An Adaptive Empirical Bayesian Thresholding Procedure for Analysing Microarray Experiments

1
An Adaptive Empirical Bayesian Thresholding
Procedure for Analysing Microarray Experiments

• Rebecca E. Walls, Stuart Barber, Mark S.
  Gilthorpe John T. Kent
• rebecca_at_maths.leeds.ac.uk
• University of Leeds

PG seminar series 6th December 2006
2
Outline

• An introduction to gene expression and
  microarrays
• ( THIS WILL BE VERY BRIEF!!)
• Empirical Bayesian methodology
• Results

3
Gene expression

• Gene expression - process in which a cell
  transfers the coded information stored in its DNA
  into proteins
• Gene expression is regulated genes will only
  express at the right time and in the right cell
  and responds to enviromental stimuli
• Interesting questions to try and answer...
• Which genes are expressed in which tissues?
• How is the expression of a gene influenced by
  external stimuli?
• What patterns of gene expression cause a disease
  or lead to disease progression?
• What patterns of gene expression influence
  response to treatment?

4
Philosophy behind microarrays

• Microarrays allow the comparison of gene
  expression between multiple samples for many
  thousands of genes simultaneously
• Previously we described gene expression as a
  process- how can we measure a process??
• Proteins notoriously hard to measure
  accurately
• Instead, we measure the abundance of
  the intermediary molecule mRNA

5
Notation

• Suppose we observe intensity measurements Tijk
  (treatment) and Cijk (control), where
• i 1, , n genes
• j 1, , m chips
• k 1, , r replicate spots
• For which of the i genes are Tijk and Cijk
  significantly different in intensity level?
• For those significant genes, can we estimate
  the level of differential expression?

6
Statistical challenges

• Data generation
• Noise
• Background
• Experimental variability (chip, samples, lab)
• Intensities not well distributed
• T-test becomes invalid!!
• Data structure
• Many obs (from 100 20k genes per chip,
  replicated)
• Multiple testing ( suppose that 10,000 genes are
  tested could incur as many as 500 false
  positives at 5 level!)
• Lack of independence (20k genes -gt 1M proteins!)

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Contrast variables

• We assume that Tijk and Cijk have been suitably
  transformed and normalised and have constant
  variance (Huber et al.) denote adjusted
  intensities Tijk and Cijk (on logarithmic
  scale)
• Define the contrast variable
• Xijk Tijk - Cijk
• We analyse the sequence of , where

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Sparse sequences

• Assume most genes are not differentially
  expressed
• sequence of will be sparse
• (0, 0, 0, -3.1, 0, 0, 0, 3.7, -2.6, 0, 0, 2.1)
• (2.2, -0.1, -1.4, -5.4, -2.6, 2.2, -1.1, 1.0,
  -1.8, 1.2, 1.0, -0.1)
• We adapt the EBayesThresh methodology of
  Johnstone and Silverman (2002), originally
  designed for thresholding wavelet coefficients

Add normally distributed noise with mean 0 and
some variance s2
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Empirical Bayesian methodology

• Suppose we have an observation Z which can be
  written in the form
• The prior on µ is a mixture of d0(µ), a point
  mass at zero, and ?(µa), a heavy-tailed Laplace
  distribution,
• in proportions according to the

mixing weight, 0 ? 1.
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Empirical Bayesian methodology

• Suppose we have an observation Z which can be
  written in the form
• The prior on µ is a mixture of d0(µ), a point
  mass at zero, and ?(µa), a heavy-tailed Laplace
  distribution,
• in proportions according to the

for small ?
mixing weight, 0 ? 1.
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Posterior distribution for µ

• The posterior distribution for µ given Z z is a
  mixture distribution with a point mass at zero
  and is given by
• where,
• and be calculated explicitly.

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Estimating µ

• Estimate µ by the posterior median
• For a fixed ?, is a
  monotonic function with a thresholding property

An observation Z will yield a non-zero µ if Z
exceeds some threshold t(?)
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Parameter estimation

• Suppose now we have a sequence of observations of
  the form
• for i 1, , n
• We need to estimate the mixing weight ?, the
  scaling parameter a, and variance s2.
• We use a maximum likelihood approach to find
  estimates for ? and a.
• To estimate s2, we employ a sum-of-squares
  approach from fitting a linear additive model
  which accounts for both variation between chip
  replicates and spot replicates nested within the
  chips

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Linear additive model

• Model each observed intensity by
• with µi gene specific mean
• between chip variation
• within chip variation
• Given estimates for sB2 and sW2, the variance of
  is given by
• Simulations show sum-of-squares approach more
  reliable as ? grows large

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Error distribution for spike-in experiment HIV
data
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Results for HIV spike-in experiment
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Data from homemade spotted array E.Coli
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Results
Table 1 Numbers of differentially expressed
genes identified by different common methods
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Conclusions

• The empirical Bayesian approach is a natural way
  to incorporate our prior belief that not many
  genes are differentially expressed.
• Making an adjustment to the variance is not
  sufficient compensation for using the incorrect
  prior distribution! Future work includes using a
  Laplace distribution for the errors.

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References

• Hedenfalk, I. et al. (2001). Gene expression
  profiles in hereditary breast cancer. The New
  England Journal of Medicine, 344 (8), 539-548.
• Huber, W. et al. (2003). Variance stabilization
  applied to microarray data calibration and to the
  quantification of differential expression.
  Bioinformatics, 18, S96-S104.
• Johnstone, I. and Silverman, B. (2004). Needles
  and straw in haystacks Empirical bayes estimates
  of possibly sparse sequences. The Annals of
  Statistics, 32, 1594-1649.
• McLachlan, G., Bean, R. and Ben-Tovim Jones, L.
  (2006). A simple implementation of a normal
  mixture approach to differential expression in
  multiclass microarrays. Bioinformatics, 22 (13),
  1608-1615.
• Smyth, G., Michaud, J. and Scott, H. (2005). Use
  of within-array replicate spots for assessing
  differential expression in microarray
  experiments. Bioinformatics, 21 (9), 2067-2075.
• Tusher, V., Tibshirani, R. and Chu, C. (2001).
  Significance analysis of microarray applied to
  transcriptional responses to ionizing radiation.
  Proc. Natn. Acad. Sci. USA, 98, 5116-5121.