probe3WayMC {semTools}R Documentation

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 = 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 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.

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_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.

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_2Z + b_3W + b_6ZW.

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_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 statistic is used for test statistic (even for objects of class lavaan.mi).

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.

Author(s)

Sunthud Pornprasertmanit (psunthud@gmail.com)

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

References

Tutorial:

Schoemann, A. M., & Jorgensen, T. D. (2021). Testing and interpreting latent variable interactions using the semTools package. Psych, 3(3), 322–335. 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. doi:10.1037/1082-989X.9.3.275

See Also

Examples


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
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

## identify latent means
x1 + x4 + x7 + x1.x4 + x1.x7 + x4.x7 + x1.x4.x7 + x10 ~ 0*1
f1 + f2 + f3 + f12 + f13 + f23 + f123 + f4 ~ NA*1
"

fitMC3way <- sem(model3, data = dat3wayMC, 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))
[Package semTools version 0.5-6 Index]