<p> Modeling Multiple Response 1</p><p>Web Appendix: MPLUS output for the data in Table 3</p><p>Mplus VERSION 6.1 MUTHEN & MUTHEN INPUT </p><p>INSTRUCTIONS</p><p>TITLE: ML estimation of Likert model</p><p>DATA: FILE IS mplusdt.out;</p><p>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;</p><p>MISSING ARE ALL (-9); </p><p>ANALYSIS: ESTIMATOR = MLR; LINK = PROBIT; INTEGRATION = GAUSSHERMITE(15); MODEL:</p><p> f1 BY x11 @ 1 x12 @ 1; f2 BY x21 @ 1 x22 @ 1; f3 BY x31 @ 1 x32 @ 1;</p><p>OUTPUT: TECH1 TECH8; PLOT: TYPE = PLOT3;</p><p>INPUT READING TERMINATED NORMALLY ML estimation of Likert model</p><p>SUMMARY OF ANALYSIS</p><p>Number of groups 1 Number of observations 1287</p><p>Number of dependent variables 6 Number of independent variables 0 Number of continuous latent variables 3</p><p>Observed dependent variables</p><p>Binary and ordered categorical (ordinal) X11 X12 X21 X22 X31 X32</p><p>Continuous latent variables F1 F2 F3</p><p>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</p><p>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</p><p>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</p><p>Input data file(s) mplusdt.out Input data format FREE</p><p>SUMMARY OF DATA</p><p>Number of missing data patterns 4</p><p>COVARIANCE COVERAGE OF DATA</p><p>Minimum covariance coverage value 0.100</p><p>PROPORTION OF DATA PRESENT FOR U </p><p>Covariance Coverage X11 X12 X21 X22 X31</p><p>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</p><p>X32 0.690</p><p>UNIVARIATE PROPORTIONS AND COUNTS FOR CATEGORICAL VARIABLES</p><p>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</p><p>THE MODEL ESTIMATION TERMINATED NORMALLY</p><p>MODEL FIT INFORMATION</p><p>Number of Free Parameters 12</p><p>Loglikelihood</p><p>H0 Value -3321.906 H0 Scaling Correction Factor 1.005 for MLR</p><p>Information Criteria Modeling Multiple Response 5</p><p>Akaike (AIC) 6667.813 Bayesian (BIC) 6729.734 Sample-Size Adjusted BIC 6691.616 (n* = (n + 2) / 24)</p><p>Chi-Square Test of Model Fit for the Binary and Ordered Categorical (Ordinal) Outcomes</p><p>Pearson Chi-Square</p><p>Value 0.000 Degrees of Freedom 51 P-Value 1.0000</p><p>Likelihood Ratio Chi-Square</p><p>Value 1.788 Degrees of Freedom 51 P-Value 1.0000</p><p>Chi-Square Test for MCAR under the Unrestricted Latent Class Indicator Model</p><p>Pearson Chi-Square</p><p>Value 1782.203 Degrees of Freedom 33 P-Value 0.0000</p><p>Likelihood Ratio Chi-Square</p><p>Value 1935.586 Degrees of Freedom 33 P-Value 0.0000 MODEL RESULTS</p><p>Two-Tailed Estimate S.E. Est./S.E. P-Value</p><p>F1 BY X11 1.000 0.000 999.000 999.000 X12 1.000 0.000 999.000 999.000</p><p>F2 BY X21 1.000 0.000 999.000 999.000 X22 1.000 0.000 999.000 999.000</p><p>F3 BY X31 1.000 0.000 999.000 999.000 X32 1.000 0.000 999.000 999.000</p><p>F2 WITH F1 -1.948 0.424 -4.599 0.000</p><p>F3 WITH F1 -3.393 0.470 -7.213 0.000 F2 2.229 0.479 4.656 0.000</p><p>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</p><p>Variances</p><p>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</p><p>QUALITY OF NUMERICAL RESULTS</p><p>Condition Number for the Information Matrix 0.281E-01 (ratio of smallest to largest eigenvalue) Modeling Multiple Response 7</p><p>TECHNICAL 1 OUTPUT</p><p>PARAMETER SPECIFICATION</p><p>TAU X11$1 X12$1 X21$1 X22$1 X31$1</p><p>1 7 8 9 10 11</p><p>TAU X32$1</p><p>1 12</p><p>NU X11 X12 X21 X22 X31</p><p>1 0 0 0 0 0</p><p>NU X32</p><p>1 0</p><p>LAMBDA F1 F2 F3</p><p>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</p><p>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</p><p>THETA</p><p>X32</p><p>ALPHA F1 F2 F3</p><p>1 0 0 0</p><p>BETA F1 F2 F3</p><p>F1 0 0 0 F2 0 0 0 F3 0 0 0</p><p>PSI</p><p>F1 F2 F3</p><p>F1 1 F2 2 3 F3 4 5 6 STARTING VALUES</p><p>TAU X11$1 X12$1 X21$1 X22$1 X31$1</p><p>1 -0.494 -0.444 0.493 0.620 -0.090</p><p>TAU X32$1</p><p>1 -0.033</p><p>NU X11 X12 X21 X22 X31</p><p>1 0.000 0.000 0.000 0.000 0.000</p><p>NU X32</p><p>1 0.000</p><p>LAMBDA F1 F2 F3</p><p>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</p><p>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</p><p>THETA X32</p><p>X32 1.000</p><p>ALPHA F1 F2 F3</p><p>1 0.000 0.000 0.000</p><p>BETA F1 F2 F3</p><p>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</p><p>F1 0.050 F2 0.000 0.050 F3 0.000 0.000 0.050</p><p>TECHNICAL 8 OUTPUT</p><p>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</p><p>SAMPLE STATISTICS FOR </p><p>ESTIMATED FACTOR SCORES </p><p>SAMPLE STATISTICS</p><p>Means F1 F2 F3</p><p>1 0.000 0.0000.000</p><p>Covariances F1 F2 F3</p><p>F1 2.000 F2 -1.549 2.898 F3 -2.712 1.911 4.734 Correlations F1 F2 F3</p><p>F1 1.000 F2 -0.643 1.000 F3 -0.881 0.516 1.000</p><p>PLOT INFORMATION</p><p>The following plots are available:</p><p>Histograms (sample values, estimated factor scores) Scatterplots (sample values, estimated factor scores) Item characteristic curves Information curves</p><p>12</p>