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)
#> Estimating baseline model (H0)
#>
#> 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
#> Akaike (AIC) 52892.45
#> Bayesian (BIC) 53189.29
#> Chi-square 2625.24
#> Degrees of Freedom (Chi-square) 79
#> P-value (Chi-square) 0.000
#> RMSEA 0.127
#>
#> 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)
#> Estimating baseline model (H0)
#>
#> 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
#> Akaike (AIC) 52892.45
#> Bayesian (BIC) 53189.29
#> Chi-square 2625.24
#> Degrees of Freedom (Chi-square) 79
#> P-value (Chi-square) 0.000
#> RMSEA 0.127
#>
#> 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|)
#> PBC =~
#> pbc1 0.930 0.004 224.45 0.000
#> pbc2 0.913 0.005 188.33 0.000
#> pbc3 0.893 0.006 160.22 0.000
#> ATT =~
#> att1 0.925 0.004 231.88 0.000
#> att2 0.914 0.004 212.56 0.000
#> att3 0.892 0.005 173.38 0.000
#> att4 0.864 0.006 139.00 0.000
#> att5 0.912 0.004 208.13 0.000
#> SN =~
#> sn1 0.920 0.007 127.42 0.000
#> sn2 0.912 0.007 124.84 0.000
#> INT =~
#> int1 0.912 0.006 162.08 0.000
#> int2 0.895 0.006 147.20 0.000
#> int3 0.866 0.007 125.35 0.000
#> BEH =~
#> b1 0.877 0.013 68.44 0.000
#> b2 0.898 0.014 64.50 0.000
#>
#> Regressions:
#> Estimate Std.Error z.value P(>|z|)
#> INT ~
#> PBC 0.241 0.032 7.42 0.000
#> ATT 0.242 0.029 8.32 0.000
#> SN 0.199 0.031 6.48 0.000
#> BEH ~
#> PBC 0.289 0.033 8.65 0.000
#> INT 0.210 0.032 6.54 0.000
#> PBC:INT 0.224 0.022 9.97 0.000
#>
#> Covariances:
#> Estimate Std.Error z.value P(>|z|)
#> PBC ~~
#> ATT 0.690 0.013 52.87 0.000
#> SN 0.693 0.014 50.95 0.000
#> ATT ~~
#> SN 0.632 0.015 42.07 0.000
#>
#> Variances:
#> Estimate Std.Error z.value P(>|z|)
#> pbc1 0.135 0.008 17.52 0.000
#> pbc2 0.167 0.009 18.71 0.000
#> pbc3 0.202 0.010 20.28 0.000
#> att1 0.144 0.007 19.83 0.000
#> att2 0.164 0.008 20.70 0.000
#> att3 0.205 0.009 22.14 0.000
#> att4 0.253 0.011 23.61 0.000
#> att5 0.169 0.008 21.01 0.000
#> sn1 0.154 0.013 11.56 0.000
#> sn2 0.168 0.013 12.52 0.000
#> int1 0.169 0.010 16.39 0.000
#> int2 0.199 0.011 18.10 0.000
#> int3 0.250 0.012 20.86 0.000
#> b1 0.231 0.022 10.30 0.000
#> b2 0.194 0.025 7.67 0.000
#> PBC 1.000
#> ATT 1.000
#> SN 1.000
#> INT 0.636 0.019 34.09 0.000
#> BEH 0.741 0.024 31.04 0.000