Multiple Comparisons for Microarray Experiments Motivation and Methods

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SPONSORED BY
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Microarrays, pathways and Thomas Bayes

• Mik Black
• Department of Statistics, The University of
  Auckland
• Otago Genomics Facility Meeting
• February 13, 2004

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Overview

• Statisticians and microarrays.
• Summary of current research.
• Statistical models for microarray data.
• False discovery rate control.
• Current and future work.

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Statisticians and microarrays

• Statisticians role
• Experimental design
• Power calculations
• Normalization
• Analysis
• Similar to any experimental situation.
• Often little scope for novel statistical research.

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Why bother?

• Build research partnerships.
• Interdisciplinary crosstalk.
• Publications.
• Increase diversity of research.

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Why bother?

• Build research partnerships.
• Interdisciplinary crosstalk.
• Publications.
• Increase diversity of research.
• Encourage good statistical practice.
• Early adoption of novel statistical methods.

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Student research

• Marcus Davy (M.Sc. thesis)
• characteristics of FDR controlling procedures for
  microarray experimentation.
• Hadley Wickham (M.Sc. project)
• spatial normalization.
• visualization techniques.
• Thomas Tiang (PGDipSci project)
• normalization strategies.
• Po-Hsun Huang (Summer studentship/Honours
  project)
• normalization for boutique arrays.

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Personal research

• Bayesian and likelihood-based models for
  microarray data.
• Semi-parametric modeling.
• False discovery rate control and extensions.

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Standard post-normalization analysis

• Calculate test statistics for each gene.
• These reflect the magnitude of observed
  differential expression relative to observed
  variability.
• Use resampling methods to obtain p-values.
• Bootstrapping
• Permutations
• Use p-values to determine genes undergoing
  significant differential expression, subject
  pre-determined level of error rate control.
• Parametric alternatives use normal distribution
  theory.

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Bayesian models for microarrays

• Newton et al. (2001) introduced a Bayesian model
  for single-array experiments based on Gamma
  distributions.
• This approach has since been extended to
  encompass multiple arrays, and to provide greater
  flexibility in terms of (non)parametric
  assumptions (Kendziorski et al. 2002 Newton and
  Kendziorski, 2002 Newton et al. 2003).
• Produces estimates of the probability of
  differential expression for each gene in the
  experiment.
• These probabilities can then be used to produce a
  list of genes undergoing significant
  differential expression.

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Which approach?

• P-values
• Advantages simple, (non-)parametric, standard.
• Disadvantages under-powered, genes considered in
  isolation, variance structure?
• Bayesian model
• Advantages intuitive output (probabilities),
  models underlying distribution of expression
  means (improved power), variance shrinkage.
• Disadvantages poor performance under model
  mis-specification, complex implementation,
  non-standard.

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Determining differential expression

• Each of the previously described methods produces
  an estimate of the likelihood of differential
  expression for each gene.
• The next step involves deciding which genes
  should be considered to have undergone
  significant differential expression.
• This decision is closely linked to the level of
  error we are willing to tolerate in our analysis.
• Although there are many options available,
  control of the false discovery rate has become a
  popular approach.

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False discovery rate control

• Introduced by Benjamini and Hochberg (1995).
• Want to control the number of incorrect
  rejections, V, as a proportion of the total
  number of rejections, R.

• Stepwise p-value adjustment guarantees

• Finner and Roters (2001) showed that

where ?0 is the proportion of true null
hypotheses.

• FDR is expectation, so control is on average.

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Adaptive control of the FDR

• Control of the FDR at level a requires estimation
  of the proportion of true null hypotheses, .
• In the microarray setting, this is the proportion
  of genes on the array which do not undergo
  differential expression.
• Estimation of this quantity is not
  straightforward. Although Storey (2002) and
  Storey and Tibshirani (2003) have proposed
  methods for this, they can produce severely
  biased estimates (Black, 2004).
• Newton et al. (2003) demonstrated that their
  Bayesian approach provides adaptive FDR control.

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Case study DNA methylation in Arabidopsis

• Two array, dye swapped design.
• ddm1 mutant versus wild-type.
• 4224 spots, 1882 features (genes).
• Multiple replicate spots per gene.
• 1523 genes with 2 spots each.

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Logged data background and foreground (array 1)
Foreground
Background
Normalized Foreground
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Logged data background and foreground (array 2)
Foreground
Background
Normalized Foreground
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Per-array loess normalization (FG only)
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Spatial check
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Simple analysis of normalized data

• Calculate two sample pooled variance t test
  statistic for each gene.
• Calculate p-value for each gene either based on
  normal probabilities, or from bootstrapping.
• Use p-values in estimation procedure.
• Use stepwise p-value adjustment to adaptively
  control the FDR at level to achieve
  FDR .

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Improved analysis

• Often small per-gene variances can make small
  fold-changes statistically significant.
• Tusher et al. (2001) proposed the SAM
  (Significance Analysis of Microarrays) method to
  overcome this problem.
• Add small fudge factor to denominator of test
  statistic.
• Usually a low quantile of the distribution of
  gene-specific standard errors.
• Functions as a shrinkage estimation procedure.

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Bayesian analysis

• Fit model of Newton and Kendziorski (2002) to the
  data.
• Use probabilities of differential expression to
  achieve adaptive FDR control.
• Hierarchical model structure allows data
  sharing across genes (effectively producing
  shrinkage).

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Summary of results

• Numbers of differentially expressed genes, and
  estimates of p0

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Bayes versus p-values gene order

• Percentage agreement on rankings of first n
  genes

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Conclusions

• Per-gene variances resulted in a large number of
  genes reported as differentially expressed under
  adaptive FDR control.
• Use of SAM procedure radically reduced this
  number through shrinkage estimation.
• Removes problem of small variances making small
  fold-changes significant.
• Bayesian model also used shrinkage estimators and
  adaptive FDR control, but detected more (and
  different) genes as differentially expressed.
• Simulations support superiority of Bayesian
  procedure assuming model is correctly specified.

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Current and future work

• Likelihood-based method which can control the
  actual (rather than average) proportion of false
  discoveries for a given set of rejections.
• e.g., For a list of differentially expressed
  genes, the probability that less than 10 of
  these are false positives is at least 95.
• Applying the Bayesian analysis approach to the
  problem of identifying differentially regulated
  pathways.
• Extension of the work of Mootha et al. (2003).

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Acknowledgements

• Rebecca Doerge (Purdue University)
• Rob Marteinssen (Cold Spring Harbor)
• Vincent Colot (URGV)
• Zach Lippman (Cold Spring Harbor)