library(modsem)
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.
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.
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:
<- 'INT ~ PBC + ATT + SN' tpb_linear
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
:
<- modsem(tpb_nonlinear, data = TPB, cov.syntax = tpb_linear,
est_lms method = "lms")
summary(est_lms)
<- modsem(tpb_nonlinear, data = TPB, cov.syntax = tpb_linear,
est_qml method = "qml")
summary(est_qml)