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

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: \[ CI_{\text{path}_1\text{vs}\text{path}_2} = \text{indirect}_{\text{path}_1} - \text{indirect}_{\text{path}_2} \]

5. C1- and C2-Measurement Coefficients

6. Summary of Regression Equations

This section summarizes all equations used in the model:

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

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

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

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

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

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

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

\[ X1_{b,i} = b_i + d_i/2 \]

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

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

Standardization and moderation

GenerateModel*() constructs the unstandardized model syntax. Request standardized estimates when fitting with wsMed(..., standardized = TRUE). PrepareData() uses condition-2 minus condition-1 differences and centered mediator averages. Differences are scaled without recentering; centering the average component changes the intercept, not the slopes above.

Let \(s_{D M_i}\), \(s_{A M_i}\) and \(s_{D Y}\) denote the marginal model-implied SDs of mediator differences, centered mediator averages and the outcome difference. Then

\[ a_i^*=a_i/s_{D M_i},\quad b_i^*=b_i s_{D M_i}/s_{D Y},\quad d_i^*=d_i s_{A M_i}/s_{D Y},\quad c'^*=c'/s_{D Y}. \]

A mediator-to-mediator coefficient uses its own source and destination SDs. Intermediate SDs cancel in a serial indirect-effect product, so every complete indirect effect satisfies \(IE^*(w)=IE(w)/s_{D Y}\). The same common marginal outcome SD applies across moderator values and categorical groups. Sum all model-implied indirect paths when computing the total indirect effect.

With moderators, use result$moderation_std for standardized conditional effects and retain result$moderation for raw effects. MP selects requested interactions and focal paths; already-fitted moderator main effects also enter conditional intercepts even when the corresponding a or cp is absent from MP. See Standardized moderated mediation for continuous and categorical examples, joint-draw intervals and plots.

The condition-specific slopes above are reconstructed on the raw scale: \(X1_b=b+d/2\) and \(X0_b=b-d/2\). Do not apply this reconstruction directly to standardized \(b^*\) and \(d^*\), whose predictor scale factors differ. A difference between two indirect effects is a contrast estimate; its confidence interval must be computed from the joint sampling distribution.