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

parallel-serial 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 (\( M_1 \)): \[ M_{1\text{diff}} = a_1 + \sum_{i=2}^{N+1} \left( b_{i1} M_{i\text{diff}} + d_{i1} M_{i\text{avg}} \right) + \epsilon_1 \] - For the parallel mediators (\( M_2, \dots, M_{N+1} \)): \[ M_{i\text{diff}} = a_i + \epsilon_i \] Where: - \( cp \): Direct effect of the independent variable. - \( b_1, b_i, b_{i1} \): Effects of the chained and parallel mediators. - \( d_1, d_i, d_{i1} \): 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. **Parallel to Chained Path Effects:** - For paths from the parallel mediators to the chained mediator: \[ \text{indirect}_{i1} = a_i \cdot b_{i1} \cdot b_1 \] 3. **Total Indirect Effect:** The total indirect effect is the sum of all individual indirect effects: \[ \text{total_indirect} = \sum_{i=1}^{N+1} \text{indirect}_i + \sum_{i=2}^{N+1} \text{indirect}_{i1} \] --- ## 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:** \[ \text{indirect}_1 = a_1 \cdot b_1 \] \[ \text{indirect}_2 = a_2 \cdot b_2 \] \[ \text{indirect}_3 = a_3 \cdot b_3 \] \[ \text{indirect}_{21} = a_2 \cdot b_{21} \cdot b_1 \] \[ \text{indirect}_{31} = a_3 \cdot b_{31} \cdot b_1 \] 2. **Comparisons:** \[ CI_{1\text{vs}2} = \text{indirect}_1 - \text{indirect}_2 \] \[ CI_{1\text{vs}3} = \text{indirect}_1 - \text{indirect}_3 \] \[ CI_{1\text{vs}21} = \text{indirect}_1 - \text{indirect}_{21} \] \[ CI_{1\text{vs}31} = \text{indirect}_1 - \text{indirect}_{31} \] \[ CI_{2\text{vs}3} = \text{indirect}_2 - \text{indirect}_3 \] \[ CI_{2\text{vs}21} = \text{indirect}_2 - \text{indirect}_{21} \] \[ CI_{3\text{vs}31} = \text{indirect}_3 - \text{indirect}_{31} \] \[ CI_{21\text{vs}31} = \text{indirect}_{21} - \text{indirect}_{31} \] --- ## 5. C1- and C2-Measurement Coefficients ### Definitions 1. **C2-Measurement Coefficient (\( X1_{b,i} \)):** \[ X1_{b,i} = b_i + d_i \] 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 \( M_1 \):** \[ X1_{b,1} = b_1 + d_1 \] \[ X0_{b,1} = X1_{b,1} - d_1 \] 2. **Mediator \( M_2 \):** \[ X1_{b,2} = b_2 + d_2 \] \[ X0_{b,2} = X1_{b,2} - d_2 \] 3. **Mediator \( M_3 \):** \[ X1_{b,3} = b_3 + d_3 \] \[ X0_{b,3} = X1_{b,3} - d_3 \] 4. **Parallel to Chained Path (\( M_2 \to M_1 \)):** \[ X1_{b,21} = b_{21} + d_{21} \] \[ X0_{b,21} = X1_{b,21} - d_{21} \] 5. **Parallel to Chained Path (\( M_3 \to M_1 \)):** \[ X1_{b,31} = b_{31} + d_{31} \] \[ X0_{b,31} = X1_{b,31} - d_{31} \] --- ## 6. Summary of Regression Equations This section summarizes all equations used in the model: \[ Y_{\text{diff}} = cp + b_1 M_{1\text{diff}} + \sum_{i=2}^{N+1} \left( b_i M_{i\text{diff}} + d_i M_{i\text{avg}} \right) + d_1 M_{1\text{avg}} + e \] \[ M_{1\text{diff}} = a_1 + \sum_{i=2}^{N+1} \left( b_{i1} M_{i\text{diff}} + d_{i1} M_{i\text{avg}} \right) + \epsilon_1 \] \[ M_{i\text{diff}} = a_i + \epsilon_i \] \[ \text{indirect}_1 = a_1 \cdot b_1 \] \[ \text{indirect}_i = a_i \cdot b_i \] \[ \text{indirect}_{i1} = a_i \cdot b_{i1} \cdot b_1 \] \[ CI_{\text{path}_1\text{vs}\text{path}_2} = \text{indirect}_{\text{path}_1} - \text{indirect}_{\text{path}_2} \] \[ X1_{b,i} = b_i + d_i \] \[ X0_{b,i} = X1_{b,i} - d_i \] --- This comprehensive approach supports models with parallel mediators influencing a chained mediator, enabling detailed analysis of their effects and interactions.