The data

For this example we are going to use the secondorder dataset present in jamovi data library after installing SEMLj. The data are generated to feature 27 observed variables for 200 cases.

The model

Assume we have three latent variables, bigX, bigM and bigY, such that bigX predicts bigY, and its effect is mediated by bigM, which functions as a mediator. Assume that each of these latent variables is measured by three facets, fx1,fx2 andfx3 for bigX, fy1,fy2 andfy3 for bigY, and fm1,fm2 andfm3 for bigM. Each facet, in turn, is measured by three observed variables, named x[i][j],y[i][j], and m[i][j] where i is the facet, j is the item indices.

Basically, we want to estimate the following model:

Syntax

Let’s start with the syntax. First, we want to define the nine facets in terms of their indicators

fx1=~x11+x12+x13
fx2=~x21+x22+x23
fx3=~x31+x32+x33

fm1=~m11+m12+m13
fm2=~m21+m22+m23
fm3=~m31+m32+m33

fy1=~y11+y12+y13
fy2=~y21+y22+y23
fy3=~y31+y32+y33

and then, we want to define the three second order latent variables as measured by the corresponding facets.

bigx=~fx1+fx2+fx3
bigm=~fm1+fm2+fm3
bigy=~fy1+fy2+fy3

We can now add the regression model among the second order latent variables:

bigm~bigx
bigy~bigm+bigx

Putting all together in the SEMLj syntax sub-module, we run this:

Interactive (GUI)

Output

As soon as we set the input, we obtain the whole set of results tables. We do not go through the results in detail, but it is interesting to see the basic results

The info table gives some detail about the estimation, and lists the syntax resulting from our setup in the interactive input panel. We can notice that the resulting syntax is identical to the one we used before in the syntax sub-module, so we can rest assured that the results are going to be the same.

The measurement model table indicates the associations between the observed variables and the facets, and then the associations between facets and second order factors. Factor loadings are all quite large, thus the model makes sense in terms of measures.

We can also ask for the reliability of the first order factors

and check they have all a pretty good alpha and omega. At the moment, reliability is not available for second order factors (we’ll see how to improve this in future versions).

The other interesting table is the regression coefficients table.

We can see that bigX has a positive and significant effect on bigM, which in turn has an effect on bigY. So, mediation is warranted.

Mediated effect

But what about the mediated effect? Well, that can be asked in the Parameters Options panel by selecting Indirect Effects.

The results are straightforward

The mediated effect amounts to 1.992, corresponding to .513 in the standardized scale.

The same result can be obtained by specifying new parameters in the Custom model settings. Let’s try: First, we ask for the parameters labels, so we can refer to the coefficients with their labels.

Then we look up in the regression table the labels of the coefficients we need

We want to multiply bigX->bigM, p37, with bigM->bigY, so p38. In addition, we add p39 to the mediated effect so we get also the total effect.

In the Custom model settings, we preceed in defining the mediated effect (ME), the total effect (TOT), and since I’m lazy, I ask to compute also the proportion of mediated effect (pME), rather than using a hand calculator:

As expected, the results are the same as the ones obtained with the indirect effects option.

PATH Diagram

If you ask for a path diagram of this model, you get a pretty ugly picture. Default settings, in fact, are not optimal for second order factors models. A better view of the model can be achieved by selecting Spring in the layout options.

The diagram is no Mona Lisa, but at least you see the model and can check that everything is as intended.

In future versions we will try to embellish the diagram for this type of models.

Alternative models

Let now assume that our mediation model has only one second order factor, the mediator, whereas the exogenous factor (bigX) and the endogenous factor (bigY) are first order latent variables. Say we want to estimate the following model:

As regards the definition of bigM, nothing changes. We defined the three facets, fm1, fm2 and fm3, and then a second order factor in the Second Order Factors panel, as we did before.

As regards the definition of bigX, it will now be defined in the input variables field, because it represents an exogenous latent variable.

As regards the definition of bigY, it will now be defined in the input variables field, but as an endogenous latent variable. This is because we later need it to be predicted by the other latent variables.

We then go to the Endogenous models and fill in the regressions as before.

We have set up a model with two latent variables and one second order mediator factor.

Related examples