10 Übungsaufgaben
Im Folgenden findet ihr verschiedene Übungsaufgaben, die ihr mit Hilfe des bisher gelernten Wissens, lösen könnt. Die Lösungen aller Aufgaben findet ihr in Kapitel 13.
10.1 Aufgabe 1
Die erste Aufgabe besteht darin, anhand eines vorgelegten Outputs herauszufinden, welches Modell auf die Daten angewandt wurde, zu begründen, woran ihr eure Lösung festmacht und ob das Modell auf die Daten passt, oder nicht. Dazu findet ihr im Folgenden fünf verschiedene Outputs.
10.1.1 Output 1
lavaan 0.6-3 ended normally after 23 iterations
Optimization method NLMINB
Number of free parameters 10
Number of equality constraints 7
Number of observations 176
Estimator ML
Model Fit Test Statistic 941.414
Degrees of freedom 41
P-value (Chi-square) 0.000
Model test baseline model:
Minimum Function Test Statistic 603.650
Degrees of freedom 28
P-value 0.000
User model versus baseline model:
Comparative Fit Index (CFI) 0.000
Tucker-Lewis Index (TLI) -0.068
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -2167.253
Loglikelihood unrestricted model (H1) -1696.546
Number of free parameters 3
Akaike (AIC) 4340.506
Bayesian (BIC) 4350.017
Sample-size adjusted Bayesian (BIC) 4340.517
Root Mean Square Error of Approximation:
RMSEA 0.353
90 Percent Confidence Interval 0.334 0.373
P-value RMSEA <= 0.05 0.000
Standardized Root Mean Square Residual:
SRMR 0.440
Parameter Estimates:
Information Expected
Information saturated (h1) model Structured
Standard Errors Standard
Latent Variables:
Estimate Std.Err z-value P(>|z|)
eta =~
y1 1.000
y2 1.000
y3 1.000
y4 1.000
y5 1.000
y6 1.000
y7 1.000
y8 1.000 0.536 0.454
Intercepts:
Estimate Std.Err z-value P(>|z|)
eta 3.712 0.049 75.516 0.000
.y1 0.000
.y2 0.000
.y3 0.000
.y4 0.000
.y5 0.000
.y6 0.000
.y7 0.000
.y8 0.000
Variances:
Estimate Std.Err z-value P(>|z|)
eta (veta) 0.287 0.046 6.287 0.000
.y1 (veps) 1.105 0.045 24.819 0.000
.y2 (veps) 1.105 0.045 24.819 0.000
.y3 (veps) 1.105 0.045 24.819 0.000
.y4 (veps) 1.105 0.045 24.819 0.000
.y5 (veps) 1.105 0.045 24.819 0.000
.y6 (veps) 1.105 0.045 24.819 0.000
.y7 (veps) 1.105 0.045 24.819 0.000
.y8 (veps) 1.105 0.045 24.819 0.000 10.1.2 Output 2
lavaan 0.6-3 ended normally after 14 iterations
Optimization method NLMINB
Number of free parameters 13
Number of equality constraints 2
Number of observations 300
Estimator ML
Model Fit Test Statistic 58.147
Degrees of freedom 16
P-value (Chi-square) 0.000
Model test baseline model:
Minimum Function Test Statistic 910.631
Degrees of freedom 15
P-value 0.000
User model versus baseline model:
Comparative Fit Index (CFI) 0.953
Tucker-Lewis Index (TLI) 0.956
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -2458.321
Loglikelihood unrestricted model (H1) -2429.247
Number of free parameters 11
Akaike (AIC) 4938.642
Bayesian (BIC) 4979.384
Sample-size adjusted Bayesian (BIC) 4944.498
Root Mean Square Error of Approximation:
RMSEA 0.094
90 Percent Confidence Interval 0.069 0.120
P-value RMSEA <= 0.05 0.003
Standardized Root Mean Square Residual:
SRMR 0.100
Parameter Estimates:
Information Expected
Information saturated (h1) model Structured
Standard Errors Standard
Latent Variables:
Estimate Std.Err z-value P(>|z|)
eta =~
y1 1.000
y2 1.000
y3 1.000
y4 1.000
y5 1.000
y6 1.000
Intercepts:
Estimate Std.Err z-value P(>|z|)
eta 0.000
.y1 -0.003 0.069 -0.048 0.962
.y2 0.417 0.072 5.790 0.000
.y3 0.350 0.072 4.894 0.000
.y4 0.667 0.068 9.778 0.000
.y5 -0.057 0.068 -0.831 0.406
.y6 0.527 0.068 7.725 0.000
Variances:
Estimate Std.Err z-value P(>|z|)
eta (veta) 0.826 0.076 10.871 0.000
.y1 (vps1) 0.619 0.059 10.405 0.000
.y2 (vps2) 0.728 0.068 10.685 0.000
.y3 (vps3) 0.709 0.067 10.642 0.000
.y4 (vps4) 0.569 0.031 18.355 0.000
.y5 (vps4) 0.569 0.031 18.355 0.000
.y6 (vps4) 0.569 0.031 18.355 0.00010.1.3 Output 3
lavaan 0.6-3 ended normally after 15 iterations
Optimization method NLMINB
Number of free parameters 7
Number of observations 200
Estimator ML
Model Fit Test Statistic 4.266
Degrees of freedom 2
P-value (Chi-square) 0.119
Model test baseline model:
Minimum Function Test Statistic 322.845
Degrees of freedom 3
P-value 0.000
User model versus baseline model:
Comparative Fit Index (CFI) 0.993
Tucker-Lewis Index (TLI) 0.989
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -1161.377
Loglikelihood unrestricted model (H1) -1159.244
Number of free parameters 7
Akaike (AIC) 2336.754
Bayesian (BIC) 2359.842
Sample-size adjusted Bayesian (BIC) 2337.665
Root Mean Square Error of Approximation:
RMSEA 0.075
90 Percent Confidence Interval 0.000 0.176
P-value RMSEA <= 0.05 0.245
Standardized Root Mean Square Residual:
SRMR 0.042
Parameter Estimates:
Information Expected
Information saturated (h1) model Structured
Standard Errors Standard
Latent Variables:
Estimate Std.Err z-value P(>|z|)
eta =~
y1 1.000
y2 1.000
y3 1.000
Intercepts:
Estimate Std.Err z-value P(>|z|)
eta 0.000
.y1 0.245 0.141 1.733 0.083
.y2 0.845 0.155 5.450 0.000
.y3 0.045 0.165 0.273 0.785
Variances:
Estimate Std.Err z-value P(>|z|)
eta (veta) 3.262 0.369 8.833 0.000
.y1 (vps1) 0.735 0.154 4.768 0.000
.y2 (vps2) 1.546 0.211 7.338 0.000
.y3 (vps3) 2.157 0.264 8.175 0.000 10.1.4 Output 4
lavaan 0.6-3 ended normally after 15 iterations
Optimization method NLMINB
Number of free parameters 18
Number of observations 499
Estimator ML
Model Fit Test Statistic 119.363
Degrees of freedom 9
P-value (Chi-square) 0.000
Model test baseline model:
Minimum Function Test Statistic 1353.079
Degrees of freedom 15
P-value 0.000
User model versus baseline model:
Comparative Fit Index (CFI) 0.918
Tucker-Lewis Index (TLI) 0.863
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -4125.719
Loglikelihood unrestricted model (H1) -4066.038
Number of free parameters 18
Akaike (AIC) 8287.439
Bayesian (BIC) 8363.266
Sample-size adjusted Bayesian (BIC) 8306.133
Root Mean Square Error of Approximation:
RMSEA 0.157
90 Percent Confidence Interval 0.132 0.182
P-value RMSEA <= 0.05 0.000
Standardized Root Mean Square Residual:
SRMR 0.056
Parameter Estimates:
Information Expected
Information saturated (h1) model Structured
Standard Errors Standard
Latent Variables:
Estimate Std.Err z-value P(>|z|)
eta =~
item_1 (la11) 0.969 0.045 21.315 0.000
item_2 (la21) 0.726 0.051 14.350 0.000
item_3 (la31) 1.166 0.048 24.139 0.000
item_4 (la41) 0.818 0.043 19.009 0.000
item_5 (la51) 0.671 0.051 13.212 0.000
item_6 (la61) 0.642 0.049 13.141 0.000
Intercepts:
Estimate Std.Err z-value P(>|z|)
eta 0.000
.item_1 (la10) 2.563 0.053 48.247 0.000
.item_2 (la20) 2.315 0.053 43.358 0.000
.item_3 (la30) 2.261 0.059 38.417 0.000
.item_4 (la40) 2.948 0.049 60.733 0.000
.item_5 (la50) 1.581 0.053 29.962 0.000
.item_6 (la60) 2.461 0.051 48.541 0.000
Variances:
Estimate Std.Err z-value P(>|z|)
eta 1.000
.item_1 (vps1) 0.468 0.040 11.758 0.000
.item_2 (vps2) 0.895 0.061 14.684 0.000
.item_3 (vps3) 0.368 0.043 8.630 0.000
.item_4 (vps4) 0.507 0.038 13.231 0.000
.item_5 (vps5) 0.940 0.063 14.896 0.000
.item_6 (vps6) 0.871 0.058 14.908 0.000 10.1.5 Output 5
lavaan 0.6-3 ended normally after 77 iterations
Optimization method NLMINB
Number of free parameters 12
Number of observations 200
Estimator ML
Model Fit Test Statistic 1.123
Degrees of freedom 2
P-value (Chi-square) 0.570
Model test baseline model:
Minimum Function Test Statistic 676.783
Degrees of freedom 6
P-value 0.000
User model versus baseline model:
Comparative Fit Index (CFI) 1.000
Tucker-Lewis Index (TLI) 1.004
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -1462.638
Loglikelihood unrestricted model (H1) -1462.077
Number of free parameters 12
Akaike (AIC) 2949.277
Bayesian (BIC) 2988.857
Sample-size adjusted Bayesian (BIC) 2950.839
Root Mean Square Error of Approximation:
RMSEA 0.000
90 Percent Confidence Interval 0.000 0.118
P-value RMSEA <= 0.05 0.705
Standardized Root Mean Square Residual:
SRMR 0.004
Parameter Estimates:
Information Expected
Information saturated (h1) model Structured
Standard Errors Standard
Latent Variables:
Estimate Std.Err z-value P(>|z|)
eta =~
y1 (la11) 1.000
y2 (la21) 1.013 0.058 17.570 0.000
y3 (la31) 1.010 0.055 18.235 0.000
y4 (la41) 1.043 0.058 18.092 0.000
Intercepts:
Estimate Std.Err z-value P(>|z|)
eta 15.205 0.159 95.754 0.000
.y1 (la10) 0.000
.y2 (la20) -0.416 0.883 -0.470 0.638
.y3 (la30) -0.220 0.848 -0.260 0.795
.y4 (la40) -0.793 0.883 -0.898 0.369
Variances:
Estimate Std.Err z-value P(>|z|)
eta (veta) 3.999 0.503 7.945 0.000
.y1 (vps1) 1.044 0.144 7.228 0.000
.y2 (vps2) 1.274 0.166 7.682 0.000
.y3 (vps3) 1.077 0.148 7.263 0.000
.y4 (vps4) 1.192 0.162 7.362 0.00010.2 Aufgabe 2
Die zweite Übungsaufgabe besteht darin, drei vorgegebene, lückenhafte Outputs zu vervollständigen. Alle Aufgaben sind eindeutig und können auf Grund eures bisherigen Vorwissens gelöst werden.
10.2.1 Output 1
Latent Variables:
Estimate Std.Err z-value P(>|z|)
eta =~
y1
y2
y3
y4
y5
y6
Intercepts:
Estimate Std.Err z-value P(>|z|)
eta 0.317 0.056 5.651 0.000
.y1
.y2
.y3
.y4
.y5
.y6
Variances:
Estimate Std.Err z-value P(>|z|)
eta (veta) 0.824 0.077 10.694 0.000
.y1 (veps) 0.709 0.026 27.386 0.000
.y2 (veps)
.y3 (veps)
.y4 (veps)
.y5 (veps)
.y6 (veps) 10.2.2 Output 2
Latent Variables:
Estimate Std.Err z-value P(>|z|)
eta =~
y1
y2
y3
y4
Intercepts:
Estimate Std.Err z-value P(>|z|)
eta
.y1 15.205 0.161 94.715 0.000
.y2 14.990 0.164 91.229 0.000
.y3 15.130 0.161 93.897 0.000
.y4 15.070 0.164 92.155 0.000
Variances:
Estimate Std.Err z-value P(>|z|)
eta (veta) 4.125 0.441 9.347 0.000
.y1 (vps1) 1.029 0.141 7.309 0.000
.y2 (vps2) 1.275 0.163 7.837 0.000
.y3 (vps3) 1.068 0.144 7.407 0.000
.y4 (vps4) 1.223 0.158 7.744 0.000 10.2.3 Output 3
Latent Variables:
Estimate Std.Err z-value P(>|z|)
eta =~
item_1 (la11)
item_2 (la21) 0.749 0.053 14.034 0.000
item_3 (la31) 1.203 0.055 22.016 0.000
item_4 (la41) 0.843 0.046 18.275 0.000
item_5 (la51) 0.692 0.053 12.965 0.000
item_6 (la61) 0.662 0.051 12.899 0.000
Intercepts:
Estimate Std.Err z-value P(>|z|)
eta 2.563 0.053 48.247 0.000
.item_1 (la10)
.item_2 (la20) 0.395 0.145 2.723 0.006
.item_3 (la30) -0.823 0.147 -5.583 0.000
.item_4 (la40) 0.786 0.125 6.280 0.000
.item_5 (la50) -0.193 0.145 -1.327 0.185
.item_6 (la60) 0.765 0.139 5.482 0.000
Variances:
Estimate Std.Err z-value P(>|z|)
eta (veta) 0.940 0.088 10.657 0.000
.item_1 (vps1) 0.468 0.040 11.758 0.000
.item_2 (vps2) 0.895 0.061 14.684 0.000
.item_3 (vps3) 0.368 0.043 8.630 0.000
.item_4 (vps4) 0.507 0.038 13.231 0.000
.item_5 (vps5) 0.940 0.063 14.896 0.000
.item_6 (vps6) 0.871 0.058 14.908 0.000 10.3 Aufgabe 3
In der dritten Übungsaufgabe habt ihr einen Output eines \(\tau\)-kongenerischen Modells vorgegeben, bei dem die Beschriftungen der Schätzer fehlen. Ergänzt bitte diese Beschriftungen der einzelnen Schätzer.
10.3.1 Output eines tau-kongenerischen Modells
Latent Variables:
Estimate Std.Err z-value P(>|z|)
eta =~
y1 1.690 0.116 14.572 0.000
y2 1.943 0.136 14.291 0.000
y3 1.905 0.147 13.002 0.000
Intercepts:
Estimate Std.Err z-value P(>|z|)
eta 0.000
.y1 0.245 0.137 1.786 0.074
.y2 0.845 0.160 5.281 0.000
.y3 0.045 0.168 0.267 0.789
Variances:
Estimate Std.Err z-value P(>|z|)
eta 1.000
.y1 0.910 0.168 5.409 0.000
.y2 1.345 0.231 5.835 0.000
.y3 2.033 0.272 7.487 0.000 10.4 Aufgabe 4
In der folgenden Aufagbe werden euch mehrere Outputs dargestellt. Eure Aufgabe ist es, aus den dargestellten Outputs abzulesen, welches Modell der Testung zu Grunde lag und dann auf dieser Grundlage, aus den Outputs die Modellgleichungen und Pfaddiagramme des Modells aufzustellen.
10.4.1 Output 1
Latent Variables:
Estimate Std.Err z-value P(>|z|)
eta =~
y1 1.000
y2 1.000
y3 1.000
y4 1.000
Intercepts:
Estimate Std.Err z-value P(>|z|)
eta 6.680 0.068 98.517 0.000
.y1 0.000
.y2 0.000
.y3 0.000
.y4 0.000
Variances:
Estimate Std.Err z-value P(>|z|)
eta (veta) 1.947 0.145 13.431 0.000
.y1 (veps) 1.295 0.048 27.221 0.000
.y2 (veps) 1.295 0.048 27.221 0.000
.y3 (veps) 1.295 0.048 27.221 0.000
.y4 (veps) 1.295 0.048 27.221 0.000 10.4.2 Output 2
Latent Variables:
Estimate Std.Err z-value P(>|z|)
eta =~
y1 1.000
y2 1.000
y3 1.000
y4 1.000
Intercepts:
Estimate Std.Err z-value P(>|z|)
eta 15.205 0.161 94.715 0.000
.y1 0.000
.y2 -0.215 0.107 -2.003 0.045
.y3 -0.075 0.102 -0.732 0.464
.y4 -0.135 0.106 -1.272 0.203
Variances:
Estimate Std.Err z-value P(>|z|)
eta (veta) 4.125 0.441 9.347 0.000
.y1 (vps1) 1.029 0.141 7.309 0.000
.y2 (vps2) 1.275 0.163 7.837 0.000
.y3 (vps3) 1.068 0.144 7.407 0.000
.y4 (vps4) 1.223 0.158 7.744 0.000 10.4.3 Output 3
Latent Variables:
Estimate Std.Err z-value P(>|z|)
eta =~
y1 (la11) 0.864 0.072 12.063 0.000
y2 (la21) 0.004 0.083 0.044 0.965
y3 (la31) 0.931 0.077 12.118 0.000
y4 (la41) 0.432 0.071 6.083 0.000
y5 (la51) 0.573 0.056 10.154 0.000
y6 (la61) 0.785 0.060 13.160 0.000
y7 (la71) 0.029 0.065 0.447 0.655
y8 (la81) 1.051 0.074 14.126 0.000
Intercepts:
Estimate Std.Err z-value P(>|z|)
eta 0.000
.y1 (la10) 3.443 0.083 41.480 0.000
.y2 (la20) 3.460 0.079 43.531 0.000
.y3 (la30) 3.989 0.089 44.760 0.000
.y4 (la40) 2.994 0.072 41.795 0.000
.y5 (la50) 3.528 0.062 56.723 0.000
.y6 (la60) 5.051 0.071 71.054 0.000
.y7 (la70) 3.966 0.062 64.312 0.000
.y8 (la80) 3.267 0.091 35.906 0.000
Variances:
Estimate Std.Err z-value P(>|z|)
eta 1.000
.y1 (vps1) 0.465 0.060 7.800 0.000
.y2 (vps2) 1.112 0.119 9.381 0.000
.y3 (vps3) 0.531 0.068 7.774 0.000
.y4 (vps4) 0.717 0.079 9.131 0.000
.y5 (vps5) 0.353 0.042 8.478 0.000
.y6 (vps6) 0.274 0.038 7.142 0.000
.y7 (vps7) 0.668 0.071 9.380 0.000
.y8 (vps8) 0.352 0.056 6.270 0.000 10.4.4 Output 4
Latent Variables:
Estimate Std.Err z-value P(>|z|)
eta =~
y1 (la11) 1.000
y2 (la21) 1.081 0.060 18.031 0.000
y3 (la31) 1.050 0.059 17.843 0.000
y4 (la41) 0.943 0.058 16.348 0.000
Intercepts:
Estimate Std.Err z-value P(>|z|)
eta 6.690 0.080 83.795 0.000
.y1 (la10) 0.000
.y2 (la20) -0.290 0.408 -0.711 0.477
.y3 (la30) -0.226 0.400 -0.566 0.571
.y4 (la40) -0.024 0.392 -0.061 0.951
Variances:
Estimate Std.Err z-value P(>|z|)
eta (veta) 1.898 0.196 9.685 0.000
.y1 (vps1) 1.251 0.105 11.965 0.000
.y2 (vps2) 1.095 0.103 10.669 0.000
.y3 (vps3) 1.107 0.101 11.009 0.000
.y4 (vps4) 1.374 0.108 12.708 0.000