Total-effect Test can Erroneously Reject Complete Mediation --- Proofs and Examples

The procedure for establishing mediation, i.e., determining whether an independent variable X influences a dependent variable Y through a mediator M, remains debated in the literature. Classic causal steps necessitate a significant “total effect”, now commonly termed statistically acknowledgment. Prior research show that the total-effect test can incorrectly reject competitive mediation and is superfluous for complementary mediation. Less attention has been paid to complete mediation (also known as full, perfect, or indirect-only mediation), where the indirect effect (ab) is significant but the direct-and-remainder effect (d) is not. Although recent work largely agrees that the total-effect test should be abandoned, its use persists across disciplines, including non-English literature. Addressing the remaining logical and empirical gaps therefore remains scientifically valuable. Accordingly, we set out four objectives for this study: 1) Demonstrating that the total-effect test can erroneously reject complete mediation, including both subtypes, assuming least squares estimation (LSE) F-test or Sobel test; 2) Verifying these results via simulations and extend them to the LAD-Z test; 3) Presenting two empirical examples illustrating each subtype; 4) Reevaluating existing concepts, theories, and techniques in mediation and related causal analyses, and proposing a broader alternative: process-and-product analysis (PAPA).