Mplus for Windows: An Introduction and Overview
Alan C. Acock
Department of HDFS
Oregon State University
7/2009
Intro to Mplus—Alan C. Acock
1
Mplus for Windows: An Introduction and Overview
Contents
Section 1: Using Mplus
1.1 Launching Mplus
1.2 Input and Output Windows
1.3 Mplus Command Structure
1.4 Selected Defaults
1.5 Commands
Section 2: Exploratory Factor Analysis
2.1 EFA with Continuous Variables
2.2 Comparing two Solutions
2.3 EFA with Categorical Outcomes
2.4 Selected Results
2.5 Comparing Two Solutions
2.6 Comparison: Categorical & Continuous
Section 3: Confirmatory Factor Analysis
3.1 CFA with Continuous Variables
3.2 Output and Interpretation
3.2.1 Missing value summary
3.2.2 Covariances and correlations
3.2.3 Model Fit
3.2.4 Model result
3.2.5 Residuals
3.2.6 Modification indices
Section 4: EFA as an Alternative to CFA
Section 5: Equality Constaints—Longitudinal CFA
5.1 Programs for testing equality constraints
5.2 Selected output
Section 6: Path Analysis
6.1 Model and programs
6.2 Indirect effects
Section 7: Putting it Together: CFA & SEM
7.1 Program
7.2 Output
7.3 Interpretation and modification indices
Section 8: Putting it Together: EFA & SEM
8.1 Program & model
8.2 Output
Section 9: Summary & Resource
Intro to Mplus—Alan C. Acock
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Section 1: Using Mplus
1.1 Launching Mplus
1.2. Input and Output Windows
 The window shown above is the input window.
 You write Mplus programs in this window to read the data to be analyzed and to
specify your model of interest.
 You then save your Mplus program and select Run Mplus from the Mplus menu
to submit your program to the Mplus engine for processing.
► File
► Open
Open ex1.inp. This is located at c:\Mplus Examples\ex1.inp.
Intro to Mplus—Alan C. Acock
3
We will utilize files from the Mplus manual for many of our examples. These typically
involve simulated data. Sometimes we well assign hypothetical variable names to
make these somewhat realistic. The Manual itself does not provide a substantial
about of explanation of the examples and the specific output so we hope the
discussion of them here will be useful at a later time when you are trying to read the
manuals.
Here is one screen of the data in ex1.dat (The ANALYSIS: command here is
not needed in the current version of Mplus)
 We have labeled missing values with a -9. Easiest to pick one value that will work
for all variables—can be any number or a dot.
 Notice we have one observation, case 13, that has a missing value on all
variables.
 The data happens to be in a fixed format.
 Could be comma delimited, cvs file from Excel.
Intro to Mplus—Alan C. Acock
4
 Could be free format, other formats possible, but more complicated
Here is the confirmatory factor analysis model we are estimating
Intro to Mplus—Alan C. Acock
5
We will explain the program in a moment, but for now we will just run it to see how the
interface works.
► Mplus
► Run Mplus
Or, you can click the Run icon.
Once Mplus has finished processing your command program, it opens an output
window.
 The output window first displays your Mplus program.
 Below the Mplus program are the Mplus model results.
 If there is an error in your Mplus program or you want to modify your Mplus
program in any way (e.g., to fit a different model to the data), you must return to
the input window and you can then modify the previous commands, save the
modified command file, and run Mplus once again to obtain new output.
1.3 Mplus Command Structure
After you have launched Mplus, you may build a command file. There are nine sets of
Mplus commands (ususally only a few of these are used, but some have numerous
subcommands) :
1.
2.
3.
4.
5.
6.
7.
8.
9.
TITLE: (optional unless you want to know what the file is intended to do)
DATA: (required),
VARIABLE: (required),
DEFINE: (some data transformations are available)
SAVEDATA: (used for specialized applications)
ANALYSIS: (for special analyses such as EFA
MODEL: (a series of equations)
OUTPUT: (many options are available)
MONTECARLO: (used for simulations, power analysis)
Rules:
1. All commands (Title, Data, etc.) must begin on a new line.
2. All command names must be followed by a colon.
3. For e.g., Title: Once you enter the colon, the key word becomes blue.
Intro to Mplus—Alan C. Acock
6
4.
5.
6.
7.
Semicolons separate command options—similar to SAS.
The records in the input setup must be no longer than 80 columns.
They can contain upper and/or lower case letters and tabs.
Only variable names are case sensitive. (Y1 and y1 are different variables)
1.4 Selected Defaults
 The current version of Mplus assumes that you either have no missing values or
are using full information maximum likelihood estimation and assuming missing
values are missing at random (MAR)
 Parameters such as loadings can be fixed
o Many loadings are fixed at 0.0 in the CFA models because the item should
not load on the factor.
o There is no path from F1 to y4 in our figure.
 Fixed parameters can be “freed,” meaning you will estimate them.
o We could add a path from to Y4 or
o Let E1 be correlated with E4
 Fixed parameters are required to stay at a specified value, such as 0.0 or 1.0.
 All free parameters are put into a vector and iterations change values of these
free parameters, until the model’s fit is optimal.
 Unless we tell it otherwise, Mplus will fix the first indicator’s loading at 1.0 as the
reference indicator (except for EFA).
o For example, F1y1 and F2y4 have fixed loadings of 1.0 by default.
o One way to change reference indicator is to reorder variables, e.g.
o F1 by y2 y1 y3 makes y2 reference indicator
o Good to pick a strong indicator as the reference indicator—don’t get a
significance test for reference indicator
1.5 Commands
The TITLE command allows you to specify a title that Mplus will print on each page of
the output file.
 This can go on and on for many lines and usually should.
 Everything is a Title until a command name appears at the start of a new line.
 I like to put the file name as the first line of a title.
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,
Intro to Mplus—Alan C. Acock
7




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 in another form (e.g., Stata, SAS, SPSS, or Excel),
you will need to convert it to one of the formats with which Mplus can work before
you read it into Mplus.
SAS and SPSS require you to write a file out an ASCII (plain text) file.
If you have the data in Stata you can use stata2mplus to set things up for you. You
can obtain it using findit stata2mplus
and install the program.
Here is the Stata session:
stata2mplus using "I:\flash\HDFS630\mplus\classnsfh", replace
This creates two files:
 classnsfh.inp that will run a basic analysis in Mplus and
 classnsfh.dat, a comma delimit ASCII file that Mplus can read with all
missing values coded/recoded as -9999.
1,1,1,1,2,1,1,1,1,1,1,1,2,1,1
3,2,2,3,2,3,2,2,2,2,2,2,2,2,2
  
1,1,1,-9999,1,2,2,1,2,2,2,2,2,1,1
3,3,3,3,3,3,3,3,3,3,3,3,3,3,3
2,2,1,2,1,2,2,1,1,2,1,-9999,1,1,1
The DATA command tells Mplus where the data is stored.
 If you store the program and the data in the same folder, you don’t need to
include the path.
 Recommended to make a separte folder for each project such.
 A long file reference can exceed the character limits per line in Mplus.
 Mplus uses all available data by default. If you want to use listwise deletion you
must specify this under the Data command. Listwise = on;
The VARIABLE command names variables.
 These must be in the identical order to the way Stata/SAS/SPSS wrote the data
file. (common mistake)
Intro to Mplus—Alan C. Acock
8
 Mplus variable names may not have more than 8 characters. Change variable
names to be 8 characters or less or you will get error messages.
 Variable names are case sensitive. Must be consistent (common mistake)
The ANALYSIS command tells Mplus what type of analysis to perform.
 Many analysis options are available.
 Some of these such as Type = EFA make additional commands unnecessary.
SECTION 2: Exploratory Factor Analysis
2.1 EFA with Continuous Variables
TITLE:
DATA:
VARIABLE:
ANALYSIS:
!
!
OUTPUT:
efa1.inp
This is an example of an exploratory
factor analysis with continuous factor
indicators
FILE IS "c:\Mplus Examples\efa1.dat";
NAMES ARE y1-y12;
TYPE = EFA 1 4;
ESTIMATOR = ml;
ROTATION = Geomin;
sampstat;
 The Type = EFA 1 4 tells Mplus to perform an exploratory factor analysis.
 The 1 and 4 following the EFA specification tells Mplus to generate all possible
factor solutions between and including 1 and 4.
 The ESTIMATOR = ml option has Mplus use the maximum likelihood estimator
to perform the factor analysis
o This provides 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
o This has an exclamation mark in front of it which makes it green. Anything
green is a comment and is ignored by the program. This subcommand is
not necessary because maximum likelihood estimation is the default.
 Mplus uses the geomin rotation which is oblique as its default. More traditional
rotations such as varimax are available. See help for a listing of options.
 We do not need a MODEL: command because the EFA 1 4 takes care of this.
One useful feature of Mplus is its ability to handle non-normal input data.
Intro to Mplus—Alan C. Acock
9
 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.
o The mlm option provides a mean-adjusted chi-square model test statistic
whereas the
o mlmv option produces a mean and variance adjusted chi-square test of
model fit.
o 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 and MODEL
commands.
 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.
OUTPUT:
sampstat ;
You can use Mplus’ Help menu to get a listing of all the options available for each
command. You might try this to see what OUTPUT options are available.
Mplus produces the
 Sample correlations,
 Root Mean Square Error of Approximation (RMSEA), and the
 Chi-square test of the one, two, three, and four factor models.
 Standard errors and z-tests for loadings and correlations of factors.
As you can see from the results, shown below, the chi-square test for a one factor
solution 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.
Intro to Mplus—Alan C. Acock
10
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. Kline indicates a .08 is
acceptable.
Run the program and interpret the results.
2.2 Comparing two Solutions
You can test whether the adding additional factors significantly improves the fit to the
data.
Model
Model
Model
Model
1
2
3
4
chi-square
chi-square
chi-square
chi-square
(54
(43
(33
(24
degrees
degrees
degrees
degrees
of
of
of
of
freedom) = 1052.089; p < .001
freedom) = 723.022; p < .001
freedom) = 341.268; p < .001
freedom)
25.799; p not sign.
Is model 4 better than model 3.
Model 3 chi-square (33 degrees of freedom) = 341.268
Model 4 chi-square (24 degrees of freedom) = 25.799
Difference chi-square (9 degrees of freedom
315.469; p < .001
This is significant at the .05 level
With Stata, you can get the probability when this is not in a table
. display 1-chi2(df,chi-square)
. display 1-chi2(9,315.469)
0
This is obviously less than .05.
Often you can’t use tables for chi-square because you have lots of degrees of freedom
and tables only show significance levels for relatively few degrees of freedom.
Estimate the model and interpret the results.
2.3. EFA with Categorical Outcomes
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
Intro to Mplus—Alan C. Acock
11
assigned a categorical value of 1.00 and all values above the median assigned a value
of 2.00.
 For categorical variables, Mplus automatically recodes the lowest value to zero
with subsequent values increasing in units of 1.00.
 While the four underlying latent factors remain continuous, the six categorical
observed variables' response values are now ordered dichotomous categories.
You may use the program that appeared in the initial exploratory factor analysis
example, with the following modifications, and the new data file that contains the
categorical variables ex4.2.dat, as shown below.
There are two estimators.
 WLSM (Weighted Least Squares) is very fast and reasonably good.
o You should use this for initial runs.
o Running this on a server used by many students, it ran in 1 sectond.
o This is the default
 MLR (Robust Maximum Liklihood). This is painfully slow, even for a simple and
well behaved example like the one we will estimate.
o Save this till you are almost done
o Use this when you need to test for the number of factors
o This took 18 minutes to run.
o Under the Analysis section you need to specify this estimator as shown
below.
TITLE:
ex4.2.inp
This is an example of an exploratory
factor analysis with categorical factor
indicators
It uses weighted least squares estimation
It computes tetrachoric correlations and does the
Factor analysis on them. The RMSEA and chi-square
Values are reported.
DATA:
FILE IS ex4.2.dat;
VARIABLE:
NAMES ARE u1-u12;
CATEGORICAL ARE u1-u12;
ANALYSIS:
TYPE = EFA 1 4;
Intro to Mplus—Alan C. Acock
12
ESTIMATOR = MLR;
PROCESSORS = 4 ;
You tell Mplus which variables are categorical with the CATEGORICAL subcommand of
the VARIABLE command, like this:
CATEGORICAL ARE u1 – u2 ;
You should also change the ESTIMATOR option for the ANALYSIS command.
 The default estimator for categorical variables is weighted least squares
o With wls it took 2 seconds
o Could use this for preliminary analysis
 I have used MLR, Maximum Likelihood Robust.
o This uses a default 7 integration points and is extremely slow to converge.
o This program took almost 20 minutes for a fairly simple model
o It makes it possible to compare models using a likelihood ratio test.
2.4 Selected results
Mplus begins with a summary of the distribution of the categorical indicators:
Next we get fit statistics for the 1 factor solution
Intro to Mplus—Alan C. Acock
13
2.5 Comparing Two Solutions
If you use Weighted Least Squares (WLSM) with categorical data you get a RMESA to help
compare the models and can do a chi-square test as described at
http://www.statmodel.com/chidiff.shtml
2.6 Comparison of Categorical and Continuous Solutions
One way of evaluating the efficacy of a categorical factor analysis is its ability to reproduce
the factors obtained when the data is continuous. The program ex4.1.inp estimates the
same factors for the 12 items when they are continuous and we can compare the results.
The low loadings on each factor are all low whether we have the continuous variables or
have dichotomized the variables. The high loads are all fairly close matches.
First, here is the result when the variables are continuous:
Intro to Mplus—Alan C. Acock
14
GEOMIN ROTATED LOADINGS
1
2
3
________
________
________
0.637
0.008
0.074
0.808
0.022
-0.005
0.631
-0.042
-0.058
0.027
0.646
-0.002
-0.029
0.760
-0.023
0.010
0.674
0.030
-0.006
0.003
0.734
-0.040
0.002
0.727
0.049
-0.007
0.707
-0.037
0.006
-0.010
0.004
0.013
0.001
0.035
-0.036
0.008
Y1
Y2
Y3
Y4
Y5
Y6
Y7
Y8
Y9
Y10
Y11
Y12
1
2
3
4
GEOMIN FACTOR CORRELATIONS
1
2
________
________
1.000
-0.039
1.000
0.007
0.029
-0.002
-0.121
3
________
1.000
-0.028
4
________
-0.021
0.041
-0.028
-0.018
0.017
-0.012
0.018
-0.016
-0.001
0.692
0.791
0.658
4
________
1.000
And here are the results for the categorical solution:
GEOMIN ROTATED LOADINGS
1
________
U1
0.628
U2
0.938
U3
0.690
U4
0.072
U5
-0.130
U6
0.015
U7
0.026
U8
-0.036
U9
0.017
U10
-0.066
U11
-0.005
U12
0.110
1
2
2
________
-0.004
0.042
-0.073
0.705
0.805
0.602
-0.014
-0.003
0.034
-0.017
0.040
-0.069
GEOMIN FACTOR CORRELATIONS
1
2
________
________
1.000
-0.026
1.000
Intro to Mplus—Alan C. Acock
3
________
0.098
-0.012
-0.019
-0.120
0.018
0.089
0.805
0.720
0.669
0.057
-0.056
0.022
4
________
-0.061
0.070
-0.036
-0.005
0.016
-0.045
0.009
-0.029
0.025
0.654
0.872
0.624
3
________
4
________
15
3
4
Item
1
2
3
4
5
6
7
8
9
10
11
12
0.025
-0.029
F1
(con)
.637
.808
.631
F1
(cat)
.628
.938
.690
0.032
-0.150
1.000
-0.059
F2
(con)
F2
(cat)
.646
.760
.674
.705
.805
.602
1.000
F3
(con)
F3
(cat)
.734
.727
.707
.805
.720
.669
F4
(con)
F4
(cat)
.692
.791
.658
.654
.872
.624
There are several notes worth keeping in mind when you perform exploratory factor analysis
with categorical outcome variables.





Although one or more of the observed variables may be categorical, any latent variables
in the model are assumed to be continuous
The analysis specification and interpretation of the output, e.g., loadings & factor
correlations, is the same whether one, a subset, or all observed variables are categorical.
Categorical observed variables may be dichotomous or ordered categorical outcomes of
more than two levels), but nominal level observed variables with more than two
categories may not be used in the analysis as outcome variables using this strategy.
Sample size requirements are somewhat more stringent than for continuous variables;
typically you want a minimum of 200 cases (preferably more) to perform any analysis
with categorical outcome variables.
Mplus provides standard errors and z-tests for all loadings and correlations.
SECTION 3: Confirmatory Factor Analysis
What if you had an a priori hypothesis that the visual perception (Y1), cubes (Y2), and
lozenges (Y3) variables belonged to a single factor whereas the paragraph (Y4),
sentence (Y5), and word meaning (Y6) variables belonged to a second factor? The
diagram shown below illustrates the model visually.
Intro to Mplus—Alan C. Acock
16
You can test this hypothesized factor structure using confirmatory factor analysis, as
shown in the next section.
The first thing you want to do is look at the correlation matrix:
Y1
Y2
Y3
Y4
Y5
Y6




Y1
1.000
0.524
0.475
-0.004
-0.029
0.023
Y2
Y3
Y4
Y5
Y6
1.000
0.533
-0.032
-0.040
-0.012
1.000
-0.007
-0.048
0.037
1.000
0.431
0.369
1.000
0.419
1.000
Y1-Y3 are highly correlated with each other so they might form a factor
Y4-Y5 are highly correlated with each other so they might form a factor
Y1-Y3 are weakly correlated with each other so there is factor separation
Y2 is slighly more negative correlated with Y4 than is Y1. Y2 is slightly more
negatively correlated with Y5 than is Y1. Y2 is slightly negatively correlated with
Y6 and Y1 is slightly positive correlated.
Intro to Mplus—Alan C. Acock
17


Compare this to the following. Y2 and Y1 have a different pattern with Y4-Y6.
The single correlation between F1 and F2 could not handle this.
The fit will not be very good.
Y1
Y2
Y3
Y4
Y5
Y6
Y1
1.000
0.524
0.475
0.200
-0.100
0.300
Y2
Y3
Y4
Y5
Y6
1.000
0.533
-0.100
0.200
0.100
1.000
-0.007
-0.048
0.037
1.000
0.431
0.369
1.000
0.419
1.000
Consider the following correlation matrix:
Y1
Y2
Y3
Y4
Y5
Y6
Y1
1.000
0.524
0.475
-0.004
-0.029
0.023
Y2
Y3
Y4
Y5
Y6
1.000
0.533
-0.032
-0.040
-0.012
1.000
0.400
-0.048
0.037
1.000
0.431
0.369
1.000
0.419
1.000
Y3 and Y4 are too correlated to be on separate factors.
 Factorial confounding will mean that Y1-Y4 load on F1 and Y3-Y4 load on F2.
 Therefore, Y3 & Y 4 are factorially confounded.


3.1 CFA with Continuous Variables
TITLE:
ex1.inp
CFA with continuous factor indicators
There are Missing values
DATA:
FILE IS
"ex1.dat" ;
VARIABLE:
NAMES ARE y1-y6;
MISSING ARE all (-9) ;
MODEL:
f1 BY y1-y3;
Intro to Mplus—Alan C. Acock
18
!
OUTPUT:
f2 BY y4-y6;
f1 WITH f2;
sampstat standardized residual patterns
mod(3.84);
When Mplus sees EFA it sets up the relationship in a certain way, but in a CFA, Mplus
needs you to provide a MODEL: to tell it how to set up the relationships that you wish
to confirm).
 The model is general in the sense that
o You must define what parameters are estimated;
o All other parameters are assumed to be fixed.
o Fixed parameters are either zero or some value you set.
 Under VARIABLE we have defined what code is used to represent missing
values.
 You do not need an ANALYSIS section, since we use the MODEL section to
specify the model and are going with the default analysis.
o This assumes full information maximum likelihood.
o To do listwise deletion we would specify this in the DATA command
 Open Help
 Under data you see that you would enter
 Listwise = on—make sure you put it under DATA
 The MODEL command allows you to specify the parameters of your model.
o The first line of the MODEL command shown above defines a latent
factor for the first factor.
o The BY keyword (an abbreviation for "measured by") is used to define
the latent variables;
o 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.
o Mplus will fix the loading for the first indicator at 1.0 unless you tell it
otherwise. Put the “best” indicator first.
 Similarly, in the second line of the MODEL: command a latent factor called
verbal has three indicators: Y1, Y2, and Y3. The third line of MODEL:
command uses the WITH keyword to correlate the F1 latent factor with the F2
latent factor.
By

Intro to Mplus—Alan C. Acock
Measured by
19
With

Correlated with
We do not need F1 with F2 because that is the default. If we wanted to
see how the model did with these fixed we would add the line F1 with
F2@0 ;
Finally, the OUTPUT command contains an added keyword, standardize. This
option instructs Mplus to output standardized parameter estimate values in addition to
the default unstandardized values. Selected output from the analysis appears below.
Why is one loading fixed at 1.0?
The default fixes the unstandardized loading of the first item after BY at 1.0
This has to do with model identification.
 In exploratory factor analysis the variance of the factor (latent variable) is
fixed at 1.0 by the program. Given this, the program estimates the loadings.
 With CFA, you need to set a variance for the latent variable because the size
of the loadings are scaled from the size of the variance.
 Setting the variance of the latent variable (factor) at 1.0 solves this problem
with EFA and is an option with CFA and you get standardized loadings. But,
Mplus suggests a more general approach in which you fix one of the loadings
of each latent variable (factor) at 1.0.
Why is this more general?
 One group might be more variable than another.
 We might find that girls not only have higher verbal skills than boys, but that
they are either more homogeneous or more heterogeneous in these skills.
 An intervention that not only improves the mean outcome, but does so in a
way that makes the distribution more homogeneous is preferred.
 In some cases we are interested in the variances of the latent variables as an
important topic and we could not study that if we fixed the variance at 1.0.
Regardless of which item you pick to fix the loading at 1, the standardized solution will
always be the same because that solution rescales the variance of the latent variable
to be 1 and the fully standardized solution also rescales the variance of each indicator
to be 1.
Intro to Mplus—Alan C. Acock
20
We should pick the strongest indicator at 1.0.
 This makes the results less confusing to readers because all of the loadings
will be less than 1.0.
 If you fixed a weak indicator at 1.0, an indicator that was twice as strong
would have a loading of 2.0 and that would be confusing to readers.
 You do not need to fix the loadings at 1, any number will identify the model
equally well.
3.2 Output and Interpret
3.2.1 Missing value summary
SUMMARY OF DATA
Number of patterns
8
SUMMARY OF MISSING DATA PATTERNS
Y1
Y2
Y3
Y4
Y5
Y6
MISSING DATA PATTERNS
1 2 3 4 5 6
x x x x x x
x x x x x x
x x x x x
x x
x
x x x x
x
x
x
x
7
x
x
x
x
x
8
x
x
x
x
x
MISSING DATA PATTERN FREQUENCIES
Pattern
Frequency
Pattern
Frequency
1
473
4
1
2
15
5
1
3
3
6
1
COVARIANCE COVERAGE OF DATA
Minimum covariance coverage value
Y1
Y2
Y3
Y4
Frequency
2
3
0.100
PROPORTION OF DATA PRESENT
Covariance Coverage
Y1
Y2
________
________
0.994
0.990
0.996
0.992
0.994
0.984
0.986
Intro to Mplus—Alan C. Acock
Pattern
7
8
Y3
________
0.998
0.988
Y4
________
Y5
________
0.990
21
Y5
Y6
Y6
0.992
0.960
0.994
0.962
0.996
0.964
0.990
0.960
0.998
0.966
Y2
________
Y3
________
Y4
________
Y5
________
1.968
1.043
-0.064
-0.091
-0.035
1.968
-0.037
-0.133
0.053
Y2
________
Y3
________
1.000
0.530
-0.032
-0.051
-0.019
1.000
-0.019
-0.073
0.029
Covariance Coverage
Y6
________
0.966
3.2.2 Covariances and correlations
Y1
Y2
Y3
Y4
Y5
Y6
Covariances
Y1
________
1.948
1.020
0.930
-0.020
-0.076
0.021
Y6
Covariances
Y6
________
1.684
Y1
Y2
Y3
Y4
Y5
Y6
Correlations
Y1
________
1.000
0.521
0.475
-0.010
-0.042
0.012
Y6
Correlations
Y6
________
1.000
Intro to Mplus—Alan C. Acock
2.070
0.810
0.710
Y4
________
1.000
0.437
0.380
1.664
0.716
Y5
________
1.000
0.428
22
3.2.3 Model Fit
THE MODEL ESTIMATION TERMINATED NORMALLY
TESTS OF MODEL FIT
Chi-Square Test of Model Fit
Value
Degrees of Freedom
P-Value
3.895
8
0.8665
Chi-Square Test of Model Fit for the Baseline Model
Value
589.067
Degrees of Freedom
15
P-Value
0.0000
CFI/TLI
CFI
TLI
Loglikelihood
H0 Value
H1 Value
Information Criteria
Number of Free Parameters
Akaike (AIC)
Bayesian (BIC)
Sample-Size Adjusted BIC
(n* = (n + 2) / 24)
1.000
1.013
-4850.279
-4848.331
19
9738.558
9818.597
9758.290
RMSEA (Root Mean Square Error Of Approximation)
Estimate
0.000
90 Percent C.I.
0.000
Probability RMSEA <= .05
0.995
0.027
SRMR (Standardized Root Mean Square Residual)
Value
0.015
3.2.4 Model result
We are usually interested in the fully standardized results but the unstandardized
results appear first.
MODEL RESULTS--Unstandardized
Intro to Mplus—Alan C. Acock
23
Two-Tailed
P-Value
Estimate
S.E.
Est./S.E.
1.000
1.123
1.019
0.000
0.098
0.088
999.000
11.430
11.532
999.000
0.000
0.000
1.000
1.032
0.869
0.000
0.129
0.105
999.000
7.972
8.316
999.000
0.000
0.000
-0.033
0.053
-0.621
0.534
Intercepts
Y1
Y2
Y3
Y4
Y5
Y6
-0.017
0.030
0.037
-0.022
-0.012
0.066
0.063
0.063
0.063
0.065
0.058
0.059
-0.267
0.478
0.590
-0.336
-0.209
1.120
0.790
0.633
0.555
0.737
0.835
0.263
Variances
F1
F2
0.912
0.786
0.125
0.138
7.308
5.677
0.000
0.000
Residual Variances
Y1
Y2
Y3
Y4
Y5
Y6
1.041
0.803
1.012
1.287
0.861
1.077
0.095
0.100
0.095
0.123
0.112
0.098
10.977
8.044
10.612
10.449
7.664
10.992
0.000
0.000
0.000
0.000
0.000
0.000
F1
BY
Y1
Y2
Y3
F2
BY
Y4
Y5
Y6
F2
WITH
F1
STANDARDIZED MODEL RESULTS
Intro to Mplus—Alan C. Acock
24
STDYX Standardization
Two-Tailed
P-Value
Estimate
S.E.
Est./S.E.
0.683
0.767
0.695
0.035
0.034
0.035
19.573
22.537
20.011
0.000
0.000
0.000
0.616
0.702
0.596
0.046
0.047
0.045
13.498
14.916
13.112
0.000
0.000
0.000
-0.039
0.062
-0.622
0.534
Intercepts
Y1
Y2
Y3
Y4
Y5
Y6
-0.012
0.021
0.026
-0.015
-0.009
0.051
0.045
0.045
0.045
0.045
0.045
0.045
-0.267
0.478
0.590
-0.336
-0.209
1.120
0.790
0.633
0.555
0.737
0.835
0.263
Variances
F1
F2
1.000
1.000
0.000
0.000
999.000
999.000
999.000
999.000
Residual Variances
Y1
Y2
Y3
Y4
Y5
Y6
0.533
0.411
0.517
0.621
0.507
0.645
0.048
0.052
0.048
0.056
0.066
0.054
11.174
7.868
10.699
11.057
7.677
11.887
0.000
0.000
0.000
0.000
0.000
0.000
F1
BY
Y1
Y2
Y3
F2
BY
Y4
Y5
Y6
F2
WITH
F1
STDY Standardization –ommitted—
Intro to Mplus—Alan C. Acock
25
R-SQUARE
Observed
Variable
Y1
Y2
Y3
Y4
Y5
Y6
Estimate
S.E.
Est./S.E.
0.466
0.582
0.483
0.387
0.496
0.369
0.049
0.054
0.050
0.056
0.065
0.055
9.483
10.849
9.700
6.883
7.605
6.747
Two-Tailed
P-Value
0.000
0.000
0.000
0.000
0.000
0.000
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.
 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 StdYX,
standardizes based on latent and observed variables' variances. This
standardized coefficient represents the amount of standardized change in an
outcome variable per standard deviation unit of a predictor variable.
 Finally, the r-square output illustrates the amount of variance accounted for in
the indicators.
3.2.5 Residuals
Mplus output reports a residual for each variance and covariance. To simplify
interpretation, it also reports a z-test (normalized residual for each variance and
covarinace.
Normalized Residuals for Covariances/Correlations/Residual Correlations
Y1
Y2
Y3
Y4
Y5
________
________
________
________
________
Y1
-0.001
Y2
0.000
0.000
Y3
0.007
-0.005
0.000
Y4
0.333
-0.071
0.159
0.000
Y5
-0.290
-0.400
-0.955
-0.027
-0.001
Y6
0.790
0.182
1.180
0.046
-0.006
Intro to Mplus—Alan C. Acock
26
Normalized Residuals for Covariances/Correlations/Residual
Correlations
Y6
________
Y6
0.000
Because there are many tests, it would not make sense to use the 1.96 value as a
significant failure. Still, we should look for a large z-score as an indicator that our
model does not explain some relationship.
3.2.6 Modification indices
Finally, Mplus reports modification indices because we specified mod(3.84).
 The 3.84 corresponds to the .05 level
 Use this with caution, especially on a large sample
 These are perameters we fixed that could improve the fit if they were free.
 We have no path from F1 to y6, for example
 The M.I. is an estimate of how much chi-square for the model would be
reduced if a single parameter is made free—one at a time.
 Nothing would improve the fit of our model.
MODEL MODIFICATION INDICES
Minimum M.I. value for printing the modification index
M.I.
E.P.C.
Std E.P.C.
3.840
StdYX E.P.C.
No modification indices above the minimum value.
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 chisquare 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.
Intro to Mplus—Alan C. Acock
27
SECTION 4: Exploratory Factor Analysis as an Alternative to CFA
Most often, when doing a CFA, a researcher uses modification indexes to modify the matrix by
allowing some fixed parameters to be free.
 We may allow an item load on more than one factor or
 We may allow two items to have correlated errors.
When we change a model this way it is no longer confirmatory, but exploratory. We are
combining the modification indexes with our own judgement to change the model.
 In Mplus 5.1 and EFA alternative was introduced that can challenge CFA.
 The rotation will find the optimal solution and this will be a better fit than we can do
by looking at a few indexes and using our own judgment.
Intro to Mplus—Alan C. Acock
28




However, if the optimal solution makes no sense, then we have a different problem.
With CFA we fix several paths at a value of 0.0.
This results in very clean factors who have a clear meaning.
However, the best guess my be a loading that is not exactly zero.
Suppose you had two latent variables measured for both the husband and wife.
 Alternatively, you might think of these as two latent variables measured for the same
person, but at two times, say one year apart.
 You believe that the first three measures are indicators of the first factor and the
second three are indicators of the second factor.
 However, it is usually unreasonable to assume that all the cross loadings are 0.000.
You expect them to be small, but there is no necessity to say they must be exactly
zero.
Intro to Mplus—Alan C. Acock
29
 Consider andolescents who have three beliefs about the certainty that they will be
caught and three beliefs about the severity of punishment if they are caught. You
measure them at two time points, at age 15 and again at age 17.
Discuss how this is different from a CFA model
What results would support your thinking?
1. The three beliefs about the certainty of being caught (Y1 – Y3) would load strongly on
factors 1 (measured the first year) and the same 3 measured a year later (Y7 – Y9), but
have weak loadings on factors 2 (measured the first year) and 4 (measured a year later).
2. Conversely, the beliefs about the severity of punishment (Y4 – Y6; Y10 – Y12) should load
strongly on factors 2 and 4 but should have relatively weak loadings on factors 1 and 3.
3. The loadings of Y1 – Y6 should be identical to the corresponding loadings of Y7 – Y12
4. The errorst E1 – E6 should be correlated with the corresponding errors in E7 – E12.
Mplus calls this exploratory factor analysis because we are not fixing values at particular
values, but clearly we are putting enormous constraints on the model.
Mplus VERSION 5.1
MUTHEN & MUTHEN
06/30/2008
6:14 PM
INPUT INSTRUCTIONS
TITLE:
DATA:
VARIABLE:
MODEL:
OUTPUT:
example3.inp
this is an example of an EFA
at two timepoints with factor loading
invariance and correlated residuals across time
FILE IS example3.dat;
NAMES ARE y1-y12;
f1-f2 BY y1-y6 (*t1 1);
f3-f4 BY y7-y12 (*t2 1);
f3-f4 WITH f1-f2;
y1-y6 PWITH y7-y12;
TECH1 STANDARDIZED;
Estimator
Rotation
Row standardization
Type of rotation
THE MODEL ESTIMATION TERMINATED NORMALLY
ML
GEOMIN
COVARIANCE
OBLIQUE
TESTS OF MODEL FIT
Chi-Square Test of Model Fit
Intro to Mplus—Alan C. Acock
30
Value
Degrees of Freedom
P-Value
43.990
42
0.3873
Chi-Square Test of Model Fit for the Baseline Model
Value
Degrees of Freedom
P-Value
1265.442
66
0.0000
CFI/TLI
CFI
TLI
0.998
0.997
Loglikelihood
H0 Value
H1 Value
-9396.403
-9374.409
Information Criteria
Number of Free Parameters
Akaike (AIC)
Bayesian (BIC)
Sample-Size Adjusted BIC
(n* = (n + 2) / 24)
48
18888.807
19091.108
18938.753
RMSEA (Root Mean Square Error Of Approximation)
Estimate
90 Percent C.I.
Probability RMSEA <= .05
0.010
0.000
1.000
0.032
SRMR (Standardized Root Mean Square Residual)
Value
0.027
MODEL RESULTS
F1
Two-Tailed
P-Value
Estimate
S.E.
Est./S.E.
0.744
0.896
0.726
0.014
-0.098
0.013
0.062
0.072
0.055
0.039
0.060
0.034
11.959
12.523
13.103
0.370
-1.619
0.398
0.000
0.000
0.000
0.712
0.106
0.690
0.052
-0.016
0.005
0.058
0.053
0.018
0.898
-0.304
0.256
0.369
0.761
0.798
BY
Y1
Y2
Y3
Y4
Y5
Y6
F2
BY
Y1
Y2
Y3
Intro to Mplus—Alan C. Acock
31
Y4
Y5
Y6
F3
0.734
0.908
0.749
0.061
0.072
0.064
12.082
12.626
11.760
0.000
0.000
0.000
0.744
0.896
0.726
0.014
-0.098
0.013
0.062
0.072
0.055
0.039
0.060
0.034
11.959
12.523
13.103
0.370
-1.619
0.398
0.000
0.000
0.000
0.712
0.106
0.690
0.052
-0.016
0.005
0.734
0.908
0.749
0.058
0.053
0.018
0.061
0.072
0.064
0.898
-0.304
0.256
12.082
12.626
11.760
0.369
0.761
0.798
0.000
0.000
0.000
0.414
0.310
0.066
0.067
6.241
4.592
0.000
0.000
0.299
0.289
0.494
0.069
0.070
0.085
4.364
4.116
5.823
0.000
0.000
0.000
0.451
0.069
6.571
0.000
0.397
0.061
6.463
0.000
0.128
0.058
2.220
0.026
Estimate
S.E.
Est./S.E.
0.584
0.707
0.569
0.011
-0.077
0.010
0.043
0.049
0.038
0.030
0.048
0.026
13.475
14.449
15.086
0.370
-1.614
0.398
BY
Y7
Y8
Y9
Y10
Y11
Y12
F4
BY
Y7
Y8
Y9
Y10
Y11
Y12
F3
WITH
F1
F2
F4
WITH
F1
F2
F3
F2
WITH
F1
Y1
WITH
Y7
Y2
WITH
Y8



STANDARDIZED MODEL RESULTS
STDYX Standardization
F1
BY
Y1
Y2
Y3
Y4
Y5
Y6
F2
Two-Tailed
P-Value
0.000
0.000
0.000
0.712
0.107
0.690
BY
Intro to Mplus—Alan C. Acock
32
Y1
Y2
Y3
Y4
Y5
Y6
F3
0.046
0.042
0.014
0.042
0.051
0.043
0.898
-0.304
0.256
13.617
14.120
13.177
0.369
0.761
0.798
0.000
0.000
0.000
0.586
0.728
0.605
0.011
-0.079
0.011
0.046
0.045
0.040
0.030
0.049
0.027
12.690
16.053
15.166
0.370
-1.621
0.398
0.000
0.000
0.000
0.711
0.105
0.691
0.041
-0.013
0.004
0.565
0.730
0.586
0.046
0.043
0.015
0.041
0.054
0.040
0.897
-0.304
0.256
13.651
13.443
14.556
0.369
0.762
0.798
0.000
0.000
0.000
0.403
0.301
0.059
0.063
6.820
4.758
0.000
0.000
BY
Y7
Y8
Y9
Y10
Y11
Y12
F4
BY
Y7
Y8
Y9
Y10
Y11
Y12
F3
WITH
F1
F2

0.041
-0.013
0.004
0.568
0.720
0.570


Beginning Time:
Ending Time:
Elapsed Time:
18:14:23
18:14:24
00:00:01
MUTHEN & MUTHEN
3463 Stoner Ave.
Los Angeles, CA 90066
Tel: (310) 391-9971
Fax: (310) 391-8971
Web: www.StatModel.com
Support: Support@StatModel.com
Copyright (c) 1998-2008 Muthen & Muthen
Section 5: Equality Constraints—Longitudinal CFA
We often need to test for equality constraints:
1. Are items truly interchangeable. Alpha assumes that all items are equally salient to the
concept being measured. That is you weight each item equally with a 1.0 weight. CFA can
extend this and test it:
 tau equivalence—All loadings are constrained to be equal.
Intro to Mplus—Alan C. Acock
33
o Compare fit of this model to a model in which they are unconstrained
 Parallel equavalence. Tau equivalence plus all error terms are equal
o Very hard to achieve and often we can proceed without this condition
2. Compare marital satisfaction of women and men
 Tau equivalence
o Women may weigh emotional support more than men
o Men may weight sexual satisfaction more than women
 If tau equivalence holds the latent variable has the same meaning in both groups.
o Without this equivalence we are compareing apples and oranges. Why compare
means if the concent has a different meaning for each group?
o Men may be more satisfied than women
3. We will focus on longitudinal CFA equivalence as this is most salient to growth models. This
section summaries the example in Brown’s book on CFA
A little algebra.
In regresson we wrote:
Y  a  bX  e
Intro to Mplus—Alan C. Acock
34
We solved for the intercept, a, using
a  M Y  bM X
rearranging this we can say
M Y  a  bM X
If we examine the figure we see that each observed variable, we will call it X, has a similar set
of equations where the latent variable is the predictor. For each X
X   X    e
Where tau is the intercept, lambda the matrix of loadings, ksi is the score on the latent variable
and theta is the error, and the mean of each X will be
M X   X   X
where kappa is the mean of the latent variable. This adds two sets of parameters that are not
shown in the figure.
 Each indicator has an intercept, tau, and
 each latent variable has a mean, kappa.
This adds 10 parameters we need to estimate, 8 intercepts and 2 latent variable means. To
estimate these 10 new parameters we need to
 Include the means along with the covariance matrix
 Make some additional restrictions because we just added 8 known means, but need to
estimate 10 new parameters.
There are two ways of identifying these parameters.
1. We could fix the latent variable means at one time at zero and estimate the latent mean at
the second, third, etc. time
2. We could fix one intercept at each wave at zero.
We will use the second approach and fix the intercept of the first indicator (A1, A5) at zero.
 This scales satisfaction at time 1 to the mean of the first indicator at time 1, A1
Intro to Mplus—Alan C. Acock
35
 This scales satisfaction at time 2 to the mean of the first indicator at time 2, B1
Instead of entering raw data (that would be fine), we will enter a row of means, a row of
standard deviations, and a correlation matrix.
1.500
1.940
1.000
0.736
0.731
0.771
0.685
0.481
0.485
0.508
1.320
2.030
1.450
2.050
1.410
1.990
6.600
2.610
6.420
2.660
6.560
2.590
6.310
2.550
1.000
0.648
0.694
0.512
0.638
0.442
0.469
1.000
0.700
0.496
0.431
0.635
0.453
1.000
0.508
0.449
0.456
0.627
1.000
0.726
0.743
0.759
1.000
0.672
0.689
1.000
0.695
1.000
 We need the means because we are estimating means of latent variables
 We need the standard deviations so Mplus can convert the correlation matrix to a
covariance matrix.
We estimate four models, each of which includes estimating means. The first model estimates
the means imposing the same form for the model at both waves. We are not restricting loadings
to be equal, intercepts to be equal, or errors to be equal. This model doesn’t make a lot of sense
because if at least the loadings aren’t equal, then we are back to comparing apples to oranges.
With unequal loadings, the very meaning of satisfaction changes over time with some indicators
becoming more salient and others less salient. This can be interesting as, for example, sexual
satisfaction may become less central and emotional supportmay become more satisfying in
more mature marriages. (a number of the comments apply to latter models)
5.1 Programs for testing equality constraints
Model 1
TITLE:
!
!
!
DATA:
VARIABLE:
ANALYSIS:
!
MPLUS PROGRAM FOR TIME1-TIME2 MSMT MODEL OF JOB SATISFACTION
This has equality constraints on everything.
This is from Brown's CFA book
Equal form
Equal factor loadings
Equal indicator intercepts
Equal indicator errors
FILE IS FIG7.2.DAT;
TYPE IS MEANS STD CORR; ! INDICATOR MEANS ALSO INPUTTED
! Raw data would work equally well
NOBS ARE 250;
NAMES ARE A1 B1 C1 D1 A2 B2 C2 D2;
ESTIMATOR=ML;
TYPE=MEANSTRUCTURE;
! ANALYSIS OF MEAN STRUCTURE this is the default
Intro to Mplus—Alan C. Acock
36
MODEL:
SATIS1 BY A1 B1 C1 D1;
SATIS2 BY A2 B2 C2 D2;
A1 WITH A2; B1 WITH B2; C1 WITH C2; D1 WITH D2; ! Correlated errors
[A1@0]; [A2@0];
! FIXES THE A INDICATOR INTERCEPTS TO ZERO
[SATIS1*]; [SATIS2*]; ! FREELY ESTIMATES FACTOR MEANS
!
[B1 B2] (4); [C1 C2] (5); [D1 D2] (6); ! Equal intercepts
!
A1 A2 (7); B1 B2 (8); C1 C2 (9); D1 D2 (10); ! Equal errors
OUTPUT:
SAMPSTAT MODINDICES(4.00) STAND RESIDUAL;
! Notes: The Model command uses numbers to create equality constraints.
! parameters followed by (1) are equal; (2) are equal, etc. Thus,
! C! equals C2 because both share a (2) and D1 = D2 because both have (3)
! The first line is confusing. The first variable is fixed at 1.0 by
! Default. Hence A1 = A2 = 1, but the (1) does not apply to either A1 or A2
! Things in [] are either means or intercepts depending on context
! A1 with A2 means correlate the errors
! A1 A2 (7) means equal error terms for A1 and A2
Next, we estimate the model imposing equal factor loadings. I consider this the minimum
equality constraint to meaningfull comparison of means. Others would disagree with many want
more constraints and with Muthén okay with what he calls “partial” invariance.
Model 2
TITLE:
MPLUS PROGRAM FOR TIME1-TIME2 MSMT MODEL OF JOB SATISFACTION
This has equality constraints on everything.
This is from Brown's CFA book
Equal form
Equal factor loadings
!
Equal indicator intercepts
!
Equal indicator errors
DATA:
FILE IS FIG7.2.DAT;
TYPE IS MEANS STD CORR; ! INDICATOR MEANS ALSO INPUTTED
! Raw data would work equally well
NOBS ARE 250;
VARIABLE:
NAMES ARE A1 B1 C1 D1 A2 B2 C2 D2;
ANALYSIS:
ESTIMATOR=ML;
!
TYPE=MEANSTRUCTURE;
! ANALYSIS OF MEAN STRUCTURE this is the default
MODEL:
SATIS1 BY A1 B1 (1);
C1 (2);
D1 (3);
SATIS2 BY A2 B2 (1)
C2 (2);
D2 (3);
A1 WITH A2; B1 WITH B2; C1 WITH C2; D1 WITH D2; ! Correlated errors
[A1@0]; [A2@0];
! FIXES THE A INDICATOR INTERCEPTS TO ZERO
[SATIS1*]; [SATIS2*]; ! FREELY ESTIMATES FACTOR MEANS
!
[B1 B2] (4); [C1 C2] (5); [D1 D2] (6); ! Equal intercepts
!
A1 A2 (7); B1 B2 (8); C1 C2 (9); D1 D2 (10); ! Equal errors
OUTPUT:
SAMPSTAT MODINDICES(4.00) STAND RESIDUAL;
! Notes: The Model command uses numbers to create equality constraints.
! parameters followed by (1) are equal; (2) are equal, etc. Thus,
! C! equals C2 because both share a (2) and D1 = D2 because both have (3)
! The first line is confusing. The first variable is fixed at 1.0 by
Intro to Mplus—Alan C. Acock
37
!
!
!
!
Default. Hence A1 = A2 = 1, but the (1) does not apply to either A1 or A2
Things in [] are either means or intercepts depending on context
A1 with A2 means correlate the errors
A1 A2 (7) means equal error terms for A1 and A2
The third model imposes equal intercepts:
TITLE:
MPLUS PROGRAM FOR TIME1-TIME2 MSMT MODEL OF JOB SATISFACTION
This has equality constraints on everything.
This is from Brown's CFA book
Equal form
Equal factor loadings
Equal indicator intercepts
!
Equal indicator errors
DATA:
FILE IS FIG7.2.DAT;
TYPE IS MEANS STD CORR; ! INDICATOR MEANS ALSO INPUTTED
! Raw data would work equally well
NOBS ARE 250;
VARIABLE:
NAMES ARE A1 B1 C1 D1 A2 B2 C2 D2;
ANALYSIS:
ESTIMATOR=ML;
!
TYPE=MEANSTRUCTURE;
! ANALYSIS OF MEAN STRUCTURE
MODEL:
SATIS1 BY A1 B1 (1)
C1 (2)
D1 (3);
SATIS2 BY A2 B2 (1)
C2 (2)
D2 (3);
A1 WITH A2; B1 WITH B2; C1 WITH C2; D1 WITH D2; ! Correlated errors
[A1@0]; [A2@0];
! FIXES THE A INDICATOR INTERCEPTS TO ZERO
[SATIS1*]; [SATIS2*]; ! FREELY ESTIMATES FACTOR MEANS
[B1 B2] (4); [C1 C2] (5); [D1 D2] (6); ! Equal intercepts
!
A1 A2 (7); B1 B2 (8); C1 C2 (9); D1 D2 (10); ! Equal errors
OUTPUT:
SAMPSTAT MODINDICES(4.00) STAND RESIDUAL;
! Notes: The Model command uses numbers to create equality constraints.
! parameters followed by (1) are equal; (2) are equal, etc. Thus,
! C! equals C2 because both share a (2) and D1 = D2 because both have (3)
! The first line is confusing. The first variable is fixed at 1.0 by
! Default. Hence A1 = A2 = 1, but the (1) does not apply to either A1 or A2
! Things in [] are either means or intercepts depending on context
! A1 with A2 means correlate the errors
! A1 A2 (7) means equal error terms for A1 and A2
The fourth model adds the final constraint of equal errors on the indicator variables. This is an
extreme level of invariance.
TITLE:
DATA:
MPLUS PROGRAM FOR TIME1-TIME2 MSMT MODEL OF JOB SATISFACTION
This has equality constraints on everything.
This is from Brown's CFA book
Equal form
Equal factor loadings
Equal indicator intercepts
Equal indicator errors
FILE IS FIG7.2.DAT;
Intro to Mplus—Alan C. Acock
38
TYPE IS MEANS STD CORR; ! INDICATOR MEANS ALSO INPUTTED
! Raw data would work equally well
NOBS ARE 250;
VARIABLE:
NAMES ARE A1 B1 C1 D1 A2 B2 C2 D2;
ANALYSIS:
ESTIMATOR=ML;
!
TYPE=MEANSTRUCTURE;
! ANALYSIS OF MEAN STRUCTURE
MODEL:
SATIS1 BY A1 B1 (1)
C1 (2)
D1 (3);
SATIS2 BY A2 B2 (1)
C2 (2)
D2 (3);
A1 WITH A2; B1 WITH B2; C1 WITH C2; D1 WITH D2; ! Correlated errors
[A1@0]; [A2@0];
! FIXES THE A INDICATOR INTERCEPTS TO ZERO
[SATIS1*]; [SATIS2*]; ! FREELY ESTIMATES FACTOR MEANS
[B1 B2] (4); [C1 C2] (5); [D1 D2] (6); ! Equal intercepts
A1 A2 (7); B1 B2 (8); C1 C2 (9); D1 D2 (10); ! Equal errors
OUTPUT:
SAMPSTAT MODINDICES(4.00) STAND RESIDUAL;
! Notes: The Model command uses numbers to create equality constraints.
! parameters followed by (1) are equal; (2) are equal, etc. Thus,
! C! equals C2 because both share a (2) and D1 = D2 because both have (3)
! The first line is confusing. The first variable is fixed at 1.0 by
! Default. Hence A1 = A2 = 1, but the (1) does not apply to either A1 or A2
! Things in [] are either means or intercepts depending on context
! A1 with A2 means correlate the errors
! A1 A2 (7) means equal error terms for A1 and A2
We can summaries these as follows:
2
Model
Equal form
Equal factor loadings
Equal intercepts
Equal error variances
2.09
3.88
7.25
90.73***
2 diff
Df
15
18
21
25
 df
1.79
3
3.37
3
83.48*** 4
RMSEA
.000
.000
.000
.103
CFI
TLI
SRMR
1.00
1.00
1.00
.96
1.01
1.01
1.01
.96
.010
.014
.026
.037
We cannot go all the way to equal indicator error variances, but we can go all the way to equal
indicator intercepts before chi-square increases significantly. Here are selected results for the
equal indicator intercepts model:
MODEL RESULTS
SATIS1
A1
B1
C1
D1
BY
SATIS2
A2
BY
Two-Tailed
P-Value
Estimate
S.E.
Est./S.E.
1.000
0.989
0.993
0.962
0.000
0.017
0.017
0.016
999.000
56.699
60.078
60.427
999.000
0.000
0.000
0.000
1.000
0.000
999.000
999.000
Intro to Mplus—Alan C. Acock
39
B2
C2
D2
0.989
0.993
0.962
0.017
0.017
0.016
56.699
60.078
60.427
0.000
0.000
0.000
2.547
0.321
7.923
0.000
0.723
0.117
6.187
0.000
1.023
0.162
6.298
0.000
1.031
0.158
6.515
0.000
D2
0.785
0.133
5.918
0.000
Means
SATIS1
SATIS2
1.500
6.617
0.121
0.160
12.421
41.422
0.000
-0.156
-0.032
-0.039
0.000
-0.156
-0.032
-0.039
0.000
0.098
0.099
0.089
0.000
0.098
0.099
0.089
999.000
-1.583
-0.327
-0.436
999.000
-1.583
-0.327
-0.436
999.000
0.113
0.743
0.663
999.000
0.113
0.743
0.663
2.936
5.013
0.291
0.499
10.099
10.050
0.000
0.000
SATIS2
WITH
SATIS1
A1
WITH
A2
B1
WITH
B2
C1
WITH
C2
D1
WITH
Intercepts
A1
B1
C1
D1
A2
B2
C2
D2
Variances
SATIS1
SATIS2
0.000 Note, these = M of A1 & A2
0.000
Residual Variances
A1
0.711
0.101
7.060
0.000
B1
1.391
0.152
9.149
0.000
C1
1.427
0.155
9.185
0.000
D1
1.070
0.124
8.655
0.000
A2
1.428
0.188
7.609
0.000
B2
2.363
0.260
9.082
0.000
C2
2.066
0.236
8.753
0.000
D2
1.839
0.214
8.610
0.000
Why would wave 2 have bigger error variances? This should be explored
STANDARDIZED MODEL RESULTS
STDYX Standardization
Estimate
SATIS1
S.E.
Est./S.E.
Two-Tailed
P-Value
BY
Intro to Mplus—Alan C. Acock
40
A1
B1
C1
D1
0.897
0.821
0.818
0.847
0.016
0.020
0.020
0.019
57.727
40.566
40.304
45.498
0.000
0.000
0.000
0.000
0.882
0.822
0.840
0.846
0.016
0.020
0.019
0.019
53.469
40.444
43.777
45.040
0.000 that all the item
0.000 variances are equal.
0.000
0.000
0.664
0.039
17.084
0.000
0.717
0.050
14.314
0.000
0.564
0.052
10.827
0.000
0.601
0.050
12.023
0.000
D2
0.560
0.055
10.149
0.000
Means
SATIS1
SATIS2
0.876
2.955
0.083
0.161
10.584
18.373
0.000
0.000
0.000
-0.075
-0.016
-0.020
0.000
-0.058
-0.012
-0.015
0.000
0.048
0.048
0.046
0.000
0.036
0.037
0.035
999.000
-1.588
-0.328
-0.436
999.000
-1.587
-0.328
-0.436
999.000
0.112
0.743
0.663
999.000
0.112
0.743
0.663
Variances
SATIS1
SATIS2
1.000
1.000
0.000
0.000
999.000
999.000
999.000
999.000
Residual Variances
A1
B1
C1
D1
A2
B2
C2
D2
0.195
0.326
0.330
0.282
0.222
0.325
0.295
0.284
0.028
0.033
0.033
0.032
0.029
0.033
0.032
0.032
6.995
9.822
9.929
8.955
7.615
9.741
9.145
8.918
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
SATIS2
A2
B2
C2
D2
Equality only meaningful
for the unstandardized
solution. These will also
be equal only in the case
BY
SATIS2
WITH
SATIS1
A1
WITH
A2
B1
WITH
B2
C1
WITH
C2
D1
WITH
Intercepts
A1
B1
C1
D1
A2
B2
C2
D2
Intro to Mplus—Alan C. Acock
41
SECTION 6: Path Analysis
6.1 Model and Program
TITLE:
DATA:
VARIABLE:
MODEL:
ex3.11
This is an example of a path analysis
with continuous dependent variables
FILE IS ex3.11.dat;
NAMES ARE y1-y3 x1-x3;
y1 y2 ON x1 x2 x3;
y3 ON y1 y2 x2;
MODEL
y2
y2
y2
y3
y3
Intro to Mplus—Alan C. Acock
indirect:
ind x1;
ind x2;
ind x3;
ind x1;
ind x2;
42
OUTPUT:
y3 ind x3;
standardized mod(3.84);
6.2. Indirect Effects
The MODEL INDIRECT: subcommand estimates indirect effects for you
 You get the Total indirect effect that combines as many specific indirect
effects as there are in the model
 Specific indirect effects of x1 go y3 include
o x1  y1  y3
o x1  y2  y3
 Tests of significant for both specific and total indirect effects
Estimate and interpret the output:
SECTION 7: Putting it Together—CFA & SEM
Interpret the figure.
Notice indirect effects.
Intro to Mplus—Alan C. Acock
43
7.1 Program
TITLE:
DATA:
VARIABLE:
MODEL:
OUTPUT:
example2cfa
This is an example of a SEM with
CFA factors with continuous factor indicators
And Indirect Effects
FILE IS example2.dat;
NAMES ARE y1-y12;
f1 BY y1-y3;
f2 by y4-y6;
f3 by y7-y9;
f4 BY y10-y12;
f3 ON f1-f2;
f4 ON f3;
MODEL INDIRECT:
f4 ind f1;
f4 ind f2;
standardized mod(3.84)
7.2 Output
Mplus VERSION 5.1
MUTHEN & MUTHEN
06/30/2008
8:12 PM
SUMMARY OF ANALYSIS
Number
Number
Number
Number
Number
of
of
of
of
of
groups
observations
dependent variables
independent variables
continuous latent variables
1
500
12
0
4
Observed dependent variables
Continuous
Y1
Y2
Y3
Y7
Y8
Y9
Y4
Y10
Continuous latent variables
F1
F2
F3
F4
Estimator
Information matrix
Maximum number of iterations
Intro to Mplus—Alan C. Acock
Y5
Y11
Y6
Y12
ML
OBSERVED
1000
44
Convergence criterion
Maximum number of steepest descent iterations
0.500D-04
20
TESTS OF MODEL FIT
Chi-Square Test of Model Fit
Value
53.492
Degrees of Freedom
50
P-Value
0.3417
Chi-Square Test of Model Fit for the Baseline Model
Value
4600.240
Degrees of Freedom
66
P-Value
0.0000
CFI/TLI
CFI
0.999
TLI
0.999
Loglikelihood
H0 Value
-6483.831
H1 Value
-6457.085
Information Criteria
Number of Free Parameters
40
Akaike (AIC)
13047.662
Bayesian (BIC)
13216.247
Sample-Size Adjusted BIC
13089.284
(n* = (n + 2) / 24)
RMSEA (Root Mean Square Error Of Approximation)
Estimate
0.012
90 Percent C.I.
0.000 0.032
Probability RMSEA <= .05
1.000
SRMR (Standardized Root Mean Square Residual)
Value
0.019
MODEL RESULTS
F1
Two-Tailed
P-Value
Estimate
S.E.
Est./S.E.
1.000
1.103
0.942
0.000
0.062
0.058
999.000
17.881
16.346
999.000
0.000
0.000
1.000
1.006
1.023
0.000
0.057
0.060
999.000
17.691
17.064
999.000
0.000
0.000
1.000
0.894
0.902
0.000
0.021
0.021
999.000
41.937
42.479
999.000
0.000
0.000
BY
Y1
Y2
Y3
F2
BY
Y4
Y5
Y6
F3
BY
Y7
Y8
Y9
Intro to Mplus—Alan C. Acock
45
F4
BY
Y10
Y11
Y12
F3
1.000
0.734
0.684
0.000
0.028
0.028
999.000
26.424
24.405
999.000
0.000
0.000
0.640
0.912
0.069
0.074
9.271
12.399
0.000
0.000
0.546
0.032
16.975
0.000
0.297
0.038
7.767
0.000
0.599
0.618
0.061
0.064
9.766
9.717
0.000
0.000
0.367
0.296
0.412
0.400
0.340
0.392
0.183
0.191
0.181
0.240
0.183
0.213
0.525
0.565
0.033
0.033
0.033
0.034
0.031
0.034
0.019
0.017
0.017
0.027
0.017
0.018
0.049
0.049
11.044
8.946
12.309
11.640
10.888
11.370
9.799
11.268
10.812
8.746
10.791
11.998
10.636
11.488
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
Estimate
S.E.
Est./S.E.
Two-Tailed
P-Value
0.787
0.843
0.751
0.023
0.020
0.025
34.084
41.362
30.614
0.000
0.000
0.000
0.779
0.805
0.789
0.023
0.021
0.022
34.055
37.480
35.305
0.000
0.000
0.000
0.948
0.006
153.038
0.000
ON
F1
F2
F4
ON
F3
F2
WITH
F1
Variances
F1
F2
Residual Variances
Y1
Y2
Y3
Y4
Y5
Y6
Y7
Y8
Y9
Y10
Y11
Y12
F3
F4
STANDARDIZED MODEL RESULTS
STDYX Standardization
F1
BY
Y1
Y2
Y3
F2
BY
Y4
Y5
Y6
F3
BY
Y7
Intro to Mplus—Alan C. Acock
46
Y8
Y9
F4
0.007
0.007
131.246
136.242
0.000
0.000
0.902
0.869
0.835
0.013
0.014
0.017
70.200
59.982
50.003
0.000
0.000
0.000
0.388
0.561
0.039
0.036
10.057
15.610
0.000
0.000
0.680
0.027
24.795
0.000
0.488
0.043
11.343
0.000
1.000
1.000
0.000
0.000
999.000
999.000
999.000
999.000
0.380
0.289
0.437
0.393
0.352
0.378
0.101
0.128
0.120
0.186
0.244
0.302
0.322
0.538
0.036
0.034
0.037
0.036
0.035
0.035
0.012
0.013
0.013
0.023
0.025
0.028
0.031
0.037
10.450
8.397
11.862
11.018
10.197
10.713
8.572
9.598
9.287
8.010
9.698
10.828
10.421
14.433
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
BY
Y10
Y11
Y12
F3
ON
F1
F2
F4
ON
F3
F2
0.934
0.938
WITH
F1
Variances
F1
F2
Residual Variances
Y1
Y2
Y3
Y4
Y5
Y6
Y7
Y8
Y9
Y10
Y11
Y12
F3
F4
TOTAL, TOTAL INDIRECT, SPECIFIC INDIRECT, AND DIRECT EFFECTS
STANDARDIZED TOTAL, TOTAL INDIRECT, SPECIFIC INDIRECT, AND DIRECT EFFECTS
STDYX Standardization
Two-Tailed
Estimate
S.E. Est./S.E.
P-Value
Effects from F1 to F4
Total
Total indirect
Specific indirect
F4
F3
F1
Effects from F2 to F4
Total
Intro to Mplus—Alan C. Acock
0.263
0.263
0.028
0.028
9.269
9.269
0.000
0.000
0.263
0.028
9.269
0.000
0.382
0.029
12.999
0.000
47
Total indirect
Specific indirect
F4
F3
F2
0.382
0.029
12.999
0.000
0.382
0.029
12.999
0.000
MODEL MODIFICATION INDICES
Minimum M.I. value for printing the modification index
M.I.
E.P.C. Std E.P.C.
BY Statements
F3
BY Y1
5.980
0.103
0.131
WITH Statements
Y3
WITH Y2
6.126
0.091
0.091
Y4
WITH Y3
5.405
0.053
0.053
Y5
WITH Y3
5.265
-0.049
-0.049
Y8
WITH Y2
5.695
-0.037
-0.037
Y9
WITH Y6
4.801
0.035
0.035
Beginning Time:
Ending Time:
Elapsed Time:
3.840
StdYX E.P.C.
0.134
0.260
0.130
-0.132
-0.155
0.133
20:12:48
20:12:49
00:00:01
7.3 Interpretation of modification indices
 We could reduce Chi-square, which now is Chi-square(50) = 53.492, by about
5.265 if we allowed the error term for Y5 to be correlated with the error term
for Y3.
 The correlation of the two errors would be about -.132—does this make
sense?
 We would do these one at a time
 We would only do it if it made sense. Say Y5 and Y3 are pen and pencil tests
and all the others are face to face interviews. There might be a method effect
that we could incorporate as an error term
 We might not have much to gain even if there is a big modification index if the
fit is already good.
 New Chi-square would be approximately Chi-square(49) = 53.492 – 5.265. A
reduction in Chi-square of 5.265 with one degree of freedom would be highly
significant. Not much need to improve on a CFI = .997; RMSEA = .012;
Intro to Mplus—Alan C. Acock
48
SECTION 8: Putting it Together—EFA & SEM
We may have a situation where we are sufficiently confident to have F3 and F4
represented by a CFA model, but not that confident about F1 and F2 for which we want
to do an EFA.
8.1 Program & model
Here are the program and results:
The (*1) in the Model line for f1-f2 by y1-y6 (*1); is included so Mplus knows
this is an EFA set. We expect y1-y3 to have strong loadings on f1 and weak loadings
on f2. We expect y4-y6 to have weak loadings on f1 and strong loadings on f2. Still,
we are not sufficiently confident of this to impose the restriction that these loadings are
exactly 0.000.
Intro to Mplus—Alan C. Acock
49
8.2 Output
TITLE:
DATA:
VARIABLE:
MODEL:
OUTPUT:
example2.inp
This is an example of a SEM with
EFA and CFA factors with continuous
factor indicators
FILE IS example2.dat;
NAMES ARE y1-y12;
f1-f2 BY y1-y6 (*1);
f3 BY y7-y9;
f4 BY y10-y12;
f3 ON f1-f2;
f4 ON f3;
MODEL INDIRECT:
f4 ind f1;
f4 ind f2;
Standardized mod(3.84)
Mplus VERSION 5.1
MUTHEN & MUTHEN
06/30/2008
8:32 PM
TESTS OF MODEL FIT
Chi-Square Test of Model Fit
Value
51.353
Degrees of Freedom
46
P-Value
0.2720
Chi-Square Test of Model Fit for the Baseline Model
Value
4600.240
Degrees of Freedom
66
P-Value
0.0000
CFI/TLI
CFI
0.999
TLI
0.998
Loglikelihood
H0 Value
-6482.762
H1 Value
-6457.085
Information Criteria
Number of Free Parameters
44
Akaike (AIC)
13053.524
Bayesian (BIC)
13238.966
Sample-Size Adjusted BIC
13099.308
(n* = (n + 2) / 24)
RMSEA (Root Mean Square Error Of Approximation)
Estimate
0.015
90 Percent C.I.
0.000 0.034
Probability RMSEA <= .05
1.000
Intro to Mplus—Alan C. Acock
50
SRMR (Standardized Root Mean Square Residual)
Value
0.018
MODEL RESULTS
F1
S.E.
Est./S.E.
0.751
0.858
0.736
0.036
-0.028
0.002
0.048
0.042
0.045
0.051
0.049
0.004
15.608
20.467
16.353
0.711
-0.568
0.627
0.000
0.000
0.000
0.477
0.570
0.530
0.034
-0.002
-0.008
0.763
0.810
0.802
0.045
0.016
0.035
0.050
0.048
0.041
0.755
-0.150
-0.220
15.367
16.837
19.461
0.450
0.881
0.826
0.000
0.000
0.000
1.000
0.894
0.902
0.000
0.021
0.021
999.000
41.937
42.479
999.000
0.000
0.000
1.000
0.734
0.684
0.000
0.028
0.028
999.000
26.424
24.405
999.000
0.000
0.000
0.493
0.721
0.058
0.057
8.461
12.752
0.000
0.000
0.546
0.032
16.975
0.000
0.479
0.053
9.094
0.000
1.000
1.000
0.000
0.000
999.000
999.000
999.000
999.000
0.376
0.290
0.406
0.408
0.329
0.393
0.183
0.191
0.034
0.035
0.034
0.035
0.033
0.035
0.019
0.017
11.064
8.239
11.817
11.742
10.046
11.073
9.796
11.269
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
BY
Y1
Y2
Y3
Y4
Y5
Y6
F2
BY
Y1
Y2
Y3
Y4
Y5
Y6
F3
BY
Y7
Y8
Y9
F4
BY
Y10
Y11
Y12
F3
ON
F1
F2
F4
ON
F3
F2
Two-Tailed
P-Value
Estimate
WITH
F1
Variances
F1
F2
Residual Variances
Y1
Y2
Y3
Y4
Y5
Y6
Y7
Y8
Intro to Mplus—Alan C. Acock
51
Y9
Y10
Y11
Y12
F3
F4
0.181
0.240
0.183
0.213
0.527
0.565
0.017
0.027
0.017
0.018
0.049
0.049
10.812
8.746
10.791
11.998
10.644
11.488
0.000
0.000
0.000
0.000
0.000
0.000
Estimate
S.E.
Est./S.E.
0.764
0.848
0.758
0.036
-0.028
0.002
0.037
0.024
0.033
0.051
0.050
0.003
20.741
34.915
23.068
0.711
-0.568
0.627
0.000
0.000
0.000
0.477
0.570
0.530
0.034
-0.002
-0.008
0.756
0.825
0.787
0.046
0.015
0.036
0.037
0.035
0.023
0.755
-0.150
-0.220
20.282
23.257
33.668
0.450
0.881
0.826
0.000
0.000
0.000
0.948
0.934
0.938
0.006
0.007
0.007
153.043
131.230
136.226
0.000
0.000
0.000
0.902
0.869
0.835
0.013
0.014
0.017
70.200
59.982
50.002
0.000
0.000
0.000
0.386
0.565
0.043
0.038
8.914
14.919
0.000
0.000
0.680
0.027
24.796
0.000
0.479
0.053
9.094
0.000
0.008
0.031
0.007
0.074
0.071
0.045
0.045
0.045
0.045
0.045
0.183
0.688
0.146
1.657
1.590
0.855
0.491
0.884
0.098
0.112
STANDARDIZED MODEL RESULTS
STDYX Standardization
F1
Two-Tailed
P-Value
BY
Y1
Y2
Y3
Y4
Y5
Y6
F2
BY
Y1
Y2
Y3
Y4
Y5
Y6
F3
BY
Y7
Y8
Y9
F4
BY
Y10
Y11
Y12
F3
ON
F1
F2
F4
ON
F3
F2
WITH
F1
Intercepts
Y1
Y2
Y3
Y4
Y5
Intro to Mplus—Alan C. Acock
52
Y6
Y7
Y8
Y9
Y10
Y11
Y12
Variances
F1
F2
Residual Variances
Y1
Y2
Y3
Y4
Y5
Y6
Y7
Y8
Y9
Y10
Y11
Y12
F3
F4
R-SQUARE
Latent
Variable
F3
F4
0.068
0.044
0.050
0.056
0.008
0.028
0.025
0.045
0.045
0.045
0.045
0.045
0.045
0.045
1.528
0.983
1.115
1.252
0.170
0.616
0.554
0.126
0.326
0.265
0.211
0.865
0.538
0.580
1.000
1.000
0.000
0.000
999.000
999.000
999.000
999.000
0.390
0.283
0.431
0.401
0.341
0.378
0.101
0.128
0.120
0.186
0.244
0.302
0.323
0.538
0.037
0.036
0.038
0.036
0.036
0.036
0.012
0.013
0.013
0.023
0.025
0.028
0.031
0.037
10.510
7.793
11.385
11.149
9.459
10.467
8.570
9.599
9.287
8.009
9.698
10.828
10.432
14.433
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
Estimate
0.677
0.462
S.E.
0.031
0.037
Est./S.E.
21.887
12.398
Two-Tailed
P-Value
0.000
0.000
STANDARDIZED TOTAL, TOTAL INDIRECT, SPECIFIC INDIRECT, AND DIRECT EFFECTS
STDYX Standardization
Two-Tailed
Estimate
S.E. Est./S.E.
P-Value
Effects from F1 to F4
Total
0.263
0.031
8.353
0.000
Total indirect
0.263
0.031
8.353
0.000
Specific indirect
F4
F3
F1
0.263
0.031
8.353
0.000
Effects from F2 to F4
Total
Intro to Mplus—Alan C. Acock
0.384
0.030
12.590
0.000
53
Total indirect
Specific indirect
F4
F3
F2
0.384
0.030
12.590
0.000
0.384
0.030
12.590
0.000
MODEL MODIFICATION INDICES
Minimum M.I. value for printing the modification index
M.I.
E.P.C. Std E.P.C.
BY Statements
F3
BY Y1
6.537
0.144
0.184
WITH Statements
Y3
WITH Y2
5.262
0.115
0.115
Y4
WITH Y1
4.954
-0.052
-0.052
Y4
WITH Y3
5.288
0.055
0.055
Y5
WITH Y3
4.367
-0.049
-0.049
Y8
WITH Y2
5.716
-0.037
-0.037
Y9
WITH Y6
4.853
0.036
0.036
Beginning Time:
Ending Time:
Elapsed Time:
3.840
StdYX E.P.C.
0.187
0.336
-0.133
0.135
-0.133
-0.157
0.133
20:32:33
20:32:33
00:00:00
Section 9: Summary & Resources
This provides a brief introduction to Mplus. We have not covered any of the
statistical theory underlying Mplus, but this should be enough for you to read the
Manual and follow more complex explications of Mplus and SEM.
Key things to remember:
1. BY means measured by and is the path (loading) between latent variables
and their indicators.
2. ON is the structural path between variables. In last example, F4 depends ON
F3, F3 depends ON both F1 and F2.
3. WITH means correlated with. Two uses include:
a. For exogenous variables WITH means the exogenous variables are
correlated. In last example, F1 is correlated WITH F2.
b. For indicators WITH means the errors/residuals are correlated. In last
examples, the modification indices suggest we might correlate the error
for Y3 WITH the error for Y5.
Intro to Mplus—Alan C. Acock
54
Additional introductory content is available at:
http://www.ats.ucla.edu/stat/mplus/
The Mplus webpage has a wealth of support that includes articles applying Mplus that
serve as models and extensive on line videos
http://www.statmodel.com
A very thorough, but gentle discussion of CFA (including sample progrms) is
Brown, Timothy A. (2006). Confirmatory Factor Analysis for Applied Research. N.Y.:
Guilford
A gentle introduction to SEM is
Kline, Rex B. (2005). Principles and practice of structural equation modeling (2nd ed.).
N.Y.: Guildford Press.
Intro to Mplus—Alan C. Acock
55