semTools (version 0.5-4)

probe3WayMC: Probing three-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

probe3WayMC(fit, nameX, nameY, modVar, valProbe1, valProbe2, group,
  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 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.

nameY

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

modVar

The name of two factors that are used as the moderators. The effect of the independent factor on each combination of the moderator variable values will be probed.

valProbe1

The values of the first moderator that will be used to probe the effect of the independent factor.

valProbe2

The values of the second moderator that will be used to probe the effect of the independent factor.

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").

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:

1. SimpleIntercept: The intercepts given each combination of moderator values. This element will be shown only if the factor intercept is estimated (e.g., not fixed at 0).

2. SimpleSlope: The 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. The sixth 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 two moderators ($Z$ and $W$) be $$Y=b_0+b_{1}X+b_{2}Z+b_{3}W+b_{4}XZ+b_{5}XW+b_{6}ZW+b_{7}XZW+r,$$ 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 $$b_{0|X=0,Z,W}=b_0+b_{2}Z+b_{3}W+b_{6}ZW.$$

The simple slope of the independent varaible at the specific values of the moderators can be obtained by $$b_{X|Z,W}=b_1+b_{3}Z+b_{4}W+b_{7}ZW.$$

The variance of the simple intercept formula is $$Var\left(b_{0|X=0,Z,W}\right)=Var\left(b_0\right)+Z^{2}Var\left(b_2\right)+W^{2}Var\left(b_3\right)+Z^2{W}^{2}Var\left(b_6\right)+2ZCov(b_0,b_2)+2WCov(b_0,b_3)+2ZWCov(b_0,b_6)+2ZWCov(b_2,b_3)+2Z^{2}WCov(b_2,b_6)+2Z{W}^{2}Cov(b_3,b_6)$$ 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 $$Var\left(b_{X|Z,W}\right)=Var\left(b_1\right)+Z^{2}Var\left(b_4\right)+W^{2}Var\left(b_5\right)+Z^2{W}^{2}Var\left(b_7\right)+2ZCov(b_1,b_4)+2WCov(b_1,b_5)+2ZWCov(b_1,b_7)+2ZWCov(b_4,b_5)+2Z^{2}WCov(b_4,b_7)+2Z{W}^{2}Cov(b_5,b_7)$$

Wald z statistic is used for test statistic (even for objects of class '>lavaan.mi).

References

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. 10.1037/1082-989X.9.3.275

See Also

• 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.

Examples

# NOT RUN {
library(lavaan)

dat3wayMC <- indProd(dat3way, 1:3, 4:6, 7:9)

model3 <- "
f1 =~ x1 + x2 + x3
f2 =~ x4 + x5 + x6
f3 =~ x7 + x8 + x9
f12 =~ x1.x4 + x2.x5 + x3.x6
f13 =~ x1.x7 + x2.x8 + x3.x9
f23 =~ x4.x7 + x5.x8 + x6.x9
f123 =~ x1.x4.x7 + x2.x5.x8 + x3.x6.x9
f4 =~ x10 + x11 + x12
f4 ~ f1 + f2 + f3 + f12 + f13 + f23 + f123
f1 ~~ 0*f12
f1 ~~ 0*f13
f1 ~~ 0*f123
f2 ~~ 0*f12
f2 ~~ 0*f23
f2 ~~ 0*f123
f3 ~~ 0*f13
f3 ~~ 0*f23
f3 ~~ 0*f123
f12 ~~ 0*f123
f13 ~~ 0*f123
f23 ~~ 0*f123
x1 ~ 0*1
x4 ~ 0*1
x7 ~ 0*1
x10 ~ 0*1
x1.x4 ~ 0*1
x1.x7 ~ 0*1
x4.x7 ~ 0*1
x1.x4.x7 ~ 0*1
f1 ~ NA*1
f2 ~ NA*1
f3 ~ NA*1
f12 ~ NA*1
f13 ~ NA*1
f23 ~ NA*1
f123 ~ NA*1
f4 ~ NA*1
"

fitMC3way <- sem(model3, data = dat3wayMC, std.lv = FALSE,
                 meanstructure = TRUE)
summary(fitMC3way)

probe3WayMC(fitMC3way, nameX = c("f1" ,"f2" ,"f3",
                                 "f12","f13","f23", # the order matters!
                                 "f123"),           # 3-way interaction
            nameY = "f4", modVar = c("f1", "f2"),
            valProbe1 = c(-1, 0, 1), valProbe2 = c(-1, 0, 1))

# }