Moderated nonlinear factor analysis lets a latent factor's item intercepts, loadings and distribution depend on covariates — a unified test of measurement invariance and differential item functioning (DIF).
Classical factor analysis assumes every item measures the latent construct the same way for everyone. MNLFA relaxes that: each item's intercept can shift with a covariate (uniform DIF) and, optionally, its loading can change too (nonuniform DIF). The factor's own mean and variance can also be modeled as functions of the covariates (impact), so genuine group differences on the construct are separated from item bias.
Continuous items use a linear factor model; binary items a two-parameter logistic (2PL) response model, so mixed item types are handled in one model. Covariates enter as linear predictors of each moderated parameter, which makes both categorical group contrasts and continuous moderators (age, time) natural.
Estimation is by marginal maximum likelihood: the latent factor is integrated out for each person with adaptive Gauss–Hermite quadrature (the integrand is re-centered at each person's posterior mode, so few nodes are needed), and the marginal likelihood is maximized over all parameters. Standard errors come from the observed information; DIF is tested with Wald statistics on the moderation coefficients. With no covariates the model reduces exactly to a single-factor model.
Identification: with the factor standardized (the default), the moderated measurement equations are identified without anchor items. Turning on factor mean/variance moderation adds impact estimation, which is only identified when some items are free of DIF and serve as anchors — the same requirement as in the wider MNLFA literature.
Numeric item columns (≥ 3) + optional covariate columns + item type (auto / continuous / binary) + toggles for moderating loadings and moderating factor mean/variance.
Parameter table (loadings, intercepts, moderation coefficients, residual variances, factor impact) with SEs / z / Wald p; per-item joint DIF tests; log-likelihood, AIC, BIC; convergence status.
A significant moderation coefficient (or per-item DIF test) means that item's behavior changes with the covariate — the item is not invariant. Non-significant moderation across items supports measurement invariance. Compare nested models by AIC/BIC to decide how much moderation to retain.