This page was created using Mplus version 5.2, the output and/or syntax may be different for other versions of Mplus.
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. You can obtain the data set by clicking here. The analysis includes 12 variables, item13 to item24.
Data:
File is m255.dat ;
Variable:
Names are
item13 item14 item15 item16 item17 item18 item19 item20 item21 item22
item23 item24;
Missing are all (-9999) ;
Analysis:
type = efa 3 3;
Comments on syntax
- Some variables in the data set have missing values for some of the cases. These are indicated in the input data file with a -9999. We use the missing statement to indicate that for all variables in the model, missing values are indicated with a -9999. Note that starting with Mplus v.5 cases with missing values on some of the variables in the model are included in the analysis by default. If we wanted to include only cases with complete data, we could use the listwise = on; option of the data command.
- We indicate the type of analysis that we would like to do, that is, exploratory factor analysis (efa), using the type option of the analysis command. 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 option, we would see the two-factor, three-factor and four-factor solution in the output.
- We have used the default geomin rotation. A number of other rotation methods, including the more traditional promax and varimax are available using the rotation option of the analysis command, for example: rotation=promax;.
- When all of the variables are continuous, as in this example, Mplus uses maximum likelihood (ML) as its method of deriving the factors by default. You may request other methods, such as unweighted least squares (ULS), using the estimator option. Note that not all methods are available for all types of 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 "Eigenvalues for exploratory factor analysis" and click on "View" to see the screen plot.
Notes and summary information
The information below printed near the top of the output, it is useful because it lets you know what Mplus did. You want to check this part of the output to make sure Mplus ran the analysis that you intended.
*** WARNINGa Data set contains cases with missing on all variables. These cases were not included in the analysis. Number of cases with missing on all variables: 1 1 WARNING(S) FOUND IN THE INPUT INSTRUCTIONS SUMMARY OF ANALYSIS Number of groups 1 Number of observationsb 1427 Number of dependent variablec 12 Number of independent variables 0 Number of continuous latent variables 0 Observed dependent variablesd Continuous ITEM13 ITEM14 ITEM15 ITEM16 ITEM17 ITEM18 ITEM19 ITEM20 ITEM21 ITEM22 ITEM23 ITEM24 Estimatore ML Rotationf GEOMIN Row standardization CORRELATION Type of rotationg OBLIQUE Epsilon value Varies Information matrix OBSERVED Maximum number of iterations 1000 Convergence criterion 0.500D-04 Maximum number of steepest descent iterations 20 Maximum number of iterations for H1 2000 Convergence criterion for H1 0.100D-03 Optimization Specifications for the Exploratory Factor Analysis Rotation Algorithm Number of random starts 30 Maximum number of iterations 10000 Derivative convergence criterion 0.100D-04 Input data file(s) m255.dat Input data format FREE
a. Warning. In this case, the output includes a warning that 1 case had missing values on all of the variables in the analysis, and hence was excluded from the model.
b. Number of observations. The number of observations used in the analysis. As mentioned above, by default, Mplus will include all cases that have at least partial data on the variables in the analysis.
c. Number of dependent variables. Gives the number of dependent (outcome) variables in the model. Note that Mplus classifies the factor indicators as dependent variables.
d. Observed dependent variables. The list of variables included in this analysis. All of the variables in our model are listed under Continuous. If the model included categorical variables, they would be listed here under the heading Categorical. If this section includes variables you did not intend to include in your analysis, you may need to use the usevariables option of the data command.
e. Estimator. The method used to estimate the model, in this case, maximum likelihood (ML).
f. Rotation. The specific rotation method used in the model.
g. Type of rotation. Rotations that allow the factors to be correlated are oblique, while rotations that force the factors to be uncorrelated are known as orthogonal. The default geomin rotation is oblique.
SUMMARY OF DATA
Number of missing data patternsh 24
COVARIANCE COVERAGE OF DATA
Minimum covariance coverage value 0.100
PROPORTION OF DATA PRESENT
Covariance Coveragei
ITEM13 ITEM14 ITEM15 ITEM16 ITEM17
________ ________ ________ ________ ________
ITEM13 0.994
ITEM14 0.994 0.998
ITEM15 0.993 0.996 0.998
ITEM16 0.991 0.994 0.994 0.995
ITEM17 0.992 0.995 0.995 0.994 0.997
ITEM18 0.992 0.996 0.996 0.994 0.996
ITEM19 0.989 0.993 0.994 0.992 0.994
ITEM20 0.974 0.977 0.977 0.975 0.976
ITEM21 0.992 0.994 0.994 0.992 0.994
ITEM22 0.986 0.989 0.989 0.987 0.989
ITEM23 0.992 0.995 0.995 0.992 0.994
ITEM24 0.989 0.992 0.992 0.989 0.991
Covariance Coverage
ITEM18 ITEM19 ITEM20 ITEM21 ITEM22
________ ________ ________ ________ ________
ITEM18 0.998
ITEM19 0.995 0.995
ITEM20 0.977 0.975 0.978
ITEM21 0.994 0.992 0.977 0.996
ITEM22 0.989 0.987 0.973 0.989 0.991
ITEM23 0.995 0.992 0.976 0.995 0.989
ITEM24 0.991 0.988 0.973 0.991 0.986
Covariance Coverage
ITEM23 ITEM24
________ ________
ITEM23 0.997
ITEM24 0.992 0.993
h. Number of missing data patterns. This gives the number of different patterns of missingness present in the variables included in the model. Large numbers of missing data patterns can result in difficulty estimating the model.
i. Covariance Coverage. If any of the variables in the model have missing values, Mplus provides information on the number and distribution of missing values. The covariance coverage matrix gives the proportion of values present for each variable individually (on the diagonal) and pairwise combinations of variables (below the diagonal). For example, 99.4% of cases have non-missing values for item13 and 99.3% of cases have valid values for item13 and item15.
RESULTS FOR EXPLORATORY FACTOR ANALYSIS
EIGENVALUES FOR SAMPLE CORRELATION MATRIXj
1 2 3 4 5
________ ________ ________ ________ ________
1 6.289 1.228 0.709 0.606 0.561
EIGENVALUES FOR SAMPLE CORRELATION MATRIX
6 7 8 9 10
________ ________ ________ ________ ________
1 0.499 0.470 0.384 0.366 0.329
EIGENVALUES FOR SAMPLE CORRELATION MATRIX
11 12
________ ________
1 0.309 0.251
EXPLORATORY FACTOR ANALYSIS WITH 3 FACTOR(S):
TESTS OF MODEL FIT
Chi-Square Test of Model Fitk
Value 137.865
Degrees of Freedom 33
P-Value 0.0000
Chi-Square Test of Model Fit for the Baseline Model
Value 9132.568
Degrees of Freedom 66
P-Value 0.0000
CFI/TLIl
CFI 0.988
TLI 0.977
Loglikelihood
H0 Value -17776.793
H1 Value -17707.861
Information Criteriam
Number of Free Parameters 57
Akaike (AIC) 35667.586
Bayesian (BIC) 35967.596
Sample-Size Adjusted BIC 35786.527
(n* = (n + 2) / 24)
RMSEA (Root Mean Square Error Of Approximation)n
Estimate 0.047
90 Percent C.I. 0.039 0.055
Probability RMSEA <= .05 0.701
SRMR (Standardized Root Mean Square Residual)
Value 0.014
MINIMUM ROTATION FUNCTION VALUE 0.31440
j. Eigenvalues for sample correlation matrix. An eigenvalue is the variance of the factor. In the initial factor solution, the first factor will account for the most variance, the second will account for the next highest amount of variance, and so on.
k. Chi-square test of model fit. Compares the fit of the model to a model with no restrictions (i.e. all variables correlated freely). Chi-square values can be used to test the difference in fit between nested models.
l. Fit indices. The Comparative Fit Index (CFI) and the Tucker Lewis Index (TLI) are measures of model fit. They have a range from 0 to 1 with higher values indicating better fit.
m. Information Criteria. The Akaike information criterion (AIC) and the Bayesian information criterion (BIC, sometimes also called the Schwarz criterion), can also be used to compare models, including non-nested models.
n. RMSEA. The root mean square error of approximation is another measure of model fit. Smaller values indicate better model fit.
GEOMIN ROTATED LOADINGSo
1 2 3
________ ________ ________
ITEM13 0.858 -0.087 0.010
ITEM14 0.832 -0.022 -0.021
ITEM15 0.724 0.085 0.007
ITEM16 0.645 0.129 -0.069
ITEM17 0.515 0.276 0.084
ITEM18 0.091 0.755 0.012
ITEM19 -0.015 0.842 -0.095
ITEM20 0.099 0.559 -0.011
ITEM21 0.221 0.408 0.199
ITEM22 0.000 0.508 0.205
ITEM23 0.123 0.011 0.795
ITEM24 -0.009 -0.008 0.804
GEOMIN FACTOR CORRELATIONSp
1 2 3
________ ________ ________
1 1.000
2 0.591 1.000
3 0.743 0.704 1.000
o. Geomin rotated loadings. The rotated loadings are the linear combination of variables that make up the factor. In addition to the factor loadings, to completely interpret an oblique rotation one needs to take into account both the factor pattern and the factor structure matrices (shown bellow) and the correlations among the factors. Note that orthogonal rotations produce only a single matrix, which gives the correlations between the variable and the factor.
p. Geomin factor correlations. The factor correlations matrix gives the correlations between the factors. For example, the correlation between factor 1 and factor 2 is 0.591.
ESTIMATED RESIDUAL VARIANCESq
ITEM13 ITEM14 ITEM15 ITEM16 ITEM17
________ ________ ________ ________ ________
1 0.332 0.354 0.388 0.543 0.388
ESTIMATED RESIDUAL VARIANCES
ITEM18 ITEM19 ITEM20 ITEM21 ITEM22
________ ________ ________ ________ ________
1 0.325 0.408 0.623 0.459 0.555
ESTIMATED RESIDUAL VARIANCES
ITEM23 ITEM24
________ ________
1 0.194 0.373
q. Estimated residual variances. These are the variances of the observed variables after accounting for all of the variance in the efa model.
Below are the standard errors for the geomin rotated loadings, factor correlations, and estimated residual variances. These values can be used to perform hypothesis tests and estimate confidence intervals.
S.E. GEOMIN ROTATED LOADINGS
1 2 3
________ ________ ________
ITEM13 0.043 0.051 0.031
ITEM14 0.030 0.035 0.040
ITEM15 0.040 0.053 0.039
ITEM16 0.073 0.060 0.099
ITEM17 0.044 0.052 0.061
ITEM18 0.043 0.033 0.029
ITEM19 0.027 0.044 0.059
ITEM20 0.045 0.039 0.040
ITEM21 0.045 0.044 0.051
ITEM22 0.022 0.048 0.057
ITEM23 0.152 0.034 0.180
ITEM24 0.024 0.090 0.104
S.E. GEOMIN FACTOR CORRELATIONS
1 2 3
________ ________ ________
1 0.000
2 0.042 0.000
3 0.054 0.030 0.000
S.E. ESTIMATED RESIDUAL VARIANCES
ITEM13 ITEM14 ITEM15 ITEM16 ITEM17
________ ________ ________ ________ ________
1 0.021 0.021 0.020 0.025 0.018
S.E. ESTIMATED RESIDUAL VARIANCES
ITEM18 ITEM19 ITEM20 ITEM21 ITEM22
________ ________ ________ ________ ________
1 0.021 0.026 0.024 0.020 0.023
S.E. ESTIMATED RESIDUAL VARIANCES
ITEM23 ITEM24
________ ________
1 0.074 0.076
Below are the z-statistics (i.e. estimate/standard error) for the geomin rotated loadings, factor correlations, and estimated residual variances. These values can be compared to a normal distribution to perform hypothesis tests.
Est./S.E. GEOMIN ROTATED LOADINGS
1 2 3
________ ________ ________
ITEM13 20.050 -1.681 0.317
ITEM14 27.757 -0.623 -0.516
ITEM15 18.307 1.618 0.165
ITEM16 8.793 2.157 -0.700
ITEM17 11.680 5.280 1.377
ITEM18 2.136 23.045 0.416
ITEM19 -0.564 19.088 -1.602
ITEM20 2.190 14.343 -0.280
ITEM21 4.966 9.225 3.884
ITEM22 -0.017 10.650 3.567
ITEM23 0.812 0.314 4.426
ITEM24 -0.357 -0.088 7.694
Est./S.E. GEOMIN FACTOR CORRELATIONS
1 2 3
________ ________ ________
1 0.000
2 14.212 0.000
3 13.724 23.775 0.000
Est./S.E. ESTIMATED RESIDUAL VARIANCES
ITEM13 ITEM14 ITEM15 ITEM16 ITEM17
________ ________ ________ ________ ________
1 16.029 16.792 19.200 21.564 21.324
Est./S.E. ESTIMATED RESIDUAL VARIANCES
ITEM18 ITEM19 ITEM20 ITEM21 ITEM22
________ ________ ________ ________ ________
1 15.574 15.812 26.444 23.392 24.618
Est./S.E. ESTIMATED RESIDUAL VARIANCES
ITEM23 ITEM24
________ ________
1 2.610 4.894
FACTOR STRUCTUREr
1 2 3
________ ________ ________
ITEM13 0.815 0.428 0.587
ITEM14 0.803 0.456 0.582
ITEM15 0.779 0.518 0.605
ITEM16 0.670 0.461 0.501
ITEM17 0.740 0.639 0.660
ITEM18 0.546 0.818 0.611
ITEM19 0.412 0.766 0.486
ITEM20 0.421 0.610 0.456
ITEM21 0.610 0.678 0.650
ITEM22 0.452 0.651 0.562
ITEM23 0.720 0.643 0.894
ITEM24 0.584 0.553 0.792
FACTOR DETERMINACIESs
1 2 3
________ ________ ________
1 0.945 0.926 0.935
r. Factor Structure. With an oblique rotation, the factor structure matrix presents the correlations between the variables and the factors. For example, the correlation between item13 and factor 1 is 0.815. As noted above, the factor structure matrix is used along with the factor loadings and factor correlations to interpret the model.
s. Factor Determinacies are the correlations between the estimated factor score and the factor.