b

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Bayesian inference in differential expression
experiments
Sylvia Richardson Natalia Bochkina Alex
Lewin Centre for Biostatistics Imperial College,
London
Biological Atlas of Insulin Resistance
www.bgx.org.uk
BBSRC
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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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Modelling differential expression
Condition 2
Condition 1
Start with given point estimates of expression

Hierarchical model of replicate variability and
array effect
Hierarchical model of replicate variability and
array effect
Posterior distribution (flat prior)
Differential expression parameter
Mixture modelling for classification
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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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I -- Bayesian hierarchical model for differential
expression (Lewin et al, Biometrics, 2006)

• Data ygcr log gene expression gene g,
  replicate r, condition c
• ?g gene effect
• dg differential effect for gene g between
  2 conditions
• ?r(g)c array effect modelled as a smooth
  (spline) function of ?g
• ?gc2 gene specific variance
• 1st level yg1r ? N(?g ½ dg ?r(g)1 ,
  ?g12)
• yg2r ? N(?g ½ dg ?r(g)2 , ?g22)
• Sr?r(g)c 0, ?r(g)c function of ?g ,
  parameters c,d
• 2nd level Flat priors for ?g , dg, c,d
• ?gc2 ? g (ac, bc)
• (lognormal or inverse-gamma)

Exchangeable variances
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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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Selecting genes that are differentially expressed

• Interested in testing the null hypothesis
• Two broad approaches have been used

P value type Mixture P(H0 ygcr)
H0 H1 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, .

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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
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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

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Posterior distributions

• Define the Bayesian T statistic
• The following conditional distributions hold

• Posterior distributions

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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

Summarise the distribution of the Tg by a tail
area
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Tail posterior probabilities 2Recall

• Corresponding distribution function involves
  numerical integration ? computationally expensive

• But
• Distribution function of
• does not involve gene specific parameters

? The percentile is easy to calculate ? Consider
the tail probability
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Key point F0 is gene independent (conjugate case)
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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 H0
• 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

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Link between p(dg,0) and the moderated t statistic
Moderated t statistic
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Histograms of measure of differential
expression Simulated data
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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

Use F0
Use Storey
Use observed proportion
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Comparison of estimated (solid line) and true
FDR (dashed line) on simulated data
p0 0.90
p0 0.70
p0 0.95
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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, 22690 genes per chip
• Data pre-processed by RMA and normalised using
  intensity dependent LOESS normalisation

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Volcano plots for muscle cell data Change
between insulin and control at 2 hours
p(tg , t (a)), a 0.05
2max p(dg ,0), 1- p(dg ,0) - 1
Cut-off 0. 925
Peaked around zero Varies steeply as a function
of
Less peaked around zero Allows better separation
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Insulin versus control
p0 0.61
p0 0.98
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Metformin versus control
Tail posterior probabilities
2 hours
12 hours
p0 0.56
p0 0.79
Estimated FDR
72 selected (FDR 0.5)
1854 selected (FDR 0.5)
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IV Extension to the analysis of multi class data

• In our case study, 3 groups (control c0, insulin
  c1, metformin c2) and 3 times points t0, t1
  ( 2 hours), t2 (12 hours) each replicated 3
  times
• ANOVA like model formulation suited to the
  analysis of such multifactorial experiments

Global variance parametrisation (borrowing
information)
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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

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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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Correlation of DE parameters and Bayesian T
statistic for insulin and metformin (2 hours)

• With joint tail posterior probabilities, and a
  cut-off of pcut 0.92, 280 selected as jointly
  perturbed at 2 hours
• Applying pairwise comparison and combining the
  lists adds another 47 genes to the list

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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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Discussion 2

• Interesting to compare performance of Tpp with
  that of mixture models
• E.g Gamma mixtures (see poster by Alex Lewin)
• dg p0d0 p1G (-x1.5, ?1) p2G (x1.5, ?2)
• H0 H1
• Dirichlet distribution for (p0, p1, p2)
• Exp(1) hyper prior for ?1 and ?2
• Also Normal and t mixtures have been considered
• dg p0d0 (1-p0) T(?,µ,t) (µ 1, t,
  ? -1 Exp(1) )
• dg p0d0 (1-p0) N(µ,t) (µ 1, t
  Exp(1) )

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Simulated data

• 3000 variables, 6 replicates, 2 conditions
• yg1r ? N(?g, ?g2)
• yg2r ? N(?g dg, ?g2)
• ?g2 0.03 LogNorm(-3.85, 0.82),
• ?g Norm(7, 25),
• dg slightly asymmetric
• 5 dg dg gt 0 h( dg),
• 10 dg dg lt 0 h(-dg),
• 85 dg N(0, 0.01),

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Comparison of mixture and tail pp

• Fit 3 mixture models (Gamma, Normal, t
  alternative) and flat model.
• Classification mixtures P H1 data, flat tail
  posterior probability.

Comparable performance, with a little edge for
the Gamma and Normal mixture
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Thanks
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
www.bgx.org.uk/ For the tail probability
paper www.bgx.org.uk/Natalia/Bochkina.ps