Introduction to DESeq and edgeR packages Peter A.C. ’t Hoen

Poisson distribution discrete probability distribution that expresses the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate and independently of the time since the last event = expected k = number of occurrences

Count process Poisson distribution Y t ~ Poisson(λ t ) with λ t = pn t t: tag λ: true expression Y: observed expression p: probability n: total number of RNA molecules Truncated Poisson distribution: zero can mean not expressed or not counted Count variance ~ λ t Murray F Freeman and John W Tukey. Ann Math Statist, 21: , (1950)

Negative binomial distribution discrete probability distribution of the number of successes in a sequence of Bernoulli trials before a specified (non-random) number r of failures occurs also arises as a continuous mixture of Poisson distributions where the mixing distribution of the Poisson rate is a gamma distribution. That is, we can view the negative binomial as a Poisson(λ) distribution, where λ is itself a random variable, distributed according to Gamma(r, p/(1 − p)).

edgeR (1) Robinson, Smyth (Biostatistics, 2008; Bioinformatics 2007) Package available from Bioconductor with very informative vignette Y ij ~ NB (  ij,  ) Var (Y ij ) =  ij ( 1 +  ij x  ) Negative binomial (gamma Poisson) with average mu Phi is overdispersion parameter (biological variation)  = 0 gives Poisson distribution

Overdispersion in our data

edgeR (2) Test per gene Y gij ~ NB (  gij,  g ) where  gij = M j x p gj Var (Y gij ) =  gij ( 1 +  ij x  g ) p gi is proportion of tags for tag g in sample i M j is library size for sample i and library j  g is dispersion parameter for tag g

edgeR (3) Estimation of common dispersion parameter by conditioning  g on the sum of counts and maximizing the common likelihood l C (  ) =  l g (  g ) Common dispersion parameter OR weighted linear combination of common and individual likelihoods WL (  g ) = l g (  g ) +  l C (  g )

edgeR (4) Exact test replacing hypergeometric probabilities with NB- derived probabilities (qCML) for single factor experiment Generalized linear models and Cox-Reid profile-adjusted likelihood (CR) method for multifactorial experiments

edgeR: what is new? Exact Test not able to work with confounders  replaced by generalized linear model with log likelihood ratio test Abundance trending in dispersion estimates

Dispersion trend dispersion abundance

Dispersion trending (after filtering for low ab) dispersion abundance

DESeq (1) Anders and Huber: Genome Biology (2010) 11:R106 Roughly same principles as edgeR No multifactorial analysis implemented yet

DESeq (2) (1)Y ij ~ NB (  ij, σ 2 ij ) (2)  ij = s j q i,ρ(j) s j scaling factor for sample j q i,ρ(j) proportional concentration of tag i in condition ρ (3)σ 2 ij =  ij + s 2 j ν i,ρ(j) ν i,ρ(j) is a smooth function depending on q i,ρ(j) (concentration) Count noise Extra variance

DESeq (3): variance trend with expression Purple: Poisson Dashed orange: edgeR (before trending) Orange: DESeq You can derive: Squared CV is 1/μ + φ

DESeq (3) Differences with edgeR: Complete shrinkage to trended dispersion; limited tagwise dispersion estimates Different variance estimates for different sample groups allowed Deals better with samples with large differences in read depth?

DESeq (4): statistical testing In analogy to initial edgeR implementation exact test on the NB probabilities in the two conditions

Conclusions edgeR and DESeq are comparable implementation of statistical tests using NB distribution edgeR and DESeq produce largely similar results Implementation of generalized linear models in edgeR allows for testing with confounders Results comparable to limma for medium – high expressed genes: modeling of stochastic effects is particularly important for low expressed genes

Comparison to limma (on sqrt scaled data)