SEM moderation = latent interaction

We want:

\[ \eta_Y = \beta_X\,\xi_X + \beta_Z\,\xi_Z + \beta_{XZ}(\xi_X\xi_Z) + \zeta \]

Problem: \((\xi_X\xi_Z)\) is not observed.

Two main families of solutions:

1. Product indicators (PI): build observed products (e.g., \((x_i z_j)\)) and model them as indicators of a latent interaction factor.
2. LMS/QML (distribution-analytic): estimate the interaction directly in the likelihood (no product indicators), but standard global fit indices typically disappear (we move away from covariance estimation).

The main technical issues

1. Nonlinearity: interaction makes the model nonlinear

2. Non-normality: product of normals is not normal

3. Fit logic changes: especially with LMS (no classical \((\chi^2)\), CFI, RMSEA for the interaction model)

Everything else is a consequence: identification, centering confusion, computational burden, interpretation…but also better estimates, correct modelling…

Difficulty 1 — The product is latent (not observed)

In regression: compute \((XZ)\).

In SEM:

• X and Z are latent
• you only have indicators
• product indicators contain extra error cross-terms

Difficulty 2 — Non-normality of the product

Even if \((X)\) and \((Z)\) are normal, \((XZ)\) is not.

par(mfrow = c(1, 2))
hist(X,  main = "Latent X (normal)")
hist(XZ, main = "Latent X*Z (non-normal)")

par(mfrow = c(1, 1))

Implication: classical normal-theory SEs / tests can misbehave, especially in small N.

Difficulty 3 — Identification + parameter explosion (PI)

Difficulty 4 — Fit indices disappear (LMS)

With LMS/QML you typically cannot report classical SEM global fit indices for the interaction model.

So evaluation becomes:

1. Fit baseline model (no interaction): report CFI/TLI/RMSEA/SRMR there
2. Add interaction: compare baseline vs interaction using log-likelihood difference test (or AIC/BIC)

This is a conceptual shift: fit ≠ truth, and sometimes fit ≠ available.

Difficulty 5 — Interpretation is conditional (probing)

Difficulty 6 — Centering: interpretation vs estimation

Interpretation: \((\beta_X)\) is the effect of X when \((Z=0)\). Centering makes “0” meaningful (often the mean).

Estimation (PI): centering indicators/products helps separate lower-order and interaction variance and stabilizes the model.

Rule of thumb:

• PI methods: use double-mean centering (e.g., dblcent)
• LMS/QML: focus on interpretability (what does Z=0 mean?) and probe slopes

Difficulty 7 — Computation & power

• interaction effects are typically smaller than main effects
• larger N needed for stable detection
• PI: bigger model → more convergence problems
• LMS/QML: numerical integration / iterative routines → slower, sometimes fragile

Evaluate: always report convergence, warnings, and sensitivity checks.

Option 0 — “Observed proxies” (baseline)

• totals / sum scores
• factor-score regression

Pros:

• simple and familiar

Cons:

• treats measurement error as absent or “handled” by factor scores
• interaction often attenuates

Option 1 — Latent interaction via product indicators (PI)

Option 2 — Latent Moderated Structural Equations (LMS)

Idea: estimate \((XZ)\) in the likelihood (no product indicators).

Pros:

• elegant, avoids product-indicator explosion
• supports **higher-order** interactions

Cons:

• classical global fit indices typically not available for interaction model
• must use baseline fit + LRT / IC

Now: fit the latent interaction with modsem

We fit the same SEM twice:

• PI dblcent (method = "dblcent")
• LMS (method = "lms") and mention QML

NOTE: You need the modsem package installed.

Latent interaction with PI dblcent

library(modsem)

model_base <- '
  intelligence =~ int1 + int2 + int3
  anxiety      =~ anx1 + anx2 + anx3
  achievement  =~ ach1 + ach2 + ach3

  achievement ~ intelligence + anxiety
'

fit_base <- modsem(model_base, data = dat)
# summary(fit_base)

model_int <- '
  intelligence =~ int1 + int2 + int3
  anxiety      =~ anx1 + anx2 + anx3
  achievement  =~ ach1 + ach2 + ach3

  achievement ~ intelligence + anxiety +
                intelligence:anxiety
'

fit_dblcent <- modsem(model_int, 
        data = dat, method = "dblcent")
# summary(fit_dblcent)

Latent interaction with LMS (and QML)

fit_lms <- modsem(model_int, data = dat, method = "lms")
fit_qml <- modsem(model_int, data = dat, method = "qml")

# summary(fit_lms)
# summary(fit_qml)

Evaluate (LMS/QML):

• baseline model: classical fit indices
• interaction model: compare log-likelihood vs baseline (LRT) and check AIC/BIC

Plots

p <- plot_interaction(x = "intelligence", z = "anxiety", 
                      y = "achievement", vals_z = c(-1, 1), 
                      model = fit_lms)
p

LMS evaluation: baseline fit + LRT

# pseudo-code: exact methods depend on the object class and available methods

fit_lms <- modsem(model_int, data = dat, method = "lms")
fit_base <- estimate_h0(fit_lms, calc.se = FALSE)

compare_fit(est_h1 = fit_lms, est_h0 = fit_base)

$D
[1] 23.78654

$df
[1] 1

$p
[1] 1.076325e-06

$diff.loglik
[1] 11.89327

Summary

Who’s best? A simple simulation

# simulation parameters
R <- 500; N <- 400; rho_xz <- 0.30
beta_x  <- 0.30; beta_z  <- 0.30 
beta_xz <- 0.20  # <-- ground 
lambda_int <- c(.80, .70, .75); 
lambda_anx <- c(.80, .70, .75)
lambda_ach <- c(.80, .70, .75)

Probing the interaction (simple slopes, JN, surface)

# Probing works directly on modsem fits

# simple_slopes(model = fit_lms, x = "intelligence", z = "anxiety", y = "achievement")
# plot_surface(model = fit_lms, x = "intelligence", z = "anxiety", y = "achievement")

plot_jn(model = fit_lms, x = "intelligence", z = "anxiety", y = "achievement")

“Under the hood”: product indicators with semTools + lavaan

library(semTools)

# matched-pair products, double-mean centered
# (x1*z1, x2*z2, x3*z3)
dat_pi <- indProd(
  data = dat,
  var1 = c("int1","int2","int3"),
  var2 = c("anx1","anx2","anx3"),
  match = TRUE,
  meanC = TRUE,
  doubleMC = TRUE
)

prod_names <- grep("^int[1-3].*anx[1-3]$", names(dat_pi), value = TRUE)

model_pi <- paste0('
  intelligence =~ int1 + int2 + int3
  anxiety      =~ anx1 + anx2 + anx3
  achievement  =~ ach1 + ach2 + ach3

  intxn =~ ', paste(prod_names, collapse = " + "), '

  achievement ~ intelligence + anxiety + intxn
')

fit_pi <- lavaan::sem(model_pi, data = dat_pi, meanstructure = TRUE)
summary(fit_pi, fit.measures = TRUE, standardized = TRUE)

Reporting checklist (minimal)

1. Measurement (baseline): CFA/SEM fit (CFI/TLI/RMSEA/SRMR) + key loadings

2. Interaction method: PI (dblcent) or LMS/QML (and why)

3. Evaluation:

• PI: report SEM fit indices (if available) + local diagnostics
• LMS/QML: baseline fit indices + LRT (ΔlogLik) or AIC/BIC

4. Effect reporting: \((\hat\beta_{XZ})\), SE/CI, probing results (simple slopes / JN)

5. Transparency: centering decisions and what “0” means for the moderator

Group comparisons (classic mg-cfa)

When the interaction term is not a continuous [latent] variable, we can run interactions using MG-CFA (slides here)

m.med=sem(model,data,group="sex",
          group.equal=c("loadings","intercepts",
                        "regressions" # <---------- #
                        )
          )

Group comparisons (handmade mg-cfa)

model_equal <- '
# measurement
intelligence =~ int1 + int2 + int3
achievement  =~ ach1 + ach2 + ach3

# structural path
achievement ~ c(b,b)*intelligence
'
fit_equal <- sem(model_equal, data = dat, 
                 group = "sex")

model_free <- '
# measurement
intelligence =~ int1 + int2 + int3
achievement  =~ ach1 + ach2 + ach3

# structural path
achievement ~ c(b_m, b_f)*intelligence
'
fit_free <- sem(model_free, data = dat, 
                group = "sex")

anova(fit_equal, fit_free)

Chi-Squared Difference Test

          Df    AIC    BIC  Chisq Chisq diff RMSEA Df diff Pr(>Chisq)
fit_free  16 7540.7 7700.9 27.024                                    
fit_equal 17 7539.6 7695.5 27.875    0.85081     0       1     0.3563

Take-home: 3 things

1. Latent interactions are hard because the product term introduces nonlinearity + non-normality.
2. PI dblcent is transparent (and often provides classic fit machinery) but can be heavy/fragile.
3. LMS/QML are elegant for estimation, but they force a new evaluation mindset: baseline fit + LRT, plus probing for interpretation.

Further reading / self-study links

• modsem documentation and vignettes (https://modsem.org/articles/modsem.html)
• modsem R package documentation (product indicator methods; LMS/QML; probing functions)
• semTools documentation for indProd()
• Aiken, L. S., & West, S. G. (1991). Multiple regression: Testing and interpreting interactions.