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BGX 1 Sylvia Richardson Natalia Bochkina Alex Lewin Centre for Biostatistics Imperial College, London Bayesian inference in differential expression experiments BBSRC Biological Atlas of Insulin Resistance

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BGX 2 Background Investigating changes of gene expression under different conditions is one of the key questions in many biological experiments Specificity of the context is –High dimensional data (ten of thousands of genes) and few samples Need to borrow information –Many sources of variability Important to adopt a flexible modelling framework Bayesian Hierarchical Modelling allows to capture important features of the data while maintaining generalisibility of the tools/ techniques developed

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BGX 3 Modelling differential expression Differential expression parameter Condition 1 Condition 2 Posterior distribution (flat prior) Mixture modelling for classification Hierarchical model of replicate variability and array effect Hierarchical model of replicate variability and array effect Start with given point estimates of expression

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BGX 4 Outline Background Bayesian hierarchical models for differential expression experiments Decision rules based on tail posterior probabilities –Comparison with existing approaches –FDR estimation for tail posterior probabilities Extension of tail posterior probabilities to analysing multiclass experiments Illustration Discussion and further work

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BGX 5 Data: y gcr = log gene expression gene g, replicate r, condition c g = gene effect δ g = differential effect for gene g between 2 conditions r(g)c = array effect – modelled as a smooth (spline) function of g gc 2 = gene specific variance 1st level y g1r N( g – ½ δ g + r(g)1, g1 2 ) y g2r N( g + ½ δ g + r(g)2, g2 2 ) Σ r r(g)c = 0, r(g)c = function of g, parameters {c,d} 2nd level Flat priors for g, δ g, {c,d} gc 2 g (a c, b c ) (lognormal or inverse-gamma) I -- Bayesian hierarchical model for differential expression (Lewin et al, Biometrics, 2006) Exchangeable variances

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BGX 6 Joint modelling of array effects and differential expression Performs normalisation simultaneously with estimation –Gives fewer false positives than plug in BHM set up allows to check some of the modelling assumptions using mixed posterior predictive checks: – the need for gene specific variances – their 2nd level distribution Found that lognormal or 2 parameter inverse gamma distribution for the variances gave similar model checks

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BGX 7 Selecting genes that are differentially expressed Interested in testing the null hypothesis Two broad approaches have been used: P value typeMixture P(H 0 | y gcr ) H0H1H0H1 U [0,1] close to 0 close to 1 close to 0 References Baldi and Long Smyth 2004, … Moderated t stat Lonnstedt & Speed 02, Newton & Kendziorski, 01, 03 Lonnstedt & Britton 05, Gottardo 06, …. H ( g ) 0 : ± g = 0 versus H ( g ) 1 : ± g 6 = 0 :

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BGX 8 Bayesian mixtures Relies on specification of prior model for : Choice of model for the alternative (see the poster by Alex Lewin) –Could influence the performance of the classification –To check how the alternative fits the data is non standard Investigate properties of Bayesian selection rules based on non informative prior for ± g ± g » ¼ 0 ± 0 + ( 1 ¡ ¼ 0 ) h ( ± g j ´ ) ± g

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BGX 9 II -- Bayesian selection rules for pairwise comparisons 1st level (no array effect): Hierarchical model Extend p value approach to consider the tail probabilities of appropriate function of parameters ® g ; ± g » 1 ; ¾ 2 gs » f ( ¾ 2 gs j µ s ) µ s » f ( µ s ) y g 1 r j ® g ; ± g ; ¾ 2 g 1 » N ( ® g ¡ ± g = 2 ; ¾ 2 g 1 ) ; r = 1 ;:::; m 1 ; y g 2 r j ® g ; ± g ; ¾ 2 g 2 » N ( ® g + ± g = 2 ; ¾ 2 g 2 ) ; r = 1 ;:::; m 2 :

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BGX 10 Posterior distributions Define the Bayesian T statistic: The following conditional distributions hold Posterior distributions: t g = ± g = w g ± g j ¹ y g ; w g » N ( ¹ y g ; w 2 g ) ; t g j ¹ y g ; w g » N ( ¹ y g = w g ; 1 ) : w h ere f ( w 2 g j s 2 gs ) d eno t es t h epos t er i or d ens i t yo f t h evar i ances

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BGX 11 Tail posterior probabilities 1 (N. Bochkina and SR, 2006) Use selection rules of the form : What statistic to choose: How to define its percentiles ? –we suppose that we could have observed data with (its expected value of under the null) –work out the percentiles using posterior distributions conditional on T g = ± g or t g ? ¹ y g = 0 ¹ y g = 0 : P f T g > T ( ® ) g j y gsr g ¸ p cu t, Summarise the distribution of the T g by a tail area

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BGX 12 Tail posterior probabilities 2 Recall Corresponding distribution function involves numerical integration computationally expensive The percentile is easy to calculate Consider the tail probability: f ( ± g j ¹ y g ; s 2 gs ) = Z w g ' (( ± g ¡ ¹ y g )= w g ) f ( w 2 g j s 2 gs ) d ( w 2 g ) But Distribution function of does not involve gene specific parameters f ( t g j ¹ y g = 0 ; s 2 gs ) f ( t g j ¹ y g = 0 ; w g ) » N ( 0 ; 1 ) t ( ® ) g = t ( ® ) = © ¡ 1 ( 1 ¡ ® = 2 ) p ( t g ; t ( ® ) ) = P fj t g j > t ( ® ) j y gsr g

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BGX 13 5%0.5% Can simulate or numerically evaluate F 0 the distribution of under H 0 p ( t g ; t ( ® ) ) Key point: F 0 is gene independent (conjugate case) D ens i t yo f p ( t g ; t ( ® ) ) f or® = 0 : 05 f or d a t agenera t e d un d er t h enu llh ypo t h es i s ¾ g 1 = ¾ g 2

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BGX 14 Another Bayesian rule A natural idea is to compare the parameter to 0, i.e. to consider : or its complementary or the 2-sided alternative : It turns out that this Bayesian selection rule behaves like a p-value: –Distribution of is uniform under H 0 –There is equivalence with frequentist testing based on the marginal distribution of under the null, in the spirit of the moderated t statistic introduced by Smyth 2004 p ( ± g ; 0 ) p ( ± g ; 0 ) = P f ± g > 0 j y gsr g max f p ( ± g ; 0 ) ; 1 ¡ p ( ± g ; 0 ) g ± g ¹ y g

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BGX 15 Link between p(δ g,0) and the moderated t statistic U n d er H 0, t h e d i s t r i b u t i ono f p ( d g ; 0 ) i sun i f orm Moderated t statistic

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BGX 16 Histograms of measure of differential expression Simulated data Under H 0 Under H 1 p(t g, t g (α) )p(δ g, 0)

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BGX 17 Tail posterior probabilities 3 Investigate the performance of selection rules based on In particular: –what is the FDR associated with each value of ? –In the conjugate case: –How does this rule compares to rules based on p cu t p ( ± g ; 0 ) FDR ( p cu t ) = ¼ 0 P f p ( t g ; t ( ® ) ) > p cu t j H 0 g P f p ( t g ; t ( ® ) ) > p cu t g ; Use Storey Use F 0 Use observed proportion p ( t g ; t ( ® ) ) = P fj t g j > t ( ® ) j y gsr g > p cu t

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BGX 18 Comparison of estimated (solid line) and true FDR (dashed line) on simulated data π 0 = 0.95 π 0 = 0.90π 0 = 0.70

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BGX 19 III-- Data Sets and Biological questions Biological Questions Understand the mechanisms of insulin resistance Cell line experiments where reaction of mouse muscle cell line to treatment by insulin or metformin (an insulin replacement drug) is observed after 2 and 12 hours Questions of interest related to simple and compound comparisons 3 replicates for each condition, Affymetrix MOE430A chip, genes per chip Data pre-processed by RMA and normalised using intensity dependent LOESS normalisation

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BGX 20 p(t g, t (α) ), α = max{ p(δ g,0), 1- p(δ g,0)} - 1 Volcano plots for muscle cell data: Change between insulin and control at 2 hours Cut-off : Less peaked around zero Allows better separation Peaked around zero Varies steeply as a function of ¹ y g

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BGX 21 Insulin versus control Tail posterior probabilities Estimated FDR 2 hours12 hours 1151 selected (FDR= 0.5%) 13 selected (FDR= 0.5%) π 0 = 0.61π 0 = 0.98

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BGX 22 Metformin versus control Tail posterior probabilities Estimated FDR 2 hours12 hours 1854 selected (FDR= 0.5%) 72 selected (FDR= 0.5%) π 0 = 0.56 π 0 = 0.79

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BGX 23 IV – Extension to the analysis of multi class data In our case study, 3 groups (control c=0, insulin c=1, metformin c=2) and 3 times points: t=0, t=1 ( 2 hours), t=2 (12 hours) each replicated 3 times ANOVA like model formulation suited to the analysis of such multifactorial experiments : y g t cr = ® g + ° g t + ± g t c + " g t cr ; " g t cr » N ( 0 ; ¾ 2 g ) ¾ ¡ 2 g » ¡ ( a ; b ) ® g ; ° g t ; ± g t c » 1 ; t ; c = 1 ; 2. Global variance parametrisation (borrowing information)

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BGX 24 Joint tail posterior probabilities Interest is in testing a compound null hypothesis, i.e. involving several differential parameters e.g. testing jointly for the effect of insulin and metformin at 2 hours In this case, we are interested in a specific alternative: Note: Rejecting the null hypothesis in an ANOVA setting corresponds to a different alternative Define joint tail posterior probabilities: where is the Bayesian T statistic for each treatment H ( g ) 0 : ± g 11 = 0 &± g 12 = 0 versus H ( g ) 1 : ± g 11 6 = 0 &± g 12 6 = 0 : p J g = P fj t g 11 j > t ( ® ) & j t g 12 j > t ( ® ) j d a t a ) t g t c = ± g t c = w g

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BGX 25 Benefits of joint posterior probabilities Takes into account correlation of the differential expression measures between the conditions induced by sharing the same variance parameter Usual practice is to: –Carry out pairwise comparisons –Select genes for each comparison using same cut-off on the pp –Intersect lists and find genes common to both lists Joint pp shown to lead to fewer false positives in this case of positive correlation (simulation study)

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BGX 26 Correlation of DE parameters and Bayesian T statistic for insulin and metformin (2 hours) With joint tail posterior probabilities, and a cut-off of p cut =0.92, 280 selected as jointly perturbed at 2 hours Applying pairwise comparison and combining the lists adds another 47 genes to the list C orre l a t i on b e t ween t g 11 an d t g 12 C orre l a t i on b e t ween ± g 11 an d± g 12

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BGX 27 Discussion 1 Tail posterior probabilities (Tpp) is a generic tool that can be used in any situations where a large number of hypotheses related in a hierarchical fashion are to be tested We have derived the distribution of the Tpp under the null and proposed a corresponding estimate of FDR This distribution requires numerical integration but is gene independent (conjugate case), so only needs to be evaluated once Tpp is a smooth function of the amount of DE with a gradient that spreads the genes, thus allowing to choose genes with desired level of uncertainty about their DE Interesting connection between Bayesian and frequentist inference for the differential expression parameter

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BGX 28 Discussion 2 Interesting to compare performance of Tpp with that of mixture models E.g Gamma mixtures (see poster by Alex Lewin) δ g ~ 0 δ G (-x|1.5, 1 ) + 2 G (x|1.5, 2 ) H 0 H 1 Dirichlet distribution for ( 0, 1, 2 ) Exp(1) hyper prior for 1 and 2 Also Normal and t mixtures have been considered: δ g ~ 0 δ 0 + (1- 0 ) T(ν,μ,τ) (μ ~ 1, τ, ν -1~ Exp(1) ) δ g ~ 0 δ 0 + (1- 0 ) N(μ,τ) (μ ~ 1, τ ~ Exp(1) )

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BGX 29 Simulated data 3000 variables, 6 replicates, 2 conditions y g1r N( g, g 2 ) y g2r N( g + δ g, g 2 ) g 2 ~ LogNorm(-3.85, 0.82), g ~ Norm(7, 25), δ g : slightly asymmetric: 5%: δ g | δ g > 0 ~ h( δ g ), 10%: δ g | δ g < 0 ~ h(-δ g ), 85%: δ g ~ N(0, 0.01), w h ere h ( j ± g jj ± g 6 = 0 ) = 0 : 2 U [ 0 ; 2 : 5 ] + 0 : 4 U [ 0 : 07 ; 0 : 7 ] + 0 : 4 N ( 0 : 7 ; 0 : 7 ).

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BGX 30 Comparison of mixture and tail pp Fit 3 mixture models (Gamma, Normal, t alternative) and flat model. Classification mixtures: P{ H 1 |data}, flat: tail posterior probability. Comparable performance, with a little edge for the Gamma and Normal mixture

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BGX 31 BBSRC Exploiting Genomics grant Wellcome Trust BAIR consortium Colleagues in the Biostatistics group: Marta Blangiardo, Anne Mette Hein, Maria de Iorio Colleagues in the Biology group at Imperial Tim Aitman, Ulrika Andersson, Dave Carling Papers and technical reports: For the tail probability paper: Thanks

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