Simultaneous directional inference

We consider the problem of inference on the signs of $n>1$ parameters. Within a simultaneous inference framework, we aim to: identify as many of the signs of the individual parameters as possible; provide confidence bounds on the number of positive (or negative) parameters on subsets of interest. Our suggestion is as follows: start by using the data to select the direction of the hypothesis test for each parameter; then, adjust the one-sided $p$-values for the selection, and use them for simultaneous inference on the selected $n$ one-sided hypotheses. The adjustment is straightforward assuming that the one-sided $p$-values are conditionally valid and mutually independent. Such assumptions are commonly satisfied in a meta-analysis, and we can apply our approach following a test of the global null hypothesis that all parameters are zero, or of the hypothesis of no qualitative interaction. We consider the use of two multiple testing principles: closed testing and partitioning. The novel procedure based on partitioning is more powerful, but slightly less informative: it only infers on positive and non-positive signs. The procedure takes at most a polynomial time, and we show its usefulness on a subgroup analysis of a medical intervention, and on a meta-analysis of an educational intervention.