Limma analysis

Linear models for microarray analysis

Linear models for microarray analysis (Limma) is a R and Bioconductor package for organising and analysing cDNA and Affymetrix microarray data. It is written by Gordon Smyth at WEHI.

Algorithm details

For a p * n matrix of expression intensities, Limma is fitting p linear models (one for each row). The lmFit function does this by calling functions such as lm.series which use lm.fit in Package:Stats. For cDNA/oligo two spotted technologies the matrix of expression intensities is usually the marix M values with respect to treatments. For Affymetrix single channel arrays the expression intensities are directly analysed comparing two treatments.

y = Xβ + ε

The linear model reduces to effectively estimating average M values using a categorical design matrix. If correlation between rows (spots) is estimated, then the function duplicateCorrelation is called. This fits a reml model on all genes to estimate a rho correlation matrix. A fisher transformation (identical to atanh(x)) is then applied to the rho matrix and an average rho calculating an mean correlation with trim=0.15 by default, which is then backtransformed to give a consensus correlation. This correlation can be utilised in lmFit by calling gls.series which fits a generalized least squares model.

List elements from lmFit

> names(fit) [1] "coefficients" "rank" "assign" "qr" [5] "df.residual" "sigma" "cov.coefficients" "stdev.unscaled" [9] "pivot" "method" "design" "Amean" [13] "genes" coefficients= estimated M values qr = qr decomposition df.residual=residual degrees of freedom for each gene Amean = Estimated unweighted A values stdev.unscaled = scaling required method = model fitting method used design=the design matrix X used genes = list of gene names

Empirical Bayes using conjugate priors is used to calculate moderated t-statistics and B statistics. The wrapper function eBayes for ebayes calculates these statistics for each row (spot).

List elements from ebayes

> names(ebayes(fit)) [1] "df.prior" "s2.prior" "s2.post" "t" "p.value" "var.prior" [7] "lods" df.prior= dg s2.prior=(so)2 s2.post=(sg)2 t=moderated t-statistics p.value = moderated t-statistics p values lods = log odds B statistics