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Mplus FAQ
Does Mplus automatically correlate the independent variables in a latent variable model?

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.

  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

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