Bayesian.SGoF(u, alpha = 0.05, gamma = 0.05, P0 = 0.5, a0 = 1, b0 = 1)gamma is random. This method has two main steps. First, Bayesian SGoF performs a pre-test at level alpha which decides if the complete null hypothesis should be rejected or not.
This Bayesian pre-test is based on lower bounds of the a posteriori probability of H0 (computed using the default a priori probability P0=.5 and a family of a priori beta distributions located at the null and indexed by a correlation factor). Second, the number of rejections is computed constructing an interval for the 'excess of significant cases', analogously to the SGoF procedure. For this, the posterior distribution of the proportion of p-values falling below gamma is used; this posterior distribution is calculated on the basis of the default priors P0=.5 and a0=b0=1, unless otherwise is indicated.
One important difference between the frequentist and the Bayesian SGoF is that the Bayesian setting induces (and hence allows for) a dependence structure among the p-values; this is very interesting for real problems where correlation is present. From a less philosophical point of view, in practice Bayesian SGoF may be more conservative than frequentist SGoF, particularly when the number of tests is small; this is due to the fact that Bayesian testing of point nulls is much more conservative than its frequentist counterpart and, therefore, the pre-test part of Bayesian SGoF may play a very important role.
Typically the choice alpha=gamma will be used for Bayesian.SGoF; this common value will be set as one of the usual significance levels (0.001, 0.01, 0.05, 0.1). Note however that alpha and gamma have different roles.
The false discovery rate is estimated by the simple method proposed by: Dalmasso , Broet , Moreau (2005), by taking n=1 in their formula.summary.Bayesian.SGoFres<-Bayesian.SGoF(Hedenfalk$x)
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