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

XXIII International Biometric Conference Montréal
July 2006
2
Microarray experiments

• 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?

3
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
• Define the contrast variable
• Xijk Tijk - Cijk
• We analyse the sequence of , where

4
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
5
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.
6
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.
7
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.

8
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(?)
9
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

10
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

11
Data from homemade spotted array
12
Leukemia dataset Golub et al.
13
Results
Table 1 Percentages of differentially expressed
genes identified by different common methods
Simulation studies show that FDR too conservative
rather than Bayesian thresholding not
conservative enough!!
14
Conclusions

• The empirical Bayesian approach is a natural way
  to incorporate our prior belief that not many
  genes are differentially expressed.
• This method is not a one-size-fits-all!
• Future work includes more reliable variance
  estimation through latent class modelling and a
  Bayesian model that can be fitted to raw data
  eliminating the need to normalise.

15
References

• Golub, T. et al. (1999). Molecular classification
  of cancer Class discovery and class prediction
  by gene expression monitoring. Science, 286,
  531-537.
• 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.