Mplus for Windows: An Introduction
This document introduces you to Mplus for Windows. It is primarily
aimed at first time users of Mplus who have prior experience
with either exploratory factor analysis (EFA), or confirmatory
factor analysis (CFA) and structural equation modeling
(SEM). The document is organized into six sections. The first
section provides a brief introduction to Mplus and describes
how to obtain access to Mplus. The second section briefly reviews
SEM assumptions and describes important and useful model fitting
features that are unique to Mplus. The third section describes
how to get started with Mplus, how to read data from an external
data file, and how to obtain descriptive sample statistics.
The fourth section explains how to fit exploratoy factor analysis
models for continuous and categorical outcomes using Mplus.
The fifth section of this document demonstrates how you can
use Mplus to test confirmatory factor analysis and structural
equation models. The sixth section presents examples of two
advanced models available in Mplus: multiple group analysis
and multilevel SEM. By the end of the course you should be able
to fit EFA and CFA/SEM models using Mplus. You will also gain
an appreciation for the types of research questions well-suited
to Mplus and some of its unique features.
2. Introduction to EFA, CFA, SEM
and Mplus
Exploratory factor analysis (EFA) is a method of data
reduction in which you may infer the presence of latent factors
that are responsible for shared variation in multiple measured
or observed variables. In EFA each observed variable in the
analysis may be related to each latent factor contained in the
analysis. By contrast, confirmatory factor analysis (CFA)
allows you to stipulate which latent factor is related to any
given observed variable. Structural equation modeling
(SEM) is a more general form of CFA in which latent factors
may be regressed onto each other. Mplus can fit EFA, CFA, and
SEM models.
To effectively use and understand the course material, you should
already know how to conduct a multiple linear regression analysis
and compute descriptive statistics such as frequency tables
using SAS, SPSS, or a similar general statistical software package.
You should also understand how to interpret the output from
a multiple linear regression analysis. This document also assumes
that you are familiar with the statistical assumptions of EFA,
CFA, and SEM, and you are comfortable using syntax-based software
programs such as SAS. If you do not have prior experience with
exploratory factor analysis, see the usage note Factor
Analysis Using SAS PROC FACTOR . If you do not have experience
with CFA or SEM, see our AMOS
tutorial for more information about SEM. Finally, you should
understand basic Microsoft Windows navigation operations: opening
files and folders, saving your work, recalling previously saved
work, etc.
3. Accessing Mplus
You may access Mplus in one of three ways:
- License a copy from Muthén
& Muthén for your own personal computer.
- Mplus is available to faculty, students, and staff at
the University of Texas at Austin via the STATS Windows
terminal server. To use the terminal server, you must obtain
an ITS computer account (an IF or departmental account)
and then validate the account for Windows NT Services. You
then download and configure client software that enables
your PC, Macintosh, or UNIX workstation to connect to the
terminal server. Finally, you connect to the server and
launch Mplus by double-clicking on the Mplus for Windows
program icon located in the STATS terminal server program
group. Details on how to obtain an ITS computer account,
account use charges, and downloading client software and
configuration instructions may be found in General
FAQ #36: Connecting to published statistical and mathematical applications
on the ITS Windows Terminal Server.
- Download the free student version of Mplus from the Muthén
& MuthénWeb
site for your own personal computer. If your models
of interest are small, the free demonstration version may
be sufficient to meet your needs. For larger models, you
will need to purchase your own copy of Mplus or access the
ITS shared copy of the software through the campus network.
The latter option is typically more cost effective, particularly
if you decide to access the other software programs available
on the server (e.g., SAS, SPSS, AMOS, etc.).
4. Getting Help with Mplus
If you have difficulties accessing Mplus on the
Windows
Terminal Server, call the ITS helpdesk at 512-475-9400 or send
e-mail to help@its.utexas.edu.
If you are able to log in to the
Windows Terminal Server
and run Mplus, but have questions about how to use Mplus or
interpret output, call the ITS helpdesk at 512-475-9400 to schedule an appointment
with an SSC statistical consultant or
send e-mail to stats@ssc.utexas.edu.
Important note: Both services are available to University
of Texas faculty, students, and staff only. See our Web
site at http://ssc.utexas.edu/consulting/free_consulting.html for more details about consulting services, as well as
frequently
asked questions and answers about EFA, CFA/SEM, Mplus, and other
topics. Non-UT and UT Mplus users will find the Muthén
& MuthénWeb
site to be a useful resource; see the Mplus
Discussion forum for frequently-asked questions and answers.
You may also post your own questions in this forum.
The Mplus User's Guide is available for check out from
the PCL general circulation desk. Alternatively, you may order
a copy from the Muthén
& Muthén Web
site.
Section 2: Latent Variable Modeling
using Mplus
1. Overview
of SEM Assumptions for Continuous Outcome Data
Before specifying and running a latent variable models, you
should give some thought to the assumptions underlying latent
variable modeling with continuous outcome variables. Several
of these assumptions are shown below:
- A theoretical basis for model specification
- A reasonable sample size
- Identified model equations
- Complete data or appropriate handling of incomplete data
- Continuously and normally distributed endogenous variables
These assumptions apply equally to all EFA and CFA/SEM software
programs. The details of these assumptions can be found in our
AMOS
tutorial, but they may be summarized as follows: Recommendations
for sample size vary depending upon the complexity of the specified
model, but typical figures range from 5 to 15 cases per estimated
parameter with overall sample size preferred to exceed N
= 200 cases. Furthermore, any model you consider should have
a theoretical basis, and substantive inferences should be drawn
based upon your ability to rule out alternative explanations
for findings, rather than on statistical considerations alone.
Like AMOS, Mplus features Full Information Maximum Likelihood
(FIML) handling of missing data, an appropriate, modern method
of missing data handling that enables Mplus to make use of all
available data points, even for cases with some missing responses.
For more details on missing data handling methods, including
FIML, see General
FAQ #25: Handling missing or incomplete data and AMOS
FAQ #5: Handling Missing Data using AMOS. One added missing
data handling feature that is unique to Mplus is its ability
to generate model modification indices for databases that are
incomplete.
2.
Categorical Outcomes and Categorical Latent Variables
Where Mplus diverges from most other SEM software packages is
in its ability to fit latent variable models to databases that
contain ordinal or dichotomous outcome variables. Note that
Mplus will not yet fit models to databases with nominal outcome
variables that contain more than two levels. Nonetheless, the
ability to fit models to variables that contain ordinal and
dichotomous categorical outcome variables is very useful. Furthermore,
Mplus will fit latent class analysis (LCA) models that
contain categorical latent variables and fit mixture models
that generate expected classifications of observations based
upon the characteristics of your specified model.
Should you use Mplus to perform EFA, CFA, and SEM analyses on
your data? In order to facilitate rapid access to both simple
and complex latent variable models, the Mplus developers have
built a streamlined set of data import and model specification
commands. All Mplus commands are specified using command syntax,
though a syntax generator is under development at the time of
this writing. If you are not comfortable with reading data and
specifying statistical models using command syntax, Mplus may
not be the optimal choice for you. On the other hand, if you
prefer to work with command syntax when you use statistical
software programs or you do not mind learning software syntax
to perform data analysis, you will probably find it useful to
learn Mplus. This is particularly true when you consider some
of the features unique to Mplus:
- The ability to build models with dichotomous and ordered
categorical outcome variables
- The capacity to build models that contain categorical
latent variables
- Optimal full information maximum likelihood (FIML) missing
data handling for both exploratory as well as CFA and SEM
models
- Modification index output, even when you invoke FIML
missing data handling
- The ability to fit multilevel or hierarchical CFA and
SEM models
Section 3: Using Mplus
1. Launching Mplus
If you are using a personal or demonstration copy of Mplus,
locate the Mplus entry in the Program Files subsection
of the Microsoft Windows Start menu.. If you are using
the STATS Windows terminal server, locate the Mplus for Windows
icon in the Citrix Program Neighborhood and double-click on
it to launch Mplus. You will then be prompted to enter your
Windows NT Services account name, your password, and the domain
name, WNT. After you have entered this information, click OK
to launch Mplus. Once you have launched Mplus, you will see
the following window appear on your computer's desktop:
2. The Input
and Output Windows
The window shown above is the input window. You write
Mplus syntax in this window to read the data to be analyzed
and to specify your model of interest. You then save your Mplus
syntax and select Run Mplus from the Mplus
menu to submit your syntax to the Mplus engine for processing:
Mplus
Run Mplus
Note: If you are using Mplus on the STATS terminal server,
do not save your work to the default Mplus directory. Instead,
save your work on one of the client disk drives or your allocated
WNTDISK server space. The server space is mounted as drive U
and has the advantage of being available to you whenever you
log in to the terminal server. There is, however, a nominal
fee associated with using this space for file storage. You may
also save your files on a local disk drive. Each local drive
is preceded by a $ (e.g., $C:) in the list of available disk
drives shown in the Save As menu option in the
File menu.
Once Mplus has finished processing your command syntax, it replaces
the input window with the output window. The output window
first displays your Mplus syntax. Below the Mplus syntax are
the Mplus model results. If there is an error in your Mplus
syntax or you want to modify your Mplus syntax in any way (e.g.,
to fit a different model to the data), you must return to the
appropriate command file by selecting that file's name from
the File menu's list of recently-accessed files.
That action returns the input window's contents to the screen
and you can then modify the previous commands, save the modified
command file, and run Mplus once again to obtain new output.
3.
Reading Data and Outputting Sample Statistics
After you have launched Mplus, you may build a command file.
There are nine Mplus commands: TITLE, DATA (required),
VARIABLE (required), DEFINE, SAVEDATA,
ANALYSIS, MODEL, OUTPUT, and MONTECARLO.
The most commonly used Mplus commands are described in this
document. According to the Mplus User's Guide, "The Mplus
commands may come in any order. The DATA and VARIABLE
commands are required for all analyses. All commands must begin
on a new line and must be followed by a colon. Semicolons separate
command options. There can be more than one option per line.
The records in the input setup must be no longer than 80 columns.
They can contain upper and/or lower case letters and tabs."
(page 1).
A description of the Mplus defaults appears in Mplus
FAQ #3: Mplus Defaults. You should review these defaults
carefully and be sure that you understand them fully prior to
analyzing data with Mplus.
The first Mplus syntax to appear in the command file is typically
a TITLE command. The TITLE command allows you
to specify a title that Mplus will print on each page of the
output file.
Following the TITLE command is the DATA command.
The DATA command specifies where Mplus will locate the
data, the format of the data, and the names of variables. At
present, Mplus will read the following file formats: tab-delimited
text, space-delimited text, and comma-delimited text. The input
data file may contain records in free field format or fixed
format. If you are using data stored in another form (e.g.,
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. See our FAQs
for information on how to convert common statistical data file
formats to plain, comma-delimited, or tab-delimited text files.
The next command is the VARIABLE command. The VARIABLE
command names the columns of data that Mplus reads using the
DATA command.
Following the VARIABLE command is the ANALYSIS
command. The ANALYSIS command tells Mplus what type of
analysis to perform. Many analysis options are available; a
number of these are shown in the examples that appear in this
document.
Consider the following example database: In 1939 Karl Holzinger
and Francis Swineford administered 26 aptitude tests to 145
students in the Grant-White School. Of the 26 tests, six are
used here: visual perception, cubes, lozenges, paragraph comprehension,
sentence completion, and word meaning. An additional variable,
gender, is included in the database, but not used in this example.
This database is available in SPSS format as one of the example
datasets used by the AMOS SEM software package. AMOS and its
example program files and datasets are available on the STATS
terminal server; a free student version of AMOS containing this
database may be downloaded from the Smallwaters
Corporation Web site. The SPSS file's name is grant.sav.
You can download this file in tab-delimited text format as grant.dat.
Then you can write the following Mplus syntax to read the data
from the file.
TITLE:
Grant-White School: Summary Statistics
DATA:
FILE IS U:\Projects\Documentation\Mplus\grant.dat ;
FORMAT IS free ;
VARIABLE:
NAMES ARE visperc
cubes
lozenges
paragrap
sentence
wordmean
gender ;
USEVARIABLES ARE visperc
cubes
lozenges
paragrap
sentence
wordmean ;
ANALYSIS:
TYPE = basic ;
In this sample program, the DATA command uses the FILE
subcommand to tell Mplus where to locate the relevant data file.
In this case, the file's location is U:\Projects\Documentation\Mplus\grant.dat.
The FORMAT subcommand uses the default free
option to let Mplus know that the data points appear in order
in the data file with the data points separated by commas, tabs,
or spaces. Alternatively, you can use FORTRAN format statements
to read data when data are in fixed columns. FORTRAN-formatted
input is recommended for large databases because it is more
efficient than the default free data field input;
see the Mplus manual for a detailed description of how to specify
FORTRAN input formats.
The next command shown is the VARIABLE command. The VARIABLE
command uses the NAMES subcommand to list the variables
contained in the Grant-White database. While it is possible
to have more than one variable name on a row of the command
file, this example lists the variables with one variable per
line becuase the appearance of the variable names in the command
file is easy to read. Becuase Mplus allows variable names to
have a maximum width of eight characters, the variable name
"paragraph" is shortened to paragrap.
Following the NAMES subcommand is the USEVARIABLES
subcommand. USEVARIABLES enables you to specify a
particular subset of variables to be used in the data analysis.
A similar subcommand, USEOBS, allows you to select subsets
of cases to be used in a particular analysis. For example, if
you wanted to limit the analysis to female participants, you
could include the subcommand
USEOBS gender EQ 1 ;
where a gender value of 1 designated female cases in
the database.
The ANALYSIS command specifies the TYPE of analysis
to be performed by Mplus. In this example the type is basic.
The basic model type does not have Mplus fit any model to the
sample data; instead Mplus will compute sample statistics only.
Using basic as the analysis type is useful during the intial
phase of building your command file because you can use the
Mplus sample statistics output to compare Mplus results to results
you obtained using SAS, SPSS, Excel, or other statistical software
programs to verify that Mplus is reading your input data correctly.
It is worth noting that Mplus has many default settings that
enable you to write compact syntax, which results in brief command
files. Once you understand the Mplus defaults fully, you may
take advantage of them to write shorter command files. For instance,
the first example shown above may be simplified:
TITLE:
Grant-White School: Summary Statistics
DATA:
FILE IS U:\Projects\Documentation\Mplus\grant.dat ;
VARIABLE:
NAMES ARE visperc
cubes
lozenges
paragrap
sentence
wordmean
gender ;
USEVARIABLES ARE visperc - wordmean ;
ANALYSIS:
TYPE = basic ;
The FORMAT is free statement has been omitted because
the default format is free-field data input. The USEVARIABLES
statement also shows a handy Mplus feature, the variable list
option. The variable list option enables you to conveniently
refer to a list of variables using a dash to separate the first
and last variables in the contiguous series of variables.
The output from the basic analysis appears below. Although Mplus
initially returns a copy of the input command file, that portion
of the output has been omitted here in the interest of saving
space.
SUMMARY OF ANALYSIS
Mplus VERSION 1.04
PAGE 2
Holzinger and Swineford Grant-White School Summary Statistics
Number of groups
1
Number of observations
145
Number of y-variables
6
Number of x-variables
0
Number of continuous latent variables
0
Observed variables in the analysis
VISPERC CUBES
LOZENGES PARAGRAP SENTENCE
WORDMEAN
Estimator
ML
Maximum number of iterations
1000
Convergence criterion
.500D-04
Input data file(s)
U:\Projects\Documentation\Mplus\grant.dat
Input data format FREE
RESULTS FOR BASIC ANALYSIS
SAMPLE STATISTICS
Means/Intercepts/Thresholds
1
2
3
4
5
________ ________
________ ________
________
1
29.579 24.800
15.966 9.952
18.848
Means/Intercepts/Thresholds
6
________
1
17.283
Covariances/Correlations/Residual Correlations
VISPERC CUBES
LOZENGES PARAGRAP
SENTENCE
________ ________
________ ________
________
VISPERC 47.801
CUBES
10.012 19.758
LOZENGES 25.798
15.417 69.172
PARAGRAP 7.973
3.421 9.207
11.393
SENTENCE 9.936
3.296 11.092
11.277 21.616
WORDMEAN 17.425
6.876 22.954
19.167 25.321
Covariances/Correlations/Residual Correlations
WORDMEAN
________
WORDMEAN 63.163
Mplus initially identifies the number of groups and observations
in the analysis, followed by the number of X (predictor) and
Y (outcome) variables and the sample (input) covariances, variances,
and means. Once you have verified that these values are correct,
you can turn your attention to fitting your model(s) of interest.
The next section continues with the same example database, but
describes how to perform an exploratory factor analysis of the
continuous variables in the Grant-White database using Mplus.
Section 4: Exploratory Factor Analysis
1. Exploratory Factor Analysis with Continuous Variables
Once you have read the data into Mplus and verified that the
sample statistics show that the data have been read correctly,
you can perform exploratory factor analysis using Mplus by altering
the ANALYSIS command as follows:
ANALYSIS:
TYPE = efa 1 2 ;
ESTIMATOR = ml ;
This syntax instructs Mplus to perform an exploratory factor
analysis of the Grant-White database. Efa tells
Mplus to perform an exploratory factor analysis. The 1 and 2
following the efa specification tells Mplus to
generate all possible factor solutions between and including
1 and 2. In this instance, one and two factor solutions will
be produced by the analysis. Finally, the ESTIMATOR = ml
option has Mplus use the maximum likelihood estimator to
perform the factor analysis and compute a chi-square goodness
of fit test that the number of hypothesized factors is sufficient
to account for the correlations among the six variables in the
analysis. This optional specification overrides the default
unweighted least-square (uls) estimator.
If your data are not joint multivariate normally distributed,
you may want to replace the ml with either the
mlm or mlmv estimators. One useful
feature of Mplus is its ability to handle non-normal input data.
Recall that the default ml estimator assumes that
the input data are distributed joint multivariate normal. If
you have reason to believe that this assumption has not been
met and your sample is reasonably large (e.g., N = 200),
you may substitute mlm or mlmv in
place of ml on the ESTIMATOR = line. The
mlm option provides a mean-adjusted chi-square
model test statistic whereas the mlmv option produces
a mean and variance adjusted chi-square test of model fit. SEM
users who are familiar with Bentler's EQS software program should
also note that the mlm chi-square test and standard
errors are equivalent to those produced by EQS in its ML;ROBUST
method.
You may also add the OUTPUT command following the ANALYSIS
command. The OUTPUT command is used to specify optional
output. For this example the keyword sampstat
tells Mplus to include sample statistics as part of its printed
output.
OUTPUT: sampstat
;
Mplus produces the sample correlations, eigenvalues, and the
chi-square test of the one factor model to the sample data.
As you can see from the results, shown below, the chi-square
test is statistically significant, so the null hypothesis that
a single factor fits the data is rejected; more factors are
required to obtain a non-significant chi-square. Since the chi-square
test is sensitive to sample size (such that large samples often
return statistically significant chi-square values) and non-normality
in the input variables, Mplus also provides the Root Mean
Square Error of Approximation (RMSEA) statistic.
The RMSEA is not as sensitive to large sample sizes. According
to Hu and Bentler (1999), RMSEA values below .06 indicate satisfactory
model fit. The RMSEA yielded a result of .162, which was consistent
with the chi-square result in suggesting that the one factor
model does not fit the data adequately.
CONTINUOUS VARIABLE CORRELATION MATRIX
VISPERC CUBES
LOZENGES PARAGRAP
SENTENCE
________ ________
________ ________
________
VISPERC
CUBES
.326
LOZENGES
.449 .417
PARAGRAP
.342 .228
.328
SENTENCE
.309 .159
.287 .719
WORDMEAN
.317 .195
.347 .714
.685
Grant-White School: Exploratory Factor Analysis
EXPLORATORY ANALYSIS WITH 1 FACTOR(S) :
EIGENVALUES FOR SAMPLE CORRELATION MATRIX
1
2
3
4
5
________ ________
________ ________
________
1
3.009 1.225
.656 .530
.311
EIGENVALUES FOR SAMPLE CORRELATION MATRIX
6
________
1
.270
EXPLORATORY ANALYSIS WITH 1 FACTOR(S) :
CHI-SQUARE VALUE
43.241
DEGREES OF FREEDOM
9
PROBABILITY VALUE
.0000
RMSEA (ROOT MEAN SQUARE ERROR OF APPROXIMATION) :
ESTIMATE (90 PERCENT C.I.) IS .162 ( .115
.212)
PROBABILITY RMSEA LE .05 IS .000
Mplus next produces the estimated factor loadings and error
variances. Notice that the visperc, cubes, and
lozenges factor loadings are low relative to the other
factor loadings displayed below. See the document Factor Analysis using SAS PROC
FACTOR for more information on interpreting factor loadings.
ESTIMATED FACTOR LOADINGS
1
________
VISPERC
.415
CUBES
.272
LOZENGES
.415
PARAGRAP
.865
SENTENCE
.818
WORDMEAN
.827
ESTIMATED ERROR VARIANCES
VISPERC CUBES
LOZENGES PARAGRAP
SENTENCE
________ ________
________ ________
________
.828 .926
.828 .252
.330
________
1
.316
The estimated correlation matrix is the correlation matrix reproduced
by Mplus under the assumption that a single factor is sufficient
to explain the sample correlations. From the model fit results
shown above, this is not the case, so it is not surprising that
this implied or model-based correlation matrix differs substantially
from the sample correlation matrix reported above.
ESTIMATED CORRELATION MATRIX
VISPERC CUBES
LOZENGES PARAGRAP
SENTENCE
________ ________
________ ________
________
VISPERC
1.000
CUBES
.113 1.000
LOZENGES
.172 .113
1.000
PARAGRAP
.359 .235
.359 1.000
SENTENCE
.339 .223
.340 .708
1.000
WORDMEAN
.343 .225
.343 .715
.677
WORDMEAN
________
WORDMEAN 1.000
The residuals matrix represents the difference between the sample
correlation matrix and the implied correlation matrix. As noted
above, since the model did not fit the observed data particularly
well, there are some values in this matrix that are non-trivial
in size. In particular, the cubes-visperc, lozenges-visperc,
and lozenges-cubes residual values are high relative
to the other values in the matrix.
RESIDUALS OBSERVED-EXPECTED
VISPERC CUBES
LOZENGES PARAGRAP
SENTENCE
________ ________
________ ________
________
VISPERC
.000
CUBES
.213 .000
LOZENGES
.276 .304
.000
PARAGRAP -.017
-.007 -.031
.000
SENTENCE -.030
-.063 -.053
.011 .000
WORDMEAN -.026
-.030
.004 .000
.009
RESIDUALS OBSERVED-EXPECTED
WORDMEAN
________
WORDMEAN
.000
The Root Mean Square Residual (RMR) is another descriptive model
fit statistic. According to Hu and Bentler (1999), RMR values
should be below .08 with lower values indicating better model
fit. The value of .1225 shown below for the one factor solution
indicates unacceptably poor model fit.
ROOT MEAN SQUARE RESIDUAL IS
.1225
In short, the one factor solution was a poor fit to the data.
In particular, the model did not account well for the correlations
among the visperc, cubes, and lozenges
variables. What about the two factor solution? Mplus reports
the two factor solution following the single factor model. The
chi-square test of model fit is non-significant, indicating
that the null hypothesis that the model fits the data cannot
be rejected (the model fits the data well). This finding is
corroborated by the RMSEA: Its estimate is zero; it's 90% confidence
interval has an upper bound value of .055, which is below the
Hu and Bentler (1999) recommended cutoff value of .06. The RMSEA
estimate and its upper bound confidence interval value should
both fall below .06 to ensure satisfactory model fit.
EXPLORATORY ANALYSIS WITH 2 FACTOR(S) :
EXPLORATORY ANALYSIS WITH 2 FACTOR(S) :
CHI-SQUARE VALUE
1.079
DEGREES OF FREEDOM
4
PROBABILITY VALUE
.8976
RMSEA (ROOT MEAN SQUARE ERROR OF APPROXIMATION) :
ESTIMATE (90 PERCENT C.I.) IS .000 ( .000
.055)
PROBABILITY RMSEA LE .05 IS .944
For exploratory factor analysis solutions with two or more factors,
Mplus reports varimax rotated loadings and promax
rotated loadings.Varimax loadings assume the two factors
are uncorrelated whereas promax loadings allow the factors to
be correlated. Directly below the promax loadings is the factor
intercorrelatrion matrix.
In this example the two factors are correlated .480. With even
a modest correlation among the two factors, you should choose
to interpret the promax rotated loadings. The loadings show
that the visperc, cubes, and lozenges variables
load onto the first factor whereas the remaining variables load
onto the second factor.
VARIMAX ROTATED LOADINGS
1
2
________ ________
VISPERC
.547 .250
CUBES
.550 .092
LOZENGES
.728 .196
PARAGRAP
.241 .830
SENTENCE
.174 .816
WORDMEAN
.247 .788
PROMAX ROTATED LOADINGS
1
2
________ ________
VISPERC
.540 .112
CUBES
.585 -.063
LOZENGES
.755 -.001
PARAGRAP
.046 .841
SENTENCE -.025
.846
WORDMEAN
.063 .794
PROMAX FACTOR CORRELATIONS
1
2
________ ________
1
1.000
2
.480 1.000
Mplus next reports estimated error variances for each observed
variable, the estimated correlation matrix, and the residual
correlation matrix. Notice that unlike the preceding one factor
solution, this dual factor solution's estimated correlation
matrix is very close in value to the original sample correlation
matrix. Accordingly, the residual correlation matrix has all
values close to zero and the RMR value of .0092 is well below
the Hu and Bentler (1999) recommended cutoff of .08.
ESTIMATED ERROR VARIANCES
VISPERC CUBES
LOZENGES PARAGRAP
SENTENCE
________ ________
________ ________
________
1
.638 .689
.431 .253
.304
ESTIMATED ERROR VARIANCES
WORDMEAN
________
1
.318
ESTIMATED CORRELATION MATRIX
VISPERC CUBES
LOZENGES PARAGRAP
SENTENCE
________ ________
________ ________
________
VISPERC
1.000
CUBES
.324 1.000
LOZENGES
.448 .419
1.000
PARAGRAP
.339 .209
.338 1.000
SENTENCE
.299 .170
.286 .719
1.000
WORDMEAN
.332 .208
.334 .714
.686
ESTIMATED CORRELATION MATRIX
WORDMEAN
________
WORDMEAN 1.000
RESIDUALS OBSERVED-EXPECTED
VISPERC CUBES
LOZENGES PARAGRAP
SENTENCE
_______ ________
________ ________
________
VISPERC
.000
CUBES
.002 .000
LOZENGES
.001 -.002
.000
PARAGRAP
.002 .019
-.010
.000
SENTENCE
.010 -.011
.000 .000
.000
WORDMEAN -.015
-.013
.013 .001
-.001
RESIDUALS OBSERVED-EXPECTED
WORDMEAN
________
WORDMEAN
.000
ROOT MEAN SQUARE RESIDUAL IS
.0092
This example assumes that the Grant-White database is complete.
In other words, there are no missing cases in the Grant-White
database. What if some cases had missing values? Often databases
have cases with incomplete data. The next section describes
a feature unique to Mplus: exploratory factor analysis of a
database with incomplete cases.
2.
Exploratory Factor Analysis with Missing Data
Suppose you altered the Grant-White database so that cases with
visperc scores that exceed 34 have missing cubes
scores and that cases with wordmean scores of 10 or below
have missing sentence values. In this instance the missing
cubes and setence completion data are said to be missing
at random (MAR) because the patterns of missing data are
explainable by the values of other variables in the database,
visual perception and word meaning. Ordinarily, if you do not
specify a missing data analysis in Mplus, Mplus performs
listwise or casewise deletion of cases with any missing
data. That is, any case with one or more missing data points
is omitted entirely from analyses. However, for exploratory
factor analysis, confirmatory factor analysis, and structural
equation modeling with continuous variables, Mplus features
a missing data option that outperforms the default listwise
deletion method. The optional method that offers superior performance
is called full information maximum likelihood (FIML); details
on FIML can be found in General FAQ #25: Handling missing or incomplete Data and in
AMOS FAQ #5: Handling missing data using AMOS.
Regardless of whether you choose to use FIML or listwise data
deletion to handle missing data, if you have missing data in
your input database, you must tell Mplus how the missing values
for each variable are represented in the database. You use the
MISSING subcommand of the VARIABLE command to accomplish
this task. In this example, missing values for cubes and sentence
are represented by -9, so the MISSING subcommand reads:
MISSING ARE all (-9) ;
The all keyword tells Mplus that all variables
in the analysis use -9 to represent missing values. If your
database contains blanks to represent missing values, you may
use the specification
MISSING = blank ;
Similarly, you may use
MISSING ARE . ;
if your database contains period symbols to represent missing
values. Other missing value specifications are available; see
the Mplus User's Guide for specifics.
If you insert the MISSING syntax into the previous exploratory
factor analysis program and specify that Mplus use the newly-created
database that contains cases with missing values, grant-missing.dat, Mplus will perform listwise
deletion of the cases with incomplete data. The Mplus command
file follows:
TITLE: Grant-White School: EFA with
Missing Data
DATA: FILE IS U:\Projects\Documentation\Mplus\grant-missing.dat
;
VARIABLE:
NAMES ARE visperc
cubes
lozenges
paragrap
sentence
wordmean
gender ;
USEVARIABLES ARE visperc - wordmean ;
MISSING ARE all (-9) ;
ANALYSIS: TYPE = efa 1 2;
ESTIMATOR = ml ;
Selected output from the analysis appears below.
Grant-White School: Exploratory Factor Analysis with Missing
Data
SUMMARY OF ANALYSIS
Number of groups
1
Number of observations
79
Number of y-variables
6
Number of x-variables
0
Number of continuous latent variables
0
Notice that Mplus considers the database to contain 79 usable
cases rather than the original 145 cases.
EXPLORATORY ANALYSIS WITH 1 FACTOR(S) :
CHI-SQUARE VALUE
14.651
DEGREES OF FREEDOM
9
PROBABILITY VALUE
.1009
RMSEA (ROOT MEAN SQUARE ERROR OF APPROXIMATION) :
ESTIMATE (90 PERCENT C.I.) IS .089 ( .000
.169)
PROBABILITY RMSEA LE .05 IS .199
The one factor solution also fits the database for the 79 useable
cases. This finding stands in direct contrast to the example
in the previous section where all 145 cases had complete data
and the one factor model was rejected. Clearly the reduction
of N from 145 to 79 has resulted in a substantial loss
of statistical power to reject false hypotheses.
Fortunately, you can use Mplus's FIML missing data handling
option to rectify the problem. Add the keyword missing
to the TYPE subcommand of the ANALYSIS command,
like this:
ANALYSIS:
TYPE = missing efa 1 2 ;
ESTIMATOR = ml ;
Run the analysis and consider the results, shown below.
Grant-White School: Exploratory Factor Analysis with Missing
Data
SUMMARY OF ANALYSIS
Number of groups
1
Number of observations
145
Number of y-variables
6
Number of x-variables
0
Number of continuous latent variables
0
Mplus now uses all 145 cases in its computations.
SUMMARY OF DATA
Number of patterns
4
COVARIANCE COVERAGE OF DATA
Minimum covariance coverage value .100
PROPORTION OF DATA PRESENT
Covariance Coverage
VISPERC CUBES
LOZENGES PARAGRAP
SENTENCE
________ ________
________ ________
________
VISPERC
1.000
CUBES
.697 .697
LOZENGES 1.000
.697 1.000
PARAGRAP 1.000
.697 1.000
1.000
SENTENCE
.821 .545
.821 .821
.821
WORDMEAN 1.000
.697 1.000
1.000
.821
Mplus futher recognizes that there are four distinct patterns
of missing data contained in the database and it displays the
amount of data used to generate each input covariance for the
analysis. From the missing data coverage matrix, you can see
that the cubes-sentence covariance has the lowest coverage
with just under 55% of cases available to build the covariance.
Mplus requires a minimum coverage value of 10% per covariance,
though you can override this default if you wish.
EXPLORATORY ANALYSIS WITH 1 FACTOR(S) :
CHI-SQUARE VALUE
29.732
DEGREES OF FREEDOM
9
PROBABILITY VALUE
.0005
RMSEA (ROOT MEAN SQUARE ERROR OF APPROXIMATION) :
ESTIMATE (90 PERCENT C.I.) IS .126 ( .078
.178)
PROBABILITY RMSEA LE .05 IS .007
Unlike the example that used listwise deletion of cases with
missing data, the chi-square test of model fit for the one factor
solution rejects the one factor model. Using FIML missing data
handling, you conclude that one factor is not sufficient to
explain the pattern of correlations among the six input variables,
just as you did in the first example from the preceding section
where Mplus used the complete database containing 145 cases.
As with the complete dataset, the two factor solution fits the
data well using the FIML method with the incomplete dataset:
EXPLORATORY ANALYSIS WITH 2 FACTOR(S) :
CHI-SQUARE VALUE
.578
DEGREES OF FREEDOM
4
PROBABILITY VALUE
.9655
RMSEA (ROOT MEAN SQUARE ERROR OF APPROXIMATION) :
ESTIMATE (90 PERCENT C.I.) IS .000 ( .000
.000)
PROBABILITY RMSEA LE .05 IS .982
3.
Exploratory factor analysis with categorical outcomes
So far, the examples shown here contained continuous outcomes.
If you have observed outcome variables that have ten or fewer
categories, and the variables' responses are dichotomous or
ordered categories, you may elect to have Mplus treat these
variables as categorical indicators. This type of model is often
sensible for analyzing Likert scale items because while the
items themselves typically are coarsely categorized on a 1 to
5 or 1 to 7 scale, the items often attempt to measure an individual's
standing on a continuous underlying unobserved variable.
For the purposes of illustration, suppose that you recode each
variable into a replacement variable where all six variables'
values at the median or below are assigned a categorical value
of 1.00 and all values above the median assigned a value of
2.00. Mplus recodes the lowest value to zero with subsequent
values increasing in units of 1.00. While the two underlying
latent factors remain continuous, the six categorical observed
variables' response values are now ordered dichotomous categories.
To analyze the modified database using Mplus, you may use the
syntax that appeared in the initial exploratory factor analysis
example, with the following modifications, and the new data
file that contains the categorical variables, grantcat.dat, as shown below.
TITLE: Grant-White School:
EFA with categorical outcomes
DATA: FILE IS U:\Projects\Documentation\Mplus\grantcat.dat
;
VARIABLE:
NAMES ARE viscat
cubescat
lozcat
paracat
sentcat
wordcat ;
USEVARIABLES ARE viscat - wordcat ;
CATEGORICAL ARE viscat - wordcat ;
ANALYSIS: TYPE = efa 1 2;
ESTIMATOR = wlsmv ;
OUTPUT: sampstat ;
First, you must change the names of the variables in the NAMES
and USEVARIABLES subcommands of the DATA command.
Next, you tell Mplus which variables are categorical with the
CATEGORICAL subcommand of the DATA command, like
this:
CATEGORICAL ARE vizcat ... wordcat ;
You should also change the ESTIMATOR option for the ANALYSIS
command. The default is unweighted least-squares (uls),
which is fast and is useful for exploratory work, but a more
optimal choice for categorical outcomes, based on the work of
Muthén, DuToit, and Spisic (1997), is weighted least-squares
with mean and variance adjustment, wlsmv.
ANALYSIS: TYPE = efa
1 2;
ESTIMATOR = wlsmv ;
Selected output from the analysis appears below. Notice that
the categorical nature of the data precludes computation of
the descriptive model fit statistics such as the RMSEA, though
Mplus does produce the familiar chi-square test of overall model
fit.
EXPLORATORY ANALYSIS WITH 2 FACTOR(S) :
CHI-SQUARE VALUE
2.823
DEGREES OF FREEDOM
4
PROBABILITY VALUE
.5875
The chi-square result for the two factor model is not significant,
which indicates that two factors are sufficient to explain the
intercorrelations among the six observed variables. The varimax
and promax rotated factor loadings appear below. The pattern
and values obtained from this analysis are consistent with the
results of the first exploratory factor analysis of the completely
continuous data discussed previously.
VARIMAX ROTATED LOADINGS
1
2
________ ________
VISCAT
.571 .332
CUBESCAT
.700 .117
LOZCAT
.667 .244
PARACAT
.473 .642
SENTCAT
.235 .847
WORDCAT
.206 .858
PROMAX ROTATED LOADINGS
1
2
________ ________
VISCAT
.559 .159
CUBESCAT
.777 -.137
LOZCAT
.698 .022
PARACAT
.347 .550
SENTCAT
.005 .876
WORDCAT
-.031
.899
PROMAX FACTOR CORRELATIONS
1
2
________ ________
1
1.000
2
.557 1.000
Although Mplus does not produce the RMSEA descriptive model
fit statistic for categorical outcomes, it does output the standardized
root mean residual, RMR:
ROOT MEAN SQUARE RESIDUAL IS
.0310
The value of .031 suggests an excellent fit of the two factor
model to the observed data.
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 (this is a property of the exploratory
factor analysis model; confirmatory factor analysis models
with categorical latent variables may be fit as mixture
models using Mplus; see the Mplus User's Guide for
more information about mixture models).
- FIML missing data handling is not available with the
analysis of categorical outcomes.
- The analysis specification and interpretation of the
output is the same whether one, a subset, or all observed
variables are categorical.
- Categorical observed variables may be dichotomous or
ordered polytymous (i.e., 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.
- 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.
Keeping these considerations in mind, Mplus provides a convenient
mechanism to perform an exploratory factor analysis of dichotmous
and ordered categorical responses. Since many exploratory factor
analyses are performed on Likert scale items that contain ordered
categories, Mplus is a useful tool for the exploration of the
factor structure of these instruments.
Section 5: Confirmatory Factor Analysis
and Structural Equation Models
The examples in the preceding section demonstrate how you can
use Mplus to fit exploratory factor analysis models to the Grant-White
database. What if you had an a priori hypothesis that the visual
perception, cubes, and lozenges variables belonged to a single
factor whereas the paragraph, sentence, and word meaning variables
belonged to a second factor? The diagram shown below illustrates
the model visually.
You can test this hypothesized factor structure using confirmatory
factor analysis, as shown in the next section.
1.
Confirmatory Factor Analysis with Continuous Variables
To set up a confirmatory factor analysis, return to the first
exploratory factor analysis example of Section
4. You may use the same Mplus syntax as that appearing in
example
1 of Section 4, with the following modifications to the
ending section of the command file:
ANALYSIS: TYPE = general
;
MODEL:
visual BY visperc@1 cubes lozenges ;
verbal BY paragrap@1 sentence wordmean ;
visual WITH verbal ;
OUTPUT: standardized
sampstat ;
The general analysis type tells Mplus that you
are fitting a general structural equation model rather than
specific model such as an exploratory factor analysis. The model
is general in the sense that you must define what parameters
are estimated; all other parameters are assumed to be fixed.
In the exploratory factor analysis context, Mplus already knows
the specifics of that model, so specifying the model is handled
automatically by Mplus. By contrast, in the confirmatory factor
analysis and structural equation modeling context each hypothesized
model is unique, so you must tell Mplus how the model is constructed.
The MODEL command allows you to specify the parameters
of your model.
The first line of the MODEL command shown above defines
a latent factor called visual. The BY keyword
(an abbreviation for "measured by") is used to define the latent
variables; the latent variable name appears on the left-hand
side of the BY keyword whereas the measured variables
appear on the right-hand side of the BY keyword. It has
three observed indicator variables: visperc, cubes,
and lozenges. Similarly, in the second line of the MODEL
command a latent factor called verbal has three indicators:
paragrap, sentence, and wordmean. The third
line of MODEL command uses the WITH keyword to
correlate the visual latent factor with the verbal
latent factor (note: this model is the same as AMOS Example
20-2r).
The visperc and paragrap variables are each followed
by @1. The @ sign tells Mplus to fix the factor
loading (regression weight) of the visual-visperc relationship
to the value that follows the @, 1.00. Similarly, the
verbal-paragrap relationship is also fixed to 1.00. The
reason you fix these two parameters is to provide a scale for
the visual and verbal latent variables' variances. If you ever
need to supply starting values for a particular parameter in
Mplus, you can specify its number after an asterisk, like this:
sentence*.5. Omitting the asterisks when you do not specify
starting values is the default. Note that each variable is separated
from the other variables in the analysis by at least one space.
Finally, the OUTPUT command contains an added keyword,
standardized. This option instructs Mplus to output
standardized parameter estimate values in addition to the default
unstandardized values. Selected output from the analysis appears
below.
Grant-White School: Confirmatory Factor Analysis
SUMMARY OF ANALYSIS
Number of groups
1
Number of observations
145
Number of y-variables
6
Number of x-variables
0
Number of continuous latent variables
2
Observed variables in the analysis
VISPERC CUBES
LOZENGES PARAGRAP SENTENCE
WORDMEAN
Continuous latent variables in the analysis
VISUAL VERBAL
The summary of analysis information tells you that there are
six continuous observed variables in the analysis and two latent
factors, visual and verbal. Mplus then displays
the input covariance matrix generated from the six observed
variables:
SAMPLE STATISTICS
Covariances/Correlations/Residual Correlations
VISPERC CUBES
LOZENGES PARAGRAP
SENTENCE
________ ________
________ ________
________
VISPERC 47.801
CUBES
10.012 19.758
LOZENGES 25.798
15.417 69.172
PARAGRAP 7.973
3.421 9.207
11.393
SENTENCE 9.936
3.296 11.092
11.277 21.616
WORDMEAN 17.425
6.876 22.954
19.167 25.321
Covariances/Correlations/Residual Correlations
WORDMEAN
________
WORDMEAN 63.163
Mplus next reports the results of fitting the hypothesized model
to the sample data.
THE MODEL ESTIMATION TERMINATED NORMALLY
TESTS OF MODEL FIT
Chi-Square Test of Model Fit
Value
3.663
Degrees
of Freedom
8
P-Value
.8861
Loglikelihood
H0
Value
-2575.128
H1
Value
-2573.297
Information Criteria
Number
of Free Parameters
13
Akaike
(AIC)
5176.256
Bayesian
(BIC)
5214.954
Sample-Size
Adjusted BIC 5173.817
(n* = (n + 2) / 24)
RMSEA (Root Mean Square Error Of Approximation)
Estimate
.000
90
Percent C.I.
.000 .046
Probability
RMSEA <= .05 .957
As was the case for the exploratory factor analysis of these
data, Mplus reports the chi-square goodness-of-fit test and
the RMSEA descriptive model fit statistic. The chi-square test
of model fit is not significant and the RMSEA value is well
below the value of .06 recommended by Hu and Bentler (1999)
as an upper boundary, so you can conclude that the proposed
model fits the data well. Mplus also reports the Akaike Information
Criterion (AIC) and the Bayesian Information Criterion
(BIC). These are descriptive indexes of model fit that you can
use to compare the goodness of model fit of two or more competing
models. Smaller values indicate better model fit.
Mplus also outputs the unstandardized coefficients (Estimates
in the output), the standard errors (abbreviated S.E. in
the output), the estimates divided by their respective standard
errors (Est./S.E.), and two standardized coefficients
for each estimated parameter in the model (Std and StdYX).
The estimate divided by the standard error tests the null hypothesis
that the parameter estimate is zero in the population from which
you drew your sample. An unstandardized estimate divided by
its standard error may be evaluated as a Z statistic,
so values that exceed +1.96 or fall below -1.96 are significant
below p = .05.
MODEL RESULTS
Estimates S.E. Est./S.E.
Std StdYX
VISUAL BY
VISPERC
1.000 .000
.000 4.358 .632
CUBES
.542 .116
4.658 2.360 .533
LOZENGES
1.392 .272
5.112 6.064 .732
VERBAL BY
PARAGRAP
1.000 .000
.000 2.920 .868
SENTENCE
1.309 .115 11.352
3.821 .825
WORDMEAN
2.247 .197 11.402
6.560 .828
VISUAL WITH
VERBAL
6.784 1.720
3.943 .533 .533
In this example, each of the estimated parameters has an estimate
to standard error ratio greater than +1.96, so each factor loading
is statistically significant, as well as the correlation between
the visual and verbal latent factors (Z
= 3.943). The variance components of the two factors, shown
in the output appearing below, are also statistically significant,
indicating that the amount of variance accounted for by each
factor is significantly different from zero.
Each unstandarized estimate represents the amount of change
in the outcome variable as a function of a single unit change
in the variable causing it. In this example, you assume that
the latent variables, in addition to some measurement error
(shown below), are responsible for the scores on the six observed
variables. For instance, for each single unit change in the
verbal latent factor, sentence scores increase
by 1.309 units.
Different measures often have different scales, so you will
often find it useful to examine the standardized coefficients
when you want to compare the relative strength of associations
across observed variables that are measured on different scales.
Mplus provides two standardized coefficients. The first, labeled
Std on the output, standardizes using the latent variables'
variances whereas the second type of standardized coefficient,
StdYX, standardizes based on latent and observed variables'
variances. This standardized coefficient represents the amount
of change in an outcome variable per standard deviation unit
of a predictor variable. In this output, you can see clearly
that the standardized coefficients of paragrap, sentence,
and wordmean are larger than those of visperc,
cubes, and lozenges. This finding suggests that
the verbal latent factor does a better job at explaining
the shared variance among paragrap, sentence,
and wordmean than does the visual latent factor
for its three indicator variables, visperc, cubes,
and lozenges.
This assertion is corroborated by the residual variances output
by Mplus. The standardized coefficients for the first three
indicators are larger than those for the remaining three indicators.
Residual Variances
Grant-White School: Confirmatory Factor Analysis
Estimates S.E. Est./S.E.
Std StdYX
VISPERC
28.485 4.739
6.011 28.485 .600
CUBES
14.050 1.978
7.105 14.050 .716
LOZENGES
31.933 7.269
4.393 31.933 .465
PARAGRAP
2.791 .584
4.775 2.791 .247
SENTENCE
6.869 1.164
5.900 6.869 .320
WORDMEAN
19.695 3.385
5.819 19.695 .314
Variances
VISUAL
18.989 5.582
3.402 1.000 1.000
VERBAL
8.525 1.376
6.196 1.000 1.000
R-SQUARE
Observed
Variable R-Square
VISPERC
.400
CUBES
.284
LOZENGES
.535
PARAGRAP
.753
SENTENCE
.680
WORDMEAN
.686
Finally, the r-square output illustrates that only modest amounts
of variance are accounted for in the first three indicators
whereas much larger amounts of variance are accounted for in
the final three indicators. As is the case with exploratory
factor analysis of continuous outcome variables, you may want
to use the mlm or mlmv estimators
in lieu of the default ml estimator if your input
data are not distributed joint multivariate normal by using
the ESTIMATOR = option on the ANALYSIS command.
The mlm option provides a mean-adjusted chi-square
model test statistic whereas the mlmv option produces
a mean and variance adjusted chi-square test of model fit; both
options also induce Mplus to produce robust standard errors
displayed in the model results table that are used to compute
Z tests of significance for individual parameter estimates.
An added advantage of the mlm option is that its
chi-square test and standard errors are equivalent to those
produced by EQS in its ML;ROBUST method. Muthén
and Muthén have placed formulas
on their Web site that allow you to use mlm-produced
chi-square values in nested model comparisons.
2. Handling Missing
Data
It is often the case that you have missing data in the context
of confirmatory factor analysis and structural equation modeling.
Using Mplus, you can employ the optimal Full Information Maximum
Likelihood (FIML) approach to handling missing data that was
described above in the section Exploratory
Factor Analysis with Missing Data in Section
4. Consider once again the same modified database, grant-missing.dat,
containing incomplete cases that was used in the earlier exploratory
factor analysis with missing data. As in the previous example,
define the missing value code to be -9 for all variables using
the MISSING subcommand in the VARIABLE command,
copy the MODEL syntax from the previous confirmatory
factor analysis example into the Mplus input window, and then
modify the ANALYSIS command so that it reads as follows:
ANALYSIS:
TYPE = general missing h1 ;
The missing keyword alerts Mplus to activate the
FIML missing data handling feature. The additional h1
keyword tells Mplus to output the chi-square goodness-of-fit
test in addition to the typical summary statistics, missing
data pattern information, parameter estimates, and standard
errors obtained in an analysis. Mplus requires that you specify
the h1 keyword because large models with many
missing data patterns can take a long time to converge. If this
describes your situation, you may want to omit the h1
option on the TYPE = line to verify that you have
specified your model correctly before invoking the h1
option to produce the chi-square test of model fit. If you elect
to remove the h1 option from the ANALYSIS TYPE
= command, be sure to omit the sampstat option
from the OUTPUT line, as well. If sampstat
is included on the OUTPUT line, Mplus automatically assumes
the h1ANALYSIS option and computes the chi-square
test of model fit, even if h1 is not included
on the ANALYSIS TYPE = line.
The command syntax for the missing data model is thus:
TITLE: Grant-White School: CFA with
missing data
DATA: FILE IS U:\Projects\Documentation\Mplus\grant-missing.dat
;
VARIABLE:
NAMES ARE visperc
cubes
lozenges
paragrap
sentence
wordmean
gender ;
USEVARIABLES ARE visperc - wordmean
;
MISSING ARE all (-9) ;
ANALYSIS: TYPE = general missing h1 ;
MODEL:
visual BY visperc@1 cubes lozenges ;
verbal BY paragrap@1 sentence wordmean ;
visual WITH verbal ;
OUTPUT: standardized sampstat
;
The chi-square test of model fit for the confirmatory factor
analysis with missing data shows that the hypothesized model
fit the data well:
TESTS OF MODEL FIT
Chi-Square Test of Model Fit
Value
2.777
Degrees
of Freedom
8
P-Value
.9476
Loglikelihood
H0
Value
-2376.312
H1
Value
-2374.923
Information Criteria
Number
of Free Parameters
19
Akaike
(AIC)
4790.623
Bayesian
(BIC)
4847.181
Sample-Size
Adjusted BIC 4787.058
(n* = (n + 2) / 24)
RMSEA (Root Mean Square Error Of Approximation)
Estimate
.000
90
Percent C.I.
.000 .011
Probability
RMSEA <= .05 .982
The Mplus parameter estimates, standard errors, and standardized
parameter estimates are similar to those found in the preceding
confirmatory factor analysis example. The only substantial difference
is the inclusion of an additional section that contains means
and intercepts for the latent factors and observed variables.
These means and intercepts are required to be estimated by the
FIML missing data handling procedure, but are otherwise not
a part of the tested model.
MODEL RESULTS
Estimates S.E. Est./S.E.
Std StdYX
VISUAL BY
VISPERC
1.000 .000
.000 4.377 .635
CUBES
.469 .127
3.679 2.051 .473
LOZENGES
1.373 .294
4.673 6.010 .725
VERBAL BY
PARAGRAP
1.000 .000
.000 2.914 .866
SENTENCE
1.187 .114 10.376
3.460 .821
WORDMEAN
2.247 .206 10.888
6.547 .827
VISUAL WITH
VERBAL
7.014 1.800
3.896 .550 .550
Residual Variances
VISPERC
28.354 5.037
5.629 28.354 .597
CUBES
14.589 2.340
6.234 14.589 .776
LOZENGES
32.642 7.938
4.112 32.642 .475
PARAGRAP
2.824 .627
4.507 2.824 .250
SENTENCE
5.781 1.070
5.401 5.781 .326
WORDMEAN
19.872 3.578
5.554 19.872 .317
Variances
VISUAL
19.158 5.859
3.270 1.000 1.000
VERBAL
8.493 1.393
6.099 1.000 1.000
Intercepts
VISPERC
29.579 .572
51.673 29.579 4.291
CUBES
24.616 .421
58.431 24.616 5.678
LOZENGES
15.965 .689
23.184 15.965 1.925
PARAGRAP
9.952 .279 35.620
9.952 2.958
SENTENCE
19.054 .366
52.057 19.054 4.522
WORDMEAN
17.283 .658
26.274 17.283 2.182
Finally, Mplus produces the r-square values for the observed
variables. Once again, these are similar to those obtained from
the original database with complete cases.
R-SQUARE
Observed
Variable R-Square
VISPERC
.403
CUBES
.224
LOZENGES
.525
PARAGRAP
.750
SENTENCE
.674
WORDMEAN
.683
If you elect to use Mplus's FIML approach to handling missing
data, be aware that the only available estimator is the maximum
likelihood option, ml. If you suspect that your
data are non-normally distributed, remember that the chi-square
test of model fit may be affected by the non-normality problem.
Depending on the severity of the non-normality problem and the
amount of missing data you have, you may want to explore other
ways of handling the missing data problem prior to performing
analyses using Mplus; see General
FAQ #25: Handling missing or incomplete data or schedule
an appointment with a consultant for details on these alternative
methods of handling missing data.
3.
Confirmatory Factor Analysis with Categorical Outcomes
Confirmatory factor analysis with dichotomous and polytomous
categorical outcomes, or confirmatory factor analysis with mixed
categorical and continuous outcomes is also possible using Mplus.
Recall the grantcat.dat database
used in the example Exploratory
Factor Analysis with Categorical Outcomes in Section
4.
Using the same database that replaces the six continuous observed
variables with a dichotomous variables, you can use the confirmatory
factor analysis syntax from the example Confirmatory
Factor Analysis With Continuous Variables with the following
modifications.
First, add the CATEGORICAL ARE vizcat ... wordcat ; statement
to the DATA command. Mplus will now treat the six observed
variables as categorical in the analysis. The entire command
syntax is shown here.
TITLE: Grant-White School: CFA with
categorical outcomes
DATA: FILE IS U:\Projects\Documentation\Mplus\grantcat.dat
;
VARIABLE:
NAMES ARE viscat
cubescat
lozcat
paracat
sentcat
wordcat ;
USEVARIABLES ARE viscat - wordcat ;
CATEGORICAL ARE viscat - wordcat ;
ANALYSIS: TYPE = general ;
MODEL:
visual BY viscat@1 cubescat lozcat ;
verbal BY paracat@1 sentcat wordcat ;
visual WITH verbal ;
OUTPUT: sampstat standardized
;
Selected results from the analysis appear below.
Chi-Square Test of Model Fit
Value
7.463*
Degrees
of Freedom
6**
P-Value
.2800
* The chi-square value for MLM, MLMV, WLSM and WLSMV
cannot be used for
chi-square difference tests.
** The degrees of freedom for MLMV and WLSMV are estimated
according to
formula 109 (page 281) in the Mplus User's
Guide.
The chi-square test of model fit is once again non-significant,
suggesting that the specified model fits the data adequately.
The default estimator for models that contain categorical outcomes
is the mean and variance-adjusted weighted least-squares method,
wlsmv. Optional estimators you may choose are
weighted least-squares (wls) and mean-adjusted
weighted least-squares (wlsm). As is the case
in the exploratory factor analysis of categorical data example,
there are no descriptive model fit statistics produced by Mplus
when it analyzes categorical outcomes. Mplus also produces a
note alerting you not to use the MLMV, WLSM, and WLSMV chi-square
values in nested model comparisons (the warning about the MLM
chi-square is not relevant as long as you use the formulas
shown on the Mplus Web site for nested model MLM chi-square
comparisons when you use the MLM estimator in the analysis of
continuous outcomes). You should not use the MLM estimator for
the analysis of intrinsically categorical outcome variables.
Mplus then outputs the model results:
MODEL RESULTS
Estimates S.E. Est./S.E.
Std StdYX
VISUAL BY
VISCAT
1.000 .000
.000 .729 .729
CUBESCAT
.831 .212
3.922 .606 .606
LOZCAT
.975 .230
4.248 .710 .710
VERBAL BY
PARACAT
1.000 .000
.000 .814 .814
SENTCAT
1.058 .134
7.920 .861 .861
WORDCAT
1.038 .127
8.154 .844 .844
VISUAL WITH
VERBAL
.397 .087
4.592 .670 .670
Variances
VISUAL
.531 .162
3.273 1.000 1.000
VERBAL
.662 .117
5.661 1.000 1.000
Thresholds
VISCAT$1
.095 .104
.913 .095 .095
CUBESCAT$1
.271 .105
2.571 .271 .271
LOZCAT$1
-.043 .104
-.415 -.043 -.043
PARACAT$1
.009 .104
.083 .009 .009
SENTCAT$1
.183 .105
1.743 .183 .183
WORDCAT$1
.043 .104
.415 .043 .043
This output is similar to that of a confirmatory factor analysis
with continuous outcomes, with one notable exception: Mplus
now produces threshold information for each categorical
variable. A threshold is the expected value of the latent variable
or factor at which an individual transitions from a value of
0 to a value of 1.00 on the categorical outcome variable when
the continuous underlying latent variable's score is zero. There
are only two categorical values for each outcome variable, so
there is only one threshold per variable. For any categorical
outcome variable with K levels, Mplus will output K-1
threshold values. For example, a five-point Likert scale item
would contain four threshold values. The first threshold would
represent the expected value at which an individual would be
most likely to transition from a value of 0 to a value of 1.00
on the Likert outcome variable. The second threshold would represent
the expected value at which an individual would be most likely
to transition from a value of 1.00 to a value of 2.00 on the
outcome variable, and so on through the fourth threshold, which
represents the expected value at which an individual would transition
from 3.00 to 4.00 on the outcome variable.
Finally, Mplus produces the r-square table output. The r-square
values are computed for the continuous latent variables underlying
the categorical outcome variables rather than the actual outcome
variables as is the case in analyses that contain continuous
outcome variables. Note that the r-square values for the categorical
outcomes cannot be interpreted as the proportion of variance
explained as is the case in the analysis of continuous outcomes.
Therefore, examining the sign and significance of the estimated
coefficients shown in the model results table above is generally
more informative than interpreting r-square values.
R-SQUARE
Observed Residual
Variable Variance R-Square
VISCAT
.469 .531
CUBESCAT
.633 .367
LOZCAT
.495 .505
PARACAT
.338 .662
SENTCAT
.259 .741
WORDCAT
.287 .713
The r-square table's residual variance output is, however, useful
for computing expected probabilities. You can use threshold
and coefficient information shown above with the residual variance
information from the r-square table to compute the expected
probability of case having a value of 0 or 1.00. Consider
following formula for computing the conditional probability
of a Y = 0 response given the factor eta.:
P(Y_ij = 0|eta_ij) = F[(tau_j - lambda_j*eta_i )*(1/square root
of theta_jj)]
where:
eta is the factor's value
F is the culmulative normal distribution fuction
tau is the measured item's threshold
lambda is the item's factor loading
theta is the residual variance of the measured item
Suppose you want to obtain the estimated probability for
sentcat = 0 at eta = 0. Using the formula, shown
above, you can compute this value:
P(Y_ij|eta_ij) = F[(.183 - 0)*(1/square root of .259)]
= F[.183*1.9649437]
= F[.3595847]
You can look up the value of .3595847 in a Z table in
a statistics textbook, or you can supply the computed value
of .3595847 to the PROBNORM function in SAS to obtain
the correct probability value. The PROBNORM function
returns the value from a cumulative normal distribution for
the inputted value. A simple SAS program such as the one shown
below enables you to obtain the final expected probability value
of .64.
DATA one ;
p = PROBNORM(.3595847)
;
RUN ;
PROC PRINT DATA = one ;
RUN ;
You may substitute other values of eta and lambda to obtain
different expected probability values. In general, the same
cautions and limitations that were discussed above in the section
Exploratory
Factor Analysis with Categorical Variables section also
apply to the analysis of categorical outcomes in the confirmatory
factor analysis and structural equation modeling contexts. In
addition, the following point is worth considering:
- Do not list indepedent (exogenous) categorical variables
in the CATEGORICAL statement. Instead, create dummy
variables (i.e., variables with values of 0 and 1 representing
group membeship status) and include them in the model as
predictors, or create a multiple group analysis based
upon category membership as described in the Multiple
Group Analysis section of this document.
4.
Structural Equation Modeling with Continuous Outcomes
In addition to exploratory and confirmatory factor analysis,
you may use Mplus to fit structural equation models that feature
causal relationships among latent variables. A ubiquituous example
of a structural equation model is that of the impact of socioeconomic
status (SES) on alienation in 1967 and 1971. A study conducted
by Wheaton, Muthén, Alwin, and Summers (1977) fit several
structural equation models to a database of 932 research participants.
The database contained the following observed, continuous variables:
Educ - Education level
SEI - Socioeconomic index
Anomia67 - Anomie in 1967
Anomia71 - Anomie in 1971
Powles67 - Powerlessness in 1967
Powles71 - Powerlessness in 1971
One of the fitted structural equation models features a latent
factor, SES, that influences Educ and SEI
scores. The SES latent variable in turn influences two
additional latent variables: Alien67 and Alien71.
Alien67 represents self-perceived alienation in 1967
and it influences responses on the anomie and powerlessness
variables measured in 1967. Similarly, Alien71 represents
self-perceived alienation in 1971 and it influences responses
on the anomie and powerlessness variables measured in 1971.
SES influences both Alien67 and Alien71
and Alien67 also influences Alien71. A figure
of this model appears below, as well as on page 31 of our AMOS
tutorial.
The dataset, wheaton-generated.dat,
used in the AMOS
tutorial example, is also used here in the analysis that
follows:
TITLE: Wheaton et al. Example 1: Full SEM
DATA:
FILE IS U:\Projects\Documentation\Mplus\wheaton-generated.dat
;
VARIABLE:
NAMES ARE educ
sei
anomia67
powles67
anomia71
powles71 ;
USEVARIABLES ARE educ - powles71
;
ANALYSIS: TYPE = general ;
MODEL: ses BY educ@1 sei ;
alien67 BY anomia67@1 powles67 ;
alien71 BY anomia71@1 powles71 ;
alien67 ON ses ;
alien71 ON ses alien67 ;
OUTPUT: standardized sampstat
;
The syntax for this analysis is similar to that of the confirmatory
factor analysis example shown in subsection
1 above. The only noteworthy difference is the use of the
ON keyword in the MODEL command to specify the
regression relationships among the latent variables; the WITH
keyword is used to specify correlations or covariances among
variables. In this example, the alien67 latent variable
is regressed on the SES latent variable. Similarly, the
alien71 latent variable is regressed on both the SES
and alien67 latent variables. The model fit statistics
appear below:
TESTS OF MODEL FIT
Chi-Square Test of Model Fit
Value
76.184
Degrees
of Freedom
6
P-Value
.0000
...output deleted...
RMSEA (Root Mean Square Error Of Approximation)
Estimate
.112
90
Percent C.I.
.090 .135
Probability
RMSEA <= .05 .000
The statistically significant chi-square test of absolute model
fit coupled with the poor RMSEA fit statistic value suggest
that this model may need some modification before it fits the
data well. The model fit and r-square tables appear below.
MODEL RESULTS
Estimates S.E. Est./S.E.
Std StdYX
SES BY
EDUC
1.000 .000
.000 2.420 .784
SEI
.592 .043 13.694
1.433 .683
ALIEN67 BY
ANOMIA67
1.000 .000
.000 2.929 .816
POWLES67
.823 .038 21.734
2.409 .793
ALIEN71 BY
ANOMIA71
1.000 .000
.000 2.989 .843
POWLES71
.825 .039 21.305
2.465 .778
ALIEN67 ON
SES
-.759 .062 -12.235
-.627 -.627
ALIEN71 ON
SES
-.172 .064 -2.689
-.139 -.139
ALIEN67
.710 .056 12.609
.696 .696
Residual Variances
EDUC
3.677 .416
8.839 3.677 .386
SEI
2.345 .172 13.651
2.345 .533
ANOMIA67
4.301 .364 11.807
4.301 .334
POWLES67
3.422 .260 13.150
3.422 .371
ANOMIA71
3.637 .369
9.849 3.637 .289
POWLES71
3.951 .289 13.681
3.951 .394
ALIEN67
5.201 .495 10.516
.606 .606
ALIEN71
3.352 .382
8.781 .375 .375
Variances
SES
5.854 .557 10.515
1.000 1.000
R-SQUARE
Observed
Variable R-Square
EDUC
.614
SEI
.467
ANOMIA67
.666
POWLES67
.629
ANOMIA71
.711
POWLES71
.606
Latent
Variable R-Square
ALIEN67
.394
ALIEN71
.625
There are several noteworthy features of these tables. First,
the model results table contains residual variance estimates
for the alien67 and alien71 latent variables.
These variables are predicted by the SES latent variable,
so it makes sense that the residual or unexplained variance
is due to factors other than SES in the model. Because
SES is not predicted by any other variables, its variance
is estimated independently and is shown in the Variances
section of the model results table. The path coefficients from
SES to alien67, from SES to alien71,
and from alien67 to alien71 and their associated
standard errors, tests of significance, and standardized coefficients
also appear in the same table.
The r-square table contains r-square values for each of the
predicted latent variables, alien67 and alien71,
as well as the observed variables. Taken as a whole, these results
suggest that the model is capturing the observed variables'
variances fairly well, though the prediction of alienation in
1967 is somewhat weak as is the variance accounted for in the
SEI variable. The model may be modified, however. When
all variables are continuous, Mplus can print modification indices
that can provide an empirical basis to aid your decision to
free additional paths, means, intercepts, or variance components
to be estimated in your model. A modification index provides
the expected drop in model fit chi-square if a parameter that
is currently not free is in fact allowed to be estimated. As
always, theory should be your first guide in the decision to
modify your model. To request modification indices, add the
following keywords to the OUTPUT line:
modindices (4)
The number shown in the parentheses is the amount of chi-square
reduction necessary for Mplus to print any given modification
index. The critical chi-square statistic is 3.84 for 1 degree
of freedom at p = .05, so this example sets the cutoff
to print modification indices at 4.00. If you do not specify
a cutoff value, Mplus supplies 10.00 as the default value. The
modification indices from this model appear below.
MODEL MODIFICATION INDICES
Minimum M.I. value for printing the modification index
4.000
M.I. E.P.C. Std E.P.C. StdYX
E.P.C.
WITH Statements
POWLES67 WITH EDUC
8.381 -.574
-.574 -.061
ANOMIA71 WITH EDUC
5.626 .533
.533 .049
ANOMIA71 WITH ANOMIA67 62.098
2.091 2.091
.164
ANOMIA71 WITH POWLES67 48.629
-1.546 -1.546
-.144
POWLES71 WITH ANOMIA67 54.470
-1.693 -1.693
-.149
POWLES71 WITH POWLES67 41.262
1.233 1.233
.128
In addition to the raw modification index value (M.I.),
Mplus also prints the understandardized expected parameter change
(E.P.C.) and standardized versions of the expected parameter
change.
You can draw several immediate conclusions about the model from
this table. First, the largest raw modification indicies are
associated with correlating the residuals of the anomie and
powerlessness variables, indicating that freeing these parameters
to be estimated will result in the largest improvement in model
fit. Second, the StdYX expected parameter change values are
comparable with each other because they are standardized coefficients.
The largest of these is the correlation of anomia67 with
anomia71 (.164). The next largest value is the correlation
of anomia67 with powles71 (-.149). However, you
must ask yourself, "Is this modification theoretically sensible
and meaningful?" about any modification you plan to undertake.
You can make a case for correlating anomia67 and anomia71,
and powles67 and powles71, because these measures
are identical instruments measured on the same people at two
different time points. It is conceivable that some method or
instrument variance is shared across time on the same measurement
instruments, but not across two distinct measurement instruments.
With this information, suppose you change the MODEL command
to add two residual covariances via the WITH statement:
anomia67 with anomie71, and powles67 and
powles71. The Mplus syntax for the MODEL command
appears below.
MODEL: ses BY educ@1 sei ;
alien67 BY anomia67@1 powles67 ;
alien71 BY anomia71@1 powles71 ;
alien67 ON ses ;
alien71 ON ses alien67 ;
anomia67 WITH anomia71 ;
powles67 WITH powles71 ;
Consider the result of this modification on the model fit statistics.
TESTS OF MODEL FIT
Chi-Square Test of Model Fit
Value
7.826
Degrees
of Freedom
4
P-Value
.0978
...output deleted...
RMSEA (Root Mean Square Error Of Approximation)
Estimate
.032
90
Percent C.I.
.000 .065
Probability
RMSEA <= .05 .782
The chi-square test of overall model fit is not signicant and
the RMSEA value is well below the recommended .06 cutoff that
indicates good model fit, so you conclude that your modified
model fits the data well (the value of .065 for the upper bound
of the 90 percent confidence interval for the RMSEA suggests
that the model could be improved even more if you wished to
pursue further model modifications). If you use them properly,
model modification indices are a powerful tool in your analytic
toolbox. The following points about model modification indices
are worth considering:
- Model modification should always be informed by theory.
- The more modifications you perform on any given model,
the more likely the results are to be sample specific (i.e.,
results won't generalize to new samples).
- Mplus model modification indices are available when you
use full information maximum likelihood (FIML) to handle
missing data.
- Mplus model modification indices are not available for
models that contain categorical outcome variables. Instead,
request tech2 on the OUTPUT to obtain
unstandardized first order derivatives that may be used
as approximate guides for modification of models containing
categorical outcomes.
Section 6: Advanced Models
Although Mplus can fit many standard models and it contains
some useful features lacking in other SEM programs at the time
of this writing (e.g., FIML missing data handling with exploratory
factor analysis, modification indices with FIML missing data
handling for structural equation and confirmatory factor analysis
models), Mplus advanced modeling features are its most distinctive
trademark. A full treatment of Mplus's advanced modeling features
is beyond the scope of this tutorial, but several representative
examples appear below.
1. Multiple Group
Analysis
Recall the first confirmatory factor analysis example that features
data from 145 students from the Grant-White School contained
in the data file grant.dat. 72 of those
students are male whereas 73 students are female. Suppose you
decide to investigate the equality of the factor structure across
the two groups of students. You can use Mplus to perform one
or more multiple group analyses in which the parameters
of your choosing are stipulated to be equal across the two groups
of children. For instance, suppose you wanted to test the equality
of the factor loading and factor variances and covariance
values for males and females. The Mplus command file shown below
performs this test.
TITLE: Grant-White School: Multiple Group CFA
DATA:
FILE IS U:\Projects\Documentation\Mplus\grant.dat
;
VARIABLE:
NAMES ARE visperc
cubes
lozenges
paragrap
sentence
wordmean
gender ;
USEVARIABLES ARE visperc - wordmean
;
GROUPING = gender (1=males 2=females);
ANALYSIS: TYPE = mgroup ;
MODEL: visual BY visperc@1
cubes lozenges ;
verbal BY paragrap@1 sentence wordmean ;
visual (1) ;
verbal (2) ;
visual WITH verbal (3) ;
OUTPUT:
standardized sampstat ;
Several new elements of this program are immediately apparent.
First, the GROUPING = option for the VARIABLE
command tells Mplus which variable in the database contains
the information about group membership. For each value of the
grouping variable, you supply a name that Mplus uses to define
separate groups in the analysis. The ANALYSIS command
contains an mgroup keyword that lets Mplus know
you are specifying a multiple group analysis. Use the GROUPING
= option for raw data; use the mgroupANALYSIS
keyword when you input summary data such as covariance matrices
for each group. Both multiple group specification methods are
included in this example for illustrative purposes, though only
the GROUPING = option is required to run the command
file because you input raw data.
By default Mplus assumes that the following specified parameter
estimates are equal across multiple groups:
- Factor loadings
- Intercepts of continuous outcome variables
- Thresholds of categorical outcome variables
That is, any model that contains factor loadings, intercepts,
or thresholds will assume their estimates are identical across
the multiple groups contained in the analysis. For instance,
in this example the four specified factor loading values are
assumed to be equal across the two groups. By contrast, parameter
estimates that are not specified in the MODEL statement
are allowed to vary across the groups. In this analysis each
of the residual variances of the six observed variables will
differ across the two groups.
The factor's variances and covariances are not assumed to be
equal across the two groups by default, so you can equate the
parameter estimate values across the two groups by using Mplus
equality constraints. You can specify which parameters you want
to be held equal across the two groups by assigning a number
in parentheses to each set of equal parameters. For example,
in the program shown above, you assigned the visual factor
variance a value of (1). Mplus thus estimates a single
factor loading common to both groups.
The output from the analysis appears below.
SUMMARY OF ANALYSIS
Number of groups
2
Grant-White School: Multiple Group CFA
Number of observations
Group MALES
72
Group FEMALES
73
Number of y-variables
6
Number of x-variables
0
Number of continuous latent variables
2
...output deleted...
TESTS OF MODEL FIT
Chi-Square Test of Model Fit
Value
22.346
Degrees
of Freedom
23
P-Value
.4994
...output deleted...
RMSEA (Root Mean Square Error Of Approximation)
Estimate
.000
90
Percent C.I.
.000 .093
Mplus initially reports the number of groups and the number
of cases within each group. Though not shown here in the interests
of conserving space, Mplus also displays the sample statistics
for each group separately. Since the obtained chi-square model
fit statistic (22.346) is smaller than its degrees of freedom
(23) and the RMSEA is well below the cutoff value of .06, you
conclude the model fits the data very well. One possible exception
to this interpretation arises from the RMSEA upper bound value
of .093, which exceeds the .06 cutoff recommended by Hu and
Bentler (1999). Overall, however, the equality of factor loadings
and factor variance-covariance structure for boys and girls
appears to be a reasonable assumption.
The model results table output by Mplus features the factor
loadings, factor variances, factor intercorrelations, and residuals
variances for each group. Notice that the factor loadings' unstandardized
regression coefficients and standard errors are identical for
the boys' group and the girls' group. The variances of the visual
and verbal factors are also identical across the two
samples, as is the covariance between the two factors. By contrast,
the residual variance estimates are not the same for the two
groups because these parameters were not listed in the model
specification.
MODEL RESULTS
Estimates S.E. Est./S.E.
Std StdYX
Group MALES
VISUAL BY
VISPERC
1.000 .000
.000 4.339 .612
CUBES
.555 .116
4.780 2.407 .527
LOZENGES
1.384 .263
5.262 6.005 .703
VERBAL BY
PARAGRAP
1.000 .000
.000 2.865 .881
SENTENCE
1.312 .116 11.344
3.759 .844
WORDMEAN
2.272 .200 11.363
6.511 .825
VISUAL WITH
VERBAL
6.896 1.698
4.060 .555 .555
Residual Variances
VISPERC
31.503 6.807
4.628 31.503 .626
CUBES
15.047 2.926
5.142 15.047 .722
LOZENGES
37.000 9.806
3.773 37.000 .506
PARAGRAP
2.366 .694
3.408 2.366 .224
SENTENCE
5.727 1.387
4.127 5.727 .288
WORDMEAN
19.950 4.513
4.421 19.950 .320
Variances
VISUAL
18.827 5.476
3.438 1.000 1.000
VERBAL
8.210 1.321
6.217 1.000 1.000
Group FEMALES
VISUAL BY
VISPERC
1.000 .000
.000 4.339 .648
CUBES
.555 .116
4.780 2.407 .558
LOZENGES
1.384 .263
5.262 6.005 .767
VERBAL BY
PARAGRAP
1.000 .000
.000 2.865 .858
SENTENCE
1.312 .116 11.344
3.759 .796
WORDMEAN
2.272 .200 11.363
6.511 .827
VISUAL WITH
VERBAL
6.896 1.698
4.060 .555 .555
Residual Variances
VISPERC
26.004 5.695
4.566 26.004 .580
CUBES
12.834 2.482
5.171 12.834 .689
LOZENGES
25.191 7.820
3.221 25.191 .411
PARAGRAP
2.947 .816
3.609 2.947 .264
SENTENCE
8.164 1.795
4.549 8.164 .366
WORDMEAN
19.614 4.749
4.130 19.614 .316
Variances
VISUAL
18.827 5.476
3.438 1.000 1.000
VERBAL
8.210 1.321
6.217 1.000 1.000
R-SQUARE
Group MALES
Observed
Variable R-Square
VISPERC
.374
CUBES
.278
LOZENGES
.494
PARAGRAP
.776
SENTENCE
.712
WORDMEAN
.680
Group FEMALES
Observed
Variable R-Square
VISPERC
.420
CUBES
.311
LOZENGES
.589
PARAGRAP
.736
SENTENCE
.634
WORDMEAN
.684
It is worth noting that you can constrain parameters to be equal
for a single group analysis in Mplus by assigning two or more
parameters listed within the MODEL command a unique
number, much as you did in the example shown above. It is therefore
possible to impose between and within-groups constraints simultaneously
using Mplus.
You can also impose equality constraints or custom model specifications
within specific groups in a multiple group analysis by referring
to the group's name. For instance, if you wanted to equate the
residual variances for the six variables for males only, you
could modify the model statement to read as follows:
MODEL: visual BY visperc@1
cubes lozenges ;
verbal BY paragrap@1 sentence wordmean ;
visual (1) ;
verbal (2) ;
visual WITH verbal (3) ;
MODEL males: visperc - wordmean
(4) ;
This model constrains the residual variance values of the six
observed variables for males to be equal, but the females' residual
variances are allowed to remain unique for each measured variable.
For more information on multiple group analysis, including cautionary
notes regarding multiple group analysis, see AMOS
FAQ #3: Multiple group analysis.
2. Multilevel Models
Investigators often draw data from sources that feature a hierarchical
or multilelvel structure such as students nested within
classrooms, patients residing in hospitals, children grouped
within a family, individuals grouped within couples, etc. In
recent years, specialized software such as HLM
and MLWin have been developed to fit regression and related-models
(e.g., ANOVA, ANCOVA, MANOVA, and MANCOVA) to such databases
because many statistical software packages such as SPSS and
SAS assume every observation is independent of the observations
that precede and follow it (some exceptions to this general
rule are the MIXED procedure in SAS and the LISREL multilevel
module, both of which may be used to fit multilevel regression
models). In situations where individuals are members of some
type of larger aggregate or cluster (e.g., families, couples,
classrooms), this independence assumption can be and
often is violated. Violations of the independence assumption
can seriously degrade the results from an analysis conducted
on multilevel data.
Although specialized software products such as HLM and related
programs permit multilevel regression analyses, Mplus features
a latent variable-based approach to multilevel modeling that
has the following benefits:
- Assessment of overall model fit using the usual maximum-likelihood
chi-square test statistic when cluster sizes are equal,
as well as the MLM and MLMV robust estimator options when
cluster sizes are not equal (the default estimator is mlm).
- Latent variables in the analysis with the concomitant
purging of measurement errors.
- The construction and testing of measurement models.
- Automatic sorting of the input data and construction
of the appropriate between and within-groups covariance
matrices used in the analysis.
- Specification of parellel process models in which multiple
sets of repeatedly-measured variables are analyzed, with
each set having its own growth parameters.
- Latent growth factors may predict other variables and
may in turn be predicted by other variables in the model.
- Separate model specifications are permissible for each
level of the analysis.
Mplus accounts for the effect of a single clustering variable
by calculating two separate covariance matrices, a between cluster
matrix and a pooled within-cluster covariance matrix. Taken
together, these matrices represent the total variation among
the observed variables included in the model. Mplus may also
be used to address the issue of cluster-sampled (i.e., non-random
sampled) data using a similar mechanism. Fortunately, as noted
above, you need only supply Mplus with the input data and the
name of the clustering variable. Mplus handles data sorting
and computation of the appropriate input matrices internally.
An example of a multilevel latent growth analysis appears below.
It is based on a more complex example that can be found on the
Mplus Web site. See
Muthén (1997) for related examples. The data
are also available for download (note: this database is sizeable;
you may want to download it via a fast Internet connection).
In this example, Y11 through Y14 are observed variables, E1
through E4 are residual variance estimates, Level1 is the random
intercept, and Trend1 is a random slope variable.
In the model diagram, the level or intercept variable is linked
to each observed variable via fixed coefficients of 1.00. The
trend or slope latent variable's first two coefficients are
fixed at 1.00 and 2.00, respectively, followed by two free parameters,
b3 and b4. The two free parameters have start
values of 2.5 and 3.5, respectively. The level and trend are
allowed to correlate. The Mplus model specification appears
next:
TITLE: Multilevel latent growth model (based on Mplus example
program)
DATA:
FILE IS
u:\projects\documentation\mplus\comp.dat;
FORMAT is
3f8 f8.4 8f8.2 3f8 2f8.2
VARIABLE:
NAMES ARE g1 g2 cluster g3
y11-y14
y21-y24
x1-x5;
USEOBS = (x1 EQ 1 AND g1 EQ 2);
MISSING = ALL (999);
USEVAR = y11-y14 ;
CLUSTER = cluster;
DEFINE: y11 = y11/5;
y12 = y12/5;
y13 = y13/5;
y14 = y14/5;
ANALYSIS: TYPE = twolevel;
MODEL:
%BETWEEN%
level1b BY y11-y14@1;
trend1b BY y11@0 y12@1 y13*2.5 y14*3.5;
[y11-y14@0];
[level1b-trend1b];
level1b WITH trend1b ;
%WITHIN%
level1w BY y11-y14@1;
trend1w BY y11@0 y12@1 y13*2.5 y14*3.5;
level1w WITH trend1w ;
OUTPUT: sampstat standardized
;
In the interests of conserving space, this program makes use
of several Mplus shortcuts. First, the DATA command illustrates
the use of the FORTRAN FORMAT statement to read the variables
from the large data file efficiently, as recommended by the
Mplus manual. The USEOBS command limits the observations
to the subset of cases of interest for this analysis.
The first multilevel analysis command is the CLUSTER
command. The CLUSTER command identifies which variable
in the database denotes group or cluster membership. In this
example, the variable's name is cluster. Following the
CLUSTER command is the DEFINE command. DEFINE
allows you to rescale the observed variables so that Mplus is
more likely to converge when it fits the multilevel model to
the database (multilevel models often have more difficultly
converging than single-level models).
The ANALYSIS command defines the type of analysis as
twolevel. This option tells Mplus that you are
fitting a two-level model to the data. At present, Mplus can
only fit multilevel models with a single clustering variable,
though Mplus can fit some three-level models if you consider
the third level of the model to consist of equally-spaced repeated
measurements of the observed variables. As mentioned previously,
you may use ml, mlm, or mlmv
as estimator options for multilevel models. If you select the
ml estimator, Mplus produces RMSEA model fit statistics
in addition to the familiar chi-square test of model fit. Use
the ml estimator option only if cluster sizes
are equal and it is reasonable to assume joint multivariate
normality of the model residuals; otherwise, use the default
mlm estimator or the optional mlmv
estimator.
The MODEL command contains the model specification statements
for the between and within-cluster components of the model.
The between-cluster model specification is listed under the
%BETWEEN% subcommand. Notice that any mean and intercept
structure specifications occur here; these occur at the between
level only. The %WITHIN% subcommand then lists the model
specification for the within-cluster model for individuals in
the dataset.
The output from this analysis appears below, with some output
deleted in the interest of conserving space. The first displayed
output is the summary of data, which displays the number of
clusters and the ID numbers contained within clusters of a given
size. For instance, two clusters contain seven cases each. These
clusters are cluster number 103 and cluster number 132.
SUMMARY OF DATA
Number of clusters
50
Size (s)
Cluster ID with Size s
2
114
3
136
6
304
7
103 132
9
102 109
10
305
11
111
14
134
15
116 106
16
118 138 110
105
17
101 128
18
133 131 122
19
303 124 146
20
147 137 307
21
129 141 145
22
144 127 142
143
23
139 308
24
119
25
120 121 112
123
26
140
27
301 108 117
29
135
34
104
35
115
40
302
41
309
Average cluster size 19.609
Mplus also displays the intraclass correlations of the observed
variables. The intraclass correlation assesses the level of
variance in the observed variable that is attributable to membership
in its cluster. Even small intraclass correlations suggest the
need for a multilevel analysis. In this analysis, the amount
of variance attributable to cluster membership ranges from 15%
to 20%, suggesting that a multilevel analysis is required.
Estimated Intraclass Correlations
for the Y Variables
Intraclass
Intraclass
Intraclass
Variable Correlation
Variable Correlation Variable Correlation
Y11
.206 Y12
.150 Y13
.167
Y14
.165
The overall test of model fit is satisfactory, as is the RMSEA
information.
TESTS OF MODEL FIT
Chi-Square Test of Model Fit
Value
7.561*
Degrees
of Freedom
4
P-Value
.1087
The model results appear below. The results are divided by level.
Mplus first outputs the results for the between-cluster portion
of the model:
MODEL RESULTS
Estimates S.E. Est./S.E.
Std StdYX
Between Level
LEVEL1B BY
Y11
1.000 .000
.000 .687 .923
Y12
1.000 .000
.000 .687 .914
Y13
1.000 .000
.000 .687 .842
Y14
1.000 .000
.000 .687 .764
TREND1B BY
Y11
.000 .000
.000 .000 .000
Y12
1.000 .000
.000 .027 .036
Y13
2.432 .173 14.026
.065 .080
Y14
3.458 .256 13.519
.092 .103
LEVEL1B WITH
TREND1B
.038 .011
3.369 2.077 2.077
Residual Variances
Y11
.082 .031
2.668 .082 .148
Y12
.016 .013
1.264 .016 .029
Y13
.005 .010
.509 .005 .007
Y14
.065 .028
2.337 .065 .080
Variances
LEVEL1B
.472 .087
5.450 1.000 1.000
TREND1B
.001 .003
.282 1.000 1.000
Means
LEVEL1B
10.557 .114
92.953 15.368 15.368
TREND1B
.522 .046 11.427
19.561 19.561
Intercepts
Y11
.000 .000
.000 .000 .000
Y12
.000 .000
.000 .000 .000
Y13
.000 .000
.000 .000 .000
Y14
.000 .000
.000 .000 .000
Mplus then displays the corresponding model results for the
within-cluster level of the model:
Within Level
LEVEL1W BY
Y11
1.000 .000
.000 1.447 .897
Y12
1.000 .000
.000 1.447 .863
Y13
1.000 .000
.000 1.447 .785
Y14
1.000 .000
.000 1.447 .689
TREND1W BY
Y11
.000 .000
.000 .000 .000
Y12
1.000 .000
.000 .193 .115
Y13
2.709 .826
3.281 .524 .284
Y14
4.237 1.417
2.991 .820 .390
LEVEL1W WITH
TREND1W
.082 .033
2.466 .294 .294
Residual Variances
Y11
.507 .052
9.791 .507 .195
Y12
.516 .038 13.567
.516 .183
Y13
.580 .045 12.885
.580 .171
Y14
.943 .167
5.646 .943 .214
Variances
LEVEL1W
2.093 .109 19.199
1.000 1.000
TREND1W
.037 .027
1.390 1.000 1.000
Though this analysis produced similar findings for the between
and within-cluster components of the model, this is not always
the case. It is often the case that you will need different
model specifications for the between versus the within-cluster
sections of the model's specification.
It is also worth noting that despite the congruence between
the within and the between-cluster components of this model,
if you fit the model as a single level model (using the mlm
estimator option), you obtain the following results:
MODEL RESULTS
Estimates S.E. Est./S.E.
Std StdYX
LEVEL BY
Y11
1.000 .000
.000 1.606 .903
Y12
1.000 .000
.000 1.606 .870
Y13
1.000 .000
.000 1.606 .797
Y14
1.000 .000
.000 1.606 .708
TREND BY
Y11
.000 .000
.000 .000 .000
Y12
1.000 .000
.000 .227 .123
Y13
2.451 .130 18.812
.556 .276
Y14
3.496 .195 17.901
.793 .350
LEVEL WITH
TREND
.124 .031
4.000 .341 .341
Residual Variances
Y11
.582 .055 10.593
.582 .184
Y12
.528 .044 12.071
.528 .155
Y13
.565 .045 12.614
.565 .139
Y14
1.061 .112
9.443 1.061 .206
Variances
LEVEL
2.580 .137 18.881
1.000 1.000
TREND
.051 .012
4.317 1.000 1.000
Means
LEVEL
10.557 .057 184.473
6.572 6.572
TREND
.517 .032 15.958
2.280 2.280
Intercepts
Y11
.000 .000
.000 .000 .000
Y12
.000 .000
.000 .000 .000
Y13
.000 .000
.000 .000 .000
Y14
.000 .000
.000 .000 .000
Although the chi-square model fit test for this model indicates
the model fits the data well (chi-square = 3.697 with 3 DF,
p = .295), you can see that all variance estimates are
statistically significant. This finding does not take into account
the non-independence of individuals who are grouped within the
same cluster; it thus stands in contrast to the more appropriate
multilevel model that shows a non-significant variance component
for the trend latent variable on both the between and within-cluster
levels.
The following notes are worth considering before you specify
a multilevel model and fit it to your data using Mplus.
- On occasion, you may need to supply starting values to
Mplus to obtain a solution that converges. Assigning
reasonable starting values to variance estimates may be
helpful. Another approach that often yields satisfactory
starting values is to fit a single-level model to the entire
sample, ingoring clustering; take the parameter estimates
from that model and supply them as the starting values for
the multilevel model.
- Each analysis should have at least 30 to 50 clusters.
- Variables measured at the group or cluster level (e.g.,
family size) may only be used at that level of the analysis.
- Variables measured at the individual or within-cluster
level exist at both levels of the analysis and need to be
considered in both the between and within-cluster model
specifications.
- FIML missing data handling is not available for multilevel
models; missing data issues must be resolved prior to the
multilevel analysis.
References
Hu, L., & Bentler, P.M. (1999). Cutoff criteria
in fix indexes in covariance structure analysis: Conventional
criteria versus new alternatives. Structural Equation Modeling,
6(1), 1-55.
Muthén, B. (1997). Latent variable modeling with longitudinal
and and multilevel data. In A. Raftery (ed.), Sociological
Methodology 1997 (pp. 453-480). Boston: Blackwell Publishers.
Muthén, B., du Toit, S.H.C., & Spisic, D. (1997).
Robust inference using weighted least squares and quadratic
estimating equations in latent variable modeling with categorical
and continous outcomes. Accepted for publication in Psychometrika.
Muthen, L.K. and Muthen, B.O. (1998). Mplus User's
Guide. Los Angeles: Muthen & Muthen.
Wheaton, B., Muthén, B., Alvin, D., & Summers,
G. (1977). Assessing reliability and stability in panel models.
In D.R. Heise (Ed.): Sociological Methodology. San
Francisco: Jossey-Bass.
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