The Structural Equation Modeling (SEM) platform in JMP Pro enables us to model complex patterns of relationships among many variables. Often, these complex patterns involve interactions (also known as moderation), when one variable influences the strength of the relation between two other variables. Figure 1 shows a conceptual model of an interaction.
Figure 2 shows the corresponding statistical model of the interaction. Note that the predictor (X), the moderator (Z), and their product term (X*Z) are all predictors of the outcome (Y).
In SEM, variables may be manifest (directly observed) or latent (not directly observed). Figures 1 and 2 show manifest variables. If we hypothesize that two manifest variables are involved in an interaction, we can use the Moderation and Mediation Add-In for easy model specification in SEM.
What if we hypothesize that two latent variables are involved in an interaction? We’ll need to specify this model manually in the SEM platform. This blog post reviews the steps involved in specifying and fitting a SEM with a latent variable interaction.
The example
We’ll work with a JMP sample data table called Job Satisfaction. To find this data table, go to Help > Sample Data Folder > Job Satisfaction.jmp. In this example, we have three latent variables: workplace leadership, workplace conflict, and job satisfaction.
We hypothesize that workplace leadership and conflict have an interactive effect on job satisfaction. Specifically, when workplace conflict is higher, the effect of leadership on job satisfaction is weaker. The conceptual model is shown in Figure 4. However, this is not a valid statistical model that can be specified in the SEM platform (we can’t have arrows pointing at other arrows!). How do we specify this as a statistical model?
Data preprocessing
Before launching the SEM platform to fit this model, there is some data preprocessing that will be required.
First, mean-center all indicators of the latent predictor and the latent moderator. In the JMP data table, highlight all the variables that need to be centered – all the indicators of Leadership and Conflict – right-click, then go to New Formula Column > Distributional > Center. New, mean-centered variables will be created.
Next, using these centered variables, we’ll create product terms. Since the predictor and the moderator are latent variables with multiple indicators, we need a set of product terms to represent their interaction. There are different ways to create this set; the approach I’ll demonstrate here is called the unconstrained product indicator (UPI) approach, and you can read more details about it in Marsh et al. (2012).
The goal of the UPI approach is to use all the indicators (i.e., make sure every indicator is involved in a product term) without reusing any. The easiest way to do this is with a matched-pairs approach, where one indicator from each latent variable is paired up, then multiplied. However, if we try to take this approach with the Job Satisfaction data, we run into two problems: (1) Leadership has four indicators whereas Conflict has three, and (2) we must decide how to pair up the indicators.
To solve Problem 1, we’ll average two indicators of Leadership to create a parcel, and use that parcel as an indicator. Then, Leadership and Conflict will have the same number of indicators. To solve Problem 2, there are many approaches we could take. In some contexts, natural pairs might arise based on the content of the items, but since that is not the case here, Marsh et al. (2012) suggest creating product terms that maximize the reliability of the latent interaction. To do this, we’ll match the most reliable indicator of Leadership with the most reliable indicator of Conflict, and so on.
To decide how to create our parcel, and then our matched pairs, let’s fit a three-factor CFA to these data (using uncentered variables for now), then select Assess Measurement Model from the red triangle menu. We’ll focus on the Indicator Reliability section of the report.
Work_L and Interact_L are the least reliable indicators of Leadership, so we’ll create a parcel of these two items to strengthen reliability. In the data table, select these two variables, then right-click > New Formula Column > Combine > Average. Then, center the averaged variable; we’ll use the centered variable later when we fit the latent interaction model.
Next, let’s rerun the three-factor CFA using the parcel in place of Work_L and Interact_L (still using uncentered variables for now).
From here, we can create our matched pairs according to indicator reliability. The parcel is the most reliable indicator of Leadership, and Inter_C is the most reliable indicator of Conflict. We’ll multiply these two columns to create a product term. The next most reliable indicators of Leadership and Conflict, respectively, are Goal_L and Person_C, so we’ll multiply these columns. Finally, we’ll multiply Support_L and Intra_C.
Be sure to create product terms with the centered variables. In the JMP data table, select the two columns to be multiplied, then right-click and go to New Formula Column > Combine > Product.
Specify and fit the model
Now we’re ready to launch the SEM platform. On launch, include:
(1) The mean-centered indicators of the predictor Leadership.
(2) The mean-centered indicators of the moderator Conflict.
(3) Their product terms.
(4) The indicators of the outcome (these can be uncentered).
The figure below shows the SEM launch window with boxes around each variable set (1), (2), (3), and (4).
To specify the model, follow these steps:
- Specify latent variables and latent interactions. For the latent predictor and moderator, use centered indicators created in preprocessing. For latent interactions, use the product variables created in preprocessing.
- Add regression paths from the latent predictor, the latent moderator, and their latent interaction to the latent outcome.
- Allow the latent predictor, latent moderator, and their latent interaction to covary.
Interpreting output
After fitting the full model, we can begin by assessing model-data fit, which the Model Comparison table suggests is good (CFI = 1, RMSEA = 0, SRMR = .0327). Next, examine the regression estimates shown below. The coefficient for the product term (Leadership*Conflict) is negative, implying that as conflict increases, the relation between leadership and job satisfaction weakens. This is consistent with our hypothesis; however, the interaction is not statistically significant.
Considerations and extensions
To use Maximum Likelihood (ML) estimation in SEM – which is the default estimator in the SEM platform in JMP Pro – a key assumption is that the variables follow a multivariate normal distribution. In latent interaction models, this assumption becomes especially important: If we create product terms using non-normally distributed variables, those product terms may follow distributions that are even further from normality. This can cause estimation problems and untrustworthy standard errors when using ML. When variables are not normally distributed, it is recommended to use the Model-Implied Instrumental Variables Two-Stage Least Squares (MIIV-2SLS) estimator instead (Kelava & Brandt, 2022), because it does not assume multivariate normality. To use the MIIV-2SLS estimator instead of ML, simply select “MIIV Two-Stage Least Squares” in the SEM launch window (JMP Pro 19 or later).
Finally, the techniques shown here can be extended to even more complex models, such as moderated mediation involving latent variables. For an example, see this JMP Community post.
Further reading
Marsh, H. W., Wen, Z., Nagengast, B., & Hau, K.-T. (2012). Structural equation models of latent interaction. In R. H. Hoyle (Ed.), Handbook of structural equation modeling (pp. 436–458). Guilford.
Kelava, A., & Brandt, H. (2022). Latent interaction effects. In R. H. Hoyle (Ed.), Handbook of structural equation modeling, 2nd ed., (pp. 427-446). Guilford.
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