--- title: "GenerateModelCP" output: rmarkdown::html_vignette vignette: > %\VignetteIndexEntry{GenerateModelCP} %\VignetteEngine{knitr::rmarkdown} %\VignetteEncoding{UTF-8} --- ## Introduction The `GenerateModelCP` function dynamically generates a Structural Equation Model (SEM) formula to analyze models with a single chained mediator and multiple parallel mediators for 'lavaan' based on the prepared dataset. This document explains the mathematical principles and the structure of the generated model.

serial-parallel within-subject mediation model

--- ## 1. Model Description ### 1.1 Regression for \( Y_{\text{diff}} \) and \( M_{\text{diff}} \) For a single chained mediator \( M_1 \) and \( N \) parallel mediators \( M_2, M_3, \dots, M_{N+1} \), the model is defined as: 1. **Outcome Difference Model (\( Y_{\text{diff}} \)):** \[ Y_{\text{diff}} = cp + b_1 M_{1\text{diff}} + \sum_{i=2}^{N+1} \left( b_i M_{i\text{diff}} + d_i M_{i\text{avg}} \right) + d_1 M_{1\text{avg}} + e \] 2. **Mediator Difference Model (\( M_{i\text{diff}} \)):** For the chained mediator (\( M_1 \)): \[ M_{1\text{diff}} = a_1 + \epsilon_1 \] For parallel mediators (\( M_2, \dots, M_{N+1} \)): \[ M_{i\text{diff}} = a_i + b_{1i} M_{1\text{diff}} + d_{1i} M_{1\text{avg}} + \epsilon_i \] Where: - \( cp \): Direct effect of the independent variable. - \( b_1, b_i \): Effects of the chained and parallel mediators. - \( d_1, d_i, d_{1i} \): Moderating effects of mediator averages. - \( \epsilon_i \): Residuals. --- ## 2. Indirect Effects For each mediator, the indirect effects are calculated as: 1. **Single-Mediator Effects:** For the chained mediator: \[ \text{indirect}_1 = a_1 \cdot b_1 \] For the parallel mediators (\( M_2, \dots, M_{N+1} \)): \[ \text{indirect}_i = a_i \cdot b_i \] 2. **Chained Path Effects:** For paths from the chained mediator through the parallel mediators: \[ \text{indirect}_{1i} = a_1 \cdot b_{1i} \cdot b_i \] 3. **Total Indirect Effect:** The total indirect effect is the sum of all individual indirect effects: \[ \text{total_indirect} = \text{indirect}_1 + \sum_{i=2}^{N+1} \left( \text{indirect}_i + \text{indirect}_{1i} \right) \] --- ## 3. Total Effect The total effect combines the direct effect and the total indirect effect: \[ \text{total_effect} = cp + \text{total_indirect} \] Where \( cp \) is the direct effect. --- ## 4. Comparison of Indirect Effects When comparing the strengths of indirect effects, the contrast between two effects is calculated as: \[ CI_{\text{path}_1\text{vs}\text{path}_2} = \text{indirect}_{\text{path}_1} - \text{indirect}_{\text{path}_2} \] ### 4.1 Example: Three Mediators (\( M_1, M_2, M_3 \)) 1. **Indirect Effects:** \[ \text{indirect}_1 = a_1 \cdot b_1 \] \[ \text{indirect}_2 = a_2 \cdot b_2 \] \[ \text{indirect}_3 = a_3 \cdot b_3 \] \[ \text{indirect}_{12} = a_1 \cdot b_{12} \cdot b_2 \] \[ \text{indirect}_{13} = a_1 \cdot b_{13} \cdot b_3 \] 2. **Comparisons:** \[ CI_{1\text{vs}2} = \text{indirect}_1 - \text{indirect}_2 \] \[ CI_{1\text{vs}3} = \text{indirect}_1 - \text{indirect}_3 \] \[ CI_{1\text{vs}12} = \text{indirect}_1 - \text{indirect}_{12} \] \[ CI_{1\text{vs}13} = \text{indirect}_1 - \text{indirect}_{13} \] \[ CI_{2\text{vs}3} = \text{indirect}_2 - \text{indirect}_3 \] \[ CI_{2\text{vs}12} = \text{indirect}_2 - \text{indirect}_{12} \] \[ CI_{2\text{vs}13} = \text{indirect}_2 - \text{indirect}_{13} \] \[ CI_{3\text{vs}12} = \text{indirect}_3 - \text{indirect}_{12} \] \[ CI_{3\text{vs}13} = \text{indirect}_3 - \text{indirect}_{13} \] \[ CI_{12\text{vs}13} = \text{indirect}_{12} - \text{indirect}_{13} \] --- ## 5. C1 and C2 Coefficients ### Definitions 1. **C2-Measurement Coefficient (\( X1_{b,i} \)):** \[ X1_{b,i} = b_i + d_i/2 \] 2. **C1-Measurement Coefficient (\( X0_{b,i} \)):** \[ X0_{b,i} = X1_{b,i} - d_i \] ### 5.1 Example: Three Mediators (\( M_1, M_2, M_3 \)) 1. **Mediator \( M_1 \):** \[ X1_{b,1} = b_1 + d_1/2 \] \[ X0_{b,1} = X1_{b,1} - d_1 \] 2. **Mediator \( M_2 \):** \[ X1_{b,2} = b_2 + d_2/2 \] \[ X0_{b,2} = X1_{b,2} - d_2 \] 3. **Mediator \( M_3 \):** \[ X1_{b,3} = b_3 + d_3/2 \] \[ X0_{b,3} = X1_{b,3} - d_3 \] 4. **Chained Path (\( M_1 \to M_2 \)):** \[ X1_{b,12} = b_{12} + d_{12}/2 \] \[ X0_{b,12} = X1_{b,12} - d_{12} \] 5. **Chained Path (\( M_1 \to M_3 \)):** \[ X1_{b,13} = b_{13} + d_{13}/2 \] \[ X0_{b,13} = X1_{b,13} - d_{13} \] --- ## 6. Summary of Regression Equations This section summarizes all equations used in the model: \[ Y_{\text{diff}} = cp + b_1 M_{1\text{diff}} + \sum_{i=2}^{N+1} \left( b_i M_{i\text{diff}} + d_i M_{i\text{avg}} \right) + d_1 M_{1\text{avg}} + e \] \[ M_{1\text{diff}} = a_1 + \epsilon_1 \] \[ M_{i\text{diff}} = a_i + b_{1i} M_{1\text{diff}} + d_{1i} M_{1\text{avg}} + \epsilon_i \] \[ \text{indirect}_1 = a_1 \cdot b_1 \] \[ \text{indirect}_i = a_i \cdot b_i \] \[ \text{indirect}_{1i} = a_1 \cdot b_{1i} \cdot b_i \] \[ CI_{\text{path}_1\text{vs}\text{path}_2} = \text{indirect}_{\text{path}_1} - \text{indirect}_{\text{path}_2} \] \[ X1_{b,i} = b_i + d_i/2 \] \[ X0_{b,i} = X1_{b,i} - d_i \] --- This comprehensive approach supports models with both chained and parallel mediators, enabling detailed analysis of their effects and interactions. ## 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](StandardizedModeratedMediation.html) 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.