Demonstration of SEM-based
IRT in Mplus
Frances M. Yang, Ph.D.1,2
Doug Tommet, M.S.1
Richard N. Jones, Sc.D.1
1Institute for Aging Research, Hebrew SeniorLife and Beth Israel Deaconess
Medical Center, Division of Gerontology, HMS and
2Department of Psychiatry, Brigham and Women’s Hospital, HMS
francesyang@hrca.harvard.edu
August 23, 2007
1
Overview
•
•
•
•
Section 1—Introduction to Mplus
Section 2—Exploratory Factor Analysis
Section 3 – Basic Assumptions of IRT
Section 4—Confirmatory Factor Analysis
– 2 PL Model
• Section 5 – Questions and Discussion
2
Section 1
Introduction to Mplus
3
• www.statmodel.com
• Used to be LISCOMP, owes lineage to LISREL
• Does just about everything other continuous
latent variable / structural equation software
implement (LISREL, EQS, AMOS, CALIS)
• Plus, very general latent variable modeling
–
–
–
–
Continuous latent variables (latent traits)
Categorical latent variables (latent classes, mixtures)
Missing data
Estimation with data from complex designs
• Expensive, demo version available
4
Formatting Data for Mplus
• Individual-level data
• Summary Data (correlations, covariances,
means, standard deviations)
• ASCII Data
– Raw text
– Fixed, Free format
5
http://www.ats.ucla.edu/stat/mplus/faq/default.htm
6
How to write a Mplus command file
•
•
•
•
Get the Users Manual
Print it, read it, live it, love it
Find a similar example
Hack the example to suit your problem
7
Mplus Commands
•
•
•
•
•
•
TITLE
DATA
VARIABLE
ANALYSIS
MODEL
OUTPUT
•
•
•
•
DEFINE
PLOT
SAVEDATA
MONTECARLO
8
Section 2
Exploratory Factor Analysis
9
Exploratory Factor Analysis in
Mplus (v.4)
• Observed outcomes variables can be:
– continuous
– binary
– ordered categorical (ordinal)
– combinations of these variable types
10
Mplus Input File
TITLE:
This is an example of an exploratory
factor analysis with dichotomous indicators
DATA:
FILE IS S:\project~1\dif\Short~1\Data\cesd.csv;
VARIABLE: NAMES =depress lonely sad effort restless
nogetgo noenergy nohappy noenjoy age gender
ethnic edu;
USEVARIABLES ARE depress-noenjoy;
CATEGORICAL=depress-noenjoy;
MISSING ARE ALL (-9999) ;
ANALYSIS: TYPE =missing efa 1 3 ;
ESTIMATOR=wlsmv;
11
Mplus Output
Mplus VERSION 4.2
MUTHEN & MUTHEN
05/29/2007
3:31 PM
INPUT INSTRUCTIONS
TITLE: This is an example of an exploratory factor analysis
with dichotomous indicators
DATA: FILE IS S:\projectdata1\dif\Short~\Data\cesd.csv;
VARIABLE: NAMES =depress lonely sad effort restless nogetgo noenergy nohappy noenjoy age
gender ethnic edu;
USEVARIABLES ARE depress-noenjoy;
CATEGORICAL=depression-noenjoy;
MISSING ARE ALL (-9999) ;
ANALYSIS: TYPE =missing efa 1 3 ;
ESTIMATOR=wlsmv;
INPUT READING TERMINATED NORMALLY
This is an example of an exploratory factor analysis with
dichotomous indicators
SUMMARY OF ANALYSIS
Number of groups
Number of observations
Number of dependent variables
Number of independent variables
Number of continuous latent variables
1
9448
9
0
0
Observed dependent variables
12
Binary and ordered categorical (ordinal)
DEPRESS
LONELY
SAD
EFFORT
NOENERGY
NOHAPPY
NOENJOY
Estimator
Maximum number of iterations
Convergence criterion
Maximum number of steepest descent iterations
RESTLESS
NOGETGO
WLSMV
1000
0.500D-04
20
Input data file(s)
S:\projectdata1\dif\Short~\Data\cesd.csv;
Input data format
FREE
SUMMARY OF DATA
Number of patterns
1
COVARIANCE COVERAGE OF DATA
Minimum covariance coverage value
0.100
PROPORTION OF DATA PRESENT
DEPRESS
LONELY
SAD
EFFORT
RESTLESS
NOGETGO
NOENERGY
NOHAPPY
NOENJOY
Covariance Coverage
DEPRESS
LONELY
________
________
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
SAD
________
EFFORT
________
RESTLESS
________
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
13
PROPORTION OF DATA PRESENT
DEPRESS
LONELY
SAD
EFFORT
RESTLESS
NOGETGO
NOENERGY
NOHAPPY
NOENJOY
Covariance Coverage
DEPRESS
LONELY
________
________
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
NOGETGO
NOENERGY
NOHAPPY
NOENJOY
Covariance Coverage
NOGETGO
NOENERGY
________
________
1.000
1.000
1.000
1.000
1.000
1.000
1.000
SAD
________
EFFORT
________
RESTLESS
________
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
NOHAPPY
________
NOENJOY
________
1.000
1.000
1.000
SUMMARY OF CATEGORICAL DATA PROPORTIONS
DEPRESS
Category
Category
LONELY
Category
Category
SAD
Category
Category
EFFORT
Category
Category
RESTLESS
Category
Category
NOGETGO
Category
Category
1
2
0.834
0.166
1
2
0.808
0.192
1
2
0.792
0.208
1
2
0.755
0.245
1
2
0.728
0.272
1
2
0.769
0.231
14
NOENERGY
Category
Category
NOHAPPY
Category
Category
NOENJOY
Category
Category
1
2
0.550
0.450
1
2
0.887
0.113
1
2
0.931
0.069
RESULTS FOR EXPLORATORY FACTOR ANALYSIS
1
1
EIGENVALUES FOR SAMPLE CORRELATION MATRIX
1
2
3
4
________
________
________
________
5.164
1.041
0.794
0.592
EIGENVALUES FOR SAMPLE CORRELATION MATRIX
6
7
8
________
________
________
0.308
0.250
0.193
5
________
0.488
9
________
0.169
EXPLORATORY ANALYSIS WITH 1 FACTOR(S) :
CHI-SQUARE VALUE
DEGREES OF FREEDOM
PROBABILITY VALUE
1518.714
22
0.0000
RMSEA (ROOT MEAN SQUARE ERROR OF APPROXIMATION) :
ESTIMATE IS 0.085
ROOT MEAN SQUARE RESIDUAL IS
DEPRESS
LONELY
SAD
EFFORT
RESTLESS
NOGETGO
NOENERGY
NOHAPPY
NOENJOY
0.0889
ESTIMATED FACTOR LOADINGS
1
________
0.860
0.736
0.837
0.712
0.563
0.688
0.544
0.810
0.837
15
ESTIMATED RESIDUAL VARIANCES
DEPRESS
LONELY
________
________
1
0.261
0.459
1
ESTIMATED RESIDUAL VARIANCES
NOGETGO
NOENERGY
________
________
0.527
0.704
1
FACTOR DETERMINACIES
1
________
0.965
SAD
________
0.299
EFFORT
________
0.493
NOHAPPY
________
0.344
NOENJOY
________
0.299
RESTLESS
________
0.683
EXPLORATORY ANALYSIS WITH 2 FACTOR(S) :
CHI-SQUARE VALUE
DEGREES OF FREEDOM
PROBABILITY VALUE
652.653
16
0.0000
RMSEA (ROOT MEAN SQUARE ERROR OF APPROXIMATION) :
ESTIMATE IS 0.065
ROOT MEAN SQUARE RESIDUAL IS
DEPRESS
LONELY
SAD
EFFORT
RESTLESS
NOGETGO
NOENERGY
NOHAPPY
NOENJOY
0.0551
VARIMAX ROTATED LOADINGS
1
2
________
________
0.770
0.400
0.723
0.256
0.842
0.270
0.406
0.639
0.374
0.438
0.238
0.846
0.228
0.585
0.752
0.338
0.750
0.387
16
DEPRESS
LONELY
SAD
EFFORT
RESTLESS
NOGETGO
NOENERGY
NOHAPPY
NOENJOY
PROMAX ROTATED LOADINGS
1
2
________
________
0.768
0.143
0.776
-0.014
0.916
-0.051
0.208
0.604
0.261
0.371
-0.095
0.937
0.012
0.620
0.774
0.074
0.749
0.136
1
2
PROMAX FACTOR CORRELATIONS
1
2
________
________
1.000
0.652
1.000
1
ESTIMATED RESIDUAL VARIANCES
DEPRESS
LONELY
________
________
0.247
0.412
SAD
________
0.219
EFFORT
________
0.427
1
ESTIMATED RESIDUAL VARIANCES
NOGETGO
NOENERGY
________
________
0.228
0.606
NOHAPPY
________
0.320
NOENJOY
________
0.288
DEPRESS
LONELY
SAD
EFFORT
RESTLESS
NOGETGO
NOENERGY
NOHAPPY
NOENJOY
FACTOR STRUCTURE
1
2
________
________
0.861
0.644
0.767
0.492
0.883
0.547
0.603
0.740
0.503
0.541
0.517
0.876
0.417
0.628
0.822
0.579
0.837
0.624
RESTLESS
________
0.668
17
FACTOR DETERMINACIES
1
________
1
0.962
2
________
0.931
EXPLORATORY ANALYSIS WITH 3 FACTOR(S) :
CHI-SQUARE VALUE
DEGREES OF FREEDOM
PROBABILITY VALUE
158.402
11
0.0000
RMSEA (ROOT MEAN SQUARE ERROR OF APPROXIMATION) :
ESTIMATE IS 0.038
ROOT MEAN SQUARE RESIDUAL IS
0.0222
DEPRESS
LONELY
SAD
EFFORT
RESTLESS
NOGETGO
NOENERGY
NOHAPPY
NOENJOY
VARIMAX ROTATED LOADINGS
1
2
________
________
0.728
0.347
0.721
0.235
0.805
0.341
0.419
0.182
0.382
0.149
0.272
0.142
0.047
0.404
0.463
0.727
0.411
0.768
3
________
0.350
0.223
0.219
0.615
0.421
0.813
0.589
0.234
0.300
DEPRESS
LONELY
SAD
EFFORT
RESTLESS
NOGETGO
NOENERGY
NOHAPPY
NOENJOY
PROMAX ROTATED LOADINGS
1
2
________
________
0.694
0.143
0.756
0.037
0.833
0.147
0.273
-0.040
0.298
-0.024
0.038
-0.096
-0.238
0.345
0.301
0.717
0.205
0.766
3
________
0.144
0.021
-0.037
0.602
0.380
0.898
0.599
-0.046
0.038
18
PROMAX FACTOR CORRELATIONS
1
2
________
________
1
1.000
2
0.563
1.000
3
0.572
0.563
3
________
1.000
1
ESTIMATED RESIDUAL VARIANCES
DEPRESS
LONELY
________
________
0.227
0.375
SAD
________
0.187
EFFORT
________
0.412
1
ESTIMATED RESIDUAL VARIANCES
NOGETGO
NOENERGY
________
________
0.244
0.488
NOHAPPY
________
0.203
NOENJOY
________
0.151
FACTOR STRUCTURE
1
2
________
________
0.857
0.615
0.790
0.475
0.894
0.594
0.595
0.453
0.502
0.358
0.498
0.431
0.299
0.549
0.679
0.861
0.658
0.903
3
________
0.622
0.475
0.522
0.736
0.537
0.866
0.658
0.530
0.587
FACTOR DETERMINACIES
1
2
________
________
0.946
0.935
3
________
0.925
DEPRESS
LONELY
SAD
EFFORT
RESTLESS
NOGETGO
NOENERGY
NOHAPPY
NOENJOY
1
Beginning Time:
Ending Time:
Elapsed Time:
RESTLESS
________
0.655
13:41:02
13:41:03
00:00:01
19
20
NOENERGY
Category
Category
NOHAPPY
Category
Category
NOENJOY
Category
Category
1
2
0.550
0.450
1
2
0.887
0.113
1
2
0.931
0.069
RESULTS FOR EXPLORATORY FACTOR ANALYSIS
1
1
EIGENVALUES FOR SAMPLE CORRELATION MATRIX
1
2
3
4
________
________
________
________
5.164
1.041
0.794
0.592
EIGENVALUES FOR SAMPLE CORRELATION MATRIX
6
7
8
________
________
________
0.308
0.250
0.193
5
________
0.488
9
________
0.169
EXPLORATORY ANALYSIS WITH 1 FACTOR(S) :
CHI-SQUARE VALUE
DEGREES OF FREEDOM
PROBABILITY VALUE
1518.714
22
0.0000
RMSEA (ROOT MEAN SQUARE ERROR OF APPROXIMATION) :
ESTIMATE IS 0.085
ROOT MEAN SQUARE RESIDUAL IS
DEPRESS
LONELY
SAD
EFFORT
RESTLESS
NOGETGO
NOENERGY
NOHAPPY
NOENJOY
0.0889
ESTIMATED FACTOR LOADINGS
1
________
0.860
0.736
0.837
0.712
0.563
0.688
0.544
0.810
0.837
21
Scree Plot with Parallel Analysis
6
5
4
3
2
1
0
1
3
5
Factor
7
9
22
A Note on Good Model Fit
• Model fit is based on how close the modelimplied covariance matrix is to the
observed covariance matrix
• Chi-Square should be low, P-value high
• CFI > 0.95 (max 1)
– Bentler. Psychol Bull, 1990; 107:238-46.
• RMSEA < .05 (min 0)
– Hu & Bentler. Psychol Meth, 1998; 4:424-53.
23
Section 3
Basic Assumptions of IRT
24
Basic Assumptions
• Unidimensionality
– In IRT models a single latent trait is sufficient to
characterize individual differences, for example
– Single common factor
– Multiple factors proportionally loading in items
• Strong local independence
– Probability of responding u is independent of other
test item responses, conditional on q
25
Section 4
Confirmatory Factor Analysis
26
Confirmatory Factor Analysis

1

*
*
*
y1
y2
y3
* 1
* 2
* 3
27
Mplus Input
TITLE:
DATA:
This is an example of a confirmatory factor analysis
(Page 47, Example 5.1)
FILE IS S:\projectdata1\dif\Short~\Data\cesd.csv;
DATA:
FILE IS
S:\project~1\dif\Short~1\Data\cesd.csv;
VARIABLE: NAMES =depress lonely sad effort restless
nogetgo noenergy nohappy noenjoy age gender
ethnic edu;
USEVARIABLES ARE depress-noenjoy;
CATEGORICAL=depress-noenjoy;
MISSING ARE ALL (-9999) ;
ANALYSIS: TYPE=missing h1;
MODEL:
OUTPUT:
f1 by depress* lonely sad;
f1 by effort* restless nogetgo noenergy;
f1 by nohappy* noenjoy;
f1@1;
Standardized ; Sampstat;
28
Mplus Output
Mplus VERSION 4.2
MUTHEN & MUTHEN
05/29/2007
3:31 PM
INPUT INSTRUCTIONS
TITLE: This is an example of an exploratory factor analysis
with dichotomous indicators
DATA: FILE IS S:\projectdata1\dif\Short~\Data\cesd.csv;
VARIABLE: NAMES =depress lonely sad effort restless nogetgo noenergy nohappy noenjoy age
gender ethnicity education;
USEVARIABLES ARE depress-noenjoy;
CATEGORICAL=depression-noenjoy;
MISSING ARE ALL (-9999) ;
ANALYSIS: TYPE =missing efa 1 3 ;
ESTIMATOR=wlsmv;
INPUT READING TERMINATED NORMALLY
This is an example of an exploratory factor analysis with
dichotomous indicators
SUMMARY OF ANALYSIS
Number of groups
Number of observations
Number of dependent variables
Number of independent variables
Number of continuous latent variables
1
9448
9
0
0
Observed dependent variables
29
Binary and ordered categorical (ordinal)
DEPRESS
LONELY
SAD
EFFORT
NOENERGY
NOHAPPY
NOENJOY
Estimator
Maximum number of iterations
Convergence criterion
Maximum number of steepest descent iterations
RESTLESS
NOGETGO
WLSMV
1000
0.500D-04
20
Input data file(s)
S:\projectdata1\dif\Short~\Data\cesd.csv;
Input data format
FREE
SUMMARY OF DATA
Number of patterns
1
COVARIANCE COVERAGE OF DATA
Minimum covariance coverage value
0.100
PROPORTION OF DATA PRESENT
DEPRESS
LONELY
SAD
EFFORT
RESTLESS
NOGETGO
NOENERGY
NOHAPPY
NOENJOY
Covariance Coverage
DEPRESS
LONELY
________
________
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
SAD
________
EFFORT
________
RESTLESS
________
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
30
PROPORTION OF DATA PRESENT
DEPRESS
LONELY
SAD
EFFORT
RESTLESS
NOGETGO
NOENERGY
NOHAPPY
NOENJOY
Covariance Coverage
DEPRESS
LONELY
________
________
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
NOGETGO
NOENERGY
NOHAPPY
NOENJOY
Covariance Coverage
NOGETGO
NOENERGY
________
________
1.000
1.000
1.000
1.000
1.000
1.000
1.000
SAD
________
EFFORT
________
RESTLESS
________
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
NOHAPPY
________
NOENJOY
________
1.000
1.000
1.000
SUMMARY OF CATEGORICAL DATA PROPORTIONS
DEPRESS
Category
Category
LONELY
Category
Category
SAD
Category
Category
EFFORT
Category
Category
RESTLESS
Category
Category
NOGETGO
Category
Category
1
2
0.834
0.166
1
2
0.808
0.192
1
2
0.792
0.208
1
2
0.755
0.245
1
2
0.728
0.272
1
2
0.769
0.231
31
NOENERGY
Category
Category
NOHAPPY
Category
Category
NOENJOY
Category
Category
1
2
0.550
0.450
1
2
0.887
0.113
1
2
0.931
0.069
SAMPLE STATISTICS
ESTIMATED SAMPLE STATISTICS
1
MEANS/INTERCEPTS/THRESHOLDS
DEPRESS$
LONELY$1
________
________
0.970
0.871
SAD$1
________
0.814
EFFORT$1
________
0.691
1
MEANS/INTERCEPTS/THRESHOLDS
NOGETGO$
NOENERGY
________
________
0.737
0.126
NOHAPPY$
________
1.208
NOENJOY$
________
1.483
CORRELATION MATRIX (WITH VARIANCES ON THE DIAGONAL)
DEPRESS
LONELY
SAD
EFFORT
________
________
________
________
DEPRESS
LONELY
SAD
EFFORT
RESTLESS
NOGETGO
NOENERGY
NOHAPPY
NOENJOY
NOENERGY
NOHAPPY
NOENJOY
0.668
0.768
0.641
0.490
0.500
0.332
0.694
0.668
0.732
0.452
0.380
0.435
0.280
0.533
0.554
0.485
0.469
0.464
0.324
0.666
0.660
0.449
0.642
0.432
0.474
0.507
CORRELATION MATRIX (WITH VARIANCES ON THE DIAGONAL)
NOGETGO
NOENERGY
NOHAPPY
NOENJOY
________
________
________
________
0.561
0.411
0.452
0.464
0.515
0.817
RESTLESS
________
0.607
RESTLESS
________
0.451
0.347
0.382
0.367
32
TESTS OF MODEL FIT
Chi-Square Test of Model Fit
Value
Degrees of Freedom
P-Value
1518.715*
22**
0.0000
*
The chi-square value for MLM, MLMV, MLR, ULS, WLSM and WLSMV cannot be used
for chi-square difference tests. MLM, MLR and WLSM chi-square difference
testing is described in the Mplus Technical Appendices at www.statmodel.com.
See chi-square difference testing in the index of the Mplus User's Guide.
**
The degrees of freedom for MLMV, ULS and WLSMV are estimated according to
a formula given in the Mplus Technical Appendices at www.statmodel.com.
See degrees of freedom in the index of the Mplus User's Guide.
Chi-Square Test of Model Fit for the Baseline Model
Value
Degrees of Freedom
P-Value
21648.365
17
0.0000
CFI/TLI
CFI
TLI
0.931
0.947
Number of Free Parameters
18
RMSEA (Root Mean Square Error Of Approximation)
Estimate
0.085
WRMR (Weighted Root Mean Square Residual)
Value
33
5.108
MODEL RESULTS
Estimates
S.E.
Est./S.E.
0.860
0.736
0.837
0.712
0.563
0.688
0.544
0.810
0.837
0.008
0.011
0.008
0.011
0.013
0.011
0.012
0.010
0.011
109.210
69.603
102.718
67.231
43.687
60.101
43.776
79.757
72.882
0.860
0.736
0.837
0.712
0.563
0.688
0.544
0.810
0.837
0.860
0.736
0.837
0.712
0.563
0.688
0.544
0.810
0.837
Thresholds
DEPRESS$1
LONELY$1
SAD$1
EFFORT$1
RESTLESS$1
NOGETGO$1
NOENERGY$1
NOHAPPY$1
NOENJOY$1
0.970
0.871
0.814
0.691
0.607
0.737
0.126
1.208
1.483
0.015
0.015
0.015
0.014
0.014
0.014
0.013
0.017
0.020
63.143
58.694
55.846
49.059
44.003
51.703
9.771
71.194
75.536
0.970
0.871
0.814
0.691
0.607
0.737
0.126
1.208
1.483
0.970
0.871
0.814
0.691
0.607
0.737
0.126
1.208
1.483
Variances
F1
1.000
0.000
0.000
1.000
1.000
F1
BY
DEPRESS
LONELY
SAD
EFFORT
RESTLESS
NOGETGO
NOENERGY
NOHAPPY
NOENJOY
Std
StdYX
34
IRT PARAMETERIZATION IN TWO-PARAMETER PROBIT METRIC
WHERE THE PROBIT IS DISCRIMINATION*(THETA - DIFFICULTY)
Item Discriminations
F1
BY
DEPRESS
LONELY
SAD
EFFORT
RESTLESS
NOGETGO
NOENERGY
NOHAPPY
NOENJOY
1.682
1.086
1.530
1.013
0.682
0.947
0.649
1.380
1.532
0.059
0.034
0.050
0.031
0.023
0.030
0.021
0.050
0.070
28.515
31.931
30.733
33.168
29.816
31.681
30.813
27.458
21.780
Item Difficulties
DEPRESS$1
LONELY$1
SAD$1
EFFORT$1
RESTLESS$1
NOGETGO$1
NOENERGY$1
NOHAPPY$1
NOENJOY$1
1.128
1.184
0.972
0.970
1.077
1.071
0.232
1.492
1.771
0.021
0.027
0.020
0.025
0.036
0.028
0.024
0.030
0.036
52.807
43.068
47.779
38.621
30.269
37.877
9.556
50.224
48.838
Variances
F1
1.000
0.000
0.000
35
R-SQUARE
Observed
Variable
Residual
Variance
R-Square
DEPRESS
LONELY
SAD
EFFORT
RESTLESS
NOGETGO
NOENERGY
NOHAPPY
NOENJOY
0.261
0.459
0.299
0.493
0.682
0.527
0.704
0.344
0.299
0.739
0.541
0.701
0.507
0.318
0.473
0.296
0.656
0.701
QUALITY OF NUMERICAL RESULTS
Condition Number for the Information Matrix
(ratio of smallest to largest eigenvalue)
Beginning Time:
Ending Time:
Elapsed Time:
0.688E-01
14:06:22
14:06:24
00:00:02
36
Factor Analysis of Binary
Variables (IRT)
1

2
3
4
y1*
y2*
y3*
y4*
1
[1]
2
[2]
3
[3]
4
[4]
y* =  + 
y1
y2
y3
y4
VAR(y*) = '
VAR() = 
assuming VAR() = 1

a=
2
1-

b=

37
Item Characteristic Curves
(ICCs)
38
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
-4
-3
-2
-1
0
1
Latent Trait Level
DEPRESS
2
3
4
39
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
-4
-3
-2
-1
0
1
Latent Trait Level
LONELY
2
3
4
40
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
-4
-3
-2
-1
0
1
Latent Trait Level
SAD
2
3
4
41
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
-4
-3
-2
-1
0
1
Latent Trait Level
2
3
4
EFFORT
42
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
-4
-3
-2
-1
0
1
Latent Trait Level
2
3
4
RESTLESS
43
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
-4
-3
-2
-1
0
1
Latent Trait Level
2
3
4
NOGETGO
44
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
-4
-3
-2
-1
0
1
Latent Trait Level
NOENERGY
2
3
4
45
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
-4
-3
-2
-1
0
1
Latent Trait Level
2
3
4
NOHAPPY
46
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
-4
-3
-2
-1
0
1
Latent Trait Level
NOENJOY
2
3
4
47
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
-4
-3
-2
-1
0
1
Latent Trait Level
DEPRESS
SAD
RESTLESS
NOENERGY
NOENJOY
2
3
4
LONELY
EFFORT
NOGETGO
NOHAPPY
48
Where to go for more information
• http://www.statmodel.com/
• http://www.ats.ucla.edu/stat/
• http://www.utexas.edu/its/rc/tutorials/stat/mplus/
• http://ourworld.compuserve.com/homepages/jsu
ebersax/lta.htm
49
Mplus Short Course
• Dates: March 2008 and August 2008
• Instructors: Bengt O. Muthén and Linda
Muthén, creators of Mplus
• www.statmodel.com
50
Section 5
Questions and
Discussion
51