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 (version 1.0.11):
#>
#> Estimator LMS
#> Optimization method EMA-NLMINB
#> Number of observations 2000
#> Number of iterations 36
#> Loglikelihood -23439.04
#> Akaike (AIC) 46986.09
#> Bayesian (BIC) 47288.54
#>
#> 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 2954.18
#> Difference test (D) 5908.36
#> Degrees of freedom (D) 1
#> P-value (D) 0.000
#>
#> R-Squared Interaction Model (H1):
#> INT 0.364
#> BEH 0.259
#> R-Squared Baseline Model (H0):
#> INT 0.367
#> BEH 0.210
#> R-Squared Change (H1 - H0):
#> INT -0.003
#> BEH 0.048
#>
#> Parameter Estimates:
#> Coefficients unstandardized
#> Information observed
#> Standard errors standard
#>
#> Latent Variables:
#> Estimate Std.Error z.value P(>|z|)
#> PBC =~
#> pbc1 1.000
#> pbc2 0.914 0.013 69.485 0.000
#> pbc3 0.802 0.012 66.040 0.000
#> ATT =~
#> att1 1.000
#> att2 0.878 0.012 71.637 0.000
#> att3 0.789 0.012 66.436 0.000
#> att4 0.695 0.011 61.044 0.000
#> att5 0.887 0.013 70.923 0.000
#> SN =~
#> sn1 1.000
#> sn2 0.889 0.017 52.718 0.000
#> INT =~
#> int1 1.000
#> int2 0.913 0.015 59.154 0.000
#> int3 0.807 0.014 55.810 0.000
#> BEH =~
#> b1 1.000
#> b2 0.959 0.030 32.035 0.000
#>
#> Regressions:
#> Estimate Std.Error z.value P(>|z|)
#> INT ~
#> PBC 0.218 0.030 7.335 0.000
#> ATT 0.214 0.026 8.175 0.000
#> SN 0.176 0.028 6.384 0.000
#> BEH ~
#> PBC 0.026 0.029 0.889 0.374
#> INT -0.015 0.030 -0.511 0.609
#> PBC:INT 0.205 0.018 11.308 0.000
#>
#> Intercepts:
#> Estimate Std.Error z.value P(>|z|)
#> .pbc2 0.072 0.018 4.094 0.000
#> .pbc3 0.191 0.016 11.582 0.000
#> .att2 0.117 0.017 6.828 0.000
#> .att3 0.217 0.017 13.036 0.000
#> .att4 0.294 0.016 18.457 0.000
#> .att5 0.093 0.017 5.318 0.000
#> .sn2 0.117 0.021 5.596 0.000
#> .int2 0.087 0.020 4.383 0.000
#> .int3 0.187 0.019 9.995 0.000
#> .b2 0.059 0.035 1.678 0.093
#> .INT 0.403 0.027 14.675 0.000
#> .BEH 0.784 0.030 26.169 0.000
#> PBC 0.988 0.024 41.100 0.000
#> ATT 1.007 0.024 41.288 0.000
#> SN 0.999 0.024 40.944 0.000
#>
#> Covariances:
#> Estimate Std.Error z.value P(>|z|)
#> PBC ~~
#> ATT 0.669 0.029 23.433 0.000
#> SN 0.669 0.029 23.109 0.000
#> ATT ~~
#> SN 0.624 0.029 21.797 0.000
#>
#> Variances:
#> Estimate Std.Error z.value P(>|z|)
#> .pbc1 0.148 0.008 19.186 0.000
#> .pbc2 0.159 0.007 21.257 0.000
#> .pbc3 0.155 0.006 23.848 0.000
#> .att1 0.167 0.007 23.535 0.000
#> .att2 0.150 0.006 24.724 0.000
#> .att3 0.159 0.006 26.387 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 12.126 0.000
#> .sn2 0.156 0.012 13.252 0.000
#> .int1 0.157 0.009 18.112 0.000
#> .int2 0.160 0.008 20.423 0.000
#> .int3 0.168 0.007 23.559 0.000
#> .b1 0.185 0.018 10.132 0.000
#> .b2 0.136 0.016 8.243 0.000
#> PBC 0.949 0.035 27.057 0.000
#> ATT 0.993 0.037 27.081 0.000
#> SN 0.982 0.039 25.364 0.000
#> .INT 0.491 0.020 24.653 0.000
#> .BEH 0.456 0.023 20.212 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 (version 1.0.11):
#>
#> Estimator LMS
#> Optimization method EMA-NLMINB
#> Number of observations 2000
#> Number of iterations 36
#> Loglikelihood -23439.04
#> Akaike (AIC) 46986.09
#> Bayesian (BIC) 47288.54
#>
#> 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 2954.18
#> Difference test (D) 5908.36
#> Degrees of freedom (D) 1
#> P-value (D) 0.000
#>
#> R-Squared Interaction Model (H1):
#> INT 0.364
#> BEH 0.259
#> R-Squared Baseline Model (H0):
#> INT 0.367
#> BEH 0.210
#> R-Squared Change (H1 - H0):
#> INT -0.003
#> BEH 0.048
#>
#> 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 0.930 1.000
#> pbc2 0.914 0.013 69.485 0.000 0.913 0.914
#> pbc3 0.802 0.012 66.040 0.000 0.893 0.802
#> ATT =~
#> att1 1.000 0.925 1.000
#> att2 0.878 0.012 71.637 0.000 0.914 0.878
#> att3 0.789 0.012 66.436 0.000 0.892 0.789
#> att4 0.695 0.011 61.044 0.000 0.864 0.695
#> att5 0.887 0.013 70.923 0.000 0.912 0.887
#> SN =~
#> sn1 1.000 0.920 1.000
#> sn2 0.889 0.017 52.718 0.000 0.912 0.889
#> 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 55.810 0.000 0.866 0.807
#> BEH =~
#> b1 1.000 0.877 1.000
#> b2 0.959 0.030 32.035 0.000 0.898 0.959
#>
#> Regressions:
#> Estimate Std.Error z.value P(>|z|) Std.all Cnt.all
#> INT ~
#> PBC 0.218 0.030 7.335 0.000 0.241 0.218
#> ATT 0.214 0.026 8.175 0.000 0.242 0.214
#> SN 0.176 0.028 6.384 0.000 0.199 0.176
#> BEH ~
#> PBC 0.026 0.029 0.889 0.374 0.289 0.232
#> INT -0.015 0.030 -0.511 0.609 0.210 0.187
#> PBC:INT 0.205 0.018 11.308 0.000 0.224 0.205
#>
#> Intercepts:
#> Estimate Std.Error z.value P(>|z|) Std.all Cnt.all
#> .pbc2 0.072 0.018 4.094 0.000
#> .pbc3 0.191 0.016 11.582 0.000
#> .att2 0.117 0.017 6.828 0.000
#> .att3 0.217 0.017 13.036 0.000
#> .att4 0.294 0.016 18.457 0.000
#> .att5 0.093 0.017 5.318 0.000
#> .sn2 0.117 0.021 5.596 0.000
#> .int2 0.087 0.020 4.383 0.000
#> .int3 0.187 0.019 9.995 0.000
#> .b2 0.059 0.035 1.678 0.093
#> .INT 0.403 0.027 14.675 0.000
#> .BEH 0.784 0.030 26.169 0.000
#> PBC 0.988 0.024 41.100 0.000
#> ATT 1.007 0.024 41.288 0.000
#> SN 0.999 0.024 40.944 0.000
#>
#> Covariances:
#> Estimate Std.Error z.value P(>|z|) Std.all Cnt.all
#> PBC ~~
#> ATT 0.669 0.029 23.433 0.000 0.690 0.669
#> SN 0.669 0.029 23.109 0.000 0.693 0.669
#> ATT ~~
#> SN 0.624 0.029 21.797 0.000 0.632 0.624
#>
#> Variances:
#> Estimate Std.Error z.value P(>|z|) Std.all Cnt.all
#> .pbc1 0.148 0.008 19.186 0.000 0.135 0.148
#> .pbc2 0.159 0.007 21.257 0.000 0.167 0.159
#> .pbc3 0.155 0.006 23.848 0.000 0.202 0.155
#> .att1 0.167 0.007 23.535 0.000 0.144 0.167
#> .att2 0.150 0.006 24.724 0.000 0.164 0.150
#> .att3 0.159 0.006 26.387 0.000 0.205 0.159
#> .att4 0.162 0.006 27.655 0.000 0.253 0.162
#> .att5 0.159 0.006 24.936 0.000 0.169 0.159
#> .sn1 0.178 0.015 12.126 0.000 0.154 0.178
#> .sn2 0.156 0.012 13.252 0.000 0.168 0.156
#> .int1 0.157 0.009 18.112 0.000 0.169 0.157
#> .int2 0.160 0.008 20.423 0.000 0.199 0.160
#> .int3 0.168 0.007 23.559 0.000 0.250 0.168
#> .b1 0.185 0.018 10.132 0.000 0.231 0.185
#> .b2 0.136 0.016 8.243 0.000 0.194 0.136
#> PBC 0.949 0.035 27.057 0.000 1.000 0.949
#> ATT 0.993 0.037 27.081 0.000 1.000 0.993
#> SN 0.982 0.039 25.364 0.000 1.000 0.982
#> .INT 0.491 0.020 24.653 0.000 0.636 0.491
#> .BEH 0.456 0.023 20.212 0.000 0.741 0.456It 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 0.914 0.013 69.485 0 0.889 0.940
#> 3 PBC =~ pbc3 0.802 0.012 66.040 0 0.778 0.826
#> 4 ATT =~ att1 1.000 NA NA NA NA NA
#> 5 ATT =~ att2 0.878 0.012 71.637 0 0.854 0.902
#> 6 ATT =~ att3 0.789 0.012 66.436 0 0.765 0.812
#> 7 ATT =~ att4 0.695 0.011 61.044 0 0.672 0.717
#> 8 ATT =~ att5 0.887 0.013 70.923 0 0.862 0.911
#> 9 SN =~ sn1 1.000 NA NA NA NA NA
#> 10 SN =~ sn2 0.889 0.017 52.718 0 0.855 0.922
#> 11 INT =~ int1 1.000 NA NA NA NA NA
#> 12 INT =~ int2 0.913 0.015 59.154 0 0.883 0.943
#> 13 INT =~ int3 0.807 0.014 55.810 0 0.779 0.835
#> 14 BEH =~ b1 1.000 NA NA NA NA NA
#> 15 BEH =~ b2 0.959 0.030 32.035 0 0.901 1.018
#> 16 INT ~ PBC 0.218 0.030 7.335 0 0.159 0.276
#> 17 INT ~ ATT 0.214 0.026 8.175 0 0.162 0.265
#> 18 INT ~ SN 0.176 0.028 6.384 0 0.122 0.231
#> 19 BEH ~ PBC 0.232 0.022 10.375 0 0.188 0.276
#> 20 BEH ~ INT 0.187 0.025 7.576 0 0.139 0.236
#> 21 BEH ~ PBC:INT 0.205 0.018 11.308 0 0.169 0.240
#> 22 pbc1 ~~ pbc1 0.148 0.008 19.186 0 0.133 0.163
#> 23 pbc2 ~~ pbc2 0.159 0.007 21.257 0 0.144 0.173
#> 24 pbc3 ~~ pbc3 0.155 0.006 23.848 0 0.142 0.167
#> 25 att1 ~~ att1 0.167 0.007 23.535 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 26.387 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 12.126 0 0.149 0.207
#> 31 sn2 ~~ sn2 0.156 0.012 13.252 0 0.133 0.179
#> 32 int1 ~~ int1 0.157 0.009 18.112 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 23.559 0 0.154 0.182
#> 35 b1 ~~ b1 0.185 0.018 10.132 0 0.149 0.221
#> 36 b2 ~~ b2 0.136 0.016 8.243 0 0.103 0.168
#> 37 PBC ~~ PBC 0.949 0.035 27.057 0 0.880 1.017
#> 38 PBC ~~ ATT 0.669 0.029 23.433 0 0.613 0.725
#> 39 PBC ~~ SN 0.669 0.029 23.109 0 0.612 0.726
#> 40 ATT ~~ ATT 0.993 0.037 27.081 0 0.921 1.065
#> 41 ATT ~~ SN 0.624 0.029 21.797 0 0.568 0.680
#> 42 SN ~~ SN 0.982 0.039 25.364 0 0.906 1.058
#> 43 INT ~~ INT 0.491 0.020 24.653 0 0.452 0.530
#> 44 BEH ~~ BEH 0.456 0.023 20.212 0 0.411 0.500
#> 45 PBC:INT ~~ PBC:INT 0.952 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