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Alan C. Acock University Distinguished Professor of Family Studies &
Knudson Chair for Family Research & Policy Oregon State University
College of Health and Human Sciences Summer Workshop Series
July 2010
Introduction to Mplus
A brief history LISREL (Joreskog and Sorbom) was being
developed in the late 1960s and released commercially in the early 1970s Originally relied on entering 8 matrices specifying
all the parameters that were being estimated or fixed at a certain value
Today has a graphic interface that generates the commands from a path diagram
Extremely capable alternative to Mplus
Alan C. Acock, July, 2010 Introduction to MPlus 1
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A brief history EQS (Bentler) was developed much later and
replaced the matrices with writing out a separate equation for each relationship It now has a nice “Diagrammer”
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A brief history AMOS (now an SPSS product) was developed
based on a graphic interface
It has the slowest introduction of new capabilities
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A brief history--Mplus Version 6 April 2010 Version 5 November 2007 Version 4 February 2006 Version 3 March 2004 Version 2 February 2001 Version 1 November 1998
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A brief history--MPlus Very rapid development Late development allowed a non graphic
interface to be highly efficient Destroys the idea that a picture is worth 1000
words—Develop statistical applications, not drawing
Need separate drawing program, but this is best for publication quality Omni Graffle (Mac) for most figures here Office Visio or Open Office Draw (PC)
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A Brief History--Mplus
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Bengt Muthén is the statistician Linda Muthén is the language/interface/business Several people have contributed programming Economy of Scale idea is reversed Microsoft has 40,000 programmers so it takes a long time to
make a useful change Mplus has a couple programmers so it rapidly adds features Many new features are added between versions
Buying Mplus
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Greatly reduced Student prices There are three modules (they apparently learned
this module idea from SPSS). You probably want all three
There is an annual maintenance and this lets you Get “free” support Get “free” updates I started with Mplus 3.0 and have only paid for
the annual maintenance fee ($175) since then
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Resources for Learning Barbara Byrne. (2010). Structural Equation
Modeling with MPlus: Basic Concepts, Applications, and Programming. (Was to be available July 1
www.statmodel.com Large, 752 page, User’s Manual as pdf file Short courses on video There are 8 of these, each is one day long Download handouts to follow videos
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Resources for Learning www.ats.ucla.edu/stat/seminars/ UCLA has several online examples and videos We will utilize files from the Mplus manual for many
of our examples. These typically involve simulated data. Sometimes we well assign hypothetical variable names to make these somewhat realistic
Brown, Timothy. (2006). Confirmatory Factor Analysis for Applied Research. N.Y.: Guilford
Kline, Rex. (2010). Principles & practices of structural equation modeling (3rd ed.). N.Y. Guilford
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The Mplus Interface
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The Mplus Interface
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The window shown above is the input window You write Mplus programs in this window to read the data to
be analyzed and to specify your model of interest You then save your Mplus program and select Run Mplus
from the Mplus menu to submit your program to the Mplus engine for processing
► File! ► Open! This is located at c:\Progrm Files\Mplus\Mplus Examples\User’s Guide Examples\!
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Mplus Command Structure
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TITLE: (optional unless you want to know what the file is intended to do)
DATA: (required), VARIABLE: (required), DEFINE: (some data transformations are available) SAVEDATA: (used for specialized applications) ANALYSIS: (for special analyses such as EFA MODEL: (a series of equations) OUTPUT: (many options are available) MONTECARLO: (used for simulations, power analysis)
Mplus Commands
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The TITLE command allows you to specify a title This can go on and on for many lines and usually should Everything is a Title until a command name appears at the start of a new
line I like to put the file name as the first line of a title
The DATA command specifies where Mplus will locate the data & the format of the data. 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., Stata, SAS, SPSS, or
Excel), you will need to convert it to one of the formats with which Mplus can work
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Here is the Data file
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Data
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We have labeled missing values with a -9. Easiest to pick one value that will work for all variables—can be any number or a dot
Notice we have one observation, case 13, that has a missing value on all variables
The data happens to be in a fixed format Could be comma delimited, cvs file from Excel Missing are Variable labels, Value labels If you have the data in Stata you can use stata2mplus
to set things up for you
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Data
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stata2mplus using example, replace !
This creates two files: classnsfh.inp that will run a basic analysis in Mplus and classnsfh.dat, a comma delimit ASCII file that Mplus can read with
all missing values coded/recoded as -9999.
1,1,1,1,2,1,1,1,1,1,1,1,2,1,1
3,2,2,3,2,3,2,2,2,2,2,2,2,2,2
. . .
1,1,1,-9999,1,2,2,1,2,2,2,2,2,1,1
3,3,3,3,3,3,3,3,3,3,3,3,3,3,3
2,2,1,2,1,2,2,1,1,2,1,-9999,1,1,1
Note, recommended to make a separate folder for each project such. Save data and Mplus programs in that folder
Mplus commands
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The VARIABLE command names variables These must be in the identical order to the way Stata/SAS/SPSS
wrote the data file (common mistake) Mplus variable names may not have more than 8 characters.
Change variable names to be 8 characters or less or you will get error messages.
The ANALYSIS command tells Mplus what type of analysis to perform. Often not needed
Many analysis options are available. Some of these such as Type = EFA make additional
commands unnecessary.
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Mplus Rules
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All commands (Title, Data, etc.) begin on a new line
All command names must be followed by a colon For e.g., Title: The key word becomes blue Semicolons separate command options—similar to SAS The record length no longer than 80 columns Variables can contain upper and/or lower case letters Only variable names are case sensitive--SAY1 ≠ say1≠ Say1≠ SaY1!
Default Assumptions
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F1 F2
x1 x2 x3 x4 x5 x6
e1 e2 e3 e4 e5 e6
1 1
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Default Assumptions
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Mplus assumes that you either have no missing values or are using full information maximum likelihood estimation and assuming missing values are missing at random (MAR)
Parameters such as loadings can be fixed Many loadings are fixed at 0.0 in the CFA models because the item
should not load on the factor. There is no path from F1 to X4 in our figure
Fixed parameters can be “freed,” meaning you will estimate them We could add a path from to X4 or Let E1 be correlated with E4!
Fixed parameters are required to stay at a specified value, such as 0.0 or 1.0
Default Assumptions
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All free parameters are put into a vector and iterations change values of these free parameters, until the model’s fit is optimal
Unless we tell it otherwise, Mplus will fix the first indicator’s loading at 1.0 as the reference indicator (except for EFA). For example, F1X1 and F2X4 have fixed loadings of 1.0 by
default. One way to change reference indicator is to reorder variables, e.g. F1 by x2 x1 x3 makes x2 reference indicator
Good to pick a strong indicator as the reference indicator—don’t get a significance test for reference indicator
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Exploratory Factor Analysis
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F1 F2
y1 y2 y3 y4 y5 y5
e1 e2 e3 e4 e5 e6
EFA with Continuous Variables
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EFA with Continuous Variables!
TITLE: !efa1.inp!
This is an example of an exploratory!
factor analysis with continuous!
! ! !factor indicators!
DATA:! !FILE IS "c:\Mplus Examples\efa4.dat";!
VARIABLE: !NAMES ARE y1-y12;!
ANALYSIS: !TYPE = EFA 1 4;!
!! ! !ESTIMATOR = ml;!
!! ! !ROTATION = Geomin;!
OUTPUT: sampstat;!
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EFA with Continuous Variables
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The Type = EFA 1 4 tells Mplus to perform exploratory factor analysis
The 1 and 4 following the EFA specification tells Mplus to generate all possible factor solutions between and including 1 and 4
The ESTIMATOR = ml option has Mplus use the maximum likelihood estimator to perform the factor analysis (default)
This provides a chi-square goodness of fit test that the number of hypothesized factors is sufficient to account for the correlations among the six variables in the analysis
This has an exclamation mark in front of it which makes it green. Anything green is a comment and is ignored by the program. This subcommand is not necessary because maximum likelihood estimation is the default
EFA with continuous variables
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Mplus uses the geomin rotation which is oblique as its default. More traditional rotations such as varimax are available. See help for a listing of options
We do not need a MODEL: EFA 1 4 takes care of this. 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
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EFA results
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Sample correlations Root Mean Square Error of Approximation (RMSEA) Chi-square test of the one, two, three, and four factor
models Sensitive to sample size (such that large samples often
return statistically significant chi-square values) Non-normality in the input variables
Standard errors and z-tests for loadings and correlations of factors
EFA results
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How many factors? Model 1 chi-square (54 degrees of freedom) = 1052.089; p < .001 Model 2 chi-square (43 degrees of freedom) = 723.022; p < .001 Model 3 chi-square (33 degrees of freedom) = 341.268; p < .001 Model 4 chi-square (24 degrees of freedom) = 25.799; p, not sign.
Is model 4 better than model 3? Model 3 chi-square (33 degrees of freedom) = 341.268 Model 4 chi-square (24 degrees of freedom) = 25.799 Difference chi-square (9 degrees of freedom 315.469; p < .001
. display 1-chi2(df,chi-square)! . display 1-chi2(9,315.469)!
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EFA Categorical Variables
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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.
For categorical variables, Mplus automatically recodes the lowest value to zero with subsequent values increasing in units of 1.00.
While the four underlying latent factors remain continuous, the six categorical observed variables' response values are now ordered dichotomous categories.
You may use the program that appeared in the initial exploratory factor analysis example, with the following modifications, and the new data file that contains the categorical variables ex4.2.dat, as shown below.
EFA Categorical Variables
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There are two estimators. WLSMV (Weighted Least Squares, mean and variance
adjustment) is very fast and reasonably good You should use this for initial runs, default
Running this on a server used by many students, it ran in 1 second
MLR (Robust Maximum Likelihood). This is painfully slow, even for a simple and well behaved example like the one we will estimate. Save this till you are almost done
Use this when you need to test for the number of factors
This took 18 minutes to run.
Under the Analysis section you need to specify this estimator as shown below.
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EFA Categorical Program
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TITLE:! !ex4.2.inp! !This is an example of an exploratory !! ! !factor analysis with categorical factor! !indicators. It uses weighted least squares!! ! !estimation it computes tetrachoric!! ! !correlations and does the Factor analysis
! !on them. The RMSEA and chi-square!! ! !values are reported.!DATA: ! ! !FILE IS ex4.2.dat;!VARIABLE: !! !NAMES ARE u1-u12;!! !CATEGORICAL ARE u1-u12;!ANALYSIS: !! !TYPE = EFA 1 4;!!! ! !ESTIMATOR = MLR;!! ! !PROCESSORS = 4 ;!
EFA Categorical Interpretation
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Run and interpret Univariate propotions for each variable
Review 4 factor solution Chi-square is a bit problematic (Difftest requires CFA) CFI RMSEA SRMR GEOMIN (correlated solution)
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EFA Categorical Interpretation
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Review 4 factor solution Factor Correlations Residual variances Standard Errors Est./S.E. = z-test
Comparison of Continuous and Categorical Solutions
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EFA at 2 waves with factor loading invariance and correlated residuals
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Does a two factor solution for measures of internal and external locus of control change after some life event
Use large, longitudinal national survey to find a subsample of people who experienced some event such as divorce and who have their internal and external locus of control measured at a wave preceding and following divorce
The figure might look like the following where items 1-3 and 7-9 measure internal locus of control and items 4-6 and 10-12 measure external locus of control
EFA at 2 waves with factor loading invariance and correlated residuals
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EFA at 2 waves with factor loading invariance and correlated residuals
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TITLE: !ex5.26.inp!! ! !this is an example of an EFA!! ! !at two time points!! ! !with factor loading!! ! !invariance and correlated! ! ! !residuals across time !DATA: !FILE IS ex5.26.dat;!VARIABLES:NAMES ARE y1-y12;!MODEL: !f1-f2 BY y1-y6 (*t1 1);!! ! !f3-f4 BY y7-y12 (*t2 1);!! ! !y1-y6 PWITH y7-y12;!OUTPUT: !TECH1 STANDARDIZED;!
EFA at 2 waves with factor loading invariance and correlated residuals
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The unstandardized loadings are equal, but the standardized loadings are not
When comparing different people, groups, or waves, you want unstandardized coefficients
Standardized coefficients are
Soooo, Beta depends on the relationship, B, and the relative standard deviations
β = BSDy
SDx
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EFA at 2 waves with factor loading invariance and correlated residuals
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The f1-f2 by y1-y6 (*t1 1); means that factors 1 and 2 are measured by y1-y6 !
The (*t1 1) means that f1 and f2 are a set of factors labeled t1 and the 1 is used to make the loadings invariant with any other set of factors that has a 1!
The f3-f4 by y7-y12 (*t2 1); means that f3 and f4 are a set of factors called t2!
The 1 after t2 means that the loadings must be the same as for t1 since that set also had a 1!
EFA at 2 waves with factor loading invariance and correlated residuals
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The y1-y6 pwith y7-y12; allows the paired errors to be correlated. It is equivalent to y1 with y7; y2 with y8; y3 with y9; y4 with y10; y5 with y11; y6 with y12;!
There is no name for the error terms so y1 with y7 is not really y1 with y7, but e1 with e7—only way to do it without naming the error terms—drawing figure first helps.
The tech1 gives a series of LISREL type matrices showing all the parameters being estimated
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Cautions when doing EFA
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Although one or more of the observed variables may be categorical, all latent variables in the model are continuous
The analysis specification and interpretation of the output, e.g., loadings & factor correlations, is the same whether one, a subset, or all observed variables are categorical
Categorical observed variables may be dichotomous or ordered categorical outcomes of more than two levels), but nominal level observed variables with more than two categories may not be used in the analysis as outcome variables using this strategy
Sample size is more stringent than for continuous variables; typically you want a minimum of 200 cases (preferably more) to perform any analysis with categorical outcome variables
Cautions when doing EFA
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Mplus provides z-tests for all loadings and correlations Manual illustrates applications with censored and count
variables Can apply mixture models with continuous indicators
Are there sub groups that have different results Variable: Names = y1 – 12;! Classes = c(2);! Type = mixture efa 1 4 ;!
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Confirmatory Factor Analysis (CFA)
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What if you had an a priori hypothesis that the visual perception (Y1), cubes (Y2), and lozenges (Y3) variables belonged to a single factor—Visual
Whereas the paragraph (Y4), sentence (Y5), and word meaning (Y6) variables belonged to a second factor--Cognitive?
F1 will not have a loading on Y4, Y5, or Y6 & F2 will not have a loading on Y1, Y2, or Y3.
Correlation of Y1 with Y4 is loading of F1 on Y1 ✕ correlation of F1 and F2 ✕ Loading of F2 on Y4!
The diagram shown below illustrates the model visually
Confirmatory Factor Analysis (CFA)
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F1 F2
y1 y2 y3 y4 y5 y5
e1 e2 e3 e4 e5 e6
1 1
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CFA means study the correlations
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Study Correlation matrix
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Compare this to the following Y2 and Y1 have a different pattern with Y4-Y6. The
single correlation between F1 and F2 could not handle this
The fit will not be very good
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MPlus CFA program
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TITLE: !ex5.1.inp!! ! !This program runs a CFA!DATA: !FILE IS ex5.1.dat; !VARIABLE:!NAMES ARE y1-y6;!MODEL: !f1 BY y1-y3; !! ! !f2 BY y4-y6;!OUTPUT: !sampstat stdyx mod(3.84);!
CFA program
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You must define what parameters are estimated; All other parameters are assumed to be fixed. Fixed parameters are either zero or some value you set.
DEFAULTS FOR ANALSYSIS, FIML To do listwise deletion we would specify this in the DATA command
Listwise = on; Put it under DATA:!
The MODEL command allows you to specify the parameters of your model The BY keyword to define the latent variables The latent variable name appears on the left-hand of the BY whereas the
measured variables appear on the right-hand side of the BY keyword Mplus will fix the loading for the first indicator at 1.0 unless you tell it
otherwise
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CFA program
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WITH keyword—the WITH keyword to correlate the F1 latent factor with the F2 latent factor is a default By Measured by With Correlated with
We do not need F1 with F2 because that is the default. If we wanted to see how the model did with these fixed we would add the line F1 with F2@0 ;!
OUTPUT: command contains an added keyword, standardize or stdyx. This option instructs Mplus to output standardized parameter estimate values in addition to the default unstandardized values
Why is one loading fixed at 1.0?
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The default fixes the unstandardized loading of the first item after BY at 1.0
This has to do with model identification In exploratory factor analysis the variance of the factor (latent
variable) is fixed at 1.0 by the program. Given this, the program estimates the loadings
With CFA, you need to set a variance for the latent variable because the size of the loadings are scaled from the size of the variance
Setting the variance of the latent variable (factor) at 1.0 solves this problem with EFA and is an option with CFA. But, Mplus suggests a more general approach in which you fix one of the loadings of each latent variable (factor) at 1.0 with CFA
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Fixed loading more general than fixed variance
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Comparing Groups: One group might be more variable than another We might find that girls not only have higher verbal skills than boys, but
that they are either more homogeneous in these skills. An intervention that not only improves the mean outcome, but does so
in a way that makes the distribution more homogeneous is preferred In some cases we are interested in the variances of the latent variables as
an important topic and we could not study that if we fixed the variance at 1.0
Regardless of which item you pick to fix the loading at 1.0, the standardized solution will always be the same because that solution rescales the variance of the latent variable to be 1.0 and the fully standardized solution also rescales the variance of each indicator to be 1.0
Run & Interpret Selected Output
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Estimator by default is ML sampstat gives us means, covariance matrix, and correlation
matrix. Good to compare to what you had in Stata, SAS, etc. Fit Statistics, Chi square, baseline chi-square, CFI, Information
criteria, RMSEA, & SRMR! Unstandardized solution (loadings, z-tests, p’s) F2 with F1! Residual Variances Standardized on all variables STDYX!
The z-tests are different Modification indices
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A Figure of CFA results
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Second Order CFA
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Conceptually, some attitudes or ideologies are generalizations Liberalism may explain more specific forms of liberalism
such as economic liberalism, social liberalism, etc. Alienation may explain value isolation, powerlessness,
normlessness etc. Such examples are second order factor analysis where a
highly general second order factor explains the relationship between several first order factors
Any correlation between the first order factors is because they have the common cause, that being the second order factor
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Second Order Factor Analysis
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F1 F2
y1 y2 y3 y4 y5 y6
e1 e2 e3 e4 e5 e6
F3 F4
y7 y8 y9 y10 y11 y12
e7 e8 e9 e10 e11 e12
F5
2nd Order Factor Analysis
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Just like we would like to have at least three indicators for a single latent variable, we would like to have three first order factors
The first indicator of each first order factor is fixed at 1.0 The first of the first order factors has its loading on the
second order factor fixed at 1.0 The first order factors are uncorrelated, i.e., their
correlations are explained by the second order factor
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2nd Order CFA
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TITLE: !ex5.6.inp!! ! !This is an example of a second!! ! !order factor analysis!DATA: !File is ex5.6.dat;!VARIABLE:!Names are y1-y12;!MODEL:!! ! !f1 by y1-y3;!! ! !f2 by y4-y6;!! ! !f3 by y7-y9;!! ! !f4 by y10-y12;!! ! !f5 by f1-f4;!OUTPUT: !sampstat stdyx mod(3.84);
2nd Order CFA Interpretation
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Correlation matrix has significant correlations of indicators of different factors, e.g., y1-y3 with y4-y12!
Standardized solution R-square for latent variables F1-F4! These models make sense conceptually, but are
rarely a reasonable fit
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Equality Constraints
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Are items truly interchangeable. Alpha assumes that all items are equally salient to the concept being measured. That is you weight each item equally with a 1.0 weight. CFA can test & extend this: tau equivalence—All loadings are constrained to be equal Compare fit of this model to a model in which they are
unconstrained
Parallel equivalence. Tau equivalence plus all error terms are equal Very hard to achieve and often we can proceed without this
condition
Equality Constraints—Marital Satisfaction of Husbands and Wives
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Lack of Tau equivalence Women may weigh emotional support more than men
Men may weight sexual satisfaction more than women
If tau equivalence holds the latent variable has the same meaning in both groups Without this equivalence we are comparing apples and oranges.
Why compare means if the concept has a different meaning for each group?
Men may be more satisfied than women
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Equality Constraints—Marital Satisfaction of Husbands and Wives
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Equality Constraints—A little algebra
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In regression we can write:
If we examine the figure we see that each observed variable, we will call it X, for each X Where tau is the intercept, lambda-x is the matrix of
loadings, kappa is the mean of the latent variable This adds 10 parameters we need to estimate, 8
intercepts and 2 latent variable means
My = a + bMx
Mx = τ x + λxκ
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Equality Constraints—Identification
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This adds 10 parameters we need to estimate, 8 intercepts and 2 latent variable means. Include the means along with the covariance matrix Make some additional restrictions We could fix one intercept at each wave at zero Now we have added 8 means and 8 new parameters to
estimate (6 intercepts and 2 latent variable means)
Equality Constraints—Data
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We can enter the means and covariances or enter the means, SDs, and correlations
1.500 1.320 1.450 1.410 6.600 6.420 6.560 6.310 1.940 2.030 2.050 1.990 2.610 2.660 2.590 2.550 1.000 0.736 1.000 0.731 0.648 1.000 0.771 0.694 0.700 1.000 0.685 0.512 0.496 0.508 1.000 0.481 0.638 0.431 0.449 0.726 1.000 0.485 0.442 0.635 0.456 0.743 0.672 1.000 0.508 0.469 0.453 0.627 0.759 0.689 0.695 1.000
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Equality Constraints—Process
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We estimate four models, each of which includes estimating means First model estimates the means imposing the same form for
the model at both waves or for both wives and husbands This model doesn’t make a lot of sense. With unequal loadings,
the meaning of satisfaction changes with some indicators becoming more salient and others less salient
This could be interesting as, for example, sexual satisfaction may be less central and emotional support may be more satisfying in more mature marriages or for wives
Equality Constraints—Program for Form
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Note, Observed variable name without () or [] refers to its error term
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Equality Constraints—Process
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Mplus puts square brackets [] around intercepts and means, defined by context [A1@0]; [A2@0]; fix first intercepts at zero [SATIS1*]; [SATIS2*] make the latent means free [B1 B2] (4); Assigning the same number to both B1 and B2
make them equal. Since these are observed variables, these refer to their intercepts being equal
A1 A2 (7) etc. make the error terms equal as in parallel equivalence, but we have not forced the loading to be equal
Next, we add restriction that loadings are equal
Equality Constraints—Equal Loadings
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Note, A1 and A2 loadings fixed at 1.0 by Mplus, B1 & B2 (1), etc.
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Equality Constraints—Equal Intercepts
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Note. [B1 B2] 4 ; makes intercepts equal. [] for latent variable is a mean; [] for observed is an intercept
Equality Constraints—Equal Errors
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Note, A1 A2 (7) means these errors are equal. No name for errors so observed variable name without () or [], actually refers to the variables error
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Summarizing Results
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Selected Results
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Selected Results
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Path Analysis
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Path Analysis: Program
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Selected result
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Interpret fit Interpret standardized result
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Selected result
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Path model: Categorical, Censored, nominal outcome variables
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Some continuous variables are Censored either above or below Marital satisfaction on a 1-7 scale has a clump at 7 who say they
are very satisfied, but there is unobserved variance among this group with some much more satisfied than others
Some categorical variables have a binary set of options (divorced, not divorced)
Some nominal categorical variables have three our more options (not employed for pay, employed part-time, employed full-time
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Path model: Categorical, Censored, nominal outcome variables
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In following model, y1 is a censored from above continuous variable u1 is a binary categorical variable
Odds ratio is odds of being in highest category
u2 is a nominal categorical variable with 3 options, 0,1, & 2 “Odds ratio” is relative risk ratio of being in category 0 versus category 2 “Odds ratio” is relative risk ratio of being in category 1 versus category 2
We cannot estimate a standardized solution or indirect effects We use maximum likelihood robust as our estimator
Y1 censored above, u1 binary, & u2 has 3-categories
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x1
x2
x3
y1
u1
u2
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Y1 censored above, u1 binary, u2 3-categories—Program
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Note, binary called categorical; more than 2 options called nominal. Censored above use the (a) to indicate this. Censored below would use (b)!
Selected Results
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Interpretations for categorical/nominal outcomes
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Odds of being in highest category on U1 is 2.803 greater for a unit increase in X1—i.e., odds are 180.3% greater of being in highest category on U1 for each unit increase in X1!
Odds of being in highest category on U1 is only .549 as great for a unit increase in X2—i.e., odds are 45.1% lower of being in highest category for each unit increase in X2!
Relative risk ratio of being in category # 1 of U2 is 1.569 times as great as being in category #3 for each unit increase in Y1!
Relative risk of being in category #2 of U2 is 5.644 times as great as being in category #3 for each unit increase in X2!
Putting it Together—SEM Model
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SEM Program
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Selected results
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Review fit statistics Review unstandardized solution Review stdyx solution Review R-square for latent variables Review total, direct, and indirect effects
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Selected results
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Review modification indices We could reduce Chi-square, which now is Chi-square(50) = 53.492,
by about 5.265 if we allowed the error term for Y5 to be correlated with the error term for Y3
The correlation of the two errors would be about -.132—does this make sense?
We would do these one at a time Say Y5 and Y3 are pen and pencil tests and all the others are face
to face interviews New Chi-square would be approximately Chi-square(49) = 53.492 –
5.265. A reduction in Chi-square of 5.265 with one degree of freedom would be highly significant. Not much need to improve on a CFI = .997; RMSEA = .012
SEM using EFA
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We can use modification indices to add parameters one at a time to improve the measurement model
We will not get the optimum measurement model Some indicators may have a very small loading and this will
provide a better fit than fixing them at zero The added loadings can make the interpretation of the latent
variables confounded
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SEM using EFA for two latent variables
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SEM using EFA
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Ideally we probably want loadings of Y1-Y3 on F2 to be very weak and the loadings of Y1 – Y3 on F1 to be very strong
Weak loadings may be more realistic than asserting that the loadings are exactly 0.0
May avoid correlating some error terms Here is the program
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SEM using EFA—Program
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Key things to remember
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BY means measured by. It is a loading of indicator on latent ON is a structural path between two variables—Direct effect WITH means correlated with. F1 with F2 or a1 with b1!
Y ind X1 X2; Indirect effects of X1 & X2 on Y! [variable] is the mean if the variable is a latent variable [variable] is an intercept if the variable is observed Variable, e.g., var1 with var2, refers to the error in var1 and var2. Errors are not given names in the program so this is the only way to show them