Not keeping the FCR means when , where is the number of true null hypotheses and is the number of rejected hypothesis. Intervals with simultaneous coverage probability can control the FCR to be bounded by .
Problems
Selection
Selection causes reduced average coverage. 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. Equivalently, the problem of selection changes the basic sense of P-values. 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. FCR is a measure of interval coverage following selection. Therefore, even though a 1−α CI does not offer selective ( conditional) coverage, the probability of constructing a no covering CI is at most α, where
![{\displaystyle Pr[\theta \in CI,CIconstructed]\leq Pr[\theta \in CI]\leq \alpha }](https://wikimedia.org/api/rest_v1/media/math/render/svg/11d0db73e2d0bb45a1543ff107611bfbead06f36)
Selection and Multiplicity
When facing both multiplicity (inference about multiple parameters) and selection, not only is the expected proportion of coverage over selected parameters at 1−α not equivalent to the expected proportion of no coverage at α, but also the latter can no longer be ensured by constructing marginal CIs for each selected parameter. FCR procedures solve this by taking the expected proportion of parameters not covered by their CIs among the selected parameters, where the proportion is 0 if no parameter is selected. 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.
Not keeping the FCR means when , where is the number of true null hypotheses and is the number of rejected hypothesis. Intervals with simultaneous coverage probability can control the FCR to be bounded by .
Problems
Selection
Selection causes reduced average coverage. 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. Equivalently, the problem of selection changes the basic sense of P-values. 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. FCR is a measure of interval coverage following selection. Therefore, even though a 1−α CI does not offer selective ( conditional) coverage, the probability of constructing a no covering CI is at most α, where
Selection and Multiplicity
When facing both multiplicity (inference about multiple parameters) and selection, not only is the expected proportion of coverage over selected parameters at 1−α not equivalent to the expected proportion of no coverage at α, but also the latter can no longer be ensured by constructing marginal CIs for each selected parameter. FCR procedures solve this by taking the expected proportion of parameters not covered by their CIs among the selected parameters, where the proportion is 0 if no parameter is selected. 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.