Probing interaction for simple intercept and simple slope for the no-centered or mean-centered latent two-way interaction
probe2WayMC(fit, nameX, nameY, modVar, valProbe, group,
omit.imps = c("no.conv", "no.se"))
character
vector of all 3 factor names used as the
predictors. The lower-order factors must be listed first, and the final
name must be the latent interaction factor.
The name of factor that is used as the dependent variable.
The name of factor that is used as a moderator. The effect of
the other independent factor will be probed at each value of the
moderator variable listed in valProbe
.
The values of the moderator that will be used to probe the effect of the focal predictor.
In multigroup models, the label of the group for which the
results will be returned. Must correspond to one of
lavInspect(fit, "group.label")
, or an integer
corresponding to which of those group labels.
character
vector specifying criteria for omitting
imputations from pooled results. Ignored unless fit
is of
class '>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.
A list with two elements:
SimpleIntercept
: The intercepts given each value of the
moderator. This element will be NULL
unless the factor intercept is
estimated (e.g., not fixed at 0).
SimpleSlope
: The slopes given each value of the moderator.
In each element, the first column represents the values of the moderators
specified in the valProbe
argument. The second column is the simple
intercept or simple slope. The third column is the SE of the simple
intercept or simple slope. The fourth column is the Wald (z)
statistic. The fifth column is the p value testing whether the simple
intercepts or slopes are different from 0.
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 the moderator (\(Z\)) be $$ Y = b_0 + b_1X + b_2Z + b_3XZ + r, $$ where \(b_0\) is the estimated intercept or the expected value of \(Y\) when both \(X\) and \(Z\) are 0, \(b_1\) is the effect of \(X\) when \(Z\) is 0, \(b_2\) is the effect of \(Z\) when \(X\) is 0, \(b_3\) is the interaction effect between \(X\) and \(Z\), and \(r\) is the residual term.
For probing two-way interaction, the simple intercept of the independent variable at each value of the moderator (Aiken & West, 1991; Cohen, Cohen, West, & Aiken, 2003; Preacher, Curran, & Bauer, 2006) can be obtained by $$ b_{0|X = 0, Z} = b_0 + b_2Z. $$
The simple slope of the independent varaible at each value of the moderator can be obtained by $$ b_{X|Z} = b_1 + b_3Z. $$
The variance of the simple intercept formula is $$ Var\left(b_{0|X = 0, Z}\right) = Var\left(b_0\right) + 2ZCov\left(b_0, b_2\right) + Z^2Var\left(b_2\right) $$ 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 $$ Var\left(b_{X|Z}\right) = Var\left(b_1\right) + 2ZCov\left(b_1, b_3\right) + Z^2Var\left(b_3\right) $$
Wald z statistic is used for test statistic (even for objects of
class '>lavaan.mi
).
Aiken, L. S., & West, S. G. (1991). Multiple regression: Testing and interpreting interactions. Newbury Park, CA: Sage.
Cohen, J., Cohen, P., West, S. G., & Aiken, L. S. (2003). Applied multiple regression/correlation analysis for the behavioral sciences (3rd ed.). New York, NY: Routledge.
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. doi:10.1037/1082-989X.9.3.275
Preacher, K. J., Curran, P. J., & Bauer, D. J. (2006). Computational tools for probing interactions in multiple linear regression, multilevel modeling, and latent curve analysis. Journal of Educational and Behavioral Statistics, 31(4), 437--448. doi:10.3102/10769986031004437
indProd
For creating the indicator products with no
centering, mean centering, double-mean centering, or residual centering.
probe3WayMC
For probing the three-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.
# NOT RUN {
library(lavaan)
dat2wayMC <- indProd(dat2way, 1:3, 4:6)
model1 <- "
f1 =~ x1 + x2 + x3
f2 =~ x4 + x5 + x6
f12 =~ x1.x4 + x2.x5 + x3.x6
f3 =~ x7 + x8 + x9
f3 ~ f1 + f2 + f12
f12 ~~0*f1
f12 ~~ 0*f2
x1 ~ 0*1
x4 ~ 0*1
x1.x4 ~ 0*1
x7 ~ 0*1
f1 ~ NA*1
f2 ~ NA*1
f12 ~ NA*1
f3 ~ NA*1
"
fitMC2way <- sem(model1, data = dat2wayMC, std.lv = FALSE,
meanstructure = TRUE)
summary(fitMC2way)
probe2WayMC(fitMC2way, nameX = c("f1", "f2", "f12"), nameY = "f3",
modVar = "f2", valProbe = c(-1, 0, 1))
# }
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