The chi-square test of model fit is non-significant, indicating that the null hypothesis that the model fits the data cannot be rejected (the model fits the data well). This finding is corroborated by the RMSEA: Its estimate is zero; it's 90% confidence interval has an upper bound value of .055, which is below the Hu and Bentler (1999) recommended cutoff value of .06. The RMSEA estimate and its upper bound confidence interval value should both fall below .06 to ensure satisfactory model fit.

EXPLORATORY ANALYSIS WITH  2 FACTOR(S) :
 

           EXPLORATORY ANALYSIS WITH  2 FACTOR(S) :
           CHI-SQUARE VALUE               1.079
           DEGREES OF FREEDOM                 4
           PROBABILITY VALUE              .8976

           RMSEA (ROOT MEAN SQUARE ERROR OF APPROXIMATION) :
           ESTIMATE (90 PERCENT C.I.) IS   .000 (  .000   .055)
           PROBABILITY RMSEA LE  .05 IS     .944

For exploratory factor analysis solutions with two or more factors, Mplus reports varimax rotated loadings and promax rotated loadings.Varimax loadings assume the two factors are uncorrelated whereas promax loadings allow the factors to be correlated. Directly below the promax loadings is the factor intercorrelatrion matrix.

In this example the two factors are correlated .480. With even a modest correlation among the two factors, you should choose to interpret the promax rotated loadings. The loadings show that the visperc, cubes, and lozenges variables load onto the first factor whereas the remaining variables load onto the second factor.

           VARIMAX ROTATED LOADINGS
                  1             2
              ________      ________
 VISPERC         .547          .250
 CUBES           .550          .092
 LOZENGES        .728          .196
 PARAGRAP        .241          .830
 SENTENCE        .174          .816
 WORDMEAN        .247          .788
 

           PROMAX ROTATED LOADINGS
                  1             2
              ________      ________
 VISPERC         .540          .112
 CUBES           .585         -.063
 LOZENGES        .755         -.001
 PARAGRAP        .046          .841
 SENTENCE       -.025          .846
 WORDMEAN        .063          .794
 

           PROMAX FACTOR CORRELATIONS
                  1             2
              ________      ________
      1         1.000
      2          .480         1.000

Mplus next reports estimated error variances for each observed variable, the estimated correlation matrix, and the residual correlation matrix. Notice that unlike the preceding one factor solution, this dual factor solution's estimated correlation matrix is very close in value to the original sample correlation matrix. Accordingly, the residual correlation matrix has all values close to zero and the RMR value of .0092 is well below the Hu and Bentler (1999) recommended cutoff of .08.

           ESTIMATED ERROR VARIANCES
              VISPERC       CUBES         LOZENGES      PARAGRAP      SENTENCE
              ________      ________      ________      ________      ________
      1          .638          .689          .431          .253          .304
 

           ESTIMATED ERROR VARIANCES
              WORDMEAN
              ________
      1          .318
 

           ESTIMATED CORRELATION MATRIX
              VISPERC       CUBES         LOZENGES      PARAGRAP      SENTENCE
              ________      ________      ________      ________      ________
 VISPERC        1.000
 CUBES           .324         1.000
 LOZENGES        .448          .419         1.000
 PARAGRAP        .339          .209          .338         1.000
 SENTENCE        .299         .170          .286          .719         1.000
 WORDMEAN        .332          .208          .334          .714          .686
 

           ESTIMATED CORRELATION MATRIX
              WORDMEAN
              ________
 WORDMEAN       1.000
 

           RESIDUALS OBSERVED-EXPECTED
              VISPERC       CUBES         LOZENGES      PARAGRAP      SENTENCE
              _______      ________      ________      ________      ________
 VISPERC         .000
 CUBES           .002          .000
 LOZENGES        .001         -.002          .000
 PARAGRAP        .002          .019         -.010          .000
 SENTENCE        .010         -.011          .000          .000          .000
 WORDMEAN       -.015         -.013          .013          .001         -.001
 

           RESIDUALS OBSERVED-EXPECTED
              WORDMEAN
             ________
 WORDMEAN        .000
 

 ROOT MEAN SQUARE RESIDUAL IS         .0092

This example assumes that the Grant-White data file  is complete. In other words, there are no missing cases in the Grant-White data file . What if some cases had missing values? Often data files have cases with incomplete data. The next section describes a feature unique to Mplus: exploratory factor analysis of a data file with incomplete cases.

2. Exploratory Factor Analysis with Missing Data

Suppose you altered the Grant-White data file  so that cases with visperc scores that exceed 34 have missing cubes scores and that cases with wordmean scores of 10 or below have missing sentence values. In this instance the missing cubes and setence completion data are said to be missing at random (MAR) because the patterns of missing data are explainable by the values of other variables in the data file , visual perception and word meaning. Ordinarily, if you do not specify a missing data analysis in Mplus, Mplus performs listwise or casewise deletion of cases with any missing data. That is, any case with one or more missing data points is omitted entirely from analyses. However, for exploratory factor analysis, confirmatory factor analysis, and structural equation modeling with continuous variables, Mplus features a missing data option that outperforms the default listwise deletion method. The optional method that offers superior performance is called full information maximum likelihood (FIML); details on FIML can be found in the UT Austin Statistical Services General FAQ #25: Handling missing or incomplete Data.

Regardless of whether you choose to use FIML or listwise data deletion to handle missing data, if you have missing data in your input data file , you must tell Mplus how the missing values for each variable are represented in the data file . You use the MISSING subcommand of the VARIABLE command to accomplish this task. In this example, missing values for cubes and sentence are represented by -9, so the MISSING subcommand reads:

MISSING ARE all (-9) ;

MISSING = blank ;

MISSING ARE . ;

TITLE:
  Grant-White School: EFA with Missing Data
DATA:
  FILE IS "c:\intromplus\grant-missing.dat" ;
VARIABLE:
  NAMES ARE cubes lozenges paragrap sentence wordmean gender ;
  USEVARIABLES ARE visperc - wordmean;
  MISSING ARE all (-9) ; 
ANALYSIS: 
  TYPE =  efa 1 2;
  ESTIMATOR = ml ;

Selected output from the analysis appears below. Grant-White School: Exploratory Factor Analysis with Missing
Data SUMMARY OF ANALYSIS

Number of groups                                1
Number of observations                         79

Number of y-variables                           6
Number of x-variables                           0
Number of continuous latent variables           0

Notice that Mplus considers the data file  to contain 79 usable cases rather than the original 145 cases.

TITLE:
  Grant-White School: EFA with Missing Data
DATA:
  FILE IS "c:\intromplus\grant-missing.dat" ;
VARIABLE:
  NAMES ARE cubes lozenges paragrap sentence wordmean gender ;
  USEVARIABLES ARE visperc - wordmean;
  MISSING ARE all (-9) ; 
ANALYSIS: 
  TYPE = missing efa 1 2 ;
  ESTIMATOR = ml ;

TITLE:
  Grant-White School: EFA with categorical outcomes 
DATA:
  FILE IS "a:\grantcat.dat" ; 
VARIABLE: 
  NAMES ARE  viscat cubescat lozcat paracat sentcat wordcat ; 
  USEVARIABLES ARE  viscat - wordcat ; 
  CATEGORICAL ARE viscat - wordcat ; 
ANALYSIS:
  TYPE = efa 1 2; 
  ESTIMATOR = wlsmv ; 
OUTPUT:
  sampstat ;

    CATEGORICAL ARE vizcat - wordcat ; 

You should also change the ESTIMATOR option for the ANALYSIS command. The default is unweighted least-squares (uls), which is fast and is useful for exploratory work, but a more optimal choice for categorical outcomes, based on the work of Muthén, DuToit, and Spisic (1997), is weighted least-squares with mean and variance adjustment, wlsmv.

    ANALYSIS:
      TYPE = efa 1 2; 
      ESTIMATOR = wlsmv ; 

Selected output from the analysis appears below. Notice that the categorical nature of the data precludes computation of the descriptive model fit statistics such as the RMSEA, though Mplus does produce the familiar chi-square test of overall model fit.

           EXPLORATORY ANALYSIS WITH  2 FACTOR(S) :
           CHI-SQUARE VALUE               2.823
           DEGREES OF FREEDOM                 4
           PROBABILITY VALUE              .5875

The chi-square result for the two factor model is not significant, which indicates that two factors are sufficient to explain the intercorrelations among the six observed variables. The varimax and promax rotated factor loadings appear below. The pattern and values obtained from this analysis are consistent with the results of the first exploratory factor analysis of the completely continuous data discussed previously.

                    VARIMAX ROTATED LOADINGS
                  1             2
              ________      ________
 VISCAT          .571          .332
 CUBESCAT        .700          .117
 LOZCAT          .667          .244
 PARACAT         .473          .642
 SENTCAT         .235          .847
 WORDCAT         .206          .858
 

           PROMAX ROTATED LOADINGS
                  1             2
              ________      ________
 VISCAT          .559          .159
 CUBESCAT        .777         -.137
 LOZCAT          .698          .022
 PARACAT         .347          .550
 SENTCAT         .005          .876
 WORDCAT        -.031          .899
 

           PROMAX FACTOR CORRELATIONS
                  1             2
              ________      ________
      1         1.000
      2          .557         1.000
 

Although Mplus does not produce the RMSEA descriptive model fit statistic for categorical outcomes, it does output the standardized root mean residual, RMR:

ROOT MEAN SQUARE RESIDUAL IS         .0310

The value of .031 suggests an excellent fit of the two factor model to the observed data. (Please note that as of version 4.2, Mplus does give the RMSEA.)
 

There are several notes worth keeping in mind when you perform exploratory factor analysis with categorical outcome variables.

• Although one or more of the observed variables may be categorical, any latent variables in the model are assumed to be continuous (this is a property of the exploratory factor analysis model; confirmatory factor analysis models with categorical latent variables may be fit as mixture models using Mplus; see the Mplus User's Guide for more information about mixture models).
• FIML missing data handling is not available with the analysis of categorical outcomes.
• The analysis specification and interpretation of the output is the same whether one, a subset, or all observed variables are categorical.
• Categorical observed variables may be dichotomous or ordered polytymous (i.e., ordered categorical outcomes of more than two levels), but nominal level observed variables with more than two categories may not be used in the analysis as outcome variables.
• Sample size requirements are somewhat more stringent than for continuous variables; typically you want a minimum of 200 cases (preferably more) to perform any analysis with categorical outcome variables.

TITLE:    
  Grant-White School:  Summary Statistics 
DATA:
  FILE IS "c:\intromplus\grant.dat" ; 
  FORMAT IS free ; 
VARIABLE: 
  NAMES ARE visperc cubes lozenges paragrap sentence
            wordmean gender ;
  USEVARIABLES ARE visperc cubes lozenges paragrap  sentence wordmean ; 
ANALYSIS:
  TYPE = general ; 
MODEL:
  visual BY visperc@1 cubes lozenges ; 
  verbal BY paragrap@1 sentence wordmean ; 
  visual WITH verbal ; 
OUTPUT:
  standardized sampstat ; 

The general analysis type tells Mplus that you are fitting a general structural equation model rather than specific model such as an exploratory factor analysis. The model is general in the sense that you must define what parameters are estimated; all other parameters are assumed to be fixed. In the exploratory factor analysis context, Mplus already knows the specifics of that model, so specifying the model is handled automatically by Mplus. By contrast, in the confirmatory factor analysis and structural equation modeling context each hypothesized model is unique, so you must tell Mplus how the model is constructed. The MODEL command allows you to specify the parameters of your model.

The first line of the MODEL command shown above defines a latent factor called visual. The BY keyword (an abbreviation for "measured by") is used to define the latent variables; the latent variable name appears on the left-hand side of the BY keyword whereas the measured variables appear on the right-hand side of the BY keyword. It has three observed indicator variables: visperc, cubes, and lozenges. Similarly, in the second line of the MODEL command a latent factor called verbal has three indicators: paragrap, sentence, and wordmean. The third line of MODEL command uses the WITH keyword to correlate the visual latent factor with the verbal latent factor.

The visperc and paragrap variables are each followed by @1. The @ sign tells Mplus to fix the factor loading (regression weight) of the visual-visperc relationship to the value that follows the @, 1.00. Similarly, the verbal-paragrap relationship is also fixed to 1.00. The reason you fix these two parameters is to provide a scale for the visual and verbal latent variables' variances. If you ever need to supply starting values for a particular parameter in Mplus, you can specify its number after an asterisk, like this: sentence*.5. Omitting the asterisks when you do not specify starting values is the default. Note that each variable is separated from the other variables in the analysis by at least one space.

Finally, the OUTPUT command contains an added keyword, standardized. This option instructs Mplus to output standardized parameter estimate values in addition to the default unstandardized values. Selected output from the analysis appears below.

Grant-White School: Confirmatory Factor Analysis

SUMMARY OF ANALYSIS

Number of groups                                1
Number of observations                        145

Number of y-variables                           6
Number of x-variables                           0
Number of continuous latent variables           2

Observed variables in the analysis
   VISPERC     CUBES       LOZENGES    PARAGRAP    SENTENCE    WORDMEAN

Continuous latent variables in the analysis
   VISUAL      VERBAL

The summary of analysis information tells you that there are six continuous observed variables in the analysis and two latent factors, visual and verbal. Mplus then displays the input covariance matrix generated from the six observed variables:

     SAMPLE STATISTICS
 

           Covariances/Correlations/Residual Correlations
              VISPERC       CUBES         LOZENGES      PARAGRAP      SENTENCE
              ________      ________      ________      ________      ________
 VISPERC       47.801
 CUBES         10.012        19.758
 LOZENGES      25.798        15.417        69.172
 PARAGRAP       7.973         3.421         9.207        11.393
 SENTENCE       9.936         3.296        11.092        11.277        21.616
 WORDMEAN      17.425         6.876        22.954        19.167        25.321

           Covariances/Correlations/Residual Correlations
              WORDMEAN
              ________
 WORDMEAN      63.163
 

Mplus next reports the results of fitting the hypothesized model to the sample data.

THE MODEL ESTIMATION TERMINATED NORMALLY

TESTS OF MODEL FIT

Chi-Square Test of Model Fit

          Value                    3.663
          Degrees of Freedom           8
          P-Value                  .8861

Loglikelihood

          H0 Value             -2575.128
          H1 Value             -2573.297

Information Criteria

          Number of Free Parameters             13
          Akaike (AIC)                    5176.256
          Bayesian (BIC)                  5214.954
          Sample-Size Adjusted BIC        5173.817
            (n* = (n + 2) / 24)

RMSEA (Root Mean Square Error Of Approximation)

          Estimate                     .000
          90 Percent C.I.              .000   .046
          Probability RMSEA <= .05     .957

As was the case for the exploratory factor analysis of these data, Mplus reports the chi-square goodness-of-fit test and the RMSEA descriptive model fit statistic. The chi-square test of model fit is not significant and the RMSEA value is well below the value of .06 recommended by Hu and Bentler (1999) as an upper boundary, so you can conclude that the proposed model fits the data well. Mplus also reports the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC). These are descriptive indexes of model fit that you can use to compare the goodness of model fit of two or more competing models. Smaller values indicate better model fit.

Mplus also outputs the unstandardized coefficients (Estimates in the output), the standard errors (abbreviated S.E. in the output), the estimates divided by their respective standard errors (Est./S.E.), and two standardized coefficients for each estimated parameter in the model (Std and StdYX). The estimate divided by the standard error tests the null hypothesis that the parameter estimate is zero in the population from which you drew your sample. An unstandardized estimate divided by its standard error may be evaluated as a Z statistic, so values that exceed +1.96 or fall below -1.96 are significant below p = .05.

MODEL RESULTS

                   Estimates     S.E.  Est./S.E.    Std     StdYX

 VISUAL   BY
    VISPERC            1.000     .000       .000    4.358     .632
    CUBES               .542     .116      4.658    2.360     .533
    LOZENGES           1.392     .272      5.112    6.064     .732

 VERBAL   BY
    PARAGRAP           1.000     .000       .000    2.920     .868
    SENTENCE           1.309     .115     11.352    3.821     .825
    WORDMEAN           2.247     .197     11.402    6.560     .828

 VISUAL   WITH
    VERBAL             6.784    1.720      3.943     .533     .533

In this example, each of the estimated parameters has an estimate to standard error ratio greater than +1.96, so each factor loading is statistically significant, as well as the correlation between the visual and verbal latent factors (Z = 3.943). The variance components of the two factors, shown in the output appearing below, are also statistically significant, indicating that the amount of variance accounted for by each factor is significantly different from zero.

Each unstandardized estimate represents the amount of change in the outcome variable as a function of a single unit change in the variable causing it. In this example, you assume that the latent variables, in addition to some measurement error (shown below), are responsible for the scores on the six observed variables. For instance, for each single unit change in the verbal latent factor, sentence scores increase by 1.309 units.

Different measures often have different scales, so you will often find it useful to examine the standardized coefficients when you want to compare the relative strength of associations across observed variables that are measured on different scales. Mplus provides two standardized coefficients. The first, labeled Std on the output, standardizes using the latent variables' variances whereas the second type of standardized coefficient, StdYX, standardizes based on latent and observed variables' variances. This standardized coefficient represents the amount of change in an outcome variable per standard deviation unit of a predictor variable. In this output, you can see clearly that the standardized coefficients of paragrap, sentence, and wordmean are larger than those of visperc, cubes, and lozenges. This finding suggests that the verbal latent factor does a better job at explaining the shared variance among paragrap, sentence, and wordmean than does the visual latent factor for its three indicator variables, visperc, cubes, and lozenges.

This assertion is corroborated by the residual variances output by Mplus. The standardized coefficients for the first three indicators are larger than those for the remaining three indicators.

Residual Variances
Grant-White School: Confirmatory Factor Analysis

                   Estimates     S.E.  Est./S.E.    Std     StdYX

    VISPERC           28.485    4.739      6.011   28.485     .600
    CUBES             14.050    1.978      7.105   14.050     .716
    LOZENGES          31.933    7.269      4.393   31.933     .465
    PARAGRAP           2.791     .584      4.775    2.791     .247
    SENTENCE           6.869    1.164      5.900    6.869     .320
    WORDMEAN          19.695    3.385      5.819   19.695     .314

 Variances
    VISUAL            18.989    5.582      3.402    1.000    1.000
    VERBAL             8.525    1.376      6.196    1.000    1.000

R-SQUARE

    Observed
    Variable  R-Square

    VISPERC       .400
    CUBES         .284
    LOZENGES      .535
    PARAGRAP      .753
    SENTENCE      .680
    WORDMEAN      .686

Finally, the r-square output illustrates that only modest amounts of variance are accounted for in the first three indicators whereas much larger amounts of variance are accounted for in the final three indicators. As is the case with exploratory factor analysis of continuous outcome variables, you may want to use the mlm or mlmv estimators in lieu of the default ml estimator if your input data are not distributed joint multivariate normal by using the ESTIMATOR = option on the ANALYSIS command. The mlm option provides a mean-adjusted chi-square model test statistic whereas the mlmv option produces a mean and variance adjusted chi-square test of model fit; both options also induce Mplus to produce robust standard errors displayed in the model results table that are used to compute Z tests of significance for individual parameter estimates. An added advantage of the mlm option is that its chi-square test and standard errors are equivalent to those produced by EQS in its ML;ROBUST method. Muthén and Muthén have placed formulas on their Web site that allow you to use mlm-produced chi-square values in nested model comparisons.

2. Handling Missing Data

It is often the case that you have missing data in the context of confirmatory factor analysis and structural equation modeling. Using Mplus, you can employ the optimal Full Information Maximum Likelihood (FIML) approach to handling missing data that was described above in the section Exploratory Factor Analysis with Missing Data in Section 4. Consider once again the same modified data file, grant-missing.dat, containing incomplete cases that was used in the earlier exploratory factor analysis with missing data. As in the previous example, define the missing value code to be -9 for all variables using the MISSING subcommand in the VARIABLE command, copy the MODEL syntax from the previous confirmatory factor analysis example into the Mplus input window, and then modify the ANALYSIS command so that it reads as follows (with the changed part in italics for emphasis).

TITLE:
  Grant-White School: CFA with missing data 
DATA:
  FILE IS "c:\intromplus\grant-missing.dat" ; 
VARIABLE: 
  NAMES ARE visperc cubes lozenges paragrap sentence wordmean gender ; 
  USEVARIABLES ARE visperc - wordmean ; 
  MISSING ARE all (-9) ; 
ANALYSIS:
  TYPE = general missing h1 ; 
MODEL: 
  visual BY visperc@1 cubes lozenges ; 
  verbal BY paragrap@1 sentence wordmean ; 
  visual WITH verbal ; 
OUTPUT:
  standardized sampstat ; 

The missing keyword alerts Mplus to activate the FIML missing data handling feature. The additional h1 keyword tells Mplus to output the chi-square goodness-of-fit test in addition to the typical summary statistics, missing data pattern information, parameter estimates, and standard errors obtained in an analysis. Mplus requires that you specify the h1 keyword because large models with many missing data patterns can take a long time to converge. If this describes your situation, you may want to omit the h1 option on the TYPE = line to verify that you have specified your model correctly before invoking the h1 option to produce the chi-square test of model fit. If you elect to remove the h1 option from the ANALYSIS TYPE = command, be sure to omit the sampstat option from the OUTPUT line, as well. If sampstat is included on the OUTPUT line, Mplus automatically assumes the h1ANALYSIS option and computes the chi-square test of model fit, even if h1 is not included on the ANALYSIS TYPE = line.

The chi-square test of model fit for the confirmatory factor analysis with missing data shows that the hypothesized model fit the data well:

TESTS OF MODEL FIT

Chi-Square Test of Model Fit

          Value                    2.777
          Degrees of Freedom           8
          P-Value                  .9476

Loglikelihood

          H0 Value             -2376.312
          H1 Value             -2374.923

Information Criteria

          Number of Free Parameters             19
          Akaike (AIC)                    4790.623
          Bayesian (BIC)                  4847.181
          Sample-Size Adjusted BIC        4787.058
            (n* = (n + 2) / 24)

RMSEA (Root Mean Square Error Of Approximation)

          Estimate                     .000
          90 Percent C.I.              .000   .011
          Probability RMSEA <= .05     .982

The Mplus parameter estimates, standard errors, and standardized parameter estimates are similar to those found in the preceding confirmatory factor analysis example. The only substantial difference is the inclusion of an additional section that contains means and intercepts for the latent factors and observed variables. These means and intercepts are required to be estimated by the FIML missing data handling procedure, but are otherwise not a part of the tested model.

MODEL RESULTS
                   Estimates     S.E.  Est./S.E.    Std     StdYX
 VISUAL   BY
    VISPERC            1.000     .000       .000    4.377     .635
    CUBES               .469     .127      3.679    2.051     .473
    LOZENGES           1.373     .294      4.673    6.010     .725

 VERBAL   BY
    PARAGRAP           1.000     .000       .000    2.914     .866
    SENTENCE           1.187     .114     10.376    3.460     .821
    WORDMEAN           2.247     .206     10.888    6.547     .827

 VISUAL   WITH
    VERBAL             7.014    1.800      3.896     .550     .550

 Residual Variances
    VISPERC           28.354    5.037      5.629   28.354     .597
    CUBES             14.589    2.340      6.234   14.589     .776
    LOZENGES          32.642    7.938      4.112   32.642     .475
    PARAGRAP           2.824     .627      4.507    2.824     .250
    SENTENCE           5.781    1.070      5.401    5.781     .326
    WORDMEAN          19.872    3.578      5.554   19.872     .317

 Variances
    VISUAL            19.158    5.859      3.270    1.000    1.000
    VERBAL             8.493    1.393      6.099    1.000    1.000

 Intercepts
    VISPERC           29.579     .572     51.673   29.579    4.291
    CUBES             24.616     .421     58.431   24.616    5.678
    LOZENGES          15.965     .689     23.184   15.965    1.925
    PARAGRAP           9.952     .279     35.620    9.952    2.958
    SENTENCE          19.054     .366     52.057   19.054    4.522
    WORDMEAN          17.283     .658     26.274   17.283    2.182

Finally, Mplus produces the r-square values for the observed variables. Once again, these are similar to those obtained from the original data file  with complete cases.

R-SQUARE

    Observed
    Variable  R-Square
    VISPERC       .403
    CUBES         .224
    LOZENGES      .525
    PARAGRAP      .750
    SENTENCE      .674
    WORDMEAN      .683

If you elect to use Mplus's FIML approach to handling missing data, be aware that the only available estimator is the maximum likelihood option, ml. If you suspect that your data are non-normally distributed, remember that the chi-square test of model fit may be affected by the non-normality problem. Depending on the severity of the non-normality problem and the amount of missing data you have, you may want to explore other ways of handling the missing data problem prior to performing analyses using Mplus; see see the UT Austin Statistical Services General FAQ #25: Handling missing or incomplete data.

3. Confirmatory Factor Analysis with Categorical Outcomes

Confirmatory factor analysis with dichotomous and polytomous categorical outcomes, or confirmatory factor analysis with mixed categorical and continuous outcomes is also possible using Mplus. Recall the grantcat.dat data file  used in the example Exploratory Factor Analysis with Categorical Outcomes in Section 4. Using the same data file  that replaces the six continuous observed variables with a dichotomous variables, you can use the confirmatory factor analysis syntax from the example Confirmatory Factor Analysis With Continuous Variables with the following modifications.

First, add the CATEGORICAL ARE vizcat ... wordcat ; statement to the DATA command. Mplus will now treat the six observed variables as categorical in the analysis. The entire command syntax is shown here.

TITLE:     
  Grant-White School: CFA with categorical outcomes 
DATA:      
  FILE IS "c:\intromplus\grantcat.dat" ; 
VARIABLE: 
  NAMES ARE  viscat cubescat lozcat paracat sentcat wordcat ; 
  USEVARIABLES ARE   viscat - wordcat ; 
  CATEGORICAL ARE viscat - wordcat ; 
ANALYSIS:
  TYPE = general  ; 
MODEL:
  visual BY viscat@1 cubescat lozcat ; 
  verbal BY paracat@1 sentcat wordcat ; 
  visual WITH verbal ; 
OUTPUT:
  sampstat standardized ; 

Selected results from the analysis appear below.

Chi-Square Test of Model Fit

          Value                    7.463*
          Degrees of Freedom           6**
          P-Value                  .2800

*  The chi-square value for MLM, MLMV, WLSM and WLSMV cannot be used for
   chi-square difference tests.

** The degrees of freedom for MLMV and WLSMV are estimated according to
   formula 109 (page 281) in the Mplus User's Guide.

The chi-square test of model fit is once again non-significant, suggesting that the specified model fits the data adequately. The default estimator for models that contain categorical outcomes is the mean and variance-adjusted weighted least-squares method, wlsmv. Optional estimators you may choose are weighted least-squares (wls) and mean-adjusted weighted least-squares (wlsm). As is the case in the exploratory factor analysis of categorical data example, there are no descriptive model fit statistics produced by Mplus when it analyzes categorical outcomes. Mplus also produces a note alerting you not to use the MLMV, WLSM, and WLSMV chi-square values in nested model comparisons (the warning about the MLM chi-square is not relevant as long as you use the formulas shown on the Mplus Web site for nested model MLM chi-square comparisons when you use the MLM estimator in the analysis of continuous outcomes). You should not use the MLM estimator for the analysis of intrinsically categorical outcome variables.

Mplus then outputs the model results:

MODEL RESULTS

                   Estimates     S.E.  Est./S.E.    Std     StdYX

 VISUAL   BY
    VISCAT             1.000     .000       .000     .729     .729
    CUBESCAT            .831     .212      3.922     .606     .606
    LOZCAT              .975     .230      4.248     .710     .710

 VERBAL   BY
    PARACAT            1.000     .000       .000     .814     .814
    SENTCAT            1.058     .134      7.920     .861     .861
    WORDCAT            1.038     .127      8.154     .844     .844

 VISUAL   WITH
    VERBAL              .397     .087      4.592     .670     .670

 Variances
    VISUAL              .531     .162      3.273    1.000    1.000
    VERBAL              .662     .117      5.661    1.000    1.000

 Thresholds
    VISCAT$1            .095     .104       .913     .095     .095
    CUBESCAT$1          .271     .105      2.571     .271     .271
    LOZCAT$1           -.043     .104      -.415    -.043    -.043
    PARACAT$1           .009     .104       .083     .009     .009
    SENTCAT$1           .183     .105      1.743     .183     .183
    WORDCAT$1           .043     .104       .415     .043     .043

This output is similar to that of a confirmatory factor analysis with continuous outcomes, with one notable exception: Mplus now produces threshold information for each categorical variable. A threshold is the expected value of the latent variable or factor at which an individual transitions from a value of 0 to a value of 1.00 on the categorical outcome variable when the continuous underlying latent variable's score is zero. There are only two categorical values for each outcome variable, so there is only one threshold per variable. For any categorical outcome variable with K levels, Mplus will output K-1 threshold values. For example, a five-point Likert scale item would contain four threshold values. The first threshold would represent the expected value at which an individual would be most likely to transition from a value of 0 to a value of 1.00 on the Likert outcome variable. The second threshold would represent the expected value at which an individual would be most likely to transition from a value of 1.00 to a value of 2.00 on the outcome variable, and so on through the fourth threshold, which represents the expected value at which an individual would transition from 3.00 to 4.00 on the outcome variable.

Finally, Mplus produces the r-square table output. The r-square values are computed for the continuous latent variables underlying the categorical outcome variables rather than the actual outcome variables as is the case in analyses that contain continuous outcome variables. Note that the r-square values for the categorical outcomes cannot be interpreted as the proportion of variance explained as is the case in the analysis of continuous outcomes. Therefore, examining the sign and significance of the estimated coefficients shown in the model results table above is generally more informative than interpreting r-square values.

R-SQUARE

    Observed  Residual
    Variable  Variance  R-Square

    VISCAT        .469      .531
    CUBESCAT      .633      .367
    LOZCAT        .495      .505
    PARACAT       .338      .662
    SENTCAT       .259      .741
    WORDCAT       .287      .713

The r-square table's residual variance output is, however, useful for computing expected probabilities. You can use threshold and coefficient information shown above with the residual variance information from the r-square table to compute the expected probability of case having a value of 0 or 1.00. Consider  following  formula for computing the conditional probability of a Y = 0 response given the factor eta.:

           P(Y_ij = 0|eta_ij) = F[(tau_j - lambda_j*eta_i )*(1/square root of theta_jj)]

where:
           eta is the factor's value
           F is the culmulative normal distribution fuction
           tau is the measured item's threshold
           lambda is the item's factor loading
           theta is the residual variance of the measured item

Suppose you want to obtain the estimated probability for sentcat = 0 at eta = 0. Using the formula, shown above, you can compute this value:

            P(Y_ij|eta_ij) = F[(.183 - 0)*(1/square root of .259)]
                                 = F[.183*1.9649437]
                                 = F[.3595847]

You can look up the value of .3595847 in a Z table in a statistics textbook, or you can supply the computed value of .3595847 to the PROBNORM function in SAS to obtain the correct probability value. The PROBNORM function returns the value from a cumulative normal distribution for the inputted value. A simple SAS program such as the one shown below enables you to obtain the final expected probability value of .64.

DATA one ; 
  p = PROBNORM(.3595847) ; 
RUN ; 

PROC PRINT DATA = one ; 
RUN ; 

You may substitute other values of eta and lambda to obtain different expected probability values. In general, the same cautions and limitations that were discussed above in the section Exploratory Factor Analysis with Categorical Variables section also apply to the analysis of categorical outcomes in the confirmatory factor analysis and structural equation modeling contexts. In addition, the following point is worth considering:

• Do not list independent (exogenous) categorical variables in the CATEGORICAL statement. Instead, create dummy variables (i.e., variables with values of 0 and 1 representing group membership status) and include them in the model as predictors, or create a multiple group analysis based upon category membership as described in the Multiple Group Analysis section of this document.

Educ - Education level
SEI - Socioeconomic index
Anomia67 - Anomie in 1967
Anomia71 - Anomie in 1971
Powles67 - Powerlessness in 1967
Powles71 - Powerlessness in 1971

One of the fitted structural equation models features a latent factor, SES, that influences Educ and SEI scores. The SES latent variable in turn influences two additional latent variables: Alien67 and Alien71. Alien67 represents self-perceived alienation in 1967 and it influences responses on the anomie and powerlessness variables measured in 1967. Similarly, Alien71 represents self-perceived alienation in 1971 and it influences responses on the anomie and powerlessness variables measured in 1971. SES influences both Alien67 and Alien71 and Alien67 also influences Alien71. 



The dataset, wheaton-generated.dat, is used in the analysis that follows:

TITLE: 
  Wheaton et al. Example 1: Full SEM 
DATA: 
  FILE IS "c:\intromplus\wheaton-generated.dat" ; 
VARIABLE: 
  NAMES ARE  educ sei anomia67 powles67 anomia71 powles71 ; 
  USEVARIABLES ARE    educ - powles71 ; 
ANALYSIS:  
  TYPE = general ; 
MODEL:      
  ses BY educ@1 sei ; 
  alien67 BY anomia67@1 powles67 ; 
  alien71 BY anomia71@1 powles71 ; 

  alien67 ON ses ; 
  alien71 ON ses alien67 ; 
OUTPUT:
  standardized sampstat ; 

The syntax for this analysis is similar to that of the confirmatory factor analysis example shown in subsection 1 above. The only noteworthy difference is the use of the ON keyword in the MODEL command to specify the regression relationships among the latent variables; the WITH keyword is used to specify correlations or covariances among variables. In this example, the alien67 latent variable is regressed on the SES latent variable. Similarly, the alien71 latent variable is regressed on both the SES and alien67 latent variables. The model fit statistics appear below:

TESTS OF MODEL FIT

Chi-Square Test of Model Fit

          Value                   76.184
          Degrees of Freedom           6
          P-Value                  .0000

<some output deleted to save space>

RMSEA (Root Mean Square Error Of Approximation)

          Estimate                     .112
          90 Percent C.I.              .090   .135
          Probability RMSEA <= .05     .000

The statistically significant chi-square test of absolute model fit coupled with the poor RMSEA fit statistic value suggest that this model may need some modification before it fits the data well. The model fit and r-square tables appear below.

MODEL RESULTS

                   Estimates     S.E.  Est./S.E.    Std     StdYX

 SES      BY
    EDUC               1.000     .000       .000    2.420     .784
    SEI                 .592     .043     13.694    1.433     .683

 ALIEN67  BY
    ANOMIA67           1.000     .000       .000    2.929     .816
    POWLES67            .823     .038     21.734    2.409     .793

 ALIEN71  BY
    ANOMIA71           1.000     .000       .000    2.989     .843
    POWLES71            .825     .039     21.305    2.465     .778

 ALIEN67  ON
    SES                -.759     .062    -12.235    -.627    -.627

 ALIEN71  ON
    SES                -.172     .064     -2.689    -.139    -.139
    ALIEN67             .710     .056     12.609     .696     .696

 Residual Variances
    EDUC               3.677     .416      8.839    3.677     .386
    SEI                2.345     .172     13.651    2.345     .533
    ANOMIA67           4.301     .364     11.807    4.301     .334
    POWLES67           3.422     .260     13.150    3.422     .371
    ANOMIA71           3.637     .369      9.849    3.637     .289
    POWLES71           3.951     .289     13.681    3.951     .394
    ALIEN67            5.201     .495     10.516     .606     .606
    ALIEN71            3.352     .382      8.781     .375     .375

 Variances
    SES                5.854     .557     10.515    1.000    1.000
 
R-SQUARE

    Observed
    Variable  R-Square

    EDUC          .614
    SEI           .467
    ANOMIA67      .666
    POWLES67      .629
    ANOMIA71      .711
    POWLES71      .606

     Latent
    Variable  R-Square

    ALIEN67       .394
    ALIEN71       .625

There are several noteworthy features of these tables. First, the model results table contains residual variance estimates for the alien67 and alien71 latent variables. These variables are predicted by the SES latent variable, so it makes sense that the residual or unexplained variance is due to factors other than SES in the model. Because SES is not predicted by any other variables, its variance is estimated independently and is shown in the Variances section of the model results table. The path coefficients from SES to alien67, from SES to alien71, and from alien67 to alien71 and their associated standard errors, tests of significance, and standardized coefficients also appear in the same table.

The r-square table contains r-square values for each of the predicted latent variables, alien67 and alien71, as well as the observed variables. Taken as a whole, these results suggest that the model is capturing the observed variables' variances fairly well, though the prediction of alienation in 1967 is somewhat weak as is the variance accounted for in the SEI variable. The model may be modified, however. When all variables are continuous, Mplus can print modification indices that can provide an empirical basis to aid your decision to free additional paths, means, intercepts, or variance components to be estimated in your model. A modification index provides the expected drop in model fit chi-square if a parameter that is currently not free is in fact allowed to be estimated. As always, theory should be your first guide in the decision to modify your model. To request modification indices, add the following keywords to the OUTPUT line:

TITLE: 
  Wheaton et al. Example 1: Full SEM 
DATA: 
  FILE IS "c:\intromplus\wheaton-generated.dat" ; 
VARIABLE: 
  NAMES ARE  educ sei anomia67 powles67 anomia71 powles71 ; 
  USEVARIABLES ARE    educ - powles71 ; 
ANALYSIS:  
  TYPE = general ; 
MODEL:      
  ses BY educ@1 sei ; 
  alien67 BY anomia67@1 powles67 ; 
  alien71 BY anomia71@1 powles71 ; 

  alien67 ON ses ; 
  alien71 ON ses alien67 ; 
OUTPUT:
  standardized sampstat modindices (4) ;

The number shown in the parentheses is the amount of chi-square reduction necessary for Mplus to print any given modification index. The critical chi-square statistic is 3.84 for 1 degree of freedom at p = .05, so this example sets the cutoff to print modification indices at 4.00. If you do not specify a cutoff value, Mplus supplies 10.00 as the default value. The modification indices from this model appear below.

MODEL MODIFICATION INDICES

Minimum M.I. value for printing the modification index     4.000

                            M.I.     E.P.C.  Std E.P.C.  StdYX E.P.C.

WITH Statements

POWLES67 WITH EDUC          8.381     -.574      -.574        -.061
ANOMIA71 WITH EDUC          5.626      .533       .533         .049
ANOMIA71 WITH ANOMIA67     62.098     2.091      2.091         .164
ANOMIA71 WITH POWLES67     48.629    -1.546     -1.546        -.144
POWLES71 WITH ANOMIA67     54.470    -1.693     -1.693        -.149
POWLES71 WITH POWLES67     41.262     1.233      1.233         .128

In addition to the raw modification index value (M.I.), Mplus also prints the unstandardized expected parameter change (E.P.C.) and standardized versions of the expected parameter change.

You can draw several immediate conclusions about the model from this table. First, the largest raw modification indicies are associated with correlating the residuals of the anomie and powerlessness variables, indicating that freeing these parameters to be estimated will result in the largest improvement in model fit. Second, the StdYX expected parameter change values are comparable with each other because they are standardized coefficients. The largest of these is the correlation of anomia67 with anomia71 (.164). The next largest value is the correlation of anomia67 with powles71 (-.149). However, you must ask yourself, "Is this modification theoretically sensible and meaningful?" about any modification you plan to undertake. You can make a case for correlating anomia67 and anomia71, and powles67 and powles71, because these measures are identical instruments measured on the same people at two different time points. It is conceivable that some method or instrument variance is shared across time on the same measurement instruments, but not across two distinct measurement instruments.

With this information, suppose you change the MODEL command to add two residual covariances via the WITH statement: anomia67 with anomie71, and powles67 and powles71. The Mplus syntax for this model is shown below, with the added part shown in italics for emphasis.

TITLE: 
  Wheaton et al. Example 1: Full SEM 
DATA: 
  FILE IS "c:\intromplus\wheaton-generated.dat" ; 
VARIABLE: 
  NAMES ARE  educ sei anomia67 powles67 anomia71 powles71 ; 
  USEVARIABLES ARE    educ - powles71 ; 
ANALYSIS:  
  TYPE = general ; 
MODEL:
  ses BY educ@1 sei ;
  alien67 BY anomia67@1 powles67 ;
  alien71 BY anomia71@1 powles71 ;

  alien67 ON ses ;
  alien71 ON ses alien67 ;

  anomia67 WITH anomia71 ;
  powles67 WITH powles71 ;
OUTPUT:
  standardized sampstat modindices (4) ;

Consider the result of this modification on the model fit statistics.

TESTS OF MODEL FIT

Chi-Square Test of Model Fit

          Value                    7.826
          Degrees of Freedom           4
          P-Value                  .0978

...output deleted...

RMSEA (Root Mean Square Error Of Approximation)

          Estimate                     .032
          90 Percent C.I.              .000   .065
          Probability RMSEA <= .05     .782

The chi-square test of overall model fit is not signicant and the RMSEA value is well below the recommended .06 cutoff that indicates good model fit, so you conclude that your modified model fits the data well (the value of .065 for the upper bound of the 90 percent confidence interval for the RMSEA suggests that the model could be improved even more if you wished to pursue further model modifications). If you use them properly, model modification indices are a powerful tool in your analytic toolbox. The following points about model modification indices are worth considering:

• Model modification should always be informed by theory.
• The more modifications you perform on any given model, the more likely the results are to be sample specific (i.e., results won't generalize to new samples).
• Mplus model modification indices are available when you use full information maximum likelihood (FIML) to handle missing data.
• Mplus model modification indices are not available for models that contain categorical outcome variables. Instead, request tech2 on the OUTPUT to obtain unstandardized first order derivatives that may be used as approximate guides for modification of models containing categorical outcomes.

Section 6: Advanced Models

Although Mplus can fit many standard models and it contains some useful features lacking in other SEM programs at the time of this writing (e.g., FIML missing data handling with exploratory factor analysis, modification indices with FIML missing data handling for structural equation and confirmatory factor analysis models), Mplus advanced modeling features are its most distinctive trademark. A full treatment of Mplus's advanced modeling features is beyond the scope of this tutorial, but several representative examples appear below.

1. Multiple Group Analysis

Recall the first confirmatory factor analysis example that features data from 145 students from the Grant-White School contained in the data file grant.dat. 72 of those students are male whereas 73 students are female. Suppose you decide to investigate the equality of the factor structure across the two groups of students. You can use Mplus to perform one or more multiple group analyses in which the parameters of your choosing are stipulated to be equal across the two groups of children. For instance, suppose you wanted to test the equality of the factor loading and factor variances and covariance values for males and females. The Mplus command file shown below performs this test.

TITLE: 
  Grant-White School: Multiple Group CFA
DATA:
  FILE IS "c:\intromplus\grant.dat" ;
VARIABLE:
  NAMES ARE  visperc cubes lozenges paragrap
             sentence wordmean gender ;
  USEVARIABLES ARE   visperc -  wordmean ;
  GROUPING = gender (1=males 2=females);
ANALYSIS:
  TYPE = mgroup ;
MODEL:
  visual BY visperc@1 cubes lozenges ;
  verbal BY paragrap@1 sentence wordmean ;
  visual (1) ;
  verbal (2) ;
  visual WITH verbal (3) ;
OUTPUT:
  standardized sampstat ;

Several new elements of this program are immediately apparent. First, the GROUPING = option for the VARIABLE command tells Mplus which variable in the data file  contains the information about group membership. For each value of the grouping variable, you supply a name that Mplus uses to define separate groups in the analysis. The ANALYSIS command contains an mgroup keyword that lets Mplus know you are specifying a multiple group analysis. Use the GROUPING = option for raw data; use the mgroupANALYSIS keyword when you input summary data such as covariance matrices for each group. Both multiple group specification methods are included in this example for illustrative purposes, though only the GROUPING = option is required to run the command file because you input raw data.

By default Mplus assumes that the following specified parameter estimates are equal across multiple groups:

• Factor loadings
• Intercepts of continuous outcome variables
• Thresholds of categorical outcome variables

TITLE: 
  Grant-White School: Multiple Group CFA
DATA:
  FILE IS "c:\intromplus\grant.dat" ;
VARIABLE:
  NAMES ARE  visperc cubes lozenges paragrap
             sentence wordmean gender ;
  USEVARIABLES ARE   visperc -  wordmean ;
  GROUPING = gender (1=males 2=females);
ANALYSIS:
  TYPE = mgroup ;
MODEL:
  visual BY visperc@1 cubes
  lozenges ; 
  verbal BY paragrap@1 sentence wordmean ;
  visual (1) ; 
  verbal (2) ;
  visual WITH verbal (3) ;
MODEL males:
  visperc - wordmean (4); 
OUTPUT:
  standardized sampstat ;

This model constrains the residual variance values of the six observed variables for males to be equal, but the females' residual variances are allowed to remain unique for each measured variable.

For more information on multiple group analysis, including cautionary notes regarding multiple group analysis, see the UT Austin Statistical Services AMOS FAQ #3: Multiple group analysis.

2. Multilevel Models

Investigators often draw data from sources that feature a hierarchical or multilelvel structure such as students nested within classrooms, patients residing in hospitals, children grouped within a family, individuals grouped within couples, etc. In recent years, specialized software such as HLM and MLWin have been developed to fit regression and related-models (e.g., ANOVA, ANCOVA, MANOVA, and MANCOVA) to such data files because many statistical software packages such as SPSS and SAS assume every observation is independent of the observations that precede and follow it (some exceptions to this general rule are the MIXED procedure in SAS and the LISREL multilevel module, both of  which may be used to fit multilevel regression models). In situations where individuals are members of some type of larger aggregate or cluster (e.g., families, couples, classrooms), this independence assumption can be and often is violated. Violations of the independence assumption can seriously degrade the results from an analysis conducted on multilevel data.

Although specialized software products such as HLM and related programs permit multilevel regression analyses, Mplus features a latent variable-based approach to multilevel modeling that has the following benefits:

• Assessment of overall model fit using the usual maximum-likelihood chi-square test statistic when cluster sizes are equal, as well as the MLM and MLMV robust estimator options when cluster sizes are not equal (the default estimator is mlm).
• Latent variables in the analysis with the concomitant purging of measurement errors.
• The construction and testing of measurement models.
• Automatic sorting of the input data and construction of the appropriate between and within-groups covariance matrices used in the analysis.
• Specification of parellel process models in which multiple sets of repeatedly-measured variables are analyzed, with each set having its own growth parameters.
• Latent growth factors may predict other variables and may in turn be predicted by other variables in the model.
• Separate model specifications are permissible for each level of the analysis.

TITLE: 
  Multilevel latent growth model (based on Mplus example program)
DATA:
  FILE IS "c:\intromplus\comp.dat"; 
VARIABLE:
  NAMES ARE g1 g2 cluster g3 y11-y14 y21-y24 x1-x5;
  USEOBS = (x1 EQ 1 AND g1 EQ 2);
  MISSING = ALL (999);
  USEVAR =  y11-y14 ;
  CLUSTER = cluster;
DEFINE:
  y11 = y11/5; 
  y12 = y12/5; 
  y13 = y13/5; 
  y14 = y14/5; 
ANALYSIS:
  TYPE = twolevel; 
MODEL:
  %BETWEEN%
    level1b BY y11-y14@1; 
    trend1b BY y11@0 y12@1 y13*2.5 y14*3.5; 

    [y11-y14@0]; 
    [level1b-trend1b]; 
    level1b WITH trend1b ; 
  %WITHIN% 
    level1w BY y11-y14@1; 
    trend1w BY y11@0 y12@1 y13*2.5 y14*3.5; 
    level1w WITH trend1w ; 
OUTPUT:
  sampstat standardized  ; 

In the interests of conserving space, this program makes use of several Mplus shortcuts. First, the DATA command illustrates the use of the FORTRAN FORMAT statement to read the variables from the large data file efficiently, as recommended by the Mplus manual. The USEOBS command limits the observations to the subset of cases of interest for this analysis.

The first multilevel analysis command is the CLUSTER command. The CLUSTER command identifies which variable in the data file  denotes group or cluster membership. In this example, the variable's name is cluster. Following the CLUSTER command is the DEFINE command. DEFINE allows you to rescale the observed variables so that Mplus is more likely to converge when it fits the multilevel model to the data file  (multilevel models often have more difficultly converging than single-level models).

The ANALYSIS command defines the type of analysis as twolevel. This option tells Mplus that you are fitting a two-level model to the data. At present, Mplus can only fit multilevel models with a single clustering variable, though Mplus can fit some three-level models if you consider the third level of the model to consist of equally-spaced repeated measurements of the observed variables. As mentioned previously, you may use ml, mlm, or mlmv as estimator options for multilevel models. If you select the ml estimator, Mplus produces RMSEA model fit statistics in addition to the familiar chi-square test of model fit. Use the ml estimator option only if cluster sizes are equal and it is reasonable to assume joint multivariate normality of the model residuals; otherwise, use the default mlm estimator or the optional mlmv estimator.

The MODEL command contains the model specification statements for the between and within-cluster components of the model. The between-cluster model specification is listed under the %BETWEEN% subcommand. Notice that any mean and intercept structure specifications occur here; these occur at the between level only. The %WITHIN% subcommand then lists the model specification for the within-cluster model for individuals in the dataset.

The output from this analysis appears below, with some output deleted in the interest of conserving space. The first displayed output is the summary of data, which displays the number of clusters and the ID numbers contained within clusters of a given size. For instance, two clusters contain seven cases each. These clusters are cluster number 103 and cluster number 132.

SUMMARY OF DATA

     Number of clusters          50

       Size (s)    Cluster ID with Size s

          2           114
          3           136
          6           304
          7           103    132
          9           102    109
         10           305
         11           111
         14           134
         15           116    106
         16           118    138    110    105
         17           101    128
         18           133    131    122
         19           303    124    146
         20           147    137    307
         21           129    141    145
         22           144    127    142    143
         23           139    308
         24           119
         25           120    121    112    123
         26           140
         27           301    108    117
         29           135
         34           104
         35           115
         40           302
         41           309

     Average cluster size  19.609

Mplus also displays the intraclass correlations of the observed variables. The intraclass correlation assesses the level of variance in the observed variable that is attributable to membership in its cluster. Even small intraclass correlations suggest the need for a multilevel analysis. In this analysis, the amount of variance attributable to cluster membership ranges from 15% to 20%, suggesting that a multilevel analysis is required.

     Estimated Intraclass Correlations for the Y Variables

               Intraclass              Intraclass              Intraclass
     Variable  Correlation   Variable  Correlation   Variable  Correlation

     Y11           .206      Y12           .150      Y13           .167
     Y14           .165

The overall test of model fit is satisfactory, as is the RMSEA information.

TESTS OF MODEL FIT

Chi-Square Test of Model Fit

          Value                    7.561*
          Degrees of Freedom           4
          P-Value                  .1087

The model results appear below. The results are divided by level. Mplus first outputs the results for the between-cluster portion of the model:

MODEL RESULTS

                   Estimates     S.E.  Est./S.E.    Std     StdYX

Between Level

 LEVEL1B  BY
    Y11                1.000     .000       .000     .687     .923
    Y12                1.000     .000       .000     .687     .914
    Y13                1.000     .000       .000     .687     .842
    Y14                1.000     .000       .000     .687     .764

 TREND1B  BY
    Y11                 .000     .000       .000     .000     .000
    Y12                1.000     .000       .000     .027     .036
    Y13                2.432     .173     14.026     .065     .080
    Y14                3.458     .256     13.519     .092     .103

 LEVEL1B  WITH
    TREND1B             .038     .011      3.369    2.077    2.077

 Residual Variances
    Y11                 .082     .031      2.668     .082     .148
    Y12                 .016     .013      1.264     .016     .029
    Y13                 .005     .010       .509     .005     .007
    Y14                 .065     .028      2.337     .065     .080

 Variances
    LEVEL1B             .472     .087      5.450    1.000    1.000
    TREND1B             .001     .003       .282    1.000    1.000

 Means
    LEVEL1B           10.557     .114     92.953   15.368   15.368
    TREND1B             .522     .046     11.427   19.561   19.561

 Intercepts
    Y11                 .000     .000       .000     .000     .000
    Y12                 .000     .000       .000     .000     .000
    Y13                 .000     .000       .000     .000     .000
    Y14                 .000     .000       .000     .000     .000

Mplus then displays the corresponding model results for the within-cluster level of the model:

Within Level

 LEVEL1W  BY
    Y11                1.000     .000       .000    1.447     .897
    Y12                1.000     .000       .000    1.447     .863
    Y13                1.000     .000       .000    1.447     .785
    Y14                1.000     .000       .000    1.447     .689

 TREND1W  BY
    Y11                 .000     .000       .000     .000     .000
    Y12                1.000     .000       .000     .193     .115
    Y13                2.709     .826      3.281     .524     .284
    Y14                4.237    1.417      2.991     .820     .390

 LEVEL1W  WITH
    TREND1W             .082     .033      2.466     .294     .294

 Residual Variances
    Y11                 .507     .052      9.791     .507     .195
    Y12                 .516     .038     13.567     .516     .183
    Y13                 .580     .045     12.885     .580     .171
    Y14                 .943     .167      5.646     .943     .214

 Variances
    LEVEL1W            2.093     .109     19.199    1.000    1.000
    TREND1W             .037     .027      1.390    1.000    1.000

Though this analysis produced similar findings for the between and within-cluster components of the model, this is not always the case. It is often the case that you will need different model specifications for the between versus the within-cluster sections of the model's specification.

It is also worth noting that despite the congruence between the within and the between-cluster components of this model, if you fit the model as a single level model (using the mlm estimator option), you obtain the following results:

MODEL RESULTS

                   Estimates     S.E.  Est./S.E.    Std     StdYX

 LEVEL    BY
    Y11                1.000     .000       .000    1.606     .903
    Y12                1.000     .000       .000    1.606     .870
    Y13                1.000     .000       .000    1.606     .797
    Y14                1.000     .000       .000    1.606     .708

 TREND    BY
    Y11                 .000     .000       .000     .000     .000
    Y12                1.000     .000       .000     .227     .123
    Y13                2.451     .130     18.812     .556     .276
    Y14                3.496     .195     17.901     .793     .350

 LEVEL    WITH
    TREND               .124     .031      4.000     .341     .341

 Residual Variances
    Y11                 .582     .055     10.593     .582     .184
    Y12                 .528     .044     12.071     .528     .155
    Y13                 .565     .045     12.614     .565     .139
    Y14                1.061     .112      9.443    1.061     .206

 Variances
    LEVEL              2.580     .137     18.881    1.000    1.000
    TREND               .051     .012      4.317    1.000    1.000

 Means
    LEVEL             10.557     .057    184.473    6.572    6.572
    TREND               .517     .032     15.958    2.280    2.280

 Intercepts
    Y11                 .000     .000       .000     .000     .000
    Y12                 .000     .000       .000     .000     .000
    Y13                 .000     .000       .000     .000     .000
    Y14                 .000     .000       .000     .000     .000

Although the chi-square model fit test for this model indicates the model fits the data well (chi-square = 3.697 with 3 DF, p = .295), you can see that all variance estimates are statistically significant. This finding does not take into account the non-independence of individuals who are grouped within the same cluster; it thus stands in contrast to the more appropriate multilevel model that shows a non-significant variance component for the trend latent variable on both the between and within-cluster levels.

The following notes are worth considering before you specify a multilevel model and fit it to your data using Mplus.

• On occasion, you may need to supply starting values to Mplus to obtain a solution that converges.  Assigning reasonable starting values to variance estimates may be helpful. Another approach that often yields satisfactory starting values is to fit a single-level model to the entire sample, ingoring clustering; take the parameter estimates from that model and supply them as the starting values for the multilevel model.
• Each analysis should have at least 30 to 50 clusters.
• Variables measured at the group or cluster level (e.g., family size) may only be used at that level of the analysis.
• Variables measured at the individual or within-cluster level exist at both levels of the analysis and need to be considered in both the between and within-cluster model specifications.
• FIML missing data handling is not available for multilevel models; missing data issues must be resolved prior to the multilevel analysis.

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

Muthén, B. (1997). Latent variable modeling with longitudinal and and multilevel data.  In A. Raftery (ed.), Sociological Methodology 1997 (pp. 453-480).  Boston: Blackwell Publishers.

Muthén, B., du Toit, S.H.C., & Spisic, D. (1997). Robust inference using weighted least squares and quadratic estimating equations in latent variable modeling with categorical and continous outcomes. Accepted for publication in Psychometrika.

Muthen, L.K. and Muthen, B.O. (1998).  Mplus User's Guide.  Los Angeles: Muthen & Muthen.

Wheaton, B., Muthén, B., Alvin, D., & Summers, G. (1977). Assessing reliability and stability in panel models. In D.R. Heise (Ed.): Sociological Methodology. San Francisco: Jossey-Bass.

This page was adapted from Mplus for Windows: An Introduction developed by the Consulting group in the Division of Statistics and Scientific Computation at UT Austin.  We are very grateful to them for their permission to copy and adapt these materials at our web site.

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