probe3WayRC {semTools}

Probing three-way interaction on the residual-centered latent interaction

Description

Probing interaction for simple intercept and simple slope for the residual-centered latent three-way interaction (Geldhof et al., 2013)

Usage

probe3WayRC(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 and residualize the products by the original indicators (Lance, 1988; Little, Bovaird, & Widaman, 2006). The process can be automated by the indProd function. Note that 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 (Geldhof et al., 2013). To use this function the model must be fit with a mean structure. 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.

The probing process on residual-centered latent interaction is based on transforming the residual-centered result into the no-centered result. See Geldhof et al. (2013) for further details. Note that this approach based on a strong assumption that the first-order latent variables are normally distributed. The probing process is applied after the no-centered result (parameter estimates and their covariance matrix among parameter estimates) has been computed. See the probe3WayMC for further details.

Value

A list with two elements:

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

2. SimpleSlope: The slopes given each value of the moderator.

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 SE 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 ([e-mail])

Terrence D. Jorgensen (University of Amsterdam; [e-mail])

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:

Geldhof, G. J., Pornprasertmanit, S., Schoemann, A., & Little, T. D. (2013). Orthogonalizing through residual centering: Extended applications and caveats. Educational and Psychological Measurement, 73(1), 27–46. doi:10.1177/0013164412445473

Lance, C. E. (1988). Residual centering, exploratory and confirmatory moderator analysis, and decomposition of effects in path models containing interactions. Applied Psychological Measurement, 12(2), 163–175. doi:10.1177/014662168801200205

Little, T. D., Bovaird, J. A., & Widaman, K. F. (2006). On the merits of orthogonalizing powered and product terms: Implications for modeling interactions. Structural Equation Modeling, 13(4), 497–519. doi:10.1207/s15328007sem1304_1

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

Pornprasertmanit, S., Schoemann, A. M., Geldhof, G. J., & Little, T. D. (submitted). Probing latent interaction estimated with a residual centering approach.

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

• 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.

• plotProbe Plot the simple intercepts and slopes of the latent interaction.

Examples

dat3wayRC <- orthogonalize(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
"

fitRC3way <- sem(model3, data = dat3wayRC, meanstructure = TRUE)
summary(fitRC3way)

probe3WayMC(fitRC3way, 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))