Installation and activation of packages

#install.packages("lavaan")
#install.packages("MIIVsem")
#install.packages("tidyverse")

library(lavaan)
library(MIIVsem)
library(tidyverse)

Data import

data_study1 <- read_csv("./Study 1 data.csv")


Starting model

This is the initial model.

Abbeviations:

cfa1 <- '
 VIRUS_WORRY =~ aw01 + aw02 +   aw03
 WORRY_EMPL =~ ewj01 + ewj02 +   ewj03
 WORRY_REC = ~ ewe01 + ewe02 +   ewe03 
 ACC =~  acc01 +  acc02 + acc03
 REJ =~  rej01 +  rej02 + rej03
 
'
fit.cfa1 <- sem(cfa1, data = data_study1, estimator="MLR", missing="FIML")
summary(fit.cfa1, fit.measures=TRUE, standardized=TRUE)
## lavaan 0.6-7 ended normally after 84 iterations
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of free parameters                         55
##                                                       
##   Number of observations                           612
##   Number of missing patterns                         3
##                                                       
## Model Test User Model:
##                                                Standard      Robust
##   Test Statistic                                218.488     192.179
##   Degrees of freedom                                 80          80
##   P-value (Chi-square)                            0.000       0.000
##   Scaling correction factor                                   1.137
##        Yuan-Bentler correction (Mplus variant)                     
## 
## Model Test Baseline Model:
## 
##   Test statistic                              8195.180    5681.932
##   Degrees of freedom                               105         105
##   P-value                                        0.000       0.000
##   Scaling correction factor                                  1.442
## 
## User Model versus Baseline Model:
## 
##   Comparative Fit Index (CFI)                    0.983       0.980
##   Tucker-Lewis Index (TLI)                       0.978       0.974
##                                                                   
##   Robust Comparative Fit Index (CFI)                         0.984
##   Robust Tucker-Lewis Index (TLI)                            0.979
## 
## Loglikelihood and Information Criteria:
## 
##   Loglikelihood user model (H0)             -14638.911  -14638.911
##   Scaling correction factor                                  1.577
##       for the MLR correction                                      
##   Loglikelihood unrestricted model (H1)     -14529.667  -14529.667
##   Scaling correction factor                                  1.316
##       for the MLR correction                                      
##                                                                   
##   Akaike (AIC)                               29387.823   29387.823
##   Bayesian (BIC)                             29630.743   29630.743
##   Sample-size adjusted Bayesian (BIC)        29456.129   29456.129
## 
## Root Mean Square Error of Approximation:
## 
##   RMSEA                                          0.053       0.048
##   90 Percent confidence interval - lower         0.045       0.040
##   90 Percent confidence interval - upper         0.062       0.056
##   P-value RMSEA <= 0.05                          0.257       0.655
##                                                                   
##   Robust RMSEA                                               0.051
##   90 Percent confidence interval - lower                     0.042
##   90 Percent confidence interval - upper                     0.060
## 
## Standardized Root Mean Square Residual:
## 
##   SRMR                                           0.038       0.038
## 
## 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
##   VIRUS_WORRY =~                                                        
##     aw01              1.000                               1.401    0.787
##     aw02              1.232    0.058   21.107    0.000    1.726    0.913
##     aw03              1.182    0.059   20.006    0.000    1.656    0.882
##   WORRY_EMPL =~                                                         
##     ewj01             1.000                               1.761    0.914
##     ewj02             1.089    0.033   33.297    0.000    1.917    0.903
##     ewj03             1.111    0.030   36.717    0.000    1.956    0.957
##   WORRY_REC =~                                                          
##     ewe01             1.000                               1.338    0.878
##     ewe02             0.878    0.035   25.244    0.000    1.175    0.877
##     ewe03             0.939    0.034   27.561    0.000    1.257    0.876
##   ACC =~                                                                
##     acc01             1.000                               1.585    0.783
##     acc02             1.056    0.036   29.293    0.000    1.674    0.905
##     acc03             1.163    0.036   32.461    0.000    1.843    0.944
##   REJ =~                                                                
##     rej01             1.000                               1.885    0.912
##     rej02             0.983    0.030   33.267    0.000    1.853    0.898
##     rej03             0.977    0.026   38.265    0.000    1.841    0.931
## 
## Covariances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##   VIRUS_WORRY ~~                                                        
##     WORRY_EMPL        0.052    0.115    0.452    0.651    0.021    0.021
##     WORRY_REC         0.455    0.099    4.599    0.000    0.243    0.243
##     ACC               1.339    0.146    9.180    0.000    0.603    0.603
##     REJ              -1.270    0.164   -7.761    0.000   -0.481   -0.481
##   WORRY_EMPL ~~                                                         
##     WORRY_REC         0.537    0.099    5.409    0.000    0.228    0.228
##     ACC              -0.535    0.129   -4.139    0.000   -0.192   -0.192
##     REJ               1.015    0.160    6.325    0.000    0.306    0.306
##   WORRY_REC ~~                                                          
##     ACC               0.073    0.111    0.659    0.510    0.034    0.034
##     REJ               0.323    0.118    2.730    0.006    0.128    0.128
##   ACC ~~                                                                
##     REJ              -2.428    0.173  -14.028    0.000   -0.813   -0.813
## 
## Intercepts:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##    .aw01              4.960    0.072   68.923    0.000    4.960    2.785
##    .aw02              3.945    0.076   51.610    0.000    3.945    2.088
##    .aw03              3.629    0.076   47.775    0.000    3.629    1.933
##    .ewj01             2.408    0.078   30.925    0.000    2.408    1.250
##    .ewj02             2.881    0.086   33.581    0.000    2.881    1.357
##    .ewj03             2.706    0.083   32.748    0.000    2.706    1.324
##    .ewe01             5.093    0.062   82.667    0.000    5.093    3.342
##    .ewe02             5.523    0.054  102.065    0.000    5.523    4.126
##    .ewe03             5.436    0.058   93.783    0.000    5.436    3.791
##    .acc01             4.296    0.082   52.523    0.000    4.296    2.123
##    .acc02             5.208    0.075   69.635    0.000    5.208    2.815
##    .acc03             5.054    0.079   64.033    0.000    5.054    2.588
##    .rej01             2.961    0.084   35.434    0.000    2.961    1.432
##    .rej02             2.920    0.083   35.012    0.000    2.920    1.415
##    .rej03             2.974    0.080   37.200    0.000    2.974    1.504
##     VIRUS_WORRY       0.000                               0.000    0.000
##     WORRY_EMPL        0.000                               0.000    0.000
##     WORRY_REC         0.000                               0.000    0.000
##     ACC               0.000                               0.000    0.000
##     REJ               0.000                               0.000    0.000
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##    .aw01              1.208    0.128    9.431    0.000    1.208    0.381
##    .aw02              0.592    0.085    6.967    0.000    0.592    0.166
##    .aw03              0.786    0.095    8.290    0.000    0.786    0.223
##    .ewj01             0.612    0.097    6.311    0.000    0.612    0.165
##    .ewj02             0.828    0.148    5.608    0.000    0.828    0.184
##    .ewj03             0.354    0.083    4.260    0.000    0.354    0.085
##    .ewe01             0.532    0.082    6.513    0.000    0.532    0.229
##    .ewe02             0.412    0.053    7.765    0.000    0.412    0.230
##    .ewe03             0.477    0.074    6.431    0.000    0.477    0.232
##    .acc01             1.582    0.122   12.960    0.000    1.582    0.386
##    .acc02             0.621    0.077    8.066    0.000    0.621    0.181
##    .acc03             0.415    0.098    4.253    0.000    0.415    0.109
##    .rej01             0.720    0.117    6.135    0.000    0.720    0.168
##    .rej02             0.823    0.143    5.753    0.000    0.823    0.193
##    .rej03             0.523    0.083    6.330    0.000    0.523    0.134
##     VIRUS_WORRY       1.963    0.184   10.672    0.000    1.000    1.000
##     WORRY_EMPL        3.100    0.224   13.823    0.000    1.000    1.000
##     WORRY_REC         1.791    0.134   13.352    0.000    1.000    1.000
##     ACC               2.512    0.179   14.030    0.000    1.000    1.000
##     REJ               3.553    0.207   17.185    0.000    1.000    1.000
miive(cfa1, data_study1)
## MIIVsem (0.5.5) results 
## 
## Number of observations                                                    610
## Number of equations                                                        10
## Estimator                                                           MIIV-2SLS
## Standard Errors                                                      standard
## Missing                                                              listwise
## 
## 
## Parameter Estimates:
## 
## 
## STRUCTURAL COEFFICIENTS:
##                    Estimate  Std.Err  z-value  P(>|z|)   Sargan   df   P(Chi)
##   ACC =~                                                                     
##     acc01             1.000                                                  
##     acc02             1.046    0.040   25.916    0.000   13.130   12    0.360
##     acc03             1.151    0.043   26.814    0.000   11.828   12    0.460
##   REJ =~                                                                     
##     rej01             1.000                                                  
##     rej02             0.967    0.028   34.542    0.000   29.478   12    0.003
##     rej03             0.956    0.025   38.315    0.000   36.037   12    0.000
##   VIRUS_WORRY =~                                                             
##     aw01              1.000                                                  
##     aw02              1.075    0.043   24.967    0.000   91.454   12    0.000
##     aw03              1.039    0.044   23.725    0.000   84.312   12    0.000
##   WORRY_EMPL =~                                                              
##     ewj01             1.000                                                  
##     ewj02             1.083    0.030   36.327    0.000   13.661   12    0.323
##     ewj03             1.104    0.027   41.613    0.000   16.909   12    0.153
##   WORRY_REC =~                                                               
##     ewe01             1.000                                                  
##     ewe02             0.871    0.031   28.304    0.000   10.442   12    0.577
##     ewe03             0.933    0.033   28.190    0.000   22.945   12    0.028
## 
## INTERCEPTS:
##                    Estimate  Std.Err  z-value  P(>|z|)   
##     acc01             0.000                              
##     acc02             0.713    0.184    3.873    0.000   
##     acc03             0.103    0.195    0.526    0.599   
##     aw01              0.000                              
##     aw02             -1.388    0.221   -6.266    0.000   
##     aw03             -1.523    0.225   -6.756    0.000   
##     ewe01             0.000                              
##     ewe02             1.088    0.161    6.770    0.000   
##     ewe03             0.685    0.173    3.958    0.000   
##     ewj01             0.000                              
##     ewj02             0.274    0.088    3.119    0.002   
##     ewj03             0.049    0.077    0.631    0.528   
##     rej01             0.000                              
##     rej02             0.051    0.097    0.522    0.602   
##     rej03             0.139    0.086    1.617    0.106

–> Result: the model does not fit according to the chisquare omnibus test as well as to the equation specific Sargan tests. Besides, the factor loadings of VIRUS_WORRY show an interesting declining pattern (.787, .913, .882). This does not have to mean something but is a bit strange as all items do not differ with regard to clearness (which would explain the pattern). For the moment, we note that without further conclusions.

The standardized residuasl represent differences between empirical covariances and model-implied covariances (i.e., those covariances that would exist if the model was true).

resid(fit.cfa1, "standardized")
## $type
## [1] "standardized"
## 
## $cov
##       aw01   aw02   aw03   ewj01  ewj02  ewj03  ewe01  ewe02  ewe03  acc01 
## aw01  -0.042                                                               
## aw02  -1.679 -0.007                                                        
## aw03  -2.348  2.044 -0.003                                                 
## ewj01 -2.536  1.034  0.545  0.000                                          
## ewj02 -1.819  1.474  2.463 -0.026  0.000                                   
## ewj03 -3.168  0.286  0.702 -0.016  0.030  0.000                            
## ewe01  0.435 -1.469 -1.154 -0.172  0.456  1.665  0.000                     
## ewe02  1.908 -0.138 -0.373 -1.403 -0.611  0.119  0.012  0.000              
## ewe03  2.454 -0.800  0.956 -1.745 -0.212  0.074 -0.057  0.053  0.000       
## acc01  3.945 -1.111 -0.859  0.081  0.912  0.842 -0.090  0.670  0.163  0.000
## acc02  5.381 -1.640 -1.929 -0.812  0.852  0.578 -0.202  0.182  1.863  0.339
## acc03  5.939 -2.204 -2.717 -1.134  0.598 -0.400 -1.456 -0.252  0.431  0.096
## rej01 -6.416  0.819  1.694  1.173  0.070 -0.155 -0.502 -1.558 -1.977 -0.544
## rej02 -4.724  1.263  2.440  1.178 -0.661  0.303  2.319  0.080 -0.332  1.611
## rej03 -5.321  2.116  2.615  0.261 -1.252 -0.453  1.444 -0.026  0.101  1.681
##       acc02  acc03  rej01  rej02  rej03 
## aw01                                    
## aw02                                    
## aw03                                    
## ewj01                                   
## ewj02                                   
## ewj03                                   
## ewe01                                   
## ewe02                                   
## ewe03                                   
## acc01                                   
## acc02  0.000                            
## acc03 -0.105  0.000                     
## rej01 -1.576 -1.795  0.000              
## rej02  0.612  0.333 -0.616  0.000       
## rej03  0.842  0.728 -0.136  0.604  0.000
## 
## $mean
##   aw01   aw02   aw03  ewj01  ewj02  ewj03  ewe01  ewe02  ewe03  acc01  acc02 
## -3.636 -1.342 -1.362  0.000  0.000  0.000  0.000  0.000  0.000  0.000  0.000 
##  acc03  rej01  rej02  rej03 
##  0.000  0.000  0.000  0.000

Most notworthy are the symmetrical and systematic differences in the first three columns. This is a typical sign that items are forced in one factor model that measure different things.


Re-specified model (without aw02 und rej03)

Based on the significant Sargan tests and strong deviation in the standardized residuals between these items and the COSMO target item, aw02 und rej03 are eliminated.

cfa2 <- '
 VIRUS_WORRY =~ aw01 + aw03
 WORRY_EMPL =~ ewj02 + ewj01 +  ewj03
 WORRY_REC = ~ ewe02 +  ewe01 + ewe03 
 ACC =~  acc01 +  acc02 + acc02
 REJ =~  rej01 +  rej02 
'
fit.cfa2 <- sem(cfa2, data = data_study1, estimator="MLR", missing="FIML")
summary(fit.cfa2, fit.measures=TRUE, standardized=TRUE)
## lavaan 0.6-7 ended normally after 78 iterations
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of free parameters                         46
##                                                       
##   Number of observations                           612
##   Number of missing patterns                         3
##                                                       
## Model Test User Model:
##                                                Standard      Robust
##   Test Statistic                                 67.362      60.001
##   Degrees of freedom                                 44          44
##   P-value (Chi-square)                            0.013       0.054
##   Scaling correction factor                                   1.123
##        Yuan-Bentler correction (Mplus variant)                     
## 
## Model Test Baseline Model:
## 
##   Test statistic                              5427.774    3695.470
##   Degrees of freedom                                66          66
##   P-value                                        0.000       0.000
##   Scaling correction factor                                  1.469
## 
## User Model versus Baseline Model:
## 
##   Comparative Fit Index (CFI)                    0.996       0.996
##   Tucker-Lewis Index (TLI)                       0.993       0.993
##                                                                   
##   Robust Comparative Fit Index (CFI)                         0.997
##   Robust Tucker-Lewis Index (TLI)                            0.995
## 
## Loglikelihood and Information Criteria:
## 
##   Loglikelihood user model (H0)             -12127.295  -12127.295
##   Scaling correction factor                                  1.500
##       for the MLR correction                                      
##   Loglikelihood unrestricted model (H1)     -12093.614  -12093.614
##   Scaling correction factor                                  1.315
##       for the MLR correction                                      
##                                                                   
##   Akaike (AIC)                               24346.589   24346.589
##   Bayesian (BIC)                             24549.759   24549.759
##   Sample-size adjusted Bayesian (BIC)        24403.718   24403.718
## 
## Root Mean Square Error of Approximation:
## 
##   RMSEA                                          0.029       0.024
##   90 Percent confidence interval - lower         0.014       0.004
##   90 Percent confidence interval - upper         0.043       0.038
##   P-value RMSEA <= 0.05                          0.996       1.000
##                                                                   
##   Robust RMSEA                                               0.026
##   90 Percent confidence interval - lower                        NA
##   90 Percent confidence interval - upper                     0.041
## 
## Standardized Root Mean Square Residual:
## 
##   SRMR                                           0.022       0.022
## 
## 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
##   VIRUS_WORRY =~                                                        
##     aw01              1.000                               1.654    0.929
##     aw03              0.819    0.053   15.401    0.000    1.354    0.721
##   WORRY_EMPL =~                                                         
##     ewj02             1.000                               1.917    0.903
##     ewj01             0.918    0.028   33.321    0.000    1.760    0.914
##     ewj03             1.020    0.020   52.044    0.000    1.956    0.957
##   WORRY_REC =~                                                          
##     ewe02             1.000                               1.175    0.878
##     ewe01             1.137    0.045   25.108    0.000    1.335    0.876
##     ewe03             1.072    0.043   25.076    0.000    1.259    0.878
##   ACC =~                                                                
##     acc01             1.000                               1.563    0.773
##     acc02             1.098    0.045   24.523    0.000    1.716    0.927
##   REJ =~                                                                
##     rej01             1.000                               1.929    0.933
##     rej02             0.924    0.029   32.198    0.000    1.782    0.864
## 
## Covariances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##   VIRUS_WORRY ~~                                                        
##     WORRY_EMPL       -0.165    0.155   -1.068    0.285   -0.052   -0.052
##     WORRY_REC         0.522    0.115    4.526    0.000    0.268    0.268
##     ACC               1.736    0.157   11.024    0.000    0.671    0.671
##     REJ              -1.845    0.182  -10.130    0.000   -0.578   -0.578
##   WORRY_EMPL ~~                                                         
##     WORRY_REC         0.513    0.095    5.424    0.000    0.228    0.228
##     ACC              -0.533    0.141   -3.771    0.000   -0.178   -0.178
##     REJ               1.155    0.173    6.678    0.000    0.312    0.312
##   WORRY_REC ~~                                                          
##     ACC               0.095    0.099    0.963    0.336    0.052    0.052
##     REJ               0.245    0.108    2.260    0.024    0.108    0.108
##   ACC ~~                                                                
##     REJ              -2.461    0.173  -14.226    0.000   -0.816   -0.816
## 
## Intercepts:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##    .aw01              4.958    0.072   68.919    0.000    4.958    2.786
##    .aw03              3.630    0.076   47.775    0.000    3.630    1.932
##    .ewj02             2.881    0.086   33.581    0.000    2.881    1.357
##    .ewj01             2.408    0.078   30.925    0.000    2.408    1.250
##    .ewj03             2.706    0.083   32.748    0.000    2.706    1.324
##    .ewe02             5.523    0.054  102.065    0.000    5.523    4.126
##    .ewe01             5.093    0.062   82.667    0.000    5.093    3.342
##    .ewe03             5.436    0.058   93.783    0.000    5.436    3.791
##    .acc01             4.296    0.082   52.523    0.000    4.296    2.123
##    .acc02             5.208    0.075   69.635    0.000    5.208    2.815
##    .rej01             2.961    0.084   35.434    0.000    2.961    1.432
##    .rej02             2.920    0.083   35.012    0.000    2.920    1.415
##     VIRUS_WORRY       0.000                               0.000    0.000
##     WORRY_EMPL        0.000                               0.000    0.000
##     WORRY_REC         0.000                               0.000    0.000
##     ACC               0.000                               0.000    0.000
##     REJ               0.000                               0.000    0.000
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##    .aw01              0.433    0.189    2.285    0.022    0.433    0.137
##    .aw03              1.696    0.134   12.628    0.000    1.696    0.481
##    .ewj02             0.828    0.148    5.609    0.000    0.828    0.184
##    .ewj01             0.613    0.097    6.323    0.000    0.613    0.165
##    .ewj03             0.354    0.083    4.268    0.000    0.354    0.085
##    .ewe02             0.412    0.054    7.652    0.000    0.412    0.230
##    .ewe01             0.540    0.082    6.607    0.000    0.540    0.232
##    .ewe03             0.471    0.074    6.381    0.000    0.471    0.229
##    .acc01             1.650    0.141   11.706    0.000    1.650    0.403
##    .acc02             0.479    0.104    4.617    0.000    0.479    0.140
##    .rej01             0.552    0.120    4.591    0.000    0.552    0.129
##    .rej02             1.080    0.169    6.396    0.000    1.080    0.254
##     VIRUS_WORRY       2.735    0.239   11.443    0.000    1.000    1.000
##     WORRY_EMPL        3.676    0.203   18.149    0.000    1.000    1.000
##     WORRY_REC         1.380    0.129   10.712    0.000    1.000    1.000
##     ACC               2.443    0.189   12.916    0.000    1.000    1.000
##     REJ               3.721    0.201   18.515    0.000    1.000    1.000
miive(cfa2, data_study1)
## MIIVsem (0.5.5) results 
## 
## Number of observations                                                    610
## Number of equations                                                         7
## Estimator                                                           MIIV-2SLS
## Standard Errors                                                      standard
## Missing                                                              listwise
## 
## 
## Parameter Estimates:
## 
## 
## STRUCTURAL COEFFICIENTS:
##                    Estimate  Std.Err  z-value  P(>|z|)   Sargan   df   P(Chi)
##   ACC =~                                                                     
##     acc01             1.000                                                  
##     acc02             1.083    0.048   22.344    0.000    9.639    9    0.380
##   REJ =~                                                                     
##     rej01             1.000                                                  
##     rej02             0.916    0.032   28.787    0.000   17.159    9    0.046
##   VIRUS_WORRY =~                                                             
##     aw01              1.000                                                  
##     aw03              0.816    0.049   16.706    0.000   16.515    9    0.057
##   WORRY_EMPL =~                                                              
##     ewj02             1.000                                                  
##     ewj01             0.915    0.025   36.334    0.000   10.077    9    0.344
##     ewj03             1.016    0.025   40.187    0.000   11.746    9    0.228
##   WORRY_REC =~                                                               
##     ewe02             1.000                                                  
##     ewe01             1.134    0.040   28.224    0.000    9.925    9    0.357
##     ewe03             1.068    0.038   28.235    0.000   10.501    9    0.311
## 
## INTERCEPTS:
##                    Estimate  Std.Err  z-value  P(>|z|)   
##     acc01             0.000                              
##     acc02             0.555    0.218    2.549    0.011   
##     aw01              0.000                              
##     aw03             -0.418    0.249   -1.681    0.093   
##     ewe01            -1.172    0.226   -5.191    0.000   
##     ewe02             0.000                              
##     ewe03            -0.463    0.212   -2.178    0.029   
##     ewj01            -0.229    0.086   -2.650    0.008   
##     ewj02             0.000                              
##     ewj03            -0.222    0.086   -2.598    0.009   
##     rej01             0.000                              
##     rej02             0.200    0.107    1.877    0.061

–> Model fits slightly. The Sargan test for awv03 is slightly n.s. (p = .057) At this time not conspicuous as p-values in Sargan test should be interpreted with regard to the number of performed tests (the same is true for rej02, that has a sign. p-value)


Structural equation models

Now, predictors are included. This approach has to goals:

  1. Test of a model with real predictors (instead of simple covariates) with their effects representing unique effects adjusted for spurious correlations due to the other predictors

  2. Generation of new, not-yet tested restrictions and implications which the model has to fulfill IF the respecified model is correct. This reduces the danger to wrongly adapt the model to a stable part of the data

Initial model (based on cfa2)

sem1 <- '
 VIRUS_WORRY =~ aw01 + aw03
 WORRY_EMPL =~ ewj02 + ewj01 +  ewj03
 WORRY_REC = ~ ewe02 +  ewe01 + ewe03 
 ACC =~  acc01 +  acc02 + acc03
 REJ =~  rej01 +  rej02 

 VIRUS_WORRY ~ health_anx_XS + worries_trait_XS + reactance_trait_XS  + lonely_XS + affected_econ_XS
 WORRY_EMPL ~  health_anx_XS + worries_trait_XS + reactance_trait_XS  + lonely_XS + affected_econ_XS
 WORRY_REC ~ health_anx_XS + worries_trait_XS + reactance_trait_XS  + lonely_XS + affected_econ_XS
 ACC ~ health_anx_XS + worries_trait_XS + reactance_trait_XS  + lonely_XS + affected_econ_XS
 REJ ~ health_anx_XS + worries_trait_XS + reactance_trait_XS  + lonely_XS + affected_econ_XS
'
fit.sem1 <- sem(sem1, data = data_study1, estimator="MLR", missing="FIML")
summary(fit.sem1, fit.measures=TRUE, standardized=TRUE)
## lavaan 0.6-7 ended normally after 109 iterations
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of free parameters                         74
##                                                       
##                                                   Used       Total
##   Number of observations                           611         612
##   Number of missing patterns                         3            
##                                                                   
## Model Test User Model:
##                                                Standard      Robust
##   Test Statistic                                150.734     138.198
##   Degrees of freedom                                 95          95
##   P-value (Chi-square)                            0.000       0.003
##   Scaling correction factor                                   1.091
##        Yuan-Bentler correction (Mplus variant)                     
## 
## Model Test Baseline Model:
## 
##   Test statistic                              7219.527    5635.481
##   Degrees of freedom                               143         143
##   P-value                                        0.000       0.000
##   Scaling correction factor                                  1.281
## 
## User Model versus Baseline Model:
## 
##   Comparative Fit Index (CFI)                    0.992       0.992
##   Tucker-Lewis Index (TLI)                       0.988       0.988
##                                                                   
##   Robust Comparative Fit Index (CFI)                         0.993
##   Robust Tucker-Lewis Index (TLI)                            0.990
## 
## Loglikelihood and Information Criteria:
## 
##   Loglikelihood user model (H0)             -12524.079  -12524.079
##   Scaling correction factor                                  1.388
##       for the MLR correction                                      
##   Loglikelihood unrestricted model (H1)     -12448.712  -12448.712
##   Scaling correction factor                                  1.221
##       for the MLR correction                                      
##                                                                   
##   Akaike (AIC)                               25196.158   25196.158
##   Bayesian (BIC)                             25522.876   25522.876
##   Sample-size adjusted Bayesian (BIC)        25287.941   25287.941
## 
## Root Mean Square Error of Approximation:
## 
##   RMSEA                                          0.031       0.027
##   90 Percent confidence interval - lower         0.021       0.017
##   90 Percent confidence interval - upper         0.040       0.036
##   P-value RMSEA <= 0.05                          1.000       1.000
##                                                                   
##   Robust RMSEA                                               0.028
##   90 Percent confidence interval - lower                     0.017
##   90 Percent confidence interval - upper                     0.038
## 
## Standardized Root Mean Square Residual:
## 
##   SRMR                                           0.024       0.024
## 
## 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
##   VIRUS_WORRY =~                                                        
##     aw01              1.000                               1.560    0.877
##     aw03              0.917    0.054   17.098    0.000    1.431    0.763
##   WORRY_EMPL =~                                                         
##     ewj02             1.000                               1.920    0.905
##     ewj01             0.922    0.027   33.952    0.000    1.770    0.918
##     ewj03             1.014    0.018   55.203    0.000    1.946    0.952
##   WORRY_REC =~                                                          
##     ewe02             1.000                               1.172    0.876
##     ewe01             1.139    0.045   25.198    0.000    1.335    0.876
##     ewe03             1.075    0.043   25.115    0.000    1.260    0.879
##   ACC =~                                                                
##     acc01             1.000                               1.587    0.784
##     acc02             1.050    0.036   29.484    0.000    1.667    0.901
##     acc03             1.166    0.035   33.208    0.000    1.850    0.948
##   REJ =~                                                                
##     rej01             1.000                               1.914    0.926
##     rej02             0.938    0.025   37.264    0.000    1.796    0.870
## 
## Regressions:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##   VIRUS_WORRY ~                                                         
##     health_anx_XS     0.339    0.048    7.034    0.000    0.217    0.344
##     worries_trt_XS    0.193    0.051    3.786    0.000    0.124    0.198
##     reactnc_trt_XS   -0.311    0.052   -5.951    0.000   -0.199   -0.307
##     lonely_XS        -0.058    0.040   -1.443    0.149   -0.037   -0.070
##     affected_cn_XS   -0.382    0.216   -1.769    0.077   -0.245   -0.066
##   WORRY_EMPL ~                                                          
##     health_anx_XS    -0.032    0.048   -0.652    0.514   -0.016   -0.026
##     worries_trt_XS    0.254    0.049    5.188    0.000    0.133    0.212
##     reactnc_trt_XS    0.028    0.041    0.691    0.490    0.015    0.022
##     lonely_XS         0.103    0.037    2.819    0.005    0.054    0.100
##     affected_cn_XS    4.144    0.264   15.713    0.000    2.158    0.585
##   WORRY_REC ~                                                           
##     health_anx_XS    -0.058    0.035   -1.650    0.099   -0.049   -0.078
##     worries_trt_XS    0.158    0.038    4.157    0.000    0.135    0.216
##     reactnc_trt_XS    0.027    0.036    0.754    0.451    0.023    0.036
##     lonely_XS         0.036    0.029    1.270    0.204    0.031    0.058
##     affected_cn_XS    0.172    0.174    0.989    0.322    0.146    0.040
##   ACC ~                                                                 
##     health_anx_XS     0.205    0.044    4.643    0.000    0.129    0.204
##     worries_trt_XS    0.100    0.047    2.136    0.033    0.063    0.101
##     reactnc_trt_XS   -0.419    0.046   -9.170    0.000   -0.264   -0.406
##     lonely_XS        -0.148    0.039   -3.768    0.000   -0.093   -0.174
##     affected_cn_XS   -0.297    0.238   -1.249    0.212   -0.187   -0.051
##   REJ ~                                                                 
##     health_anx_XS    -0.133    0.050   -2.682    0.007   -0.070   -0.110
##     worries_trt_XS   -0.082    0.052   -1.573    0.116   -0.043   -0.068
##     reactnc_trt_XS    0.650    0.049   13.171    0.000    0.340    0.523
##     lonely_XS         0.159    0.045    3.542    0.000    0.083    0.155
##     affected_cn_XS    1.076    0.269    4.002    0.000    0.562    0.153
## 
## Covariances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##  .VIRUS_WORRY ~~                                                        
##    .WORRY_EMPL       -0.082    0.087   -0.949    0.343   -0.045   -0.045
##    .WORRY_REC         0.485    0.093    5.209    0.000    0.322    0.322
##    .ACC               1.158    0.118    9.833    0.000    0.647    0.647
##    .REJ              -1.083    0.128   -8.464    0.000   -0.553   -0.553
##  .WORRY_EMPL ~~                                                         
##    .WORRY_REC         0.253    0.068    3.722    0.000    0.162    0.162
##    .ACC              -0.244    0.083   -2.955    0.003   -0.132   -0.132
##    .REJ               0.318    0.094    3.382    0.001    0.157    0.157
##  .WORRY_REC ~~                                                          
##    .ACC               0.136    0.081    1.690    0.091    0.089    0.089
##    .REJ               0.090    0.082    1.100    0.271    0.053    0.053
##  .ACC ~~                                                                
##    .REJ              -1.588    0.122  -12.974    0.000   -0.792   -0.792
## 
## Intercepts:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##    .aw01              4.509    0.217   20.758    0.000    4.509    2.534
##    .aw03              3.214    0.211   15.255    0.000    3.214    1.714
##    .ewj02             1.109    0.177    6.257    0.000    1.109    0.522
##    .ewj01             0.774    0.157    4.921    0.000    0.774    0.402
##    .ewj03             0.909    0.175    5.200    0.000    0.909    0.445
##    .ewe02             4.968    0.163   30.506    0.000    4.968    3.712
##    .ewe01             4.461    0.180   24.749    0.000    4.461    2.928
##    .ewe03             4.840    0.169   28.557    0.000    4.840    3.376
##    .acc01             5.143    0.191   26.966    0.000    5.143    2.540
##    .acc02             6.094    0.197   30.909    0.000    6.094    3.294
##    .acc03             6.038    0.218   27.749    0.000    6.038    3.093
##    .rej01             0.969    0.193    5.010    0.000    0.969    0.469
##    .rej02             1.051    0.194    5.424    0.000    1.051    0.509
##    .VIRUS_WORRY       0.000                               0.000    0.000
##    .WORRY_EMPL        0.000                               0.000    0.000
##    .WORRY_REC         0.000                               0.000    0.000
##    .ACC               0.000                               0.000    0.000
##    .REJ               0.000                               0.000    0.000
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##    .aw01              0.734    0.154    4.770    0.000    0.734    0.232
##    .aw03              1.467    0.132   11.109    0.000    1.467    0.417
##    .ewj02             0.819    0.144    5.683    0.000    0.819    0.182
##    .ewj01             0.582    0.090    6.446    0.000    0.582    0.157
##    .ewj03             0.393    0.084    4.682    0.000    0.393    0.094
##    .ewe02             0.417    0.054    7.728    0.000    0.417    0.233
##    .ewe01             0.539    0.082    6.570    0.000    0.539    0.232
##    .ewe03             0.468    0.074    6.354    0.000    0.468    0.227
##    .acc01             1.582    0.121   13.029    0.000    1.582    0.386
##    .acc02             0.645    0.076    8.533    0.000    0.645    0.188
##    .acc03             0.389    0.092    4.225    0.000    0.389    0.102
##    .rej01             0.610    0.102    5.994    0.000    0.610    0.143
##    .rej02             1.032    0.166    6.233    0.000    1.032    0.242
##    .VIRUS_WORRY       1.750    0.200    8.764    0.000    0.719    0.719
##    .WORRY_EMPL        1.874    0.147   12.714    0.000    0.508    0.508
##    .WORRY_REC         1.295    0.118   10.966    0.000    0.942    0.942
##    .ACC               1.833    0.142   12.869    0.000    0.728    0.728
##    .REJ               2.190    0.162   13.497    0.000    0.598    0.598
miive(sem1, data_study1)
## MIIVsem (0.5.5) results 
## 
## Number of observations                                                    609
## Number of equations                                                        13
## Estimator                                                           MIIV-2SLS
## Standard Errors                                                      standard
## Missing                                                              listwise
## 
## 
## Parameter Estimates:
## 
## 
## STRUCTURAL COEFFICIENTS:
##                    Estimate  Std.Err  z-value  P(>|z|)   Sargan   df   P(Chi)
##   ACC =~                                                                     
##     acc01             1.000                                                  
##     acc02             1.039    0.040   25.984    0.000   17.301   15    0.301
##     acc03             1.147    0.042   27.010    0.000   14.687   15    0.474
##   REJ =~                                                                     
##     rej01             1.000                                                  
##     rej02             0.930    0.030   31.098    0.000   22.696   15    0.091
##   VIRUS_WORRY =~                                                             
##     aw01              1.000                                                  
##     aw03              0.885    0.047   19.035    0.000   48.136   15    0.000
##   WORRY_EMPL =~                                                              
##     ewj02             1.000                                                  
##     ewj01             0.916    0.025   36.497    0.000   15.844   15    0.392
##     ewj03             1.011    0.025   40.457    0.000   16.639   15    0.341
##   WORRY_REC =~                                                               
##     ewe02             1.000                                                  
##     ewe01             1.133    0.040   28.193    0.000   12.614   15    0.632
##     ewe03             1.067    0.038   28.177    0.000   16.915   15    0.324
##                                                          
##   ACC ~                                                  
##     health_anx_XS     0.267    0.056    4.793    0.000   
##     worries_trt_XS    0.075    0.059    1.278    0.201   
##     reactnc_trt_XS   -0.406    0.053   -7.608    0.000   
##     lonely_XS        -0.134    0.046   -2.918    0.004   
##     affected_cn_XS   -0.222    0.289   -0.769    0.442   
##   REJ ~                                                  
##     health_anx_XS    -0.133    0.051   -2.625    0.009   
##     worries_trt_XS   -0.105    0.053   -1.966    0.049   
##     reactnc_trt_XS    0.635    0.049   13.094    0.000   
##     lonely_XS         0.170    0.042    4.095    0.000   
##     affected_cn_XS    1.128    0.262    4.298    0.000   
##   VIRUS_WORRY ~                                          
##     health_anx_XS     0.302    0.048    6.282    0.000   
##     worries_trt_XS    0.164    0.051    3.235    0.001   
##     reactnc_trt_XS   -0.336    0.046   -7.284    0.000   
##     lonely_XS        -0.060    0.040   -1.519    0.129   
##     affected_cn_XS   -0.386    0.250   -1.545    0.122   
##   WORRY_EMPL ~                                           
##     health_anx_XS    -0.043    0.049   -0.878    0.380   
##     worries_trt_XS    0.259    0.052    4.992    0.000   
##     reactnc_trt_XS    0.010    0.047    0.201    0.841   
##     lonely_XS         0.126    0.041    3.117    0.002   
##     affected_cn_XS    4.118    0.256   16.103    0.000   
##   WORRY_REC ~                                            
##     health_anx_XS    -0.030    0.039   -0.765    0.444   
##     worries_trt_XS    0.154    0.042    3.714    0.000   
##     reactnc_trt_XS    0.025    0.038    0.651    0.515   
##     lonely_XS         0.028    0.032    0.862    0.389   
##     affected_cn_XS    0.177    0.205    0.867    0.386   
## 
## INTERCEPTS:
##                    Estimate  Std.Err  z-value  P(>|z|)   
##     ACC               4.947    0.223   22.193    0.000   
##     acc01             0.000                              
##     acc02             0.737    0.183    4.040    0.000   
##     acc03             0.120    0.193    0.622    0.534   
##     aw01              0.000                              
##     aw03             -0.765    0.237   -3.222    0.001   
##     ewe01            -1.163    0.226   -5.156    0.000   
##     ewe02             0.000                              
##     ewe03            -0.458    0.213   -2.153    0.031   
##     ewj01            -0.232    0.086   -2.691    0.007   
##     ewj02             0.000                              
##     ewj03            -0.208    0.085   -2.450    0.014   
##     REJ               1.038    0.203    5.126    0.000   
##     rej01             0.000                              
##     rej02             0.160    0.102    1.575    0.115   
##     VIRUS_WORRY       4.795    0.193   24.883    0.000   
##     WORRY_EMPL        1.124    0.197    5.691    0.000   
##     WORRY_REC         4.927    0.158   31.201    0.000

The results show that adding predictors again causes misfit. As the predictors were added as composites with a saturated pattern of effects (every effect is estimated), the reason can only be in the measurement model. Let’s look onto the standardized residuals:

resid(fit.sem1, "standardized")
## $type
## [1] "standardized"
## 
## $cov
##                    aw01   aw03   ewj02  ewj01  ewj03  ewe02  ewe01  ewe03 
## aw01               -0.003                                                 
## aw03               -0.022 -0.016                                          
## ewj02              -0.558  3.459  0.000                                   
## ewj01              -1.651  1.928 -0.543  0.000                            
## ewj03              -3.084  2.245  0.304 -0.019  0.000                     
## ewe02               0.562 -0.298 -0.572 -1.444  0.252  0.000              
## ewe01              -1.939 -0.951  0.542 -0.165  1.897  0.235  0.000       
## ewe03               1.364  0.787 -0.208 -1.844  0.150 -0.048 -0.175  0.000
## acc01               0.018 -0.901  1.033  0.107  0.922  0.823 -0.031  0.254
## acc02               1.669 -1.920  1.063 -0.958  0.733  0.217 -0.218  2.136
## acc03               1.855 -2.848  0.893 -1.383 -0.562 -0.313 -1.810  0.573
## rej01              -3.332  3.529 -0.556  0.860 -1.015 -1.241  0.033 -1.772
## rej02              -0.924  2.344 -0.753  1.306  0.467  0.790  3.044  0.329
## health_anx_XS      -5.352  4.123 -0.261  1.039 -0.756  1.464 -0.197 -1.128
## worries_trait_XS   -5.649  4.194  0.111  0.850 -0.991  0.484  0.371 -0.840
## reactance_trait_XS -2.926  2.923 -0.341  0.762 -0.586 -0.358  1.834 -1.748
## lonely_XS          -2.542  2.251  0.876  0.136 -1.069 -0.590  0.599 -0.068
## affected_econ_XS   -1.572  1.400 -0.050  1.273 -1.743 -0.053  0.900 -0.923
##                    acc01  acc02  acc03  rej01  rej02  hl__XS wr__XS rc__XS
## aw01                                                                      
## aw03                                                                      
## ewj02                                                                     
## ewj01                                                                     
## ewj03                                                                     
## ewe02                                                                     
## ewe01                                                                     
## ewe03                                                                     
## acc01               0.000                                                 
## acc02               0.928  0.000                                          
## acc03              -0.007 -0.248  0.000                                   
## rej01               0.718 -0.940 -0.902  0.000                            
## rej02               2.054  0.741  0.719  0.000  0.000                     
## health_anx_XS       1.719 -1.026 -0.129 -0.488  0.560  0.000              
## worries_trait_XS    0.548 -0.405  0.023 -1.497  1.677  0.000  0.000       
## reactance_trait_XS  0.982 -0.258 -0.197 -1.608  1.394  0.000  0.000  0.000
## lonely_XS           0.980  1.313 -1.290  0.012 -0.012  0.000  0.000  0.000
## affected_econ_XS    0.855 -1.973  1.401  0.175 -0.167  0.000  0.000  0.000
##                    lnl_XS af__XS
## aw01                            
## aw03                            
## ewj02                           
## ewj01                           
## ewj03                           
## ewe02                           
## ewe01                           
## ewe03                           
## acc01                           
## acc02                           
## acc03                           
## rej01                           
## rej02                           
## health_anx_XS                   
## worries_trait_XS                
## reactance_trait_XS              
## lonely_XS           0.000       
## affected_econ_XS    0.000  0.000
## 
## $mean
##               aw01               aw03              ewj02              ewj01 
##              0.090             -1.588              0.000              0.000 
##              ewj03              ewe02              ewe01              ewe03 
##              0.000              0.000              0.000              0.000 
##              acc01              acc02              acc03              rej01 
##              0.000              0.000              0.000              0.000 
##              rej02      health_anx_XS   worries_trait_XS reactance_trait_XS 
##              0.000              0.000              0.000              0.000 
##          lonely_XS   affected_econ_XS 
##              0.000              0.000

This model shows the same pattern as in the initial CFA: the residuals show quite clearly that both virus-related worries indicators do not work in a synchronous fashion. This is most striking in the case of health_anxiety and trait worries, whose correlations with the COSMO target item (aw01) is greatly overestimated, while that with the infection-oriented item (aw03) is strongly underestimated. This is evidence that a) a single latent variable as the response-generating mechanism is wrong and that the first item is not as health-related as the second.

This fits with the initial observation of the varying pattern of descending loadings.


Final model: Single indicator aw01

It is unfortunate but we have to eliminate a further indicator of virus-related worries. As an alternative it would possible to break virus-related worries in two latent variables–one with infection-related worries as intended initially and one with overall worries stemming from the COSMO survey. As the goal of study 1, however, consisted in validating the COSMO study, we decided to eliminate infection-related worries.

Having one indicator, however, poses the challenge how to identify the measurement model. Hence, the error variance has to be fixed to some value. The routine scheme is to fix it to zero but this would equate the indicator with the latent variable which also implies perfect measurement. Better, hence is to fix the error variance to a reasonable value.

We decided to fix the error variance of the latent virus-related/overall worry single-indicator variable to the variance of the indicator times one minus its reliability (Lin et al.). From a strict perspective, using Cronbach’s alpha in this case is not exactly correct as both items on which alpha was calculated are no convergent measures of the same factor. But at least it is a lower boundary for the error variance.

Lin, C.-H., Sher, P. J., & Shih, H.-Y. (2005). Past progress and future directions in conceptualizing customer perceived value. International Journal of Service Industry Management, 16(4), 318-336.

data_study1 %>% 
  summarise(var = var(aw01, na.rm=TRUE)) %>% 
  mutate(fixed.error = var*(1-.80))
## # A tibble: 1 x 2
##     var fixed.error
##   <dbl>       <dbl>
## 1  3.18       0.635
sem2 <- '
 VIRUS_WORRY =~ aw01 
 WORRY_EMPL =~ ewj02 + ewj01 +  ewj03
 WORRY_REC = ~ ewe02 +  ewe01 + ewe03
 ACC =~  acc01 +  acc02 + acc03
 REJ =~  rej01 +  rej02 
 
 VIRUS_WORRY ~ health_anx_XS + worries_trait_XS + reactance_trait_XS  + lonely_XS + affected_econ_XS
 WORRY_EMPL ~  health_anx_XS + worries_trait_XS + reactance_trait_XS  + lonely_XS + affected_econ_XS
 WORRY_REC ~ health_anx_XS + worries_trait_XS + reactance_trait_XS  + lonely_XS + affected_econ_XS
 ACC ~ health_anx_XS + worries_trait_XS + reactance_trait_XS  + lonely_XS + affected_econ_XS
 REJ ~ health_anx_XS + worries_trait_XS + reactance_trait_XS  + lonely_XS + affected_econ_XS

 #Error correction 
 aw01~~.635*aw01
'
fit.sem2 <- sem(sem2, data = data_study1, estimator="MLR", missing="FIML")
summary(fit.sem2, fit.measures=TRUE, standardized=TRUE)
## lavaan 0.6-7 ended normally after 103 iterations
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of free parameters                         70
##                                                       
##                                                   Used       Total
##   Number of observations                           611         612
##   Number of missing patterns                         2            
##                                                                   
## Model Test User Model:
##                                                Standard      Robust
##   Test Statistic                                100.241      92.280
##   Degrees of freedom                                 80          80
##   P-value (Chi-square)                            0.063       0.164
##   Scaling correction factor                                   1.086
##        Yuan-Bentler correction (Mplus variant)                     
## 
## Model Test Baseline Model:
## 
##   Test statistic                              6770.375    5235.021
##   Degrees of freedom                               126         126
##   P-value                                        0.000       0.000
##   Scaling correction factor                                  1.293
## 
## User Model versus Baseline Model:
## 
##   Comparative Fit Index (CFI)                    0.997       0.998
##   Tucker-Lewis Index (TLI)                       0.995       0.996
##                                                                   
##   Robust Comparative Fit Index (CFI)                         0.998
##   Robust Tucker-Lewis Index (TLI)                            0.997
## 
## Loglikelihood and Information Criteria:
## 
##   Loglikelihood user model (H0)             -11474.299  -11474.299
##   Scaling correction factor                                  1.396
##       for the MLR correction                                      
##   Loglikelihood unrestricted model (H1)     -11424.178  -11424.178
##   Scaling correction factor                                  1.231
##       for the MLR correction                                      
##                                                                   
##   Akaike (AIC)                               23088.598   23088.598
##   Bayesian (BIC)                             23397.655   23397.655
##   Sample-size adjusted Bayesian (BIC)        23175.420   23175.420
## 
## Root Mean Square Error of Approximation:
## 
##   RMSEA                                          0.020       0.016
##   90 Percent confidence interval - lower         0.000       0.000
##   90 Percent confidence interval - upper         0.032       0.028
##   P-value RMSEA <= 0.05                          1.000       1.000
##                                                                   
##   Robust RMSEA                                               0.017
##   90 Percent confidence interval - lower                     0.000
##   90 Percent confidence interval - upper                     0.030
## 
## Standardized Root Mean Square Residual:
## 
##   SRMR                                           0.015       0.015
## 
## 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
##   VIRUS_WORRY =~                                                        
##     aw01              1.000                               1.591    0.894
##   WORRY_EMPL =~                                                         
##     ewj02             1.000                               1.920    0.904
##     ewj01             0.922    0.027   33.964    0.000    1.770    0.918
##     ewj03             1.014    0.018   55.219    0.000    1.946    0.952
##   WORRY_REC =~                                                          
##     ewe02             1.000                               1.173    0.876
##     ewe01             1.138    0.045   25.159    0.000    1.335    0.876
##     ewe03             1.074    0.043   25.151    0.000    1.260    0.878
##   ACC =~                                                                
##     acc01             1.000                               1.586    0.783
##     acc02             1.050    0.036   29.450    0.000    1.666    0.901
##     acc03             1.167    0.035   33.197    0.000    1.851    0.948
##   REJ =~                                                                
##     rej01             1.000                               1.916    0.927
##     rej02             0.937    0.025   37.050    0.000    1.795    0.870
## 
## Regressions:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##   VIRUS_WORRY ~                                                         
##     health_anx_XS     0.300    0.049    6.154    0.000    0.188    0.298
##     worries_trt_XS    0.166    0.053    3.126    0.002    0.104    0.167
##     reactnc_trt_XS   -0.335    0.054   -6.180    0.000   -0.210   -0.324
##     lonely_XS        -0.061    0.043   -1.425    0.154   -0.038   -0.071
##     affected_cn_XS   -0.387    0.236   -1.638    0.101   -0.243   -0.066
##   WORRY_EMPL ~                                                          
##     health_anx_XS    -0.032    0.048   -0.653    0.514   -0.016   -0.026
##     worries_trt_XS    0.254    0.049    5.188    0.000    0.133    0.212
##     reactnc_trt_XS    0.028    0.040    0.691    0.490    0.015    0.022
##     lonely_XS         0.103    0.037    2.818    0.005    0.054    0.100
##     affected_cn_XS    4.144    0.264   15.711    0.000    2.158    0.585
##   WORRY_REC ~                                                           
##     health_anx_XS    -0.058    0.035   -1.647    0.099   -0.049   -0.078
##     worries_trt_XS    0.158    0.038    4.156    0.000    0.135    0.216
##     reactnc_trt_XS    0.027    0.036    0.754    0.451    0.023    0.036
##     lonely_XS         0.036    0.029    1.269    0.204    0.031    0.058
##     affected_cn_XS    0.172    0.174    0.990    0.322    0.146    0.040
##   ACC ~                                                                 
##     health_anx_XS     0.205    0.044    4.644    0.000    0.129    0.204
##     worries_trt_XS    0.100    0.047    2.137    0.033    0.063    0.101
##     reactnc_trt_XS   -0.418    0.046   -9.170    0.000   -0.264   -0.406
##     lonely_XS        -0.148    0.039   -3.772    0.000   -0.093   -0.174
##     affected_cn_XS   -0.296    0.238   -1.246    0.213   -0.187   -0.051
##   REJ ~                                                                 
##     health_anx_XS    -0.133    0.050   -2.681    0.007   -0.070   -0.110
##     worries_trt_XS   -0.082    0.052   -1.579    0.114   -0.043   -0.068
##     reactnc_trt_XS    0.650    0.049   13.167    0.000    0.339    0.522
##     lonely_XS         0.159    0.045    3.544    0.000    0.083    0.155
##     affected_cn_XS    1.078    0.269    4.004    0.000    0.563    0.153
## 
## Covariances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##  .VIRUS_WORRY ~~                                                        
##    .WORRY_EMPL       -0.126    0.091   -1.379    0.168   -0.066   -0.066
##    .WORRY_REC         0.508    0.100    5.062    0.000    0.322    0.322
##    .ACC               1.211    0.115   10.541    0.000    0.646    0.646
##    .REJ              -1.150    0.125   -9.161    0.000   -0.560   -0.560
##  .WORRY_EMPL ~~                                                         
##    .WORRY_REC         0.253    0.068    3.722    0.000    0.162    0.162
##    .ACC              -0.244    0.082   -2.959    0.003   -0.132   -0.132
##    .REJ               0.318    0.094    3.381    0.001    0.157    0.157
##  .WORRY_REC ~~                                                          
##    .ACC               0.136    0.081    1.687    0.092    0.088    0.088
##    .REJ               0.090    0.082    1.094    0.274    0.053    0.053
##  .ACC ~~                                                                
##    .REJ              -1.588    0.122  -12.966    0.000   -0.792   -0.792
## 
## Intercepts:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##    .aw01              4.793    0.215   22.299    0.000    4.793    2.694
##    .ewj02             1.109    0.177    6.257    0.000    1.109    0.522
##    .ewj01             0.775    0.157    4.923    0.000    0.775    0.402
##    .ewj03             0.909    0.175    5.199    0.000    0.909    0.445
##    .ewe02             4.968    0.163   30.486    0.000    4.968    3.712
##    .ewe01             4.461    0.180   24.756    0.000    4.461    2.928
##    .ewe03             4.840    0.169   28.566    0.000    4.840    3.376
##    .acc01             5.143    0.191   26.973    0.000    5.143    2.540
##    .acc02             6.094    0.197   30.915    0.000    6.094    3.294
##    .acc03             6.039    0.218   27.749    0.000    6.039    3.093
##    .rej01             0.969    0.194    5.006    0.000    0.969    0.469
##    .rej02             1.054    0.194    5.435    0.000    1.054    0.511
##    .VIRUS_WORRY       0.000                               0.000    0.000
##    .WORRY_EMPL        0.000                               0.000    0.000
##    .WORRY_REC         0.000                               0.000    0.000
##    .ACC               0.000                               0.000    0.000
##    .REJ               0.000                               0.000    0.000
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##    .aw01              0.635                               0.635    0.201
##    .ewj02             0.819    0.144    5.688    0.000    0.819    0.182
##    .ewj01             0.583    0.090    6.448    0.000    0.583    0.157
##    .ewj03             0.392    0.084    4.667    0.000    0.392    0.094
##    .ewe02             0.415    0.054    7.681    0.000    0.415    0.232
##    .ewe01             0.539    0.082    6.615    0.000    0.539    0.232
##    .ewe03             0.469    0.074    6.365    0.000    0.469    0.228
##    .acc01             1.584    0.122   13.035    0.000    1.584    0.386
##    .acc02             0.647    0.076    8.525    0.000    0.647    0.189
##    .acc03             0.386    0.091    4.252    0.000    0.386    0.101
##    .rej01             0.604    0.102    5.948    0.000    0.604    0.141
##    .rej02             1.037    0.166    6.240    0.000    1.037    0.244
##    .VIRUS_WORRY       1.918    0.140   13.734    0.000    0.758    0.758
##    .WORRY_EMPL        1.874    0.147   12.718    0.000    0.509    0.509
##    .WORRY_REC         1.297    0.118   10.960    0.000    0.942    0.942
##    .ACC               1.832    0.143   12.851    0.000    0.728    0.728
##    .REJ               2.195    0.163   13.502    0.000    0.598    0.598
miive(sem2, data_study1)
## MIIVsem (0.5.5) results 
## 
## Number of observations                                                    610
## Number of equations                                                        12
## Estimator                                                           MIIV-2SLS
## Standard Errors                                                      standard
## Missing                                                              listwise
## 
## 
## Parameter Estimates:
## 
## 
## STRUCTURAL COEFFICIENTS:
##                    Estimate  Std.Err  z-value  P(>|z|)   Sargan   df   P(Chi)
##   ACC =~                                                                     
##     acc01             1.000                                                  
##     acc02             1.040    0.040   25.936    0.000   16.965   14    0.258
##     acc03             1.148    0.043   26.950    0.000   13.870   14    0.459
##   REJ =~                                                                     
##     rej01             1.000                                                  
##     rej02             0.929    0.030   30.853    0.000   21.553   14    0.088
##   WORRY_EMPL =~                                                              
##     ewj02             1.000                                                  
##     ewj01             0.918    0.025   36.551    0.000   12.480   14    0.568
##     ewj03             1.014    0.025   40.477    0.000   13.711   14    0.471
##   WORRY_REC =~                                                               
##     ewe02             1.000                                                  
##     ewe01             1.134    0.040   28.267    0.000   12.006   14    0.606
##     ewe03             1.068    0.038   28.252    0.000   15.134   14    0.369
##                                                          
##   ACC ~                                                  
##     health_anx_XS     0.270    0.056    4.858    0.000   
##     worries_trt_XS    0.070    0.059    1.204    0.228   
##     reactnc_trt_XS   -0.407    0.053   -7.611    0.000   
##     lonely_XS        -0.132    0.046   -2.880    0.004   
##     affected_cn_XS   -0.213    0.289   -0.738    0.460   
##   REJ ~                                                  
##     health_anx_XS    -0.132    0.050   -2.613    0.009   
##     worries_trt_XS   -0.106    0.053   -1.991    0.046   
##     reactnc_trt_XS    0.635    0.048   13.102    0.000   
##     lonely_XS         0.171    0.042    4.110    0.000   
##     affected_cn_XS    1.130    0.262    4.311    0.000   
##   VIRUS_WORRY ~                                          
##     health_anx_XS     0.301    0.048    6.262    0.000   
##     worries_trt_XS    0.166    0.051    3.281    0.001   
##     reactnc_trt_XS   -0.336    0.046   -7.284    0.000   
##     lonely_XS        -0.061    0.040   -1.540    0.124   
##     affected_cn_XS   -0.389    0.249   -1.562    0.118   
##   WORRY_EMPL ~                                           
##     health_anx_XS    -0.041    0.049   -0.836    0.403   
##     worries_trt_XS    0.256    0.052    4.949    0.000   
##     reactnc_trt_XS    0.009    0.047    0.197    0.844   
##     lonely_XS         0.128    0.041    3.146    0.002   
##     affected_cn_XS    4.124    0.256   16.133    0.000   
##   WORRY_REC ~                                            
##     health_anx_XS    -0.032    0.039   -0.816    0.415   
##     worries_trt_XS    0.157    0.041    3.782    0.000   
##     reactnc_trt_XS    0.025    0.038    0.656    0.512   
##     lonely_XS         0.027    0.032    0.832    0.406   
##     affected_cn_XS    0.172    0.204    0.842    0.400   
## 
## INTERCEPTS:
##                    Estimate  Std.Err  z-value  P(>|z|)   
##     ACC               4.941    0.223   22.162    0.000   
##     acc01             0.000                              
##     acc02             0.736    0.183    4.027    0.000   
##     acc03             0.117    0.194    0.606    0.545   
##     ewe01            -1.170    0.225   -5.193    0.000   
##     ewe02             0.000                              
##     ewe03            -0.461    0.212   -2.173    0.030   
##     ewj01            -0.237    0.086   -2.750    0.006   
##     ewj02             0.000                              
##     ewj03            -0.215    0.085   -2.528    0.011   
##     REJ               1.037    0.202    5.124    0.000   
##     rej01             0.000                              
##     rej02             0.169    0.103    1.644    0.100   
##     VIRUS_WORRY       4.797    0.193   24.913    0.000   
##     WORRY_EMPL        1.120    0.197    5.676    0.000   
##     WORRY_REC         4.930    0.158   31.228    0.000
resid(fit.sem2, "standardized")
## $type
## [1] "standardized"
## 
## $cov
##                    aw01   ewj02  ewj01  ewj03  ewe02  ewe01  ewe03  acc01 
## aw01               -0.001                                                 
## ewj02               1.256  0.000                                          
## ewj01              -0.012 -1.250  0.000                                   
## ewj03              -1.075  0.555 -0.036  0.000                            
## ewe02               0.525 -0.576 -1.494  0.255  0.000                     
## ewe01              -2.242  0.536 -0.165  1.985  0.198  0.000              
## ewe03               1.475 -0.203 -1.902  0.160 -0.069 -0.127  0.000       
## acc01              -0.803  1.049  0.105  0.934  0.831 -0.028  0.261  0.000
## acc02               0.415  1.083 -0.974  0.760  0.220 -0.213  2.132  0.998
## acc03               0.044  0.928 -1.416 -0.587 -0.314 -1.826  0.588 -0.007
## rej01              -0.734 -0.585  0.902 -1.148 -1.248  0.042 -1.769  0.729
## rej02               0.825 -0.745  1.344  0.500  0.818  3.136  0.354  2.073
## health_anx_XS      -0.004 -0.277  1.112 -0.940  1.587 -0.217 -1.241  1.784
## worries_trait_XS    0.001  0.121  0.919 -1.292  0.496  0.394 -0.927  0.565
## reactance_trait_XS  0.008 -0.351  0.798 -0.692 -0.367  1.882 -1.826  0.990
## lonely_XS           0.004  0.898  0.144 -1.196 -0.608  0.618 -0.065  0.990
## affected_econ_XS    0.004 -0.045  1.311 -1.989 -0.059  0.918 -0.940  0.855
##                    acc02  acc03  rej01  rej02  hl__XS wr__XS rc__XS lnl_XS
## aw01                                                                      
## ewj02                                                                     
## ewj01                                                                     
## ewj03                                                                     
## ewe02                                                                     
## ewe01                                                                     
## ewe03                                                                     
## acc01                                                                     
## acc02               0.000                                                 
## acc03              -0.284  0.000                                          
## rej01              -0.988 -0.898  0.000                                   
## rej02               0.707  0.733  0.000  0.000                            
## health_anx_XS      -1.104 -0.161 -0.569  0.590  0.000                     
## worries_trait_XS   -0.426  0.026 -1.838  1.759  0.000  0.000              
## reactance_trait_XS -0.277 -0.190 -1.788  1.477  0.000  0.000  0.000       
## lonely_XS           1.354 -1.345 -0.010  0.009  0.000  0.000  0.000  0.000
## affected_econ_XS   -2.042  1.421  0.167 -0.156  0.000  0.000  0.000  0.000
##                    af__XS
## aw01                     
## ewj02                    
## ewj01                    
## ewj03                    
## ewe02                    
## ewe01                    
## ewe03                    
## acc01                    
## acc02                    
## acc03                    
## rej01                    
## rej02                    
## health_anx_XS            
## worries_trait_XS         
## reactance_trait_XS       
## lonely_XS                
## affected_econ_XS    0.000
## 
## $mean
##               aw01              ewj02              ewj01              ewj03 
##             -0.586              0.000              0.000              0.000 
##              ewe02              ewe01              ewe03              acc01 
##              0.000              0.000              0.000              0.000 
##              acc02              acc03              rej01              rej02 
##              0.000              0.000              0.000              0.000 
##      health_anx_XS   worries_trait_XS reactance_trait_XS          lonely_XS 
##              0.000              0.000              0.000              0.000 
##   affected_econ_XS 
##              0.000

This model cleanly fits from the perspective of the chisquare test as well as the Sargan tests. The test for rej02 is now non-significant (albeit not as ideal). We decide to keep it due to the multi-test implications and capitilization on chance