Given a linear model fit from
lmFit, compute moderated t-statistics, moderated F-statistic, and log-odds of differential expression by empirical Bayes moderation of the standard errors towards a global value.
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numeric value between 0 and 1, assumed proportion of genes which are differentially expressed
numeric vector of length 2, assumed lower and upper limits for the standard deviation of log2-fold-changes for differentially expressed genes
logical, should an intensity-trend be allowed for the prior variance? Default is that the prior variance is constant.
logical, should the estimation of
numeric vector of length 1 or 2, giving left and right tail proportions of
the minimum log2-fold-change that is considered scientifically meaningful
These functions are used to rank genes in order of evidence for differential expression. They use an empirical Bayes method to squeeze the genewise-wise residual variances towards a common value (or towards a global trend) (Smyth, 2004; Phipson et al, 2016). The degrees of freedom for the individual variances are increased to reflect the extra information gained from the empirical Bayes moderation, resulting in increased statistical power to detect differential expression.
Theese functions accept as input an
MArrayLM fitted model object
fit produced by
The columns of
fit define a set of contrasts which are to be tested equal to zero.
The fitted model object may have been processed by
contrasts.fit before being passed to
eBayes to convert the coefficients of the original design matrix into an arbitrary number of contrasts.
The empirical Bayes moderated t-statistics test each individual contrast equal to zero. For each gene (row), the moderated F-statistic tests whether all the contrasts are zero. The F-statistic is an overall test computed from the set of t-statistics for that probe. This is exactly analogous the relationship between t-tests and F-statistics in conventional anova, except that the residual mean squares have been moderated between genes.
df.prior are computed by
s2.post is the weighted average of
sigma^2 with weights proportional to
The log-odds of differential expression
lods was called the B-statistic by Loennstedt and Speed (2002).
F are computed by
eBayes does not compute ordinary t-statistics because they always have worse performance than the moderated versions.
The ordinary (unmoderated) t-statistics can, however, can be easily extracted from the linear model output for comparison purposes—see the example code below.
treat computes empirical Bayes moderated-t p-values relative to a minimum meaningful fold-change threshold.
Instead of testing for genes that have true log-fold-changes different from zero, it tests whether the true log2-fold-change is greater than
lfc in absolute value (McCarthy and Smyth, 2009).
In other words, it uses an interval null hypothesis, where the interval is [-lfc,lfc].
When the number of DE genes is large,
treat is often useful for giving preference to larger fold-changes and for prioritizing genes that are biologically important.
treat is concerned with p-values rather than posterior odds, so it does not compute the B-statistic
The idea of thresholding doesn't apply to F-statistics in a straightforward way, so moderated F-statistics are also not computed.
treat is identical to
eBayes, except that F-statistics and B-statistics are not computed.
lfc threshold is usually chosen relatively small, because significantly DE genes must all have fold changes substantially greater than the testing threshold.
Typical values for
The top genes chosen by
treat can be examined using
Note that the
lfc testing threshold used by
treat to the define the null hypothesis is not the same as a log2-fold-change cutoff, as the observed log2-fold-change needs to substantially larger than
lfc for the gene to be called as significant.
In practice, modest values for
lfc such as
log2(1.5) are usually the most useful.
In practice, setting
lfc=log2(1.5) will usually cause most differentially expressed genes to have estimated fold-changes of 2-fold or greater, depending on the sample size and precision of the experiment.
The use of
trend=TRUE is known as the limma-trend method (Law et al, 2014; Phipson et al, 2016).
With this option, an intensity-dependent trend is fitted to the prior variances
squeezeVar is called with the
covariate equal to
Amean, the average log2-intensity for each gene.
The trend that is fitted can be examined by
limma-trend is useful for processing expression values that show a mean-variance relationship.
This is often useful for microarray data, and it can also be applied to RNA-seq counts that have been converted to log2-counts per million (logCPM) values (Law et al, 2014).
When applied to RNA-seq logCPM values, limma-trend give similar results to the
The voom method incorporates the mean-variance trend into the precision weights, whereas limma-trend incorporates the trend into the empirical Bayes moderation.
limma-trend is somewhat simpler than
voom because it assumes that the sequencing depths (library sizes) are not wildly different between the samples and it applies the mean-variance trend on a genewise basis instead to individual observations.
limma-trend is recommended for RNA-seq analysis when the library sizes are reasonably consistent (less than 3-fold difference from smallest to largest) because of its simplicity and speed.
robust=TRUE then the robust empirical Bayes procedure of Phipson et al (2016) is used.
This is frequently useful to protect the empirical Bayes procedure against hyper-variable or hypo-variable genes, especially when analysing RNA-seq data.
squeezeVar for more details.
eBayes produces an object of class
MArrayLM-class) containing everything found in
fit plus the following added components:
numeric matrix of moderated t-statistics.
numeric matrix of two-sided p-values corresponding to the t-statistics.
numeric matrix giving the log-odds of differential expression (on the natural log scale).
estimated prior value for
degrees of freedom associated with
row-wise numeric vector giving the total degrees of freedom associated with the t-statistics for each gene. Equal to
row-wise numeric vector giving the posterior values for
column-wise numeric vector giving estimated prior values for the variance of the log2-fold-changes for differentially expressed gene for each constrast. Used for evaluating
row-wise numeric vector of moderated F-statistics for testing all contrasts defined by the columns of
row-wise numeric vector giving p-values corresponding to
lods have the same dimensions as the input object
fit, with rows corresponding to genes and columns to coefficients or contrasts.
F.p.value correspond to rows, with length equal to the number of genes.
var.prior corresponds to columns, with length equal to the number of contrasts.
df.prior have length 1, then the same value applies to all genes.
var.prior contain empirical Bayes hyperparameters used to obtain
treat a produces an
MArrayLM object similar to that from
eBayes but without
The algorithm used by
robust=TRUE was revised slightly in limma 3.27.6.
df.prior returned may be slightly smaller than previously.
Gordon Smyth and Davis McCarthy
Law, CW, Chen, Y, Shi, W, Smyth, GK (2014). Voom: precision weights unlock linear model analysis tools for RNA-seq read counts. Genome Biology 15, R29. http://genomebiology.com/2014/15/2/R29
Loennstedt, I., and Speed, T. P. (2002). Replicated microarray data. Statistica Sinica 12, 31-46.
McCarthy, D. J., and Smyth, G. K. (2009). Testing significance relative to a fold-change threshold is a TREAT. Bioinformatics 25, 765-771. http://bioinformatics.oxfordjournals.org/content/25/6/765
Phipson, B, Lee, S, Majewski, IJ, Alexander, WS, and Smyth, GK (2016). Robust hyperparameter estimation protects against hypervariable genes and improves power to detect differential expression. Annals of Applied Statistics 10, 946-963. http://projecteuclid.org/euclid.aoas/1469199900
Smyth, G. K. (2004). Linear models and empirical Bayes methods for assessing differential expression in microarray experiments. Statistical Applications in Genetics and Molecular Biology 3, Article 3. http://www.statsci.org/smyth/pubs/ebayes.pdf
An overview of linear model functions in limma is given by 06.LinearModels.
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# See also lmFit examples # Simulate gene expression data, # 6 microarrays and 100 genes with one gene differentially expressed set.seed(2016) sigma2 <- 0.05 / rchisq(100, df=10) * 10 y <- matrix(rnorm(100*6,sd=sqrt(sigma2)),100,6) design <- cbind(Intercept=1,Group=c(0,0,0,1,1,1)) y[1,4:6] <- y[1,4:6] + 1 fit <- lmFit(y,design) # Moderated t-statistic fit <- eBayes(fit) topTable(fit,coef=2) # Ordinary t-statistic ordinary.t <- fit$coef[,2] / fit$stdev.unscaled[,2] / fit$sigma # Treat relative to a 10% fold-change tfit <- treat(fit,lfc=log2(1.1)) topTreat(tfit,coef=2)
logFC AveExpr t P.Value adj.P.Val B 1 1.0261201 0.613239029 5.133348 1.067577e-05 0.001067577 3.255898 65 -0.4142890 0.132351578 -2.023881 5.065654e-02 0.984178668 -4.656835 86 -0.3849375 -0.062785795 -1.993078 5.408190e-02 0.984178668 -4.712776 6 -0.3726952 -0.030427277 -1.854613 7.207864e-02 0.984178668 -4.955723 48 -0.3678583 0.124175287 -1.797670 8.084747e-02 0.984178668 -5.051492 72 0.3688488 -0.023410981 1.727803 9.282497e-02 0.984178668 -5.165605 31 -0.3376838 -0.142492004 -1.665522 1.047219e-01 0.984178668 -5.264117 3 0.3432493 0.006803296 1.651775 1.075114e-01 0.984178668 -5.285447 90 -0.3070586 -0.256501945 -1.566593 1.261933e-01 0.984178668 -5.414233 41 -0.3205739 -0.082905186 -1.563103 1.270119e-01 0.984178668 -5.419385 logFC AveExpr t P.Value adj.P.Val 1 1.0261201 0.613239029 4.4454621 4.281708e-05 0.004281708 65 -0.4142890 0.132351578 -1.3521499 9.785258e-02 0.995284171 86 -0.3849375 -0.062785795 -1.2811304 1.095148e-01 0.995284171 6 -0.3726952 -0.030427277 -1.1703653 1.327319e-01 0.995284171 48 -0.3678583 0.124175287 -1.1257102 1.432378e-01 0.995284171 72 0.3688488 -0.023410981 1.0836934 1.546077e-01 0.995284171 31 -0.3376838 -0.142492004 -0.9873279 1.775689e-01 0.995284171 3 0.3432493 0.006803296 0.9900846 1.778003e-01 0.995284171 41 -0.3205739 -0.082905186 -0.8926422 2.050634e-01 0.995284171 90 -0.3070586 -0.256501945 -0.8650592 2.112381e-01 0.995284171
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