Before using this function, researchers need to make the products of the
indicators between the first-order factors using mean centering (Marsh, Wen,
& Hau, 2004). Note that the double-mean centering may not be appropriate for
probing interaction if researchers are interested in simple intercepts. The
mean or double-mean centering can be done by the indProd
function. The indicator products can be made for all possible combination or
matched-pair approach (Marsh et al., 2004). Next, the hypothesized model
with the regression with latent interaction will be used to fit all original
indicators and the product terms. See the example for how to fit the product
term below. Once the lavaan result is obtained, this function will be used
to probe the interaction.
Let that the latent interaction model regressing the dependent variable
(\(Y\)) on the independent varaible (\(X\)) and two moderators (\(Z\)
and \(W\)) be $$ Y = b_0 + b_1X + b_2Z + b_3W + b_4XZ + b_5XW + b_6ZW
+ b_7XZW + r, $$ where \(b_0\) is the estimated intercept or the expected
value of \(Y\) when \(X\), \(Z\), and \(W\) are 0, \(b_1\) is the
effect of \(X\) when \(Z\) and \(W\) are 0, \(b_2\) is the effect of
\(Z\) when \(X\) and \(W\) is 0, \(b_3\) is the effect of \(W\)
when \(X\) and \(Z\) are 0, \(b_4\) is the interaction effect between
\(X\) and \(Z\) when \(W\) is 0, \(b_5\) is the interaction effect
between \(X\) and \(W\) when \(Z\) is 0, \(b_6\) is the interaction
effect between \(Z\) and \(W\) when \(X\) is 0, \(b_7\) is the
three-way interaction effect between \(X\), \(Z\), and \(W\), and
\(r\) is the residual term.
For probing three-way interaction, the simple intercept of the independent
variable at the specific values of the moderators (Aiken & West, 1991) can
be obtained by $$ b_{0|X = 0, Z, W} = b_0 + b_2Z + b_3W + b_6ZW. $$
The simple slope of the independent varaible at the specific values of the
moderators can be obtained by $$ b_{X|Z, W} = b_1 + b_3Z + b_4W + b_7ZW.
$$
The variance of the simple intercept formula is $$ Var\left(b_{0|X = 0,
Z, W}\right) = Var\left(b_0\right) + Z^2Var\left(b_2\right) +
W^2Var\left(b_3\right) + Z^2W^2Var\left(b_6\right) + 2ZCov\left(b_0,
b_2\right) + 2WCov\left(b_0, b_3\right) + 2ZWCov\left(b_0, b_6\right) +
2ZWCov\left(b_2, b_3\right) + 2Z^2WCov\left(b_2, b_6\right) +
2ZW^2Cov\left(b_3, b_6\right) $$ where \(Var\) denotes the variance of a
parameter estimate and \(Cov\) denotes the covariance of two parameter
estimates.
The variance of the simple slope formula is $$ Var\left(b_{X|Z,
W}\right) = Var\left(b_1\right) + Z^2Var\left(b_4\right) +
W^2Var\left(b_5\right) + Z^2W^2Var\left(b_7\right) + 2ZCov\left(b_1,
b_4\right) + 2WCov\left(b_1, b_5\right) + 2ZWCov\left(b_1, b_7\right) +
2ZWCov\left(b_4, b_5\right) + 2Z^2WCov\left(b_4, b_7\right) +
2ZW^2Cov\left(b_5, b_7\right) $$
Wald z statistic is used for test statistic (even for objects of
class '>lavaan.mi).