quadratic effects
quadratic.RmdQuadratic Effects and Interaction Effects
Quadratic effects are essentially a special case of interaction
effects—where a variable interacts with itself. As such, all of the
methods in modsem can also be used to estimate quadratic
effects.
Below is a simple example using the LMS approach.
library(modsem)
m1 <- '
# Outer Model
X =~ x1 + x2 + x3
Y =~ y1 + y2 + y3
Z =~ z1 + z2 + z3
# Inner model
Y ~ X + Z + Z:X + X:X
'
est1Lms <- modsem(m1, data = oneInt, method = "lms")
summary(est1Lms)
#>
#> modsem (version 1.0.4):
#> Estimator LMS
#> Optimization method EM-NLMINB
#> Number of observations 2000
#> Number of iterations 115
#> Loglikelihood -14687.59
#> Akaike (AIC) 29439.17
#> Bayesian (BIC) 29618.4
#>
#> 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 3144.29
#> Difference test (D) 6288.58
#> Degrees of freedom (D) 2
#> P-value (D) 0.000
#>
#> R-Squared:
#> Y 0.595
#> R-Squared Null-Model (H0):
#> Y 0.395
#> R-Squared Change:
#> Y 0.200
#>
#> 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 0.015 52.327 0.000
#> x3 0.915 0.014 63.559 0.000
#> Z =~
#> z1 1.000
#> z2 0.810 0.014 59.634 0.000
#> z3 0.881 0.014 61.614 0.000
#> Y =~
#> y1 1.000
#> y2 0.798 0.012 66.606 0.000
#> y3 0.899 0.008 108.526 0.000
#>
#> Regressions:
#> Estimate Std.Error z.value P(>|z|)
#> Y ~
#> X 0.672 0.040 16.768 0.000
#> Z 0.569 0.030 19.134 0.000
#> X:X -0.007 0.026 -0.249 0.803
#> X:Z 0.715 0.034 21.210 0.000
#>
#> Intercepts:
#> Estimate Std.Error z.value P(>|z|)
#> x1 1.022 0.023 44.828 0.000
#> x2 1.215 0.021 57.737 0.000
#> x3 0.919 0.020 45.714 0.000
#> z1 1.012 0.033 30.780 0.000
#> z2 1.206 0.027 44.861 0.000
#> z3 0.916 0.030 30.400 0.000
#> y1 1.044 0.041 25.480 0.000
#> y2 1.225 0.030 41.401 0.000
#> y3 0.960 0.037 25.942 0.000
#> Y 0.000
#> X 0.000
#> Z 0.000
#>
#> Covariances:
#> Estimate Std.Error z.value P(>|z|)
#> X ~~
#> Z 0.199 0.029 6.931 0.000
#>
#> Variances:
#> Estimate Std.Error z.value P(>|z|)
#> x1 0.160 0.008 18.897 0.000
#> x2 0.163 0.008 21.142 0.000
#> x3 0.165 0.008 20.508 0.000
#> z1 0.167 0.012 14.015 0.000
#> z2 0.160 0.009 16.948 0.000
#> z3 0.158 0.009 18.302 0.000
#> y1 0.160 0.010 15.274 0.000
#> y2 0.154 0.007 20.884 0.000
#> y3 0.163 0.008 19.504 0.000
#> X 0.972 0.042 23.289 0.000
#> Z 1.017 0.049 20.703 0.000
#> Y 0.983 0.047 20.796 0.000In this example, we have a simple model with two quadratic effects
and one interaction effect. We estimate the model using both the
QML and double-centering approaches, with data from a
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
'
est2Dblcent <- modsem(m2, data = jordan)
est2Qml <- modsem(m2, data = jordan, method = "qml")
#> Warning: SE's for some coefficients could not be computed.
summary(est2Qml)
#>
#> modsem (version 1.0.4):
#> Estimator QML
#> Optimization method NLMINB
#> Number of observations 6038
#> Number of iterations 86
#> Loglikelihood -110520.22
#> Akaike (AIC) 221142.45
#> Bayesian (BIC) 221484.45
#>
#> 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 1.06
#> 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 50.584 0.000
#> enjoy3 0.894 0.020 43.669 0.000
#> enjoy4 0.999 0.021 48.225 0.000
#> enjoy5 1.047 0.021 50.399 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 41.829 0.000
#> academic5 1.113 0.029 38.649 0.000
#> academic6 1.198 0.030 40.357 0.000
#> CAREER =~
#> career1 1.000
#> career2 1.040 0.016 65.181 0.000
#> career3 0.952 0.016 57.839 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 26.293 0.000
#> SC 0.467 0.023 19.899 0.000
#> ENJ:ENJ 0.026 0.022 1.221 0.222
#> ENJ:SC -0.040 0.046 -0.874 0.382
#> SC:SC -0.002 0.034 -0.049 0.961
#>
#> Intercepts:
#> Estimate Std.Error z.value P(>|z|)
#> enjoy1 0.000
#> enjoy2 0.000 0.005 0.032 0.975
#> enjoy3 0.000 0.009 -0.035 0.972
#> enjoy4 0.000
#> enjoy5 0.000 0.006 0.059 0.953
#> academic1 0.000 0.007 -0.016 0.988
#> academic2 0.000 0.011 -0.014 0.989
#> academic3 0.000 0.011 -0.039 0.969
#> academic4 0.000 0.010 -0.022 0.983
#> academic5 -0.001 0.011 -0.061 0.951
#> academic6 0.001 0.011 0.064 0.949
#> career1 -0.004 0.015 -0.246 0.806
#> career2 -0.005 0.015 -0.299 0.765
#> career3 -0.004 0.015 -0.255 0.798
#> career4 -0.004 0.014 -0.269 0.788
#> 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 44.332 0.000
#> enjoy2 0.488 0.011 44.407 0.000
#> enjoy3 0.596 0.012 48.234 0.000
#> enjoy4 0.488 0.011 44.561 0.000
#> enjoy5 0.442 0.010 42.472 0.000
#> academic1 0.644 0.013 49.816 0.000
#> academic2 0.566 0.012 47.862 0.000
#> academic3 0.474 0.011 44.316 0.000
#> academic4 0.455 0.010 43.580 0.000
#> academic5 0.565 0.012 47.696 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 37.018 0.000
#> career3 0.436 0.010 43.274 0.000
#> career4 0.576 0.012 48.373 0.000
#> ENJ 0.500 0.017 29.546 0.000
#> SC 0.338 0.015 23.196 0.000
#> CAREER 0.302 0.010 29.710 0.000Note: The other approaches (e.g., LMS
and constrained methods) can also be used but may be slower depending on
the number of interaction effects, especially for the LMS
and constrained approaches.