The FCR gives a simultaneous coverage at a (1 − α)×100% level for all of the parameters considered in the problem.
Both methods address the problem of multiple comparisons, FCR from confidence intervals (CIs) and FDR from P-value's point of view.
There are many FCR procedures which can be used depending on the length of the CI – Bonferroni-selected–Bonferroni-adjusted, [citation needed] Adjusted BH-Selected CIs (Benjamini and Yekutieli 2005[1]).
The incentive of choosing one procedure over another is to ensure that the CI is as narrow as possible and to keep the FCR.
For microarray experiments and other modern applications, there are a huge number of parameters, often tens of thousands or more and it is very important to choose the most powerful procedure.
Summing each type of outcome over all Hi yields the following random variables: In m hypothesis tests of which
Selection can be presented as conditioning on an event defined by the data and may affect the coverage probability of a CI for a single parameter.
FCR procedures consider that the goal of conditional coverage following any selection rule for any set of (unknown) values for the parameters is impossible to achieve.
A weaker property when it comes to selective CIs is possible and will avoid false coverage statements.
This false coverage-statement rate (FCR) is a property of any procedure that is defined by the way in which parameters are selected and the way in which the multiple intervals are constructed.
Simultaneous CIs with Bonferroni procedure when we have m parameters, each marginal CI constructed at the 1 − α/m level.
Without selection, these CIs offer simultaneous coverage, in the sense that the probability that all CIs cover their respective parameters is at least 1 − α. unfortunately, even such a strong property does not ensure the conditional confidence property following selection.
The Bonferroni–Bonferroni procedure cannot offer conditional coverage, however it does control the FCR at <α In fact it does so too well, in the sense that the FCR is much too close to 0 for large values of θ. Intervals selection is based on Bonferroni testing, and Bonferroni CIs are then constructed.
In BH procedure for FDR after sorting the p values P(1) ≤ • • • ≤ P(m) and calculating R = max{ j : P( j) ≤ j • q/m}, the R null hypotheses for which P(i) ≤ R • q/m are rejected.
If testing is done using the Bonferroni procedure, then the lower bound of the FCR may drop well below the desired level q, implying that the intervals are too long.