lavTestLRT: LRT test

In lavaan: Latent Variable Analysis

LRT test

Description

LRT test for comparing (nested) lavaan models.

Usage

lavTestLRT(object, ..., method = "default", test = "default",
           a_method = "delta", scaled_shifted = TRUE,
           check_nested = TRUE,
           type = "Chisq", model_names = NULL)
anova(object, ...)

Arguments

object	An object of class lavaan.
...	additional objects of class lavaan.
method	Character string. The possible options are "satorra.bentler.2001", "satorra.bentler.2010", "satorra.2000", and "standard". See details. Note that the robust options ("satorra.bentler.2001", "satorra.bentler.2010" and "satorra.2000") require that the models were fitted with a robust (scaled) test statistic (for example, using estimator = "MLM" or test = "satorra.bentler"). If none of the models were fitted with a robust test statistic, but a robust method= is requested, a warning is issued, the method= argument is ignored, and a standard (regular) chi-squared difference test is computed instead.
test	Character string specifying which scaled test statistics to use, in case multiple scaled test= options were requested when fitting the model(s). See details. FMG nested tests can be requested using names such as "pall", "all", "peba4", or "pols3"; the convenience value "fmg" resolves to the recommended nested default.
a_method	Character string. The possible options are "exact" and "delta". This is only used when method = "satorra.2000". It determines how the Jacobian of the constraint function (the matrix A) will be computed. Note that if a_method = "exact", the models must be nested in the parameter sense, while if a_method = "delta", they only need to be nested in the covariance matrix sense.
scaled_shifted	Logical. Only used when method = "satorra.2000". If TRUE, we use a scaled and shifted test statistic. If FALSE, the statistic depends on the test that was used to fit the models: when the models were fitted with a mean-and-variance adjusted test (test = "mean.var.adjusted" or "scaled_shifted"), a mean and variance adjusted (Satterthwaite style) difference test is used; when they were fitted with a (Satorra-)Bentler scaled test (test = "satorra.bentler", "yuan.bentler" or "yuan.bentler.mplus"), a scaled (non-shifted) difference test is used. The title printed above the anova table reflects the statistic that is actually used.
check_nested	Logical. If TRUE (the default), the nesting of each pair of models (in the covariance sense) is verified with the NET procedure (Bentler & Satorra, 2010; Asparouhov & Muthen, 2019): the less restricted model must be able to exactly reproduce the model-implied moments of the more restricted model. If a pair of models is found not to be nested, a warning is issued (the difference test is still computed). For a pair of models with equal degrees of freedom, equivalence (nesting in both directions) is checked instead, and the warning states whether the models are equivalent (the difference test is meaningless) or not equivalent (the difference test is invalid). Pairs that cannot be checked are skipped silently; see lavTestNET for the details of the procedure (and for its limitations). Independently of this argument, several sanity checks are carried out to verify that all models were fitted to the same data (see the Details section).
type	Character. If "Chisq", the test statistic for each model is the (scaled or unscaled) model fit test statistic. If "Cf", the test statistic for each model is computed by the lavTablesFitCf function. If "browne.residual.adf" (alias "browne") or "browne.residual.nt", the standard chi-squared difference is calculated from each model's residual-based statistic.
model_names	Character vector. If provided, use these model names in the first column of the anova table.

Details

The anova function for lavaan objects simply calls the lavTestLRT function, which has a few additional arguments.

The chi-squared difference test is only valid when all models are (nested and) fitted to the same set of sample statistics. Several sanity checks are therefore carried out (in the spirit of semTools::net()): the number of groups, the group labels, the (group-specific) numbers of observations, and the observed sample statistics must all be identical across models. If any of these checks fails, a warning is issued (but the table is still computed). In particular, a single-group (pooled) model cannot be compared directly with a multiple-group model: their test statistics are computed relative to different (saturated) models. Instead, refit the pooled model as a multiple-group model in which all parameters are constrained to be equal across groups (group_equal = "all"), and compare that model with the unrestricted multiple-group model. See the Examples section for an illustration.

The only test= options that currently have actual consequences are "satorra.bentler", "yuan.bentler", or "yuan.bentler.mplus" because "mean.var.adjusted" and "scaled_shifted" are currently distinguished by the scaled_shifted argument. See lavOptions for details about the test= options implied by robust estimator= options. The "default" is to select the first available scaled statistic, if any. To check which test(s) were computed when fitting your model(s), use lavInspect(fit, "options")$test.

If type = "Chisq" and the test statistics are scaled, a special scaled difference test statistic is computed. If method is "satorra.bentler.2001", a simple approximation is used described in Satorra & Bentler (2001). In some settings, this can lead to a negative test statistic. To ensure a positive test statistic, we can use the method proposed by Satorra & Bentler (2010). Alternatively, when method="satorra.2000", the original formulas of Satorra (2000) are used. The latter is used for model comparison when ... contains additional (nested) models. Even when test statistics are scaled in object or ..., users may request the method="standard" test statistic, without a robust adjustment.

Conversely, a robust method= ("satorra.bentler.2001", "satorra.bentler.2010" or "satorra.2000") can only be applied when the models were fitted with a robust (scaled) test statistic. If none of the models in object or ... were fitted with a robust test statistic, a robust method= cannot be honored: in that case a warning is issued, the method= argument is ignored, and a standard (regular) chi-squared difference test is returned.

FMG nested tests are available when type = "Chisq" and test is an FMG test name. The supported nested FMG methods are "pall", "all", "peba" and "pols" variants. The convenience value "fmg" resolves to "pall_ug_rls" for single-group comparisons and "pall_ug_ml" for multiple-group comparisons. They use the Satorra (2000) UGamma projection; method may be left at "default" or set to "standard" or "satorra.2000". Other nested LRT methods are not used for FMG tests.

Value

An object of class anova. When given a single argument, it simply returns the test statistic of this model. When given a sequence of objects, this function tests the models against one another, after reordering the models according to their degrees of freedom.

The object carries a few (hidden) attributes that record which difference test was computed (intended for developers; new in 0.7-3): "type" (the type of test, in canonical spelling, e.g. "chisq" or "browne.residual.nt.model"), "method" (the method that was used for a scaled difference test, e.g. "satorra.bentler.2001"; "standard" for an unscaled difference test), "test" (the name of the scaled test statistic the difference test was based on, e.g. "satorra.bentler"; "standard" if unscaled) and "scaled" (logical). For scaled difference tests, the attributes "scale" (and "shift") contain the scaling factors (and shift parameters).

Note

If there is a lavaan model stored in object@external$h1.model, it will be added to ...

References

Satorra, A. (2000). Scaled and adjusted restricted tests in multi-sample analysis of moment structures. In Heijmans, R.D.H., Pollock, D.S.G. & Satorra, A. (eds.), Innovations in multivariate statistical analysis: A Festschrift for Heinz Neudecker (pp.233-247). London, UK: Kluwer Academic Publishers.

Satorra, A., & Bentler, P. M. (2001). A scaled difference chi-square test statistic for moment structure analysis. Psychometrika, 66(4), 507-514. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/BF02296192")}

Satorra, A., & Bentler, P. M. (2010). Ensuring positiveness of the scaled difference chi-square test statistic. Psychometrika, 75(2), 243-248. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/s11336-009-9135-y")}

Foldnes, N., Moss, J., & Gronneberg, S. (2024). Improved goodness of fit procedures for structural equation models. Structural Equation Modeling: A Multidisciplinary Journal, 1-13. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/10705511.2024.2372028")}

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")}

Foldnes, N., Gronneberg, S., & Moss, J. (2026). Penalized eigenvalue block averaging: Extension to nested model comparison and Monte Carlo evaluations. Behavior Research Methods, 58(4). \Sexpr[results=rd]{tools:::Rd_expr_doi("10.3758/s13428-026-02968-4")}

Examples

HS.model <- '
    visual  =~ x1 + b1*x2 + x3
    textual =~ x4 + b2*x5 + x6
    speed   =~ x7 + b3*x8 + x9
'
fit1 <- cfa(HS.model, data = HolzingerSwineford1939)
fit0 <- cfa(HS.model, data = HolzingerSwineford1939,
            orthogonal = TRUE)
lavTestLRT(fit1, fit0)


## A single-group (pooled) model can NOT be compared directly with a
## multiple-group model; the warning explains why the test is invalid:
HS <- '
    visual  =~ x1 + x2 + x3
    textual =~ x4 + x5 + x6
    speed   =~ x7 + x8 + x9
'
fit.pooled <- cfa(HS, data = HolzingerSwineford1939, meanstructure = TRUE)
fit.groups <- cfa(HS, data = HolzingerSwineford1939, group = "school")
lavTestLRT(fit.pooled, fit.groups) # invalid comparison (warning)

## A valid alternative: refit the pooled model as a multiple-group model
## with all parameters constrained to be equal across groups
fit.pooled2 <- cfa(HS, data = HolzingerSwineford1939, group = "school",
                   group_equal = "all")
lavTestLRT(fit.pooled2, fit.groups) # valid comparison


## When multiple test statistics are selected when the model is fitted,
## use the type= and test= arguments to select a test for comparison.

## refit models, requesting 6 test statistics (in addition to "standard")
t6.1 <- cfa(HS.model, data = HolzingerSwineford1939,
            test = c("browne.residual.adf","scaled.shifted","mean.var.adjusted",
                     "satorra.bentler", "yuan.bentler", "yuan.bentler.mplus"))
t6.0 <- cfa(HS.model, data = HolzingerSwineford1939, orthogonal = TRUE,
            test = c("browne.residual.adf","scaled.shifted","mean.var.adjusted",
                     "satorra.bentler", "yuan.bentler", "yuan.bentler.mplus"))

## By default (test="default", type="Chisq"), the first scaled statistic
## requested will be used. Here, that is "scaled.shifted"
lavTestLRT(t6.1, t6.0)
## But even if "satorra.bentler" were requested first, method="satorra.2000"
## provides the scaled-shifted chi-squared difference test:
lavTestLRT(t6.1, t6.0, method = "satorra.2000")
## == lavTestLRT(update(t6.1, test = "scaled.shifted"), update(t6.0, test = "scaled.shifted"))

## The mean- and variance-adjusted (Satterthwaite) statistic implies
## scaled_shifted = FALSE
lavTestLRT(t6.1, t6.0, method = "satorra.2000", scaled_shifted = FALSE)

## Because "satorra.bentler" is not the first scaled test in the list,
## we MUST request it explicitly:
lavTestLRT(t6.1, t6.0, test = "satorra.bentler") # method="satorra.bentler.2001"
## == lavTestLRT(update(t6.1, test = "satorra.bentler"),
##               update(t6.0, test = "satorra.bentler"))
## The "strictly-positive test" is necessary when the above test is < 0:
lavTestLRT(t6.1, t6.0, test = "satorra.bentler", method = "satorra.bentler.2010")

## Likewise, other scaled statistics can be selected:
lavTestLRT(t6.1, t6.0, test = "yuan.bentler")
## == lavTestLRT(update(t6.1, test = "yuan.bentler"),
##               update(t6.0, test = "yuan.bentler"))
lavTestLRT(t6.1, t6.0, test = "yuan.bentler.mplus")
## == lavTestLRT(update(t6.1, test = "yuan.bentler.mplus"),
##               update(t6.0, test = "yuan.bentler.mplus"))

## To request the difference between Browne's (1984) residual-based statistics,
## rather than statistics based on the fitted model's discrepancy function,
## use the type= argument:
lavTestLRT(t6.1, t6.0, type = "browne.residual.adf")

## Despite requesting multiple robust tests, it is still possible to obtain
## the standard chi-squared difference test (i.e., without a robust correction)
lavTestLRT(t6.1, t6.0, method = "standard")
## == lavTestLRT(update(t6.1, test = "standard"), update(t6.0, test = "standard"))

## FMG nested p-values use the Satorra (2000) UGamma projection
lavTestLRT(fit1, fit0, method = "satorra.2000", test = "pall")

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