UCLA Academic Technology Services HomeServicesClassesContactJobs
Stat Computing > Seminars > Introduction to Mplus: Featuring CFA
Search

Statistical Computing Seminars
Intoduction to Mplus: Featuring Confirmatory Factor Analysis

!This page is under construction!


This page was adapted from Mplus for Windows: An Introduction developed by the Statistical Support group, a division of Research Consulting at ITS at UT Austin.  We are very grateful to them for their permission to copy and adapt these materials at our web site.


Section 1: Introduction
        1. About this Document
        2. Introduction to SEM and Mplus
        3. Accessing Mplus
        4. Getting Help with Mplus
 Section 2: Latent Variable Modeling Using Mplus
        1. Overview of SEM Assumptions
        2. Categorical Outcomes and Categorical Latent Variables
        3. Should you use Mplus?
 Section 3: Using Mplus
        1. Launching Mplus
        2. The Command and Output Windows
        3. Reading Data and Outputting Sample Statistics
 Section 4: Exploratory Factor Analysis
        1. Exploratory Factor Analysis with Continuous Variables
        2. Exploratory Factor Analysis with Missing Data
        3. Exploratory Factor Analysis with Categorical Outcomes
 Section 5: Confirmatory Factor Analysis and Structural Equation Models
        1. Confirmatory Factor Analysis with Continuous Variables
        2. Handling Missing Data
        3. Confirmatory Factor Analysis with Categorical Outcomes
        4. Structural Equation Modeling with Continuous Outcomes
 Section 6: Advanced Models
        1. Multiple Group Analysis
        2. Multilevel Models
 References

Section 1: Introduction

1. About this Document

This document introduces you to Mplus for Windows. It is primarily aimed at first time users of Mplus who have prior experience with either exploratory factor analysis (EFA), or confirmatory factor analysis (CFA) and structural equation modeling (SEM). The document is organized into six sections. The first section provides a brief introduction to Mplus and describes how to obtain access to Mplus. The second section briefly reviews SEM assumptions and describes important and useful model fitting features that are unique to Mplus. The third section describes how to get started with Mplus, how to read data from an external data file, and how to obtain descriptive sample statistics. The fourth section explains how to fit exploratory factor analysis models for continuous and categorical outcomes using Mplus. The fifth section of this document demonstrates how you can use Mplus to test confirmatory factor analysis and structural equation models. The sixth section presents examples of two advanced models available in Mplus: multiple group analysis and multilevel SEM. By the end of the course you should be able to fit EFA and CFA/SEM models using Mplus. You will also gain an appreciation for the types of research questions well-suited to Mplus and some of its unique features.

2. Introduction to EFA, CFA, SEM and Mplus

Exploratory factor analysis (EFA) is a method of data reduction in which you may infer the presence of latent factors that are responsible for shared variation in multiple measured or observed variables. In EFA each observed variable in the analysis may be related to each latent factor contained in the analysis. By contrast, confirmatory factor analysis (CFA) allows you to stipulate which latent factor is related to any given observed variable. Structural equation modeling (SEM) is a more general form of CFA in which latent factors may be regressed onto each other. Mplus can fit EFA, CFA, and SEM models, among other models.

To effectively use and understand the course material, you should already know how to conduct a multiple linear regression analysis and compute descriptive statistics such as frequency tables using SAS, Stata, SPSS, or a similar general statistical software package. You should also understand how to interpret the output from a multiple linear regression analysis. This document also assumes that you are familiar with the statistical assumptions of EFA, CFA, and SEM, and you are comfortable using syntax-based software programs. If you do not have prior experience with exploratory factor analysis, we would recommend seeing our Stat Books for Loan under the section on Factor Analysis and Structural Equation Modeling for more information about Factor Analysis and SEM. Finally, you should understand basic Microsoft Windows navigation operations: opening files and folders, saving your work, recalling previously saved work, etc.

3. Accessing Mplus

You may access Mplus in one of three ways:

  1. License a copy from Muthén &  Muthén for your own personal computer.
  2. Access it from the CLICC lab in the Powell Library or as part of visiting us in Statistical Consulting.
  3. Download the free student version of Mplus from the Muthén &  MuthénWeb site for your own personal computer. If your models of interest are small, the free demonstration version may be sufficient to meet your needs.
4. Getting Help with Mplus

Important note: Our Statistical Consulting services are available only to researchers in the UCLA community. Non-UCLA researchers will find the Muthén &  MuthénWeb site to be a useful resource; also see the Mplus Discussion forum for frequently-asked questions and answers. You may also post your own questions in this forum.

Section 2: Latent Variable Modeling using Mplus

1. Overview of SEM Assumptions for Continuous Outcome Data

Before specifying and running a latent variable models, you should give some thought to the assumptions underlying latent variable modeling with continuous outcome variables. Several of these assumptions are shown below:

These assumptions apply equally to all EFA and CFA/SEM software programs. The details of these assumptions can be found in the UT Austin AMOS tutorial, but they may be summarized as follows: Recommendations for sample size vary depending upon the complexity of the specified model, but typical figures range from 5 to 15 cases per estimated parameter with overall sample size preferred to exceed N = 200 cases. Furthermore, any model you consider should have a theoretical basis, and substantive inferences should be drawn based upon your ability to rule out alternative explanations for findings, rather than on statistical considerations alone.

Like AMOS, Mplus features Full Information Maximum Likelihood (FIML) handling of missing data, an appropriate, modern method of missing data handling that enables Mplus to make use of all available data points, even for cases with some missing responses. For more details on missing data handling methods, including FIML, see the UT Austin Statistical Services General FAQ #25: Handling missing or incomplete data. One added missing data handling feature that is unique to Mplus is its ability to generate model modification indices for data files  that are incomplete.

2. Categorical Outcomes and Categorical Latent Variables

Where Mplus diverges from most other SEM software packages is in its ability to fit latent variable models to data files that contain ordinal or dichotomous outcome variables. Note that Mplus will not yet fit models to data files with nominal outcome variables????? that contain more than two levels. Nonetheless, the ability to fit models to variables that contain ordinal and dichotomous categorical outcome variables is very useful. Furthermore, Mplus will fit latent class analysis (LCA) models that contain categorical latent variables and fit mixture models that generate expected classifications of observations based upon the characteristics of your specified model.

3. Should you use Mplus?

Should you use Mplus to perform EFA, CFA, and SEM analyses on your data? In order to facilitate rapid access to both simple and complex latent variable models, the Mplus developers have built a streamlined set of data import and model specification commands. All Mplus commands are specified using command syntax. If you are not comfortable with reading data and specifying statistical models using command syntax, Mplus may not be the optimal choice for you. On the other hand, if you prefer to work with command syntax when you use statistical software programs or you do not mind learning software syntax to perform data analysis, you will probably find it useful to learn Mplus. This is particularly true when you consider some of the features unique to Mplus:
Section 3: Using Mplus

1. Launching Mplus

If you are using a personal or demonstration copy of Mplus, locate the Mplus entry in the Program Files subsection of the Microsoft Windows Start menu. Once you have launched Mplus, you will see the following window appear on your computer's desktop: This is the window where you can open or enter an Mplus program.

Mplus command
window

2. The Input and Output Windows

The window shown above is the input window. You write Mplus syntax in this window to read the data to be analyzed and to specify your model of interest. You then save your Mplus syntax and select Run Mplus from the Mplus menu to submit your syntax to the Mplus engine for processing:

Mplus
        Run Mplus

Once Mplus has finished processing your command syntax, it replaces the input window with the output window. The output window first displays your Mplus syntax. Below the Mplus syntax are the Mplus model results. If there is an error in your Mplus syntax or you want to modify your Mplus syntax in any way (e.g., to fit a different model to the data), you must return to the appropriate command file by selecting that file's name from the File menu's list of recently-accessed files. That action returns the input window's contents to the screen and you can then modify the previous commands, save the modified command file, and run Mplus once again to obtain new output.

3. Reading Data and Outputting Sample Statistics

After you have launched Mplus, you may build a command file. There are nine Mplus commands: TITLE, DATA (required), VARIABLE (required), DEFINE, SAVEDATA, ANALYSIS, MODEL, OUTPUT, and MONTECARLO. The most commonly used Mplus commands are described in this document. According to the Mplus User's Guide, The Mplus commands may come in any order. The DATA and VARIABLE commands are required for all analyses. All commands must begin on a new line and must be followed by a colon. Semicolons separate command options. There can be more than one option per line. The records in the input setup must be no longer than 80 columns. They can contain upper and/or lower case letters and tabs. A description of the Mplus defaults appears in the UT Austin Mplus FAQ #3: Mplus Defaults

The first Mplus syntax to appear in the command file is typically a TITLE command. The TITLE command allows you to specify a title that Mplus will print on each page of the output file.

Following the TITLE command is the DATA command. The DATA command specifies where Mplus will locate the data, the format of the data, and the names of variables. At present, Mplus will read the following file formats: tab-delimited text, space-delimited text, and comma-delimited text. The input data file may contain records in free field format or fixed format. If you are using data stored as a SAS, Stata or SPSS file, you can see our Mplus Frequently Asked Questions for tips on how to convert these data files for use in Mplus.

The next command is the VARIABLE command. The VARIABLE command names the columns of data that Mplus reads using the DATA command. The can be combined with the USEVARIABLES command to select a subset of the variables for analysis.

Following the VARIABLE command is the ANALYSIS command. The ANALYSIS command tells Mplus what type of analysis to perform. Many analysis options are available; a number of these are shown in the examples that appear in this document.

Consider the following example data file: In 1939 Karl Holzinger and Francis Swineford administered 26 aptitude tests to 145 students in the Grant-White School. Of the 26 tests, six are used here: visual perception, cubes, lozenges, paragraph comprehension, sentence completion, and word meaning. An additional variable, gender, is included in the data file, but not used in this example. You can download this file as a tab-separated file The SPSS file's name is grant.sav. You can download this file in tab-delimited text format as grant.dat. Then you can write the following Mplus syntax to read the data from the file.
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 = basic ;
In this sample program, the DATA command uses the FILE subcommand to tell Mplus where to locate the relevant data file. In this case, the file's location is c:\intromplus\grant.dat. The FORMAT subcommand uses the default free option to let Mplus know that the data points appear in order in the data file with the data points separated by commas, tabs, or spaces. 

The next command shown is the VARIABLE command. The VARIABLE command uses the NAMES subcommand to list the variables contained in the Grant-White data file . Note the variable names span two lines; all commands can span across multiple lines.  Because Mplus restricts variable names to have a maximum width of eight characters, the variable name paragraph is shortened to paragrap.

Following the NAMES subcommand is the USEVARIABLES subcommand. USEVARIABLES enables you to specify a particular subset of variables to be used in the data analysis.

A similar subcommand, USEOBS, allows you to select subsets of cases to be used in a particular analysis. The example below shows how you could limit the analysis to female participants, selecting just those where gender=1.  It also shows how you can use the dash notation to specify a group of variables in the USEVARIABLES statement, indicating all of the variables contigously between visperc to wordmean.

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-wordmean ; 
  USEOBS gender EQ 1 ;
ANALYSIS:
  TYPE = basic ;

The ANALYSIS command specifies the TYPE of analysis to be performed by Mplus. In this example the type is basic. The basic model type does not fit any model to the sample data; instead Mplus will compute sample statistics only. Using basic as the analysis type is useful during the initial phase of building your command file because you can use the Mplus sample statistics output to compare Mplus results to results you obtained using SAS, SPSS, Excel, or other statistical software programs to verify that Mplus is reading your input data correctly.

Running the program above with the data grant.dat yields the output from this basic analysis below. Although Mplus initially returns a copy of the input command file, that portion of the output has been omitted here in the interest of saving space.

Grant-White School:  Summary Statistics

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                     0

Observed variables in the analysis
   VISPERC     CUBES       LOZENGES    PARAGRAP    SENTENCE    WORDMEAN

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

Input data file(s)
  c:\IntroMplus\grant.dat

Input data format  FREE

RESULTS FOR BASIC ANALYSIS

     SAMPLE STATISTICS

           Means
              VISPERC       CUBES         LOZENGES      PARAGRAP      SENTENCE
              ________      ________      ________      ________      ________
      1        29.579        24.800        15.966         9.952        18.848

           Means
              WORDMEAN
              ________
      1        17.283

           Covariances
              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
              WORDMEAN
              ________
 WORDMEAN      63.163

           Correlations
              VISPERC       CUBES         LOZENGES      PARAGRAP      SENTENCE
              ________      ________      ________      ________      ________
 VISPERC        1.000
 CUBES          0.326         1.000
 LOZENGES       0.449         0.417         1.000
 PARAGRAP       0.342         0.228         0.328         1.000
 SENTENCE       0.309         0.159         0.287         0.719         1.000
 WORDMEAN       0.317         0.195         0.347         0.714         0.685

           Correlations
              WORDMEAN
              ________
 WORDMEAN       1.000


Mplus initially identifies the number of groups and observations in the analysis, followed by the number of X (predictor) and Y (outcome) variables and the sample (input) covariances, variances, and means. Once you have verified that these values are correct, you can turn your attention to fitting your model(s) of interest. The next section continues with the same example data file , but describes how to perform an exploratory factor analysis of the continuous variables in the Grant-White data file using Mplus.

Section 4: Exploratory Factor Analysis

1. Exploratory Factor Analysis with Continuous Variables

Once you have read the data into Mplus and verified that the sample statistics show that the data have been read correctly, you can perform exploratory factor analysis using Mplus by altering the ANALYSIS command as follows:

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 = efa 1 2 ; 
  ESTIMATOR = ml ;
OUTPUT:
  sampstat ; 
This syntax instructs Mplus to perform an exploratory factor analysis of the Grant-White data file. Efa tells Mplus to perform an exploratory factor analysis. The 1 and 2 following the efa specification tells Mplus to generate all possible factor solutions between and including 1 and 2. In this instance, one and two factor solutions will be produced by the analysis. Finally, the ESTIMATOR = ml option has Mplus use the maximum likelihood estimator to perform the factor analysis and compute a chi-square goodness of fit test that the number of hypothesized factors is sufficient to account for the correlations among the six variables in the analysis. This optional specification overrides the default unweighted least-square (uls) estimator.

If your data are not joint multivariate normally distributed, you may want to replace the ml with either the mlm or mlmv estimators. One useful feature of Mplus is its ability to handle non-normal input data. Recall that the default ml estimator assumes that the input data are distributed joint multivariate normal. If you have reason to believe that this assumption has not been met and your sample is reasonably large (e.g., N = 200), you may substitute mlm or mlmv in place of ml on the ESTIMATOR = line. 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. SEM users who are familiar with Bentler's EQS software program should also note that the mlm chi-square test and standard errors are equivalent to those produced by EQS in its ML;ROBUST method.

You may also add the OUTPUT command following the ANALYSIS command. The OUTPUT command is used to specify optional output. For this example the keyword sampstat tells Mplus to include sample statistics as part of its printed output.

 

Mplus produces the sample correlations, eigenvalues, and the chi-square test of the one factor model to the sample data. As you can see from the results, shown below, the chi-square test is statistically significant, so the null hypothesis that a single factor fits the data is rejected; more factors are required to obtain a non-significant chi-square. Since the chi-square test is sensitive to sample size (such that large samples often return statistically significant chi-square values) and non-normality in the input variables, Mplus also provides the Root Mean Square Error of Approximation (RMSEA) statistic. The RMSEA is not as sensitive to large sample sizes. According to Hu and Bentler (1999), RMSEA values below .06 indicate satisfactory model fit. The RMSEA yielded a result of .162, which was consistent with the chi-square result in suggesting that the one factor model does not fit the data adequately.

           CONTINUOUS VARIABLE CORRELATION MATRIX
              VISPERC       CUBES         LOZENGES      PARAGRAP      SENTENCE
              ________      ________      ________      ________      ________
 VISPERC
 CUBES           .326
 LOZENGES        .449          .417
 PARAGRAP        .342          .228          .328
 SENTENCE        .309          .159          .287          .719
 WORDMEAN        .317          .195          .347          .714          .685

           EXPLORATORY ANALYSIS WITH  1 FACTOR(S) :
           EIGENVALUES FOR SAMPLE CORRELATION MATRIX
                  1             2             3             4             5
             ________      ________      ________      ________      ________
      1         3.009         1.225          .656          .530          .311
 
           EIGENVALUES FOR SAMPLE CORRELATION MATRIX
                  6
              ________
      1          .270
 
           EXPLORATORY ANALYSIS WITH  1 FACTOR(S) :
           CHI-SQUARE VALUE              43.241
           DEGREES OF FREEDOM                 9
           PROBABILITY VALUE              .0000

           RMSEA (ROOT MEAN SQUARE ERROR OF APPROXIMATION) :
           ESTIMATE (90 PERCENT C.I.) IS   .162 (  .115   .212)
           PROBABILITY RMSEA LE  .05 IS     .000


Mplus next produces the estimated factor loadings and error variances. Notice that the visperc, cubes, and lozenges factor loadings are low relative to the other factor loadings displayed below. See Factor Analysis Using SAS PROC FACTOR (courtesy of The University of Texas at Austin Statistical Services) for more information on interpreting factor loadings.

           ESTIMATED FACTOR LOADINGS
                  1
              ________
 VISPERC         .415
 CUBES           .272
 LOZENGES        .415
 PARAGRAP        .865
 SENTENCE        .818
 WORDMEAN        .827  

           ESTIMATED ERROR VARIANCES
              VISPERC       CUBES         LOZENGES      PARAGRAP      SENTENCE
              ________      ________      ________      ________      ________
                .828          .926          .828          .252          .330

              ________
      1          .316


The estimated correlation matrix is the correlation matrix reproduced by Mplus under the assumption that a single factor is sufficient to explain the sample correlations. From the model fit results shown above, this is not the case, so it is not surprising that this implied or model-based correlation matrix differs substantially from the sample correlation matrix reported above.

           ESTIMATED CORRELATION MATRIX
              VISPERC       CUBES         LOZENGES      PARAGRAP      SENTENCE
              ________      ________      ________      ________      ________
 VISPERC        1.000
 CUBES           .113         1.000
 LOZENGES        .172          .113         1.000
 PARAGRAP        .359          .235          .359         1.000
 SENTENCE        .339          .223          .340          .708         1.000
 WORDMEAN        .343          .225          .343          .715          .677
 

              WORDMEAN
              ________
 WORDMEAN       1.000


The residuals matrix represents the difference between the sample correlation matrix and the implied correlation matrix. As noted above, since the model did not fit the observed data particularly well, there are some values in this matrix that are non-trivial in size. In particular, the cubes-visperc, lozenges-visperc, and lozenges-cubes residual values are high relative to the other values in the matrix.

           RESIDUALS OBSERVED-EXPECTED
              VISPERC       CUBES         LOZENGES      PARAGRAP      SENTENCE
              ________      ________      ________      ________      ________
 VISPERC         .000
 CUBES           .213          .000
 LOZENGES        .276          .304          .000
 PARAGRAP       -.017         -.007         -.031          .000
 SENTENCE       -.030         -.063         -.053          .011          .000
 WORDMEAN       -.026         -.030          .004          .000          .009
 

           RESIDUALS OBSERVED-EXPECTED
              WORDMEAN
              ________
 WORDMEAN        .000


The Root Mean Square Residual (RMR) is another descriptive model fit statistic. According to Hu and Bentler (1999), RMR values should be below .08 with lower values indicating better model fit. The value of .1225 shown below for the one factor solution indicates unacceptably poor model fit.

 ROOT MEAN SQUARE RESIDUAL IS         .1225

In short, the one factor solution was a poor fit to the data. In particular, the model did not account well for the correlations among the visperc, cubes, and lozenges variables. What about the two factor solution? Mplus reports the two factor solution following the single factor model.

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) ;
The all keyword tells Mplus that all variables in the analysis use -9 to represent missing values. If your data file  contains blanks to represent missing values, you may use the specification
MISSING = blank ;
Similarly, you may use
MISSING ARE . ;
if your data file  contains period symbols to represent missing values. Other missing value specifications are available; see the Mplus User's Guide for specifics.

If you insert the MISSING syntax into the previous exploratory factor analysis program and specify that Mplus use the newly-created data file  that contains cases with missing values, grant-missing.dat, Mplus will perform listwise deletion of the cases with incomplete data. The Mplus command file follows:
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.

           EXPLORATORY ANALYSIS WITH  1 FACTOR(S) :
           CHI-SQUARE VALUE              14.651
           DEGREES OF FREEDOM                 9
           PROBABILITY VALUE              .1009

           RMSEA (ROOT MEAN SQUARE ERROR OF APPROXIMATION) :
           ESTIMATE (90 PERCENT C.I.) IS   .089 (  .000   .169)
           PROBABILITY RMSEA LE  .05 IS     .199


The one factor solution also fits the data file  for the 79 useable cases. This finding stands in direct contrast to the example in the previous section where all 145 cases had complete data and the one factor model was rejected. Clearly the reduction of N from 145 to 79 has resulted in a substantial loss of statistical power to reject false hypotheses.

Fortunately, you can use Mplus's FIML missing data handling option to rectify the problem. Add the keyword missing to the TYPE subcommand of the ANALYSIS command, like this:
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 ;
Run the analysis and consider the results, shown below.

Grant-White School: Exploratory Factor Analysis with Missing Data

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           0


Mplus now uses all 145 cases in its computations.

SUMMARY OF DATA

     Number of patterns           4

COVARIANCE COVERAGE OF DATA

Minimum covariance coverage value    .100
 

     PROPORTION OF DATA PRESENT
 

           Covariance Coverage
              VISPERC       CUBES         LOZENGES      PARAGRAP      SENTENCE
              ________      ________      ________      ________      ________
 VISPERC        1.000
 CUBES           .697          .697
 LOZENGES       1.000          .697         1.000
 PARAGRAP       1.000          .697         1.000         1.000
 SENTENCE        .821          .545          .821          .821          .821
 WORDMEAN       1.000          .697         1.000         1.000          .821


Mplus futher recognizes that there are four distinct patterns of missing data contained in the data file  and it displays the amount of data used to generate each input covariance for the analysis. From the missing data coverage matrix, you can see that the cubes-sentence covariance has the lowest coverage with just under 55% of cases available to build the covariance. Mplus requires a minimum coverage value of 10% per covariance, though you can override this default if you wish.

           EXPLORATORY ANALYSIS WITH  1 FACTOR(S) :
           CHI-SQUARE VALUE              29.732
           DEGREES OF FREEDOM                 9
           PROBABILITY VALUE              .0005

           RMSEA (ROOT MEAN SQUARE ERROR OF APPROXIMATION) :
           ESTIMATE (90 PERCENT C.I.) IS   .126 (  .078   .178)
           PROBABILITY RMSEA LE  .05 IS     .007


Unlike the example that used listwise deletion of cases with missing data, the chi-square test of model fit for the one factor solution rejects the one factor model. Using FIML missing data handling, you conclude that one factor is not sufficient to explain the pattern of correlations among the six input variables, just as you did in the first example from the preceding section where Mplus used the complete data file  containing 145 cases. As with the complete dataset, the two factor solution fits the data well using the FIML method with the incomplete dataset:

           EXPLORATORY ANALYSIS WITH  2 FACTOR(S) :
           CHI-SQUARE VALUE                .578
           DEGREES OF FREEDOM                 4
           PROBABILITY VALUE              .9655

           RMSEA (ROOT MEAN SQUARE ERROR OF APPROXIMATION) :
           ESTIMATE (90 PERCENT C.I.) IS   .000 (  .000   .000)
           PROBABILITY RMSEA LE  .05 IS     .982


3. Exploratory factor analysis with categorical outcomes

So far, the examples shown here contained continuous outcomes. If you have observed outcome variables that have ten or fewer categories, and the variables' responses are dichotomous or ordered categories, you may elect to have Mplus treat these variables as categorical indicators. This type of model is often sensible for analyzing Likert scale items because while the items themselves typically are coarsely categorized on a 1 to 5 or 1 to 7 scale, the items often attempt to measure an individual's standing on a continuous underlying unobserved variable.

For the purposes of illustration, suppose that you recode each variable into a replacement variable where all six variables' values at the median or below are assigned a categorical value of 1.00 and all values above the median assigned a value of 2.00. Mplus recodes the lowest value to zero with subsequent values increasing in units of 1.00. While the two underlying latent factors remain continuous, the six categorical observed variables' response values are now ordered dichotomous categories. To analyze the modified data file  using Mplus, you may use the syntax that appeared in the initial exploratory factor analysis example, with the following modifications, and the new data file that contains the categorical variables, grantcat.dat, as shown below.
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 ;
First, you must change the names of the variables in the NAMES and USEVARIABLES subcommands of the DATA command. Next, you tell Mplus which variables are categorical with the CATEGORICAL subcommand of the DATA command, like this:
    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.

Keeping these considerations in mind, Mplus provides a convenient mechanism to perform an exploratory factor analysis of dichotomous and ordered categorical responses. Since many exploratory factor analyses are performed on Likert scale items that contain ordered categories, Mplus is a useful tool for the exploration of the factor structure of these instruments.

Section 5: Confirmatory Factor Analysis and Structural Equation Models

The examples in the preceding section demonstrate how you can use Mplus to fit exploratory factor analysis models to the Grant-White data file . What if you had an a priori hypothesis that the visual perception, cubes, and lozenges variables belonged to a single factor whereas the paragraph, sentence, and word meaning variables belonged to a second factor? The diagram shown below illustrates the model visually.

Confirmatory factor analysis model with
two factors and three indicators per factor

You can test this hypothesized factor structure using confirmatory factor analysis, as shown in the next section.

1. Confirmatory Factor Analysis with Continuous Variables

Below we show an example running the confirmatory factor analysis from above.  It uses the same TITLE, DATA, and VARIABLE statements from the exploratory factor analysis shown in Section 4, but adds/changes the ANALYSIS, MODEL, and OUTPUT statements as shown below, with the changes shown in italics for emphasis.
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:

4. Structural Equation Modeling with Continuous Outcomes

In addition to exploratory and confirmatory factor analysis, you may use Mplus to fit structural equation models that feature causal relationships among latent variables. An ubiquitous example of a structural equation model is that of the impact of socioeconomic status (SES) on alienation in 1967 and 1971. A study conducted by Wheaton, Muthén, Alwin, and Summers (1977) fit several structural equation models to a data file  of 932 research participants. The data file  contained the following observed, continuous variables:

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. 

Stability of
Alienation Structural Equation Model Diagram

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 th