Standardized Absolute Proportion of Mediation (SAPM)

• The paper’s main contribution is the introduction of SAPM as a standardized measure that compares indirect versus direct pathway strengths using signal-to-noise ratios.
• SAPM addresses sign pathologies and sensitivity to estimation noise by using absolute values and standardized errors, ensuring robust mediation testing in both continuous and binary settings.
• SAPM integrates with Complete Mediation Test frameworks by applying practical thresholds and simulation-backed insights to improve specificity and reduce Type I error in mediation analysis.

Standardized Absolute Proportion of Mediation (SAPM) is a mediation-based dominance measure proposed for complete mediation testing. It is designed to evaluate whether the indirect pathway through a mediator meaningfully dominates any direct pathway from exposure to outcome, while avoiding two failure modes of conventional proportion-mediated criteria: sign pathologies under competitive mediation and sensitivity to estimation noise. In the linear setting, SAPM is defined as $$\mathrm{SAPM} = |Z_{\mathrm{IE}}| / (|Z_{\mathrm{IE}}| + |Z_{\mathrm{DE}}|)$$, where $Z_{\mathrm{IE}}$ and $Z_{\mathrm{DE}}$ are standardized indirect and direct effects; in binary generalized linear settings it is defined analogously on the log-odds scale using standardized natural indirect and direct effects (Tsai et al., 18 Jul 2025).

1. Conceptual basis and motivation

SAPM was introduced in the context of the Complete Mediation Test (CMT), where complete mediation means that the exposure $A$ affects the outcome $Y$ only through the mediator $M$, with no direct effect. Conventional CMT procedures relied on two significance-based conditions: the indirect effect is statistically significant and the direct effect is not statistically significant. The reported concern is that these criteria can have severely inflated Type I error, reaching up to $25\%$ in some non-complete-mediation scenarios. A third criterion based on a large mediation proportion had also been used heuristically, but thresholds such as $\mathrm{PM} > 0.8$ were not rigorously validated and can be misleading when indirect and direct effects have opposite signs (Tsai et al., 18 Jul 2025).

SAPM was proposed to address two specific problems. First, taking absolute values removes sign-related anomalies, including mediation proportions greater than $1$ under competitive mediation. Second, standardization by standard errors changes the comparison from raw effect magnitudes to effect-to-noise ratios. This matters because a direct effect can appear “insignificant” merely because its standard error is large, not because the direct pathway is substantively negligible. SAPM therefore functions as a standardized dominance criterion layered on top of conventional mediation significance testing (Tsai et al., 18 Jul 2025).

2. Formal definitions on continuous and binary scales

In the continuous mediator and continuous outcome setting, under linear-Gaussian structural equations with covariates $C$ and no $Z_{\mathrm{IE}}$0–$Z_{\mathrm{IE}}$1 interaction,

$Z_{\mathrm{IE}}$2

with mean-zero errors. The indirect, direct, and total effects are

$Z_{\mathrm{IE}}$3

To avoid competitive-mediation pathologies, the paper defines the Absolute Proportion of Mediation (APM) as

$Z_{\mathrm{IE}}$4

Standardized test statistics are

$Z_{\mathrm{IE}}$5

and SAPM is

$Z_{\mathrm{IE}}$6

The dominance threshold $Z_{\mathrm{IE}}$7 admits the equivalent inequality

$Z_{\mathrm{IE}}$8

“Absolute” refers to the use of absolute values to bound the proportion in $Z_{\mathrm{IE}}$9; “standardized” refers to the use of $Z_{\mathrm{DE}}$0-scale effects, so that indirect and direct pathways are compared on a signal-to-noise scale rather than on raw coefficients (Tsai et al., 18 Jul 2025).

For binary mediator and outcome models with no $Z_{\mathrm{DE}}$1–$Z_{\mathrm{DE}}$2 interaction, the paper quantifies natural indirect and direct effects on the odds-ratio scale. Denoting these by $Z_{\mathrm{DE}}$3 and $Z_{\mathrm{DE}}$4, the total effect satisfies

$Z_{\mathrm{DE}}$5

The absolute proportion on the logit scale is

$Z_{\mathrm{DE}}$6

with standardized statistics

$Z_{\mathrm{DE}}$7

and

$Z_{\mathrm{DE}}$8

The same threshold equivalence holds on this scale. The paper explicitly cautions that mediation proportions in binary generalized linear models are scale-dependent; SAPM is defined on the log-odds scale, and conclusions can differ from risk-difference or risk-ratio formulations (Tsai et al., 18 Jul 2025).

3. SAPM within complete mediation testing

The paper organizes complete mediation testing into three criteria families. The first uses only significance conditions; the second adds an unstandardized absolute dominance criterion; the third replaces that criterion with SAPM. The resulting framework is summarized below (Tsai et al., 18 Jul 2025).

Criterion	Conditions	Role
CMT1	$Z_{\mathrm{DE}}$9 and $A$0	Significance-only complete mediation test
CMT2($A$1)	CMT1 plus $A$2	Adds unstandardized absolute dominance
CMT3($A$3)	CMT1 plus $A$4	Adds standardized absolute dominance

At $A$5, $A$6. The central recommendation is CMT3($A$7), which augments the presence/absence logic of indirect and direct pathways with a requirement that the indirect pathway dominate the direct pathway on a standardized scale. This changes the test from a purely significance-based filter into a combined significance-and-dominance decision rule (Tsai et al., 18 Jul 2025).

Recommended thresholds depend on data type. For continuous mediator and outcome models, $A$8 or $A$9 yielded the highest Youden indices across multiple scenarios. For binary mediator and outcome models on the log-odds scale, $Y$0 or $Y$1 was preferred; the paper attributes the lower optimal thresholds to the tendency of PM and APM to under-estimate mediation with binary data, so that using $Y$2 in binary settings materially reduces power (Tsai et al., 18 Jul 2025).

4. Estimation workflow and practical computation

In the continuous linear case, the estimation workflow is product-based. One fits the mediator model $Y$3 to estimate $Y$4 and $Y$5, and fits the outcome model $Y$6 to estimate $Y$7, $Y$8, their variances, and $Y$9 if available. The indirect effect estimate is $M$0, with delta-method standard error

$M$1

SAPM is then computed from $M$2 and $M$3 (Tsai et al., 18 Jul 2025).

In the binary case, the workflow uses causal mediation procedures under no $M$4–$M$5 interaction to estimate $M$6 and $M$7 together with standard errors on the log scale. The paper specifically refers to CAUSALMED for this purpose. After forming $M$8 and $M$9, SAPM is computed as the standardized absolute dominance ratio on the log-odds scale. Inference for SAPM itself is not given in closed form; the stated practical options are nonparametric bootstrap, including percentile or BCa intervals, or a delta-method approximation after mapping to the $25\%$0 space. When many mediators or exposure analogues are tested, such as many SNPs in Mendelian randomization, significance thresholds for the first two CMT conditions should be adjusted; Bonferroni correction is illustrated (Tsai et al., 18 Jul 2025).

A related implementation context appears in earlier product-method work on natural indirect effects and mediation proportion. Cheng, Spiegelman, and Li developed closed-form interval estimators via estimating equations and the multivariate delta method for four common data types, introduced exact expressions for binary outcomes without the rare outcome assumption, and released the R package mediateP for point and variance estimation of $25\%$1, $25\%$2, $25\%$3, and $25\%$4 (Cheng et al., 2021). This suggests that SAPM inherits much of its computational substrate from the same mediation-effect estimators, but replaces signed raw proportions with absolute standardized contrasts.

5. Theoretical properties and empirical performance

Under a bivariate normal approximation for the standardized statistics,

$25\%$5

the paper analyzes the behavior of CMT1. In the complete mediation case $25\%$6, power increases with $25\%$7; specifically, when $25\%$8, power is approximately $25\%$9, while $\mathrm{PM} > 0.8$0 and $\mathrm{PM} > 0.8$1 correspond to power greater than $\mathrm{PM} > 0.8$2 and $\mathrm{PM} > 0.8$3, respectively. In this setting, correlation $\mathrm{PM} > 0.8$4 has negligible effect when $\mathrm{PM} > 0.8$5. In non-complete-mediation settings with equal means $\mathrm{PM} > 0.8$6, the Type I error of CMT1 is maximized at $\mathrm{PM} > 0.8$7 and $\mathrm{PM} > 0.8$8, attaining $\mathrm{PM} > 0.8$9 (Tsai et al., 18 Jul 2025).

The paper reports that adding SAPM can materially reduce this error. Numerical integration under the same BVN setup showed that imposing SAPM with $1$0 or $1$1 reduced Type I error below $1$2 in non-complete-mediation scenarios such as $1$3 or $1$4, whereas CMT1 misclassified such cases at rates from $1$5 to $1$6. The point is not merely that SAPM is more stringent, but that the dominance criterion removes many cases in which the direct pathway is statistically inconclusive only because it is noisy rather than small (Tsai et al., 18 Jul 2025).

Simulation results reinforce this interpretation. In continuous settings, CMT1 had high sensitivity, approximately $1$7, but poor specificity, about $1$8 to $1$9, implying substantial false classification of partial mediation as complete mediation. CMT2 improved specificity but under-estimated the mediation proportion and had worse Youden performance than CMT3. CMT3 with $C$0 or $C$1 delivered the best overall sensitivity-specificity trade-off. In binary settings, CMT1 again had the highest sensitivity and lowest specificity, while CMT3 with $C$2 achieved superior Youden performance; using $C$3 in binary settings harmed power (Tsai et al., 18 Jul 2025).

The paper also discusses an alternative dominance metric, the proportion of squared standardized effects,

$C$4

Although this can be linked to a non-central Beta distribution under independence and known noncentrality, the paper argues that the needed independence assumption is unrealistic and the noncentrality parameter is unknown in practice. Simulations found PSSE highly right-skewed and sensitive to thresholding, with no advantage over SAPM (Tsai et al., 18 Jul 2025).

6. Relation to earlier mediation measures, applications, and limitations

SAPM is closely related to, but distinct from, the mediation proportion literature. Earlier product-method work defined the mediation proportion as

$C$5

with effects evaluated on the link scale of the outcome model. For continuous outcomes with identity links and no exposure–mediator interaction, Cheng, Spiegelman, and Li obtained

$C$6

so that

$C$7

For binary outcomes, they provided exact natural effect expressions on the log-odds-ratio scale without the rare outcome assumption, and showed that approximate rare-outcome formulas can be substantially biased when the outcome is common, especially with a binary mediator (Cheng et al., 2021). This suggests that SAPM should be interpreted as a standardized absolute dominance measure built on top of the same causal effect scales rather than as a replacement for $C$8, $C$9, or $Z_{\mathrm{IE}}$00.

The Mendelian randomization application in the SAPM paper makes this distinction operational. There the goal is SNP-level assessment of non-pleiotropy, corresponding to complete mediation of the SNP effect on disease through an exposure. With binary exposure $Z_{\mathrm{IE}}$01 (insomnia), binary outcome $Z_{\mathrm{IE}}$02 (CHD), and genotype coded $Z_{\mathrm{IE}}$03–$Z_{\mathrm{IE}}$04, the procedure estimated $Z_{\mathrm{IE}}$05 and $Z_{\mathrm{IE}}$06 on the odds-ratio scale, formed $Z_{\mathrm{IE}}$07 and $Z_{\mathrm{IE}}$08, and applied CMT3 with binary thresholds $Z_{\mathrm{IE}}$09. Among $Z_{\mathrm{IE}}$10 SNPs previously judged valid by omnibus MR tests such as MR-Egger and Cochran’s $Z_{\mathrm{IE}}$11, CMT3 achieved near-perfect sensitivity, while CMT1 falsely classified one of three invalid SNPs as completely mediated. The paper’s detailed examples include rs12094039_T, with $Z_{\mathrm{IE}}$12, $Z_{\mathrm{IE}}$13, and $Z_{\mathrm{IE}}$14, and rs112270607_T, with $Z_{\mathrm{IE}}$15, $Z_{\mathrm{IE}}$16, and $Z_{\mathrm{IE}}$17, which fails the direct-effect condition and, under a strict $Z_{\mathrm{IE}}$18 rule, also fails the SAPM condition at $Z_{\mathrm{IE}}$19 (Tsai et al., 18 Jul 2025).

The limitations are correspondingly structural. SAPM depends on correct mediation models, correct standard errors, and the scale on which natural effects are defined. The SAPM paper studied only matched data types, both continuous or both binary; mixed-type mediation was left for future work. The generalized linear formulation assumes no exposure–mediator interaction. The paper also notes that measurement error in $Z_{\mathrm{IE}}$20 or $Z_{\mathrm{IE}}$21 can inflate standard errors, thereby lowering SAPM by reducing signal-to-noise even when an indirect pathway exists. Related mediation-effect work further emphasizes that exact binary-outcome formulas are preferable to rare-outcome approximations when outcome prevalence is common, and that mediation-effect estimation rests on standard counterfactual mediation assumptions such as consistency, composition, no unmeasured confounding given covariates, and cross-world independence (Tsai et al., 18 Jul 2025, Cheng et al., 2021).