GenerateModelCN function dynamically generates a
Structural Equation Model (SEM) formula to analyze chained or nested
mediation for ‘lavaan’ based on the prepared dataset. This document
explains the mathematical principles and the structure of the generated
model.
For \(N\) mediators \(M_1, M_2, \dots, M_N\), the difference model is defined as:
Outcome Difference Model (\(Y_{\text{diff}}\)): \[ Y_{\text{diff}} = cp + \sum_{i=1}^N \left( b_i M_{\text{diff},i} + d_i M_{\text{avg},i} \right) + e \]
Mediator Difference Model (\(M_{\text{diff},i}\)): \[ M_{\text{diff},i} = a_i + \sum_{j<i} \left( b_{ji} M_{\text{diff},j} + d_{ji} M_{\text{avg},j} \right) + \epsilon_i \]
Where: - \(cp\): Intercept term for the outcome difference model. - \(b_i\): Average effect of mediator \(M_i\) on \(Y_{\text{diff}}\). - \(d_i\): Moderator effect for \(M_{\text{avg},i}\) in \(Y_{\text{diff}}\). - \(b_{ji}\) and \(d_{ji}\): Regression coefficients for \(M_{\text{diff},j}\) and \(M_{\text{avg},j}\) on \(M_{\text{diff},i}\), respectively. - \(\epsilon_i\): Residual for \(M_{\text{diff},i}\).
For each mediator \(M_i\), the indirect effect is defined as: \[ \text{indirect}_i = a_i \cdot b_i \]
For chained mediators, the indirect effects follow the paths through the mediators: 1. For a single mediator \(M_i\): \[ \text{indirect}_i = a_i \cdot b_i \]
The total indirect effect is: \[ \text{total_indirect} = \sum_{\text{all paths}} \text{indirect}_{\text{path}} \]
For three mediators \(M_1 \to M_2 \to M_3\), the indirect effects include:
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.
When there are multiple mediators or pathways, comparing their indirect effects provides insights into the relative influence of each mediator or chain.
The contrast between two indirect effects, \(\text{indirect}_{\text{path}_1}\) and \(\text{indirect}_{\text{path}_2}\), is calculated as: \[ CI_{\text{path}_1\text{vs}\text{path}_2} = \text{indirect}_{\text{path}_1} - \text{indirect}_{\text{path}_2} \]
For three mediators, the following indirect effects are defined:
Direct Path Effects: \[ \text{indirect}_1 = a_1 \cdot b_1 \] \[ \text{indirect}_2 = a_2 \cdot b_2 \] \[ \text{indirect}_3 = a_3 \cdot b_3 \]
Chained Path Effects: \[ \text{indirect}_{12} = a_1 \cdot b_{12} \cdot b_2 \] \[ \text{indirect}_{23} = a_2 \cdot b_{23} \cdot b_3 \] \[ \text{indirect}_{123} = a_1 \cdot b_{12} \cdot b_{23} \cdot b_3 \]
The indirect effects are compared as follows: \[ CI_{1\text{vs}2} = \text{indirect}_1 - \text{indirect}_2 \] \[ CI_{1\text{vs}3} = \text{indirect}_1 - \text{indirect}_3 \] \[ CI_{2\text{vs}3} = \text{indirect}_2 - \text{indirect}_3 \] \[ CI_{1\text{vs}12} = \text{indirect}_1 - \text{indirect}_{12} \] \[ CI_{1\text{vs}23} = \text{indirect}_1 - \text{indirect}_{23} \] \[ CI_{1\text{vs}123} = \text{indirect}_1 - \text{indirect}_{123} \] \[ CI_{2\text{vs}12} = \text{indirect}_2 - \text{indirect}_{12} \] \[ CI_{2\text{vs}23} = \text{indirect}_2 - \text{indirect}_{23} \] \[ CI_{2\text{vs}123} = \text{indirect}_2 - \text{indirect}_{123} \] \[ CI_{3\text{vs}12} = \text{indirect}_3 - \text{indirect}_{12} \] \[ CI_{3\text{vs}23} = \text{indirect}_3 - \text{indirect}_{23} \] \[ CI_{3\text{vs}123} = \text{indirect}_3 - \text{indirect}_{123} \] \[ CI_{12\text{vs}23} = \text{indirect}_{12} - \text{indirect}_{23} \] \[ CI_{12\text{vs}123} = \text{indirect}_{12} - \text{indirect}_{123} \] \[ CI_{23\text{vs}123} = \text{indirect}_{23} - \text{indirect}_{123} \] —
For C1- and C2-measurement conditions, the coefficients are calculated as follows:
C2-Measurement Coefficient (\(X1_{b,i}\)): \[ X1_{b,i} = b_i + d_i \]
C1-Measurement Coefficient (\(X0_{b,i}\)): \[ X0_{b,i} = X1_{b,i} - d_i \]
For chained pathways: 1. C2-Measurement Coefficient (\(X1_{b,ij}\)): \[ X1_{b,ij} = b_{ij} + d_{ij} \]
C1-Measurement Coefficient (\(X0_{b,ij}\)): \[ X0_{b,ij} = X1_{b,ij} - d_{ij} \] — For three mediators \(M_1, M_2, M_3\), the coefficients are calculated as follows:
C2-Measurement Coefficient: \[ X1_{b,1} = b_1 + d_1 \]
C1-Measurement Coefficient: \[ X0_{b,1} = X1_{b,1} - d_1 \]
C2-Measurement Coefficient: \[ X1_{b,2} = b_2 + d_2 \]
C1-Measurement Coefficient: \[ X0_{b,2} = X1_{b,2} - d_2 \]
C2-Measurement Coefficient: \[ X1_{b,3} = b_3 + d_3 \]
C1-Measurement Coefficient: \[ X0_{b,3} = X1_{b,3} - d_3 \]
C2-Measurement Coefficient: \[ X1_{b,12} = b_{12} + d_{12} \]
C1-Measurement Coefficient: \[ X0_{b,12} = X1_{b,12} - d_{12} \] —
This section summarizes all the regression equations:
Outcome Difference Model (\(Y_{\text{diff}}\)): \[ Y_{\text{diff}} = cp + \sum_{i=1}^N \left( b_i M_{\text{diff},i} + d_i M_{\text{avg},i} \right) + e \]
Mediator Difference Model (\(M_{\text{diff},i}\)): \[ M_{\text{diff},i} = a_i + \sum_{j<i} \left( b_{ji} M_{\text{diff},j} + d_{ji} M_{\text{avg},j} \right) + \epsilon_i \]
Indirect Effects: \[ \text{indirect}_{1 \dots k} = a_1 \cdot b_{12} \cdot b_{23} \cdot \dots \cdot b_k \]
Comparison of Indirect Effects \[ CI_{\text{path}_1\text{vs}\text{path}_2} = \text{indirect}_{\text{path}_1} - \text{indirect}_{\text{path}_2} \]
C1- and C2-Measurement Coefficients: \[ X1_{b,i} = b_i + d_i, \quad X0_{b,i} = X1_{b,i} - d_i \] \[ X1_{b,ij} = b_{ij} + d_{ij}, \quad X0_{b,ij} = X1_{b,ij} - d_{ij} \]
By combining these equations, the GenerateModelCN
function supports chained mediation analysis with flexibility in
handling nested pathways.