Simulations & Illustration 4 Conclusions and future work S. Robin (AgroParisTech / INRA) Dispersion modeling for RNAseq differential analysis IBC, Victoria 9 / 20
17 Statistical inference & Test statistics A two-hidden layer model Parameter θ = {ω = (ω k ), α = (α k ), λ = (λ ij )}; Hidden variables H = {Z = (Z i ), U = (U ijr )}; Observed variables Y = (Y ijr ). S. Robin (AgroParisTech / INRA) Dispersion modeling for RNAseq differential analysis IBC, Victoria 10 / 20
18 Statistical inference & Test statistics A two-hidden layer model Parameter θ = {ω = (ω k ), α = (α k ), λ = (λ ij )}; Hidden variables H = {Z = (Z i ), U = (U ijr )}; Observed variables Y = (Y ijr ). Likelihood decomposition. [4] log p θ (Y ) = log p θ (Y, H)dH S. Robin (AgroParisTech / INRA) Dispersion modeling for RNAseq differential analysis IBC, Victoria 10 / 20
19 Statistical inference & Test statistics A two-hidden layer model Parameter θ = {ω = (ω k ), α = (α k ), λ = (λ ij )}; Hidden variables H = {Z = (Z i ), U = (U ijr )}; Observed variables Y = (Y ijr ). Likelihood decomposition. [4] log p θ (Y ) = log p θ (Y, H)dH = E [log p θ (Y, H) Y ] + H [p θ (H Y )] S. Robin (AgroParisTech / INRA) Dispersion modeling for RNAseq differential analysis IBC, Victoria 10 / 20
20 Statistical inference & Test statistics A two-hidden layer model Parameter θ = {ω = (ω k ), α = (α k ), λ = (λ ij )}; Hidden variables H = {Z = (Z i ), U = (U ijr )}; Observed variables Y = (Y ijr ). Likelihood decomposition. [4] log p θ (Y ) = log p θ (Y, H)dH = E [log p θ (Y, H) Y ] + H [p θ (H Y )] = E [log p ω (Z) + log p α (U Z) + log p λ (Y U) Y ] + H [p θ (H Y )] S. Robin (AgroParisTech / INRA) Dispersion modeling for RNAseq differential analysis IBC, Victoria 10 / 20
21 Statistical inference & Test statistics EM algorithm Aim: Find θ = arg max θ log p θ (Y ). S. Robin (AgroParisTech / INRA) Dispersion modeling for RNAseq differential analysis IBC, Victoria 11 / 20
22 Statistical inference & Test statistics EM algorithm Aim: Find θ = arg max θ log p θ (Y ). E step: Compute conditional moments Z Y : multinomial; U Y : mixture of Gammas. S. Robin (AgroParisTech / INRA) Dispersion modeling for RNAseq differential analysis IBC, Victoria 11 / 20
23 Statistical inference & Test statistics EM algorithm Aim: Find θ = arg max θ log p θ (Y ). E step: Compute conditional moments Z Y : multinomial; U Y : mixture of Gammas. M step: Estimate the parameters ω k : explicit; α k : numerical via quasi-newton (or fix-point); λ ij : explicit. S. Robin (AgroParisTech / INRA) Dispersion modeling for RNAseq differential analysis IBC, Victoria 11 / 20
24 Statistical inference & Test statistics Three contrasts Test: H 0 = {λ i1 = λ i2 } vs H 1 = {λ i1 λ i2 }. S. Robin (AgroParisTech / INRA) Dispersion modeling for RNAseq differential analysis IBC, Victoria 12 / 20
25 Statistical inference & Test statistics Three contrasts Test: H 0 = {λ i1 = λ i2 } vs H 1 = {λ i1 λ i2 }. Considered contrasts: Difference = λ i1 λ i2 Ratio = λ i1 / λ i2 ) Log-ratio = ln ( λi1 / λ i2 S. Robin (AgroParisTech / INRA) Dispersion modeling for RNAseq differential analysis IBC, Victoria 12 / 20
26 Statistical inference & Test statistics Three contrasts Test: H 0 = {λ i1 = λ i2 } vs H 1 = {λ i1 λ i2 }. Considered contrasts: Difference = λ i1 λ i2 Ratio = λ i1 / λ i2 ) Log-ratio = ln ( λi1 / λ i2 Constrast variance: First-order approximation derived via the -method. Test statistic = Contrast / V(Contrast) S. Robin (AgroParisTech / INRA) Dispersion modeling for RNAseq differential analysis IBC, Victoria 12 / 20
27 Simulations & Illustration Outline 1 Modeling dispersion in RNAseq experiments 2 Statistical inference & Test statistics 3 Simulations & Illustration 4 Conclusions and future work S. Robin (AgroParisTech / INRA) Dispersion modeling for RNAseq differential analysis IBC, Victoria 13 / 20
28 Simulations & Illustration Simulation design compcoder package [16]: Independent realistic RNAseq simulation. p = 1000, 5000 genes (inc. 10% down- and 10% up-regulated); d = 2 conditions; n j = 3, 5, 10; Library size = Evaluation criteria Type-I error control; ROC curve Estimation of the dispersion: ( ) V(Y ijr ) = λ ij 1 + λ ij E θ(z ik Y )/ α k k S. Robin (AgroParisTech / INRA) Dispersion modeling for RNAseq differential analysis IBC, Victoria 14 / 20
29 Simulations & Illustration Type-I error control (p = 1000) 0.15 nj= MixtNB DESeq edger DSS DEXUS NBPSeq TSPM SAMseq PoisSeq QuasiSeq 0.15 nj= MixtNB DESeq edger DSS DEXUS NBPSeq TSPM SAMseq PoisSeq QuasiSeq 0.15 nj= MixtNB DESeq edger DSS DEXUS NBPSeq TSPM SAMseq PoisSeq QuasiSeq S. Robin (AgroParisTech / INRA) Dispersion modeling for RNAseq differential analysis IBC, Victoria 15 / 20
30 Simulations & Illustration ROC curves (p = 5000) ROC curve ROC curve True positive rate Difference Ratio Log Ratio True positive rate MixtNB DESeq edger DSS False positive rate False positive rate ROC curve ROC curve True positive rate False positive rate MixtNB DEXUS NBPSeq TSPM True positive rate False positive rate MixtNB SAMseq PoisSeq QuasiSeq S. Robin (AgroParisTech / INRA) Dispersion modeling for RNAseq differential analysis IBC, Victoria 16 / 20
31 Simulations & Illustration Estimation of dispersion (p = 5000) relative error relative error Common DESeq2 DEXUS DSS edger MixtNB NBPSeq number of mixture components Estimation error of V(Y ) as a function of K Precision with the different methods S. Robin (AgroParisTech / INRA) Dispersion modeling for RNAseq differential analysis IBC, Victoria 17 / 20
32 Simulations & Illustration Illustration: data from [9] p = genes, d = 2 conditions (treated/control), n j = 3, 4 K = 3 dispersion groups (BIC). Number of declared differentially expressed genes: α 5% 1%.1% Difference Ratio Log Ratio DESeq edger DSS S. Robin (AgroParisTech / INRA) Dispersion modeling for RNAseq differential analysis IBC, Victoria 18 / 20
33 Conclusions and future work Conclusions Summary. A generic framework for RNAseq differential analysis Can account for the library size via an offset µ jr. Flexible modeling of over-dispersion via mixture model A genuine EM algorithm taking advantage of the Poisson-Gamma representation State-of-the art accuracy + control of the type-i error + estimation of the dispersion Published in Biometrics (2015) [3] + R CRAN package MixtNB S. Robin (AgroParisTech / INRA) Dispersion modeling for RNAseq differential analysis IBC, Victoria 19 / 20
34 Conclusions and future work Future works Generalize to more complex designs (taking advantage of the GLM framework) Negative binomial latent-block model (LBM) for metagenomics: Y ijr = number of reads from species i in medium j (rep. r) Simultaneous clustering of species and medium using a variational EM algorithm for Poisson-Gamma. S. Robin (AgroParisTech / INRA) Dispersion modeling for RNAseq differential analysis IBC, Victoria 20 / 20
35 Appendix S. Anders and W. Huber. Differential expression analysis for sequence count data. Genome biology, 11(10):R106, P.L. Auer and R.W. Doerge. A two-stage poisson model for testing RNA-seq data. Statistical applications in genetics and molecular biology, 10(1):1 26, E. Bonafede, F. Picard, S. Robin, and C. Viroli. Modeling overdispersion heterogeneity in differential expression analysis using mixtures. Biometrics, pages n/a n/a, A. P. Dempster, N. M. Laird, and D. B. Rubin. Maximum likelihood from incomplete data via the EM algorithm. J. R. Statist. Soc. B, 39:1 38, Y. Di, D.W. Schafer, J.S. Cumbie, and J.H. Chang. The NBP negative binomial model for assessing differential gene expression from RNA-Seq. Statistical Applications in Genetics and Molecular Biology, 10(1):1 28, T. Hardcastle and K. Kelly. BaySeq: Empirical Bayesian methods for identifying differential expression in sequence count data. BMC Bioinformatics, 11(422):1 15, G. Klambauer, T. Unterthiner, and S. Hochreiter. DEXUS: identifying differential expression in RNA-Seq studies with unknown condtions. Nucleics Acids Research, 42(21):1 11, C.W. Law, Y. Chen, W. Shi, and G.K. Smyth. Voom: precision weights unlock linear model analysis tools for RNA-seq read counts. Genome Biology, 15:R29, H. Li, M. T. Lovci, Y. S. Kwon, M. G. Rosenfeld, X. D. Fu, and G. W. Yeo. Determination of tag density required for digital transcriptome analysis: application to an androgen-sensitive prostate cancer model. Proceedings of the National Academy of Sciences, 105(51): , J. Li and R. Tibshirani. S. Robin (AgroParisTech / INRA) Dispersion modeling for RNAseq differential analysis IBC, Victoria 20 / 20
36 Appendix Finding consistent patterns: a nonparametric approach for identifying differential expression in RNA-Seq data. Statistical Methods in Medical Research, 22(5): , M.I. Love, W. Huber, and S. Anders. Moderated estimation of fold change and dispersion for RNA-seq data with DESeq2. Genome Biology, 15:550, S.P. Lund, D. Nettleton, D.J. McCarthy, and G.K. Smyth. Detecting differential expression in RNA-sequence data using quasi-likelihood with shrunken dispersion estimates. Statistical applications in genetics and molecular biology, 11(5):8, D.J. McCarthy, Y. Chen, and G.K. Smyth. Differential expression analysis of multifactor RNA-seq experiments with respect to biological variation. Nucleics Acids Research, 40(10): , M. D. Robinson, D. J. McCarthy, and G. K. Smyth. edger: a bioconductor package for differential expression analysis of digital gene expression data. Bioinformatics, 26(1): , M. D. Robinson and G. K. Smyth. Small-sample estimation of negative binomial dispersion, with application to SAGE data. Biostatistics, 9: , C. Soneson. compcoder - an R package for benchmarking differential expression methods for RNA-seq data. Bioinformatics, 30(17): , S. Tarazona, F. García-Alcalde, J. Dopazo, A. Ferrer, and A. Conesa. Differential expression in RNA-seq: a matter of depth. Genome research, 21(12): , H. Wu, C. Wang, and Z. Wu. A new shrinkage estimator for dispersion improves differential expression detection in RNA-seq data. Biostatistics, 14(2): , S. Robin (AgroParisTech / INRA) Dispersion modeling for RNAseq differential analysis IBC, Victoria 20 / 20
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