Directional replicability: When can the factor of two be omitted

Directional replicability addresses the question of whether an effect studied across <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math> independent studies is present with the same direction in at least <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>r</mml:mi></mml:math> of them, for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>r</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn></mml:math> . 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi></mml:math> -values for both directions (left and right), and then doubling the smaller of the two resulting combined <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi></mml:math> -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.