The Basic Syntax
modsem basically introduces a new feature to the lavaan-syntax – the semicolon operator (“:”). The semicolon operator works the same way as in the lm()-function. In order to specify an interaction effect between two variables, you join them by Var1:Var2, Models can either be estimated using the one of the product indicator approaches (“ca”, “rca”, “dblcent”, “pind”) or by using the latent moderated structural equations approach (“lms”), or the quasi maximum likelihood approach (“qml”). The product indicator approaches are estimated via lavaan, whilst the lms and qml approaches are estimated via modsem itself.
A Simple Example
Here we can see a simple example of how to specify an interaction effect between two latent variables in lavaan.
m1 <- '
# Outer Model
X =~ x1 + x2 +x3
Y =~ y1 + y2 + y3
Z =~ z1 + z2 + z3
# Inner model
Y ~ X + Z + X:Z
'
est1 <- modsem(m1, oneInt)
summary(est1)
#> modsem:
#> Method = dblcent
#> lavaan 0.6-18 ended normally after 159 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 ~~
#> .x2z2 0.000
#> .x2z3 0.000
#> .x3z2 0.000
#> .x3z3 0.000
#> .x2z1 ~~
#> .x1z2 0.000
#> .x1z2 ~~
#> .x2z3 0.000
#> .x3z1 ~~
#> .x1z2 0.000
#> .x1z2 ~~
#> .x3z3 0.000
#> .x2z1 ~~
#> .x1z3 0.000
#> .x2z2 ~~
#> .x1z3 0.000
#> .x3z1 ~~
#> .x1z3 0.000
#> .x3z2 ~~
#> .x1z3 0.000
#> .x2z1 ~~
#> .x3z2 0.000
#> .x3z3 0.000
#> .x3z1 ~~
#> .x2z2 0.000
#> .x2z2 ~~
#> .x3z3 0.000
#> .x3z1 ~~
#> .x2z3 0.000
#> .x3z2 ~~
#> .x2z3 0.000
#> .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 0.000By default the model is estimated using the “dblcent” method. If you want to use another method, but the method can be changed using the method argument.
est1 <- modsem(m1, oneInt, method = "lms")
summary(est1)
#> Estimating null model
#> EM: Iteration = 1, LogLik = -17831.87, Change = -17831.875
#> EM: Iteration = 2, LogLik = -17831.87, Change = 0.000
#>
#> modsem (version 1.0.2):
#> 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 unstandardized
#> Information expected
#> Standard errors standard
#>
#> Latent Variables:
#> Estimate Std.Error z.value P(>|z|)
#> X =~
#> x1 1.000
#> x2 0.804 0.013 63.7 0.000
#> x3 0.915 0.014 67.4 0.000
#> Z =~
#> z1 1.000
#> z2 0.810 0.012 66.2 0.000
#> z3 0.881 0.013 67.7 0.000
#> Y =~
#> y1 1.000
#> y2 0.799 0.008 105.3 0.000
#> y3 0.899 0.008 111.2 0.000
#>
#> Regressions:
#> Estimate Std.Error z.value P(>|z|)
#> Y ~
#> X 0.676 0.031 21.7 0.000
#> Z 0.572 0.029 20.0 0.000
#> X:Z 0.712 0.027 25.9 0.000
#>
#> Intercepts:
#> Estimate Std.Error z.value P(>|z|)
#> x1 1.025 0.019 53.2 0.000
#> x2 1.218 0.017 73.5 0.000
#> x3 0.922 0.018 50.8 0.000
#> z1 1.016 0.024 42.0 0.000
#> z2 1.209 0.020 59.6 0.000
#> z3 0.920 0.022 42.6 0.000
#> y1 1.046 0.031 33.7 0.000
#> y2 1.227 0.025 48.5 0.000
#> y3 0.962 0.028 34.2 0.000
#> Y 0.000
#> X 0.000
#> Z 0.000
#>
#> Covariances:
#> Estimate Std.Error z.value P(>|z|)
#> X ~~
#> Z 0.198 0.024 8.3 0.000
#>
#> Variances:
#> Estimate Std.Error z.value P(>|z|)
#> x1 0.160 0.008 19.3 0.000
#> x2 0.163 0.007 23.5 0.000
#> x3 0.165 0.008 21.5 0.000
#> z1 0.166 0.009 18.4 0.000
#> z2 0.160 0.007 22.6 0.000
#> z3 0.158 0.008 20.9 0.000
#> y1 0.160 0.009 17.8 0.000
#> y2 0.154 0.007 22.7 0.000
#> y3 0.163 0.008 20.7 0.000
#> X 0.972 0.016 60.6 0.000
#> Z 1.017 0.019 54.9 0.000
#> Y 0.984 0.037 26.3 0.000Interactions Between two Observed Variables
modsem does not only allow you to estimate interactions between latent variables, but also interactions between observed variables. Here we first run a regression with only observed variables, where there is an interaction between x1 and z2, and then run an equivalent model using modsem().
Regression
reg1 <- lm(y1 ~ x1*z1, oneInt)
summary(reg1)
#>
#> Call:
#> lm(formula = y1 ~ x1 * z1, data = oneInt)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -3.7155 -0.8087 -0.0367 0.8078 4.6531
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 0.51422 0.04618 11.135 <2e-16 ***
#> x1 0.05477 0.03387 1.617 0.1060
#> z1 -0.06575 0.03461 -1.900 0.0576 .
#> x1:z1 0.54361 0.02272 23.926 <2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 1.184 on 1996 degrees of freedom
#> Multiple R-squared: 0.4714, Adjusted R-squared: 0.4706
#> F-statistic: 593.3 on 3 and 1996 DF, p-value: < 2.2e-16Using modsem() In general, when you have interactions between observed variables it is recommended that you use method = “pind”. Interaction effects with observed variables is not supported by the LMS- and QML-approach. In certain circumstances, you can define a latent variabale with a single indicator to estimate the interaction effect between two observed variables, in the LMS and QML approach, but it is generally not recommended.
# Here we use "pind" as the method (see chapter 3)
est2 <- modsem('y1 ~ x1 + z1 + x1:z1', data = oneInt, method = "pind")
summary(est2)
#> modsem:
#> Method = pind
#> lavaan 0.6-18 ended normally after 1 iteration
#>
#> Estimator ML
#> Optimization method NLMINB
#> Number of model parameters 4
#>
#> Number of observations 2000
#>
#> Model Test User Model:
#>
#> Test statistic 0.000
#> Degrees of freedom 0
#>
#> Parameter Estimates:
#>
#> Standard errors Standard
#> Information Expected
#> Information saturated (h1) model Structured
#>
#> Regressions:
#> Estimate Std.Err z-value P(>|z|)
#> y1 ~
#> x1 0.055 0.034 1.619 0.105
#> z1 -0.066 0.035 -1.902 0.057
#> x1z1 0.544 0.023 23.950 0.000
#>
#> Variances:
#> Estimate Std.Err z-value P(>|z|)
#> .y1 1.399 0.044 31.623 0.000Interactions between Latent and Observed Variables
modsem also allows you to estimate interaction effects between latent and observed variables. To do so, you just join a latent and an observed variable by a colon, e.g., ‘latent:observer’. As with interactions between observed variables, it is generally recommended that you use method = “pind” for estimating the effect between observed x latent
m3 <- '
# Outer Model
X =~ x1 + x2 +x3
Y =~ y1 + y2 + y3
# Inner model
Y ~ X + z1 + X:z1
'
est3 <- modsem(m3, oneInt, method = "pind")
summary(est3)
#> modsem:
#> Method = pind
#> lavaan 0.6-18 ended normally after 45 iterations
#>
#> Estimator ML
#> Optimization method NLMINB
#> Number of model parameters 22
#>
#> Number of observations 2000
#>
#> Model Test User Model:
#>
#> Test statistic 4468.171
#> Degrees of freedom 32
#> P-value (Chi-square) 0.000
#>
#> 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 0.803 0.013 63.697 0.000
#> x3 0.915 0.014 67.548 0.000
#> Y =~
#> y1 1.000
#> y2 0.798 0.007 115.375 0.000
#> y3 0.899 0.007 120.783 0.000
#> Xz1 =~
#> x1z1 1.000
#> x2z1 0.947 0.010 96.034 0.000
#> x3z1 0.902 0.009 99.512 0.000
#>
#> Regressions:
#> Estimate Std.Err z-value P(>|z|)
#> Y ~
#> X 0.021 0.034 0.614 0.540
#> z1 -0.185 0.023 -8.096 0.000
#> Xz1 0.646 0.017 37.126 0.000
#>
#> Covariances:
#> Estimate Std.Err z-value P(>|z|)
#> X ~~
#> Xz1 1.243 0.055 22.523 0.000
#>
#> Variances:
#> Estimate Std.Err z-value P(>|z|)
#> .x1 0.158 0.009 17.976 0.000
#> .x2 0.164 0.007 23.216 0.000
#> .x3 0.162 0.008 20.325 0.000
#> .y1 0.158 0.009 17.819 0.000
#> .y2 0.154 0.007 22.651 0.000
#> .y3 0.164 0.008 20.744 0.000
#> .x1z1 0.315 0.017 18.328 0.000
#> .x2z1 0.428 0.019 22.853 0.000
#> .x3z1 0.337 0.016 21.430 0.000
#> X 0.982 0.036 26.947 0.000
#> .Y 1.112 0.040 27.710 0.000
#> Xz1 3.965 0.136 29.217 0.000Quadratic Effects
In essence, quadratic effects are just a special case of interaction effects. Thus modsem can also be used to estimate quadratic effects.
m4 <- '
# Outer Model
X =~ x1 + x2 + x3
Y =~ y1 + y2 + y3
Z =~ z1 + z2 + z3
# Inner model
Y ~ X + Z + Z:X + X:X
'
est4 <- modsem(m4, oneInt, "qml")
summary(est4)
#> Estimating null model
#> Starting M-step
#>
#> modsem (version 1.0.2):
#> Estimator QML
#> Optimization method NLMINB
#> Number of observations 2000
#> Number of iterations 123
#> Loglikelihood -17496.2
#> Akaike (AIC) 35056.4
#> Bayesian (BIC) 35235.63
#>
#> 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 335.68
#> Difference test (D) 671.35
#> Degrees of freedom (D) 2
#> P-value (D) 0.000
#>
#> R-Squared:
#> Y 0.607
#> R-Squared Null-Model (H0):
#> Y 0.395
#> R-Squared Change:
#> Y 0.212
#>
#> 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 63.961 0.000
#> x3 0.914 0.013 67.797 0.000
#> Z =~
#> z1 1.000
#> z2 0.810 0.012 65.124 0.000
#> z3 0.881 0.013 67.621 0.000
#> Y =~
#> y1 1.000
#> y2 0.798 0.007 107.567 0.000
#> y3 0.899 0.008 112.542 0.000
#>
#> Regressions:
#> Estimate Std.Error z.value P(>|z|)
#> Y ~
#> X 0.674 0.032 20.888 0.000
#> Z 0.566 0.030 18.948 0.000
#> X:X -0.005 0.023 -0.207 0.836
#> X:Z 0.713 0.029 24.554 0.000
#>
#> Intercepts:
#> Estimate Std.Error z.value P(>|z|)
#> x1 1.023 0.024 42.894 0.000
#> x2 1.216 0.020 60.996 0.000
#> x3 0.919 0.022 41.484 0.000
#> z1 1.012 0.024 41.576 0.000
#> z2 1.206 0.020 59.271 0.000
#> z3 0.916 0.022 42.063 0.000
#> y1 1.042 0.038 27.684 0.000
#> y2 1.224 0.030 40.159 0.000
#> y3 0.958 0.034 28.101 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 8.239 0.000
#>
#> Variances:
#> Estimate Std.Error z.value P(>|z|)
#> x1 0.158 0.009 18.145 0.000
#> x2 0.162 0.007 23.188 0.000
#> x3 0.165 0.008 20.821 0.000
#> z1 0.166 0.009 18.341 0.000
#> z2 0.159 0.007 22.622 0.000
#> z3 0.158 0.008 20.714 0.000
#> y1 0.159 0.009 17.975 0.000
#> y2 0.154 0.007 22.670 0.000
#> y3 0.164 0.008 20.711 0.000
#> X 0.983 0.036 26.994 0.000
#> Z 1.019 0.038 26.951 0.000
#> Y 0.943 0.038 24.820 0.000More Complicated Examples
Here we can see a more complicated example using the model for the theory of planned 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 + INT:PBC
'
# the double centering apporach
est_tpb <- modsem(tpb, TPB)
# using the lms approach
est_tpb_lms <- modsem(tpb, TPB, method = "lms")
#> Warning: It is recommended that you have at least 32 nodes for interaction
#> effects between exogenous and endogenous variables in the lms approach 'nodes =
#> 24'
summary(est_tpb_lms)
#> Estimating null model
#> EM: Iteration = 1, LogLik = -26393.22, Change = -26393.223
#> EM: Iteration = 2, LogLik = -26393.22, Change = 0.000
#>
#> modsem (version 1.0.2):
#> Estimator LMS
#> Optimization method EM-NLMINB
#> Number of observations 2000
#> Number of iterations 103
#> Loglikelihood -23463.37
#> Akaike (AIC) 47034.74
#> Bayesian (BIC) 47337.19
#>
#> Numerical Integration:
#> Points of integration (per dim) 24
#> Dimensions 1
#> Total points of integration 24
#>
#> 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 2929.85
#> Difference test (D) 5859.70
#> Degrees of freedom (D) 1
#> P-value (D) 0.000
#>
#> R-Squared:
#> INT 0.361
#> BEH 0.248
#> R-Squared Null-Model (H0):
#> INT 0.367
#> BEH 0.210
#> R-Squared Change:
#> INT -0.006
#> BEH 0.038
#>
#> Parameter Estimates:
#> Coefficients unstandardized
#> Information expected
#> Standard errors standard
#>
#> Latent Variables:
#> Estimate Std.Error z.value P(>|z|)
#> PBC =~
#> pbc1 1.000
#> pbc2 0.911 0.013 68.99 0.000
#> pbc3 0.802 0.012 65.62 0.000
#> ATT =~
#> att1 1.000
#> att2 0.877 0.013 69.86 0.000
#> att3 0.789 0.012 65.22 0.000
#> att4 0.695 0.012 60.27 0.000
#> att5 0.887 0.013 69.73 0.000
#> SN =~
#> sn1 1.000
#> sn2 0.889 0.017 51.86 0.000
#> INT =~
#> int1 1.000
#> int2 0.913 0.016 58.72 0.000
#> int3 0.807 0.014 55.68 0.000
#> BEH =~
#> b1 1.000
#> b2 0.961 0.032 29.57 0.000
#>
#> Regressions:
#> Estimate Std.Error z.value P(>|z|)
#> INT ~
#> PBC 0.217 0.029 7.42 0.000
#> ATT 0.213 0.026 8.14 0.000
#> SN 0.177 0.028 6.43 0.000
#> BEH ~
#> PBC 0.228 0.022 10.16 0.000
#> INT 0.182 0.025 7.40 0.000
#> PBC:INT 0.204 0.019 10.78 0.000
#>
#> Intercepts:
#> Estimate Std.Error z.value P(>|z|)
#> pbc1 0.959 0.018 52.50 0.000
#> pbc2 0.950 0.017 54.98 0.000
#> pbc3 0.960 0.016 61.44 0.000
#> att1 0.987 0.022 45.30 0.000
#> att2 0.983 0.019 50.50 0.000
#> att3 0.995 0.018 55.40 0.000
#> att4 0.980 0.016 59.46 0.000
#> att5 0.969 0.020 49.18 0.000
#> sn1 0.979 0.022 44.88 0.000
#> sn2 0.987 0.020 50.19 0.000
#> int1 0.995 0.020 49.29 0.000
#> int2 0.995 0.019 52.46 0.000
#> int3 0.990 0.017 56.99 0.000
#> b1 0.989 0.021 47.62 0.000
#> b2 1.008 0.019 51.73 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 0.658 0.020 32.52 0.000
#> SN 0.657 0.021 31.50 0.000
#> ATT ~~
#> SN 0.616 0.019 33.05 0.000
#>
#> Variances:
#> Estimate Std.Error z.value P(>|z|)
#> pbc1 0.147 0.008 19.34 0.000
#> pbc2 0.164 0.007 22.31 0.000
#> pbc3 0.154 0.006 23.99 0.000
#> att1 0.167 0.007 23.50 0.000
#> att2 0.150 0.006 24.33 0.000
#> att3 0.159 0.006 26.54 0.000
#> att4 0.163 0.006 27.59 0.000
#> att5 0.159 0.006 25.12 0.000
#> sn1 0.178 0.015 12.07 0.000
#> sn2 0.156 0.012 13.13 0.000
#> int1 0.157 0.009 18.25 0.000
#> int2 0.160 0.008 20.68 0.000
#> int3 0.168 0.007 23.73 0.000
#> b1 0.186 0.019 9.68 0.000
#> b2 0.135 0.017 7.73 0.000
#> PBC 0.933 0.015 61.26 0.000
#> ATT 0.985 0.014 68.94 0.000
#> SN 0.974 0.015 63.45 0.000
#> INT 0.491 0.020 24.95 0.000
#> BEH 0.456 0.023 19.61 0.000Here is an example included two quadratic- and one interaction
effect, using the included dataset jordan. The dataset is
subset of the PISA 2006 dataset.
m2 <- '
ENJ =~ enjoy1 + enjoy2 + enjoy3 + enjoy4 + enjoy5
CAREER =~ career1 + career2 + career3 + career4
SC =~ academic1 + academic2 + academic3 + academic4 + academic5 + academic6
CAREER ~ ENJ + SC + ENJ:ENJ + SC:SC + ENJ:SC
'
est_jordan <- modsem(m2, data = jordan)
est_jordan_qml <- modsem(m2, data = jordan, method = "qml")
summary(est_jordan_qml)
#> Estimating null model
#> Starting M-step
#>
#> modsem (version 1.0.2):
#> Estimator QML
#> Optimization method NLMINB
#> Number of observations 6038
#> Number of iterations 101
#> Loglikelihood -110520.22
#> Akaike (AIC) 221142.45
#> Bayesian (BIC) 221484.44
#>
#> Fit Measures for H0:
#> Loglikelihood -110521
#> Akaike (AIC) 221138.58
#> Bayesian (BIC) 221460.46
#> Chi-square 1016.34
#> Degrees of Freedom (Chi-square) 87
#> P-value (Chi-square) 0.000
#> RMSEA 0.005
#>
#> Comparative fit to H0 (no interaction effect)
#> Loglikelihood change 1.07
#> Difference test (D) 2.13
#> Degrees of freedom (D) 3
#> P-value (D) 0.546
#>
#> R-Squared:
#> CAREER 0.512
#> R-Squared Null-Model (H0):
#> CAREER 0.510
#> R-Squared Change:
#> CAREER 0.002
#>
#> Parameter Estimates:
#> Coefficients unstandardized
#> Information observed
#> Standard errors standard
#>
#> Latent Variables:
#> Estimate Std.Error z.value P(>|z|)
#> ENJ =~
#> enjoy1 1.000
#> enjoy2 1.002 0.020 50.587 0.000
#> enjoy3 0.894 0.020 43.669 0.000
#> enjoy4 0.999 0.021 48.227 0.000
#> enjoy5 1.047 0.021 50.400 0.000
#> SC =~
#> academic1 1.000
#> academic2 1.104 0.028 38.946 0.000
#> academic3 1.235 0.030 41.720 0.000
#> academic4 1.254 0.030 41.828 0.000
#> academic5 1.113 0.029 38.647 0.000
#> academic6 1.198 0.030 40.356 0.000
#> CAREER =~
#> career1 1.000
#> career2 1.040 0.016 65.180 0.000
#> career3 0.952 0.016 57.838 0.000
#> career4 0.818 0.017 48.358 0.000
#>
#> Regressions:
#> Estimate Std.Error z.value P(>|z|)
#> CAREER ~
#> ENJ 0.523 0.020 26.286 0.000
#> SC 0.467 0.023 19.884 0.000
#> ENJ:ENJ 0.026 0.022 1.206 0.228
#> ENJ:SC -0.039 0.046 -0.851 0.395
#> SC:SC -0.002 0.035 -0.058 0.953
#>
#> Intercepts:
#> Estimate Std.Error z.value P(>|z|)
#> enjoy1 0.000 0.013 -0.008 0.994
#> enjoy2 0.000 0.015 0.010 0.992
#> enjoy3 0.000 0.013 -0.023 0.982
#> enjoy4 0.000 0.014 0.008 0.993
#> enjoy5 0.000 0.014 0.025 0.980
#> academic1 0.000 0.016 -0.009 0.993
#> academic2 0.000 0.014 -0.009 0.993
#> academic3 0.000 0.015 -0.028 0.978
#> academic4 0.000 0.016 -0.015 0.988
#> academic5 -0.001 0.014 -0.044 0.965
#> academic6 0.001 0.015 0.048 0.962
#> career1 -0.004 0.017 -0.204 0.838
#> career2 -0.004 0.018 -0.248 0.804
#> career3 -0.004 0.017 -0.214 0.830
#> career4 -0.004 0.016 -0.232 0.816
#> CAREER 0.000
#> ENJ 0.000
#> SC 0.000
#>
#> Covariances:
#> Estimate Std.Error z.value P(>|z|)
#> ENJ ~~
#> SC 0.218 0.009 25.477 0.000
#>
#> Variances:
#> Estimate Std.Error z.value P(>|z|)
#> enjoy1 0.487 0.011 44.335 0.000
#> enjoy2 0.488 0.011 44.406 0.000
#> enjoy3 0.596 0.012 48.233 0.000
#> enjoy4 0.488 0.011 44.561 0.000
#> enjoy5 0.442 0.010 42.470 0.000
#> academic1 0.645 0.013 49.813 0.000
#> academic2 0.566 0.012 47.864 0.000
#> academic3 0.473 0.011 44.319 0.000
#> academic4 0.455 0.010 43.579 0.000
#> academic5 0.565 0.012 47.695 0.000
#> academic6 0.502 0.011 45.434 0.000
#> career1 0.373 0.009 40.392 0.000
#> career2 0.328 0.009 37.019 0.000
#> career3 0.436 0.010 43.272 0.000
#> career4 0.576 0.012 48.372 0.000
#> ENJ 0.500 0.017 29.547 0.000
#> SC 0.338 0.015 23.195 0.000
#> CAREER 0.302 0.010 29.711 0.000Note: The other approaches work as well, but might be quite slow depending on the number of interaction effects (particularly for the LMS- and constrained approach).