non-centered interaction terms (LMS and QML)
meanstructure_lms_qml.RmdNon-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 (1.0.14) ended normally after 86 iterations
#>
#> Estimator LMS
#> Optimization method EMA-NLMINB
#> Number of model parameters 54
#>
#> Number of observations 2000
#>
#> Loglikelihood and Information Criteria:
#> Loglikelihood -26326.02
#> Akaike (AIC) 52760.04
#> Bayesian (BIC) 53062.49
#>
#> Fit Measures for Baseline Model (H0):
#> Standard
#> Chi-square 66.27
#> Degrees of Freedom (Chi-square) 82
#> P-value (Chi-square) 0.897
#> RMSEA 0.000
#>
#> Loglikelihood -26393.22
#> Akaike (AIC) 52892.45
#> Bayesian (BIC) 53189.29
#>
#> Comparative Fit to H0 (LRT test):
#> Loglikelihood change 67.20
#> Difference test (D) 134.40
#> 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|)
#> ATT =~
#> att1 1.000
#> att2 0.878 0.012 71.635 0.000
#> att3 0.789 0.012 66.429 0.000
#> att4 0.695 0.011 61.040 0.000
#> att5 0.887 0.013 70.920 0.000
#> SN =~
#> sn1 1.000
#> sn2 0.888 0.017 52.697 0.000
#> PBC =~
#> pbc1 1.000
#> pbc2 0.912 0.013 69.394 0.000
#> pbc3 0.801 0.012 66.023 0.000
#> INT =~
#> int1 1.000
#> int2 0.913 0.015 59.154 0.000
#> int3 0.807 0.014 55.811 0.000
#> BEH =~
#> b1 1.000
#> b2 0.960 0.030 32.031 0.000
#>
#> Regressions:
#> Estimate Std.Error z.value P(>|z|)
#> INT ~
#> ATT 0.213 0.026 8.165 0.000
#> SN 0.176 0.028 6.388 0.000
#> PBC 0.217 0.030 7.348 0.000
#> BEH ~
#> PBC 0.025 0.029 0.859 0.390
#> INT -0.015 0.030 -0.511 0.610
#> INT:PBC 0.205 0.018 11.323 0.000
#>
#> Intercepts:
#> Estimate Std.Error z.value P(>|z|)
#> .pbc2 0.075 0.018 4.230 0.000
#> .pbc3 0.192 0.016 11.717 0.000
#> .att2 0.117 0.017 6.817 0.000
#> .att3 0.217 0.017 13.027 0.000
#> .att4 0.294 0.016 18.446 0.000
#> .att5 0.093 0.017 5.308 0.000
#> .sn2 0.117 0.021 5.605 0.000
#> .int2 0.087 0.020 4.384 0.000
#> .int3 0.187 0.019 9.996 0.000
#> .b2 0.058 0.035 1.646 0.100
#> ATT 1.014 0.024 42.006 0.000
#> SN 1.005 0.024 41.655 0.000
#> PBC 0.997 0.024 42.405 0.000
#> .INT 0.403 0.027 14.692 0.000
#> .BEH 0.785 0.030 26.219 0.000
#>
#> Covariances:
#> Estimate Std.Error z.value P(>|z|)
#> ATT ~~
#> SN 0.629 0.029 21.949 0.000
#> PBC ~~
#> ATT 0.677 0.029 23.718 0.000
#> SN 0.677 0.029 23.338 0.000
#>
#> Variances:
#> Estimate Std.Error z.value P(>|z|)
#> .pbc1 0.144 0.008 18.402 0.000
#> .pbc2 0.160 0.007 21.450 0.000
#> .pbc3 0.155 0.006 23.901 0.000
#> .att1 0.167 0.007 23.537 0.000
#> .att2 0.150 0.006 24.723 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 12.096 0.000
#> .sn2 0.157 0.012 13.269 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 10.173 0.000
#> .b2 0.135 0.016 8.195 0.000
#> ATT 0.998 0.037 27.148 0.000
#> SN 0.987 0.039 25.418 0.000
#> PBC 0.961 0.035 27.249 0.000
#> .INT 0.491 0.020 24.656 0.000
#> .BEH 0.455 0.023 20.213 0.000Comparing 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 (1.0.14) ended normally after 86 iterations
#>
#> Estimator LMS
#> Optimization method EMA-NLMINB
#> Number of model parameters 54
#>
#> Number of observations 2000
#>
#> Loglikelihood and Information Criteria:
#> Loglikelihood -26326.02
#> Akaike (AIC) 52760.04
#> Bayesian (BIC) 53062.49
#>
#> Fit Measures for Baseline Model (H0):
#> Standard
#> Chi-square 66.27
#> Degrees of Freedom (Chi-square) 82
#> P-value (Chi-square) 0.897
#> RMSEA 0.000
#>
#> Loglikelihood -26393.22
#> Akaike (AIC) 52892.45
#> Bayesian (BIC) 53189.29
#>
#> Comparative Fit to H0 (LRT test):
#> Loglikelihood change 67.20
#> Difference test (D) 134.40
#> 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
#> ATT =~
#> att1 1.000 0.925 1.000
#> att2 0.878 0.012 71.635 0.000 0.915 0.878
#> att3 0.789 0.012 66.429 0.000 0.892 0.789
#> att4 0.695 0.011 61.040 0.000 0.865 0.695
#> att5 0.887 0.013 70.920 0.000 0.912 0.887
#> SN =~
#> sn1 1.000 0.920 1.000
#> sn2 0.888 0.017 52.697 0.000 0.912 0.888
#> PBC =~
#> pbc1 1.000 0.933 1.000
#> pbc2 0.912 0.013 69.394 0.000 0.913 0.912
#> pbc3 0.801 0.012 66.023 0.000 0.894 0.801
#> INT =~
#> int1 1.000 0.912 1.000
#> int2 0.913 0.015 59.154 0.000 0.895 0.913
#> int3 0.807 0.014 55.811 0.000 0.866 0.807
#> BEH =~
#> b1 1.000 0.877 1.000
#> b2 0.960 0.030 32.031 0.000 0.899 0.960
#>
#> Regressions:
#> Estimate Std.Error z.value P(>|z|) Std.all Cnt.all
#> INT ~
#> ATT 0.213 0.026 8.165 0.000 0.242 0.213
#> SN 0.176 0.028 6.388 0.000 0.199 0.176
#> PBC 0.217 0.030 7.348 0.000 0.242 0.217
#> BEH ~
#> PBC 0.025 0.029 0.859 0.390 0.290 0.232
#> INT -0.015 0.030 -0.511 0.610 0.212 0.189
#> INT:PBC 0.205 0.018 11.323 0.000 0.225 0.205
#>
#> Intercepts:
#> Estimate Std.Error z.value P(>|z|) Std.all Cnt.all
#> .pbc2 0.075 0.018 4.230 0.000
#> .pbc3 0.192 0.016 11.717 0.000
#> .att2 0.117 0.017 6.817 0.000
#> .att3 0.217 0.017 13.027 0.000
#> .att4 0.294 0.016 18.446 0.000
#> .att5 0.093 0.017 5.308 0.000
#> .sn2 0.117 0.021 5.605 0.000
#> .int2 0.087 0.020 4.384 0.000
#> .int3 0.187 0.019 9.996 0.000
#> .b2 0.058 0.035 1.646 0.100
#> ATT 1.014 0.024 42.006 0.000
#> SN 1.005 0.024 41.655 0.000
#> PBC 0.997 0.024 42.405 0.000
#> .INT 0.403 0.027 14.692 0.000
#> .BEH 0.785 0.030 26.219 0.000
#>
#> Covariances:
#> Estimate Std.Error z.value P(>|z|) Std.all Cnt.all
#> ATT ~~
#> SN 0.629 0.029 21.949 0.000 0.634 0.629
#> PBC ~~
#> ATT 0.677 0.029 23.718 0.000 0.692 0.677
#> SN 0.677 0.029 23.338 0.000 0.695 0.677
#>
#> Variances:
#> Estimate Std.Error z.value P(>|z|) Std.all Cnt.all
#> .pbc1 0.144 0.008 18.402 0.000 0.130 0.144
#> .pbc2 0.160 0.007 21.450 0.000 0.166 0.160
#> .pbc3 0.155 0.006 23.901 0.000 0.201 0.155
#> .att1 0.167 0.007 23.537 0.000 0.144 0.167
#> .att2 0.150 0.006 24.723 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 12.096 0.000 0.153 0.178
#> .sn2 0.157 0.012 13.269 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 10.173 0.000 0.231 0.186
#> .b2 0.135 0.016 8.195 0.000 0.192 0.135
#> ATT 0.998 0.037 27.148 0.000 1.000 0.998
#> SN 0.987 0.039 25.418 0.000 1.000 0.987
#> PBC 0.961 0.035 27.249 0.000 1.000 0.961
#> .INT 0.491 0.020 24.656 0.000 0.634 0.491
#> .BEH 0.455 0.023 20.213 0.000 0.738 0.455It 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 group est std.error z.value p.value ci.lower
#> 1 ATT =~ att1 1 1.000 NA NA NA NA
#> 2 ATT =~ att2 1 0.878 0.012 71.635 0 0.854
#> 3 ATT =~ att3 1 0.789 0.012 66.429 0 0.765
#> 4 ATT =~ att4 1 0.695 0.011 61.040 0 0.672
#> 5 ATT =~ att5 1 0.887 0.013 70.920 0 0.862
#> 6 SN =~ sn1 1 1.000 NA NA NA NA
#> 7 SN =~ sn2 1 0.888 0.017 52.697 0 0.855
#> 8 PBC =~ pbc1 1 1.000 NA NA NA NA
#> 9 PBC =~ pbc2 1 0.912 0.013 69.394 0 0.886
#> 10 PBC =~ pbc3 1 0.801 0.012 66.023 0 0.777
#> 11 INT =~ int1 1 1.000 NA NA NA NA
#> 12 INT =~ int2 1 0.913 0.015 59.154 0 0.883
#> 13 INT =~ int3 1 0.807 0.014 55.811 0 0.779
#> 14 BEH =~ b1 1 1.000 NA NA NA NA
#> 15 BEH =~ b2 1 0.960 0.030 32.031 0 0.901
#> 16 INT ~ ATT 1 0.213 0.026 8.165 0 0.162
#> 17 INT ~ SN 1 0.176 0.028 6.388 0 0.122
#> 18 INT ~ PBC 1 0.217 0.030 7.348 0 0.159
#> 19 BEH ~ PBC 1 0.232 0.022 10.404 0 0.189
#> 20 BEH ~ INT 1 0.189 0.025 7.667 0 0.141
#> 21 BEH ~ INT:PBC 1 0.205 0.018 11.323 0 0.169
#> 22 pbc1 ~~ pbc1 1 0.144 0.008 18.402 0 0.129
#> 23 pbc2 ~~ pbc2 1 0.160 0.007 21.450 0 0.145
#> 24 pbc3 ~~ pbc3 1 0.155 0.006 23.901 0 0.142
#> 25 att1 ~~ att1 1 0.167 0.007 23.537 0 0.153
#> 26 att2 ~~ att2 1 0.150 0.006 24.723 0 0.138
#> 27 att3 ~~ att3 1 0.159 0.006 26.389 0 0.148
#> 28 att4 ~~ att4 1 0.162 0.006 27.655 0 0.151
#> 29 att5 ~~ att5 1 0.159 0.006 24.936 0 0.146
#> 30 sn1 ~~ sn1 1 0.178 0.015 12.096 0 0.149
#> 31 sn2 ~~ sn2 1 0.157 0.012 13.269 0 0.133
#> 32 int1 ~~ int1 1 0.157 0.009 18.111 0 0.140
#> 33 int2 ~~ int2 1 0.160 0.008 20.423 0 0.145
#> 34 int3 ~~ int3 1 0.168 0.007 23.560 0 0.154
#> 35 b1 ~~ b1 1 0.186 0.018 10.173 0 0.150
#> 36 b2 ~~ b2 1 0.135 0.016 8.195 0 0.103
#> 37 ATT ~~ ATT 1 0.998 0.037 27.148 0 0.926
#> 38 ATT ~~ SN 1 0.629 0.029 21.949 0 0.573
#> 39 SN ~~ SN 1 0.987 0.039 25.418 0 0.911
#> 40 PBC ~~ ATT 1 0.677 0.029 23.718 0 0.621
#> 41 PBC ~~ SN 1 0.677 0.029 23.338 0 0.620
#> 42 PBC ~~ PBC 1 0.961 0.035 27.249 0 0.892
#> 43 PBC ~~ INT:PBC 1 0.000 NA NA NA NA
#> 44 INT ~~ INT 1 0.491 0.020 24.656 0 0.452
#> 45 INT ~~ INT:PBC 1 0.000 NA NA NA NA
#> 46 BEH ~~ BEH 1 0.455 0.023 20.213 0 0.411
#> 47 INT:PBC ~~ INT:PBC 1 0.969 NA NA NA NA
#> ci.upper
#> 1 NA
#> 2 0.902
#> 3 0.812
#> 4 0.717
#> 5 0.911
#> 6 NA
#> 7 0.921
#> 8 NA
#> 9 0.938
#> 10 0.824
#> 11 NA
#> 12 0.943
#> 13 0.835
#> 14 NA
#> 15 1.019
#> 16 0.265
#> 17 0.231
#> 18 0.275
#> 19 0.276
#> 20 0.238
#> 21 0.240
#> 22 0.160
#> 23 0.174
#> 24 0.168
#> 25 0.181
#> 26 0.162
#> 27 0.171
#> 28 0.174
#> 29 0.171
#> 30 0.207
#> 31 0.180
#> 32 0.174
#> 33 0.176
#> 34 0.182
#> 35 0.222
#> 36 0.167
#> 37 1.070
#> 38 0.685
#> 39 1.063
#> 40 0.733
#> 41 0.734
#> 42 1.031
#> 43 NA
#> 44 0.530
#> 45 NA
#> 46 0.499
#> 47 NA