# GenerateModelPC

## 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.

`

## 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.