In this simple example we have an observed dependent variable (y), predicted by latent variables (x1, x2 and x3). Each of the three latent variables is associated with a set of observed variables. Assume that all of the variables are continuous. To fit this model we use the Mplus input file below. The Model section of the input file contains the commands for estimating the latent variables (e.g., x1 by a1 a2 a3). The Model section of the input also contains the command y on x1 x2 x3, which specifies that y should be regressed on the three x variables. Note that we have not specified correlations between the x variables. We have included tech1 under Output, this will allow us to see a listing of all parameters estimated in the model. The dataset can be downloaded here.
Data: File is D:\data\mydata.dat ; Variable: Names are a1 a2 a3 b1 b2 b3 c1 c2 c3 y female; Analysis: Type = general ; Model: x1 by a1 a2 a3; x2 by b1 b2 b3 b4; x3 by c1 c2 c3; y on x1 x2 x3; Output: tech1;
Below is the output for this model. Some of the output has been omitted, the entire output can be viewed by clicking here. Looking at the output below, under MODEL RESULTS we see the path loadings for the latent variables X1, X2, and X3 (indicated with the BY). Next the coefficients for the regression of y on the three latent variables (X1, X2 and X3). Next we see the correlations between the three latent variables, first X1 with X2 and X3, and then X2 with X3 (indicated by WITH). Mplus included the correlations between the latent independent (predictor) variables, without us having to specifically request them (i.e., by default). It is worth noting that had we run just the measurement portion of the model, i.e., omitting the y on x1 x2 x3 but leaving the model otherwise the same, Mplus would have correlated the three latent variables by default.
MODEL RESULTS Two-Tailed Estimate S.E. Est./S.E. P-Value X1 BY A1 1.000 0.000 999.000 999.000 A2 0.937 0.023 40.581 0.000 A3 0.773 0.027 29.137 0.000 X2 BY B1 1.000 0.000 999.000 999.000 B2 1.182 0.109 10.859 0.000 B3 0.070 0.020 3.415 0.001 B4 0.026 0.010 2.631 0.009 X3 BY C1 1.000 0.000 999.000 999.000 C2 2.192 0.316 6.933 0.000 C3 1.814 0.250 7.249 0.000 Y ON X1 0.019 0.008 2.376 0.018 X2 0.041 0.011 3.646 0.000 X3 1.377 0.228 6.033 0.000 X2 WITH X1 0.018 0.096 0.190 0.850 X3 WITH X1 -0.003 0.009 -0.386 0.699 X2 0.012 0.007 1.809 0.070