Principles in harmony: Closed testing meets the partitioning principle for computational efficiency

We explore and utilize the algorithmic relationship between the closed testing principle for multiple tests with family-wise error rate (FWER) control and the partitioning principle for the construction of simultaneous confidence intervals.

Starting with the simple observation that a multiple test with FWER control is formally equivalent to a one-sided simultaneous confidence interval for the vector of binary parameter indicating whether the null or alternative hypothesis is true, we show that the closed testing and partitioning principles follow the same computational approach.

We will then utilise this relationship to extend concepts of consonance for closed tests to the partitioning principle, with the aim of deriving computationally feasible and efficient algorithms for the calculation of simultaneous confidence intervals.

We will also utilize the relationship between closed testing and partitioning principle to extend common closed testing procedures to simultaneous confidence intervals, referencing the existing literature on informative simultaneous confidence intervals.

The relationships and extensions will be illustrated by simple, instructive examples.