demonstration of sem-based irt in mplus

1

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, [email protected]

August 23, 2007

mailto:[email protected]

2

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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• 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

http://www.statmodel.com/

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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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http://www.ats.ucla.edu/stat/mplus/faq/default.htm

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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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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 FileTITLE: 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 VERSION 4.2MUTHEN & MUTHEN05/29/2007 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 1Number of observations 9448

Number of dependent variables 9Number of independent variables 0Number of continuous latent variables 0

Observed dependent variables

Mplus Output

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Binary and ordered categorical (ordinal) DEPRESS LONELY SAD EFFORT RESTLESS NOGETGO NOENERGY NOHAPPY NOENJOY

Estimator WLSMVMaximum number of iterations 1000Convergence criterion 0.500D-04Maximum number of steepest descent iterations 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

Covariance Coverage DEPRESS LONELY SAD EFFORT RESTLESS ________ ________ ________ ________ ________ DEPRESS 1.000 LONELY 1.000 1.000 SAD 1.000 1.000 1.000 EFFORT 1.000 1.000 1.000 1.000 RESTLESS 1.000 1.000 1.000 1.000 1.000 NOGETGO 1.000 1.000 1.000 1.000 1.000 NOENERGY 1.000 1.000 1.000 1.000 1.000 NOHAPPY 1.000 1.000 1.000 1.000 1.000 NOENJOY 1.000 1.000 1.000 1.000 1.000

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Covariance Coverage NOGETGO NOENERGY NOHAPPY NOENJOY ________ ________ ________ ________ NOGETGO 1.000 NOENERGY 1.000 1.000 NOHAPPY 1.000 1.000 1.000 NOENJOY 1.000 1.000 1.000 1.000

SUMMARY OF CATEGORICAL DATA PROPORTIONS

DEPRESS Category 1 0.834 Category 2 0.166 LONELY Category 1 0.808 Category 2 0.192 SAD Category 1 0.792 Category 2 0.208 EFFORT Category 1 0.755 Category 2 0.245 RESTLESS Category 1 0.728 Category 2 0.272 NOGETGO Category 1 0.769 Category 2 0.231

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NOENERGY Category 1 0.550 Category 2 0.450 NOHAPPY Category 1 0.887 Category 2 0.113 NOENJOY Category 1 0.931 Category 2 0.069

RESULTS FOR EXPLORATORY FACTOR ANALYSIS

EIGENVALUES FOR SAMPLE CORRELATION MATRIX 1 2 3 4 5 ________ ________ ________ ________ ________ 1 5.164 1.041 0.794 0.592 0.488

EIGENVALUES FOR SAMPLE CORRELATION MATRIX 6 7 8 9 ________ ________ ________ ________ 1 0.308 0.250 0.193 0.169

EXPLORATORY ANALYSIS WITH 1 FACTOR(S) :

CHI-SQUARE VALUE 1518.714 DEGREES OF FREEDOM 22 PROBABILITY VALUE 0.0000

RMSEA (ROOT MEAN SQUARE ERROR OF APPROXIMATION) : ESTIMATE IS 0.085

ROOT MEAN SQUARE RESIDUAL IS 0.0889

ESTIMATED FACTOR LOADINGS 1 ________ DEPRESS 0.860 LONELY 0.736 SAD 0.837 EFFORT 0.712 RESTLESS 0.563 NOGETGO 0.688 NOENERGY 0.544 NOHAPPY 0.810 NOENJOY 0.837

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ESTIMATED RESIDUAL VARIANCES DEPRESS LONELY SAD EFFORT RESTLESS ________ ________ ________ ________ ________ 1 0.261 0.459 0.299 0.493 0.683

ESTIMATED RESIDUAL VARIANCES NOGETGO NOENERGY NOHAPPY NOENJOY ________ ________ ________ ________ 1 0.527 0.704 0.344 0.299

FACTOR DETERMINACIES 1 ________ 1 0.965





VARIMAX ROTATED LOADINGS 1 2 ________ ________ DEPRESS 0.770 0.400 LONELY 0.723 0.256 SAD 0.842 0.270 EFFORT 0.406 0.639 RESTLESS 0.374 0.438 NOGETGO 0.238 0.846 NOENERGY 0.228 0.585 NOHAPPY 0.752 0.338 NOENJOY 0.750 0.387

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PROMAX ROTATED LOADINGS 1 2 ________ ________ DEPRESS 0.768 0.143 LONELY 0.776 -0.014 SAD 0.916 -0.051 EFFORT 0.208 0.604 RESTLESS 0.261 0.371 NOGETGO -0.095 0.937 NOENERGY 0.012 0.620 NOHAPPY 0.774 0.074 NOENJOY 0.749 0.136

PROMAX FACTOR CORRELATIONS 1 2 ________ ________ 1 1.000 2 0.652 1.000



FACTOR STRUCTURE 1 2 ________ ________ DEPRESS 0.861 0.644 LONELY 0.767 0.492 SAD 0.883 0.547 EFFORT 0.603 0.740 RESTLESS 0.503 0.541 NOGETGO 0.517 0.876 NOENERGY 0.417 0.628 NOHAPPY 0.822 0.579 NOENJOY 0.837 0.624

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FACTOR DETERMINACIES 1 2 ________ ________ 1 0.962 0.931





VARIMAX ROTATED LOADINGS 1 2 3 ________ ________ ________ DEPRESS 0.728 0.347 0.350 LONELY 0.721 0.235 0.223 SAD 0.805 0.341 0.219 EFFORT 0.419 0.182 0.615 RESTLESS 0.382 0.149 0.421 NOGETGO 0.272 0.142 0.813 NOENERGY 0.047 0.404 0.589 NOHAPPY 0.463 0.727 0.234 NOENJOY 0.411 0.768 0.300

PROMAX ROTATED LOADINGS 1 2 3 ________ ________ ________ DEPRESS 0.694 0.143 0.144 LONELY 0.756 0.037 0.021 SAD 0.833 0.147 -0.037 EFFORT 0.273 -0.040 0.602 RESTLESS 0.298 -0.024 0.380 NOGETGO 0.038 -0.096 0.898 NOENERGY -0.238 0.345 0.599 NOHAPPY 0.301 0.717 -0.046 NOENJOY 0.205 0.766 0.038

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PROMAX FACTOR CORRELATIONS 1 2 3 ________ ________ ________ 1 1.000 2 0.563 1.000 3 0.572 0.563 1.000



FACTOR STRUCTURE 1 2 3 ________ ________ ________ DEPRESS 0.857 0.615 0.622 LONELY 0.790 0.475 0.475 SAD 0.894 0.594 0.522 EFFORT 0.595 0.453 0.736 RESTLESS 0.502 0.358 0.537 NOGETGO 0.498 0.431 0.866 NOENERGY 0.299 0.549 0.658 NOHAPPY 0.679 0.861 0.530 NOENJOY 0.658 0.903 0.587

FACTOR DETERMINACIES 1 2 3 ________ ________ ________ 1 0.946 0.935 0.925

Beginning Time: 13:41:02 Ending Time: 13:41:03 Elapsed Time: 00:00:01

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RESULTS FOR EXPLORATORY FACTOR ANALYSIS

EIGENVALUES FOR SAMPLE CORRELATION MATRIX 1 2 3 4 5 ________ ________ ________ ________ ________ 1 5.164 1.041 0.794 0.592 0.488

EIGENVALUES FOR SAMPLE CORRELATION MATRIX 6 7 8 9 ________ ________ ________ ________ 1 0.308 0.250 0.193 0.169





ESTIMATED FACTOR LOADINGS 1 ________ DEPRESS 0.860 LONELY 0.736 SAD 0.837 EFFORT 0.712 RESTLESS 0.563 NOGETGO 0.688 NOENERGY 0.544 NOHAPPY 0.810 NOENJOY 0.837

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Scree Plot with Parallel Analysis

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enva

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1 3 5 7 9Factor

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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; 107:238-46.• RMSEA < .05 (min 0)

– Hu & Bentler. Psychol Meth, 1998; 4:424-53.

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Section 3

Basic Assumptions of IRT

25

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

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Section 4

Confirmatory Factor Analysis

27

Confirmatory Factor Analysis

y

y *

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Mplus InputTITLE: 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;

f1@1; OUTPUT: Standardized ; Sampstat;

29

Mplus VERSION 4.2MUTHEN & MUTHEN05/29/2007 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 1Number of observations 9448

Number of dependent variables 9Number of independent variables 0Number of continuous latent variables 0

Observed dependent variables

Mplus Output

30

Binary and ordered categorical (ordinal) DEPRESS LONELY SAD EFFORT RESTLESS NOGETGO NOENERGY NOHAPPY NOENJOY

Estimator WLSMVMaximum number of iterations 1000Convergence criterion 0.500D-04Maximum number of steepest descent iterations 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



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Covariance Coverage NOGETGO NOENERGY NOHAPPY NOENJOY ________ ________ ________ ________ NOGETGO 1.000 NOENERGY 1.000 1.000 NOHAPPY 1.000 1.000 1.000 NOENJOY 1.000 1.000 1.000 1.000

SUMMARY OF CATEGORICAL DATA PROPORTIONS

DEPRESS Category 1 0.834 Category 2 0.166 LONELY Category 1 0.808 Category 2 0.192 SAD Category 1 0.792 Category 2 0.208 EFFORT Category 1 0.755 Category 2 0.245 RESTLESS Category 1 0.728 Category 2 0.272 NOGETGO Category 1 0.769 Category 2 0.231

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SAMPLE STATISTICS

ESTIMATED SAMPLE STATISTICS

MEANS/INTERCEPTS/THRESHOLDS DEPRESS$ LONELY$1 SAD$1 EFFORT$1 RESTLESS ________ ________ ________ ________ ________ 1 0.970 0.871 0.814 0.691 0.607

MEANS/INTERCEPTS/THRESHOLDS NOGETGO$ NOENERGY NOHAPPY$ NOENJOY$ ________ ________ ________ ________ 1 0.737 0.126 1.208 1.483

CORRELATION MATRIX (WITH VARIANCES ON THE DIAGONAL) DEPRESS LONELY SAD EFFORT RESTLESS ________ ________ ________ ________ ________ DEPRESS LONELY 0.668 SAD 0.768 0.732 EFFORT 0.641 0.452 0.485 RESTLESS 0.490 0.380 0.469 0.449 NOGETGO 0.500 0.435 0.464 0.642 0.451 NOENERGY 0.332 0.280 0.324 0.432 0.347 NOHAPPY 0.694 0.533 0.666 0.474 0.382 NOENJOY 0.668 0.554 0.660 0.507 0.367

CORRELATION MATRIX (WITH VARIANCES ON THE DIAGONAL) NOGETGO NOENERGY NOHAPPY NOENJOY ________ ________ ________ ________ NOENERGY 0.561 NOHAPPY 0.411 0.452 NOENJOY 0.464 0.515 0.817

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TESTS OF MODEL FIT

Chi-Square Test of Model Fit

Value 1518.715* Degrees of Freedom 22** P-Value 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 21648.365 Degrees of Freedom 17 P-Value 0.0000

CFI/TLI

CFI 0.931 TLI 0.947

Number of Free Parameters 18

RMSEA (Root Mean Square Error Of Approximation)

Estimate 0.085

WRMR (Weighted Root Mean Square Residual)

Value 5.108

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MODEL RESULTS

Estimates S.E. Est./S.E. Std StdYX

F1 BY DEPRESS 0.860 0.008 109.210 0.860 0.860 LONELY 0.736 0.011 69.603 0.736 0.736 SAD 0.837 0.008 102.718 0.837 0.837 EFFORT 0.712 0.011 67.231 0.712 0.712 RESTLESS 0.563 0.013 43.687 0.563 0.563 NOGETGO 0.688 0.011 60.101 0.688 0.688 NOENERGY 0.544 0.012 43.776 0.544 0.544 NOHAPPY 0.810 0.010 79.757 0.810 0.810 NOENJOY 0.837 0.011 72.882 0.837 0.837

Thresholds DEPRESS$1 0.970 0.015 63.143 0.970 0.970 LONELY$1 0.871 0.015 58.694 0.871 0.871 SAD$1 0.814 0.015 55.846 0.814 0.814 EFFORT$1 0.691 0.014 49.059 0.691 0.691 RESTLESS$1 0.607 0.014 44.003 0.607 0.607 NOGETGO$1 0.737 0.014 51.703 0.737 0.737 NOENERGY$1 0.126 0.013 9.771 0.126 0.126 NOHAPPY$1 1.208 0.017 71.194 1.208 1.208 NOENJOY$1 1.483 0.020 75.536 1.483 1.483

Variances F1 1.000 0.000 0.000 1.000 1.000

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IRT PARAMETERIZATION IN TWO-PARAMETER PROBIT METRICWHERE THE PROBIT IS DISCRIMINATION*(THETA - DIFFICULTY)

Item Discriminations

F1 BY DEPRESS 1.682 0.059 28.515 LONELY 1.086 0.034 31.931 SAD 1.530 0.050 30.733 EFFORT 1.013 0.031 33.168 RESTLESS 0.682 0.023 29.816 NOGETGO 0.947 0.030 31.681 NOENERGY 0.649 0.021 30.813 NOHAPPY 1.380 0.050 27.458 NOENJOY 1.532 0.070 21.780

Item Difficulties DEPRESS$1 1.128 0.021 52.807 LONELY$1 1.184 0.027 43.068 SAD$1 0.972 0.020 47.779 EFFORT$1 0.970 0.025 38.621 RESTLESS$1 1.077 0.036 30.269 NOGETGO$1 1.071 0.028 37.877 NOENERGY$1 0.232 0.024 9.556 NOHAPPY$1 1.492 0.030 50.224 NOENJOY$1 1.771 0.036 48.838

Variances F1 1.000 0.000 0.000

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R-SQUARE

Observed Residual Variable Variance R-Square

DEPRESS 0.261 0.739 LONELY 0.459 0.541 SAD 0.299 0.701 EFFORT 0.493 0.507 RESTLESS 0.682 0.318 NOGETGO 0.527 0.473 NOENERGY 0.704 0.296 NOHAPPY 0.344 0.656 NOENJOY 0.299 0.701

QUALITY OF NUMERICAL RESULTS

Condition Number for the Information Matrix 0.688E-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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y* = + VAR(y*) = '

VAR() =

assuming VAR() = 1

a = 1-2

b =

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Item Characteristic Curves(ICCs)

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NOHAPPY

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DEPRESS LONELYSAD EFFORTRESTLESS NOGETGONOENERGY NOHAPPYNOENJOY

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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/jsuebersax/lta.htm

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Mplus Short Course

• Dates: March 2008 and August 2008

• Instructors: Bengt O. Muthén and Linda Muthén, creators of Mplus

• www.statmodel.com

http://www.statmodel.com/

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Section 5

Questions andDiscussion

demonstration of sem-based irt in mplus

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