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 1.0.8):
#> 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 0.014 55.948 0.000
#> x3 0.915 0.016 55.893 0.000
#> Z =~
#> z1 1.000
#> z2 0.810 0.017 47.672 0.000
#> z3 0.881 0.016 55.160 0.000
#> Y =~
#> y1 1.000
#> y2 0.798 0.009 90.080 0.000
#> y3 0.899 0.011 81.861 0.000
#>
#> Regressions:
#> Estimate Std.Error z.value P(>|z|)
#> Y ~
#> X 0.672 0.038 17.528 0.000
#> Z 0.569 0.033 17.367 0.000
#> X:X -0.007 0.024 -0.271 0.787
#> X:Z 0.715 0.035 20.354 0.000
#>
#> Intercepts:
#> Estimate Std.Error z.value P(>|z|)
#> x1 1.022 0.023 44.849 0.000
#> x2 1.215 0.022 56.299 0.000
#> x3 0.919 0.022 42.355 0.000
#> z1 1.012 0.035 28.955 0.000
#> z2 1.206 0.024 49.720 0.000
#> z3 0.916 0.029 31.176 0.000
#> y1 1.044 0.037 28.452 0.000
#> y2 1.225 0.028 43.895 0.000
#> y3 0.960 0.036 26.470 0.000
#> Y 0.000
#> X 0.000
#> Z 0.000
#>
#> Covariances:
#> Estimate Std.Error z.value P(>|z|)
#> X ~~
#> Z 0.199 0.031 6.517 0.000
#>
#> Variances:
#> Estimate Std.Error z.value P(>|z|)
#> x1 0.160 0.009 17.922 0.000
#> x2 0.163 0.007 21.806 0.000
#> x3 0.165 0.009 19.308 0.000
#> z1 0.167 0.011 15.723 0.000
#> z2 0.160 0.009 18.506 0.000
#> z3 0.158 0.010 16.188 0.000
#> y1 0.160 0.011 14.413 0.000
#> y2 0.154 0.009 16.941 0.000
#> y3 0.163 0.010 16.464 0.000
#> X 0.972 0.041 23.658 0.000
#> Z 1.017 0.046 22.038 0.000
#> Y 0.983 0.042 23.366 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 1.0.8):
#> Estimator QML
#> Optimization method NLMINB
#> Number of observations 6038
#> Number of iterations 71
#> Loglikelihood -110520.23
#> 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 0.513
#> R-Squared Null-Model (H0):
#> CAREER 0.510
#> R-Squared Change:
#> CAREER 0.003
#>
#> 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 50.583 0.000
#> enjoy3 0.894 0.020 43.667 0.000
#> enjoy4 0.999 0.021 48.224 0.000
#> enjoy5 1.047 0.021 50.398 0.000
#> SC =~
#> academic1 1.000
#> academic2 1.104 0.028 38.949 0.000
#> academic3 1.235 0.030 41.723 0.000
#> academic4 1.254 0.030 41.832 0.000
#> academic5 1.113 0.029 38.651 0.000
#> academic6 1.198 0.030 40.360 0.000
#> CAREER =~
#> career1 1.000
#> career2 1.040 0.016 65.180 0.000
#> career3 0.952 0.016 57.839 0.000
#> career4 0.818 0.017 48.357 0.000
#>
#> Regressions:
#> Estimate Std.Error z.value P(>|z|)
#> CAREER ~
#> ENJ 0.524 0.020 26.422 0.000
#> SC 0.467 0.023 19.886 0.000
#> ENJ:ENJ 0.027 0.022 1.225 0.220
#> ENJ:SC -0.040 0.046 -0.873 0.383
#> SC:SC -0.001 0.035 -0.043 0.966
#>
#> Intercepts:
#> Estimate Std.Error z.value P(>|z|)
#> enjoy1 0.000
#> enjoy2 0.000
#> enjoy3 0.000
#> enjoy4 0.000
#> enjoy5 0.000
#> academic1 0.000 0.012 -0.011 0.991
#> academic2 0.000
#> academic3 0.000 0.011 -0.039 0.969
#> academic4 0.000 0.009 -0.027 0.979
#> academic5 -0.001 0.011 -0.056 0.955
#> academic6 0.001 0.008 0.086 0.931
#> career1 -0.004
#> career2 -0.005
#> career3 -0.004
#> career4 -0.004 0.006 -0.643 0.520
#> 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 44.404 0.000
#> enjoy3 0.596 0.012 48.232 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 49.812 0.000
#> academic2 0.566 0.012 47.864 0.000
#> academic3 0.473 0.011 44.318 0.000
#> academic4 0.455 0.010 43.579 0.000
#> academic5 0.565 0.012 47.696 0.000
#> academic6 0.502 0.011 45.435 0.000
#> career1 0.373 0.009 40.391 0.000
#> career2 0.328 0.009 37.018 0.000
#> career3 0.436 0.010 43.271 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 23.198 0.000
#> CAREER 0.302 0.010 29.712 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.