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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For a single chained mediator \(M_1\) and \(N\) parallel mediators \(M_2, M_3, \dots, M_{N+1}\), the model is defined as:
Outcome Difference Model (\(Y_{\text{diff}}\)): \[ Y_{\text{diff}} = cp + b_1 M_{1\text{diff}} + \sum_{i=2}^{N+1} \left( b_i M_{i\text{diff}} + d_i M_{i\text{avg}} \right) + d_1 M_{1\text{avg}} + e \]
Mediator Difference Model (\(M_{i\text{diff}}\)):
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.
For each mediator, the indirect effects are calculated as:
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 comparing the strengths of indirect effects, the contrast between two effects is calculated as: \[ CI_{\text{path}_1\text{vs}\text{path}_2} = \text{indirect}_{\text{path}_1} - \text{indirect}_{\text{path}_2} \]
Indirect Effects:
\[ \text{indirect}_1 = a_1 \cdot b_1 \]
\[ \text{indirect}_2 = a_2 \cdot b_2 \]
\[ \text{indirect}_3 = a_3 \cdot b_3 \]
\[ \text{indirect}_{21} = a_2 \cdot b_{21} \cdot b_1 \]
\[ \text{indirect}_{31} = a_3 \cdot b_{31} \cdot b_1 \]
Comparisons:
\[ CI_{1\text{vs}2} = \text{indirect}_1 - \text{indirect}_2 \]
\[ CI_{1\text{vs}3} = \text{indirect}_1 - \text{indirect}_3 \]
\[ CI_{1\text{vs}21} = \text{indirect}_1 - \text{indirect}_{21} \]
\[ CI_{1\text{vs}31} = \text{indirect}_1 - \text{indirect}_{31} \]
\[ CI_{2\text{vs}3} = \text{indirect}_2 - \text{indirect}_3 \]
\[ CI_{2\text{vs}21} = \text{indirect}_2 - \text{indirect}_{21} \]
\[ CI_{3\text{vs}31} = \text{indirect}_3 - \text{indirect}_{31} \]
\[ CI_{21\text{vs}31} = \text{indirect}_{21} - \text{indirect}_{31} \]
C2-Measurement Coefficient (\(X1_{b,i}\)): \[ X1_{b,i} = b_i + d_i/2 \]
C1-Measurement Coefficient (\(X0_{b,i}\)): \[ X0_{b,i} = X1_{b,i} - d_i \]
Mediator \(M_1\):
\[ X1_{b,1} = b_1 + d_1/2 \]
\[ X0_{b,1} = X1_{b,1} - d_1 \]
Mediator \(M_2\):
\[ X1_{b,2} = b_2 + d_2/2 \]
\[ X0_{b,2} = X1_{b,2} - d_2 \]
Mediator \(M_3\):
\[ X1_{b,3} = b_3 + d_3/2 \]
\[ X0_{b,3} = X1_{b,3} - d_3 \]
Parallel to Chained Path (\(M_2 \to M_1\)):
\[ X1_{b,21} = b_{21} + d_{21}/2 \]
\[ X0_{b,21} = X1_{b,21} - d_{21} \]
Parallel to Chained Path (\(M_3 \to M_1\)):
\[ X1_{b,31} = b_{31} + d_{31}/2 \]
\[ X0_{b,31} = X1_{b,31} - d_{31} \]
This section summarizes all equations used in the model:
\[ Y_{\text{diff}} = cp + b_1 M_{1\text{diff}} + \sum_{i=2}^{N+1} \left( b_i M_{i\text{diff}} + d_i M_{i\text{avg}} \right) + d_1 M_{1\text{avg}} + e \]
\[ M_{1\text{diff}} = a_1 + \sum_{i=2}^{N+1} \left( b_{i1} M_{i\text{diff}} + d_{i1} M_{i\text{avg}} \right) + \epsilon_1 \]
\[ M_{i\text{diff}} = a_i + \epsilon_i \]
\[ \text{indirect}_1 = a_1 \cdot b_1 \]
\[ \text{indirect}_i = a_i \cdot b_i \]
\[ \text{indirect}_{i1} = a_i \cdot b_{i1} \cdot b_1 \]
\[ CI_{\text{path}_1\text{vs}\text{path}_2} = \text{indirect}_{\text{path}_1} - \text{indirect}_{\text{path}_2} \]
\[ X1_{b,i} = b_i + d_i/2 \]
\[ X0_{b,i} = X1_{b,i} - d_i \]
This comprehensive approach supports models with parallel mediators influencing a chained mediator, enabling detailed analysis of their effects and interactions.
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.