Search for noninvariant parameters across groups.

The function implements the sequential selection method similar to that discussed in Yoon and Millsap (2007). The function proceeds in the order of metric, scalar (threshold), and strict invariance. In each stage, invariance constraints in all items, and the constraint associated with the biggest test statistic above a predefined threshold is freed, before recomputing the test statistic for the next constraint to free.

Usage

pinSearch(
  config_mod,
  ...,
  type = c("loadings", "intercepts", "thresholds", "residuals", "residual.covariances"),
  inv_test = c("mod", "score", "lrt"),
  sig_level = 0.05,
  control_fdr = FALSE,
  effect_size = FALSE,
  progress = FALSE
)

Arguments

config_mod: Syntax of a configural invariance model to be passed to lavaan::cfa().
...: Additonal arguments passed to lavaan::cfa().
type: Character variable indicating the stage of invariance to be searched. Currently supported options are (a) for continuous indicators, "loadings", "intercepts", "residuals", and "residual.covariances", and (b) "loadings", "thresholds", "residual.covariances", in an increasingly strict order. A stricter model (e.g., "residual.covariances") will have constraints of all previous stages.
inv_test: Character variable indicating the statistical test to be used for specification search. Currently supported options are "mod" for modification index using lavaan::modindices(), "score" for score test statistic using lavaan::lavTestScore(), and (experimental) "lrt" for likelihood ratio test statistic using lavaan::lavTestLRT().
sig_level: Significance level used to determine whether the parameter associated with the highest modification index should be removed. Default is .05.
control_fdr: Logical; whether to use adjust for false discovery rate for multiple testing. If TRUE, the method by Benjamini & Gavrilov (2009) will be used.
effect_size: Logical; whether to compute dmacs (two groups) or fmacs (> two groups) effect size or not (default). This is an experimental feature.
progress: Logical; an experimental feature of showing a progress bar if TRUE. Because the number of steps is unknown until the stopping criteria are reached, the progress bar may be inaccurate.

Value

A list of three elements:

Partial Invariance Fit: A lavaan::lavaan object containing the final partial invariance model.
Non-Invariant Items: A data frame of non-invariant parameters.
effect_size: Effect size statistics obtained from pin_effsize().

Details

Note that when an item has a non-invariant loading, the corresponding intercept constraint will automatically be freed, as intercept difference across groups is sensitive to the location of the zero point for the latent variable and the item.

For a particular stage of invariance constraints, the Benjamini & Gavrilov method uses an adjusted $alpha$ level of $$iq / [m + 1 - i(1 - q)]$$ where $i$ is the step index in the search, $m$ is the maximum number of constraints that can be freed, and $q$ is the desirable significance level.

References

Yoon, M., & Millsap, R. E. (2007). Detecting violations of factorial invariance using data-based specification searches: A Monte Carlo study. Structural Equation Modeling: A Multidisciplinary Journal, 14(3), 435-463.

Examples

library(lavaan)
#> This is lavaan 0.6-19
#> lavaan is FREE software! Please report any bugs.
library(MASS)
# Simulate random data
set.seed(14) # for reproducible results
mod_ninv <- "f =~ c(1, 0.8, .6)*y1 + c(0.8, 1.2, 1.0)*y2"
mod_inv <- paste0("1*y", 3:6, collapse = " + ")
sim_mod <- paste(
    paste(mod_ninv, "+", mod_inv),
    "f ~~ c(1, 1.3, 1.5)*f
     f ~ c(0, 0.5, 1.0)*1
     y1 ~ c(0, 0.3, 0)*1
     y3 ~ c(0.3, 0, -0.3)*1
     y1 ~~ c(1, .5, 1)*y1",
    sep = "\n"
)
# The uniqueness of each item is assumed to be 1.0
dat_sim <- simulateData(sim_mod, sample.nobs = c(100, 100, 100))
# Fit configural model:
sam_config <- paste(
    paste0("f =~ ", paste0("y", 1:6, collapse = " + "))
)
pinSearch(sam_config,
    data = dat_sim, group = "group",
    type = "intercepts"
)
#> $`Partial Invariance Fit`
#> lavaan 0.6-19 ended normally after 55 iterations
#> 
#>   Estimator                                         ML
#>   Optimization method                           NLMINB
#>   Number of model parameters                        58
#>   Number of equality constraints                    23
#> 
#>   Number of observations per group:                   
#>     1                                              100
#>     2                                              100
#>     3                                              100
#> 
#> Model Test User Model:
#>                                                       
#>   Test statistic                                55.735
#>   Degrees of freedom                                46
#>   P-value (Chi-square)                           0.154
#>   Test statistic for each group:
#>     1                                           23.938
#>     2                                           15.880
#>     3                                           15.917
#> 
#> $`Non-Invariant Items`
#>   group lhs rhs       type
#> 1     2  y1     intercepts
#>