Estimate interaction effects in structural equation models (SEMs)
modsem.Rdmodsem() is a function for estimating interaction effects between latent variables
in structural equation models (SEMs).
Methods for estimating interaction effects in SEMs can basically be split into two frameworks:
Product Indicator (PI) based approaches (
"dblcent","rca","uca","ca","pind")Distributionally (DA) based approaches (
"lms","qml").
For the product indicator-based approaches, modsem() is essentially a fancy wrapper for lavaan::sem() which generates the
necessary syntax and variables for the estimation of models with latent product indicators.
The distributionally based approaches are implemented separately and are
not estimated using lavaan::sem(), but rather using custom functions (largely
written in C++ for performance reasons). For greater control, it is advised that
you use one of the sub-functions (modsem_pi, modsem_da, modsem_mplus) directly,
as passing additional arguments to them via modsem() can lead to unexpected behavior.
Arguments
- model.syntax
lavaansyntax- data
dataframe
- method
method to use:
"dblcent"double centering approach (passed to
lavaan)."ca"constrained approach (passed to
lavaan)."rca"residual centering approach (passed to
lavaan)."uca"unconstrained approach (passed to
lavaan)."pind"prod ind approach, with no constraints or centering (passed to
lavaan)."lms"latent moderated structural equations (not passed to
lavaan)."qml"quasi maximum likelihood estimation (not passed to
lavaan)."custom"use parameters specified in the function call (passed to
lavaan)."mplus"estimate model through
Mplus.
- ...
arguments passed to other functions depending on the method (see
modsem_pi,modsem_da, andmodsem_mplus)
Value
modsem object with class modsem_pi, modsem_da, or modsem_mplus
Examples
library(modsem)
# For more examples, check README and/or GitHub.
# One interaction
m1 <- '
# Outer Model
X =~ x1 + x2 +x3
Y =~ y1 + y2 + y3
Z =~ z1 + z2 + z3
# Inner model
Y ~ X + Z + X:Z
'
# Double centering approach
est1 <- modsem(m1, oneInt)
summary(est1)
#> Estimating baseline model (H0)
#> modsem (version 1.0.11, approach = dblcent):
#>
#> Interaction Model Fit Measures (H1):
#> Loglikelihood -26807.61
#> Akaike (AIC) 53735.22
#> Bayesian (BIC) 54071.28
#> Chi-square 122.92
#> Degrees of Freedom 111
#> P-value (Chi-square) 0.207
#> RMSEA 0.007
#> CFI 1.000
#> SRMR 0.008
#>
#> Fit Measures for Baseline Model (H0):
#> Loglikelihood -27137.74
#> Akaike (AIC) 54393.48
#> Bayesian (BIC) 54723.93
#> Chi-square 783.18
#> Degrees of Freedom 112
#> P-value (Chi-square) 0.000
#> RMSEA 0.055
#> CFI 0.987
#> SRMR 0.141
#>
#> Comparative Fit to H0 (LRT test):
#> Chi-square diff 660.257
#> Degrees of freedom diff 1
#> P-value (LRT) 0.000
#>
#> R-Squared Interaction Model (H1):
#> Y 0.602
#> R-Squared Baseline Model (H0):
#> Y 0.397
#> R-Squared Change (H1 - H0):
#> Y 0.204
#>
#> lavaan 0.6-19 ended normally after 161 iterations
#>
#> Estimator ML
#> Optimization method NLMINB
#> Number of model parameters 60
#>
#> Number of observations 2000
#>
#> Model Test User Model:
#>
#> Test statistic 122.924
#> Degrees of freedom 111
#> P-value (Chi-square) 0.207
#>
#> Parameter Estimates:
#>
#> Standard errors Standard
#> Information Expected
#> Information saturated (h1) model Structured
#>
#> Latent Variables:
#> Estimate Std.Err z-value P(>|z|)
#> X =~
#> x1 1.000
#> x2 0.804 0.013 63.612 0.000
#> x3 0.916 0.014 67.144 0.000
#> Y =~
#> y1 1.000
#> y2 0.798 0.007 107.428 0.000
#> y3 0.899 0.008 112.453 0.000
#> Z =~
#> z1 1.000
#> z2 0.812 0.013 64.763 0.000
#> z3 0.882 0.013 67.014 0.000
#> XZ =~
#> x1z1 1.000
#> x2z1 0.805 0.013 60.636 0.000
#> x3z1 0.877 0.014 62.680 0.000
#> x1z2 0.793 0.013 59.343 0.000
#> x2z2 0.646 0.015 43.672 0.000
#> x3z2 0.706 0.016 44.292 0.000
#> x1z3 0.887 0.014 63.700 0.000
#> x2z3 0.716 0.016 45.645 0.000
#> x3z3 0.781 0.017 45.339 0.000
#>
#> Regressions:
#> Estimate Std.Err z-value P(>|z|)
#> Y ~
#> X 0.675 0.027 25.379 0.000
#> Z 0.561 0.026 21.606 0.000
#> XZ 0.702 0.027 26.360 0.000
#>
#> Covariances:
#> Estimate Std.Err z-value P(>|z|)
#> .x1z1 ~~
#> .x1z2 0.115 0.008 14.802 0.000
#> .x1z3 0.114 0.008 13.947 0.000
#> .x2z1 0.125 0.008 16.095 0.000
#> .x3z1 0.140 0.009 16.135 0.000
#> .x1z2 ~~
#> .x1z3 0.103 0.007 14.675 0.000
#> .x2z2 0.128 0.006 20.850 0.000
#> .x3z2 0.146 0.007 21.243 0.000
#> .x1z3 ~~
#> .x2z3 0.116 0.007 17.818 0.000
#> .x3z3 0.135 0.007 18.335 0.000
#> .x2z1 ~~
#> .x2z2 0.135 0.006 20.905 0.000
#> .x2z3 0.145 0.007 21.145 0.000
#> .x3z1 0.114 0.007 16.058 0.000
#> .x2z2 ~~
#> .x2z3 0.117 0.006 20.419 0.000
#> .x3z2 0.116 0.006 20.586 0.000
#> .x2z3 ~~
#> .x3z3 0.109 0.006 18.059 0.000
#> .x3z1 ~~
#> .x3z2 0.138 0.007 19.331 0.000
#> .x3z3 0.158 0.008 20.269 0.000
#> .x3z2 ~~
#> .x3z3 0.131 0.007 19.958 0.000
#> X ~~
#> Z 0.201 0.024 8.271 0.000
#> XZ 0.016 0.025 0.628 0.530
#> Z ~~
#> XZ 0.062 0.025 2.449 0.014
#>
#> Variances:
#> Estimate Std.Err z-value P(>|z|)
#> .x1 0.160 0.009 17.871 0.000
#> .x2 0.162 0.007 22.969 0.000
#> .x3 0.163 0.008 20.161 0.000
#> .y1 0.159 0.009 17.896 0.000
#> .y2 0.154 0.007 22.640 0.000
#> .y3 0.164 0.008 20.698 0.000
#> .z1 0.168 0.009 18.143 0.000
#> .z2 0.158 0.007 22.264 0.000
#> .z3 0.158 0.008 20.389 0.000
#> .x1z1 0.311 0.014 22.227 0.000
#> .x2z1 0.292 0.011 27.287 0.000
#> .x3z1 0.327 0.012 26.275 0.000
#> .x1z2 0.290 0.011 26.910 0.000
#> .x2z2 0.239 0.008 29.770 0.000
#> .x3z2 0.270 0.009 29.117 0.000
#> .x1z3 0.272 0.012 23.586 0.000
#> .x2z3 0.245 0.009 27.979 0.000
#> .x3z3 0.297 0.011 28.154 0.000
#> X 0.981 0.036 26.895 0.000
#> .Y 0.990 0.038 25.926 0.000
#> Z 1.016 0.038 26.856 0.000
#> XZ 1.045 0.044 24.004 0.000
#>
# \dontrun{
# The Constrained Approach
est1_ca <- modsem(m1, oneInt, method = "ca")
summary(est1_ca)
#> Estimating baseline model (H0)
#> modsem (version 1.0.11, approach = ca):
#>
#> Interaction Model Fit Measures (H1):
#> Loglikelihood -24339.32
#> Akaike (AIC) 48746.65
#> Bayesian (BIC) 48937.08
#> Chi-square 60.40
#> Degrees of Freedom 56
#> P-value (Chi-square) 0.320
#> RMSEA 0.006
#> CFI 1.000
#> SRMR 0.020
#>
#> Fit Measures for Baseline Model (H0):
#> Loglikelihood -24669.20
#> Akaike (AIC) 49404.40
#> Bayesian (BIC) 49589.23
#> Chi-square 720.15
#> Degrees of Freedom 57
#> P-value (Chi-square) 0.000
#> RMSEA 0.076
#> CFI 0.972
#> SRMR 0.124
#>
#> Comparative Fit to H0 (LRT test):
#> Chi-square diff 659.749
#> Degrees of freedom diff 1
#> P-value (LRT) 0.000
#>
#> R-Squared Interaction Model (H1):
#> Y 0.594
#> R-Squared Baseline Model (H0):
#> Y 0.393
#> R-Squared Change (H1 - H0):
#> Y 0.201
#>
#> lavaan 0.6-19 ended normally after 286 iterations
#>
#> Estimator ML
#> Optimization method NLMINB
#> Number of model parameters 43
#> Row rank of the constraints matrix 9
#>
#> Number of observations 2000
#>
#> Model Test User Model:
#>
#> Test statistic 60.401
#> Degrees of freedom 56
#> P-value (Chi-square) 0.320
#>
#> Parameter Estimates:
#>
#> Standard errors Standard
#> Information Expected
#> Information saturated (h1) model Structured
#>
#> Latent Variables:
#> Estimate Std.Err z-value P(>|z|)
#> X =~
#> x1 (l_1_X) 1.000
#> x2 (l_2_X) 0.805 0.011 73.101 0.000
#> x3 (l_3_X) 0.912 0.012 76.636 0.000
#> Y =~
#> y1 (l_1_Y) 1.000
#> y2 (l_2_Y) 0.798 0.008 106.156 0.000
#> y3 (l_3_Y) 0.899 0.008 111.106 0.000
#> Z =~
#> z1 (l_1_Z) 1.000
#> z2 (l_2_Z) 0.813 0.011 74.076 0.000
#> z3 (l_3_Z) 0.878 0.011 76.574 0.000
#> XZ =~
#> x1z1 (l_11) 1.000
#> x2z2 (l_22) 0.654 0.010 62.981 0.000
#> x3z3 (l_33) 0.801 0.012 65.421 0.000
#>
#> Regressions:
#> Estimate Std.Err z-value P(>|z|)
#> Y ~
#> X (G_X_) 0.678 0.026 25.602 0.000
#> Z (G_Z_) 0.565 0.026 21.868 0.000
#> XZ (G_XZ) 0.715 0.026 27.094 0.000
#>
#> Covariances:
#> Estimate Std.Err z-value P(>|z|)
#> X ~~
#> Z (C_X_Z) 0.198 0.022 9.042 0.000
#> XZ (C_X_X) 0.000 NA
#> Z ~~
#> XZ (C_Z_) 0.000
#> .x1z1 ~~
#> .x2z2 0.000
#> .x3z3 0.000
#> .x2z2 ~~
#> .x3z3 0.000
#>
#> Intercepts:
#> Estimate Std.Err z-value P(>|z|)
#> XZ (M_XZ) 0.198 0.022 9.042 0.000
#> .x1 1.023 0.024 43.002 0.000
#> .x2 1.215 0.020 60.936 0.000
#> .x3 0.919 0.022 41.603 0.000
#> .y1 1.039 0.040 26.293 0.000
#> .y2 1.222 0.032 38.225 0.000
#> .y3 0.955 0.036 26.709 0.000
#> .z1 1.011 0.024 41.726 0.000
#> .z2 1.206 0.020 59.286 0.000
#> .z3 0.916 0.022 42.229 0.000
#> .x1z1 0.012 0.034 0.352 0.725
#> .x2z2 0.001 0.023 0.031 0.975
#> .x3z3 -0.007 0.028 -0.262 0.793
#>
#> Variances:
#> Estimate Std.Err z-value P(>|z|)
#> X (Vr_X) 0.976 0.029 33.284 0.000
#> .Y (Zt_Y) 0.987 0.038 25.711 0.000
#> Z (Vr_Z) 1.011 0.030 33.294 0.000
#> XZ (V_XZ) 1.026 0.032 32.214 0.000
#> .x1 (Vr_x1) 0.156 0.008 19.174 0.000
#> .x2 (Vr_x2) 0.163 0.006 25.240 0.000
#> .x3 (Vr_x3) 0.165 0.007 22.239 0.000
#> .y1 (Vr_y1) 0.159 0.009 17.881 0.000
#> .y2 (Vr_y2) 0.154 0.007 22.631 0.000
#> .y3 (Vr_y3) 0.164 0.008 20.686 0.000
#> .z1 (Vr_z1) 0.164 0.008 19.449 0.000
#> .z2 (Vr_z2) 0.159 0.007 24.247 0.000
#> .z3 (Vr_z3) 0.160 0.007 22.514 0.000
#> .x1z1 (V_11) 0.343 0.011 30.345 0.000
#> .x2z2 (V_22) 0.236 0.006 37.239 0.000
#> .x3z3 (V_33) 0.285 0.008 34.438 0.000
#>
#> Constraints:
#> |Slack|
#> Var_XZ - ((Var_X)*(Var_Z)+(Cov_X_Z)^2) 0.000
#> Cov_X_XZ - 0 0.000
#> Cov_Z_XZ - 0 0.000
#> V_11-(_1_X^2*(V_X)*V_1+_1_Z^2*(V_Z)*V_1+V 0.000
#> V_22-(_2_X^2*(V_X)*V_2+_2_Z^2*(V_Z)*V_2+V 0.000
#> V_33-(_3_X^2*(V_X)*V_3+_3_Z^2*(V_Z)*V_3+V 0.000
#> lambda_x1z1_XZ-(lambda_x1_X*lambda_z1_Z) 0.000
#> lambda_x2z2_XZ-(lambda_x2_X*lambda_z2_Z) 0.000
#> lambda_x3z3_XZ-(lambda_x3_X*lambda_z3_Z) 0.000
#> Mean_XZ - ((Cov_X_Z)) 0.000
#>
# LMS approach
est1_lms <- modsem(m1, oneInt, method = "lms")
summary(est1_lms)
#>
#> modsem (version 1.0.11):
#>
#> Estimator LMS
#> Optimization method EMA-NLMINB
#> Number of observations 2000
#> Number of iterations 46
#> Loglikelihood -14661.61
#> Akaike (AIC) 29385.22
#> Bayesian (BIC) 29558.85
#>
#> Numerical Integration:
#> Points of integration (per dim) 24
#> Dimensions 1
#> Total points of integration 24
#>
#> Fit Measures for Baseline Model (H0):
#> Loglikelihood -17831.87
#> 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 (LRT test):
#> Loglikelihood change 3170.26
#> Difference test (D) 6340.53
#> Degrees of freedom (D) 1
#> P-value (D) 0.000
#>
#> R-Squared Interaction Model (H1):
#> Y 0.599
#> R-Squared Baseline Model (H0):
#> Y 0.395
#> R-Squared Change (H1 - H0):
#> Y 0.204
#>
#> 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 63.91 0.000
#> x3 0.914 0.013 67.73 0.000
#> Z =~
#> z1 1.000
#> z2 0.810 0.012 65.09 0.000
#> z3 0.881 0.013 67.62 0.000
#> Y =~
#> y1 1.000
#> y2 0.798 0.007 107.55 0.000
#> y3 0.899 0.008 112.58 0.000
#>
#> Regressions:
#> Estimate Std.Error z.value P(>|z|)
#> Y ~
#> X 0.672 0.031 21.66 0.000
#> Z 0.570 0.030 18.75 0.000
#> X:Z 0.718 0.028 25.84 0.000
#>
#> Intercepts:
#> Estimate Std.Error z.value P(>|z|)
#> .x1 1.024 0.024 42.85 0.000
#> .x2 1.217 0.020 60.92 0.000
#> .x3 0.921 0.022 41.45 0.000
#> .z1 1.011 0.024 41.54 0.000
#> .z2 1.205 0.020 59.24 0.000
#> .z3 0.915 0.022 42.03 0.000
#> .y1 1.037 0.033 31.40 0.000
#> .y2 1.220 0.027 45.42 0.000
#> .y3 0.954 0.030 31.80 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 18.17 0.000
#> .x2 0.162 0.007 23.16 0.000
#> .x3 0.164 0.008 20.76 0.000
#> .z1 0.167 0.009 18.50 0.000
#> .z2 0.160 0.007 22.68 0.000
#> .z3 0.158 0.008 20.78 0.000
#> .y1 0.160 0.009 18.01 0.000
#> .y2 0.154 0.007 22.69 0.000
#> .y3 0.164 0.008 20.68 0.000
#> X 0.981 0.036 27.04 0.000
#> Z 1.018 0.038 26.93 0.000
#> .Y 0.980 0.038 25.94 0.000
#>
# QML approach
est1_qml <- modsem(m1, oneInt, method = "qml")
summary(est1_qml)
#>
#> modsem (version 1.0.11):
#>
#> Estimator QML
#> Optimization method NLMINB
#> Number of observations 2000
#> Number of iterations 98
#> Loglikelihood -17501.61
#> Akaike (AIC) 35065.21
#> Bayesian (BIC) 35238.84
#>
#> Fit Measures for Baseline Model (H0):
#> Loglikelihood -17831.87
#> 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 (LRT test):
#> Loglikelihood change 330.27
#> Difference test (D) 660.54
#> Degrees of freedom (D) 1
#> P-value (D) 0.000
#>
#> R-Squared Interaction Model (H1):
#> Y 0.617
#> R-Squared Baseline Model (H0):
#> Y 0.395
#> R-Squared Change (H1 - H0):
#> Y 0.222
#>
#> 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 0.913 0.013 67.77 0.000
#> Z =~
#> z1 1.000
#> z2 0.810 0.012 65.15 0.000
#> z3 0.881 0.013 67.66 0.000
#> Y =~
#> y1 1.000
#> y2 0.798 0.007 107.56 0.000
#> y3 0.899 0.008 112.52 0.000
#>
#> Regressions:
#> Estimate Std.Error z.value P(>|z|)
#> Y ~
#> X 0.673 0.031 21.89 0.000
#> Z 0.566 0.030 18.82 0.000
#> X:Z 0.713 0.027 26.50 0.000
#>
#> Intercepts:
#> Estimate Std.Error z.value P(>|z|)
#> .x1 1.023 0.024 42.84 0.000
#> .x2 1.216 0.020 60.92 0.000
#> .x3 0.920 0.022 41.44 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 1.037 0.033 31.31 0.000
#> .y2 1.221 0.027 45.29 0.000
#> .y3 0.954 0.030 31.72 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 18.12 0.000
#> .x2 0.162 0.007 23.18 0.000
#> .x3 0.165 0.008 20.83 0.000
#> .z1 0.166 0.009 18.46 0.000
#> .z2 0.160 0.007 22.69 0.000
#> .z3 0.158 0.008 20.80 0.000
#> .y1 0.159 0.009 17.98 0.000
#> .y2 0.154 0.007 22.66 0.000
#> .y3 0.164 0.008 20.69 0.000
#> X 0.983 0.036 26.99 0.000
#> Z 1.018 0.038 26.95 0.000
#> .Y 0.901 0.040 22.70 0.000
#>
# }
# Theory Of Planned 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
'
# Double centering approach
est_tpb <- modsem(tpb, data = TPB)
summary(est_tpb)
#> Estimating baseline model (H0)
#> modsem (version 1.0.11, approach = dblcent):
#>
#> Interaction Model Fit Measures (H1):
#> Loglikelihood -34958.88
#> Akaike (AIC) 70073.77
#> Bayesian (BIC) 70510.64
#> Chi-square 207.61
#> Degrees of Freedom 222
#> P-value (Chi-square) 0.747
#> RMSEA 0.000
#> CFI 1.000
#> SRMR 0.010
#>
#> Fit Measures for Baseline Model (H0):
#> Loglikelihood -35025.66
#> Akaike (AIC) 70205.32
#> Bayesian (BIC) 70636.59
#> Chi-square 341.17
#> Degrees of Freedom 223
#> P-value (Chi-square) 0.000
#> RMSEA 0.016
#> CFI 0.998
#> SRMR 0.048
#>
#> Comparative Fit to H0 (LRT test):
#> Chi-square diff 133.551
#> Degrees of freedom diff 1
#> P-value (LRT) 0.000
#>
#> R-Squared Interaction Model (H1):
#> INT 0.367
#> BEH 0.278
#> R-Squared Baseline Model (H0):
#> INT 0.367
#> BEH 0.211
#> R-Squared Change (H1 - H0):
#> INT -0.000
#> BEH 0.067
#>
#> lavaan 0.6-19 ended normally after 173 iterations
#>
#> Estimator ML
#> Optimization method NLMINB
#> Number of model parameters 78
#>
#> Number of observations 2000
#>
#> Model Test User Model:
#>
#> Test statistic 207.615
#> Degrees of freedom 222
#> P-value (Chi-square) 0.747
#>
#> Parameter Estimates:
#>
#> Standard errors Standard
#> Information Expected
#> Information saturated (h1) model Structured
#>
#> Latent Variables:
#> Estimate Std.Err z-value P(>|z|)
#> ATT =~
#> att1 1.000
#> att2 0.878 0.012 71.509 0.000
#> att3 0.789 0.012 66.368 0.000
#> att4 0.695 0.011 61.017 0.000
#> att5 0.887 0.013 70.884 0.000
#> SN =~
#> sn1 1.000
#> sn2 0.889 0.017 52.553 0.000
#> PBC =~
#> pbc1 1.000
#> pbc2 0.912 0.013 69.500 0.000
#> pbc3 0.801 0.012 65.830 0.000
#> INT =~
#> int1 1.000
#> int2 0.914 0.016 58.982 0.000
#> int3 0.808 0.015 55.547 0.000
#> BEH =~
#> b1 1.000
#> b2 0.960 0.030 31.561 0.000
#> INTPBC =~
#> int1pbc1 1.000
#> int2pbc1 0.931 0.015 63.809 0.000
#> int3pbc1 0.774 0.013 60.107 0.000
#> int1pbc2 0.893 0.013 68.173 0.000
#> int2pbc2 0.826 0.017 48.845 0.000
#> int3pbc2 0.690 0.015 45.300 0.000
#> int1pbc3 0.799 0.012 67.008 0.000
#> int2pbc3 0.738 0.015 47.809 0.000
#> int3pbc3 0.622 0.014 45.465 0.000
#>
#> Regressions:
#> Estimate Std.Err z-value P(>|z|)
#> INT ~
#> ATT 0.213 0.026 8.170 0.000
#> SN 0.177 0.028 6.416 0.000
#> PBC 0.217 0.030 7.340 0.000
#> BEH ~
#> INT 0.191 0.024 7.817 0.000
#> PBC 0.230 0.022 10.507 0.000
#> INTPBC 0.204 0.018 11.425 0.000
#>
#> Covariances:
#> Estimate Std.Err z-value P(>|z|)
#> .int1pbc1 ~~
#> .int1pbc2 0.126 0.009 14.768 0.000
#> .int1pbc3 0.102 0.007 13.794 0.000
#> .int2pbc1 0.104 0.007 14.608 0.000
#> .int3pbc1 0.091 0.006 14.109 0.000
#> .int1pbc2 ~~
#> .int1pbc3 0.095 0.007 13.852 0.000
#> .int2pbc2 0.128 0.007 19.320 0.000
#> .int3pbc2 0.119 0.006 19.402 0.000
#> .int1pbc3 ~~
#> .int2pbc3 0.110 0.006 19.911 0.000
#> .int3pbc3 0.097 0.005 19.415 0.000
#> .int2pbc1 ~~
#> .int2pbc2 0.152 0.008 18.665 0.000
#> .int2pbc3 0.138 0.007 18.779 0.000
#> .int3pbc1 0.082 0.006 13.951 0.000
#> .int2pbc2 ~~
#> .int2pbc3 0.121 0.007 18.361 0.000
#> .int3pbc2 0.104 0.005 19.047 0.000
#> .int2pbc3 ~~
#> .int3pbc3 0.087 0.005 19.180 0.000
#> .int3pbc1 ~~
#> .int3pbc2 0.139 0.007 21.210 0.000
#> .int3pbc3 0.123 0.006 21.059 0.000
#> .int3pbc2 ~~
#> .int3pbc3 0.114 0.005 21.021 0.000
#> ATT ~~
#> SN 0.629 0.029 21.977 0.000
#> PBC 0.678 0.029 23.721 0.000
#> INTPBC 0.086 0.024 3.519 0.000
#> SN ~~
#> PBC 0.678 0.029 23.338 0.000
#> INTPBC 0.055 0.025 2.230 0.026
#> PBC ~~
#> INTPBC 0.087 0.024 3.609 0.000
#>
#> Variances:
#> Estimate Std.Err z-value P(>|z|)
#> .att1 0.167 0.007 23.528 0.000
#> .att2 0.150 0.006 24.693 0.000
#> .att3 0.160 0.006 26.378 0.000
#> .att4 0.163 0.006 27.649 0.000
#> .att5 0.159 0.006 24.930 0.000
#> .sn1 0.178 0.015 12.110 0.000
#> .sn2 0.156 0.012 13.221 0.000
#> .pbc1 0.145 0.008 18.440 0.000
#> .pbc2 0.160 0.007 21.547 0.000
#> .pbc3 0.154 0.007 23.716 0.000
#> .int1 0.158 0.009 18.152 0.000
#> .int2 0.160 0.008 20.345 0.000
#> .int3 0.167 0.007 23.414 0.000
#> .b1 0.186 0.018 10.058 0.000
#> .b2 0.135 0.017 8.080 0.000
#> .int1pbc1 0.266 0.013 20.971 0.000
#> .int2pbc1 0.292 0.012 24.421 0.000
#> .int3pbc1 0.251 0.010 26.305 0.000
#> .int1pbc2 0.290 0.012 24.929 0.000
#> .int2pbc2 0.269 0.010 26.701 0.000
#> .int3pbc2 0.253 0.009 29.445 0.000
#> .int1pbc3 0.223 0.009 24.431 0.000
#> .int2pbc3 0.234 0.008 27.633 0.000
#> .int3pbc3 0.203 0.007 29.288 0.000
#> ATT 0.998 0.037 27.138 0.000
#> SN 0.987 0.039 25.394 0.000
#> PBC 0.962 0.035 27.260 0.000
#> .INT 0.490 0.020 24.638 0.000
#> .BEH 0.455 0.023 20.068 0.000
#> INTPBC 1.020 0.041 24.612 0.000
#>
# \dontrun{
# The Constrained Approach
est_tpb_ca <- modsem(tpb, data = TPB, method = "ca")
summary(est_tpb_ca)
#> Estimating baseline model (H0)
#> modsem (version 1.0.11, approach = ca):
#>
#> Interaction Model Fit Measures (H1):
#> Loglikelihood -32849.60
#> Akaike (AIC) 65817.20
#> Bayesian (BIC) 66147.66
#> Chi-square 124.54
#> Degrees of Freedom 130
#> P-value (Chi-square) 0.619
#> RMSEA 0.000
#> CFI 1.000
#> SRMR 0.024
#>
#> Fit Measures for Baseline Model (H0):
#> Loglikelihood -32918.35
#> Akaike (AIC) 65952.70
#> Bayesian (BIC) 66277.56
#> Chi-square 262.04
#> Degrees of Freedom 131
#> P-value (Chi-square) 0.000
#> RMSEA 0.022
#> CFI 0.996
#> SRMR 0.046
#>
#> Comparative Fit to H0 (LRT test):
#> Chi-square diff 137.501
#> Degrees of freedom diff 1
#> P-value (LRT) 0.000
#>
#> R-Squared Interaction Model (H1):
#> INT 0.369
#> BEH 0.268
#> R-Squared Baseline Model (H0):
#> INT 0.369
#> BEH 0.211
#> R-Squared Change (H1 - H0):
#> INT -0.000
#> BEH 0.057
#>
#> lavaan 0.6-19 ended normally after 273 iterations
#>
#> Estimator ML
#> Optimization method NLMINB
#> Number of model parameters 67
#> Row rank of the constraints matrix 8
#>
#> Number of observations 2000
#>
#> Model Test User Model:
#>
#> Test statistic 124.540
#> Degrees of freedom 130
#> P-value (Chi-square) 0.619
#>
#> Parameter Estimates:
#>
#> Standard errors Standard
#> Information Expected
#> Information saturated (h1) model Structured
#>
#> Latent Variables:
#> Estimate Std.Err z-value P(>|z|)
#> ATT =~
#> att1 (l_1_A) 1.000
#> att2 (l_2_A) 0.878 0.012 71.494 0.000
#> att3 (l_3_A) 0.789 0.012 66.355 0.000
#> att4 (l_4_) 0.695 0.011 61.005 0.000
#> att5 (l_5_) 0.887 0.013 70.871 0.000
#> SN =~
#> sn1 (l_1_S) 1.000
#> sn2 (l_2_S) 0.889 0.017 52.693 0.000
#> PBC =~
#> pbc1 (l_1_P) 1.000
#> pbc2 (l_2_P) 0.913 0.011 80.296 0.000
#> pbc3 (l_3_P) 0.795 0.011 75.450 0.000
#> INT =~
#> int1 (l_1_I) 1.000
#> int2 (l_2_I) 0.913 0.013 71.295 0.000
#> int3 (l_3_I) 0.797 0.012 66.791 0.000
#> BEH =~
#> b1 (l_1_B) 1.000
#> b2 (l_2_B) 0.962 0.031 30.667 0.000
#> INTPBC =~
#> int1p1 (l_11) 1.000
#> int2p2 (l_22) 0.833 0.012 66.642 0.000
#> int3p3 (l_33) 0.634 0.010 62.661 0.000
#>
#> Regressions:
#> Estimate Std.Err z-value P(>|z|)
#> INT ~
#> ATT (G_AT) 0.214 0.026 8.193 0.000
#> SN (G_SN) 0.177 0.028 6.427 0.000
#> PBC (G_PBC_I) 0.220 0.029 7.615 0.000
#> BEH ~
#> INT (G_INT_) 0.189 0.024 7.810 0.000
#> PBC (G_PBC_B) 0.231 0.022 10.631 0.000
#> INTP (G_INTP) 0.209 0.018 11.587 0.000
#>
#> Covariances:
#> Estimate Std.Err z-value P(>|z|)
#> ATT ~~
#> SN (C_ATT_S) 0.630 0.026 24.699 0.000
#> PBC (C_ATT_P) 0.680 0.023 29.948 0.000
#> INTP (C_ATT_I) 0.028 0.018 1.551 0.121
#> SN ~~
#> PBC (C_SN_P) 0.683 0.023 29.055 0.000
#> INTP (C_SN_I) -0.004 0.019 -0.199 0.843
#> PBC ~~
#> INTP (C_PB) 0.000 NA
#> .int1pbc1 ~~
#> .in22 0.000
#> .in33 0.000
#> .int2pbc2 ~~
#> .in33 0.000
#>
#> Intercepts:
#> Estimate Std.Err z-value P(>|z|)
#> INTPBC (M_IN) 0.480 0.014 33.491 0.000
#> .att1 1.014 0.024 41.999 0.000
#> .att2 1.007 0.021 46.960 0.000
#> .att3 1.016 0.020 51.448 0.000
#> .att4 0.999 0.018 55.647 0.000
#> .att5 0.992 0.022 45.664 0.000
#> .sn1 1.005 0.024 41.594 0.000
#> .sn2 1.010 0.022 46.638 0.000
#> .pbc1 0.997 0.024 42.323 0.000
#> .pbc2 0.985 0.022 44.751 0.000
#> .pbc3 0.991 0.020 50.566 0.000
#> .int1 1.014 0.022 46.839 0.000
#> .int2 1.012 0.020 50.167 0.000
#> .int3 1.005 0.018 54.971 0.000
#> .b1 0.997 0.022 45.295 0.000
#> .b2 1.015 0.021 49.087 0.000
#> .int1pb1 -0.014 0.029 -0.478 0.633
#> .int2pb2 0.011 0.025 0.450 0.652
#> .int3pb3 -0.005 0.020 -0.269 0.788
#>
#> Variances:
#> Estimate Std.Err z-value P(>|z|)
#> ATT (V_AT) 0.998 0.034 29.072 0.000
#> SN (V_SN) 0.990 0.037 26.938 0.000
#> PBC (V_PB) 0.970 0.026 37.161 0.000
#> .INT (Z_IN) 0.495 0.019 26.260 0.000
#> .BEH (Z_BE) 0.453 0.023 19.820 0.000
#> INTPBC (V_IN) 0.992 0.032 31.229 0.000
#> .att1 (Vr_t1) 0.167 0.007 23.528 0.000
#> .att2 (Vr_t2) 0.150 0.006 24.693 0.000
#> .att3 (Vr_t3) 0.160 0.006 26.378 0.000
#> .att4 (Vr_4) 0.163 0.006 27.649 0.000
#> .att5 (Vr_5) 0.159 0.006 24.929 0.000
#> .sn1 (Vr_s1) 0.178 0.015 12.118 0.000
#> .sn2 (Vr_s2) 0.156 0.012 13.235 0.000
#> .pbc1 (Vr_p1) 0.141 0.007 19.079 0.000
#> .pbc2 (Vr_p2) 0.160 0.007 22.885 0.000
#> .pbc3 (Vr_p3) 0.156 0.006 25.493 0.000
#> .int1 (Vr_n1) 0.152 0.008 19.481 0.000
#> .int2 (Vr_n2) 0.161 0.007 22.582 0.000
#> .int3 (Vr_n3) 0.169 0.006 26.510 0.000
#> .b1 (Vr_b1) 0.187 0.019 9.943 0.000
#> .b2 (Vr_b2) 0.134 0.017 7.874 0.000
#> .int1p1 (V_11) 0.279 0.010 29.251 0.000
#> .int2p2 (V_22) 0.260 0.008 33.952 0.000
#> .int3p3 (V_33) 0.208 0.005 37.854 0.000
#>
#> Constraints:
#> |Slack|
#> V_INTPBC-((2*C_ATT_PBC*G_ATT_INT*G_PBC_IN 0.000
#> Cov_PBC_INTPBC - 0 0.000
#> V_11-(_1_INT^2*(2*C_ATT_PBC*G_ATT_INT*G_P 0.000
#> V_22-(_2_INT^2*(2*C_ATT_PBC*G_ATT_INT*G_P 0.000
#> V_33-(_3_INT^2*(2*C_ATT_PBC*G_ATT_INT*G_P 0.000
#> lmbd_nt1pbc1_INTPBC-(lmbd_nt1_INT*_1_PBC) 0.000
#> lmbd_nt2pbc2_INTPBC-(lmbd_nt2_INT*_2_PBC) 0.000
#> lmbd_nt3pbc3_INTPBC-(lmbd_nt3_INT*_3_PBC) 0.000
#> M_INTPBC-((C_ATT_PBC*G_ATT_INT+C_SN_PBC*G 0.000
#>
# LMS approach
est_tpb_lms <- modsem(tpb, data = TPB, method = "lms", nodes = 32)
summary(est_tpb_lms)
#>
#> modsem (version 1.0.11):
#>
#> Estimator LMS
#> Optimization method EMA-NLMINB
#> Number of observations 2000
#> Number of iterations 32
#> Loglikelihood -23423.24
#> Akaike (AIC) 46954.49
#> Bayesian (BIC) 47256.94
#>
#> 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 2969.98
#> Difference test (D) 5939.96
#> Degrees of freedom (D) 1
#> P-value (D) 0.000
#>
#> R-Squared Interaction Model (H1):
#> INT 0.366
#> BEH 0.263
#> R-Squared Baseline Model (H0):
#> INT 0.367
#> BEH 0.210
#> R-Squared Change (H1 - H0):
#> INT -0.001
#> BEH 0.053
#>
#> Parameter Estimates:
#> Coefficients unstandardized
#> Information observed
#> Standard errors standard
#>
#> Latent Variables:
#> Estimate Std.Error z.value P(>|z|)
#> PBC =~
#> pbc1 1.000
#> pbc2 0.912 0.013 69.46 0.000
#> pbc3 0.801 0.012 66.09 0.000
#> ATT =~
#> att1 1.000
#> att2 0.878 0.012 71.56 0.000
#> att3 0.789 0.012 66.38 0.000
#> att4 0.695 0.011 61.00 0.000
#> att5 0.887 0.013 70.85 0.000
#> SN =~
#> sn1 1.000
#> sn2 0.888 0.017 52.60 0.000
#> INT =~
#> int1 1.000
#> int2 0.913 0.015 59.06 0.000
#> int3 0.807 0.014 55.74 0.000
#> BEH =~
#> b1 1.000
#> b2 0.960 0.030 31.75 0.000
#>
#> Regressions:
#> Estimate Std.Error z.value P(>|z|)
#> INT ~
#> PBC 0.217 0.030 7.35 0.000
#> ATT 0.213 0.026 8.16 0.000
#> SN 0.176 0.028 6.38 0.000
#> BEH ~
#> PBC 0.232 0.022 10.41 0.000
#> INT 0.190 0.025 7.68 0.000
#> PBC:INT 0.205 0.018 11.31 0.000
#>
#> Intercepts:
#> Estimate Std.Error z.value P(>|z|)
#> .pbc1 0.998 0.024 42.46 0.000
#> .pbc2 0.985 0.022 44.98 0.000
#> .pbc3 0.992 0.020 50.50 0.000
#> .att1 1.014 0.024 42.00 0.000
#> .att2 1.007 0.021 46.97 0.000
#> .att3 1.016 0.020 51.46 0.000
#> .att4 0.999 0.018 55.66 0.000
#> .att5 0.992 0.022 45.67 0.000
#> .sn1 1.006 0.024 41.68 0.000
#> .sn2 1.010 0.022 46.73 0.000
#> .int1 1.014 0.022 46.97 0.000
#> .int2 1.012 0.020 50.41 0.000
#> .int3 1.005 0.018 54.81 0.000
#> .b1 1.001 0.021 46.99 0.000
#> .b2 1.019 0.020 51.07 0.000
#>
#> Covariances:
#> Estimate Std.Error z.value P(>|z|)
#> PBC ~~
#> ATT 0.678 0.029 23.74 0.000
#> SN 0.677 0.029 23.36 0.000
#> ATT ~~
#> SN 0.629 0.029 21.95 0.000
#>
#> Variances:
#> Estimate Std.Error z.value P(>|z|)
#> .pbc1 0.144 0.008 18.39 0.000
#> .pbc2 0.160 0.007 21.44 0.000
#> .pbc3 0.155 0.006 23.90 0.000
#> .att1 0.167 0.007 23.54 0.000
#> .att2 0.150 0.006 24.72 0.000
#> .att3 0.159 0.006 26.39 0.000
#> .att4 0.162 0.006 27.66 0.000
#> .att5 0.159 0.006 24.93 0.000
#> .sn1 0.178 0.015 12.09 0.000
#> .sn2 0.157 0.012 13.26 0.000
#> .int1 0.157 0.009 18.11 0.000
#> .int2 0.160 0.008 20.41 0.000
#> .int3 0.168 0.007 23.55 0.000
#> .b1 0.186 0.018 10.12 0.000
#> .b2 0.135 0.017 8.14 0.000
#> PBC 0.962 0.035 27.31 0.000
#> ATT 0.998 0.037 27.14 0.000
#> SN 0.987 0.039 25.41 0.000
#> .INT 0.491 0.020 24.65 0.000
#> .BEH 0.455 0.023 20.14 0.000
#>
# QML approach
est_tpb_qml <- modsem(tpb, data = TPB, method = "qml")
summary(est_tpb_qml)
#>
#> modsem (version 1.0.11):
#>
#> Estimator QML
#> Optimization method NLMINB
#> Number of observations 2000
#> Number of iterations 68
#> Loglikelihood -26326.42
#> Akaike (AIC) 52760.83
#> Bayesian (BIC) 53063.28
#>
#> 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 66.81
#> Difference test (D) 133.61
#> Degrees of freedom (D) 1
#> P-value (D) 0.000
#>
#> R-Squared Interaction Model (H1):
#> BEH 0.261
#> INT 0.366
#> R-Squared Baseline Model (H0):
#> BEH 0.210
#> INT 0.367
#> R-Squared Change (H1 - H0):
#> BEH 0.051
#> INT 0.000
#>
#> Parameter Estimates:
#> Coefficients unstandardized
#> Information observed
#> Standard errors standard
#>
#> Latent Variables:
#> Estimate Std.Error z.value P(>|z|)
#> INT =~
#> int1 1.000
#> int2 0.913 0.015 59.05 0.000
#> int3 0.807 0.014 55.73 0.000
#> ATT =~
#> att1 1.000
#> att2 0.878 0.012 71.56 0.000
#> att3 0.789 0.012 66.37 0.000
#> att4 0.695 0.011 61.00 0.000
#> att5 0.887 0.013 70.85 0.000
#> SN =~
#> sn1 1.000
#> sn2 0.888 0.017 52.62 0.000
#> PBC =~
#> pbc1 1.000
#> pbc2 0.912 0.013 69.48 0.000
#> pbc3 0.801 0.012 66.10 0.000
#> BEH =~
#> b1 1.000
#> b2 0.961 0.030 31.69 0.000
#>
#> Regressions:
#> Estimate Std.Error z.value P(>|z|)
#> BEH ~
#> INT 0.189 0.025 7.68 0.000
#> PBC 0.232 0.022 10.40 0.000
#> INT:PBC 0.206 0.018 11.36 0.000
#> INT ~
#> ATT 0.213 0.026 8.17 0.000
#> SN 0.176 0.028 6.38 0.000
#> PBC 0.217 0.030 7.35 0.000
#>
#> Intercepts:
#> Estimate Std.Error z.value P(>|z|)
#> .int1 1.014 0.022 46.96 0.000
#> .int2 1.012 0.020 50.40 0.000
#> .int3 1.005 0.018 54.80 0.000
#> .att1 1.014 0.024 42.01 0.000
#> .att2 1.007 0.021 46.97 0.000
#> .att3 1.016 0.020 51.45 0.000
#> .att4 0.999 0.018 55.65 0.000
#> .att5 0.992 0.022 45.67 0.000
#> .sn1 1.006 0.024 41.66 0.000
#> .sn2 1.010 0.022 46.71 0.000
#> .pbc1 0.998 0.024 42.41 0.000
#> .pbc2 0.985 0.022 44.93 0.000
#> .pbc3 0.991 0.020 50.45 0.000
#> .b1 1.001 0.021 46.85 0.000
#> .b2 1.019 0.020 50.91 0.000
#>
#> Covariances:
#> Estimate Std.Error z.value P(>|z|)
#> ATT ~~
#> SN 0.629 0.029 21.70 0.000
#> PBC 0.678 0.029 23.45 0.000
#> SN ~~
#> PBC 0.678 0.029 23.08 0.000
#>
#> Variances:
#> Estimate Std.Error z.value P(>|z|)
#> .int1 0.157 0.009 18.11 0.000
#> .int2 0.160 0.008 20.41 0.000
#> .int3 0.168 0.007 23.55 0.000
#> .att1 0.167 0.007 23.53 0.000
#> .att2 0.150 0.006 24.71 0.000
#> .att3 0.160 0.006 26.38 0.000
#> .att4 0.162 0.006 27.64 0.000
#> .att5 0.159 0.006 24.93 0.000
#> .sn1 0.178 0.015 12.09 0.000
#> .sn2 0.157 0.012 13.26 0.000
#> .pbc1 0.144 0.008 18.39 0.000
#> .pbc2 0.160 0.007 21.43 0.000
#> .pbc3 0.155 0.006 23.89 0.000
#> .b1 0.186 0.018 10.13 0.000
#> .b2 0.135 0.017 8.08 0.000
#> .BEH 0.459 0.023 20.20 0.000
#> .INT 0.491 0.020 24.64 0.000
#> ATT 0.998 0.037 26.93 0.000
#> SN 0.988 0.039 25.22 0.000
#> PBC 0.963 0.036 27.07 0.000
#>
# }