Introduction

This vignette demonstrates how to evaluate and compare model fit for latent interaction models estimated via

• LMS (Latent Moderated Structural Equations)
• QML (Quasi-Maximum Likelihood)

using the modsem package (v≥1.0.8). Because standard Chi-square statistics are not available under LMS/QML, we assess fit by:

1. Examining fit indices for the baseline (no-interaction) model.
2. Conducting a likelihood-ratio difference test to compare the baseline and interaction models (Klein & Moosbrugger, 2000; Klein & Múthen, 2007).

If the baseline model shows acceptable fit and adding the latent interaction significantly improves fit, the interaction model can also be deemed well-fitting.

Example

We define a model with three latent variables (X, Y, Z) and their interaction (X:Z):

m1 <- "
# Outer (measurement) model
X =~ x1 + x2 + x3
Y =~ y1 + y2 + y3
Z =~ z1 + z2 + z3

# Inner (structural) model
Y ~ X + Z + X:Z
"

# Estimate the full (H1) model via LMS
est_h1 <- modsem(m1, oneInt, method = "lms")

# Estimate the baseline (H0) model without interaction
est_h0 <- estimate_h0(est_h1, calc.se = FALSE) # std.errors are not needed

Fit measures baseline model

To get fit measures for the baseline model you can use the fit_modsem_da() function.

fit_modsem_da(est_h0)

It can also be used to get fit measures for the full model, but should be pared with chisq = FALSE to avoid the Chi-square test. If it is set to TRUE it will calculate the Chi-square test while ignoring the interaction terms in the model.

fit_modsem_da(est_h1, chisq = FALSE)

Difference Test of Fit

Compare H0 vs. H1 using a log-likelihood ratio test:

compare_fit(est_h0, est_h1)

A significant p-value indicates the latent interaction term significantly improves model fit.

Inspecting Fit Indices

For convenience, you can also use the modsem_inspect() function with what = "fit" to get fit indices for both models, and comparative fit in one go.

modsem_inspect(est_h1, what = "fit")

References

Klein, A., & Moosbrugger, H. (2000). 
  <doi:10.1007/BF02296338>.
  "Maximum likelihood estimation of latent interaction effects with the LMS method."
Klein, A. G., & Muthén, B. O. (2007). 
  <doi:10.1080/00273170701710205>.
  "Quasi-maximum likelihood estimation of structural equation models with multiple interaction and quadratic effects."