The Latent Moderated Structural Equations (LMS) and the Quasi Maximum Likelihood (QML) approach
Both the LMS- and QML approach works on most models, but interaction effects with endogenous can be a bit tricky to estimate (see the vignette. Both approaches (particularily the LMS approach) are quite computationally intensive, and are thus partly implemented in C++ (using Rcpp and RcppArmadillo). Additionally starting parameters are estimated using the double centering approach (and the means of the observed variables) are used to generate good starting parameters for faster convergence. If you want to see the progress of the estimation process you can use ´verbose = TRUE´.
A Simple Example
Here you can see an example of the LMS approach for a simple model. By default the summary function calculates fit measures compared to a null model (i.e., the same model without an interaction term).
library(modsem)
m1 <- '
# Outer Model
X =~ x1
X =~ x2 + x3
Z =~ z1 + z2 + z3
Y =~ y1 + y2 + y3
# Inner model
Y ~ X + Z
Y ~ X:Z
'
lms1 <- modsem(m1, oneInt, method = "lms")
summary(lms1, standardized = TRUE) # standardized estimates
#> Estimating null model
#> EM: Iteration = 1, LogLik = -17831.87, Change = -17831.875
#> EM: Iteration = 2, LogLik = -17831.87, Change = 0.000
#>
#> modsem (version 1.0.3):
#> Estimator LMS
#> Optimization method EM-NLMINB
#> Number of observations 2000
#> Number of iterations 92
#> Loglikelihood -14687.85
#> Akaike (AIC) 29437.71
#> Bayesian (BIC) 29611.34
#>
#> Numerical Integration:
#> Points of integration (per dim) 24
#> Dimensions 1
#> Total points of integration 24
#>
#> Fit Measures for H0:
#> Loglikelihood -17832
#> Akaike (AIC) 35723.75
#> Bayesian (BIC) 35891.78
#> Chi-square 17.52
#> Degrees of Freedom (Chi-square) 24
#> P-value (Chi-square) 0.826
#> RMSEA 0.000
#>
#> Comparative fit to H0 (no interaction effect)
#> Loglikelihood change 3144.02
#> Difference test (D) 6288.04
#> Degrees of freedom (D) 1
#> P-value (D) 0.000
#>
#> R-Squared:
#> Y 0.596
#> R-Squared Null-Model (H0):
#> Y 0.395
#> R-Squared Change:
#> Y 0.201
#>
#> Parameter Estimates:
#> Coefficients standardized
#> Information expected
#> Standard errors standard
#>
#> Latent Variables:
#> Estimate Std.Error z.value P(>|z|)
#> X =~
#> x1 0.926
#> x2 0.891 0.014 64.39 0.000
#> x3 0.912 0.013 67.69 0.000
#> Z =~
#> z1 0.927
#> z2 0.898 0.014 64.59 0.000
#> z3 0.913 0.013 67.87 0.000
#> Y =~
#> y1 0.969
#> y2 0.954 0.009 105.92 0.000
#> y3 0.961 0.009 111.95 0.000
#>
#> Regressions:
#> Estimate Std.Error z.value P(>|z|)
#> Y ~
#> X 0.427 0.020 21.79 0.000
#> Z 0.370 0.018 20.16 0.000
#> X:Z 0.454 0.017 26.28 0.000
#>
#> Covariances:
#> Estimate Std.Error z.value P(>|z|)
#> X ~~
#> Z 0.199 0.024 8.43 0.000
#>
#> Variances:
#> Estimate Std.Error z.value P(>|z|)
#> x1 0.142 0.007 19.27 0.000
#> x2 0.206 0.009 23.86 0.000
#> x3 0.169 0.008 21.31 0.000
#> z1 0.141 0.008 18.34 0.000
#> z2 0.193 0.009 22.39 0.000
#> z3 0.167 0.008 20.52 0.000
#> y1 0.061 0.003 17.93 0.000
#> y2 0.090 0.004 22.72 0.000
#> y3 0.077 0.004 20.69 0.000
#> X 1.000 0.016 61.06 0.000
#> Z 1.000 0.018 55.21 0.000
#> Y 0.404 0.015 26.54 0.000Here you can see the same example using the QML approach.
qml1 <- modsem(m1, oneInt, method = "qml")
summary(qml1)
#> Estimating null model
#> Starting M-step
#>
#> modsem (version 1.0.3):
#> Estimator QML
#> Optimization method NLMINB
#> Number of observations 2000
#> Number of iterations 111
#> Loglikelihood -17496.22
#> Akaike (AIC) 35054.43
#> Bayesian (BIC) 35228.06
#>
#> Fit Measures for H0:
#> Loglikelihood -17832
#> Akaike (AIC) 35723.75
#> Bayesian (BIC) 35891.78
#> Chi-square 17.52
#> Degrees of Freedom (Chi-square) 24
#> P-value (Chi-square) 0.826
#> RMSEA 0.000
#>
#> Comparative fit to H0 (no interaction effect)
#> Loglikelihood change 335.66
#> Difference test (D) 671.32
#> Degrees of freedom (D) 1
#> P-value (D) 0.000
#>
#> R-Squared:
#> Y 0.607
#> R-Squared Null-Model (H0):
#> Y 0.395
#> R-Squared Change:
#> Y 0.211
#>
#> Parameter Estimates:
#> Coefficients unstandardized
#> Information observed
#> Standard errors standard
#>
#> Latent Variables:
#> Estimate Std.Error z.value P(>|z|)
#> X =~
#> x1 1.000
#> x2 0.803 0.013 63.96 0.000
#> x3 0.914 0.013 67.80 0.000
#> Z =~
#> z1 1.000
#> z2 0.810 0.012 65.12 0.000
#> z3 0.881 0.013 67.62 0.000
#> Y =~
#> y1 1.000
#> y2 0.798 0.007 107.57 0.000
#> y3 0.899 0.008 112.55 0.000
#>
#> Regressions:
#> Estimate Std.Error z.value P(>|z|)
#> Y ~
#> X 0.674 0.032 20.94 0.000
#> Z 0.566 0.030 18.96 0.000
#> X:Z 0.712 0.028 25.45 0.000
#>
#> Intercepts:
#> Estimate Std.Error z.value P(>|z|)
#> x1 1.023 0.024 42.89 0.000
#> x2 1.215 0.020 60.99 0.000
#> x3 0.919 0.022 41.48 0.000
#> z1 1.012 0.024 41.57 0.000
#> z2 1.206 0.020 59.27 0.000
#> z3 0.916 0.022 42.06 0.000
#> y1 1.038 0.033 31.45 0.000
#> y2 1.221 0.027 45.49 0.000
#> y3 0.955 0.030 31.86 0.000
#> Y 0.000
#> X 0.000
#> Z 0.000
#>
#> Covariances:
#> Estimate Std.Error z.value P(>|z|)
#> X ~~
#> Z 0.200 0.024 8.24 0.000
#>
#> Variances:
#> Estimate Std.Error z.value P(>|z|)
#> x1 0.158 0.009 18.14 0.000
#> x2 0.162 0.007 23.19 0.000
#> x3 0.165 0.008 20.82 0.000
#> z1 0.166 0.009 18.34 0.000
#> z2 0.159 0.007 22.62 0.000
#> z3 0.158 0.008 20.71 0.000
#> y1 0.159 0.009 17.98 0.000
#> y2 0.154 0.007 22.67 0.000
#> y3 0.164 0.008 20.71 0.000
#> X 0.983 0.036 26.99 0.000
#> Z 1.019 0.038 26.95 0.000
#> Y 0.943 0.038 24.87 0.000A more complicated example
Here you can see an example of a more complicated example using the
model from the theory of planned behaviour (TPB), where there are two
endogenous variables, where there is an interaction between an
endogenous and exogenous variable. When estimating more complicated
models with the LMS-approach, it is recommended that you increase the
number of nodes used for numerical integration. By default the number of
nodes is set to 16, and can be increased using the nodes argument. The
argument has no effect on the QML approach. When there is an interaction
effect between an endogenous and exogenous variable, it is recommended
that you use at least 32 nodes for the LMS-approach. You can also get
robust standard errors by setting robust.se = TRUE in the
modsem() function.
Note: If you want the lms-approach to give as similar results as
possible to mplus, you would have to increase the number of nodes (e.g.,
nodes = 100).
# ATT = Attitude,
# PBC = Perceived Behavioural Control
# INT = Intention
# SN = Subjective Norms
# BEH = Behaviour
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
'
lms2 <- modsem(tpb, TPB, method = "lms", nodes = 32)
summary(lms2)
#> Estimating null model
#> EM: Iteration = 1, LogLik = -26393.22, Change = -26393.223
#> EM: Iteration = 2, LogLik = -26393.22, Change = 0.000
#>
#> modsem (version 1.0.3):
#> Estimator LMS
#> Optimization method EM-NLMINB
#> Number of observations 2000
#> Number of iterations 70
#> Loglikelihood -23439.02
#> Akaike (AIC) 46986.04
#> Bayesian (BIC) 47288.49
#>
#> Numerical Integration:
#> Points of integration (per dim) 32
#> Dimensions 1
#> Total points of integration 32
#>
#> Fit Measures for H0:
#> Loglikelihood -26393
#> 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 (no interaction effect)
#> Loglikelihood change 2954.20
#> Difference test (D) 5908.41
#> Degrees of freedom (D) 1
#> P-value (D) 0.000
#>
#> R-Squared:
#> INT 0.364
#> BEH 0.259
#> R-Squared Null-Model (H0):
#> INT 0.367
#> BEH 0.210
#> R-Squared Change:
#> INT -0.003
#> BEH 0.049
#>
#> Parameter Estimates:
#> Coefficients unstandardized
#> Information expected
#> Standard errors standard
#>
#> Latent Variables:
#> Estimate Std.Error z.value P(>|z|)
#> PBC =~
#> pbc1 1.000
#> pbc2 0.914 0.013 68.52 0.000
#> pbc3 0.802 0.012 65.02 0.000
#> ATT =~
#> att1 1.000
#> att2 0.878 0.012 70.81 0.000
#> att3 0.789 0.012 65.77 0.000
#> att4 0.695 0.011 61.09 0.000
#> att5 0.887 0.013 70.26 0.000
#> SN =~
#> sn1 1.000
#> sn2 0.889 0.017 52.00 0.000
#> INT =~
#> int1 1.000
#> int2 0.913 0.016 58.38 0.000
#> int3 0.807 0.015 55.37 0.000
#> BEH =~
#> b1 1.000
#> b2 0.959 0.033 29.28 0.000
#>
#> Regressions:
#> Estimate Std.Error z.value P(>|z|)
#> INT ~
#> PBC 0.218 0.030 7.36 0.000
#> ATT 0.214 0.026 8.19 0.000
#> SN 0.176 0.027 6.43 0.000
#> BEH ~
#> PBC 0.233 0.022 10.35 0.000
#> INT 0.188 0.025 7.62 0.000
#> PBC:INT 0.205 0.019 10.90 0.000
#>
#> Intercepts:
#> Estimate Std.Error z.value P(>|z|)
#> pbc1 0.990 0.022 45.57 0.000
#> pbc2 0.978 0.020 48.28 0.000
#> pbc3 0.985 0.018 53.86 0.000
#> att1 1.009 0.023 43.19 0.000
#> att2 1.002 0.021 48.19 0.000
#> att3 1.012 0.019 53.21 0.000
#> att4 0.995 0.017 56.95 0.000
#> att5 0.988 0.021 46.75 0.000
#> sn1 1.001 0.023 42.73 0.000
#> sn2 1.006 0.021 48.06 0.000
#> int1 1.010 0.021 47.81 0.000
#> int2 1.009 0.020 51.14 0.000
#> int3 1.002 0.018 56.02 0.000
#> b1 0.999 0.021 47.31 0.000
#> b2 1.017 0.020 51.50 0.000
#> INT 0.000
#> BEH 0.000
#> PBC 0.000
#> ATT 0.000
#> SN 0.000
#>
#> Covariances:
#> Estimate Std.Error z.value P(>|z|)
#> PBC ~~
#> ATT 0.668 0.021 31.78 0.000
#> SN 0.668 0.022 30.52 0.000
#> ATT ~~
#> SN 0.623 0.019 32.90 0.000
#>
#> Variances:
#> Estimate Std.Error z.value P(>|z|)
#> pbc1 0.148 0.008 18.81 0.000
#> pbc2 0.159 0.007 21.62 0.000
#> pbc3 0.155 0.007 23.64 0.000
#> att1 0.167 0.007 23.64 0.000
#> att2 0.150 0.006 24.73 0.000
#> att3 0.159 0.006 26.68 0.000
#> att4 0.162 0.006 27.71 0.000
#> att5 0.159 0.006 25.11 0.000
#> sn1 0.178 0.015 11.97 0.000
#> sn2 0.156 0.012 13.20 0.000
#> int1 0.157 0.009 18.25 0.000
#> int2 0.160 0.008 20.48 0.000
#> int3 0.168 0.007 24.27 0.000
#> b1 0.185 0.020 9.46 0.000
#> b2 0.136 0.018 7.71 0.000
#> PBC 0.947 0.017 55.23 0.000
#> ATT 0.992 0.014 69.87 0.000
#> SN 0.981 0.015 64.37 0.000
#> INT 0.491 0.020 24.97 0.000
#> BEH 0.456 0.023 19.46 0.000
qml2 <- modsem(tpb, TPB, method = "qml")
summary(qml2, standardized = TRUE) # standardized estimates
#> Estimating null model
#> Starting M-step
#>
#> modsem (version 1.0.3):
#> Estimator QML
#> Optimization method NLMINB
#> Number of observations 2000
#> Number of iterations 73
#> Loglikelihood -26326.25
#> Akaike (AIC) 52760.5
#> Bayesian (BIC) 53062.95
#>
#> Fit Measures for H0:
#> Loglikelihood -26393
#> 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 (no interaction effect)
#> Loglikelihood change 66.97
#> Difference test (D) 133.95
#> Degrees of freedom (D) 1
#> P-value (D) 0.000
#>
#> R-Squared:
#> INT 0.366
#> BEH 0.263
#> R-Squared Null-Model (H0):
#> INT 0.367
#> BEH 0.210
#> R-Squared Change:
#> INT 0.000
#> BEH 0.053
#>
#> Parameter Estimates:
#> Coefficients standardized
#> Information observed
#> Standard errors standard
#>
#> Latent Variables:
#> Estimate Std.Error z.value P(>|z|)
#> PBC =~
#> pbc1 0.933
#> pbc2 0.913 0.013 69.47 0.000
#> pbc3 0.894 0.014 66.10 0.000
#> ATT =~
#> att1 0.925
#> att2 0.915 0.013 71.56 0.000
#> att3 0.892 0.013 66.38 0.000
#> att4 0.865 0.014 61.00 0.000
#> att5 0.912 0.013 70.85 0.000
#> SN =~
#> sn1 0.921
#> sn2 0.913 0.017 52.61 0.000
#> INT =~
#> int1 0.912
#> int2 0.895 0.015 59.05 0.000
#> int3 0.866 0.016 55.73 0.000
#> BEH =~
#> b1 0.877
#> b2 0.900 0.028 31.71 0.000
#>
#> Regressions:
#> Estimate Std.Error z.value P(>|z|)
#> INT ~
#> PBC 0.243 0.033 7.35 0.000
#> ATT 0.242 0.030 8.16 0.000
#> SN 0.199 0.031 6.37 0.000
#> BEH ~
#> PBC 0.289 0.028 10.37 0.000
#> INT 0.212 0.028 7.69 0.000
#> PBC:INT 0.227 0.020 11.32 0.000
#>
#> Covariances:
#> Estimate Std.Error z.value P(>|z|)
#> PBC ~~
#> ATT 0.692 0.030 23.45 0.000
#> SN 0.695 0.030 23.08 0.000
#> ATT ~~
#> SN 0.634 0.029 21.70 0.000
#>
#> Variances:
#> Estimate Std.Error z.value P(>|z|)
#> pbc1 0.130 0.007 18.39 0.000
#> pbc2 0.166 0.008 21.43 0.000
#> pbc3 0.201 0.008 23.89 0.000
#> att1 0.144 0.006 23.53 0.000
#> att2 0.164 0.007 24.71 0.000
#> att3 0.204 0.008 26.38 0.000
#> att4 0.252 0.009 27.65 0.000
#> att5 0.168 0.007 24.93 0.000
#> sn1 0.153 0.013 12.09 0.000
#> sn2 0.167 0.013 13.26 0.000
#> int1 0.168 0.009 18.11 0.000
#> int2 0.199 0.010 20.41 0.000
#> int3 0.249 0.011 23.55 0.000
#> b1 0.231 0.023 10.12 0.000
#> b2 0.191 0.024 8.10 0.000
#> PBC 1.000 0.037 27.07 0.000
#> ATT 1.000 0.037 26.93 0.000
#> SN 1.000 0.040 25.22 0.000
#> INT 0.634 0.026 24.64 0.000
#> BEH 0.737 0.037 20.17 0.000