Linear Models and Empirical Bayes Methods for Assessing Differential Expression in Microarray Experiments
Abstract
The problem of identifying differentially expressed genes in designed microarray experiments is considered. Lonnstedt and Speed (2002) derived an expression for the posterior odds of differential expression in a replicated two-color experiment using a simple hierarchical parametric model. The purpose of this paper is to develop the hierarchical model of Lonnstedt and Speed (2002) into a practical approach for general microarray experiments with arbitrary numbers of treatments and RNA samples. The model is reset in the context of general linear models with arbitrary coefficients and contrasts of interest. The approach applies equally well to both single channel and two color microarray experiments. Consistent, closed form estimators are derived for the hyperparameters in the model. The estimators proposed have robust behavior even for small numbers of arrays and allow for incomplete data arising from spot filtering or spot quality weights. The posterior odds statistic is reformulated in terms of a moderated t-statistic in which posterior residual standard deviations are used in place of ordinary standard deviations. The empirical Bayes approach is equivalent to shrinkage of the estimated sample variances towards a pooled estimate, resulting in far more stable inference when the number of arrays is small. The use of moderated t-statistics has the advantage over the posterior odds that the number of hyperparameters which need to estimated is reduced; in particular, knowledge of the non-null prior for the fold changes are not required. The moderated t-statistic is shown to follow a t-distribution with augmented degrees of freedom. The moderated t inferential approach extends to accommodate tests of composite null hypotheses through the use of moderated F-statistics. The performance of the methods is demonstrated in a simulation study. Results are presented for two publicly available data sets.
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The significance of apparent differential expression was assessed by a Bayesian approach, adjusting p-values to control false discovery rate.
“the significance of the apparent differential expression was assessed by a Bayesian approach [89], adjusting the p-values to control the false discovery rate.”
From Genome-wide transcriptional changes induced by phagocytosis or growth on bacteria in Dictyostelium (2008) · cites this paper as [89]
Fitting the same linear model to each gene allows borrowing strength between genes to moderate residual variances.
“The fact that the same linear model is fitted to each gene allows us to borrow strength between genes in order to moderate the residual variances (16).”
From limma powers differential expression analyses for RNA-sequencing and microarray studies (2015) · cites this paper as [16]
It was an innovation of limma to show that exact small-sample inference could be conducted using empirical Bayes posterior variance estimators.
“It was an innovation of the limma package to show that exact small-sample inference could be conducted using the empirical Bayes posterior variance estimators (16).”
From limma powers differential expression analyses for RNA-sequencing and microarray studies (2015) · cites this paper as [16]
The core component of limma is the ability to fit gene-wise linear models to gene expression data to assess DE.
“The core component of the limma package is the ability to fit gene-wise linear models to gene expression data in order to assess DE (16).”
From limma powers differential expression analyses for RNA-sequencing and microarray studies (2015) · cites this paper as [16]
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“the empirical Bayes log-odds of differential PSI/PSU/PIRs (Smyth, 2004) (as implemented in “ebayes”, from the limma package in R).”
Method
- Authors
- Smyth, G. K.
- Affiliations
- Walter and Eliza Hall Institute of Medical Research (AU)
- Corresponding author
- Smyth, Gordon K
- ORCID
- Smyth, Gordon K
- Venue
- Statistical Applications in Genetics and Molecular Biology
- Publisher
- Walter de Gruyter GmbH
- Published
- 2004-01-12
- Volume
- 3
- Issue
- 1
- Pages
- 1-25
- ISSN
- 1544-6115 (ISSN-L)
- OpenAlex
- W2081098333
- Cited by
- 11,829
- References
- 30
- Citations by year
- 2026: 214 · 2025: 252 · 2024: 324 · 2023: 427 · 2022: 374 · 2021: 464 · 2020: 534 · 2019: 510 · 2018: 550 · 2017: 634 · 2016: 718 · 2015: 892 · 2014: 901 · 2013: 917 · 2012: 982
- FWCI
- 78.27
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Statistical Applications in Genetics and Molecular Biology (publishedVersion)
- PubMed (publishedVersion)
- Topics
- Life Sciences › Biochemistry, Genetics and Molecular Biology › Molecular Biology › Gene expression and cancer classification
- Life Sciences › Biochemistry, Genetics and Molecular Biology › Molecular Biology › Bioinformatics and Genomic Networks
- Life Sciences › Biochemistry, Genetics and Molecular Biology › Molecular Biology › Genomics and Phylogenetic Studies
- Keywords
- differential gene expression, microarray experiments, empirical Bayes, hierarchical models, posterior odds, hyperparameter estimation, composite null hypothesis, two-color microarrays
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
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