Wald test

Wald test for testing a linear hypothesis about the parameters of a fitted lavaan object.

Usage

lavTestWald(object, constraints = NULL, verbose = FALSE)

Arguments

object

An object of class lavaan.

constraints

A character string (typically between single quotes) containing one or more equality constraints. See examples for more details.

verbose

Logical. If TRUE, print out the restriction matrix and the estimated restricted values.

Details

The constraints are specified using the "==" operator. Both the left-hand side and the right-hand side of the equality can contain a linear combination of model parameters, or a constant (like zero). The model parameters must be specified by their user-specified labels. Names of defined parameters (using the ":=" operator) can be included too.

The test statistic is based on the variance-covariance matrix of the estimated parameters (vcov) of the fitted object. Therefore, if robust standard errors were requested for the fitted object (e.g., se = "robust.sem" or se = "robust.huber.white"), the main test statistic is already the generalized (‘robust’) Wald test described in Satorra (2000). In that case (or when a scaled test statistic was requested, so that the ingredients for a sandwich-type covariance matrix are available), additional versions of the Wald test statistic are computed as well: the standard (normal-theory) statistic (stat.standard), a mean-scaled statistic (stat.scaled), a mean-and-variance adjusted statistic with fractional degrees of freedom (stat.adjusted), and the generalized statistic (stat.robust); see Satorra (2000).

Value

A list containing at least three elements: the Wald test statistic (stat), the degrees of freedom (df), and a p-value under the chi-square distribution (p.value). If robust versions could be computed, the list also contains the standard, scaled, adjusted and generalized versions of the test statistic, together with their p-values (see Details).

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: Kluwer Academic Publishers.

Examples

HS.model <- '
    visual  =~ x1 + b1*x2 + x3
    textual =~ x4 + b2*x5 + x6
    speed   =~ x7 + b3*x8 + x9
'

fit <- cfa(HS.model, data=HolzingerSwineford1939)

# test 1: test about a single parameter
# this is the 'chi-square' version of the 
# z-test from the summary() output
lavTestWald(fit, constraints = "b1 == 0")
#> $stat
#> [1] 30.84248
#> 
#> $df
#> [1] 1
#> 
#> $p.value
#> [1] 2.79844e-08
#> 
#> $se
#> [1] "standard"
#> 

# test 2: several constraints
con = '
   2*b1 == b3
   b2 - b3 == 0
'
lavTestWald(fit, constraints = con)
#> $stat
#> [1] 0.147265
#> 
#> $df
#> [1] 2
#> 
#> $p.value
#> [1] 0.9290131
#> 
#> $se
#> [1] "standard"
#>