Web Appendix: MPLUS Output for the Data in Table 3

Modeling Multiple Response 1

Web Appendix: MPLUS output for the data in Table 3

Mplus VERSION 6.1 MUTHEN & MUTHEN INPUT

INSTRUCTIONS

TITLE: ML estimation of Likert model

DATA: FILE IS mplusdt.out;

VARIABLE: NAMES ARE x11 x12 x21 x22 x31 x32 age edu; CATEGORICAL ARE x11 x12 x21 x22 x31 x32; USEVARIABLES ARE x11 x12 x21 x22 x31 x32;

MISSING ARE ALL (-9);

ANALYSIS: ESTIMATOR = MLR; LINK = PROBIT; INTEGRATION = GAUSSHERMITE(15); MODEL:

f1 BY x11 @ 1 x12 @ 1; f2 BY x21 @ 1 x22 @ 1; f3 BY x31 @ 1 x32 @ 1;

OUTPUT: TECH1 TECH8; PLOT: TYPE = PLOT3;

INPUT READING TERMINATED NORMALLY ML estimation of Likert model

SUMMARY OF ANALYSIS

Number of groups 1 Number of observations 1287

Number of dependent variables 6 Number of independent variables 0 Number of continuous latent variables 3

Observed dependent variables

Binary and ordered categorical (ordinal) X11 X12 X21 X22 X31 X32

Continuous latent variables F1 F2 F3

Estimator MLR Information matrix OBSERVED Optimization Specifications for the Quasi-Newton Algorithm for Continuous Outcomes Maximum number of iterations 100 Convergence criterion 0.100D-05 Optimization Specifications for the EM Algorithm Maximum number of iterations 500 Convergence criteria Loglikelihood change 0.100D-02 Relative loglikelihood change 0.100D-05 Derivative 0.100D-02 Optimization Specifications for the M step of the EM Algorithm for Categorical Latent variables Number of M step iterations 1 M step convergence criterion 0.100D-02 Basis for M step termination ITERATION Optimization Specifications for the M step of the EM Algorithm for Censored, Binary or Ordered Categorical (Ordinal), Unordered Categorical (Nominal) and Count Outcomes

Number of M step iterations 1 M step convergence criterion 0.100D-02 Basis for M step termination ITERATION Maximum value for logit thresholds 10 Minimum value for logit thresholds -10 Modeling Multiple Response 3

Minimum expected cell size for chi-square 0.100D-01 Maximum number of iterations for H1 2000 Convergence criterion for H1 0.100D-03 Optimization algorithm EMA Integration Specifications Type GAUSSHERMITE Number of integration points 15 Dimensions of numerical integration 3 Adaptive quadrature ON Link PROBIT Cholesky ON

Input data file(s) mplusdt.out Input data format FREE

SUMMARY OF DATA

Number of missing data patterns 4

COVARIANCE COVERAGE OF DATA

Minimum covariance coverage value 0.100

PROPORTION OF DATA PRESENT FOR U

Covariance Coverage X11 X12 X21 X22 X31

X11 1.000 X12 1.000 1.000 X21 0.709 0.709 0.709 X22 0.690 0.690 0.600 0.690 X31 0.709 0.709 0.709 0.600 0.709 X32 0.690 0.690 0.600 0.690 0.600 Covariance Coverage X32

X32 0.690

UNIVARIATE PROPORTIONS AND COUNTS FOR CATEGORICAL VARIABLES

X11 Category 1 0.291 375.000 Category 2 0.709 912.000 X12 Category 1 0.310 399.000 Category 2 0.690 888.000 X21 Category 1 0.708 646.000 Category 2 0.292 266.000 X22 Category 1 0.753 669.000 Category 2 0.247 219.000 X31 Category 1 0.459 419.000 Category 2 0.541 493.000 X32 Category 1 0.485 431.000 Category 2 0.515 457.000

THE MODEL ESTIMATION TERMINATED NORMALLY

MODEL FIT INFORMATION

Number of Free Parameters 12

Loglikelihood

H0 Value -3321.906 H0 Scaling Correction Factor 1.005 for MLR

Information Criteria Modeling Multiple Response 5

Akaike (AIC) 6667.813 Bayesian (BIC) 6729.734 Sample-Size Adjusted BIC 6691.616 (n* = (n + 2) / 24)

Chi-Square Test of Model Fit for the Binary and Ordered Categorical (Ordinal) Outcomes

Pearson Chi-Square

Value 0.000 Degrees of Freedom 51 P-Value 1.0000

Likelihood Ratio Chi-Square

Chi-Square Test for MCAR under the Unrestricted Latent Class Indicator Model

Pearson Chi-Square

Likelihood Ratio Chi-Square

Value 1935.586 Degrees of Freedom 33 P-Value 0.0000 MODEL RESULTS

Two-Tailed Estimate S.E. Est./S.E. P-Value

F1 BY X11 1.000 0.000 999.000 999.000 X12 1.000 0.000 999.000 999.000

F2 BY X21 1.000 0.000 999.000 999.000 X22 1.000 0.000 999.000 999.000

F3 BY X31 1.000 0.000 999.000 999.000 X32 1.000 0.000 999.000 999.000

F2 WITH F1 -1.948 0.424 -4.599 0.000

F3 WITH F1 -3.393 0.470 -7.213 0.000 F2 2.229 0.479 4.656 0.000

Thresholds X11$1 -1.103 0.091 -12.085 0.000 X12$1 -0.996 0.088 -11.283 0.000 X21$1 0.926 0.149 6.233 0.000 X22$1 1.174 0.157 7.462 0.000 X31$1 -1.053 0.148 -7.118 0.000 X32$1 -0.934 0.146 -6.410 0.000

Variances

F1 3.032 0.426 7.119 0.000 F2 5.631 1.114 5.054 0.000 F3 7.402 1.305 5.672 0.000

QUALITY OF NUMERICAL RESULTS

Condition Number for the Information Matrix 0.281E-01 (ratio of smallest to largest eigenvalue) Modeling Multiple Response 7

TECHNICAL 1 OUTPUT

PARAMETER SPECIFICATION

TAU X11$1 X12$1 X21$1 X22$1 X31$1

1 7 8 9 10 11

TAU X32$1

NU X11 X12 X21 X22 X31

1 0 0 0 0 0

NU X32

LAMBDA F1 F2 F3

X11 0 0 0 X12 0 0 0 X21 0 0 0 X22 0 0 0 X31 0 0 0 X32 0 0 0 THETA X11 X12 X21 X22 X31

X11 0 X12 0 0 X21 0 0 0 X22 0 0 0 0 X31 0 0 0 0 0 X32 0 0 0 0 0

ALPHA F1 F2 F3

1 0 0 0

BETA F1 F2 F3

F1 0 0 0 F2 0 0 0 F3 0 0 0

F1 F2 F3

F1 1 F2 2 3 F3 4 5 6 STARTING VALUES

TAU X11$1 X12$1 X21$1 X22$1 X31$1

1 -0.494 -0.444 0.493 0.620 -0.090

TAU X32$1

1 -0.033

NU X11 X12 X21 X22 X31

1 0.000 0.000 0.000 0.000 0.000

NU X32

1 0.000

LAMBDA F1 F2 F3

X11 1.000 0.000 0.000 X12 1.000 0.000 0.000 X21 0.000 1.000 0.000 X22 0.000 1.000 0.000 X31 0.000 0.000 1.000 X32 0.000 0.000 1.000 THETA X11 X12 X21 X22 X31

X11 1.000 X12 0.000 1.000 X21 0.000 0.000 1.000 X22 0.000 0.000 0.000 1.000 X31 0.000 0.000 0.000 0.000 1.000 X32 0.000 0.000 0.000 0.000 0.000

THETA X32

X32 1.000

ALPHA F1 F2 F3

1 0.000 0.000 0.000

BETA F1 F2 F3

F1 0.000 0.000 0.000 F2 0.000 0.000 0.000 F3 0.000 0.000 0.000 PSI F1 F2 F3

F1 0.050 F2 0.000 0.050 F3 0.000 0.000 0.050

TECHNICAL 8 OUTPUT

E STEP ITER LOGLIKELIHOOD ABS CHANGE REL CHANGE ALGORITHM 1 -0.38261107D+04 0.0000000 0.0000000 EM 2 -0.34037539D+04 422.3567415 0.1103880 FS 3 -0.33282446D+04 75.5093879 0.0221841 FS 4 -0.33220036D+04 6.2409246 0.0018751 FS 5 -0.33219065D+04 0.0971098 0.0000292 FS 6 -0.33219065D+04 0.0000292 0.0000000 FS

SAMPLE STATISTICS FOR

ESTIMATED FACTOR SCORES

SAMPLE STATISTICS

Means F1 F2 F3

1 0.000 0.0000.000

Covariances F1 F2 F3

F1 2.000 F2 -1.549 2.898 F3 -2.712 1.911 4.734 Correlations F1 F2 F3

F1 1.000 F2 -0.643 1.000 F3 -0.881 0.516 1.000

PLOT INFORMATION

The following plots are available:

Histograms (sample values, estimated factor scores) Scatterplots (sample values, estimated factor scores) Item characteristic curves Information curves