---
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.
 <p align="center">
    <img src="Wd.png" alt="parallel-serial within-subject mediation model" width="60%">
  </p>`

---

## 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.
