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library(modsem)
#> This is modsem (1.0.11). Please report any bugs!

Non-centered interaction terms

Using the LMS and QML approaches it is possible to estimate interaction terms where the means of the latent variables are not centered (i.e., they have non-zero means).

Here we can see an example using the TPB dataset:

tpb <- ' 
# Outer Model (Based on Hagger et al., 2007)
  ATT =~ att1 + att2 + att3 + att4 + att5
  SN =~ sn1 + sn2
  PBC =~ pbc1 + pbc2 + pbc3
  INT =~ int1 + int2 + int3
  BEH =~ b1 + b2

# Inner Model (Based on Steinmetz et al., 2011)
  INT ~ ATT + SN + PBC
  BEH ~ INT + PBC 
  BEH ~ INT:PBC

# Adding Latent Intercepts
  INT ~ 1
  BEH ~ 1
  PBC ~ 1
  SN  ~ 1
  ATT ~ 1
'

est <- modsem(tpb, TPB, method = "lms", nodes = 32)
summary(est)
#> 
#> modsem (version 1.0.11):
#> 
#>   Estimator                                         LMS
#>   Optimization method                        EMA-NLMINB
#>   Number of observations                           2000
#>   Number of iterations                               39
#>   Loglikelihood                               -23423.26
#>   Akaike (AIC)                                 46954.53
#>   Bayesian (BIC)                               47256.98
#>  
#> Numerical Integration:
#>   Points of integration (per dim)                    32
#>   Dimensions                                          1
#>   Total points of integration                        32
#>  
#> Fit Measures for Baseline Model (H0):
#>   Loglikelihood                               -26393.22
#>   Akaike (AIC)                                 52892.45
#>   Bayesian (BIC)                               53189.29
#>   Chi-square                                      66.27
#>   Degrees of Freedom (Chi-square)                    82
#>   P-value (Chi-square)                            0.897
#>   RMSEA                                           0.000
#>  
#> Comparative Fit to H0 (LRT test):
#>   Loglikelihood change                          2969.96
#>   Difference test (D)                           5939.92
#>   Degrees of freedom (D)                              1
#>   P-value (D)                                     0.000
#>  
#> R-Squared Interaction Model (H1):
#>   INT                                             0.366
#>   BEH                                             0.262
#> R-Squared Baseline Model (H0):
#>   INT                                             0.367
#>   BEH                                             0.210
#> R-Squared Change (H1 - H0):
#>   INT                                            -0.001
#>   BEH                                             0.052
#> 
#> Parameter Estimates:
#>   Coefficients                           unstandardized
#>   Information                                  observed
#>   Standard errors                              standard
#>  
#> Latent Variables:
#>                  Estimate  Std.Error  z.value  P(>|z|)
#>   PBC =~        
#>     pbc1            1.000                             
#>     pbc2            0.912      0.013   69.536    0.000
#>     pbc3            0.801      0.012   66.151    0.000
#>   ATT =~        
#>     att1            1.000                             
#>     att2            0.878      0.012   71.639    0.000
#>     att3            0.789      0.012   66.432    0.000
#>     att4            0.695      0.011   61.043    0.000
#>     att5            0.887      0.013   70.924    0.000
#>   SN =~         
#>     sn1             1.000                             
#>     sn2             0.888      0.017   52.698    0.000
#>   INT =~        
#>     int1            1.000                             
#>     int2            0.913      0.015   59.156    0.000
#>     int3            0.807      0.014   55.812    0.000
#>   BEH =~        
#>     b1              1.000                             
#>     b2              0.960      0.030   32.029    0.000
#> 
#> Regressions:
#>                  Estimate  Std.Error  z.value  P(>|z|)
#>   INT ~         
#>     PBC             0.217      0.030    7.350    0.000
#>     ATT             0.213      0.026    8.164    0.000
#>     SN              0.176      0.028    6.386    0.000
#>   BEH ~         
#>     PBC             0.025      0.029    0.869    0.385
#>     INT            -0.015      0.030   -0.499    0.618
#>     PBC:INT         0.205      0.018   11.316    0.000
#> 
#> Intercepts:
#>                  Estimate  Std.Error  z.value  P(>|z|)
#>    .pbc2            0.075      0.018    4.238    0.000
#>    .pbc3            0.192      0.016   11.731    0.000
#>    .att2            0.117      0.017    6.826    0.000
#>    .att3            0.217      0.017   13.036    0.000
#>    .att4            0.294      0.016   18.456    0.000
#>    .att5            0.093      0.017    5.317    0.000
#>    .sn2             0.117      0.021    5.612    0.000
#>    .int2            0.087      0.020    4.385    0.000
#>    .int3            0.187      0.019    9.997    0.000
#>    .b2              0.058      0.035    1.641    0.101
#>    .INT             0.403      0.027   14.696    0.000
#>    .BEH             0.784      0.030   26.211    0.000
#>     PBC             0.996      0.024   42.361    0.000
#>     ATT             1.013      0.024   41.961    0.000
#>     SN              1.004      0.024   41.612    0.000
#> 
#> Covariances:
#>                  Estimate  Std.Error  z.value  P(>|z|)
#>   PBC ~~        
#>     ATT             0.678      0.029   23.740    0.000
#>     SN              0.677      0.029   23.359    0.000
#>   ATT ~~        
#>     SN              0.629      0.029   21.948    0.000
#> 
#> Variances:
#>                  Estimate  Std.Error  z.value  P(>|z|)
#>    .pbc1            0.144      0.008   18.398    0.000
#>    .pbc2            0.160      0.007   21.447    0.000
#>    .pbc3            0.155      0.006   23.900    0.000
#>    .att1            0.167      0.007   23.536    0.000
#>    .att2            0.150      0.006   24.724    0.000
#>    .att3            0.159      0.006   26.389    0.000
#>    .att4            0.162      0.006   27.655    0.000
#>    .att5            0.159      0.006   24.936    0.000
#>    .sn1             0.178      0.015   12.094    0.000
#>    .sn2             0.157      0.012   13.270    0.000
#>    .int1            0.157      0.009   18.111    0.000
#>    .int2            0.160      0.008   20.423    0.000
#>    .int3            0.168      0.007   23.560    0.000
#>    .b1              0.186      0.018   10.178    0.000
#>    .b2              0.135      0.016    8.190    0.000
#>     PBC             0.962      0.035   27.315    0.000
#>     ATT             0.998      0.037   27.147    0.000
#>     SN              0.987      0.039   25.418    0.000
#>    .INT             0.491      0.020   24.655    0.000
#>    .BEH             0.455      0.023   20.212    0.000

Comparing this to the estimates we get when PBC and INT have zero means, we see that the coefficients BEH~PBC and BEH~INT are drastically changed. This is not a bug, and is a function of the interaction effect rescaling the coefficients, when not centered at zero. When using the standardized_estimates function, or summary(est, standardized = TRUE) the interaction effect is centered, and we can see that the coefficients BEH~PBC and BEH~INT are rescaled once again.

summary(est, standardized = TRUE, centered = TRUE)
#> 
#> modsem (version 1.0.11):
#> 
#>   Estimator                                         LMS
#>   Optimization method                        EMA-NLMINB
#>   Number of observations                           2000
#>   Number of iterations                               39
#>   Loglikelihood                               -23423.26
#>   Akaike (AIC)                                 46954.53
#>   Bayesian (BIC)                               47256.98
#>  
#> Numerical Integration:
#>   Points of integration (per dim)                    32
#>   Dimensions                                          1
#>   Total points of integration                        32
#>  
#> Fit Measures for Baseline Model (H0):
#>   Loglikelihood                               -26393.22
#>   Akaike (AIC)                                 52892.45
#>   Bayesian (BIC)                               53189.29
#>   Chi-square                                      66.27
#>   Degrees of Freedom (Chi-square)                    82
#>   P-value (Chi-square)                            0.897
#>   RMSEA                                           0.000
#>  
#> Comparative Fit to H0 (LRT test):
#>   Loglikelihood change                          2969.96
#>   Difference test (D)                           5939.92
#>   Degrees of freedom (D)                              1
#>   P-value (D)                                     0.000
#>  
#> R-Squared Interaction Model (H1):
#>   INT                                             0.366
#>   BEH                                             0.262
#> R-Squared Baseline Model (H0):
#>   INT                                             0.367
#>   BEH                                             0.210
#> R-Squared Change (H1 - H0):
#>   INT                                            -0.001
#>   BEH                                             0.052
#> 
#> Parameter Estimates:
#>   Coefficients                             standardized
#>   Information                                  observed
#>   Standard errors                              standard
#>  
#> Latent Variables:
#>                  Estimate  Std.Error  z.value  P(>|z|)  Std.all  Cnt.all
#>   PBC =~        
#>     pbc1            1.000                                 0.933    1.000
#>     pbc2            0.912      0.013   69.536    0.000    0.913    0.912
#>     pbc3            0.801      0.012   66.151    0.000    0.894    0.801
#>   ATT =~        
#>     att1            1.000                                 0.925    1.000
#>     att2            0.878      0.012   71.639    0.000    0.915    0.878
#>     att3            0.789      0.012   66.432    0.000    0.892    0.789
#>     att4            0.695      0.011   61.043    0.000    0.865    0.695
#>     att5            0.887      0.013   70.924    0.000    0.912    0.887
#>   SN =~         
#>     sn1             1.000                                 0.921    1.000
#>     sn2             0.888      0.017   52.698    0.000    0.912    0.888
#>   INT =~        
#>     int1            1.000                                 0.912    1.000
#>     int2            0.913      0.015   59.156    0.000    0.895    0.913
#>     int3            0.807      0.014   55.812    0.000    0.866    0.807
#>   BEH =~        
#>     b1              1.000                                 0.877    1.000
#>     b2              0.960      0.030   32.029    0.000    0.899    0.960
#> 
#> Regressions:
#>                  Estimate  Std.Error  z.value  P(>|z|)  Std.all  Cnt.all
#>   INT ~         
#>     PBC             0.217      0.030    7.350    0.000    0.242    0.217
#>     ATT             0.213      0.026    8.164    0.000    0.242    0.213
#>     SN              0.176      0.028    6.386    0.000    0.199    0.176
#>   BEH ~         
#>     PBC             0.025      0.029    0.869    0.385    0.290    0.232
#>     INT            -0.015      0.030   -0.499    0.618    0.212    0.189
#>     PBC:INT         0.205      0.018   11.316    0.000    0.225    0.205
#> 
#> Intercepts:
#>                  Estimate  Std.Error  z.value  P(>|z|)  Std.all  Cnt.all
#>    .pbc2            0.075      0.018    4.238    0.000                  
#>    .pbc3            0.192      0.016   11.731    0.000                  
#>    .att2            0.117      0.017    6.826    0.000                  
#>    .att3            0.217      0.017   13.036    0.000                  
#>    .att4            0.294      0.016   18.456    0.000                  
#>    .att5            0.093      0.017    5.317    0.000                  
#>    .sn2             0.117      0.021    5.612    0.000                  
#>    .int2            0.087      0.020    4.385    0.000                  
#>    .int3            0.187      0.019    9.997    0.000                  
#>    .b2              0.058      0.035    1.641    0.101                  
#>    .INT             0.403      0.027   14.696    0.000                  
#>    .BEH             0.784      0.030   26.211    0.000                  
#>     PBC             0.996      0.024   42.361    0.000                  
#>     ATT             1.013      0.024   41.961    0.000                  
#>     SN              1.004      0.024   41.612    0.000                  
#> 
#> Covariances:
#>                  Estimate  Std.Error  z.value  P(>|z|)  Std.all  Cnt.all
#>   PBC ~~        
#>     ATT             0.678      0.029   23.740    0.000    0.692    0.678
#>     SN              0.677      0.029   23.359    0.000    0.695    0.677
#>   ATT ~~        
#>     SN              0.629      0.029   21.948    0.000    0.634    0.629
#> 
#> Variances:
#>                  Estimate  Std.Error  z.value  P(>|z|)  Std.all  Cnt.all
#>    .pbc1            0.144      0.008   18.398    0.000    0.130    0.144
#>    .pbc2            0.160      0.007   21.447    0.000    0.166    0.160
#>    .pbc3            0.155      0.006   23.900    0.000    0.201    0.155
#>    .att1            0.167      0.007   23.536    0.000    0.144    0.167
#>    .att2            0.150      0.006   24.724    0.000    0.164    0.150
#>    .att3            0.159      0.006   26.389    0.000    0.204    0.159
#>    .att4            0.162      0.006   27.655    0.000    0.252    0.162
#>    .att5            0.159      0.006   24.936    0.000    0.168    0.159
#>    .sn1             0.178      0.015   12.094    0.000    0.153    0.178
#>    .sn2             0.157      0.012   13.270    0.000    0.167    0.157
#>    .int1            0.157      0.009   18.111    0.000    0.168    0.157
#>    .int2            0.160      0.008   20.423    0.000    0.199    0.160
#>    .int3            0.168      0.007   23.560    0.000    0.249    0.168
#>    .b1              0.186      0.018   10.178    0.000    0.232    0.186
#>    .b2              0.135      0.016    8.190    0.000    0.192    0.135
#>     PBC             0.962      0.035   27.315    0.000    1.000    0.962
#>     ATT             0.998      0.037   27.147    0.000    1.000    0.998
#>     SN              0.987      0.039   25.418    0.000    1.000    0.987
#>    .INT             0.491      0.020   24.655    0.000    0.634    0.491
#>    .BEH             0.455      0.023   20.212    0.000    0.738    0.455

It is also possible to get the centered solution using the centered_estimates() function. Note, that centered_estimates() removes the mean structure of the model all together.

centered_estimates(est)
#>        lhs op     rhs label   est std.error z.value p.value ci.lower ci.upper
#> 1      PBC =~    pbc1       1.000        NA      NA      NA       NA       NA
#> 2      PBC =~    pbc2       0.912     0.013  69.536       0    0.886    0.938
#> 3      PBC =~    pbc3       0.801     0.012  66.151       0    0.777    0.824
#> 4      ATT =~    att1       1.000        NA      NA      NA       NA       NA
#> 5      ATT =~    att2       0.878     0.012  71.639       0    0.854    0.902
#> 6      ATT =~    att3       0.789     0.012  66.432       0    0.765    0.812
#> 7      ATT =~    att4       0.695     0.011  61.043       0    0.672    0.717
#> 8      ATT =~    att5       0.887     0.013  70.924       0    0.862    0.911
#> 9       SN =~     sn1       1.000        NA      NA      NA       NA       NA
#> 10      SN =~     sn2       0.888     0.017  52.698       0    0.855    0.921
#> 11     INT =~    int1       1.000        NA      NA      NA       NA       NA
#> 12     INT =~    int2       0.913     0.015  59.156       0    0.883    0.943
#> 13     INT =~    int3       0.807     0.014  55.812       0    0.779    0.835
#> 14     BEH =~      b1       1.000        NA      NA      NA       NA       NA
#> 15     BEH =~      b2       0.960     0.030  32.029       0    0.902    1.019
#> 16     INT  ~     PBC       0.217     0.030   7.350       0    0.159    0.275
#> 17     INT  ~     ATT       0.213     0.026   8.164       0    0.162    0.265
#> 18     INT  ~      SN       0.176     0.028   6.386       0    0.122    0.231
#> 19     BEH  ~     PBC       0.232     0.022  10.404       0    0.189    0.276
#> 20     BEH  ~     INT       0.189     0.025   7.666       0    0.141    0.237
#> 21     BEH  ~ PBC:INT       0.205     0.018  11.316       0    0.169    0.240
#> 22    pbc1 ~~    pbc1       0.144     0.008  18.398       0    0.129    0.160
#> 23    pbc2 ~~    pbc2       0.160     0.007  21.447       0    0.145    0.174
#> 24    pbc3 ~~    pbc3       0.155     0.006  23.900       0    0.142    0.168
#> 25    att1 ~~    att1       0.167     0.007  23.536       0    0.153    0.181
#> 26    att2 ~~    att2       0.150     0.006  24.724       0    0.138    0.162
#> 27    att3 ~~    att3       0.159     0.006  26.389       0    0.148    0.171
#> 28    att4 ~~    att4       0.162     0.006  27.655       0    0.151    0.174
#> 29    att5 ~~    att5       0.159     0.006  24.936       0    0.146    0.171
#> 30     sn1 ~~     sn1       0.178     0.015  12.094       0    0.149    0.207
#> 31     sn2 ~~     sn2       0.157     0.012  13.270       0    0.133    0.180
#> 32    int1 ~~    int1       0.157     0.009  18.111       0    0.140    0.174
#> 33    int2 ~~    int2       0.160     0.008  20.423       0    0.145    0.176
#> 34    int3 ~~    int3       0.168     0.007  23.560       0    0.154    0.182
#> 35      b1 ~~      b1       0.186     0.018  10.178       0    0.150    0.222
#> 36      b2 ~~      b2       0.135     0.016   8.190       0    0.103    0.167
#> 37     PBC ~~     PBC       0.962     0.035  27.315       0    0.893    1.031
#> 38     PBC ~~     ATT       0.678     0.029  23.740       0    0.622    0.733
#> 39     PBC ~~      SN       0.677     0.029  23.359       0    0.620    0.734
#> 40     ATT ~~     ATT       0.998     0.037  27.147       0    0.926    1.070
#> 41     ATT ~~      SN       0.629     0.029  21.948       0    0.573    0.685
#> 42      SN ~~      SN       0.987     0.039  25.418       0    0.911    1.063
#> 43     INT ~~     INT       0.491     0.020  24.655       0    0.452    0.530
#> 44     BEH ~~     BEH       0.455     0.023  20.212       0    0.411    0.499
#> 45 PBC:INT ~~ PBC:INT       0.969        NA      NA      NA       NA       NA
#> 46     INT ~~ PBC:INT       0.000        NA      NA      NA       NA       NA
#> 47     PBC ~~ PBC:INT       0.000        NA      NA      NA       NA       NA