# GenerateModelP

## Introduction

The `GenerateModelP` function dynamically generates a Structural Equation Model (SEM) formula to analysis parallel mediation for ‘lavaan’ based on the prepared dataset. This document explains the mathematical principles and the structure of the generated model.

### 1.2 Difference Model for \(Y_{\text{diff}}\)

Taking the difference between the two conditions: \[ Y_{\text{diff}} = Y_2 - Y_1 = (b_{20} - b_{10}) + \sum_{i=1}^N b_{i2} M_{i2} - \sum_{i=1}^N b_{i1} M_{i1} + (e_2 - e_1) \]

Define: - \(\Delta b_0 = b_{20} - b_{10}\): Difference in intercepts. - \(e = e_2 - e_1\): Difference in residuals.

Substitute mediator difference and average: 1. Mediator difference: \[ M_{\text{diff},i} = M_{i2} - M_{i1} \]

2. Mediator average: \[ M_{\text{avg},i} = \frac{M_{i1} + M_{i2}}{2} \]

Substitute \(M_{i2} = M_{\text{avg},i} + \frac{M_{\text{diff},i}}{2}\) and \(M_{i1} = M_{\text{avg},i} - \frac{M_{\text{diff},i}}{2}\) into the equation: \[ Y_{\text{diff}} = \Delta b_0 + \sum_{i=1}^N \left( \frac{b_{i1} + b_{i2}}{2} \cdot M_{\text{diff},i} + (b_{i2} - b_{i1}) \cdot M_{\text{avg},i} \right) + e \]

Define: - \(b_i = \frac{b_{i1} + b_{i2}}{2}\): Average effect of the \(i\)-th mediator. - \(d_i = b_{i2} - b_{i1}\): Difference in the effect of the \(i\)-th mediator.

The final equation becomes: \[ Y_{\text{diff}} = \Delta b_0 + \sum_{i=1}^N \left( b_i M_{\text{diff},i} + d_i M_{\text{avg},i} \right) + e \]

### 1.3 Regression for \(M_{\text{diff}}\)

Each mediator difference \(M_{\text{diff},i}\) is modeled as: \[ M_{\text{diff},i} = a_i + \epsilon_i \]

Where: - \(a_i\): Intercept term for the \(i\)-th mediator difference. - \(\epsilon_i\): Residual for \(M_{\text{diff},i}\).

## 2. Indirect Effects

For each mediator \(M_i\), the indirect effect is defined as: \[ \text{indirect}_i = a_i \cdot b_i \]

Where: - \(a_i\): Effect of the independent variable on mediator \(M_i\). - \(b_i\): Average effect of mediator \(M_i\) on the dependent variable.

The total indirect effect is: \[ \text{total_indirect} = \sum_{i=1}^N \text{indirect}_i \]

The contrast between indirect effects of two mediators \(M_i\) and \(M_j\) is: \[ CI_{i,j} = \text{indirect}_i - \text{indirect}_j \]

## 3. Total Effect

The total effect combines the direct effect and the total indirect effect: \[ \text{total_effect} = c_p + \text{total_indirect} \]

Where \(c_p\) is the direct effect of the independent variable on the dependent variable.

## 4. Comparison of Indirect Effects

When there are multiple mediators (\(M_1, M_2, \dots, M_N\)), comparing their indirect effects provides insights into the relative influence of each mediator. This section details the formulas and interpretations for such comparisons.

### 4.1 Indirect Effect Definition

For a mediator \(M_i\), the indirect effect is defined as: \[ \text{indirect}_i = a_i \cdot b_i \]

Where: - \(a_i\): Effect of the independent variable on mediator \(M_i\). - \(b_i\): Average effect of mediator \(M_i\) on the dependent variable.

### 4.2 Comparing Indirect Effects

To compare the indirect effects of two mediators \(M_i\) and \(M_j\), we calculate the contrast: \[ CI_{i,j} = \text{indirect}_i - \text{indirect}_j \]

#### Interpretation

1. \(CI_{i,j} > 0\):
   - Mediator \(M_i\) has a stronger indirect effect than \(M_j\).
2. \(CI_{i,j} < 0\):
   - Mediator \(M_j\) has a stronger indirect effect than \(M_i\).
3. \(CI_{i,j} = 0\):
   - Both mediators contribute equally to the indirect effect.

## 5. C1- and C2-Measurement Coefficients

To compute C1- and C2-measurement coefficients \(X1_{b,i}\) and \(X0_{b,i}\), consider two mediators \(M_1\) and \(M_2\):

### 5.1 Difference Model with Two Mediators

From the difference model: \[ Y_{\text{diff}} = \Delta b_0 + \left(\frac{b_{11} + b_{21}}{2}\right) M_{\text{diff}} + \left(b_{21} - b_{11}\right) M_{\text{avg}} + e \]

Define: - \(b = \frac{b_{11} + b_{21}}{2}\): Average effect. - \(d = b_{21} - b_{11}\): Difference in effect.

### 5.2 C2-Measurement Coefficients

The C2-measurement coefficient \(X1_{b,i}\) is defined as: \[ X1_{b,i} = b + d \]

Substitute \(b\) and \(d\): \[ X1_{b,i} = \frac{b_{11} + b_{21}}{2} + (b_{21} - b_{11}) = b_{21} \]

Thus, \(X1_{b,i}\) is the effect of \(M_i\) under Condition 2.

### 5.3 C1-Measurement Coefficients

The C1-measurement coefficient \(X0_{b,i}\) is defined as: \[ X0_{b,i} = X1_{b,i} - d \]

Substitute \(X1_{b,i} = b_{21}\) and \(d = b_{21} - b_{11}\): \[ X0_{b,i} = b_{21} - (b_{21} - b_{11}) = b_{11} \]

Thus, \(X0_{b,i}\) is the effect of \(M_i\) under Condition 1.

Additional Interpretation: The coefficient \(d_i = b_{2i} - b_{1i}\) reflects the moderating effect of the within-subject variable X, capturing how the mediator’s influence differs across conditions.

## 6. Summary of Regression Equations

This section summarizes all the regression equations used in the analysis, including the difference model, indirect effects, mediator comparisons, and C1- and C2-measurement coefficients.

### 6.1 Difference Model

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

\[ M_{\text{diff},i} = a_i + \epsilon_i \]

### 6.2 Defined parameters

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

\[ \text{total_indirect} = \sum_{i=1}^N \text{indirect}_i \]

\[ CI_{i,j} = \text{indirect}_i - \text{indirect}_j \]

\[ X1_{b,i} = b_i + d_i \]

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

### Summary

By combining these equations: 1. The difference model \(Y_{\text{diff}}\) decomposes into contributions from mediator differences (\(M_{\text{diff}}\)) and averages (\(M_{\text{avg}}\)). 2. Indirect effects and their contrasts provide insights into the mediators’ relative importance. 3. C1- and C2-measurement coefficients quantify the effects in specific conditions.