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
'
est1_lms <- modsem(m1, data = oneInt, method = "lms")
summary(est1_lms)
#>
#> modsem (1.0.13) ended normally after 50 iterations
#>
#> Estimator LMS
#> Optimization method EMA-NLMINB
#> Number of model parameters 32
#>
#> Number of observations 2000
#>
#> Loglikelihood and Information Criteria:
#> Loglikelihood -17493.58
#> Akaike (AIC) 35051.16
#> Bayesian (BIC) 35230.39
#>
#> Numerical Integration:
#> Points of integration (per dim) 24
#> Dimensions 1
#> Total points of integration 24
#>
#> Fit Measures for Baseline Model (H0):
#> Standard
#> Chi-square 17.52
#> Degrees of Freedom (Chi-square) 24
#> P-value (Chi-square) 0.826
#> RMSEA 0.000
#>
#> Loglikelihood -17831.87
#> Akaike (AIC) 35723.75
#> Bayesian (BIC) 35891.78
#>
#> Comparative Fit to H0 (LRT test):
#> Loglikelihood change 338.29
#> Difference test (D) 676.59
#> Degrees of freedom (D) 2
#> 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 0.804 0.013 63.813 0.000
#> x3 0.914 0.014 67.610 0.000
#> Z =~
#> z1 1.000
#> z2 0.810 0.012 65.085 0.000
#> z3 0.881 0.013 67.613 0.000
#> Y =~
#> y1 1.000
#> y2 0.798 0.007 107.549 0.000
#> y3 0.899 0.008 112.580 0.000
#>
#> Regressions:
#> Estimate Std.Error z.value P(>|z|)
#> Y ~
#> X 0.673 0.031 21.654 0.000
#> Z 0.569 0.030 18.712 0.000
#> X:X -0.004 0.021 -0.194 0.846
#> Z:X 0.720 0.029 24.855 0.000
#>
#> Intercepts:
#> Estimate Std.Error z.value P(>|z|)
#> .x1 1.023 0.024 42.796 0.000
#> .x2 1.215 0.020 60.867 0.000
#> .x3 0.919 0.022 41.391 0.000
#> .z1 1.011 0.024 41.565 0.000
#> .z2 1.206 0.020 59.262 0.000
#> .z3 0.916 0.022 42.054 0.000
#> .y1 1.040 0.038 27.463 0.000
#> .y2 1.222 0.031 39.874 0.000
#> .y3 0.956 0.034 27.884 0.000
#>
#> Covariances:
#> Estimate Std.Error z.value P(>|z|)
#> X ~~
#> Z 0.200 0.024 8.240 0.000
#>
#> Variances:
#> Estimate Std.Error z.value P(>|z|)
#> .x1 0.158 0.009 18.172 0.000
#> .x2 0.162 0.007 23.157 0.000
#> .x3 0.164 0.008 20.759 0.000
#> .z1 0.167 0.009 18.507 0.000
#> .z2 0.160 0.007 22.681 0.000
#> .z3 0.158 0.008 20.780 0.000
#> .y1 0.160 0.009 18.012 0.000
#> .y2 0.154 0.007 22.685 0.000
#> .y3 0.164 0.008 20.683 0.000
#> X 0.981 0.036 26.975 0.000
#> Z 1.017 0.038 26.934 0.000
#> .Y 0.979 0.038 25.929 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
'
est2_dca <- modsem(m2, data = jordan)
est2_qml <- modsem(m2, data = jordan, method = "qml")
summary(est2_qml)
#>
#> modsem (1.0.13) ended normally after 48 iterations
#>
#> Estimator QML
#> Optimization method NLMINB
#> Number of model parameters 51
#>
#> Number of observations 6038
#>
#> Loglikelihood and Information Criteria:
#> Loglikelihood -110519.99
#> Akaike (AIC) 221141.98
#> Bayesian (BIC) 221483.98
#>
#> Fit Measures for Baseline Model (H0):
#> Standard
#> Chi-square 1016.34
#> Degrees of Freedom (Chi-square) 87
#> P-value (Chi-square) 0.000
#> RMSEA 0.042
#>
#> Loglikelihood -110521.29
#> Akaike (AIC) 221138.58
#> Bayesian (BIC) 221460.46
#>
#> Comparative Fit to H0 (LRT test):
#> Loglikelihood change 1.30
#> Difference test (D) 2.59
#> Degrees of freedom (D) 3
#> P-value (D) 0.458
#>
#> R-Squared Interaction Model (H1):
#> CAREER 0.513
#> R-Squared Baseline Model (H0):
#> CAREER 0.510
#> R-Squared Change (H1 - H0):
#> 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 50.574 0.000
#> enjoy3 0.894 0.020 43.665 0.000
#> enjoy4 0.999 0.021 48.219 0.000
#> enjoy5 1.047 0.021 50.391 0.000
#> SC =~
#> academic1 1.000
#> academic2 1.104 0.028 38.951 0.000
#> academic3 1.235 0.030 41.727 0.000
#> academic4 1.254 0.030 41.836 0.000
#> academic5 1.113 0.029 38.653 0.000
#> academic6 1.198 0.030 40.364 0.000
#> CAREER =~
#> career1 1.000
#> career2 1.040 0.016 65.185 0.000
#> career3 0.952 0.016 57.843 0.000
#> career4 0.818 0.017 48.361 0.000
#>
#> Regressions:
#> Estimate Std.Error z.value P(>|z|)
#> CAREER ~
#> ENJ 0.526 0.020 26.309 0.000
#> SC 0.464 0.023 20.004 0.000
#> ENJ:ENJ 0.029 0.022 1.341 0.180
#> ENJ:SC -0.046 0.045 -1.015 0.310
#> SC:SC 0.001 0.036 0.025 0.980
#>
#> Intercepts:
#> Estimate Std.Error z.value P(>|z|)
#> .enjoy1 0.000 0.011 -0.010 0.992
#> .enjoy2 0.000 0.013 0.011 0.991
#> .enjoy3 0.000 0.017 -0.018 0.986
#> .enjoy4 0.000 0.016 0.005 0.996
#> .enjoy5 0.000 0.016 0.020 0.984
#> .academic1 0.000 0.014 -0.011 0.991
#> .academic2 0.000 0.013 -0.011 0.992
#> .academic3 0.000 0.013 -0.034 0.973
#> .academic4 0.000 0.014 -0.018 0.986
#> .academic5 -0.001 0.013 -0.049 0.961
#> .academic6 0.001 0.015 0.046 0.964
#> .career1 -0.005 0.020 -0.231 0.817
#> .career2 -0.005 0.020 -0.270 0.787
#> .career3 -0.005 0.019 -0.240 0.811
#> .career4 -0.005 0.018 -0.257 0.798
#>
#> Covariances:
#> Estimate Std.Error z.value P(>|z|)
#> ENJ ~~
#> SC 0.218 0.009 25.479 0.000
#>
#> Variances:
#> Estimate Std.Error z.value P(>|z|)
#> .enjoy1 0.487 0.011 44.350 0.000
#> .enjoy2 0.489 0.011 44.421 0.000
#> .enjoy3 0.596 0.012 48.234 0.000
#> .enjoy4 0.488 0.011 44.568 0.000
#> .enjoy5 0.442 0.010 42.478 0.000
#> .academic1 0.644 0.013 49.812 0.000
#> .academic2 0.566 0.012 47.864 0.000
#> .academic3 0.473 0.011 44.320 0.000
#> .academic4 0.455 0.010 43.582 0.000
#> .academic5 0.565 0.012 47.684 0.000
#> .academic6 0.502 0.011 45.441 0.000
#> .career1 0.373 0.009 40.396 0.000
#> .career2 0.328 0.009 37.017 0.000
#> .career3 0.436 0.010 43.275 0.000
#> .career4 0.576 0.012 48.375 0.000
#> ENJ 0.500 0.017 29.545 0.000
#> SC 0.338 0.015 23.199 0.000
#> .CAREER 0.302 0.010 29.725 0.000NOTE: We can also use the LMS approach to estimate
this model, but it will be a lot slower, since we have to integrate
along both ENJ and SC. In the first example it
is sufficient to only integrate along X, but the addition
of the SC:SC term means that we have to explicitly model
SC as a moderator. This means that we (by default) have to
integrate along 24^2=576 nodes. This both affects the the
optimization process, but also dramatically affects the computation time
of the standard errors. To make the estimation process it is possible to
reduce the number of quadrature nodes. Additionally, we can also pass
mean.observed = FALSE, constraining the intercepts of the
indicators to zero.
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
'
est2_lms <- modsem(m2, data = jordan, method = "lms",
nodes = 15, mean.observed = FALSE)
summary(est2_lms)
#>
#> modsem (1.0.13) ended normally after 22 iterations
#>
#> Estimator LMS
#> Optimization method EMA-NLMINB
#> Number of model parameters 37
#>
#> Number of observations 6038
#>
#> Loglikelihood and Information Criteria:
#> Loglikelihood -110520.02
#> Akaike (AIC) 221114.03
#> Bayesian (BIC) 221362.15
#>
#> Numerical Integration:
#> Points of integration (per dim) 15
#> Dimensions 2
#> Total points of integration 225
#>
#> Fit Measures for Baseline Model (H0):
#> Standard
#> Chi-square 1016.34
#> Degrees of Freedom (Chi-square) 87
#> P-value (Chi-square) 0.000
#> RMSEA 0.042
#>
#> Loglikelihood -110521.29
#> Akaike (AIC) 221108.58
#> Bayesian (BIC) 221329.87
#>
#> Comparative Fit to H0 (LRT test):
#> Loglikelihood change 1.27
#> Difference test (D) 2.55
#> Degrees of freedom (D) 4
#> P-value (D) 0.636
#>
#> R-Squared Interaction Model (H1):
#> CAREER 0.512
#> R-Squared Baseline Model (H0):
#> CAREER 0.510
#> R-Squared Change (H1 - H0):
#> 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 50.379 0.000
#> enjoy3 0.894 0.021 43.536 0.000
#> enjoy4 0.998 0.021 48.014 0.000
#> enjoy5 1.047 0.021 50.180 0.000
#> SC =~
#> academic1 1.000
#> academic2 1.105 0.029 38.558 0.000
#> academic3 1.236 0.030 41.239 0.000
#> academic4 1.255 0.030 41.331 0.000
#> academic5 1.114 0.029 38.257 0.000
#> academic6 1.199 0.030 39.900 0.000
#> CAREER =~
#> career1 1.000
#> career2 1.040 0.016 65.193 0.000
#> career3 0.952 0.016 57.847 0.000
#> career4 0.818 0.017 48.365 0.000
#>
#> Regressions:
#> Estimate Std.Error z.value P(>|z|)
#> CAREER ~
#> ENJ 0.525 0.020 26.396 0.000
#> SC 0.465 0.023 20.168 0.000
#> ENJ:ENJ 0.028 0.021 1.321 0.187
#> ENJ:SC -0.049 0.042 -1.160 0.246
#> SC:SC 0.003 0.032 0.081 0.935
#>
#> Intercepts:
#> Estimate Std.Error z.value P(>|z|)
#> .CAREER -0.004 0.013 -0.330 0.742
#>
#> Covariances:
#> Estimate Std.Error z.value P(>|z|)
#> ENJ ~~
#> SC 0.216 0.009 25.383 0.000
#>
#> Variances:
#> Estimate Std.Error z.value P(>|z|)
#> .enjoy1 0.487 0.011 44.349 0.000
#> .enjoy2 0.488 0.011 44.393 0.000
#> .enjoy3 0.596 0.012 48.227 0.000
#> .enjoy4 0.488 0.011 44.630 0.000
#> .enjoy5 0.442 0.010 42.529 0.000
#> .academic1 0.645 0.013 49.799 0.000
#> .academic2 0.566 0.012 47.870 0.000
#> .academic3 0.473 0.011 44.311 0.000
#> .academic4 0.455 0.010 43.588 0.000
#> .academic5 0.565 0.012 47.684 0.000
#> .academic6 0.502 0.011 45.432 0.000
#> .career1 0.373 0.009 40.396 0.000
#> .career2 0.328 0.009 37.021 0.000
#> .career3 0.436 0.010 43.282 0.000
#> .career4 0.576 0.012 48.380 0.000
#> ENJ 0.498 0.017 29.579 0.000
#> SC 0.337 0.015 22.823 0.000
#> .CAREER 0.302 0.010 29.736 0.000