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.

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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.

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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: $$\text{total_indirect} = \sum_{i=1}^{N+1} \text{indirect}_i + \sum_{i=2}^{N+1} \text{indirect}_{i1}$$

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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.

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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:

   $$CI_{1\text{vs}2} = \text{indirect}_1 - \text{indirect}_2$$

   $$CI_{1\text{vs}3} = \text{indirect}_1 - \text{indirect}_3$$

   $$CI_{1\text{vs}21} = \text{indirect}_1 - \text{indirect}_{21}$$

   $$CI_{1\text{vs}31} = \text{indirect}_1 - \text{indirect}_{31}$$

   $$CI_{2\text{vs}3} = \text{indirect}_2 - \text{indirect}_3$$

   $$CI_{2\text{vs}21} = \text{indirect}_2 - \text{indirect}_{21}$$

   $$CI_{3\text{vs}31} = \text{indirect}_3 - \text{indirect}_{31}$$

   $$CI_{21\text{vs}31} = \text{indirect}_{21} - \text{indirect}_{31}$$

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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 ($M_3 \to M_1$):

   $$X1_{b,31} = b_{31} + d_{31}$$

   $$X0_{b,31} = X1_{b,31} - d_{31}$$

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6. Summary of Regression Equations

This section summarizes all equations used in the model:

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

$$M_{1\text{diff}} = a_1 + \sum_{i=2}^{N+1} \left( b_{i1} M_{i\text{diff}} + d_{i1} M_{i\text{avg}} \right) + \epsilon_1$$

$$M_{i\text{diff}} = a_i + \epsilon_i$$

$$\text{indirect}_1 = a_1 \cdot b_1$$

$$\text{indirect}_i = a_i \cdot b_i$$

$$\text{indirect}_{i1} = a_i \cdot b_{i1} \cdot b_1$$

$$CI_{\text{path}_1\text{vs}\text{path}_2} = \text{indirect}_{\text{path}_1} - \text{indirect}_{\text{path}_2}$$

$$X1_{b,i} = b_i + d_i$$

$$X0_{b,i} = X1_{b,i} - d_i$$

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