GenerateModelP

Introduction

The GenerateModelP function dynamically generates a Structural Equation Model (SEM) formula to analysis parallel mediation for ‘lavaan’ based on the prepared dataset. This document explains the mathematical principles and the structure of the generated model.

parallel within-subject mediation model

1.2 Difference Model for $Y_{\text{diff}}$

Taking the difference between the two conditions: \[ Y_{\text{diff}} = Y_2 - Y_1 = (b_{20} - b_{10}) + \sum_{i=1}^N b_{i2} M_{i2} - \sum_{i=1}^N b_{i1} M_{i1} + (e_2 - e_1) \]

Define: - $\Delta b_0 = b_{20} - b_{10}$: Difference in intercepts. - $e = e_2 - e_1$: Difference in residuals.

Substitute mediator difference and average: 1. Mediator difference: \[ M_{\text{diff},i} = M_{i2} - M_{i1} \]

Mediator average: \[ M_{\text{avg},i} = \frac{M_{i1} + M_{i2}}{2} \]

Substitute $M_{i2} = M_{\text{avg},i} + \frac{M_{\text{diff},i}}{2}$ and $M_{i1} = M_{\text{avg},i} - \frac{M_{\text{diff},i}}{2}$ into the equation: \[ Y_{\text{diff}} = \Delta b_0 + \sum_{i=1}^N \left( \frac{b_{i1} + b_{i2}}{2} \cdot M_{\text{diff},i} + (b_{i2} - b_{i1}) \cdot M_{\text{avg},i} \right) + e \]

Define: - $b_i = \frac{b_{i1} + b_{i2}}{2}$: Average effect of the $i$-th mediator. - $d_i = b_{i2} - b_{i1}$: Difference in the effect of the $i$-th mediator.

The final equation becomes: \[ Y_{\text{diff}} = \Delta b_0 + \sum_{i=1}^N \left( b_i M_{\text{diff},i} + d_i M_{\text{avg},i} \right) + e \]

1.3 Regression for $M_{\text{diff}}$

Each mediator difference $M_{\text{diff},i}$ is modeled as: \[ M_{\text{diff},i} = a_i + \epsilon_i \]

Where: - $a_i$: Intercept term for the $i$-th mediator difference. - $\epsilon_i$: Residual for $M_{\text{diff},i}$.

2. Indirect Effects

For each mediator $M_i$, the indirect effect is defined as: \[ \text{indirect}_i = a_i \cdot b_i \]

Where: - $a_i$: Effect of the independent variable on mediator $M_i$. - $b_i$: Average effect of mediator $M_i$ on the dependent variable.

The total indirect effect is: \[ \text{total_indirect} = \sum_{i=1}^N \text{indirect}_i \]

The contrast between indirect effects of two mediators $M_i$ and $M_j$ is: \[ CI_{i,j} = \text{indirect}_i - \text{indirect}_j \]

3. Total Effect

The total effect combines the direct effect and the total indirect effect: \[ \text{total_effect} = c_p + \text{total_indirect} \]

Where $c_p$ is the direct effect of the independent variable on the dependent variable.

4. Comparison of Indirect Effects

When there are multiple mediators ($M_1, M_2, \dots, M_N$), comparing their indirect effects provides insights into the relative influence of each mediator. This section details the formulas and interpretations for such comparisons.

4.1 Indirect Effect Definition

For a mediator $M_i$, the indirect effect is defined as: \[ \text{indirect}_i = a_i \cdot b_i \]

Where: - $a_i$: Effect of the independent variable on mediator $M_i$. - $b_i$: Average effect of mediator $M_i$ on the dependent variable.

4.2 Comparing Indirect Effects

To compare the indirect effects of two mediators $M_i$ and $M_j$, we calculate the contrast: \[ CI_{i,j} = \text{indirect}_i - \text{indirect}_j \]

Interpretation

$CI_{i,j} > 0$:
- Mediator $M_i$ has a stronger indirect effect than $M_j$.
$CI_{i,j} < 0$:
- Mediator $M_j$ has a stronger indirect effect than $M_i$.
$CI_{i,j} = 0$:
- Both mediators contribute equally to the indirect effect.

5. C1- and C2-Measurement Coefficients

To compute C1- and C2-measurement coefficients $X1_{b,i}$ and $X0_{b,i}$, consider two mediators $M_1$ and $M_2$:

5.1 Difference Model with Two Mediators

From the difference model: \[ Y_{\text{diff}} = \Delta b_0 + \left(\frac{b_{11} + b_{21}}{2}\right) M_{\text{diff}} + \left(b_{21} - b_{11}\right) M_{\text{avg}} + e \]

Define: - $b = \frac{b_{11} + b_{21}}{2}$: Average effect. - $d = b_{21} - b_{11}$: Difference in effect.

5.2 C2-Measurement Coefficients

The C2-measurement coefficient $X1_{b,i}$ is defined as: \[ X1_{b,i} = b + d/2 \]

Substitute $b$ and $d$: \[ X1_{b,i} = \frac{b_{11} + b_{21}}{2} + \frac{b_{21} - b_{11}}{2} = b_{21} \]

Thus, $X1_{b,i}$ is the effect of $M_i$ under Condition 2.

5.3 C1-Measurement Coefficients

The C1-measurement coefficient $X0_{b,i}$ is defined as: \[ X0_{b,i} = X1_{b,i} - d \]

Substitute $X1_{b,i} = b_{21}$ and $d = b_{21} - b_{11}$: \[ X0_{b,i} = b_{21} - (b_{21} - b_{11}) = b_{11} \]

Thus, $X0_{b,i}$ is the effect of $M_i$ under Condition 1.

Additional Interpretation: The coefficient $d_i = b_{2i} - b_{1i}$ reflects the moderating effect of the within-subject variable X, capturing how the mediator’s influence differs across conditions.

6. Summary of Regression Equations

This section summarizes all the regression equations used in the analysis, including the difference model, indirect effects, mediator comparisons, and C1- and C2-measurement coefficients.

6.1 Difference Model

\[ Y_{\text{diff}} = cp + \sum_{i=1}^N \left( b_i M_{\text{diff},i} + d_i M_{\text{avg},i} \right) + e \]

\[ M_{\text{diff},i} = a_i + \epsilon_i \]

6.2 Defined parameters

\[ \text{indirect}_i = a_i \cdot b_i \]

\[ \text{total_indirect} = \sum_{i=1}^N \text{indirect}_i \]

\[ CI_{i,j} = \text{indirect}_i - \text{indirect}_j \]

\[ X1_{b,i} = b_i + d_i/2 \]

\[ X0_{b,i} = X1_{b,i} - d_i \]

Summary

By combining these equations: 1. The difference model $Y_{\text{diff}}$ decomposes into contributions from mediator differences ($M_{\text{diff}}$) and averages ($M_{\text{avg}}$). 2. Indirect effects and their contrasts provide insights into the mediators’ relative importance. 3. C1- and C2-measurement coefficients quantify the effects in specific conditions.

Standardization and moderation

GenerateModel*() constructs the unstandardized model syntax. Request standardized estimates when fitting with wsMed(..., standardized = TRUE). PrepareData() uses condition-2 minus condition-1 differences and centered mediator averages. Differences are scaled without recentering; centering the average component changes the intercept, not the slopes above.

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

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

A mediator-to-mediator coefficient uses its own source and destination SDs. Intermediate SDs cancel in a serial indirect-effect product, so every complete indirect effect satisfies $IE^*(w)=IE(w)/s_{D Y}$. The same common marginal outcome SD applies across moderator values and categorical groups. Sum all model-implied indirect paths when computing the total indirect effect.

With moderators, use result$moderation_std for standardized conditional effects and retain result$moderation for raw effects. MP selects requested interactions and focal paths; already-fitted moderator main effects also enter conditional intercepts even when the corresponding a or cp is absent from MP. See Standardized moderated mediation for continuous and categorical examples, joint-draw intervals and plots.

The condition-specific slopes above are reconstructed on the raw scale: $X1_b=b+d/2$ and $X0_b=b-d/2$. Do not apply this reconstruction directly to standardized $b^*$ and $d^*$, whose predictor scale factors differ. A difference between two indirect effects is a contrast estimate; its confidence interval must be computed from the joint sampling distribution.