SEM in Mplus

Jeremy Albright

Posted on
SEM Mplus

This page describes how to set up code in Mplus to fit a full structural equation model with latent variables. The model consists of three latent variables and eleven manifest variables, as described here. Mplus only reads data in text format, see this post for details on how to prepare a data file for Mplus. The data can be accessed from Github. To review, the model to be fit is the following:

The post on CFA in Mplus described the steps towards fitting and testing the measurement model for the two measures of democracy. Here we are going to move from fitting a measurement model to actually testing structural relationships between variables. The model will keep both latent variables from the measurement model, which represented democracy measured in 1960 (\(\eta_1\)) and democracy measured in 1965 (\(\eta_2\)). We will also add a latent variable measuring industrialization in 1960 (\(\xi_1\)). The model expects that democracy in 1965 will be associated with democracy in 1960 as well as industrialization in 1960.

The Mplus syntax to run the model is the following:


TITLE: Bollen's (1989, chapter 8) SEM Example;

DATA: FILE IS sem-bollen.dat;

VARIABLE: NAMES ARE y1 y2 y3 y4 y5 y6 y7 y8 x1 x2 x3;

MODEL:
    eta_1 BY y1  
             y2 (l2) 
             y3 (l3) 
             y4 (l4);
    eta_2 BY y5  
             y6 (l2) 
             y7 (l3) 
             y8 (l4);
    xi_1  BY x1 x2 x3;

    eta_1 ON xi_1;
    eta_2 ON eta_1 xi_1;

    y1 WITH y5;
    y2 WITH y4;
    y2 WITH y6;
    y3 WITH y7;
    y4 WITH y8;
    y6 WITH y8;

    
OUTPUT: STANDARDIZED;

The optional TITLE command labels the model. The title here indicates that we are replicating the model described in chapter 8 (pg. 324) of from Bollen (1989). Note that every command must end with a semicolon. Also keep in mind that the number of characters in any row of the input file cannot exceed 80.

The DATA command points to where the data are located. In this example, it is assumed that the data are in the same folder as this input file. If not, fuller pathnames to the data file would need to be used, such as "C:\Users\you\Documents\mplus-files\sem-bollen.dat".

The VARIABLE command lists the variables in the order in which they appear in the data file. The model will be using all of the variables in the data file. If this were not the case, we would add a second line specifying the USEVARIABLES, or the variables that will be used in the analysis. Mplus would then ignore any columns that were not listed after USEVARIABLES.

Note that there are no missing values in this file. If there were missing, we would add a line after the NAMES ARE statement like the following:

MISSING ARE ALL (-999)

This of course assumes missing values have all been recoded as -999. The choice of numeric value for missing is up to the user who prepares the data.

Mplus will by default use maximum likelihood estimation (specifically, Full Information Maximum Likelihood, or FIML, which is robust to data that have values missing at random). The default is also to report the conventional chi-square test and maximum likelihood standard errors. The optional ANALYSIS command can be used to change the estimator for some or all statistics. For example, adding

ANALYSIS: ESTIMATOR = MLM

to the input file will tell Mplus to still use maximum likelihood estimation for model parameters and standard errors but to report the Satorra-Bentler chi-square statistic that is more robust to non-normality in the data. Alternatively,

ANALYSIS: ESTIMATOR = MLR

will use maximum likelihood to estimate the parameters as well as cluster-robust standard errors based on the sandwich estimator. The full list of estimators can be found in the Mplus User’s Guide, see the ANALYSIS COMMAND chapter. ESTIMATOR = ML is the default and does not need to be specified if that is the estimator the user desires.

The MODEL command describes the model. The syntax for latent variables lists the name of the latent variable, followed by the word BY, followed by a list of the observed variables. Here this syntax specifies three latent variables. The first is \(\eta_1\) (Greek letter pronounced “eta”) and is measured with the variables \(y_1-y_4\). The second is \(\eta_2\) and is measured with the variables \(y_5-y_8\). The last is \(\xi_1\) (Greek letter pronounced “xi”) and is measured by the observed variables \(x_1-x_3\).

The syntax retains all of the constraints described in the tutorial on CFA in Mplus. That is, the respective loadings for the 1960 and 1965 democracy indicators are constrained to be equal, and certain covariances between the observed variable error terms are free parameters to be estimated. The equality constraints are specified with the labels l2, l3, and l4 in parentheses after each observed variable is listed. The WITH statements introduce the covariances.

The primary difference from the CFA example is that now there are structural relationships between the latent variables. These are captured with the ON statements, which are used to specify regression-type linear associations. The dependent variable is listed first, followed by ON, followed by the independent variables. We have the following latent variable regressions:

  • eta_1 ON xi_1; specifies that 1960 democracy (\(\eta_1\)) is associated with 1960 industrialization (\(\xi_1\)).
  • eta_2 ON eta_1 xi_1; specifies that 1965 democracy (\(\eta_2\)) is associated with 1960 democracy (\(\eta_1\)) and 1960 industrialization (\(\xi_1\)).

Finally, because latent variables are unobserved and hence have an arbitrary scaling, it is preferable to present standardized estimates rather than the unstandardized parameters. We can get this by adding the optional OUTPUT: STANDARDIZED command, which will produce three types of standardization in the output file: STDYX, STDY, and STD. In most cases, STDYX will be the section of interest, as it standardizes the output to be interpreted in standard deviation units (just like standardized regression coefficients). STDY would be of interest if we had a binary covariate in the model, as it only converts the outcome to standard deviation units (standard deviations of dummy variables are not usually useful). Since we only have continuous latent variables and no observed binary variables, we can focus on STDXY.

With our syntax ready we can now save the file and then click the red Run button in the toolbar to get the estimates. Doing so yields the following:


Mplus VERSION 8
MUTHEN & MUTHEN
06/26/2019   3:28 PM

INPUT INSTRUCTIONS

  TITLE: Bollens (1989, chapter 8) SEM Example;

  DATA: FILE IS sem-bollen.dat;

  VARIABLE: NAMES ARE y1 y2 y3 y4 y5 y6 y7 y8 x1 x2 x3;

  MODEL:
      eta_1 BY y1
               y2 (l2)
               y3 (l3)
               y4 (l4);
      eta_2 BY y5
               y6 (l2)
               y7 (l3)
               y8 (l4);
      xi_1  BY x1 x2 x3;

      eta_1 ON xi_1;
      eta_2 ON eta_1 xi_1;

      y1 WITH y5;
      y2 WITH y4;
      y2 WITH y6;
      y3 WITH y7;
      y4 WITH y8;
      y6 WITH y8;


  OUTPUT: STANDARDIZED;




INPUT READING TERMINATED NORMALLY



Bollens (1989, chapter 8) SEM Example;

SUMMARY OF ANALYSIS

Number of groups                                                 1
Number of observations                                          75

Number of dependent variables                                   11
Number of independent variables                                  0
Number of continuous latent variables                            3

Observed dependent variables

  Continuous
   Y1          Y2          Y3          Y4          Y5          Y6
   Y7          Y8          X1          X2          X3

Continuous latent variables
   ETA_1       ETA_2       XI_1


Estimator                                                       ML
Information matrix                                        OBSERVED
Maximum number of iterations                                  1000
Convergence criterion                                    0.500D-04
Maximum number of steepest descent iterations                   20

Input data file(s)
  sem-bollen.dat

Input data format  FREE



UNIVARIATE SAMPLE STATISTICS


     UNIVARIATE HIGHER-ORDER MOMENT DESCRIPTIVE STATISTICS

         Variable/         Mean/     Skewness/   Minimum/ % with                Percentiles
        Sample Size      Variance    Kurtosis    Maximum  Min/Max      20%/60%    40%/80%    Median

     Y1                    5.465      -0.093       1.250   10.67%       2.500      5.000      5.400
              75.000       6.787      -1.104      10.000    6.67%       6.900      7.500
     Y2                    4.256       0.325       0.000   34.67%       0.000      3.333      3.333
              75.000      15.372      -1.426      10.000   21.33%       4.800     10.000
     Y3                    6.563      -0.606       0.000   10.67%       3.333      6.667      6.667
              75.000      10.621      -0.657      10.000    1.33%       6.667     10.000
     Y4                    4.453       0.120       0.000   22.67%       0.000      3.333      3.333
              75.000      11.069      -1.164      10.000   10.67%       6.667      6.667
     Y5                    5.136      -0.233       0.000    6.67%       2.500      5.000      5.000
              75.000       6.735      -0.718      10.000    2.67%       6.250      7.500
     Y6                    2.978       0.911       0.000   40.00%       0.000      0.000      2.233
              75.000      11.224      -0.400      10.000   10.67%       3.333      6.667
     Y7                    6.196      -0.565       0.000   13.33%       3.333      6.667      6.667
              75.000      10.655      -0.672      10.000   26.67%       6.667     10.000
     Y8                    4.043       0.455       0.000   16.00%       0.368      3.333      3.333
              75.000      10.393      -0.906      10.000   12.00%       3.333      6.667
     X1                    5.054       0.259       3.784    1.33%       4.317      4.727      5.075
              75.000       0.530      -0.693       6.737    1.33%       5.226      5.631
     X2                    4.792      -0.353       1.386    1.33%       3.466      4.595      4.963
              75.000       2.252      -0.505       7.872    1.33%       5.236      6.203
     X3                    3.558       0.086       1.002    1.33%       2.113      3.167      3.568
              75.000       1.950      -0.880       6.425    1.33%       3.977      4.586


THE MODEL ESTIMATION TERMINATED NORMALLY



MODEL FIT INFORMATION

Number of Free Parameters                       39

Loglikelihood

          H0 Value                       -1548.818
          H1 Value                       -1528.728

Information Criteria

          Akaike (AIC)                    3175.636
          Bayesian (BIC)                  3266.018
          Sample-Size Adjusted BIC        3143.100
            (n* = (n + 2) / 24)

Chi-Square Test of Model Fit

          Value                             40.179
          Degrees of Freedom                    38
          P-Value                           0.3739

RMSEA (Root Mean Square Error Of Approximation)

          Estimate                           0.028
          90 Percent C.I.                    0.000  0.087
          Probability RMSEA <= .05           0.665

CFI/TLI

          CFI                                0.997
          TLI                                0.995

Chi-Square Test of Model Fit for the Baseline Model

          Value                            730.654
          Degrees of Freedom                    55
          P-Value                           0.0000

SRMR (Standardized Root Mean Square Residual)

          Value                              0.048



MODEL RESULTS

                                                    Two-Tailed
                    Estimate       S.E.  Est./S.E.    P-Value

 ETA_1    BY
    Y1                 1.000      0.000    999.000    999.000
    Y2                 1.191      0.142      8.407      0.000
    Y3                 1.175      0.120      9.804      0.000
    Y4                 1.251      0.123     10.177      0.000

 ETA_2    BY
    Y5                 1.000      0.000    999.000    999.000
    Y6                 1.191      0.142      8.407      0.000
    Y7                 1.175      0.120      9.804      0.000
    Y8                 1.251      0.123     10.177      0.000

 XI_1     BY
    X1                 1.000      0.000    999.000    999.000
    X2                 2.180      0.139     15.691      0.000
    X3                 1.818      0.152     11.952      0.000

 ETA_1    ON
    XI_1               1.471      0.391      3.758      0.000

 ETA_2    ON
    ETA_1              0.865      0.076     11.430      0.000
    XI_1               0.600      0.238      2.520      0.012

 Y1       WITH
    Y5                 0.583      0.364      1.598      0.110

 Y2       WITH
    Y4                 1.440      0.691      2.084      0.037
    Y6                 2.183      0.731      2.986      0.003

 Y3       WITH
    Y7                 0.712      0.619      1.149      0.251

 Y4       WITH
    Y8                 0.363      0.461      0.787      0.431

 Y6       WITH
    Y8                 1.372      0.578      2.373      0.018

 Intercepts
    Y1                 5.465      0.299     18.282      0.000
    Y2                 4.256      0.439      9.696      0.000
    Y3                 6.563      0.394     16.658      0.000
    Y4                 4.453      0.380     11.729      0.000
    Y5                 5.136      0.304     16.871      0.000
    Y6                 2.978      0.392      7.590      0.000
    Y7                 6.196      0.364     17.004      0.000
    Y8                 4.043      0.375     10.772      0.000
    X1                 5.054      0.084     60.127      0.000
    X2                 4.792      0.173     27.657      0.000
    X3                 3.558      0.161     22.066      0.000

 Variances
    XI_1               0.449      0.087      5.171      0.000

 Residual Variances
    Y1                 1.855      0.457      4.058      0.000
    Y2                 7.581      1.345      5.637      0.000
    Y3                 4.956      0.961      5.156      0.000
    Y4                 3.224      0.742      4.347      0.000
    Y5                 2.313      0.483      4.785      0.000
    Y6                 4.968      0.895      5.554      0.000
    Y7                 3.560      0.738      4.824      0.000
    Y8                 3.308      0.713      4.640      0.000
    X1                 0.081      0.020      4.131      0.000
    X2                 0.120      0.070      1.723      0.085
    X3                 0.467      0.089      5.236      0.000
    ETA_1              3.875      0.889      4.361      0.000
    ETA_2              0.164      0.233      0.705      0.481


STANDARDIZED MODEL RESULTS


STDYX Standardization

                                                    Two-Tailed
                    Estimate       S.E.  Est./S.E.    P-Value

 ETA_1    BY
    Y1                 0.850      0.043     19.906      0.000
    Y2                 0.690      0.061     11.330      0.000
    Y3                 0.758      0.050     15.154      0.000
    Y4                 0.838      0.043     19.394      0.000

 ETA_2    BY
    Y5                 0.817      0.044     18.704      0.000
    Y6                 0.755      0.052     14.459      0.000
    Y7                 0.802      0.049     16.262      0.000
    Y8                 0.829      0.043     19.414      0.000

 XI_1     BY
    X1                 0.920      0.023     39.711      0.000
    X2                 0.973      0.017     58.931      0.000
    X3                 0.872      0.031     28.318      0.000

 ETA_1    ON
    XI_1               0.448      0.104      4.285      0.000

 ETA_2    ON
    ETA_1              0.884      0.052     16.849      0.000
    XI_1               0.187      0.073      2.547      0.011

 Y1       WITH
    Y5                 0.281      0.145      1.942      0.052

 Y2       WITH
    Y4                 0.291      0.116      2.517      0.012
    Y6                 0.356      0.097      3.658      0.000

 Y3       WITH
    Y7                 0.169      0.138      1.230      0.219

 Y4       WITH
    Y8                 0.111      0.134      0.831      0.406

 Y6       WITH
    Y8                 0.338      0.112      3.032      0.002

 Intercepts
    Y1                 2.111      0.203     10.395      0.000
    Y2                 1.120      0.141      7.923      0.000
    Y3                 1.924      0.188     10.245      0.000
    Y4                 1.354      0.156      8.708      0.000
    Y5                 1.948      0.191     10.195      0.000
    Y6                 0.876      0.135      6.489      0.000
    Y7                 1.963      0.189     10.411      0.000
    Y8                 1.244      0.150      8.277      0.000
    X1                 6.943      0.579     12.001      0.000
    X2                 3.194      0.285     11.199      0.000
    X3                 2.548      0.238     10.709      0.000

 Variances
    XI_1               1.000      0.000    999.000    999.000

 Residual Variances
    Y1                 0.277      0.073      3.809      0.000
    Y2                 0.525      0.084      6.250      0.000
    Y3                 0.426      0.076      5.616      0.000
    Y4                 0.298      0.072      4.123      0.000
    Y5                 0.333      0.071      4.663      0.000
    Y6                 0.430      0.079      5.461      0.000
    Y7                 0.357      0.079      4.524      0.000
    Y8                 0.313      0.071      4.423      0.000
    X1                 0.154      0.043      3.602      0.000
    X2                 0.053      0.032      1.665      0.096
    X3                 0.239      0.054      4.455      0.000
    ETA_1              0.800      0.094      8.550      0.000
    ETA_2              0.035      0.050      0.710      0.478


STDY Standardization

                                                    Two-Tailed
                    Estimate       S.E.  Est./S.E.    P-Value

 ETA_1    BY
    Y1                 0.850      0.043     19.906      0.000
    Y2                 0.690      0.061     11.330      0.000
    Y3                 0.758      0.050     15.154      0.000
    Y4                 0.838      0.043     19.394      0.000

 ETA_2    BY
    Y5                 0.817      0.044     18.704      0.000
    Y6                 0.755      0.052     14.459      0.000
    Y7                 0.802      0.049     16.262      0.000
    Y8                 0.829      0.043     19.414      0.000

 XI_1     BY
    X1                 0.920      0.023     39.711      0.000
    X2                 0.973      0.017     58.931      0.000
    X3                 0.872      0.031     28.318      0.000

 ETA_1    ON
    XI_1               0.448      0.104      4.285      0.000

 ETA_2    ON
    ETA_1              0.884      0.052     16.849      0.000
    XI_1               0.187      0.073      2.547      0.011

 Y1       WITH
    Y5                 0.281      0.145      1.942      0.052

 Y2       WITH
    Y4                 0.291      0.116      2.517      0.012
    Y6                 0.356      0.097      3.658      0.000

 Y3       WITH
    Y7                 0.169      0.138      1.230      0.219

 Y4       WITH
    Y8                 0.111      0.134      0.831      0.406

 Y6       WITH
    Y8                 0.338      0.112      3.032      0.002

 Intercepts
    Y1                 2.111      0.203     10.395      0.000
    Y2                 1.120      0.141      7.923      0.000
    Y3                 1.924      0.188     10.245      0.000
    Y4                 1.354      0.156      8.708      0.000
    Y5                 1.948      0.191     10.195      0.000
    Y6                 0.876      0.135      6.489      0.000
    Y7                 1.963      0.189     10.411      0.000
    Y8                 1.244      0.150      8.277      0.000
    X1                 6.943      0.579     12.001      0.000
    X2                 3.194      0.285     11.199      0.000
    X3                 2.548      0.238     10.709      0.000

 Variances
    XI_1               1.000      0.000    999.000    999.000

 Residual Variances
    Y1                 0.277      0.073      3.809      0.000
    Y2                 0.525      0.084      6.250      0.000
    Y3                 0.426      0.076      5.616      0.000
    Y4                 0.298      0.072      4.123      0.000
    Y5                 0.333      0.071      4.663      0.000
    Y6                 0.430      0.079      5.461      0.000
    Y7                 0.357      0.079      4.524      0.000
    Y8                 0.313      0.071      4.423      0.000
    X1                 0.154      0.043      3.602      0.000
    X2                 0.053      0.032      1.665      0.096
    X3                 0.239      0.054      4.455      0.000
    ETA_1              0.800      0.094      8.550      0.000
    ETA_2              0.035      0.050      0.710      0.478


STD Standardization

                                                    Two-Tailed
                    Estimate       S.E.  Est./S.E.    P-Value

 ETA_1    BY
    Y1                 2.201      0.241      9.140      0.000
    Y2                 2.621      0.345      7.602      0.000
    Y3                 2.586      0.301      8.584      0.000
    Y4                 2.754      0.298      9.236      0.000

 ETA_2    BY
    Y5                 2.154      0.239      9.017      0.000
    Y6                 2.565      0.327      7.853      0.000
    Y7                 2.530      0.295      8.581      0.000
    Y8                 2.694      0.294      9.172      0.000

 XI_1     BY
    X1                 0.670      0.065     10.343      0.000
    X2                 1.460      0.128     11.420      0.000
    X3                 1.218      0.128      9.489      0.000

 ETA_1    ON
    XI_1               0.448      0.104      4.285      0.000

 ETA_2    ON
    ETA_1              0.884      0.052     16.849      0.000
    XI_1               0.187      0.073      2.547      0.011

 Y1       WITH
    Y5                 0.583      0.364      1.598      0.110

 Y2       WITH
    Y4                 1.440      0.691      2.084      0.037
    Y6                 2.183      0.731      2.986      0.003

 Y3       WITH
    Y7                 0.712      0.619      1.149      0.251

 Y4       WITH
    Y8                 0.363      0.461      0.787      0.431

 Y6       WITH
    Y8                 1.372      0.578      2.373      0.018

 Intercepts
    Y1                 5.465      0.299     18.282      0.000
    Y2                 4.256      0.439      9.696      0.000
    Y3                 6.563      0.394     16.658      0.000
    Y4                 4.453      0.380     11.729      0.000
    Y5                 5.136      0.304     16.871      0.000
    Y6                 2.978      0.392      7.590      0.000
    Y7                 6.196      0.364     17.004      0.000
    Y8                 4.043      0.375     10.772      0.000
    X1                 5.054      0.084     60.127      0.000
    X2                 4.792      0.173     27.657      0.000
    X3                 3.558      0.161     22.066      0.000

 Variances
    XI_1               1.000      0.000    999.000    999.000

 Residual Variances
    Y1                 1.855      0.457      4.058      0.000
    Y2                 7.581      1.345      5.637      0.000
    Y3                 4.956      0.961      5.156      0.000
    Y4                 3.224      0.742      4.347      0.000
    Y5                 2.313      0.483      4.785      0.000
    Y6                 4.968      0.895      5.554      0.000
    Y7                 3.560      0.738      4.824      0.000
    Y8                 3.308      0.713      4.640      0.000
    X1                 0.081      0.020      4.131      0.000
    X2                 0.120      0.070      1.723      0.085
    X3                 0.467      0.089      5.236      0.000
    ETA_1              0.800      0.094      8.550      0.000
    ETA_2              0.035      0.050      0.710      0.478


R-SQUARE

    Observed                                        Two-Tailed
    Variable        Estimate       S.E.  Est./S.E.    P-Value

    Y1                 0.723      0.073      9.953      0.000
    Y2                 0.475      0.084      5.665      0.000
    Y3                 0.574      0.076      7.577      0.000
    Y4                 0.702      0.072      9.697      0.000
    Y5                 0.667      0.071      9.352      0.000
    Y6                 0.570      0.079      7.230      0.000
    Y7                 0.643      0.079      8.131      0.000
    Y8                 0.687      0.071      9.707      0.000
    X1                 0.846      0.043     19.855      0.000
    X2                 0.947      0.032     29.465      0.000
    X3                 0.761      0.054     14.159      0.000

     Latent                                         Two-Tailed
    Variable        Estimate       S.E.  Est./S.E.    P-Value

    ETA_1              0.200      0.094      2.143      0.032
    ETA_2              0.965      0.050     19.305      0.000


QUALITY OF NUMERICAL RESULTS

     Condition Number for the Information Matrix              0.282E-03
       (ratio of smallest to largest eigenvalue)


DIAGRAM INFORMATION

  Use View Diagram under the Diagram menu in the Mplus Editor to view the diagram.
  If running Mplus from the Mplus Diagrammer, the diagram opens automatically.

  Diagram output
    c:\users\jeremy\documents\mplus-data\bollen-sem.dgm

     Beginning Time:  15:28:27
        Ending Time:  15:28:27
       Elapsed Time:  00:00:00



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The first part of the output reiterates the code. We then see that INPUT READING TERMINATED NORMALLY. If there were syntax errors, Mplus would alert us at this point, and we would want to go back and check our syntax and data.

The next section describes the model and estimator, followed by a table of descriptive statistics for the observed variables.

Next, the output states THE MODEL ESTIMATION TERMINATED NORMALLY. This is important information. If the model were not identified and/or convergence did not occur after, Mplus would tell us here. Output that does not say that the estimation terminated normally should not ever be reported.

We are then presented with model fit information. We look for a non-significant \(\chi^2\) test, a RMSEA less than 0.05, CFI/TLI above 0.90 to 0.95, and SRMR less than 0.08. Consult Hu and Bentler (1999) for fuller details on interpretation.

The next section presents the parameter estimates. The unstandardized results are presented first, followed by the standardized results. Note that the estimates for the loadings are the same for both latent democracy variables, which is what we imposed by labeling the respective parameters in the syntax. Note also that there are estimates corresponding to the error covariances, as we specified in our WITH statements.

Our interest is in the structural relationships between the latent variables. Since we do not know what a “unit” of democracy is, we should look at the results under the STDXY heading. Here we see the following:

  1. A standard deviation increase in 1960 industrialization is associated with a .448 standard deviation increase in 1960 democracy, \(p < 0.001\).
  2. A standard deviation increase in 1960 industrialization is associated with a .187 standard deviation increase in 1965 democracy, \(p = 0.011\), holding 1960 democracy constant.
  3. A standard deviation increase in 1960 democracy is associated with a .884 standard deviation increase in 1965 democracy, \(p < 0.001\), holding 1960 industrialization constant.

To view a path diagram of the model, click on Diagram \(\rightarrow\) View Diagram in Mplus. This will open a new application that shows the model, such as the following:

The user can toggle between unstandardized parameter estimates (shown) and the different standardizations. In addition, some formatting can be performed to get the image in better shape for publication.

Citations

Bollen, K.A. (1989). Structural Equations with Latent Variables. New York, NY: Wiley.

Hu, L., & Bentler, P. M. (1999). Cutoff criteria for fit indexes in covariance structure analysis: Conventional criteria versus new alternatives. Structural Equation Modeling, 6, 1–55.