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"))

Value

A list with two elements:

SimpleIntercept: The simple intercepts given each value of the moderator.
SimpleSlope: The simple slopes given each value of the moderator.

In each element, the first column represents the values of the moderator 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, and the fifth column is the associated p value testing the null hypothesis that each simple intercept or slope is 0.

Arguments

fit: A fitted lavaan::lavaan or lavaan.mi::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::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.

Author

Sunthud Pornprasertmanit (psunthud@gmail.com)

Terrence D. Jorgensen (University of Amsterdam; TJorgensen314@gmail.com)

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 variable ($X$) 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.

To probe a 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 $$ b_{0|X = 0, Z} = b_0 + b_2 Z. $$

The simple slope of the independent varaible at each value of the moderator can be obtained by $$ b_{X|Z} = b_1 + b_3 Z. $$

The variance of the simple intercept formula is $$ Var\left(b_{0|X = 0, Z}\right) = Var\left(b_0\right) + 2Z \times Cov\left(b_0, b_2\right) + Z^2 \times Var\left(b_2\right) $$, 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 $$ Var\left(b_{X|Z}\right) = Var\left(b_1\right) + 2Z \times Cov\left(b_1, b_3\right) + Z^2 \times Var\left(b_3\right) $$

Wald z statistic is used for test statistic (even for objects of class lavaan.mi::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. 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.

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. tools:::Rd_expr_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. tools:::Rd_expr_doi("10.3102/10769986031004437")

Examples

Run this code


dat2wayMC <- indProd(dat2way, 1:3, 4:6) # double mean centered by default

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 # not necessary, but implied by double mean centering
"

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
"
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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