Stat Computing > Mplus > Output

Annotated Mplus Output
Exploratory Factor Analysis

This page shows an example exploratory factor analysis with footnotes explaining the output.  The data used in this example were collected on 1428 college students (complete data on 1365 observations) and are responses to items on a survey.  We will use some dichotomous variables, including faculty sex and faculty nationality (US citizen or foreign citizen) (facsex, facnat); ordered categorical variables, including faculty rank, student rank and grade (A, B, C, etc.) (facrank, studrnk1, grade); and some continuous variables (faculty salary, years teaching, years teaching at the University of Texas, and number of students in the class) (salary, yrsteach, yrsut, nstud) in this analysis.  You can obtain the data set by clicking here.  We do not claim that these variables are the best to select for a factor analysis.  Rather, we selected them to have a representation of each type of variable (dichotomous, ordered categorical and continuous) in our analysis.  We are using Mplus version 3.1.

title:  Exploratory factor analysis with 1/2, categorical and continuous 
        variables.
data:  file is "d:\m255_for_mplus1.dat";
variable:  names are facsex facnat facrank studrnk1 grade 
             salary yrsteach yrsut nstud sample;
           usevar are facsex facnat facrank studrnk1 grade 
             salary, yrsteach yrsut nstud;
           missing are all (-9);
           categorical are facsex facnat facrank studrnk1 
             grade;
analysis:  type = efa 3 3;

Comments on syntax

• We have more variables in the data file than we would like to use in the factor analysis, so we use the usevar statement to indicate which variables that we would like to use in the analysis. 
• Some variables in the data set have missing values for some of the cases.  These are indicated in the input data file with a -9, not a dot (.).  We use the missing statement to indicate that for all variables listed on the usevar statement, missing values are indicated with a -9. 
• All categorical variables, both dichotomous and ordered categorical, are listed on the categorical statement. 
• We indicate the type of analysis that we would like to do, exploratory factor analysis, on the analysis statement.  The numbers at the end of the statement indicate the minimum and maximum number of factors to be extracted.  By using 3 3, we are saying that we only want a three-factor solution.  We have done this to save space.  We suggest that you use a reasonable range here, and each solution will be shown in the output.  For example, if we had 2 4 at the end of the statement, we would see the two-factor, three-factor and four-factor solution in the output. 
• All solutions are shown with a varimax and a promax rotation.  You do not have the option of seeing the unrotated solution. 
• By default, Mplus uses unweighted least squares (ULS) as its method of deriving the factors.  You may select other methods, such as weight least squares (WLS), by using the estimator statement.  Note that you cannot use certain methods with certain types of variables, for example, you cannot use maximum likelihood estimation if you have categorical variables.
• If you would like to get a scree plot, you can use the plot command and indicate plot2.  For example:
       plot: type = plot2;
  To see the graph, you need to click on "Graph" at the top of Mplus, and select "View Graphs".  You then select the scree plot and click on "View" to see the graph.

Notes and summary information

The information below is very useful because it lets you know what Mplus did.  You can use this part of the output to make sure that Mplus ran the analysis that you intended.

INPUT READING TERMINATED NORMALLY

Exploratory factor analysis with 1/2, categorical and continuous
variable.

SUMMARY OF ANALYSIS

Number of groups                                                 1
Number of observations                                        1057

Number of dependent variables                                    9
Number of independent variables                                  0
Number of continuous latent variables                            0

Observed dependent variables

  Continuous
   SALARY      YRSTEACH    YRSUT       NSTUD

  Binary and ordered categorical (ordinal)
   FACSEX      FACNAT      FACRANK     STUDRNK1    GRADE


Estimator                                                      ULS
Maximum number of iterations                                  1000
Convergence criterion                                    0.500D-04
Maximum number of steepest descent iterations                   20

Input data file(s)
  d:\m255_for_mplus1.dat

Input data format  FREE

SUMMARY OF CATEGORICAL DATA PROPORTIONS

    FACSEX
      Category 1    0.633
      Category 2    0.367
    FACNAT
      Category 1    0.949
      Category 2    0.051
    FACRANK
      Category 1    0.088
      Category 2    0.265
      Category 3    0.445
      Category 4    0.202
    STUDRNK1
      Category 1    0.188
      Category 2    0.197
      Category 3    0.250
      Category 4    0.231
      Category 5    0.134
    GRADE
      Category 1    0.006
      Category 2    0.022
      Category 3    0.193
      Category 4    0.470
      Category 5    0.309

Looking at the proportions in some of the categories, it might be worth recoding some of the variables so that small categories are combined into larger ones.  For example, Category 1 of grade might be combined into one of the other categories.

           THE INPUT SAMPLE CORRELATION MATRIX IS NOT POSITIVE DEFINITE.
           THE ESTIMATES GIVEN BELOW ARE STILL VALID.

RESULTS FOR EXPLORATORY FACTOR ANALYSIS


           EIGENVALUES FOR SAMPLE CORRELATION MATRIX
                  1             2             3             4             5
              ________      ________      ________      ________      ________
      1         2.961         2.012         1.293         1.066         0.833


           EIGENVALUES FOR SAMPLE CORRELATION MATRIX
                  6             7             8             9
              ________      ________      ________      ________
      1         0.529         0.286         0.126        -0.107

Factor analysis solutions

Below are the varimax and promax rotated loadings.  These loadings are the correlations between the variable and the factor.  For example, 0.236 is the correlation between the variable facsex and the first factor.  For the varimax loadings, the range is +1 to -1.  In some cases, you may get a loading that is outside of this range.  If this happens, the solution is said to be "inadmissible."  You will probably not want to rely on those results.  Rather, you might try rerunning the factor analysis and extract fewer factors.  Also, remember that when you have categorical variables, the correlation matrix is not a Pearson product-moment correlation matrix, but rather a polychoric correlation matrix.  A polychoric correlation matrix requires an even greater sample size than does a Pearson correlation matrix.  If the sample size is too small, you will likely get negative variances for your variables.  We have a SAS FAQ on polychoric correlations and a Stata FAQ on polychoric correlations.  In our example, are correlation matrix is a combination of tetrachoric correlations (two dichotomous variables), polychoric correlations (two categorical variables), and Pearson correlations (two continuous variables).

           EXPLORATORY ANALYSIS WITH 3 FACTOR(S) :

           ROOT MEAN SQUARE RESIDUAL IS        0.0604

           VARIMAX ROTATED LOADINGS
                  1             2             3
              ________      ________      ________
 FACSEX         0.236         0.855        -0.470
 FACNAT        -0.046         0.192         0.984
 FACRANK       -0.799        -0.184         0.040
 STUDRNK1      -0.011         0.288         0.058
 GRADE          0.041         0.116         0.091
 SALARY        -0.665        -0.159         0.037
 YRSTEACH      -0.827         0.112         0.023
 YRSUT         -0.871         0.181        -0.068
 NSTUD          0.182        -0.954        -0.333

For promax rotated solutions, the loadings might be slightly less than -1 or slightly greater than +1.  This is because the factors are not orthogonal with an oblique rotation.

           PROMAX ROTATED LOADINGS
                  1             2             3
              ________      ________      ________
 FACSEX         0.112         0.890        -0.440
 FACNAT         0.112         0.111         1.010
 FACRANK       -0.791        -0.170        -0.006
 STUDRNK1      -0.016         0.284         0.068
 GRADE          0.051         0.108         0.099
 SALARY        -0.657        -0.148        -0.002
 YRSTEACH      -0.838         0.127        -0.015
 YRSUT         -0.902         0.206        -0.109
 NSTUD          0.178        -0.931        -0.362

The promax factor correlations listed below give the correlations between the factors.  For example, the correlation between factor 1 and factor 2 is 0.077.

           PROMAX FACTOR CORRELATIONS
                  1             2             3
              ________      ________      ________
      1         1.000
      2         0.077         1.000
      3        -0.211         0.035         1.000

The estimated residual variances given below are the variances of the variables after accounting for all of the variance in the efa model.  Notice that studrnk1 and grade have high variances (as compared to the other variables in the analysis).  This makes sense when you look at the factor loadings, because neither studrnk1 nor grade load highly on any factor; hence, their residual variances are high.  Although it may seem strange to see a negative variance, we are not worried in this case because the values are so close to zero.

           ESTIMATED RESIDUAL VARIANCES
              FACSEX        FACNAT        FACRANK       STUDRNK1      GRADE
              ________      ________      ________      ________      ________
      1        -0.008        -0.008         0.327         0.913         0.977

           ESTIMATED RESIDUAL VARIANCES
              SALARY        YRSTEACH      YRSUT         NSTUD
              ________      ________      ________      ________
      1         0.531         0.304         0.205        -0.054

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