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 statistic is used for test statistic.