The Latent Moderated Structural Equations (LMS) and the Quasi Maximum Likelihood (QML) Approach
Both the LMS and QML approaches work on
most models, but interaction effects with endogenous variables can be
tricky to estimate (see the vignette).
Both approaches, particularly the LMS approach, are
computationally intensive and are partially 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 monitor the
progress of the estimation process, you can use
verbose = TRUE.
A Simple Example
Here is 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
#>
#> modsem (version 1.0.4):
#> 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 65.27 0.000
#> x3 0.912 0.013 68.77 0.000
#> Z =~
#> z1 0.927
#> z2 0.898 0.014 65.55 0.000
#> z3 0.913 0.013 69.05 0.000
#> Y =~
#> y1 0.969
#> y2 0.954 0.009 106.30 0.000
#> y3 0.961 0.009 112.47 0.000
#>
#> Regressions:
#> Estimate Std.Error z.value P(>|z|)
#> Y ~
#> X 0.427 0.020 20.89 0.000
#> Z 0.370 0.019 19.53 0.000
#> X:Z 0.454 0.017 26.72 0.000
#>
#> Covariances:
#> Estimate Std.Error z.value P(>|z|)
#> X ~~
#> Z 0.199 0.028 7.23 0.000
#>
#> Variances:
#> Estimate Std.Error z.value P(>|z|)
#> x1 0.142 0.007 19.63 0.000
#> x2 0.206 0.009 24.10 0.000
#> x3 0.169 0.008 21.53 0.000
#> z1 0.141 0.008 18.74 0.000
#> z2 0.193 0.009 22.64 0.000
#> z3 0.167 0.008 20.78 0.000
#> y1 0.061 0.003 17.98 0.000
#> y2 0.090 0.004 22.74 0.000
#> y3 0.077 0.004 20.73 0.000
#> X 1.000 0.032 30.84 0.000
#> Z 1.000 0.038 26.66 0.000
#> Y 0.404 0.015 27.12 0.000Here is the same example using the QML approach:
qml1 <- modsem(m1, oneInt, method = "qml")
summary(qml1)
#>
#> modsem (version 1.0.4):
#> 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
Below is an example of a more complex model based on the theory of
planned behavior (TPB), which includes two endogenous variables and an
interaction between an endogenous and exogenous variable. When
estimating more complex models with the LMS approach, it is
recommended to increase the number of nodes used for numerical
integration. By default, the number of nodes is set to 16, but this can
be increased using the nodes argument. The
nodes argument has no effect on the QML
approach.
When there is an interaction effect between an endogenous and
exogenous variable, it is recommended to use at least 32 nodes for the
LMS approach. You can also obtain robust standard errors by
setting robust.se = TRUE in the modsem()
function.
Note: If you want the LMS approach to
produce results as similar as possible to Mplus, you should increase the
number of nodes (e.g., nodes = 100).
# ATT = Attitude
# PBC = Perceived Behavioral Control
# INT = Intention
# SN = Subjective Norms
# BEH = Behavior
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)
#>
#> modsem (version 1.0.4):
#> 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 69.02 0.000
#> pbc3 0.802 0.012 65.40 0.000
#> ATT =~
#> att1 1.000
#> att2 0.878 0.012 70.82 0.000
#> att3 0.789 0.012 65.78 0.000
#> att4 0.695 0.011 61.10 0.000
#> att5 0.887 0.013 70.28 0.000
#> SN =~
#> sn1 1.000
#> sn2 0.889 0.017 52.14 0.000
#> INT =~
#> int1 1.000
#> int2 0.913 0.015 58.98 0.000
#> int3 0.807 0.014 55.84 0.000
#> BEH =~
#> b1 1.000
#> b2 0.959 0.030 31.77 0.000
#>
#> Regressions:
#> Estimate Std.Error z.value P(>|z|)
#> INT ~
#> PBC 0.218 0.030 7.32 0.000
#> ATT 0.214 0.026 8.18 0.000
#> SN 0.176 0.027 6.42 0.000
#> BEH ~
#> PBC 0.233 0.023 10.24 0.000
#> INT 0.188 0.025 7.59 0.000
#> PBC:INT 0.205 0.019 11.04 0.000
#>
#> Intercepts:
#> Estimate Std.Error z.value P(>|z|)
#> pbc1 0.990 0.023 43.17 0.000
#> pbc2 0.978 0.021 45.79 0.000
#> pbc3 0.985 0.019 51.31 0.000
#> att1 1.009 0.024 41.46 0.000
#> att2 1.002 0.022 46.30 0.000
#> att3 1.012 0.020 51.24 0.000
#> att4 0.995 0.018 55.01 0.000
#> att5 0.988 0.022 44.94 0.000
#> sn1 1.001 0.024 40.93 0.000
#> sn2 1.006 0.022 46.05 0.000
#> int1 1.010 0.022 44.93 0.000
#> int2 1.009 0.021 48.14 0.000
#> int3 1.002 0.019 52.81 0.000
#> b1 0.999 0.022 45.57 0.000
#> b2 1.017 0.021 49.38 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 0.028 23.93 0.000
#> SN 0.668 0.029 23.44 0.000
#> ATT ~~
#> SN 0.623 0.029 21.63 0.000
#>
#> Variances:
#> Estimate Std.Error z.value P(>|z|)
#> pbc1 0.148 0.008 18.92 0.000
#> pbc2 0.159 0.007 21.70 0.000
#> pbc3 0.155 0.007 23.70 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 25.12 0.000
#> sn1 0.178 0.015 11.99 0.000
#> sn2 0.156 0.012 13.23 0.000
#> int1 0.157 0.009 18.36 0.000
#> int2 0.160 0.008 20.57 0.000
#> int3 0.168 0.007 24.33 0.000
#> b1 0.185 0.019 9.96 0.000
#> b2 0.136 0.017 8.13 0.000
#> PBC 0.947 0.034 28.02 0.000
#> ATT 0.992 0.037 27.11 0.000
#> SN 0.981 0.039 24.99 0.000
#> INT 0.491 0.020 25.00 0.000
#> BEH 0.456 0.023 20.13 0.000
qml2 <- modsem(tpb, TPB, method = "qml")
summary(qml2, standardized = TRUE) # Standardized estimates
#>
#> modsem (version 1.0.4):
#> 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