lavTestNET: Nesting and Equivalence Testing (NET)

In lavaan: Latent Variable Analysis

Nesting and Equivalence Testing (NET)

Description

Checks – for every pair of fitted models – whether the more restricted model is nested within the less restricted model in the covariance sense, using the NET procedure of Bentler & Satorra (2010); for pairs of models with equal degrees of freedom, it checks whether the models are equivalent.

Usage

lavTestNET(object, ..., crit = 1e-7, npoints = 1L,
           iseed = 12345L, model_names = NULL)

Arguments

object	An object of class lavaan.
...	additional (at least one) objects of class lavaan.
crit	Numeric. The criterion applied to the minimum of the fit function when the less restricted model is fitted to the model-implied moments of the more restricted model: values below crit imply nesting (or equivalence, if the degrees of freedom are equal), values above 10 * crit imply non-nesting, and values in between are inconclusive (following Asparouhov & Muthen, 2019).
npoints	Integer. The number of parameter points of the restricted model where the check is carried out. The first point is always the vector of parameter estimates. If npoints > 1L, additional randomly perturbed parameter values are used (respecting any equality and inequality constraints). This protects against ‘data specific’ conclusions: two models may accidentally appear to be nested (or equivalent) at the estimated parameter values only. See the discussion in Asparouhov & Muthen (2019).
iseed	Integer. The seed for the random perturbations (only used if npoints > 1L). The state of the random generator of the user is left untouched.
model_names	Character vector. If provided, use these model names in the rows and columns of the result.

Details

The procedure works as follows: the model-implied (first and) second order moments of the more restricted model are treated as if they were sample statistics, and the less restricted model is fitted to these moments. If (and only if) the minimum of the fit function is zero, the less restricted model can exactly reproduce these moments, and the more restricted model is nested within the less restricted model at this parameter point. If both models have the same degrees of freedom and each model is nested within the other, the models are equivalent.

Whenever the nesting can be established parametrically (the parameter table of the restricted model equals the parameter table of the less restricted model with some free parameters fixed and/or constrained to be equal), no refitting is needed, and the (numerically exact) verdict is based on the parameter tables alone.

Note that nesting in the covariance sense does not by itself guarantee that the usual chi-squared difference test behaves as a chi-square variate: if the nesting only holds on the boundary of the parameter space (for example, a variance fixed to zero), the standard regularity conditions fail.

The procedure is not available for multilevel models, and (currently) not for categorical models with conditional_x = TRUE, correlation structures, or models involving auxiliary variables.

Value

A list of class lavaan.net with elements nested (a logical matrix; entry [i, j] indicates whether the model in row i is nested within the model in column j; NA means the pair could not be checked, or the result was inconclusive), fx (the corresponding fit function minima), reason (a character matrix explaining – for pairs that could not be checked – why the check was skipped; these reasons are also shown by the print method), df (the degrees of freedom of the models, in the order used in the matrix), crit and npoints.

References

Bentler, P. M., & Satorra, A. (2010). Testing model nesting and equivalence. Psychological Methods, 15(2), 111-123. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1037/a0019625")}

Asparouhov, T., & Muthen, B. (2019). Nesting and equivalence testing for structural equation models. Structural Equation Modeling: A Multidisciplinary Journal, 26(2), 302-309. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/10705511.2018.1513795")}

See Also

lavTestLRT, which uses this machinery (see its check_nested argument) to warn when a chi-squared difference test is requested for non-nested models.

Examples

HS.model <- ' visual  =~ x1 + x2 + x3
              textual =~ x4 + x5 + x6
              speed   =~ x7 + x8 + x9 '
fit1 <- cfa(HS.model, data = HolzingerSwineford1939)
# nested: the same model with orthogonal factors
fit0 <- cfa(HS.model, data = HolzingerSwineford1939, orthogonal = TRUE)
# equivalent: the same model with a different scaling of the factors
fit1b <- cfa(HS.model, data = HolzingerSwineford1939, std_lv = TRUE)
# not nested
model2 <- ' f1 =~ x1 + x2 + x3 + x4
            f2 =~ x5 + x6 + x7 + x8 + x9 '
fit2 <- cfa(model2, data = HolzingerSwineford1939)

lavTestNET(fit1, fit0, fit1b, fit2,
           model_names = c("3f", "3f-orth", "3f-std.lv", "2f"))

lavaan documentation built on Oct. 8, 2026, 5:06 p.m.