29.3 Saturazione sul fattore sbagliato
Brown (2015) considera anche il caso opposto, ovvero quello nel quale il ricercatore ipotizza una saturazione spuria. Per i dati in discussione, si può avere la situazione presente.
model4 <- "
copingm =~ x1 + x2 + x3 + x4
socialm =~ x4 +x5 + x6 + x7 + x8 + x12
enhancem =~ x9 + x10 + x11
"Adattiamo il modello ai dati.
Esaminiamo la soluzione ottenuta.
summary(fit4, fit.measures = TRUE)
#> lavaan 0.6.15 ended normally after 59 iterations
#>
#> Estimator ML
#> Optimization method NLMINB
#> Number of model parameters 40
#>
#> Number of observations 500
#>
#> Model Test User Model:
#>
#> Test statistic 212.717
#> Degrees of freedom 50
#> P-value (Chi-square) 0.000
#>
#> Model Test Baseline Model:
#>
#> Test statistic 1664.026
#> Degrees of freedom 66
#> P-value 0.000
#>
#> User Model versus Baseline Model:
#>
#> Comparative Fit Index (CFI) 0.898
#> Tucker-Lewis Index (TLI) 0.866
#>
#> Loglikelihood and Information Criteria:
#>
#> Loglikelihood user model (H0) -12010.051
#> Loglikelihood unrestricted model (H1) -11903.692
#>
#> Akaike (AIC) 24100.101
#> Bayesian (BIC) 24268.685
#> Sample-size adjusted Bayesian (SABIC) 24141.723
#>
#> Root Mean Square Error of Approximation:
#>
#> RMSEA 0.081
#> 90 Percent confidence interval - lower 0.070
#> 90 Percent confidence interval - upper 0.092
#> P-value H_0: RMSEA <= 0.050 0.000
#> P-value H_0: RMSEA >= 0.080 0.554
#>
#> Standardized Root Mean Square Residual:
#>
#> SRMR 0.073
#>
#> Parameter Estimates:
#>
#> Standard errors Standard
#> Information Expected
#> Information saturated (h1) model Structured
#>
#> Latent Variables:
#> Estimate Std.Err z-value P(>|z|)
#> copingm =~
#> x1 1.000
#> x2 0.741 0.093 7.925 0.000
#> x3 0.932 0.118 7.906 0.000
#> x4 0.699 0.117 5.995 0.000
#> socialm =~
#> x4 1.000
#> x5 1.725 0.260 6.634 0.000
#> x6 2.098 0.305 6.879 0.000
#> x7 2.717 0.401 6.775 0.000
#> x8 2.619 0.382 6.848 0.000
#> x12 0.900 0.236 3.818 0.000
#> enhancem =~
#> x9 1.000
#> x10 0.638 0.076 8.408 0.000
#> x11 0.767 0.094 8.153 0.000
#>
#> Covariances:
#> Estimate Std.Err z-value P(>|z|)
#> copingm ~~
#> socialm 0.410 0.072 5.663 0.000
#> enhancem 0.661 0.148 4.456 0.000
#> socialm ~~
#> enhancem 0.347 0.089 3.902 0.000
#>
#> Intercepts:
#> Estimate Std.Err z-value P(>|z|)
#> .x1 0.000 0.092 0.000 1.000
#> .x2 0.000 0.068 0.000 1.000
#> .x3 0.000 0.086 0.000 1.000
#> .x4 0.000 0.063 0.000 1.000
#> .x5 0.000 0.077 0.000 1.000
#> .x6 0.000 0.079 0.000 1.000
#> .x7 0.000 0.111 0.000 1.000
#> .x8 0.000 0.101 0.000 1.000
#> .x12 0.000 0.119 0.000 1.000
#> .x9 0.000 0.120 0.000 1.000
#> .x10 0.000 0.078 0.000 1.000
#> .x11 0.000 0.115 0.000 1.000
#> copingm 0.000
#> socialm 0.000
#> enhancem 0.000
#>
#> Variances:
#> Estimate Std.Err z-value P(>|z|)
#> .x1 3.106 0.230 13.478 0.000
#> .x2 1.686 0.125 13.449 0.000
#> .x3 2.698 0.200 13.477 0.000
#> .x4 0.463 0.069 6.719 0.000
#> .x5 1.805 0.130 13.886 0.000
#> .x6 1.378 0.115 12.022 0.000
#> .x7 3.255 0.248 13.143 0.000
#> .x8 2.418 0.194 12.455 0.000
#> .x12 6.740 0.430 15.673 0.000
#> .x9 3.891 0.436 8.933 0.000
#> .x10 1.724 0.183 9.435 0.000
#> .x11 4.662 0.371 12.579 0.000
#> copingm 1.129 0.218 5.170 0.000
#> socialm 0.397 0.111 3.566 0.000
#> enhancem 3.277 0.524 6.258 0.000È chiaro che il modello model4 è inadeguato. Il problema emerge chiaramente anche esaminando i MI.
modindices(fit4)
#> lhs op rhs mi epc sepc.lv sepc.all sepc.nox
#> 47 copingm =~ x5 0.090 0.036 0.038 0.022 0.022
#> 48 copingm =~ x6 0.554 0.090 0.096 0.054 0.054
#> 49 copingm =~ x7 0.107 0.055 0.059 0.024 0.024
#> 50 copingm =~ x8 3.919 -0.306 -0.325 -0.143 -0.143
#> 51 copingm =~ x12 6.109 0.499 0.530 0.199 0.199
#> 52 copingm =~ x9 0.390 -0.096 -0.102 -0.038 -0.038
#> 53 copingm =~ x10 0.027 -0.016 -0.017 -0.010 -0.010
#> 54 copingm =~ x11 0.823 0.123 0.131 0.051 0.051
#> 55 socialm =~ x1 1.990 -0.398 -0.251 -0.122 -0.122
#> 56 socialm =~ x2 0.638 0.166 0.105 0.069 0.069
#> 57 socialm =~ x3 0.372 0.160 0.101 0.053 0.053
#> 58 socialm =~ x9 0.315 -0.130 -0.082 -0.031 -0.031
#> 59 socialm =~ x10 1.423 0.179 0.113 0.064 0.064
#> 60 socialm =~ x11 0.520 -0.150 -0.094 -0.037 -0.037
#> 61 enhancem =~ x1 1.029 0.067 0.121 0.059 0.059
#> 62 enhancem =~ x2 0.232 0.023 0.042 0.028 0.028
#> 63 enhancem =~ x3 0.153 -0.024 -0.043 -0.023 -0.023
#> 64 enhancem =~ x4 0.745 -0.031 -0.056 -0.040 -0.040
#> 65 enhancem =~ x5 0.343 -0.028 -0.050 -0.029 -0.029
#> 66 enhancem =~ x6 0.103 0.015 0.027 0.015 0.015
#> 67 enhancem =~ x7 2.752 -0.110 -0.198 -0.080 -0.080
#> 68 enhancem =~ x8 0.129 -0.021 -0.038 -0.017 -0.017
#> 69 enhancem =~ x12 116.781 0.916 1.658 0.624 0.624
#> 70 x1 ~~ x2 1.709 0.177 0.177 0.077 0.077
#> 71 x1 ~~ x3 2.273 -0.257 -0.257 -0.089 -0.089
#> 72 x1 ~~ x4 0.850 0.103 0.103 0.086 0.086
#> 73 x1 ~~ x5 0.292 -0.064 -0.064 -0.027 -0.027
#> 74 x1 ~~ x6 0.188 -0.048 -0.048 -0.023 -0.023
#> 75 x1 ~~ x7 0.023 -0.025 -0.025 -0.008 -0.008
#> 76 x1 ~~ x8 0.419 -0.093 -0.093 -0.034 -0.034
#> 77 x1 ~~ x12 0.025 -0.034 -0.034 -0.007 -0.007
#> 78 x1 ~~ x9 0.011 0.020 0.020 0.006 0.006
#> 79 x1 ~~ x10 0.004 0.008 0.008 0.003 0.003
#> 80 x1 ~~ x11 1.804 0.259 0.259 0.068 0.068
#> 81 x2 ~~ x3 0.071 -0.034 -0.034 -0.016 -0.016
#> 82 x2 ~~ x4 2.979 -0.143 -0.143 -0.162 -0.162
#> 83 x2 ~~ x5 2.403 0.135 0.135 0.077 0.077
#> 84 x2 ~~ x6 0.551 0.060 0.060 0.040 0.040
#> 85 x2 ~~ x7 0.457 -0.081 -0.081 -0.035 -0.035
#> 86 x2 ~~ x8 0.012 0.011 0.011 0.006 0.006
#> 87 x2 ~~ x12 0.134 -0.058 -0.058 -0.017 -0.017
#> 88 x2 ~~ x9 1.033 0.145 0.145 0.056 0.056
#> 89 x2 ~~ x10 1.140 -0.100 -0.100 -0.058 -0.058
#> 90 x2 ~~ x11 0.323 0.081 0.081 0.029 0.029
#> 91 x3 ~~ x4 1.472 0.127 0.127 0.113 0.113
#> 92 x3 ~~ x5 0.140 0.041 0.041 0.019 0.019
#> 93 x3 ~~ x6 0.717 0.087 0.087 0.045 0.045
#> 94 x3 ~~ x7 0.317 0.086 0.086 0.029 0.029
#> 95 x3 ~~ x8 3.121 -0.237 -0.237 -0.093 -0.093
#> 96 x3 ~~ x12 0.001 0.006 0.006 0.001 0.001
#> 97 x3 ~~ x9 0.000 0.003 0.003 0.001 0.001
#> 98 x3 ~~ x10 4.165 -0.241 -0.241 -0.111 -0.111
#> 99 x3 ~~ x11 3.806 0.350 0.350 0.099 0.099
#> 100 x4 ~~ x5 0.316 -0.036 -0.036 -0.039 -0.039
#> 101 x4 ~~ x6 0.052 -0.015 -0.015 -0.019 -0.019
#> 102 x4 ~~ x7 1.182 0.099 0.099 0.081 0.081
#> 103 x4 ~~ x8 0.062 -0.021 -0.021 -0.020 -0.020
#> 104 x4 ~~ x12 0.033 0.020 0.020 0.011 0.011
#> 105 x4 ~~ x9 1.418 -0.115 -0.115 -0.086 -0.086
#> 106 x4 ~~ x10 0.914 0.061 0.061 0.068 0.068
#> 107 x4 ~~ x11 0.517 -0.068 -0.068 -0.047 -0.047
#> 108 x5 ~~ x6 0.611 0.073 0.073 0.046 0.046
#> 109 x5 ~~ x7 0.115 -0.045 -0.045 -0.019 -0.019
#> 110 x5 ~~ x8 0.079 0.034 0.034 0.016 0.016
#> 111 x5 ~~ x12 3.265 -0.302 -0.302 -0.087 -0.087
#> 112 x5 ~~ x9 0.203 0.066 0.066 0.025 0.025
#> 113 x5 ~~ x10 0.000 0.002 0.002 0.001 0.001
#> 114 x5 ~~ x11 2.312 -0.224 -0.224 -0.077 -0.077
#> 115 x6 ~~ x7 2.239 -0.200 -0.200 -0.094 -0.094
#> 116 x6 ~~ x8 0.073 0.033 0.033 0.018 0.018
#> 117 x6 ~~ x12 0.478 0.109 0.109 0.036 0.036
#> 118 x6 ~~ x9 1.251 -0.153 -0.153 -0.066 -0.066
#> 119 x6 ~~ x10 0.784 0.079 0.079 0.051 0.051
#> 120 x6 ~~ x11 0.370 0.083 0.083 0.033 0.033
#> 121 x7 ~~ x8 1.644 0.219 0.219 0.078 0.078
#> 122 x7 ~~ x12 0.433 -0.152 -0.152 -0.032 -0.032
#> 123 x7 ~~ x9 0.005 -0.015 -0.015 -0.004 -0.004
#> 124 x7 ~~ x10 1.836 -0.179 -0.179 -0.076 -0.076
#> 125 x7 ~~ x11 0.348 -0.119 -0.119 -0.031 -0.031
#> 126 x8 ~~ x12 2.680 -0.335 -0.335 -0.083 -0.083
#> 127 x8 ~~ x9 0.676 0.147 0.147 0.048 0.048
#> 128 x8 ~~ x10 0.337 0.068 0.068 0.033 0.033
#> 129 x8 ~~ x11 3.437 -0.330 -0.330 -0.098 -0.098
#> 130 x12 ~~ x9 7.051 0.713 0.713 0.139 0.139
#> 131 x12 ~~ x10 6.960 0.465 0.465 0.136 0.136
#> 132 x12 ~~ x11 68.717 2.238 2.238 0.399 0.399
#> 133 x9 ~~ x10 0.081 0.138 0.138 0.053 0.053
#> 134 x9 ~~ x11 0.166 0.209 0.209 0.049 0.049
#> 135 x10 ~~ x11 0.423 -0.211 -0.211 -0.075 -0.075Il MI relativo alla saturazione di x12 su enhancem è uguale a 116.781. Chiaramente, in una revisione del modello, questo problema dovrebbe deve essere affrontato.
References
Brown, Timothy A. 2015. Confirmatory Factor Analysis for Applied Research. Guilford publications.