Complete Mediation Test (CMT)

• Complete Mediation Test (CMT) is a collection of procedures that determine whether an exposure’s effect is entirely transmitted through mediators across different model frameworks.
• It distinguishes methodologies by employing linear path tests, intersection–union regression, natural-effects formulations, conditional-independence with DML, high-dimensional models, and interventionist approaches.
• CMT methods provide practical insights for causal mediation analysis by refining testing rules, addressing total-effect limitations, and clarifying underlying assumptions.

Searching arXiv for the cited CMT-related papers to ground the article in the current literature. {"queries":[{"query":"id:(Han et al., 2023)"},{"query":"id:(Huber et al., 4 Mar 2026)"},{"query":"id:(Hillier et al., 2024)"},{"query":"id:(Tsai et al., 18 Jul 2025)"},{"query":"id:(Robins et al., 2020)"},{"query":"id:(Zhou et al., 2019)"},{"query":"id:(Kwon et al., 2024)"}]} Complete Mediation Test (CMT) denotes a family of procedures for determining whether the effect of an exposure, treatment, or instrument on an outcome operates entirely through one or more mediators. Across the recent literature, the label is used for several non-identical targets: indirect-only mediation in linear path models, the hypothesis $c'=0$ together with $ab\neq 0$ in regression-based mediation, the conditional-independence hypothesis $Y \perp T \mid (M,X)$ in semiparametric causal analysis, and the sharp null $Y(d,m)=Y(m)$ or $SDE=0$ in mechanism-testing and interventionist formulations (Han et al., 2023, Hillier et al., 2024, Huber et al., 4 Mar 2026, Kwon et al., 2024, Robins et al., 2020, Zhou et al., 2019, Tsai et al., 18 Jul 2025). This suggests that CMT is best understood as a class of tests for complete or full mediation rather than a single canonical statistic.

1. Major formalizations of complete mediation

The literature distinguishes several definitions of complete mediation. In the linear single-mediator framework used in the debate over the total-effect test, indirect-only mediation is the case in which the indirect path is statistically acknowledged while the direct-and-remainder path is statistically inconclusive. Formally, $a\times b$ is statistically acknowledged but $d$ is not, with $d=0$ in the classical idealization and statistically inconclusive in finite samples (Han et al., 2023). In the causal-effects formulation used for natural effects, complete mediation is the hypothesis $NDE=0$ with $NIE\neq 0$ (Tsai et al., 18 Jul 2025). In the high-dimensional linear model, complete mediation means $ab\neq 0$0, so the total effect $ab\neq 0$1 equals the indirect effect $ab\neq 0$2 (Zhou et al., 2019). In the conditional-independence formulation, full mediation plus mediator exogeneity implies $ab\neq 0$3 (Huber et al., 4 Mar 2026). In sharp-null mechanism testing, full mediation is the assertion that $ab\neq 0$4 almost surely for all $ab\neq 0$5 and $ab\neq 0$6 (Kwon et al., 2024).

Framework	Formal criterion	Characteristic test object
Linear path decomposition	$ab\neq 0$7 acknowledged and $ab\neq 0$8 inconclusive	$ab\neq 0$9, $Y \perp T \mid (M,X)$0, $Y \perp T \mid (M,X)$1, and sometimes $Y \perp T \mid (M,X)$2 (Han et al., 2023)
Intersection–union regression test	$Y \perp T \mid (M,X)$3 and $Y \perp T \mid (M,X)$4	augmented LR for $Y \perp T \mid (M,X)$5, Wald or LR for $Y \perp T \mid (M,X)$6 (Hillier et al., 2024)
Natural-effects formulation	$Y \perp T \mid (M,X)$7 with $Y \perp T \mid (M,X)$8	$Y \perp T \mid (M,X)$9, $Y(d,m)=Y(m)$0, SAPM (Tsai et al., 18 Jul 2025)
Conditional-independence CMT	$Y(d,m)=Y(m)$1	orthogonal moments and $Y(d,m)=Y(m)$2 (Huber et al., 4 Mar 2026)
High-dimensional linear mediation	$Y(d,m)=Y(m)$3, hence $Y(d,m)=Y(m)$4	de-biased Wald test for $Y(d,m)=Y(m)$5 (Zhou et al., 2019)
Sharp-null mechanism testing	$Y(d,m)=Y(m)$6 or $Y(d,m)=Y(m)$7	IV inequalities or interventional contrasts (Kwon et al., 2024, Robins et al., 2020)

Within the path-analytic typology, three mediation types are distinguished. Complementary mediation requires that both the indirect and direct-and-remainder paths pass statistical partition tests and have the same sign, so $Y(d,m)=Y(m)$8 and $Y(d,m)=Y(m)$9 are statistically acknowledged and $SDE=0$0. Competitive mediation also requires both paths to pass, but with opposite signs, so $SDE=0$1. Indirect-only mediation is the case in which the indirect path passes a statistical partition test while the direct-and-remainder path fails (Han et al., 2023). That literature further separates indirect-only mediation into directionally complementary indirect-only mediation (“d-plementary IO”), where $SDE=0$2 and $SDE=0$3 share the same sign but only $SDE=0$4 is acknowledged, and directionally competitive indirect-only mediation (“d-petitive IO”), where $SDE=0$5 and $SDE=0$6 have opposite signs and only $SDE=0$7 is acknowledged (Han et al., 2023).

2. Regression-path foundations and the critique of the total-effect gatekeeper

The classical linear mediation model underlying much of the modern discussion is

$SDE=0$8

$SDE=0$9

and the reorganized outcome equation

$a\times b$0

with $a\times b$1 (Han et al., 2023). In this setup, $a\times b$2 is the total effect, $a\times b$3 is the direct effect conditional on $a\times b$4 and is the paper’s “direct-and-remainder” path, and $a\times b$5 is the indirect effect (Han et al., 2023).

A central result of the recent literature is that testing the total effect $a\times b$6 is not a valid gatekeeper for complete mediation. Under least-squares estimation with transformed data of rank $a\times b$7, the estimators become

$a\times b$8

with rejection regions expressed in the transformed $a\times b$9 coordinates for $d$0, $d$1, $d$2, $d$3, and the Sobel test for $d$4 (Han et al., 2023). On that basis, the paper proves that, for indirect-only mediation, the intersection

$d$5

under LSE-F, and that for sufficiently large $d$6,

$d$7

under LSE-Sobel (Han et al., 2023). These results mean that the indirect effect can be statistically acknowledged while both the direct-and-remainder path and the total effect are statistically inconclusive.

The same paper also sharpens the older result for complementary mediation. Under LSE-F, prior work proved

$d$8

whenever $d$9, so the total-effect test adds nothing once $d=0$0, $d=0$1, and $d=0$2 are acknowledged and of the same sign. Under LSE-Sobel, the asymptotic result

$d=0$3

shows that total-effect testing is likewise superfluous in complementary mediation (Han et al., 2023).

The simulation evidence in that line of work is explicitly quantitative. Data were generated with $d=0$4, $d=0$5, $d=0$6, $d=0$7, and $d=0$8 independent datasets. Under LSE-F, for $d=0$9, the proportion of cases with $NDE=0$0 given $NDE=0$1 and $NDE=0$2 exceeds $NDE=0$3. Erroneous judgments are more frequent in directionally competitive indirect-only mediation than in directionally complementary indirect-only mediation. LSE-Sobel and LAD-Z show similar patterns (Han et al., 2023).

In response, that paper proposes process-and-product analysis (PAPA), which treats mediation as a process $NDE=0$4 producing a product $NDE=0$5 and assigns three tasks: testing effect hypotheses, classifying effect types, and analyzing effect sizes. A plausible implication is that, within this framework, CMT is not merely a decision rule but part of a broader decomposition strategy that separates process-level path acknowledgment from product-level aggregation (Han et al., 2023).

3. Decision rules, calibrated criteria, and competing operational CMTs

One operational CMT follows directly from the indirect-only framework. The recommended protocol is: fit the mediator and outcome models, estimate $NDE=0$6, $NDE=0$7, and $NDE=0$8, compute $NDE=0$9, test $NIE\neq 0$0 and $NIE\neq 0$1, and do not test $NIE\neq 0$2 to qualify mediation status. Under LSE-F, $NIE\neq 0$3 and $NIE\neq 0$4 are tested via F-tests and $NIE\neq 0$5 is acknowledged if both pass; under LSE-Sobel, $NIE\neq 0$6 is tested with

$NIE\neq 0$7

rejecting when $NIE\neq 0$8; under LAD-Z, one uses $NIE\neq 0$9 for $ab\neq 0$00. If $ab\neq 0$01 is statistically acknowledged and $ab\neq 0$02 is statistically inconclusive, the procedure declares indirect-only, or complete, mediation; the sign of $ab\neq 0$03 relative to $ab\neq 0$04 then determines whether the case is d-plementary IO or d-petitive IO (Han et al., 2023).

A distinct CMT architecture is the intersection–union framework developed for the hypothesis of complete mediation in a single-mediator regression model. There, the hypotheses are

$ab\neq 0$05

The indirect-effect component is difficult because $ab\neq 0$06 is nonregular. The classical LR test based on $ab\neq 0$07 can be severely conservative, with null rejection probability satisfying $ab\neq 0$08; at $ab\neq 0$09 it can be near $ab\neq 0$10. The Sobel/Wald test is worse: at $ab\neq 0$11 and $ab\neq 0$12 its null rejection probability can be approximately $ab\neq 0$13 (Hillier et al., 2024). To remedy this, the paper proposes the simply-augmented LR test with critical region

$ab\neq 0$14

so $ab\neq 0$15 is rejected if $ab\neq 0$16 or $ab\neq 0$17. Reported values include $ab\neq 0$18, $ab\neq 0$19, $ab\neq 0$20, with corresponding $ab\neq 0$21 values $ab\neq 0$22, $ab\neq 0$23, and $ab\neq 0$24. Complete mediation is concluded only if the augmented LR rejects $ab\neq 0$25 and the direct-effect test fails to reject $ab\neq 0$26 (Hillier et al., 2024).

A third operational line begins from the critique of “significance-only” complete mediation rules. In that formulation, conventional CMT1 declares complete mediation if the indirect effect is significant and the direct effect is non-significant, using $ab\neq 0$27 and $ab\neq 0$28. Theoretical analysis shows that the Type I error of this rule can reach $ab\neq 0$29, with

$ab\neq 0$30

at $ab\neq 0$31 and $ab\neq 0$32 (Tsai et al., 18 Jul 2025). To address this, the paper evaluates two proportion-based extensions. The absolute proportion of mediation is

$ab\neq 0$33

and the proposed standardized absolute proportion of mediation is

$ab\neq 0$34

The preferred rule, CMT3$ab\neq 0$35, requires three conditions: significant indirect effect, non-significant direct effect, and $ab\neq 0$36. Recommended thresholds are $ab\neq 0$37–$ab\neq 0$38 for continuous mediator and outcome, and $ab\neq 0$39–$ab\neq 0$40 for binary mediator and/or outcome (Tsai et al., 18 Jul 2025).

These three constructions share a common aim but impose materially different decision logics. One emphasizes direct testing of $ab\neq 0$41 and $ab\neq 0$42 while discarding $ab\neq 0$43 as a prerequisite; one uses intersection–union logic with a calibrated indirect-effect test and a null direct-effect test; and one augments significance criteria with a standardized dominance condition based on SAPM. A plausible implication is that the name “CMT” does not uniquely determine the inferential target unless the underlying framework is stated explicitly (Han et al., 2023, Hillier et al., 2024, Tsai et al., 18 Jul 2025).

4. Conditional-independence and double-machine-learning formulations

A substantially different CMT is developed for treatment effects that may be fully mediated by observed intermediate outcomes. The observed data are $ab\neq 0$44, where $ab\neq 0$45 is the outcome, $ab\neq 0$46 the treatment, $ab\neq 0$47 the vector of mediators or surrogate outcomes, and $ab\neq 0$48 pre-treatment covariates. Under the mean version of full mediation, $ab\neq 0$49 does not depend on $ab\neq 0$50, equivalently $ab\neq 0$51; under full mediation and identifiability of causal mechanisms with conditionally randomized treatment, the key testable implication is

$ab\neq 0$52

(Huber et al., 4 Mar 2026).

This implication generates conditional moment restrictions. In randomized or conditionally randomized settings, with $ab\neq 0$53 and $ab\neq 0$54,

$ab\neq 0$55

In observational settings, where treatment may depend on $ab\neq 0$56 given $ab\neq 0$57, $ab\neq 0$58 is replaced by $ab\neq 0$59:

$ab\neq 0$60

The DML implementation estimates $ab\neq 0$61 and either $ab\neq 0$62 or $ab\neq 0$63, forms residuals $ab\neq 0$64 and $ab\neq 0$65, computes

$ab\neq 0$66

stacks $ab\neq 0$67, estimates $ab\neq 0$68, and uses the quadratic form

$ab\neq 0$69

Under $ab\neq 0$70 and regularity, $ab\neq 0$71 (Huber et al., 4 Mar 2026).

The framework is explicitly orthogonal. For

$ab\neq 0$72

the Gateaux derivative at the truth is zero, and sample splitting with cross-fitting is used to mitigate overfitting bias and deliver $ab\neq 0$73 asymptotics. Flexible learners such as lasso, boosting, random forests, and neural nets may be used for nuisance estimation, provided the product of $ab\neq 0$74 errors is $ab\neq 0$75; a sufficient condition is that each nuisance be estimated at $ab\neq 0$76 (Huber et al., 4 Mar 2026).

This CMT also distinguishes randomized from non-randomized treatment assignment. Under conditional randomization, full mediation and mediator exogeneity imply both testability and identifiability of causal mechanisms. In observational settings, full mediation remains testable through $ab\neq 0$77, but identifiability of indirect mechanisms is no longer guaranteed because treatment–mediator confounding may persist (Huber et al., 4 Mar 2026). The paper further states that its DML framework is root-$ab\neq 0$78 consistent and asymptotically normal under specific regularity conditions, accommodates high-dimensional covariates, and has good finite-sample performance in simulations (Huber et al., 4 Mar 2026).

5. High-dimensional, interventionist, and sharp-null extensions

In high-dimensional linear mediation, the mediator vector $ab\neq 0$79 may satisfy $ab\neq 0$80. The structural equations are

$ab\neq 0$81

with indirect effect $ab\neq 0$82, direct effect $ab\neq 0$83, and total effect $ab\neq 0$84. Complete mediation means $ab\neq 0$85, hence $ab\neq 0$86 (Zhou et al., 2019). The paper constructs a de-biased estimator for $ab\neq 0$87 under complete mediation,

$ab\neq 0$88

and a Wald statistic

$ab\neq 0$89

which is asymptotically standard normal under $ab\neq 0$90 (Zhou et al., 2019). A key efficiency result is that, under complete mediation, the asymptotic variance of the OLS estimator of the total effect minus the asymptotic variance of $ab\neq 0$91 is positive semidefinite, so the indirect-effect-based CMT is more powerful than directly testing the total effect (Zhou et al., 2019).

A different extension comes from the interventionist, separable-treatment approach. Instead of relying on nested counterfactuals such as $ab\neq 0$92, treatment is decomposed into distinct components that act along different causal pathways. In the canonical example, $ab\neq 0$93 is decomposed into $ab\neq 0$94 and $ab\neq 0$95, where $ab\neq 0$96 affects $ab\neq 0$97 but not $ab\neq 0$98, and $ab\neq 0$99 affects $Y \perp T \mid (M,X)$00 but not $Y \perp T \mid (M,X)$01, with deterministic linkage $Y \perp T \mid (M,X)$02 in the observed world. Complete mediation is then the null that the separable direct effect vanishes:

$Y \perp T \mid (M,X)$03

Under the NPSEM-IE for the expanded graph, this contrast equals the pure direct effect. The framework provides sufficient conditions for identifying “four-arm” interventional distributions from “two-arm” observed data and yields a sound and complete graph-theoretic algorithm based on edge-expanded graphs, SWIGs, and the recanting district criterion (Robins et al., 2020).

Sharp-null mechanism testing develops yet another CMT. Here the null is

$Y \perp T \mid (M,X)$04

With binary $Y \perp T \mid (M,X)$05 and binary $Y \perp T \mid (M,X)$06, independence and monotonicity imply the instrumental inequalities

$Y \perp T \mid (M,X)$07

$Y \perp T \mid (M,X)$08

for all Borel sets $Y \perp T \mid (M,X)$09 (Kwon et al., 2024). For multi-valued or multi-dimensional mediators, the test is characterized by the feasibility of a linear program over type shares $Y \perp T \mid (M,X)$10 subject to linear restrictions. That framework also provides lower bounds on the prevalence of alternative mechanisms through quantities such as

$Y \perp T \mid (M,X)$11

and on the principal-strata average direct effect $Y \perp T \mid (M,X)$12 (Kwon et al., 2024). Relative to traditional mediation analysis, its stated advantage is that it does not require stringent assumptions about how $Y \perp T \mid (M,X)$13 is assigned, while focusing on the sharp null rather than estimating average direct and indirect effects (Kwon et al., 2024).

6. Applications, assumptions, and continuing controversies

The empirical illustrations attached to CMT differ with the framework. In the HINTS 5 Cycle 4 analyses, one model examined caregiving $Y \perp T \mid (M,X)$14 smoking via psychological distress and produced $Y \perp T \mid (M,X)$15 with $Y \perp T \mid (M,X)$16, $Y \perp T \mid (M,X)$17 with $Y \perp T \mid (M,X)$18, $Y \perp T \mid (M,X)$19 with $Y \perp T \mid (M,X)$20, and $Y \perp T \mid (M,X)$21 with $Y \perp T \mid (M,X)$22, giving a directionally competitive indirect-only pattern. A second model examined employment $Y \perp T \mid (M,X)$23 physical activity via psychological distress and produced $Y \perp T \mid (M,X)$24 with $Y \perp T \mid (M,X)$25, $Y \perp T \mid (M,X)$26 with $Y \perp T \mid (M,X)$27, $Y \perp T \mid (M,X)$28 with $Y \perp T \mid (M,X)$29, and $Y \perp T \mid (M,X)$30 with $Y \perp T \mid (M,X)$31, giving a directionally complementary indirect-only pattern. Both are used to illustrate erroneous rejection by the total-effect test (Han et al., 2023). The augmented-LR intersection–union framework is illustrated with an entrepreneurial attitudes study in which $Y \perp T \mid (M,X)$32, $Y \perp T \mid (M,X)$33, and $Y \perp T \mid (M,X)$34; at $Y \perp T \mid (M,X)$35, $Y \perp T \mid (M,X)$36, so the augmented LR rejects $Y \perp T \mid (M,X)$37 while the direct-effect test fails to reject $Y \perp T \mid (M,X)$38, leading to a complete-mediation conclusion for that subgroup (Hillier et al., 2024). The SAPM-based framework is applied in Mendelian Randomization to test non-pleiotropy, with UK Biobank analyses of insomnia and coronary heart disease reporting that CMT3$Y \perp T \mid (M,X)$39 improved specificity relative to CMT1 while maintaining near-perfect sensitivity for valid SNPs at $Y \perp T \mid (M,X)$40–$Y \perp T \mid (M,X)$41 (Tsai et al., 18 Jul 2025).

The DML conditional-independence formulation is illustrated with randomized experiments on maternal mental health and social norms. In the Pakistan CBT application, CMT rejects the joint null with p-values approximately $Y \perp T \mid (M,X)$42–$Y \perp T \mid (M,X)$43, indicating either a residual direct effect or mediator exogeneity failure. In the Saudi Arabia social norms application, CMT strongly rejects with $Y \perp T \mid (M,X)$44, consistent with a direct or additional channel beyond sign-up and/or mediator exogeneity violations. The simulation design also shows near-nominal size of approximately $Y \perp T \mid (M,X)$45 under the null, strong power by $Y \perp T \mid (M,X)$46, high power against mediator-outcome confounding, and the fact that treatment–mediator confounding does not inflate false rejections because the test is not designed to detect it (Huber et al., 4 Mar 2026). The sharp-null mechanism-testing literature revisits the same Saudi Arabia study and also the Pakistan CBT study using IV-style inequalities, reaching rejection for individual mechanisms such as sign-up, grandmother presence, and relationship quality, but not necessarily for some joint mechanisms under element-wise monotonicity (Kwon et al., 2024).

The assumptions required by CMT depend on the formulation but are substantial in every case. Regression-based path models require linearity, standard mediation assumptions such as no unmeasured confounding between $Y \perp T \mid (M,X)$47 and $Y \perp T \mid (M,X)$48, and, for the geometric proofs, $Y \perp T \mid (M,X)$49 (Han et al., 2023). The natural-effects and SAPM framework requires consistency, SUTVA, temporal ordering, no unmeasured confounding of the $Y \perp T \mid (M,X)$50–$Y \perp T \mid (M,X)$51, $Y \perp T \mid (M,X)$52–$Y \perp T \mid (M,X)$53, and $Y \perp T \mid (M,X)$54–$Y \perp T \mid (M,X)$55 relations conditional on covariates, positivity, and correct model specification; for natural effects it also requires that no mediator–outcome confounders are affected by $Y \perp T \mid (M,X)$56 (Tsai et al., 18 Jul 2025). The DML conditional-independence formulation requires SUTVA, faithfulness, positivity, correct conditioning on pre-treatment covariates, and mediator exogeneity $Y \perp T \mid (M,X)$57; in observational settings it does not guarantee identification of the indirect effect even if the CMT does not reject (Huber et al., 4 Mar 2026). High-dimensional CMT additionally relies on sparsity, tail conditions, and matrix regularity conditions (Zhou et al., 2019). Interventionist formulations require separability of treatment components and fail in the presence of recanting witnesses or recanting districts (Robins et al., 2020).

Several controversies recur across these strands. One concerns the status of the total effect as a prerequisite: one paper proves it is superfluous for complementary mediation and can erroneously reject both competitive and complete mediation under LSE-F and LSE-Sobel, with similar simulation evidence under LAD-Z (Han et al., 2023). Another concerns the evidential meaning of a non-significant direct effect: significance-only rules can have worst-case Type I error $Y \perp T \mid (M,X)$58, which motivates intersection–union calibration or SAPM thresholds (Tsai et al., 18 Jul 2025, Hillier et al., 2024). A third concerns target mismatch. Conditional-independence CMTs, sharp-null mechanism tests, and interventionist SDE tests do not ask exactly the same question as classical path-coefficient tests. This suggests that any use of the term “Complete Mediation Test” is interpretable only relative to its estimand—indirect-only path structure, $Y \perp T \mid (M,X)$59 with $Y \perp T \mid (M,X)$60, $Y \perp T \mid (M,X)$61, $Y \perp T \mid (M,X)$62, or $Y \perp T \mid (M,X)$63—and the assumptions under which that estimand is meaningful (Han et al., 2023, Huber et al., 4 Mar 2026, Kwon et al., 2024, Robins et al., 2020).