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.

*** WARNING^{a}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 observations^{b}1427 Number of dependent variable^{c}12 Number of independent variables 0 Number of continuous latent variables 0 Observed dependent variables^{d}Continuous ITEM13 ITEM14 ITEM15 ITEM16 ITEM17 ITEM18 ITEM19 ITEM20 ITEM21 ITEM22 ITEM23 ITEM24 Estimator^{e}ML Rotation^{f}GEOMIN Row standardization CORRELATION Type of rotation^{g}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 patterns^{h}24 COVARIANCE COVERAGE OF DATA Minimum covariance coverage value 0.100 PROPORTION OF DATA PRESENT Covariance Coverage^{i}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 MATRIX^{j}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 Fit^{k}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/TLI^{l}CFI 0.988 TLI 0.977 Loglikelihood H0 Value -17776.793 H1 Value -17707.861 Information Criteria^{m}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 LOADINGS^{o}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 CORRELATIONS^{p}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 VARIANCES^{q}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 STRUCTURE1 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 DETERMINACIES^{r}^{s}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.