Model equations: parallel-serial mediation

Difference scores and centered averages

For mediator \(i\), \(M_{i1}\) and \(M_{i2}\) are its measurements in conditions 1 and 2. Define

\[ M_{\mathrm{diff},i}=M_{i2}-M_{i1},\qquad \widetilde M_{\mathrm{avg},i}=\frac{M_{i1}+M_{i2}}{2},\qquad M_{\mathrm{avg},i}=\widetilde M_{\mathrm{avg},i}-\overline{\widetilde M}_{\mathrm{avg},i}. \]

The bar denotes the sample mean used for centering. The outcome difference is \(Y_{\mathrm{diff}}=Y_2-Y_1\). PrepareData() constructs these variables: diff is not centered, whereas avg is centered.

Regression equations

The equations below show the regression part of the model without covariates or an external moderator. All \(N\) mediators predict the outcome difference:

\[ Y_{\mathrm{diff}}=c'+\sum_{i=1}^{N}\left(b_iM_{\mathrm{diff},i}+d_iM_{\mathrm{avg},i}\right)+e. \]

The mediator equations determine the structure:

\[ \begin{aligned}M_{\mathrm{diff},1}&=a_1+\sum_{i=2}^{N}\left(b_{i1}M_{\mathrm{diff},i}+d_{i1}M_{\mathrm{avg},i}\right)+\epsilon_1,\\M_{\mathrm{diff},i}&=a_i+\epsilon_i,\quad i=2,\ldots,N.\end{aligned} \]

The intercepts \(a_i\) and \(c'\) represent direct condition effects in their respective equations; the model syntax labels \(c'\) as cp. Each \(b_i\) is the coefficient of a mediator difference, and \(d_i\) is the coefficient of its centered average. For a mediator-to-mediator connection, \(b_{ji}\) and \(d_{ji}\) are the corresponding coefficients from mediator \(j\) to mediator \(i\) (syntax labels b_j_i and d_j_i).

The \(d\) coefficients capture differences between condition-specific slopes. The average component is not itself an external moderator. Moderation by a separate variable \(W\) requires specifying W and MP.

Indirect pathways

Every mediator has a single-mediator indirect pathway \(IE_i=a_i b_i\). A longer directed pathway is the product of its starting \(a\) coefficient, its mediator-to-mediator \(b\) coefficients and its final outcome \(b\) coefficient. The centered-average \(d\) coefficients remain in the fitted regressions but do not enter these indirect-effect products.

For three mediators, the complete set is:

Mediator route to Y	Indirect effect	Result identifier
M1 → Y	\(a_1 b_1\)	indirect_1
M2 → Y	\(a_2 b_2\)	indirect_2
M3 → Y	\(a_3 b_3\)	indirect_3
M2 → M1 → Y	\(a_2 b_{21} b_1\)	indirect_2_1
M3 → M1 → Y	\(a_3 b_{31} b_1\)	indirect_3_1

Total effects and contrasts

Sum every complete indirect pathway to obtain the total indirect effect:

\[ IE_{\mathrm{total}}=\sum_{p\in\mathcal P}IE_p,\qquad TE=c'+IE_{\mathrm{total}}. \]

Here \(\mathcal P\) is the set of directed condition-to-outcome pathways through one or more mediators. To compare two such pathways, define a signed contrast:

\[ \Delta_{p,q}=IE_p-IE_q. \]

A positive contrast means that the first indirect effect is numerically larger; it does not necessarily have a larger absolute magnitude. For example, \(-0.1-(-0.3)=0.2\), although \(|-0.1|<|-0.3|\). A contrast estimate alone does not establish a difference. Compute its CI from the difference of the effects within each matched joint draw, using wsmed_contrasts(); do not subtract the endpoints of two separate CIs. See Read and export results.

Condition-specific slopes

On the unstandardized scale, the outcome slopes in conditions 1 and 2 are

\[ g_{i1}=b_i-d_i/2,\qquad g_{i2}=b_i+d_i/2. \]

The same relation applies to each included mediator-to-mediator connection, using \(b_{ji}\) and \(d_{ji}\). These are path coefficients, not additional indirect effects.

Standardization and moderation

GenerateModel*() constructs unstandardized model syntax. With a fitted wsMed result, use summary(result, standardized = TRUE) to request standardized effects and paths, or set standardized = TRUE in wsMed() to make that scale the reporting default.

Let \(s_{D M_i}\), \(s_{A M_i}\) and \(s_{D Y}\) be the marginal model-implied SDs of the mediator differences, centered averages and outcome difference. Then

\[ a_i^*=a_i/s_{D M_i},\quad b_i^*=b_i s_{D M_i}/s_{D Y},\quad d_i^*=d_i s_{A M_i}/s_{D Y},\quad c'^*=c'/s_{D Y}. \]

Mediator-to-mediator slopes use the corresponding source and destination SDs. Intermediate scales cancel in each complete indirect-effect product, giving \(IE_p^*=IE_p/s_{D Y}\). With moderation, evaluate the conditional paths first: \(IE_p^*(w)=IE_p(w)/s_{D Y}\). All moderator values or groups share the same marginal endpoint scale. Continuous moderator probes retain their raw units.

Standardized CIs transform coefficients and scales jointly in each draw. Do not divide raw CI endpoints by a single fitted SD, or reconstruct condition-specific slopes as \(b_i^*\pm d_i^*/2\): the two standardized coefficients use different predictor scales. See Standardized effects for conditional effects, interaction coefficients and inference.