# GenerateModelCP

## 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.

## 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. 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. Chained Path (\(M_1 \to M_2\)):

   \[ X1_{b,12} = b_{12} + d_{12} \]

   \[ X0_{b,12} = X1_{b,12} - d_{12} \]

5. Chained Path (\(M_1 \to M_3\)):

   \[ X1_{b,13} = b_{13} + d_{13} \]

   \[ X0_{b,13} = X1_{b,13} - d_{13} \]

## 6. Summary of Regression Equations

This section summarizes all equations used in the model:

\[ Y_{\text{diff}} = cp + b_1 M_{1\text{diff}} + \sum_{i=2}^{N+1} \left( b_i M_{i\text{diff}} + d_i M_{i\text{avg}} \right) + d_1 M_{1\text{avg}} + e \]

\[ M_{1\text{diff}} = a_1 + \epsilon_1 \]

\[ M_{i\text{diff}} = a_i + b_{1i} M_{1\text{diff}} + d_{1i} M_{1\text{avg}} + \epsilon_i \]

\[ \text{indirect}_1 = a_1 \cdot b_1 \]

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

\[ \text{indirect}_{1i} = a_1 \cdot b_{1i} \cdot b_i \]

\[ CI_{\text{path}_1\text{vs}\text{path}_2} = \text{indirect}_{\text{path}_1} - \text{indirect}_{\text{path}_2} \] \[ X1_{b,i} = b_i + d_i \]

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

This comprehensive approach supports models with both chained and parallel mediators, enabling detailed analysis of their effects and interactions.