probe2WayMC: Probing two-way interaction on the no-centered or mean-centered latent interaction

Description

Probing interaction for simple intercept and simple slope for the no-centered or mean-centered latent two-way interaction

Usage

probe2WayMC(fit, nameX, nameY, modVar, valProbe, group = 1L,
  omit.imps = c("no.conv", "no.se"))

Arguments

fit

A fitted '>lavaan or '>lavaan.mi object with a latent 2-way interaction.

nameX

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.

nameY

The name of factor that is used as the dependent variable.

modVar

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.

valProbe

The values of the moderator that will be used to probe the effect of the focal predictor.

group

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.

omit.imps

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.

Value

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.

Details

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).

References

Tutorial:

Schoemann, A. M., & Jorgensen, T. D. (2021). Testing and interpreting latent variable interactions using the semTools package. Psych, 3(3), 322--335. 10.3390/psych3030024

Background literature:

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. 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. 10.3102/10769986031004437

Examples

Run this code

# NOT RUN {
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 + 0*f2
x1 + x4 + x1.x4 + x7 ~ 0*1 # identify latent means
f1 + f2 + f12 + f3 ~ NA*1
"

fitMC2way <- sem(model1, data = dat2wayMC, meanstructure = TRUE)
summary(fitMC2way)

probe2WayMC(fitMC2way, nameX = c("f1", "f2", "f12"), nameY = "f3",
            modVar = "f2", valProbe = c(-1, 0, 1))


## can probe multigroup models, one group at a time
dat2wayMC$g <- 1:2

model2 <- "
f1 =~ x1 + x2 + x3
f2 =~ x4 + x5 + x6
f12 =~ x1.x4 + x2.x5 + x3.x6
f3 =~ x7 + x8 + x9
f3 ~ c(b1.g1, b1.g2)*f1 + c(b2.g1, b2.g2)*f2 + c(b12.g1, b12.g2)*f12
f12 ~~ 0*f1 + 0*f2
x1 + x4 + x1.x4 + x7 ~ 0*1 # identify latent means
f1 + f2 + f12 ~ NA*1
f3 ~ NA*1 + c(b0.g1, b0.g2)*1
"
fit2 <- sem(model2, data = dat2wayMC, group = "g")
probe2WayMC(fit2, nameX = c("f1", "f2", "f12"), nameY = "f3",
            modVar = "f2", valProbe = c(-1, 0, 1)) # group = 1 by default
probe2WayMC(fit2, nameX = c("f1", "f2", "f12"), nameY = "f3",
            modVar = "f2", valProbe = c(-1, 0, 1), group = 2)

# }

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