interaction effects between endogenous variables
interaction_two_etas.RmdThe Problem
Interaction effects between two endogenous (i.e., dependent)
variables work as you would expect for the product indicator methods
("dblcent", "rca", "ca",
"uca"). However, for the lms and
qml approaches, it is not as straightforward.
The lms and qml approaches can (by default)
handle interaction effects between endogenous and exogenous (i.e.,
independent) variables, but they do not natively support interaction
effects between two endogenous variables. When an interaction effect
exists between two endogenous variables, the equations cannot easily be
written in “reduced” form, meaning that normal estimation procedures
won’t work.
The Solution
Despite these limitations, there is a workaround for both the
lms and qml approaches. Essentially, the model
can be split into two parts: one linear and one non-linear. You can
replace the covariance matrix used in the estimation of the non-linear
model with the model-implied covariance matrix from a linear model. This
allows you to treat an endogenous variable as if it were
exogenous—provided that it can be expressed in a linear model.
Example
Let’s consider the theory of planned behavior (TPB), where we wish to
estimate the quadratic effect of INT on BEH
(INT:INT). We can use the following model:
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
BEH ~ INT:INT
'Since INT is an endogenous variable, its quadratic term
(i.e., an interaction effect with itself) would involve two endogenous
variables. As a result, we would normally not be able to estimate this
model using the lms or qml approaches.
However, we can split the model into two parts: one linear and one
non-linear.
While INT is an endogenous variable, it can be expressed
in a linear model since it is not affected by any interaction terms:
tpb_linear <- 'INT ~ PBC + ATT + SN'We can then remove this part from the original model, giving us:
tpb_nonlinear <- '
# 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)
BEH ~ INT + PBC
BEH ~ INT:INT
'Now, we can estimate the non-linear model since INT is
treated as an exogenous variable. However, this would not incorporate
the structural model for INT. To address this, we can
instruct modsem to replace the covariance matrix
(phi) of (INT, PBC,
ATT, SN) with the model-implied covariance
matrix from the linear model while estimating both models
simultaneously. To achieve this, we use the cov.syntax
argument in modsem:
est_lms <- modsem(tpb_nonlinear, data = TPB, cov.syntax = tpb_linear,
method = "lms")
#> Warning: It is recommended that you have at least 32 nodes for interaction
#> effects between endogenous variables in the lms approach 'nodes = 24'
summary(est_lms)
#>
#> modsem (1.0.12) ended normally after 29 iterations
#>
#> Estimator LMS
#> Optimization method EMA-NLMINB
#> Number of model parameters 54
#>
#> Number of observations 2000
#>
#> Loglikelihood and Information Criteria:
#> Loglikelihood -26360.48
#> Akaike (AIC) 52828.96
#> Bayesian (BIC) 53131.41
#>
#> Numerical Integration:
#> Points of integration (per dim) 24
#> Dimensions 1
#> Total points of integration 24
#>
#> Fit Measures for Baseline Model (H0):
#> Loglikelihood -26393.22
#> 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 (LRT test):
#> Loglikelihood change 32.74
#> Difference test (D) 65.49
#> Degrees of freedom (D) 1
#> P-value (D) 0.000
#>
#> R-Squared Interaction Model (H1):
#> INT 0.369
#> BEH 0.239
#> R-Squared Baseline Model (H0):
#> INT 0.367
#> BEH 0.210
#> R-Squared Change (H1 - H0):
#> INT 0.003
#> BEH 0.029
#>
#> Parameter Estimates:
#> Coefficients unstandardized
#> Information observed
#> Standard errors standard
#>
#> Latent Variables:
#> Estimate Std.Error z.value P(>|z|)
#> ATT =~
#> att1 1.000
#> att2 0.878 0.012 71.582 0.000
#> att3 0.789 0.012 66.390 0.000
#> att4 0.695 0.011 61.010 0.000
#> att5 0.887 0.013 70.870 0.000
#> SN =~
#> sn1 1.000
#> sn2 0.888 0.017 52.613 0.000
#> PBC =~
#> pbc1 1.000
#> pbc2 0.913 0.013 69.390 0.000
#> pbc3 0.801 0.012 66.087 0.000
#> INT =~
#> int1 1.000
#> int2 0.914 0.015 58.995 0.000
#> int3 0.807 0.015 55.616 0.000
#> BEH =~
#> b1 1.000
#> b2 0.960 0.032 29.910 0.000
#>
#> Regressions:
#> Estimate Std.Error z.value P(>|z|)
#> INT ~
#> ATT 0.213 0.026 8.155 0.000
#> SN 0.175 0.028 6.320 0.000
#> PBC 0.222 0.030 7.482 0.000
#> BEH ~
#> PBC 0.239 0.023 10.588 0.000
#> INT 0.197 0.025 7.756 0.000
#> INT:INT 0.128 0.016 7.897 0.000
#>
#> Intercepts:
#> Estimate Std.Error z.value P(>|z|)
#> .int1 1.014 0.022 46.966 0.000
#> .int2 1.012 0.020 50.412 0.000
#> .int3 1.005 0.018 54.811 0.000
#> .att1 1.014 0.024 42.012 0.000
#> .att2 1.007 0.021 46.978 0.000
#> .att3 1.016 0.020 51.467 0.000
#> .att4 0.999 0.018 55.665 0.000
#> .att5 0.992 0.022 45.682 0.000
#> .sn1 1.006 0.024 41.678 0.000
#> .sn2 1.011 0.022 46.728 0.000
#> .pbc1 0.998 0.024 42.429 0.000
#> .pbc2 0.985 0.022 44.952 0.000
#> .pbc3 0.991 0.020 50.473 0.000
#> .b1 0.999 0.023 42.629 0.000
#> .b2 1.017 0.022 46.239 0.000
#>
#> Covariances:
#> Estimate Std.Error z.value P(>|z|)
#> ATT ~~
#> SN 0.629 0.029 21.710 0.000
#> PBC ~~
#> ATT 0.677 0.029 23.466 0.000
#> SN 0.677 0.029 23.099 0.000
#>
#> Variances:
#> Estimate Std.Error z.value P(>|z|)
#> .int1 0.158 0.009 18.229 0.000
#> .int2 0.160 0.008 20.381 0.000
#> .int3 0.168 0.007 23.640 0.000
#> .att1 0.167 0.007 23.534 0.000
#> .att2 0.150 0.006 24.719 0.000
#> .att3 0.159 0.006 26.388 0.000
#> .att4 0.162 0.006 27.650 0.000
#> .att5 0.159 0.006 24.931 0.000
#> .sn1 0.178 0.015 12.084 0.000
#> .sn2 0.157 0.012 13.261 0.000
#> .pbc1 0.145 0.008 18.445 0.000
#> .pbc2 0.160 0.007 21.429 0.000
#> .pbc3 0.154 0.006 23.804 0.000
#> .b1 0.186 0.020 9.423 0.000
#> .b2 0.135 0.018 7.594 0.000
#> ATT 0.998 0.037 26.957 0.000
#> SN 0.987 0.039 25.241 0.000
#> PBC 0.961 0.036 27.064 0.000
#> .INT 0.488 0.020 24.577 0.000
#> .BEH 0.475 0.024 19.740 0.000
est_qml <- modsem(tpb_nonlinear, data = TPB, cov.syntax = tpb_linear,
method = "qml")
summary(est_qml)
#>
#> modsem (1.0.12) ended normally after 61 iterations
#>
#> Estimator QML
#> Optimization method NLMINB
#> Number of model parameters 54
#>
#> Number of observations 2000
#>
#> Loglikelihood and Information Criteria:
#> Loglikelihood -26360.52
#> Akaike (AIC) 52829.04
#> Bayesian (BIC) 53131.49
#>
#> Fit Measures for Baseline Model (H0):
#> Loglikelihood -26393.22
#> 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 (LRT test):
#> Loglikelihood change 32.70
#> Difference test (D) 65.41
#> Degrees of freedom (D) 1
#> P-value (D) 0.000
#>
#> R-Squared Interaction Model (H1):
#> INT 0.370
#> BEH 0.239
#> R-Squared Baseline Model (H0):
#> INT 0.367
#> BEH 0.210
#> R-Squared Change (H1 - H0):
#> INT 0.003
#> BEH 0.029
#>
#> Parameter Estimates:
#> Coefficients unstandardized
#> Information observed
#> Standard errors standard
#>
#> Latent Variables:
#> Estimate Std.Error z.value P(>|z|)
#> ATT =~
#> att1 1.000
#> att2 0.878 0.012 71.563 0.000
#> att3 0.789 0.012 66.373 0.000
#> att4 0.695 0.011 60.999 0.000
#> att5 0.887 0.013 70.851 0.000
#> SN =~
#> sn1 1.000
#> sn2 0.888 0.017 52.623 0.000
#> PBC =~
#> pbc1 1.000
#> pbc2 0.913 0.013 69.384 0.000
#> pbc3 0.801 0.012 66.079 0.000
#> INT =~
#> int1 1.000
#> int2 0.914 0.015 59.036 0.000
#> int3 0.807 0.015 55.653 0.000
#> BEH =~
#> b1 1.000
#> b2 0.960 0.032 29.902 0.000
#>
#> Regressions:
#> Estimate Std.Error z.value P(>|z|)
#> INT ~
#> ATT 0.213 0.026 8.173 0.000
#> SN 0.175 0.028 6.333 0.000
#> PBC 0.222 0.030 7.507 0.000
#> BEH ~
#> PBC 0.239 0.023 10.586 0.000
#> INT 0.197 0.025 7.758 0.000
#> INT:INT 0.128 0.016 7.879 0.000
#>
#> Intercepts:
#> Estimate Std.Error z.value P(>|z|)
#> .int1 1.014 0.022 46.963 0.000
#> .int2 1.012 0.020 50.404 0.000
#> .int3 1.005 0.018 54.804 0.000
#> .att1 1.014 0.024 42.007 0.000
#> .att2 1.007 0.021 46.967 0.000
#> .att3 1.016 0.020 51.455 0.000
#> .att4 0.999 0.018 55.654 0.000
#> .att5 0.992 0.022 45.672 0.000
#> .sn1 1.006 0.024 41.658 0.000
#> .sn2 1.010 0.022 46.707 0.000
#> .pbc1 0.998 0.024 42.414 0.000
#> .pbc2 0.985 0.022 44.934 0.000
#> .pbc3 0.991 0.020 50.454 0.000
#> .b1 0.999 0.023 42.638 0.000
#> .b2 1.017 0.022 46.249 0.000
#>
#> Covariances:
#> Estimate Std.Error z.value P(>|z|)
#> ATT ~~
#> SN 0.629 0.029 21.696 0.000
#> PBC ~~
#> ATT 0.678 0.029 23.447 0.000
#> SN 0.678 0.029 23.081 0.000
#>
#> Variances:
#> Estimate Std.Error z.value P(>|z|)
#> .int1 0.158 0.009 18.219 0.000
#> .int2 0.160 0.008 20.377 0.000
#> .int3 0.168 0.007 23.630 0.000
#> .att1 0.167 0.007 23.530 0.000
#> .att2 0.150 0.006 24.712 0.000
#> .att3 0.160 0.006 26.379 0.000
#> .att4 0.162 0.006 27.646 0.000
#> .att5 0.159 0.006 24.925 0.000
#> .sn1 0.178 0.015 12.086 0.000
#> .sn2 0.157 0.012 13.263 0.000
#> .pbc1 0.145 0.008 18.444 0.000
#> .pbc2 0.160 0.007 21.425 0.000
#> .pbc3 0.154 0.006 23.799 0.000
#> .b1 0.185 0.020 9.417 0.000
#> .b2 0.135 0.018 7.596 0.000
#> ATT 0.998 0.037 26.933 0.000
#> SN 0.988 0.039 25.225 0.000
#> PBC 0.962 0.036 27.041 0.000
#> .INT 0.488 0.020 24.585 0.000
#> .BEH 0.475 0.024 19.736 0.000It is also possible to make modsem attempt to split up
the model syntax automatically using the auto.split.syntax
argument:
est_lms <- modsem(tpb, data = TPB, method = "lms", auto.split.syntax = TRUE)
#> Warning: It is recommended that you have at least 32 nodes for interaction
#> effects between endogenous variables in the lms approach 'nodes = 24'
summary(est_lms)
#>
#> modsem (1.0.12) ended normally after 29 iterations
#>
#> Estimator LMS
#> Optimization method EMA-NLMINB
#> Number of model parameters 54
#>
#> Number of observations 2000
#>
#> Loglikelihood and Information Criteria:
#> Loglikelihood -26360.48
#> Akaike (AIC) 52828.96
#> Bayesian (BIC) 53131.41
#>
#> Numerical Integration:
#> Points of integration (per dim) 24
#> Dimensions 1
#> Total points of integration 24
#>
#> Fit Measures for Baseline Model (H0):
#> Loglikelihood -26393.22
#> 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 (LRT test):
#> Loglikelihood change 32.74
#> Difference test (D) 65.49
#> Degrees of freedom (D) 1
#> P-value (D) 0.000
#>
#> R-Squared Interaction Model (H1):
#> INT 0.369
#> BEH 0.239
#> R-Squared Baseline Model (H0):
#> INT 0.367
#> BEH 0.210
#> R-Squared Change (H1 - H0):
#> INT 0.003
#> BEH 0.029
#>
#> Parameter Estimates:
#> Coefficients unstandardized
#> Information observed
#> Standard errors standard
#>
#> Latent Variables:
#> Estimate Std.Error z.value P(>|z|)
#> ATT =~
#> att1 1.000
#> att2 0.878 0.012 71.582 0.000
#> att3 0.789 0.012 66.390 0.000
#> att4 0.695 0.011 61.010 0.000
#> att5 0.887 0.013 70.870 0.000
#> SN =~
#> sn1 1.000
#> sn2 0.888 0.017 52.613 0.000
#> PBC =~
#> pbc1 1.000
#> pbc2 0.913 0.013 69.390 0.000
#> pbc3 0.801 0.012 66.087 0.000
#> INT =~
#> int1 1.000
#> int2 0.914 0.015 58.995 0.000
#> int3 0.807 0.015 55.616 0.000
#> BEH =~
#> b1 1.000
#> b2 0.960 0.032 29.910 0.000
#>
#> Regressions:
#> Estimate Std.Error z.value P(>|z|)
#> INT ~
#> ATT 0.213 0.026 8.155 0.000
#> SN 0.175 0.028 6.320 0.000
#> PBC 0.222 0.030 7.482 0.000
#> BEH ~
#> PBC 0.239 0.023 10.588 0.000
#> INT 0.197 0.025 7.756 0.000
#> INT:INT 0.128 0.016 7.897 0.000
#>
#> Intercepts:
#> Estimate Std.Error z.value P(>|z|)
#> .int1 1.014 0.022 46.966 0.000
#> .int2 1.012 0.020 50.412 0.000
#> .int3 1.005 0.018 54.811 0.000
#> .att1 1.014 0.024 42.012 0.000
#> .att2 1.007 0.021 46.978 0.000
#> .att3 1.016 0.020 51.467 0.000
#> .att4 0.999 0.018 55.665 0.000
#> .att5 0.992 0.022 45.682 0.000
#> .sn1 1.006 0.024 41.678 0.000
#> .sn2 1.011 0.022 46.728 0.000
#> .pbc1 0.998 0.024 42.429 0.000
#> .pbc2 0.985 0.022 44.952 0.000
#> .pbc3 0.991 0.020 50.473 0.000
#> .b1 0.999 0.023 42.629 0.000
#> .b2 1.017 0.022 46.239 0.000
#>
#> Covariances:
#> Estimate Std.Error z.value P(>|z|)
#> ATT ~~
#> SN 0.629 0.029 21.710 0.000
#> PBC 0.677 0.029 23.466 0.000
#> SN ~~
#> PBC 0.677 0.029 23.099 0.000
#>
#> Variances:
#> Estimate Std.Error z.value P(>|z|)
#> .int1 0.158 0.009 18.229 0.000
#> .int2 0.160 0.008 20.381 0.000
#> .int3 0.168 0.007 23.640 0.000
#> .att1 0.167 0.007 23.534 0.000
#> .att2 0.150 0.006 24.719 0.000
#> .att3 0.159 0.006 26.388 0.000
#> .att4 0.162 0.006 27.650 0.000
#> .att5 0.159 0.006 24.931 0.000
#> .sn1 0.178 0.015 12.084 0.000
#> .sn2 0.157 0.012 13.261 0.000
#> .pbc1 0.145 0.008 18.445 0.000
#> .pbc2 0.160 0.007 21.429 0.000
#> .pbc3 0.154 0.006 23.804 0.000
#> .b1 0.186 0.020 9.423 0.000
#> .b2 0.135 0.018 7.594 0.000
#> ATT 0.998 0.037 26.957 0.000
#> SN 0.987 0.039 25.241 0.000
#> PBC 0.961 0.036 27.064 0.000
#> .INT 0.488 0.020 24.577 0.000
#> .BEH 0.475 0.024 19.740 0.000