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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 August 23, 2007
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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
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Section 1 Introduction to Mplus
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Used to be LISCOMP, owes lineage to LISREL
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
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Formatting Data for Mplus
Individual-level data Summary Data (correlations, covariances, means, standard deviations) ASCII Data Raw text Fixed, Free format
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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
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Mplus Commands TITLE DATA VARIABLE ANALYSIS MODEL OUTPUT DEFINE PLOT
SAVEDATA MONTECARLO
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Exploratory Factor Analysis
Section 2 Exploratory Factor Analysis
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Exploratory Factor Analysis in Mplus (v.4)
Observed outcomes variables can be: continuous binary ordered categorical (ordinal) combinations of these variable types
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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;
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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 Observed dependent variables
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Binary and ordered categorical (ordinal)
DEPRESS LONELY SAD EFFORT RESTLESS NOGETGO NOENERGY NOHAPPY NOENJOY Estimator WLSMV Maximum number of iterations Convergence criterion D-04 Maximum number of steepest descent iterations Input data file(s) S:\projectdata1\dif\Short~\Data\cesd.csv; Input data format FREE SUMMARY OF DATA Number of patterns COVARIANCE COVERAGE OF DATA Minimum covariance coverage value PROPORTION OF DATA PRESENT Covariance Coverage DEPRESS LONELY SAD EFFORT RESTLESS ________ ________ ________ ________ ________ DEPRESS LONELY SAD EFFORT RESTLESS NOGETGO NOENERGY NOHAPPY NOENJOY
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PROPORTION OF DATA PRESENT
Covariance Coverage DEPRESS LONELY SAD EFFORT RESTLESS ________ ________ ________ ________ ________ DEPRESS LONELY SAD EFFORT RESTLESS NOGETGO NOENERGY NOHAPPY NOENJOY NOGETGO NOENERGY NOHAPPY NOENJOY ________ ________ ________ ________ NOGETGO NOENERGY NOHAPPY NOENJOY SUMMARY OF CATEGORICAL DATA PROPORTIONS DEPRESS Category Category LONELY Category Category SAD Category Category EFFORT Category Category RESTLESS Category Category NOGETGO Category Category
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NOENERGY Category 1 0.550 Category 2 0.450 NOHAPPY Category 1 0.887
NOENJOY Category Category RESULTS FOR EXPLORATORY FACTOR ANALYSIS EIGENVALUES FOR SAMPLE CORRELATION MATRIX ________ ________ ________ ________ ________ ________ ________ ________ ________ EXPLORATORY ANALYSIS WITH 1 FACTOR(S) : CHI-SQUARE VALUE DEGREES OF FREEDOM PROBABILITY VALUE RMSEA (ROOT MEAN SQUARE ERROR OF APPROXIMATION) : ESTIMATE IS ROOT MEAN SQUARE RESIDUAL IS ESTIMATED FACTOR LOADINGS 1 ________ DEPRESS LONELY SAD EFFORT RESTLESS NOGETGO NOENERGY NOHAPPY NOENJOY
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ESTIMATED RESIDUAL VARIANCES
DEPRESS LONELY SAD EFFORT RESTLESS ________ ________ ________ ________ ________ NOGETGO NOENERGY NOHAPPY NOENJOY ________ ________ ________ ________ FACTOR DETERMINACIES 1 ________ EXPLORATORY ANALYSIS WITH 2 FACTOR(S) : CHI-SQUARE VALUE DEGREES OF FREEDOM PROBABILITY VALUE RMSEA (ROOT MEAN SQUARE ERROR OF APPROXIMATION) : ESTIMATE IS ROOT MEAN SQUARE RESIDUAL IS VARIMAX ROTATED LOADINGS ________ ________ DEPRESS LONELY SAD EFFORT RESTLESS NOGETGO NOENERGY NOHAPPY NOENJOY
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PROMAX ROTATED LOADINGS
________ ________ DEPRESS LONELY SAD EFFORT RESTLESS NOGETGO NOENERGY NOHAPPY NOENJOY PROMAX FACTOR CORRELATIONS ESTIMATED RESIDUAL VARIANCES DEPRESS LONELY SAD EFFORT RESTLESS ________ ________ ________ ________ ________ NOGETGO NOENERGY NOHAPPY NOENJOY ________ ________ ________ ________ FACTOR STRUCTURE DEPRESS LONELY SAD EFFORT RESTLESS NOGETGO NOENERGY NOHAPPY NOENJOY
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FACTOR DETERMINACIES 1 2 ________ ________ 1 0.962 0.931
________ ________ EXPLORATORY ANALYSIS WITH 3 FACTOR(S) : CHI-SQUARE VALUE DEGREES OF FREEDOM PROBABILITY VALUE RMSEA (ROOT MEAN SQUARE ERROR OF APPROXIMATION) : ESTIMATE IS ROOT MEAN SQUARE RESIDUAL IS VARIMAX ROTATED LOADINGS ________ ________ ________ DEPRESS LONELY SAD EFFORT RESTLESS NOGETGO NOENERGY NOHAPPY NOENJOY PROMAX ROTATED LOADINGS DEPRESS LONELY SAD EFFORT RESTLESS NOGETGO NOENERGY NOHAPPY NOENJOY
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PROMAX FACTOR CORRELATIONS
________ ________ ________ ESTIMATED RESIDUAL VARIANCES DEPRESS LONELY SAD EFFORT RESTLESS ________ ________ ________ ________ ________ NOGETGO NOENERGY NOHAPPY NOENJOY ________ ________ ________ ________ FACTOR STRUCTURE DEPRESS LONELY SAD EFFORT RESTLESS NOGETGO NOENERGY NOHAPPY NOENJOY FACTOR DETERMINACIES Beginning Time: 13:41:02 Ending Time: 13:41:03 Elapsed Time: 00:00:01
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NOENERGY Category 1 0.550 Category 2 0.450 NOHAPPY Category 1 0.887
NOENJOY Category Category RESULTS FOR EXPLORATORY FACTOR ANALYSIS EIGENVALUES FOR SAMPLE CORRELATION MATRIX ________ ________ ________ ________ ________ ________ ________ ________ ________ EXPLORATORY ANALYSIS WITH 1 FACTOR(S) : CHI-SQUARE VALUE DEGREES OF FREEDOM PROBABILITY VALUE RMSEA (ROOT MEAN SQUARE ERROR OF APPROXIMATION) : ESTIMATE IS ROOT MEAN SQUARE RESIDUAL IS ESTIMATED FACTOR LOADINGS 1 ________ DEPRESS LONELY SAD EFFORT RESTLESS NOGETGO NOENERGY NOHAPPY NOENJOY
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Scree Plot with Parallel Analysis
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A Note on Good Model Fit Model fit is based on how close the model-implied 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: RMSEA < .05 (min 0) Hu & Bentler. Psychol Meth, 1998; 4:
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Section 3 Basic Assumptions of IRT
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Basic Assumptions Unidimensionality Strong local independence
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
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Confirmatory Factor Analysis
Section 4 Confirmatory Factor Analysis
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Confirmatory Factor Analysis
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Mplus Input DATA: FILE IS S:\project~1\dif\Short~1\Data\cesd.csv;
TITLE: This is an example of a confirmatory factor analysis (Page 47, Example 5.1) DATA: 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: f1 by depress* lonely sad; f1 by effort* restless nogetgo noenergy; f1 by nohappy* noenjoy; OUTPUT: Standardized ; Sampstat;
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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 Observed dependent variables
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Binary and ordered categorical (ordinal)
DEPRESS LONELY SAD EFFORT RESTLESS NOGETGO NOENERGY NOHAPPY NOENJOY Estimator WLSMV Maximum number of iterations Convergence criterion D-04 Maximum number of steepest descent iterations Input data file(s) S:\projectdata1\dif\Short~\Data\cesd.csv; Input data format FREE SUMMARY OF DATA Number of patterns COVARIANCE COVERAGE OF DATA Minimum covariance coverage value PROPORTION OF DATA PRESENT Covariance Coverage DEPRESS LONELY SAD EFFORT RESTLESS ________ ________ ________ ________ ________ DEPRESS LONELY SAD EFFORT RESTLESS NOGETGO NOENERGY NOHAPPY NOENJOY
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PROPORTION OF DATA PRESENT
Covariance Coverage DEPRESS LONELY SAD EFFORT RESTLESS ________ ________ ________ ________ ________ DEPRESS LONELY SAD EFFORT RESTLESS NOGETGO NOENERGY NOHAPPY NOENJOY NOGETGO NOENERGY NOHAPPY NOENJOY ________ ________ ________ ________ NOGETGO NOENERGY NOHAPPY NOENJOY SUMMARY OF CATEGORICAL DATA PROPORTIONS DEPRESS Category Category LONELY Category Category SAD Category Category EFFORT Category Category RESTLESS Category Category NOGETGO Category Category
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NOENERGY Category 1 0.550 Category 2 0.450 NOHAPPY Category 1 0.887
NOENJOY Category Category SAMPLE STATISTICS ESTIMATED SAMPLE STATISTICS MEANS/INTERCEPTS/THRESHOLDS DEPRESS$ LONELY$ SAD$ EFFORT$ RESTLESS ________ ________ ________ ________ ________ NOGETGO$ NOENERGY NOHAPPY$ NOENJOY$ ________ ________ ________ ________ CORRELATION MATRIX (WITH VARIANCES ON THE DIAGONAL) DEPRESS LONELY SAD EFFORT RESTLESS DEPRESS LONELY SAD EFFORT RESTLESS NOGETGO NOENERGY NOHAPPY NOENJOY NOGETGO NOENERGY NOHAPPY NOENJOY NOENERGY NOHAPPY NOENJOY
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TESTS OF MODEL FIT Chi-Square Test of Model Fit Value * Degrees of Freedom ** P-Value * 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 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 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 CFI/TLI CFI TLI Number of Free Parameters RMSEA (Root Mean Square Error Of Approximation) Estimate WRMR (Weighted Root Mean Square Residual) Value
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MODEL RESULTS Estimates S.E. Est./S.E. Std StdYX F BY DEPRESS LONELY SAD EFFORT RESTLESS NOGETGO NOENERGY NOHAPPY NOENJOY Thresholds DEPRESS$ LONELY$ SAD$ EFFORT$ RESTLESS$ NOGETGO$ NOENERGY$ NOHAPPY$ NOENJOY$ Variances F
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IRT PARAMETERIZATION IN TWO-PARAMETER PROBIT METRIC
WHERE THE PROBIT IS DISCRIMINATION*(THETA - DIFFICULTY) Item Discriminations F BY DEPRESS LONELY SAD EFFORT RESTLESS NOGETGO NOENERGY NOHAPPY NOENJOY Item Difficulties DEPRESS$ LONELY$ SAD$ EFFORT$ RESTLESS$ NOGETGO$ NOENERGY$ NOHAPPY$ NOENJOY$ Variances F
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R-SQUARE Observed Residual Variable Variance R-Square DEPRESS LONELY SAD EFFORT RESTLESS NOGETGO NOENERGY NOHAPPY NOENJOY QUALITY OF NUMERICAL RESULTS Condition Number for the Information Matrix E-01 (ratio of smallest to largest eigenvalue) Beginning Time: 14:06:22 Ending Time: 14:06:24 Elapsed Time: 00:00:02
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Factor Analysis of Binary Variables (IRT)
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Item Characteristic Curves
(ICCs)
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Where to go for more information
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Mplus Short Course Dates: March 2008 and August 2008
Instructors: Bengt O. Muthén and Linda Muthén, creators of Mplus
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Questions and Discussion
Section 5 Questions and Discussion
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