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TOC(n = 10, p0 = 0.95, sigma = 1, D, F0, F1, n1 = n, n2 = n, paired = FALSE, plot = TRUE, local.show=FALSE, alpha.show = TRUE, sensitivity.show = TRUE,
nplot = 100, xlim, ylim = c(0, 1), main, legend.show = FALSE, ...)n, but can be set to different values via n1 and n2.sigma), by default 1; this is a shortcut for specifying a simple symmetrical alternative hypothesis through F1.sigma, but mixtures of normals can be specified, see Details and Examples.D and -D and standard deviation sigma; mixture of normals are again possible, see Details and Examples.plotnplot rows whose columns contain the information for the individual curves. The number of columns and their names will depend on the number and value of the p0 specified, and whether alpha and sensitivity are displayed. Additionally, the returned data frame has an attribute param, which is a list with all the non-plotting arguments to the function.
The underlying model assumes that a proportion p0 of genes are not differentially expressed between groups, and that 1-p0 are. The logarithmized gene expression values are assumed to be generated by mixtures of normal distributions. Both null and alternative hypothesis are specified through the means of the respective mixture components; these means can be interpreted as average log2 fold changes in units of the standard deviation sigma.
Note that the model does not assume that all genes have the same standard deviation sigma, only that the mean log2 fold change for all regulated genes is proportional to their individual variability (standard deviation). sigma generally does not need to be specified explicitly and can be left at its default value of one, so that D can be interpreted straightforward as log2 fold change between groups.
The default null distribution of the log2 expression values is a single normal distribution with mean zero (and standard deviation sigma); the default alternative distribution is is an equal mixture of two normals with means D and -D (and again standard deviation sigma). However, general mixtures of normals can be specified for both null and alternative distribution through F0 and F1, respectively: both are lists with two elements:
D is the vector of means (i.e. log2 fold changes),
p is the vector of mixing proportions for the means.
If present, p must be the same length as D; its elements do not
need to be normalized, i.e. sum to one; if absent, equal mixing is assumed, see Examples. A wide (mixture) null hypothesis, or an empirical null hypothesis as outlined by Efron (2004), can be used if genes with log fold changes close to zero are thought to be of no biological interest, and are counted as effectively not regulated. Similarly, the alternative hypothesis can be any mixture of large and small effects, symmetric or non-symmetric, depending on the expected regulation patterns, see Examples.
As a consequence, both the null distribution of the t-statistics (for the unregulated genes) and their alternative distribution (for the regulated genes) are mixtures of (generally non-central) t-distributions, see FDR.
Sample size n and standard deviation sigma are atomic values, but multiple p0 can be specified, resulting in multiple curves. Additionally, the usual significance level and sensitivity for a classical one-hypothesis can be displayed.
Efron, B. (2004) Large-Scale Simultaneous Hypothesis Testing: The Choice of a Null Hypothesis. JASA, 99, 96-104.
FDR, samplesize, EOC# Default null and alternative distributions, assuming different proportions
# of regulated genes
TOC(p0=c(0.90, 0.95, 0.99), legend.show=TRUE)
# The effect of sample size and effect size
par(mfrow=c(2,2))
TOC(p0=c(0.90, 0.95, 0.99), n=5, D=1)
TOC(p0=c(0.90, 0.95, 0.99), n=30, D=1)
TOC(p0=c(0.90, 0.95, 0.99), n=5, D=2)
TOC(p0=c(0.90, 0.95, 0.99), n=30, D=2)
# A wide null distribution that allows to disregard genes of small effect
# unspecified p means equal mixing proportions
ret = TOC(F0=list(D=c(-0.25,0,0.25)), main="Wide F0")
attr(ret,"param")$F0 # the null hypothesis
# An extended (and unsymmetric) alternative
ret = TOC(F1=list(D=c(-2,-1,1), p=c(1,2,2)), p0=0.95, main="Unsymmetric F1")
attr(ret,"param")$F1 # F1$p is normalized
# Unequal sample sizes
TOC(n1=10, n2=30)
# Curves for a paired t-test
TOC(paired=TRUE)
# The output contains all the x- and y-coordinates
ret = TOC(p0=c(0.90, 0.95, 0.99), main="Default settings")
dim(ret)
colnames(ret)
ret[1:10,]
# Additionally, the list of arguments that determine the experiment
attr(ret,"param")
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