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 (version 1.0.11):
#>
#> Estimator LMS
#> Optimization method EMA-NLMINB
#> Number of observations 2000
#> Number of iterations 43
#> Loglikelihood -14687.6
#> Akaike (AIC) 29439.21
#> Bayesian (BIC) 29618.43
#>
#> 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 3144.27
#> Difference test (D) 6288.54
#> Degrees of freedom (D) 2
#> P-value (D) 0.000
#>
#> R-Squared Interaction Model (H1):
#> Y 0.595
#> R-Squared Baseline Model (H0):
#> Y 0.395
#> R-Squared Change (H1 - H0):
#> Y 0.199
#>
#> 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 0.012 64.321 0.000
#> x3 0.915 0.013 67.946 0.000
#> Z =~
#> z1 1.000
#> z2 0.810 0.012 65.093 0.000
#> z3 0.881 0.013 67.607 0.000
#> Y =~
#> y1 1.000
#> y2 0.798 0.007 107.545 0.000
#> y3 0.899 0.008 112.576 0.000
#>
#> Regressions:
#> Estimate Std.Error z.value P(>|z|)
#> Y ~
#> X 0.672 0.031 21.680 0.000
#> Z 0.569 0.030 19.028 0.000
#> X:X -0.005 0.021 -0.258 0.796
#> X:Z 0.715 0.029 24.732 0.000
#>
#> Intercepts:
#> Estimate Std.Error z.value P(>|z|)
#> .x1 1.022 0.021 47.661 0.000
#> .x2 1.215 0.018 67.203 0.000
#> .x3 0.919 0.020 45.924 0.000
#> .z1 1.011 0.024 41.694 0.000
#> .z2 1.205 0.020 59.449 0.000
#> .z3 0.915 0.022 42.181 0.000
#> .y1 1.041 0.037 28.058 0.000
#> .y2 1.223 0.030 40.701 0.000
#> .y3 0.957 0.034 28.478 0.000
#>
#> Covariances:
#> Estimate Std.Error z.value P(>|z|)
#> X ~~
#> Z 0.199 0.024 8.323 0.000
#>
#> Variances:
#> Estimate Std.Error z.value P(>|z|)
#> .x1 0.160 0.008 19.299 0.000
#> .x2 0.163 0.007 23.886 0.000
#> .x3 0.165 0.008 21.226 0.000
#> .z1 0.167 0.009 18.477 0.000
#> .z2 0.160 0.007 22.672 0.000
#> .z3 0.158 0.008 20.798 0.000
#> .y1 0.160 0.009 18.011 0.000
#> .y2 0.154 0.007 22.680 0.000
#> .y3 0.164 0.008 20.680 0.000
#> X 0.972 0.033 29.873 0.000
#> Z 1.017 0.038 26.946 0.000
#> .Y 0.984 0.038 25.985 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")
#> Warning: The variance-covariance matrix of the estimated parameters
#> (vcov) does not appear to be positive definite! The smallest
#> eigenvalue (= -2.838495e-04) is smaller than zero. This may
#> be a symptom that the model is not identified.
summary(est2_qml)
#>
#> modsem (version 1.0.11):
#>
#> Estimator QML
#> Optimization method NLMINB
#> Number of observations 6038
#> Number of iterations 65
#> Loglikelihood -110516.71
#> Akaike (AIC) 221135.42
#> Bayesian (BIC) 221477.42
#>
#> Fit Measures for Baseline Model (H0):
#> Loglikelihood -110521.29
#> 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 0.042
#>
#> Comparative Fit to H0 (LRT test):
#> Loglikelihood change 4.58
#> Difference test (D) 9.16
#> Degrees of freedom (D) 3
#> P-value (D) 0.027
#>
#> R-Squared Interaction Model (H1):
#> CAREER 0.524
#> R-Squared Baseline Model (H0):
#> CAREER 0.510
#> R-Squared Change (H1 - H0):
#> CAREER 0.014
#>
#> 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 50.555 0.000
#> enjoy3 0.894 0.020 43.654 0.000
#> enjoy4 0.999 0.021 48.220 0.000
#> enjoy5 1.047 0.021 50.376 0.000
#> SC =~
#> academic1 1.000
#> academic2 1.104 0.028 38.950 0.000
#> academic3 1.235 0.030 41.718 0.000
#> academic4 1.254 0.030 41.827 0.000
#> academic5 1.113 0.029 38.646 0.000
#> academic6 1.199 0.030 40.365 0.000
#> CAREER =~
#> career1 1.000
#> career2 1.040 0.016 65.180 0.000
#> career3 0.952 0.016 57.859 0.000
#> career4 0.818 0.017 48.363 0.000
#>
#> Regressions:
#> Estimate Std.Error z.value P(>|z|)
#> CAREER ~
#> ENJ 0.543 0.021 26.141 0.000
#> SC 0.461 0.023 19.660 0.000
#> ENJ:ENJ 0.051 0.020 2.539 0.011
#> ENJ:SC -0.016 0.037 -0.435 0.664
#> SC:SC -0.003 0.030 -0.102 0.918
#>
#> Intercepts:
#> Estimate Std.Error z.value P(>|z|)
#> .enjoy1 0.000 0.007 -0.015 0.988
#> .enjoy2 0.000 0.007 0.022 0.983
#> .enjoy3 0.000 0.008 -0.036 0.971
#> .enjoy4 0.000 0.007 0.011 0.991
#> .enjoy5 0.000 0.009 0.039 0.969
#> .academic1 0.000 0.006 -0.022 0.983
#> .academic2 0.000 0.006 -0.021 0.983
#> .academic3 0.000 0.007 -0.063 0.950
#> .academic4 0.000 0.005 -0.043 0.966
#> .academic5 -0.001 0.010 -0.067 0.947
#> .academic6 0.001 0.008 0.082 0.935
#> .career1 -0.021 0.014 -1.473 0.141
#> .career2 -0.022 0.014 -1.567 0.117
#> .career3 -0.020 0.014 -1.430 0.153
#> .career4 -0.018 0.014 -1.300 0.193
#>
#> Covariances:
#> Estimate Std.Error z.value P(>|z|)
#> ENJ ~~
#> SC 0.218 0.009 25.476 0.000
#>
#> Variances:
#> Estimate Std.Error z.value P(>|z|)
#> .enjoy1 0.487 0.011 44.354 0.000
#> .enjoy2 0.489 0.011 44.444 0.000
#> .enjoy3 0.596 0.012 48.252 0.000
#> .enjoy4 0.487 0.011 44.570 0.000
#> .enjoy5 0.442 0.010 42.490 0.000
#> .academic1 0.645 0.013 49.810 0.000
#> .academic2 0.566 0.012 47.857 0.000
#> .academic3 0.473 0.011 44.316 0.000
#> .academic4 0.455 0.010 43.576 0.000
#> .academic5 0.565 0.012 47.690 0.000
#> .academic6 0.501 0.011 45.425 0.000
#> .career1 0.374 0.009 40.413 0.000
#> .career2 0.328 0.009 37.034 0.000
#> .career3 0.436 0.010 43.269 0.000
#> .career4 0.576 0.012 48.378 0.000
#> ENJ 0.500 0.017 29.535 0.000
#> SC 0.338 0.015 23.191 0.000
#> .CAREER 0.299 0.010 29.422 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, and calculate standard errors
using the outer product of the score function, instead of the negative
of the hessian matrix. 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, OFIM.hessian = FALSE,
mean.observed = FALSE)
#> Warning: It is recommended that you have at least 16 nodes for interaction
#> effects between exogenous variables in the lms approach 'nodes = 15'
summary(est2_lms)
#>
#> modsem (version 1.0.11):
#>
#> Estimator LMS
#> Optimization method EMA-NLMINB
#> Number of observations 6038
#> Number of iterations 51
#> Loglikelihood -100036.82
#> Akaike (AIC) 200147.64
#> Bayesian (BIC) 200395.75
#>
#> Numerical Integration:
#> Points of integration (per dim) 15
#> Dimensions 2
#> Total points of integration 225
#>
#> Fit Measures for Baseline Model (H0):
#> Loglikelihood -110521.29
#> Akaike (AIC) 221108.58
#> Bayesian (BIC) 221329.87
#> Chi-square 1016.34
#> Degrees of Freedom (Chi-square) 87
#> P-value (Chi-square) 0.000
#> RMSEA 0.042
#>
#> Comparative Fit to H0 (LRT test):
#> Loglikelihood change 10484.47
#> Difference test (D) 20968.94
#> Degrees of freedom (D) 4
#> P-value (D) 0.000
#>
#> R-Squared Interaction Model (H1):
#> CAREER 0.509
#> R-Squared Baseline Model (H0):
#> CAREER 0.510
#> R-Squared Change (H1 - H0):
#> CAREER -0.001
#>
#> 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 0.021 47.917 0.000
#> enjoy3 0.895 0.020 45.290 0.000
#> enjoy4 0.997 0.019 52.613 0.000
#> enjoy5 1.047 0.019 55.454 0.000
#> SC =~
#> academic1 1.000
#> academic2 1.105 0.028 39.155 0.000
#> academic3 1.236 0.029 43.117 0.000
#> academic4 1.255 0.029 44.024 0.000
#> academic5 1.115 0.027 40.605 0.000
#> academic6 1.199 0.028 42.786 0.000
#> CAREER =~
#> career1 1.000
#> career2 1.040 0.019 54.673 0.000
#> career3 0.952 0.019 49.620 0.000
#> career4 0.818 0.017 47.318 0.000
#>
#> Regressions:
#> Estimate Std.Error z.value P(>|z|)
#> CAREER ~
#> ENJ 0.525 0.021 24.881 0.000
#> SC 0.465 0.024 19.423 0.000
#> ENJ:ENJ 0.025 0.018 1.437 0.151
#> ENJ:SC -0.050 0.029 -1.715 0.086
#> SC:SC 0.003 0.023 0.113 0.910
#>
#> Intercepts:
#> Estimate Std.Error z.value P(>|z|)
#> .CAREER -0.004 0.013 -0.277 0.782
#>
#> Covariances:
#> Estimate Std.Error z.value P(>|z|)
#> ENJ ~~
#> SC 0.213 0.007 29.509 0.000
#>
#> Variances:
#> Estimate Std.Error z.value P(>|z|)
#> .enjoy1 0.488 0.009 56.029 0.000
#> .enjoy2 0.490 0.009 55.913 0.000
#> .enjoy3 0.595 0.011 53.584 0.000
#> .enjoy4 0.490 0.009 56.846 0.000
#> .enjoy5 0.444 0.009 50.759 0.000
#> .academic1 0.645 0.012 52.068 0.000
#> .academic2 0.566 0.010 55.059 0.000
#> .academic3 0.473 0.009 50.901 0.000
#> .academic4 0.455 0.009 49.260 0.000
#> .academic5 0.565 0.010 55.869 0.000
#> .academic6 0.502 0.010 52.330 0.000
#> .career1 0.373 0.008 46.727 0.000
#> .career2 0.328 0.007 45.884 0.000
#> .career3 0.436 0.009 46.215 0.000
#> .career4 0.576 0.011 54.848 0.000
#> ENJ 0.490 0.014 34.422 0.000
#> SC 0.336 0.013 26.607 0.000
#> .CAREER 0.301 0.010 29.673 0.000