Probing interaction for simple intercept and simple slope for the no-centered or mean-centered latent two-way interaction
probe3WayMC(fit, nameX, nameY, modVar, valProbe1, valProbe2, group = 1L,
omit.imps = c("no.conv", "no.se"))A list with two elements:
SimpleIntercept: The model-implied intercepts given each
combination of moderator values.
SimpleSlope: The model-implied slopes given each combination
of moderator values.
In each element, the first column represents values of the first moderator
specified in the valProbe1 argument. The second column represents
values of the second moderator specified in the valProbe2 argument.
The third column is the simple intercept or simple slope. The fourth column
is the standard error of the simple intercept or simple slope. The fifth
column is the Wald (z) statistic, and the sixth column is its
associated p value to test the null hypothesis that each simple
intercept or simple slope equals 0.
A fitted lavaan::lavaan or lavaan.mi::lavaan.mi object with a latent 2-way interaction.
character vector of all 7 factor names used as the
predictors. The 3 lower-order factors must be listed first, followed by
the 3 second-order factors (specifically, the 4th element must be the
interaction between the factors listed first and second, the 5th element
must be the interaction between the factors listed first and third, and
the 6th element must be the interaction between the factors listed second
and third). The final name will be the factor representing the 3-way
interaction.
The name of factor that is used as the dependent variable.
The name of two factors that are used as the moderators. The effect of the independent factor will be probed at each combination of the moderator variables' chosen values.
The values of the first moderator that will be used to probe the effect of the independent factor.
The values of the second moderator that will be used to probe the effect of the independent factor.
In multigroup models, the label of the group for which the
results will be returned. Must correspond to one of
lavInspect(fit, "group.label").
character vector specifying criteria for omitting
imputations from pooled results. Ignored unless fit is of
class lavaan.mi::lavaan.mi. Can include any of
c("no.conv", "no.se", "no.npd"), the first 2 of which are the
default setting, which excludes any imputations that did not
converge or for which standard errors could not be computed. The
last option ("no.npd") would exclude any imputations which
yielded a nonpositive definite covariance matrix for observed or
latent variables, which would include any "improper solutions" such
as Heywood cases. NPD solutions are not excluded by default because
they are likely to occur due to sampling error, especially in small
samples. However, gross model misspecification could also cause
NPD solutions, users can compare pooled results with and without
this setting as a sensitivity analysis to see whether some
imputations warrant further investigation.
Sunthud Pornprasertmanit (psunthud@gmail.com)
Terrence D. Jorgensen (University of Amsterdam; TJorgensen314@gmail.com)
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 variable (\(X\)) 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.
To probe a 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 variable 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 statistics are calculated (even for objects of class lavaan.mi::lavaan.mi) to test null hypotheses that simple intercepts or slopes are 0.
Tutorial:
Schoemann, A. M., & Jorgensen, T. D. (2021). Testing and interpreting
latent variable interactions using the semTools package.
Psych, 3(3), 322--335. tools:::Rd_expr_doi("10.3390/psych3030024")
Background literature:
Aiken, L. S., & West, S. G. (1991). Multiple regression: Testing and interpreting interactions. Newbury Park, CA: Sage.
Marsh, H. W., Wen, Z., & Hau, K. T. (2004). Structural equation models of latent interactions: Evaluation of alternative estimation strategies and indicator construction. Psychological Methods, 9(3), 275--300. tools:::Rd_expr_doi("10.1037/1082-989X.9.3.275")
indProd() For creating the indicator products with no
centering, mean centering, double-mean centering, or residual centering.
probe2WayMC() For probing the two-way latent interaction
when the results are obtained from mean-centering, or double-mean centering
probe2WayRC() For probing the two-way latent interaction
when the results are obtained from residual-centering approach.
probe3WayRC() For probing the two-way latent interaction
when the results are obtained from residual-centering approach.
plotProbe() Plot the simple intercepts and slopes of the
latent interaction.
dat3wayMC <- indProd(dat3way, 1:3, 4:6, 7:9)
model3 <- " ## define latent variables
f1 =~ x1 + x2 + x3
f2 =~ x4 + x5 + x6
f3 =~ x7 + x8 + x9
## 2-way interactions
f12 =~ x1.x4 + x2.x5 + x3.x6
f13 =~ x1.x7 + x2.x8 + x3.x9
f23 =~ x4.x7 + x5.x8 + x6.x9
## 3-way interaction
f123 =~ x1.x4.x7 + x2.x5.x8 + x3.x6.x9
## outcome variable
f4 =~ x10 + x11 + x12
## latent regression model
f4 ~ b1*f1 + b2*f2 + b3*f3 + b12*f12 + b13*f13 + b23*f23 + b123*f123
## orthogonal terms among predictors
## (not necessary, but implied by double mean centering)
f1 ~~ 0*f12 + 0*f13 + 0*f123
f2 ~~ 0*f12 + 0*f23 + 0*f123
f3 ~~ 0*f13 + 0*f23 + 0*f123
f12 + f13 + f23 ~~ 0*f123
"
fitMC3way <- sem(model3, data = dat3wayMC, meanstructure = TRUE)
summary(fitMC3way)
probe3WayMC(fitMC3way, nameX = c("f1" ,"f2" ,"f3",
"f12","f13","f23", # this order matters!
"f123"), # 3-way interaction
nameY = "f4", modVar = c("f1", "f2"),
valProbe1 = c(-1, 0, 1), valProbe2 = c(-1, 0, 1))
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