Get ready by loading some packages:
library(tidyverse)
library(psych)
library(lavaan)
library(semPlot)
library(semTools)This tutorial focuses on interpreting the results of a CFA and exploring areas of localized strain to subsequently compare alternative models.
d <- read_csv("cfa_data.csv")
summary(d)## SPRt_s WMTt OPr EIr
## Min. :-0.60412 Min. :-0.67012 Min. :0.3750 Min. :0.700
## 1st Qu.:-0.17407 1st Qu.:-0.12850 1st Qu.:0.7265 1st Qu.:1.300
## Median : 0.07328 Median : 0.06882 Median :0.8400 Median :1.550
## Mean : 0.05501 Mean : 0.04815 Mean :0.8176 Mean :1.529
## 3rd Qu.: 0.24032 3rd Qu.: 0.20718 3rd Qu.:0.9384 3rd Qu.:1.800
## Max. : 0.92267 Max. : 0.66606 Max. :1.0000 Max. :2.100
## NA's :26 NA's :31 NA's :28 NA's :44
## TAGJT TWGJT UAGJT_ug UWGJT_ug
## Min. :0.2083 Min. :0.1250 Min. :0.0000 Min. :0.0000
## 1st Qu.:0.4896 1st Qu.:0.5417 1st Qu.:0.3333 1st Qu.:0.4167
## Median :0.5417 Median :0.5833 Median :0.4167 Median :0.6667
## Mean :0.5687 Mean :0.6083 Mean :0.4476 Mean :0.6194
## 3rd Qu.:0.6250 3rd Qu.:0.7083 3rd Qu.:0.5833 3rd Qu.:0.7708
## Max. :1.0000 Max. :0.9583 Max. :1.0000 Max. :1.0000
## NA's :8 NA's :36 NA's :8 NA's :36
## MKT
## Min. :0.2083
## 1st Qu.:0.5417
## Median :0.6667
## Mean :0.6401
## 3rd Qu.:0.7500
## Max. :1.0000
## NA's :21Factorability:
KMO(d)## Kaiser-Meyer-Olkin factor adequacy
## Call: KMO(r = d)
## Overall MSA = 0.77
## MSA for each item =
## SPRt_s WMTt OPr EIr TAGJT TWGJT UAGJT_ug UWGJT_ug
## 0.70 0.51 0.81 0.83 0.79 0.77 0.78 0.75
## MKT
## 0.76cortest.bartlett(cor(d, use = "pairwise.complete.obs"), n = 156)## $chisq
## [1] 243.5961
##
## $p.value
## [1] 1.183665e-32
##
## $df
## [1] 36Two models - copy and pasted from the last tutorial.
One <- '
# latent variable definitions
language =~ EIr + OPr + UWGJT_ug + MKT + TWGJT + MKT + SPRt_s + WMTt + TAGJT + UAGJT_ug
'
Two <- '
# latent variable definitions
implicit =~ EIr + OPr + TWGJT + TAGJT + WMTt + SPRt_s
explicit =~ UWGJT_ug + UAGJT_ug + MKT
# covariances
implicit~~explicit
'And then fit them both:
fitOne <- cfa(One, data = d, missing = "fiml", estimator = "MLR")
fitTwo <- cfa(Two, data = d, missing = "fiml", estimator = "MLR")And compare…
lavaan::summary(fitOne, estimates = T, standardized = T, fit.measures = T)## lavaan 0.6-7 ended normally after 84 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of free parameters 27
##
## Number of observations 156
## Number of missing patterns 26
##
## Model Test User Model:
## Standard Robust
## Test Statistic 30.458 29.942
## Degrees of freedom 27 27
## P-value (Chi-square) 0.294 0.317
## Scaling correction factor 1.017
## Yuan-Bentler correction (Mplus variant)
##
## Model Test Baseline Model:
##
## Test statistic 183.271 176.306
## Degrees of freedom 36 36
## P-value 0.000 0.000
## Scaling correction factor 1.040
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.977 0.979
## Tucker-Lewis Index (TLI) 0.969 0.972
##
## Robust Comparative Fit Index (CFI) 0.979
## Robust Tucker-Lewis Index (TLI) 0.973
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) 274.605 274.605
## Scaling correction factor 1.010
## for the MLR correction
## Loglikelihood unrestricted model (H1) 289.834 289.834
## Scaling correction factor 1.014
## for the MLR correction
##
## Akaike (AIC) -495.211 -495.211
## Bayesian (BIC) -412.865 -412.865
## Sample-size adjusted Bayesian (BIC) -498.328 -498.328
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.029 0.026
## 90 Percent confidence interval - lower 0.000 0.000
## 90 Percent confidence interval - upper 0.071 0.070
## P-value RMSEA <= 0.05 0.753 0.774
##
## Robust RMSEA 0.027
## 90 Percent confidence interval - lower 0.000
## 90 Percent confidence interval - upper 0.070
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.059 0.059
##
## Parameter Estimates:
##
## Standard errors Sandwich
## Information bread Observed
## Observed information based on Hessian
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## language =~
## EIr 1.000 0.183 0.591
## OPr 0.433 0.091 4.785 0.000 0.079 0.526
## UWGJT_ug 0.723 0.167 4.325 0.000 0.132 0.556
## MKT 0.459 0.101 4.550 0.000 0.084 0.491
## TWGJT 0.594 0.099 6.005 0.000 0.109 0.787
## SPRt_s 0.385 0.179 2.150 0.032 0.071 0.234
## WMTt 0.104 0.133 0.782 0.434 0.019 0.074
## TAGJT 0.453 0.095 4.765 0.000 0.083 0.565
## UAGJT_ug 0.348 0.131 2.664 0.008 0.064 0.287
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .EIr 1.529 0.028 54.086 0.000 1.529 4.936
## .OPr 0.818 0.013 62.349 0.000 0.818 5.424
## .UWGJT_ug 0.617 0.021 28.925 0.000 0.617 2.590
## .MKT 0.640 0.015 43.732 0.000 0.640 3.738
## .TWGJT 0.606 0.012 50.090 0.000 0.606 4.381
## .SPRt_s 0.052 0.026 1.991 0.046 0.052 0.173
## .WMTt 0.046 0.023 2.000 0.046 0.046 0.180
## .TAGJT 0.567 0.012 47.599 0.000 0.567 3.861
## .UAGJT_ug 0.446 0.018 24.347 0.000 0.446 2.008
## language 0.000 0.000 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .EIr 0.062 0.010 6.195 0.000 0.062 0.650
## .OPr 0.016 0.002 7.497 0.000 0.016 0.723
## .UWGJT_ug 0.039 0.006 6.864 0.000 0.039 0.691
## .MKT 0.022 0.003 7.190 0.000 0.022 0.759
## .TWGJT 0.007 0.002 3.471 0.001 0.007 0.381
## .SPRt_s 0.086 0.010 8.765 0.000 0.086 0.945
## .WMTt 0.065 0.008 8.034 0.000 0.065 0.994
## .TAGJT 0.015 0.003 5.669 0.000 0.015 0.681
## .UAGJT_ug 0.045 0.006 7.231 0.000 0.045 0.918
## language 0.034 0.010 3.385 0.001 1.000 1.000lavaan::summary(fitTwo, estimates = T, standardized = T, fit.measures = T)## lavaan 0.6-7 ended normally after 95 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of free parameters 28
##
## Number of observations 156
## Number of missing patterns 26
##
## Model Test User Model:
## Standard Robust
## Test Statistic 21.043 21.508
## Degrees of freedom 26 26
## P-value (Chi-square) 0.740 0.715
## Scaling correction factor 0.978
## Yuan-Bentler correction (Mplus variant)
##
## Model Test Baseline Model:
##
## Test statistic 183.271 176.306
## Degrees of freedom 36 36
## P-value 0.000 0.000
## Scaling correction factor 1.040
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 1.000 1.000
## Tucker-Lewis Index (TLI) 1.047 1.044
##
## Robust Comparative Fit Index (CFI) 1.000
## Robust Tucker-Lewis Index (TLI) 1.042
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) 279.313 279.313
## Scaling correction factor 1.047
## for the MLR correction
## Loglikelihood unrestricted model (H1) 289.834 289.834
## Scaling correction factor 1.014
## for the MLR correction
##
## Akaike (AIC) -502.625 -502.625
## Bayesian (BIC) -417.229 -417.229
## Sample-size adjusted Bayesian (BIC) -505.858 -505.858
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.000 0.000
## 90 Percent confidence interval - lower 0.000 0.000
## 90 Percent confidence interval - upper 0.047 0.049
## P-value RMSEA <= 0.05 0.961 0.953
##
## Robust RMSEA 0.000
## 90 Percent confidence interval - lower 0.000
## 90 Percent confidence interval - upper 0.048
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.052 0.052
##
## Parameter Estimates:
##
## Standard errors Sandwich
## Information bread Observed
## Observed information based on Hessian
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## implicit =~
## EIr 1.000 0.182 0.586
## OPr 0.436 0.089 4.906 0.000 0.079 0.524
## TWGJT 0.626 0.116 5.392 0.000 0.114 0.821
## TAGJT 0.465 0.101 4.618 0.000 0.084 0.575
## WMTt 0.107 0.131 0.813 0.416 0.019 0.076
## SPRt_s 0.367 0.175 2.099 0.036 0.067 0.221
## explicit =~
## UWGJT_ug 1.000 0.180 0.752
## UAGJT_ug 0.328 0.179 1.830 0.067 0.059 0.265
## MKT 0.607 0.123 4.944 0.000 0.109 0.636
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## implicit ~~
## explicit 0.022 0.006 4.006 0.000 0.685 0.685
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .EIr 1.530 0.028 54.065 0.000 1.530 4.935
## .OPr 0.819 0.013 62.390 0.000 0.819 5.423
## .TWGJT 0.607 0.012 50.156 0.000 0.607 4.384
## .TAGJT 0.567 0.012 47.636 0.000 0.567 3.862
## .WMTt 0.046 0.023 2.005 0.045 0.046 0.180
## .SPRt_s 0.052 0.026 1.994 0.046 0.052 0.173
## .UWGJT_ug 0.616 0.021 29.071 0.000 0.616 2.583
## .UAGJT_ug 0.446 0.018 24.223 0.000 0.446 2.008
## .MKT 0.638 0.015 43.185 0.000 0.638 3.725
## implicit 0.000 0.000 0.000
## explicit 0.000 0.000 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .EIr 0.063 0.011 5.922 0.000 0.063 0.657
## .OPr 0.017 0.002 7.407 0.000 0.017 0.725
## .TWGJT 0.006 0.002 2.838 0.005 0.006 0.325
## .TAGJT 0.014 0.003 5.644 0.000 0.014 0.669
## .WMTt 0.065 0.008 8.008 0.000 0.065 0.994
## .SPRt_s 0.086 0.010 8.767 0.000 0.086 0.951
## .UWGJT_ug 0.025 0.007 3.355 0.001 0.025 0.434
## .UAGJT_ug 0.046 0.007 6.657 0.000 0.046 0.930
## .MKT 0.018 0.003 5.083 0.000 0.018 0.596
## implicit 0.033 0.010 3.247 0.001 1.000 1.000
## explicit 0.032 0.010 3.365 0.001 1.000 1.000Sometimes it is useful to extract factor scores for individuals associated with latent variables - these are trait/ability estimates that account for error, unlike simple composites of raw indicator scores. We can use predict() or lavPredict() to extract these:
lavPredict(fitTwo)## implct explct
## [1,] -0.026 -0.074
## [2,] -0.071 -0.037
## [3,] -0.272 -0.326
## [4,] -0.138 -0.150
## [5,] 0.198 0.224
## [6,] -0.161 -0.323
## [7,] 0.081 0.107
## [8,] -0.147 -0.129
## [9,] -0.026 -0.072
## [10,] 0.134 0.189
## [11,] 0.036 0.156
## [12,] -0.044 -0.011
## [13,] 0.137 0.096
## [14,] -0.194 -0.221
## [15,] 0.244 0.176
## [16,] 0.057 0.166
## [17,] -0.079 -0.050
## [18,] -0.123 -0.116
## [19,] 0.023 0.077
## [20,] -0.052 -0.044
## [21,] 0.134 0.159
## [22,] -0.033 0.070
## [23,] 0.006 -0.001
## [24,] 0.039 -0.008
## [25,] -0.001 0.061
## [26,] -0.146 -0.199
## [27,] 0.044 0.139
## [28,] -0.185 -0.124
## [29,] -0.113 -0.001
## [30,] -0.151 0.039
## [31,] 0.206 0.229
## [32,] -0.256 -0.265
## [33,] -0.255 -0.140
## [34,] 0.094 0.135
## [35,] -0.270 -0.037
## [36,] -0.124 -0.145
## [37,] 0.021 0.016
## [38,] -0.296 -0.050
## [39,] -0.166 -0.059
## [40,] 0.080 0.156
## [41,] 0.187 0.101
## [42,] -0.100 -0.184
## [43,] -0.022 0.012
## [44,] -0.125 -0.124
## [45,] -0.171 -0.064
## [46,] -0.071 0.046
## [47,] -0.040 -0.039
## [48,] 0.011 -0.006
## [49,] -0.355 -0.208
## [50,] -0.268 -0.228
## [51,] 0.268 0.146
## [52,] -0.040 -0.007
## [53,] -0.066 -0.070
## [54,] -0.071 0.011
## [55,] -0.151 -0.109
## [56,] 0.045 0.053
## [57,] -0.111 -0.187
## [58,] -0.115 -0.088
## [59,] -0.080 -0.090
## [60,] 0.056 0.072
## [61,] -0.041 -0.060
## [62,] -0.114 -0.077
## [63,] -0.022 0.087
## [64,] -0.174 -0.275
## [65,] -0.116 -0.122
## [66,] -0.265 -0.229
## [67,] -0.205 -0.253
## [68,] -0.087 -0.047
## [69,] -0.187 -0.177
## [70,] 0.022 0.031
## [71,] 0.061 0.045
## [72,] 0.001 0.067
## [73,] 0.209 0.174
## [74,] 0.059 0.091
## [75,] -0.001 -0.017
## [76,] -0.099 0.015
## [77,] -0.104 -0.077
## [78,] -0.092 0.041
## [79,] 0.194 0.208
## [80,] 0.011 -0.127
## [81,] 0.298 0.131
## [82,] -0.136 -0.240
## [83,] 0.319 0.222
## [84,] -0.151 -0.173
## [85,] 0.146 0.080
## [86,] 0.247 0.203
## [87,] -0.016 0.094
## [88,] 0.203 0.190
## [89,] 0.278 0.226
## [90,] -0.007 0.034
## [91,] 0.294 0.280
## [92,] 0.013 -0.062
## [93,] -0.027 -0.026
## [94,] -0.167 -0.005
## [95,] 0.096 0.047
## [96,] -0.196 -0.323
## [97,] -0.032 0.005
## [98,] 0.323 0.314
## [99,] 0.112 0.082
## [100,] -0.019 -0.135
## [101,] 0.110 0.058
## [102,] 0.047 0.025
## [103,] -0.094 0.079
## [104,] 0.269 0.218
## [105,] 0.078 -0.150
## [106,] 0.239 0.263
## [107,] -0.013 0.109
## [108,] -0.190 -0.285
## [109,] 0.210 0.266
## [110,] 0.080 0.083
## [111,] 0.291 0.206
## [112,] 0.189 0.205
## [113,] 0.136 0.064
## [114,] 0.215 0.112
## [115,] 0.012 -0.006
## [116,] 0.190 0.127
## [117,] 0.164 -0.008
## [118,] -0.353 -0.270
## [119,] 0.281 0.282
## [120,] -0.115 -0.096
## [121,] -0.025 -0.047
## [122,] 0.074 0.075
## [123,] 0.200 0.084
## [124,] 0.157 0.052
## [125,] -0.145 -0.124
## [126,] 0.188 -0.041
## [127,] -0.040 0.103
## [128,] 0.055 -0.239
## [129,] -0.098 -0.005
## [130,] 0.120 0.049
## [131,] -0.075 -0.217
## [132,] -0.046 -0.223
## [133,] -0.257 -0.283
## [134,] 0.176 0.002
## [135,] -0.082 -0.106
## [136,] 0.231 0.122
## [137,] -0.088 -0.034
## [138,] 0.134 0.127
## [139,] 0.085 -0.074
## [140,] 0.009 0.025
## [141,] -0.178 -0.143
## [142,] -0.125 -0.063
## [143,] 0.064 0.088
## [144,] 0.001 0.111
## [145,] 0.432 0.352
## [146,] 0.129 0.186
## [147,] 0.043 -0.065
## [148,] -0.130 0.003
## [149,] -0.010 -0.010
## [150,] -0.111 -0.115
## [151,] -0.028 -0.031
## [152,] 0.069 0.089
## [153,] 0.044 0.045
## [154,] 0.057 0.053
## [155,] -0.087 0.006
## [156,] -0.095 0.075To join these to the main dataset:
d <- bind_cols(d, as_tibble(lavPredict(fitTwo)))Coefficient omega is a factor-analytic analogue to Cronbach’s alpha. Its formula involves a comparison of the amount of variance in the indicator variables explained by the latent variable to the total amount of variance among the indicator variables.
Let’s look at the one factor model first, using the reliability() function from semTools:
reliability(fitOne)## language
## alpha 0.6443558
## omega 0.6544694
## omega2 0.6544694
## omega3 0.6421481
## avevar 0.2051856You’ll notice that the Cronbach’s alpha is nearly identical to the omega values provided. This is generally to be expected in a one factor model.
And now the 2-factor model:
reliability(fitTwo)## implicit explicit
## alpha 0.5391454 0.5014376
## omega 0.5409959 0.5780287
## omega2 0.5409959 0.5780287
## omega3 0.5170551 0.5914536
## avevar 0.2028521 0.3506341These numbers don’t necessairly look great, but there are some interesting things going on here - for one, the omega values indicate that the Explicit factor is actually a bit more reliable than a basic alpha calculation would suggest. Second, considering that the number of indicators per factor is substantially lower in the 2 factor model (6 and 3, instead of 9), the drop in reliability seems modest.
resid(fitOne, type="standardized")## $type
## [1] "standardized"
##
## $cov
## EIr OPr UWGJT_ MKT TWGJT SPRt_s WMTt TAGJT UAGJT_
## EIr 0.174
## OPr 1.278 0.311
## UWGJT_ug -1.239 -0.641 0.073
## MKT 0.521 -1.328 2.339 0.122
## TWGJT -0.375 0.282 -0.122 -1.761 -0.573
## SPRt_s 0.226 -0.253 -0.046 0.537 -1.402 -0.046
## WMTt 0.016 -0.278 -1.872 -0.852 -0.470 1.888 0.055
## TAGJT -0.464 -0.595 -1.629 -1.230 1.443 0.186 0.155 0.000
## UAGJT_ug -0.431 0.983 0.076 -0.218 -0.907 0.919 1.119 0.229 0.041
##
## $mean
## EIr OPr UWGJT_ug MKT TWGJT SPRt_s WMTt TAGJT
## -0.602 -0.194 0.927 -1.079 1.177 1.034 1.457 1.222
## UAGJT_ug
## -1.097The standardized residuals indicate that the model accounts for most of the relationships among variables - except perhaps the relationship between the MKT and untimed written GJT (UWGJT).
modindices(fitOne)## lhs op rhs mi epc sepc.lv sepc.all sepc.nox
## 30 EIr ~~ OPr 2.053 0.005 0.005 0.164 0.164
## 31 EIr ~~ UWGJT_ug 0.969 -0.006 -0.006 -0.119 -0.119
## 32 EIr ~~ MKT 0.410 0.003 0.003 0.073 0.073
## 33 EIr ~~ TWGJT 0.115 -0.001 -0.001 -0.057 -0.057
## 34 EIr ~~ SPRt_s 0.165 0.003 0.003 0.046 0.046
## 35 EIr ~~ WMTt 0.104 0.002 0.002 0.038 0.038
## 36 EIr ~~ TAGJT 0.224 -0.002 -0.002 -0.057 -0.057
## 37 EIr ~~ UAGJT_ug 0.207 -0.003 -0.003 -0.049 -0.049
## 38 OPr ~~ UWGJT_ug 0.459 -0.002 -0.002 -0.075 -0.075
## 39 OPr ~~ MKT 1.163 -0.002 -0.002 -0.113 -0.113
## 40 OPr ~~ TWGJT 0.061 0.000 0.000 0.037 0.037
## 41 OPr ~~ SPRt_s 0.055 -0.001 -0.001 -0.024 -0.024
## 42 OPr ~~ WMTt 0.088 -0.001 -0.001 -0.032 -0.032
## 43 OPr ~~ TAGJT 0.350 -0.001 -0.001 -0.064 -0.064
## 44 OPr ~~ UAGJT_ug 1.070 0.003 0.003 0.102 0.102
## 45 UWGJT_ug ~~ MKT 9.625 0.010 0.010 0.335 0.335
## 46 UWGJT_ug ~~ TWGJT 0.011 0.000 0.000 0.017 0.017
## 47 UWGJT_ug ~~ SPRt_s 0.002 0.000 0.000 0.004 0.004
## 48 UWGJT_ug ~~ WMTt 2.122 -0.008 -0.008 -0.160 -0.160
## 49 UWGJT_ug ~~ TAGJT 1.441 -0.003 -0.003 -0.134 -0.134
## 50 UWGJT_ug ~~ UAGJT_ug 0.103 0.001 0.001 0.033 0.033
## 51 MKT ~~ TWGJT 1.632 -0.002 -0.002 -0.186 -0.186
## 52 MKT ~~ SPRt_s 0.325 0.002 0.002 0.056 0.056
## 53 MKT ~~ WMTt 0.365 -0.002 -0.002 -0.062 -0.062
## 54 MKT ~~ TAGJT 0.868 -0.002 -0.002 -0.097 -0.097
## 55 MKT ~~ UAGJT_ug 0.038 -0.001 -0.001 -0.019 -0.019
## 56 TWGJT ~~ SPRt_s 1.336 -0.004 -0.004 -0.155 -0.155
## 57 TWGJT ~~ WMTt 0.004 0.000 0.000 0.009 0.009
## 58 TWGJT ~~ TAGJT 4.417 0.003 0.003 0.331 0.331
## 59 TWGJT ~~ UAGJT_ug 0.841 -0.002 -0.002 -0.118 -0.118
## 60 SPRt_s ~~ WMTt 3.947 0.015 0.015 0.194 0.194
## 61 SPRt_s ~~ TAGJT 0.054 0.001 0.001 0.023 0.023
## 62 SPRt_s ~~ UAGJT_ug 0.857 0.005 0.005 0.084 0.084
## 63 WMTt ~~ TAGJT 0.174 0.001 0.001 0.042 0.042
## 64 WMTt ~~ UAGJT_ug 1.172 0.005 0.005 0.099 0.099
## 65 TAGJT ~~ UAGJT_ug 0.037 0.000 0.000 0.018 0.018For this one factor model, modification indices will only be able to suggest freeing correlated error parameters. It looks like correlating the timed GJTs (written and aural), the RT measures, and the untimed written GJT and the MKT might lead to better fit.
Technically, it is probably better to try these out one by one, starting out with the largest MI and informed by the standardized residuals, and see how it then affects localized strain indices. However, we’ll just do all of them for illustration purposes.
First, we’ll need to specify a new model:
One.err <- '
# latent variable definitions
language =~ EIr + OPr + UWGJT_ug + MKT + TWGJT + MKT + SPRt_s + WMTt + TAGJT + UAGJT_ug
# correlated errors
UWGJT_ug ~~ MKT
TWGJT ~~ TAGJT
SPRt_s ~~ WMTt
'Now fit and examine:
fitOne.err <- cfa(One.err, data = d, missing = "fiml", estimator = "MLR")
lavaan::summary(fitOne.err, estimates = T, standardized = T, fit.measures = T)## lavaan 0.6-7 ended normally after 99 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of free parameters 30
##
## Number of observations 156
## Number of missing patterns 26
##
## Model Test User Model:
## Standard Robust
## Test Statistic 12.746 12.808
## Degrees of freedom 24 24
## P-value (Chi-square) 0.970 0.969
## Scaling correction factor 0.995
## Yuan-Bentler correction (Mplus variant)
##
## Model Test Baseline Model:
##
## Test statistic 183.271 176.306
## Degrees of freedom 36 36
## P-value 0.000 0.000
## Scaling correction factor 1.040
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 1.000 1.000
## Tucker-Lewis Index (TLI) 1.115 1.120
##
## Robust Comparative Fit Index (CFI) 1.000
## Robust Tucker-Lewis Index (TLI) 1.115
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) 283.461 283.461
## Scaling correction factor 1.029
## for the MLR correction
## Loglikelihood unrestricted model (H1) 289.834 289.834
## Scaling correction factor 1.014
## for the MLR correction
##
## Akaike (AIC) -506.923 -506.923
## Bayesian (BIC) -415.427 -415.427
## Sample-size adjusted Bayesian (BIC) -510.387 -510.387
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.000 0.000
## 90 Percent confidence interval - lower 0.000 0.000
## 90 Percent confidence interval - upper 0.000 0.000
## P-value RMSEA <= 0.05 0.998 0.998
##
## Robust RMSEA 0.000
## 90 Percent confidence interval - lower 0.000
## 90 Percent confidence interval - upper 0.000
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.039 0.039
##
## Parameter Estimates:
##
## Standard errors Sandwich
## Information bread Observed
## Observed information based on Hessian
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## language =~
## EIr 1.000 0.193 0.623
## OPr 0.437 0.092 4.733 0.000 0.084 0.559
## UWGJT_ug 0.638 0.166 3.843 0.000 0.123 0.517
## MKT 0.391 0.092 4.229 0.000 0.076 0.440
## TWGJT 0.540 0.093 5.841 0.000 0.104 0.755
## SPRt_s 0.364 0.172 2.112 0.035 0.070 0.234
## WMTt 0.082 0.130 0.629 0.529 0.016 0.061
## TAGJT 0.378 0.088 4.301 0.000 0.073 0.497
## UAGJT_ug 0.340 0.128 2.660 0.008 0.066 0.296
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .UWGJT_ug ~~
## .MKT 0.011 0.005 2.413 0.016 0.011 0.351
## .TWGJT ~~
## .TAGJT 0.003 0.002 1.481 0.139 0.003 0.236
## .SPRt_s ~~
## .WMTt 0.014 0.007 2.043 0.041 0.014 0.182
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .EIr 1.530 0.028 54.011 0.000 1.530 4.928
## .OPr 0.818 0.013 62.306 0.000 0.818 5.419
## .UWGJT_ug 0.617 0.021 29.025 0.000 0.617 2.584
## .MKT 0.639 0.015 43.309 0.000 0.639 3.725
## .TWGJT 0.607 0.012 50.269 0.000 0.607 4.383
## .SPRt_s 0.053 0.026 2.012 0.044 0.053 0.175
## .WMTt 0.048 0.023 2.070 0.038 0.048 0.186
## .TAGJT 0.568 0.012 47.811 0.000 0.568 3.867
## .UAGJT_ug 0.446 0.018 24.338 0.000 0.446 2.008
## language 0.000 0.000 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .EIr 0.059 0.010 5.732 0.000 0.059 0.612
## .OPr 0.016 0.002 6.847 0.000 0.016 0.687
## .UWGJT_ug 0.042 0.006 6.992 0.000 0.042 0.733
## .MKT 0.024 0.003 7.265 0.000 0.024 0.806
## .TWGJT 0.008 0.002 3.467 0.001 0.008 0.431
## .SPRt_s 0.086 0.010 8.760 0.000 0.086 0.945
## .WMTt 0.065 0.008 8.018 0.000 0.065 0.996
## .TAGJT 0.016 0.003 5.475 0.000 0.016 0.753
## .UAGJT_ug 0.045 0.006 7.031 0.000 0.045 0.913
## language 0.037 0.010 3.563 0.000 1.000 1.000Compare to the original model (the original is nested in the new model):
anova(fitOne, fitOne.err)## Scaled Chi-Squared Difference Test (method = "satorra.bentler.2001")
##
## lavaan NOTE:
## The "Chisq" column contains standard test statistics, not the
## robust test that should be reported per model. A robust difference
## test is a function of two standard (not robust) statistics.
##
## Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
## fitOne.err 24 -506.92 -415.43 12.746
## fitOne 27 -495.21 -412.86 30.458 14.836 3 0.001962 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1This definitely appears to be an improvement.
Let’s check out the reliability:
reliability(fitOne.err)## language
## alpha 0.6443558
## omega 0.6097852
## omega2 0.6097852
## omega3 0.6133908
## avevar 0.2016788Omega estimates drop because now some additional variance in indicators is being accounted for by the correlated errors.
Time to revisit localized strain:
resid(fitOne.err, type="standardized")## $type
## [1] "standardized"
##
## $cov
## EIr OPr UWGJT_ MKT TWGJT SPRt_s WMTt TAGJT UAGJT_
## EIr -0.022
## OPr 0.924 0.332
## UWGJT_ug -1.084 -0.642 0.028
## MKT 0.731 -1.192 0.428 0.168
## TWGJT -0.596 0.063 1.149 -0.126 -0.696
## SPRt_s 0.162 -0.403 0.077 0.686 -1.247 0.012
## WMTt 0.076 -0.236 -1.760 -0.733 -0.272 -0.337 0.115
## TAGJT -0.012 -0.285 -0.261 0.122 -0.504 0.391 0.337 0.083
## UAGJT_ug -0.716 0.799 0.159 -0.078 -0.914 0.905 1.137 0.532 0.047
##
## $mean
## EIr OPr UWGJT_ug MKT TWGJT SPRt_s WMTt TAGJT
## -0.687 -0.216 1.213 0.123 0.484 0.753 1.306 -0.109
## UAGJT_ug
## -0.966This all looks pretty good.
MIs:
modindices(fitOne.err)## lhs op rhs mi epc sepc.lv sepc.all sepc.nox
## 33 EIr ~~ OPr 0.859 0.004 0.004 0.126 0.126
## 34 EIr ~~ UWGJT_ug 1.910 -0.008 -0.008 -0.162 -0.162
## 35 EIr ~~ MKT 1.578 0.005 0.005 0.138 0.138
## 36 EIr ~~ TWGJT 0.126 -0.001 -0.001 -0.064 -0.064
## 37 EIr ~~ SPRt_s 0.058 0.002 0.002 0.028 0.028
## 38 EIr ~~ WMTt 0.262 0.004 0.004 0.062 0.062
## 39 EIr ~~ TAGJT 0.001 0.000 0.000 0.004 0.004
## 40 EIr ~~ UAGJT_ug 0.431 -0.004 -0.004 -0.075 -0.075
## 41 OPr ~~ UWGJT_ug 0.150 -0.001 -0.001 -0.041 -0.041
## 42 OPr ~~ MKT 0.909 -0.002 -0.002 -0.095 -0.095
## 43 OPr ~~ TWGJT 0.026 0.000 0.000 0.025 0.025
## 44 OPr ~~ SPRt_s 0.159 -0.002 -0.002 -0.043 -0.043
## 45 OPr ~~ WMTt 0.012 0.000 0.000 -0.012 -0.012
## 46 OPr ~~ TAGJT 0.081 0.000 0.000 -0.031 -0.031
## 47 OPr ~~ UAGJT_ug 0.835 0.003 0.003 0.094 0.094
## 48 UWGJT_ug ~~ TWGJT 2.574 0.004 0.004 0.202 0.202
## 49 UWGJT_ug ~~ SPRt_s 0.015 0.001 0.001 0.012 0.012
## 50 UWGJT_ug ~~ WMTt 1.615 -0.007 -0.007 -0.126 -0.126
## 51 UWGJT_ug ~~ TAGJT 0.330 -0.001 -0.001 -0.054 -0.054
## 52 UWGJT_ug ~~ UAGJT_ug 0.115 0.001 0.001 0.032 0.032
## 53 MKT ~~ TWGJT 0.337 -0.001 -0.001 -0.069 -0.069
## 54 MKT ~~ SPRt_s 0.618 0.003 0.003 0.070 0.070
## 55 MKT ~~ WMTt 0.081 -0.001 -0.001 -0.027 -0.027
## 56 MKT ~~ TAGJT 0.079 0.000 0.000 0.025 0.025
## 57 MKT ~~ UAGJT_ug 0.051 -0.001 -0.001 -0.020 -0.020
## 58 TWGJT ~~ SPRt_s 1.244 -0.004 -0.004 -0.138 -0.138
## 59 TWGJT ~~ WMTt 0.026 0.000 0.000 0.020 0.020
## 60 TWGJT ~~ UAGJT_ug 0.824 -0.002 -0.002 -0.109 -0.109
## 61 SPRt_s ~~ TAGJT 0.295 0.002 0.002 0.049 0.049
## 62 SPRt_s ~~ UAGJT_ug 0.634 0.004 0.004 0.071 0.071
## 63 WMTt ~~ TAGJT 0.119 0.001 0.001 0.031 0.031
## 64 WMTt ~~ UAGJT_ug 0.700 0.004 0.004 0.076 0.076
## 65 TAGJT ~~ UAGJT_ug 0.307 0.001 0.001 0.048 0.048Again, looking good - nothing over 3.84 or 4.0. And at least two of the method effects make sense from a test-task specification perspective: RTs tend to be correlated within individuals (faster people are faster across tasks) and timed GJTs both have time pressure.
e <- read_csv("EIT_learner_single_scores.csv")eit <- 'OP =~ I01+I02+I03+I04+I05+I06+I07+I08+I09+I10+
I11+I12+I13+I14+I15+I16+I17+I18+I19+I20+I21+
I22+I23+I24+I25+I26+I27+I28+I29+I30'eitFit <- cfa(eit, data = e, ordered = TRUE)
lavaan::summary(eitFit, estimates = T, standardized = T, fit.measures = T)## lavaan 0.6-7 ended normally after 69 iterations
##
## Estimator DWLS
## Optimization method NLMINB
## Number of free parameters 150
##
## Used Total
## Number of observations 317 318
##
## Model Test User Model:
## Standard Robust
## Test Statistic 437.022 754.276
## Degrees of freedom 405 405
## P-value (Chi-square) 0.131 0.000
## Scaling correction factor 0.757
## Shift parameter 177.158
## simple second-order correction
##
## Model Test Baseline Model:
##
## Test statistic 156352.395 31511.507
## Degrees of freedom 435 435
## P-value 0.000 0.000
## Scaling correction factor 5.017
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 1.000 0.989
## Tucker-Lewis Index (TLI) 1.000 0.988
##
## Robust Comparative Fit Index (CFI) NA
## Robust Tucker-Lewis Index (TLI) NA
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.016 0.052
## 90 Percent confidence interval - lower 0.000 0.046
## 90 Percent confidence interval - upper 0.026 0.058
## P-value RMSEA <= 0.05 1.000 0.257
##
## Robust RMSEA NA
## 90 Percent confidence interval - lower NA
## 90 Percent confidence interval - upper NA
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.041 0.041
##
## Parameter Estimates:
##
## Standard errors Robust.sem
## Information Expected
## Information saturated (h1) model Unstructured
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## OP =~
## I01 1.000 0.534 0.534
## I02 1.204 0.126 9.548 0.000 0.643 0.643
## I03 1.354 0.137 9.896 0.000 0.723 0.723
## I04 1.314 0.138 9.516 0.000 0.701 0.701
## I05 1.492 0.138 10.806 0.000 0.797 0.797
## I06 1.535 0.144 10.691 0.000 0.820 0.820
## I07 1.448 0.141 10.243 0.000 0.773 0.773
## I08 1.569 0.149 10.560 0.000 0.838 0.838
## I09 1.566 0.146 10.700 0.000 0.836 0.836
## I10 1.501 0.144 10.442 0.000 0.801 0.801
## I11 1.473 0.139 10.623 0.000 0.787 0.787
## I12 1.522 0.142 10.698 0.000 0.813 0.813
## I13 1.473 0.146 10.075 0.000 0.786 0.786
## I14 1.510 0.144 10.503 0.000 0.806 0.806
## I15 1.522 0.147 10.384 0.000 0.812 0.812
## I16 1.631 0.155 10.557 0.000 0.871 0.871
## I17 1.601 0.153 10.487 0.000 0.855 0.855
## I18 1.654 0.154 10.719 0.000 0.883 0.883
## I19 1.466 0.145 10.079 0.000 0.783 0.783
## I20 1.644 0.151 10.897 0.000 0.878 0.878
## I21 1.673 0.155 10.769 0.000 0.893 0.893
## I22 1.555 0.146 10.625 0.000 0.830 0.830
## I23 1.713 0.158 10.808 0.000 0.915 0.915
## I24 1.666 0.153 10.895 0.000 0.890 0.890
## I25 1.634 0.152 10.769 0.000 0.872 0.872
## I26 1.631 0.148 11.020 0.000 0.871 0.871
## I27 1.577 0.146 10.775 0.000 0.842 0.842
## I28 1.592 0.147 10.801 0.000 0.850 0.850
## I29 1.551 0.147 10.534 0.000 0.828 0.828
## I30 1.504 0.145 10.379 0.000 0.803 0.803
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .I01 0.000 0.000 0.000
## .I02 0.000 0.000 0.000
## .I03 0.000 0.000 0.000
## .I04 0.000 0.000 0.000
## .I05 0.000 0.000 0.000
## .I06 0.000 0.000 0.000
## .I07 0.000 0.000 0.000
## .I08 0.000 0.000 0.000
## .I09 0.000 0.000 0.000
## .I10 0.000 0.000 0.000
## .I11 0.000 0.000 0.000
## .I12 0.000 0.000 0.000
## .I13 0.000 0.000 0.000
## .I14 0.000 0.000 0.000
## .I15 0.000 0.000 0.000
## .I16 0.000 0.000 0.000
## .I17 0.000 0.000 0.000
## .I18 0.000 0.000 0.000
## .I19 0.000 0.000 0.000
## .I20 0.000 0.000 0.000
## .I21 0.000 0.000 0.000
## .I22 0.000 0.000 0.000
## .I23 0.000 0.000 0.000
## .I24 0.000 0.000 0.000
## .I25 0.000 0.000 0.000
## .I26 0.000 0.000 0.000
## .I27 0.000 0.000 0.000
## .I28 0.000 0.000 0.000
## .I29 0.000 0.000 0.000
## .I30 0.000 0.000 0.000
## OP 0.000 0.000 0.000
##
## Thresholds:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## I01|t1 -1.529 0.110 -13.853 0.000 -1.529 -1.529
## I01|t2 -1.480 0.107 -13.811 0.000 -1.480 -1.480
## I01|t3 -0.647 0.076 -8.502 0.000 -0.647 -0.647
## I01|t4 -0.252 0.071 -3.529 0.000 -0.252 -0.252
## I02|t1 -1.371 0.101 -13.608 0.000 -1.371 -1.371
## I02|t2 -1.030 0.086 -11.989 0.000 -1.030 -1.030
## I02|t3 -0.553 0.075 -7.413 0.000 -0.553 -0.553
## I02|t4 -0.107 0.071 -1.514 0.130 -0.107 -0.107
## I03|t1 -1.504 0.109 -13.835 0.000 -1.504 -1.504
## I03|t2 -1.176 0.091 -12.861 0.000 -1.176 -1.176
## I03|t3 -0.155 0.071 -2.186 0.029 -0.155 -0.155
## I03|t4 0.326 0.072 4.534 0.000 0.326 0.326
## I04|t1 -1.776 0.130 -13.631 0.000 -1.776 -1.776
## I04|t2 -1.504 0.109 -13.835 0.000 -1.504 -1.504
## I04|t3 -0.991 0.085 -11.706 0.000 -0.991 -0.991
## I04|t4 -0.376 0.072 -5.202 0.000 -0.376 -0.376
## I05|t1 -1.457 0.106 -13.780 0.000 -1.457 -1.457
## I05|t2 -1.058 0.087 -12.173 0.000 -1.058 -1.058
## I05|t3 -0.428 0.073 -5.869 0.000 -0.428 -0.428
## I05|t4 -0.012 0.071 -0.168 0.866 -0.012 -0.012
## I06|t1 -0.813 0.080 -10.203 0.000 -0.813 -0.813
## I06|t2 -0.507 0.074 -6.864 0.000 -0.507 -0.507
## I06|t3 -0.091 0.071 -1.290 0.197 -0.091 -0.091
## I06|t4 0.480 0.074 6.533 0.000 0.480 0.480
## I07|t1 -1.145 0.090 -12.697 0.000 -1.145 -1.145
## I07|t2 -0.824 0.080 -10.306 0.000 -0.824 -0.824
## I07|t3 -0.293 0.072 -4.088 0.000 -0.293 -0.293
## I07|t4 0.235 0.071 3.306 0.001 0.235 0.235
## I08|t1 -1.259 0.095 -13.238 0.000 -1.259 -1.259
## I08|t2 -0.697 0.077 -9.041 0.000 -0.697 -0.697
## I08|t3 -0.260 0.071 -3.641 0.000 -0.260 -0.260
## I08|t4 0.599 0.075 7.960 0.000 0.599 0.599
## I09|t1 -1.044 0.086 -12.081 0.000 -1.044 -1.044
## I09|t2 -0.657 0.076 -8.611 0.000 -0.657 -0.657
## I09|t3 -0.260 0.071 -3.641 0.000 -0.260 -0.260
## I09|t4 0.428 0.073 5.869 0.000 0.428 0.428
## I10|t1 -1.192 0.092 -12.940 0.000 -1.192 -1.192
## I10|t2 -0.759 0.078 -9.679 0.000 -0.759 -0.759
## I10|t3 -0.351 0.072 -4.868 0.000 -0.351 -0.351
## I10|t4 0.260 0.071 3.641 0.000 0.260 0.260
## I11|t1 -0.525 0.074 -7.084 0.000 -0.525 -0.525
## I11|t2 0.099 0.071 1.402 0.161 0.099 0.099
## I11|t3 0.717 0.078 9.254 0.000 0.717 0.717
## I11|t4 1.208 0.093 13.018 0.000 1.208 1.208
## I12|t1 -1.192 0.092 -12.940 0.000 -1.192 -1.192
## I12|t2 -0.697 0.077 -9.041 0.000 -0.697 -0.697
## I12|t3 0.004 0.071 0.056 0.955 0.004 0.004
## I12|t4 0.571 0.075 7.632 0.000 0.571 0.571
## I13|t1 -0.059 0.071 -0.841 0.400 -0.059 -0.059
## I13|t2 0.507 0.074 6.864 0.000 0.507 0.507
## I13|t3 1.176 0.091 12.861 0.000 1.176 1.176
## I13|t4 1.582 0.114 13.865 0.000 1.582 1.582
## I14|t1 -0.738 0.078 -9.467 0.000 -0.738 -0.738
## I14|t2 -0.187 0.071 -2.634 0.008 -0.187 -0.187
## I14|t3 0.334 0.072 4.646 0.000 0.334 0.334
## I14|t4 1.176 0.091 12.861 0.000 1.176 1.176
## I15|t1 -0.195 0.071 -2.746 0.006 -0.195 -0.195
## I15|t2 0.402 0.073 5.536 0.000 0.402 0.402
## I15|t3 0.916 0.082 11.120 0.000 0.916 0.916
## I15|t4 1.371 0.101 13.608 0.000 1.371 1.371
## I16|t1 -0.075 0.071 -1.065 0.287 -0.075 -0.075
## I16|t2 0.376 0.072 5.202 0.000 0.376 0.376
## I16|t3 0.813 0.080 10.203 0.000 0.813 0.813
## I16|t4 1.192 0.092 12.940 0.000 1.192 1.192
## I17|t1 -0.044 0.071 -0.617 0.537 -0.044 -0.044
## I17|t2 0.437 0.073 5.980 0.000 0.437 0.437
## I17|t3 0.802 0.079 10.098 0.000 0.802 0.802
## I17|t4 1.582 0.114 13.865 0.000 1.582 1.582
## I18|t1 -0.437 0.073 -5.980 0.000 -0.437 -0.437
## I18|t2 -0.012 0.071 -0.168 0.866 -0.012 -0.012
## I18|t3 0.480 0.074 6.533 0.000 0.480 0.480
## I18|t4 1.259 0.095 13.238 0.000 1.259 1.259
## I19|t1 -0.260 0.071 -3.641 0.000 -0.260 -0.260
## I19|t2 0.562 0.075 7.523 0.000 0.562 0.562
## I19|t3 1.086 0.088 12.352 0.000 1.086 1.086
## I19|t4 1.457 0.106 13.780 0.000 1.457 1.457
## I20|t1 -0.628 0.076 -8.286 0.000 -0.628 -0.628
## I20|t2 -0.028 0.071 -0.393 0.695 -0.028 -0.028
## I20|t3 0.489 0.074 6.643 0.000 0.489 0.489
## I20|t4 0.991 0.085 11.706 0.000 0.991 0.991
## I21|t1 -0.824 0.080 -10.306 0.000 -0.824 -0.824
## I21|t2 -0.376 0.072 -5.202 0.000 -0.376 -0.376
## I21|t3 0.099 0.071 1.402 0.161 0.099 0.099
## I21|t4 0.791 0.079 9.994 0.000 0.791 0.791
## I22|t1 -0.147 0.071 -2.074 0.038 -0.147 -0.147
## I22|t2 0.553 0.075 7.413 0.000 0.553 0.553
## I22|t3 0.978 0.084 11.610 0.000 0.978 0.978
## I22|t4 1.480 0.107 13.811 0.000 1.480 1.480
## I23|t1 -0.657 0.076 -8.611 0.000 -0.657 -0.657
## I23|t2 -0.171 0.071 -2.410 0.016 -0.171 -0.171
## I23|t3 0.368 0.072 5.091 0.000 0.368 0.368
## I23|t4 0.813 0.080 10.203 0.000 0.813 0.813
## I24|t1 -0.553 0.075 -7.413 0.000 -0.553 -0.553
## I24|t2 -0.203 0.071 -2.858 0.004 -0.203 -0.203
## I24|t3 0.326 0.072 4.534 0.000 0.326 0.326
## I24|t4 1.224 0.094 13.093 0.000 1.224 1.224
## I25|t1 -0.445 0.073 -6.090 0.000 -0.445 -0.445
## I25|t2 0.075 0.071 1.065 0.287 0.075 0.075
## I25|t3 0.562 0.075 7.523 0.000 0.562 0.562
## I25|t4 1.100 0.088 12.441 0.000 1.100 1.100
## I26|t1 -0.916 0.082 -11.120 0.000 -0.916 -0.916
## I26|t2 -0.235 0.071 -3.306 0.001 -0.235 -0.235
## I26|t3 0.211 0.071 2.970 0.003 0.211 0.211
## I26|t4 1.058 0.087 12.173 0.000 1.058 1.058
## I27|t1 -1.004 0.085 -11.801 0.000 -1.004 -1.004
## I27|t2 -0.203 0.071 -2.858 0.004 -0.203 -0.203
## I27|t3 0.301 0.072 4.200 0.000 0.301 0.301
## I27|t4 0.869 0.081 10.717 0.000 0.869 0.869
## I28|t1 -0.571 0.075 -7.632 0.000 -0.571 -0.571
## I28|t2 -0.123 0.071 -1.738 0.082 -0.123 -0.123
## I28|t3 0.544 0.074 7.304 0.000 0.544 0.544
## I28|t4 1.671 0.121 13.811 0.000 1.671 1.671
## I29|t1 -0.953 0.083 -11.416 0.000 -0.953 -0.953
## I29|t2 -0.351 0.072 -4.868 0.000 -0.351 -0.351
## I29|t3 0.235 0.071 3.306 0.001 0.235 0.235
## I29|t4 1.332 0.099 13.497 0.000 1.332 1.332
## I30|t1 -1.130 0.090 -12.613 0.000 -1.130 -1.130
## I30|t2 -0.813 0.080 -10.203 0.000 -0.813 -0.813
## I30|t3 -0.227 0.071 -3.194 0.001 -0.227 -0.227
## I30|t4 0.869 0.081 10.717 0.000 0.869 0.869
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .I01 0.715 0.715 0.715
## .I02 0.587 0.587 0.587
## .I03 0.477 0.477 0.477
## .I04 0.508 0.508 0.508
## .I05 0.365 0.365 0.365
## .I06 0.328 0.328 0.328
## .I07 0.402 0.402 0.402
## .I08 0.298 0.298 0.298
## .I09 0.301 0.301 0.301
## .I10 0.358 0.358 0.358
## .I11 0.381 0.381 0.381
## .I12 0.339 0.339 0.339
## .I13 0.382 0.382 0.382
## .I14 0.350 0.350 0.350
## .I15 0.340 0.340 0.340
## .I16 0.242 0.242 0.242
## .I17 0.270 0.270 0.270
## .I18 0.220 0.220 0.220
## .I19 0.387 0.387 0.387
## .I20 0.230 0.230 0.230
## .I21 0.203 0.203 0.203
## .I22 0.311 0.311 0.311
## .I23 0.164 0.164 0.164
## .I24 0.209 0.209 0.209
## .I25 0.239 0.239 0.239
## .I26 0.242 0.242 0.242
## .I27 0.291 0.291 0.291
## .I28 0.277 0.277 0.277
## .I29 0.314 0.314 0.314
## .I30 0.356 0.356 0.356
## OP 0.285 0.053 5.423 0.000 1.000 1.000
##
## Scales y*:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## I01 1.000 1.000 1.000
## I02 1.000 1.000 1.000
## I03 1.000 1.000 1.000
## I04 1.000 1.000 1.000
## I05 1.000 1.000 1.000
## I06 1.000 1.000 1.000
## I07 1.000 1.000 1.000
## I08 1.000 1.000 1.000
## I09 1.000 1.000 1.000
## I10 1.000 1.000 1.000
## I11 1.000 1.000 1.000
## I12 1.000 1.000 1.000
## I13 1.000 1.000 1.000
## I14 1.000 1.000 1.000
## I15 1.000 1.000 1.000
## I16 1.000 1.000 1.000
## I17 1.000 1.000 1.000
## I18 1.000 1.000 1.000
## I19 1.000 1.000 1.000
## I20 1.000 1.000 1.000
## I21 1.000 1.000 1.000
## I22 1.000 1.000 1.000
## I23 1.000 1.000 1.000
## I24 1.000 1.000 1.000
## I25 1.000 1.000 1.000
## I26 1.000 1.000 1.000
## I27 1.000 1.000 1.000
## I28 1.000 1.000 1.000
## I29 1.000 1.000 1.000
## I30 1.000 1.000 1.000reliability(eitFit)## For constructs with categorical indicators, the alpha and the average variance extracted are calculated from polychoric (polyserial) correlations, not from Pearson correlations.## OP
## alpha 0.9824348
## omega 0.9775452
## omega2 0.9775452
## omega3 0.9842079
## avevar 0.6638463