Directional replicability: When can the factor of two be omitted

Directional replicability addresses the question of whether an effect studied across $n$ independent studies is present with the same direction in at least $r$ of them, for $r\ge 2$ . When the expected direction of the effect is not specified in advance, the state of the art recommends assessing replicability separately by combining one-sided $p$ -values for both directions (left and right), and then doubling the smaller of the two resulting combined $p$ -values to account for multiple testing. In this work, we show that this multiplicative correction is not always necessary, and give a sufficient and necessary condition under which it can be safely omitted.