Score test — lavTestScore • lavaan

Score test (or Lagrange Multiplier test) for releasing one or more fixed or constrained parameters in the model.

Usage

lavTestScore(object, add = NULL, release = NULL,
             univariate = TRUE, cumulative = FALSE,
             epc = FALSE, standardized = epc, cov_std = epc,
             verbose = FALSE, warn = TRUE, information = "expected", ...)

Arguments

object: An object of class lavaan.
add: Either a character string (typically between single quotes) or a parameter table containing additional (currently fixed-to-zero) parameters for which the score test must be computed.
release: Vector of integers. The indices of the constraints that should be released. The indices correspond to the order in which the equality constraints appear in the parameter table.
univariate: Logical. If TRUE, compute the univariate score statistics, one for each constraint.
cumulative: Logical. If TRUE, order the univariate score statistics from large to small, and compute a series of multivariate score statistics, each time adding an additional constraint.
epc: Logical. If TRUE, and we are releasing existing constraints, compute the expected parameter changes for the existing (free) parameters, for each released constraint.
standardized: If TRUE, two extra columns (sepc.lv and sepc.all) in the $epc table will contain standardized values for the EPCs. In the first column (sepc.lv), standardization is based on the variances of the (continuous) latent variables. In the second column (sepc.all), standardization is based on the variances of both (continuous) observed and latent variables. (Residual) covariances are standardized using (residual) variances.
cov_std: Logical. See standardizedSolution.
verbose: Logical. Not used for now.
warn: Logical. If TRUE, print out warnings if they occur.
information: character indicating the type of information matrix to use (check lavInspect for available options). "expected" information is the default, which provides better control of Type I errors.
...: to allow use of old argumentname cov_std

Details

This function can be used to compute both multivariate and univariate score tests. There are two modes: 1) releasing fixed-to-zero parameters (using the add argument), and 2) releasing existing equality constraints (using the release argument). The two modes cannot be used simultaneously.

When adding new parameters, they should not already be part of the model (i.e. not listed in the parameter table). If you want to test for a parameter that was explicitly fixed to a constant (say to zero), it is better to label the parameter, and use an explicit equality constraint.

In addition to the standard (normal-theory) score test, robust versions are computed whenever the fitted object provides the ingredients for a sandwich-type covariance matrix: either because robust standard errors were requested (e.g., se = "robust.sem" or se = "robust.huber.white"), or because a scaled test statistic was requested (e.g., test = "satorra.bentler"). Following Satorra (2000), three robust versions are reported: a mean-scaled statistic (score.scaled), a mean-and-variance adjusted statistic with fractional degrees of freedom (score.adjusted), and the generalized, asymptotically distribution-free statistic (score.robust). For a single restriction, the three versions coincide. The univariate (and cumulative) score tests are then accompanied by scaled counterparts in the X2.scaled column. Note that for estimators that are not asymptotically efficient (e.g., DWLS, ULS and PML), only the robust versions have the correct asymptotic distribution.

Value

A list containing at least one data.frame:

$test: The total score test, with columns for the score test statistic (X2), the degrees of freedom (df), and a p value under the $\chi^2$ distribution (p.value). If robust versions could be computed, three additional rows are included: score.scaled, score.adjusted and score.robust (see Details).
$uni: Optional (if univariate=TRUE). Each 1-df score test, equivalent to modification indices. If robust versions could be computed, the scaled statistics are provided in the X2.scaled and p.value.scaled columns. If epc=TRUE when adding parameters (not when releasing constraints), an unstandardized EPC is provided for each added parameter, as would be returned by modificationIndices.
$cumulative: Optional (if cumulative=TRUE). Cumulative score tests, with scaled counterparts if robust versions could be computed.
$epc: Optional (if epc=TRUE). Parameter estimates, expected parameter changes, and expected parameter values if all the tested constraints were freed.

References

Bentler, P. M., & Chou, C. P. (1993). Some new covariance structure model improvement statistics. Sage Focus Editions, 154, 235-255.

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

    b1 == b2
    b2 == b3
'
fit <- cfa(HS.model, data=HolzingerSwineford1939)

# test 1: release both two equality constraints
lavTestScore(fit, cumulative = TRUE)
#> $test
#> 
#> total score test:
#> 
#>    test     X2 df p.value
#> 1 score 12.306  2   0.002
#> 
#> $uni
#> 
#> univariate score tests:
#> 
#>   lhs op rhs     X2 df p.value
#> 1  b1 ==  b2 12.060  1   0.001
#> 2  b2 ==  b3  0.947  1   0.330
#> 
#> $cumulative
#> 
#> cumulative score tests:
#> 
#>   lhs op rhs     X2 df p.value
#> 1  b1 ==  b2 12.060  1   0.001
#> 2  b2 ==  b3 12.306  2   0.002
#> 

# test 2: the score test for adding two (currently fixed
# to zero) cross-loadings
newpar = '
    visual =~ x9
    textual =~ x3
'
lavTestScore(fit, add = newpar)
#> $test
#> 
#> total score test:
#> 
#>    test     X2 df p.value
#> 1 score 46.061  2       0
#> 
#> $uni
#> 
#> univariate score tests:
#> 
#>           lhs op rhs     X2 df p.value
#> 1  visual=~x9 ==   0 35.117  1   0.000
#> 2 textual=~x3 ==   0 10.938  1   0.001
#> 

# equivalently, "add" can be a parameter table specifying parameters to free,
# but must include some additional information:
PT.add <- data.frame(lhs = c("visual","textual"),
                     op = c("=~","=~"),
                     rhs = c("x9","x3"),
                     user = 10L, # needed to identify new parameters
                     free = 1, # arbitrary numbers > 0
                     start = 0) # null-hypothesized value
PT.add
#>       lhs op rhs user free start
#> 1  visual =~  x9   10    1     0
#> 2 textual =~  x3   10    1     0
lavTestScore(fit, add = PT.add) # same result as above
#> $test
#> 
#> total score test:
#> 
#>    test     X2 df p.value
#> 1 score 46.061  2       0
#> 
#> $uni
#> 
#> univariate score tests:
#> 
#>           lhs op rhs     X2 df p.value
#> 1  visual=~x9 ==   0 35.117  1   0.000
#> 2 textual=~x3 ==   0 10.938  1   0.001
#>