Statistica Sinica 31 (2021), 1961-1983

TOTAL-EFFECT TEST IS SUPERFLUOUS FOR

ESTABLISHING COMPLEMENTARY MEDIATION

Yingkai Jiang^{1}, Xinshu Zhao^{2}, Lixing Zhu^{3}, Jun S. Liu^{4} and Ke Deng^{1}

Abstract: Mediation, which means that an independent variable *X* affects a dependent variable *Y* through a mediator *M*, is a key concept in causal inference. For establishing mediation, there is a long debate on whether to require the "total effect" of *X* on *Y* to be statistically significant. It has been shown that total-effect test can erroneously reject "competitive mediation". For "complementary mediation", however, the situation becomes more complicated. This article provides an explicit proof that the total effect is statistically significant whenever mediated effect and direct effect bear the same sign and are both significant, as long as the least square estimation (LSE) and *F*-tests are used to estimate and test mediation effects. We also show that the similar result can be obtained when the Sobel test is used. Our results support the growing agreement that total-effect test is unnecessary for establishing any type of mediation.

Key words and phrases: Complementary mediation, hypothesis testing, linear model, mediation analysis, percentage coefficient, percentage scale, total-effect test.