Measurement Invariance of EGA Structure

Estimates configural invariance using bootEGA on all data (across groups) first. After configural variance is established, then metric invariance is tested using the community structure that established configural invariance (see Details for more information on this process)

Usage

invariance(
  data,
  groups,
  structure = NULL,
  iter = 500,
  configural.threshold = 0.7,
  configural.type = c("parametric", "resampling"),
  corr = c("auto", "cor_auto", "pearson", "spearman"),
  na.data = c("pairwise", "listwise"),
  model = c("BGGM", "glasso", "TMFG"),
  algorithm = c("leiden", "louvain", "walktrap"),
  uni.method = c("expand", "LE", "louvain"),
  ncores,
  seed = NULL,
  verbose = TRUE,
  ...
)

Arguments

data

Matrix or data frame. Should consist only of variables to be used in the analysis

groups

Numeric or character vector (length = nrow(data)). Group membership corresponding to each case in data

structure

Numeric or character vector (length = ncol(data)). A vector representing the structure (numbers or labels for each item). Can be theoretical factors or the structure detected by EGA. If supplied, then configural invariance check is skipped (i.e., configural invariance is assumed based on the given structure)

iter

Numeric (length = 1). Number of iterations to perform for the permutation. Defaults to 500 (recommended)

configural.threshold

Numeric (length = 1). Value to use a threshold in itemStability to determine which items should be removed during configural invariance (see Details for more information). Defaults to 0.70 (recommended)

configural.type

Character (length = 1). Type of bootstrap to use for configural invariance in bootEGA. Defaults to "parametric"

corr

Character (length = 1). Method to compute correlations. Defaults to "auto". Available options:

"auto" — Automatically computes appropriate correlations for the data using Pearson's for continuous, polychoric for ordinal, tetrachoric for binary, and polyserial/biserial for ordinal/binary with continuous. To change the number of categories that are considered ordinal, use ordinal.categories (see polychoric.matrix for more details)
"cor_auto" — Uses cor_auto to compute correlations. Arguments can be passed along to the function
"pearson" — Pearson's correlation is computed for all variables regardless of categories
"spearman" — Spearman's rank-order correlation is computed for all variables regardless of categories

For other similarity measures, compute them first and input them into data with the sample size (n)

na.data

Character (length = 1). How should missing data be handled? Defaults to "pairwise". Available options:

"pairwise" — Computes correlation for all available cases between two variables
"listwise" — Computes correlation for all complete cases in the dataset

model

Character (length = 1). Defaults to "glasso". Available options:

"BGGM" — Computes the Bayesian Gaussian Graphical Model. Set argument ordinal.categories to determine levels allowed for a variable to be considered ordinal. See ?BGGM::estimate for more details
"glasso" — Computes the GLASSO with EBIC model selection. See EBICglasso.qgraph for more details
"TMFG" — Computes the TMFG method. See TMFG for more details

algorithm

Character or cluster_* function (length = 1). Defaults to "walktrap". Three options are listed below but all are available (see community.detection for other options):

"leiden" — See cluster_leiden for more details
"louvain" — By default, "louvain" will implement the Louvain algorithm using the consensus clustering method (see community.consensus for more information). This function will implement consensus.method = "most_common" and consensus.iter = 1000 unless specified otherwise
"walktrap" — See cluster_walktrap for more details

uni.method

Character (length = 1). What unidimensionality method should be used? Defaults to "louvain". Available options:

"expand" — Expands the correlation matrix with four variables correlated 0.50. If number of dimension returns 2 or less in check, then the data are unidimensional; otherwise, regular EGA with no matrix expansion is used. This method was used in the Golino et al.'s (2020) Psychological Methods simulation
"LE" — Applies the Leading Eigenvector algorithm (cluster_leading_eigen) on the empirical correlation matrix. If the number of dimensions is 1, then the Leading Eigenvector solution is used; otherwise, regular EGA is used. This method was used in the Christensen et al.'s (2023) Behavior Research Methods simulation
"louvain" — Applies the Louvain algorithm (cluster_louvain) on the empirical correlation matrix. If the number of dimensions is 1, then the Louvain solution is used; otherwise, regular EGA is used. This method was validated Christensen's (2022) PsyArXiv simulation. Consensus clustering can be used by specifying either "consensus.method" or "consensus.iter"

ncores

Numeric (length = 1). Number of cores to use in computing results. Defaults to ceiling(parallel::detectCores() / 2) or half of your computer's processing power. Set to 1 to not use parallel computing

If you're unsure how many cores your computer has, then type: parallel::detectCores()

seed

Numeric (length = 1). Defaults to NULL or random results. Set for reproducible results. See Reproducibility and PRNG for more details on random number generation in

verbose

Boolean (length = 1). Should progress be displayed? Defaults to TRUE. Set to FALSE to not display progress

...

Additional arguments that can be passed on to auto.correlate, network.estimation, community.detection, community.consensus, EGA, bootEGA, and net.loads

Value

Returns a list containing:

configural.results

bootEGA results from the final run that produced configural invariance. This output will be output on the final run of unsuccessful configural invariance runs

memberships

Original memberships provided in structure or from EGA if structure = NULL

EGA

Original EGA results for the full sample

groups

A list containing:

EGA — EGA results for each group
loadings — Network loadings (net.loads) for each group
loadingsDifference — Difference between the dominant loadings of each group

permutation

A list containing:

groups — Permutated groups acorss iterations
loadings — Network loadings (net.loads) for each group for each permutation
loadingsDifference — Difference between the dominant loadings of each group for each permutation

results

Data frame of the results (which are printed)

Details

In traditional psychometrics, measurement invariance is performed in sequential testing from more flexible (more free parameters) to more rigid (fewer free parameters) structures. Measurement invariance in network psychometrics is no different.

Configural Invariance

To establish configural invariance, the data are collapsed across groups and a common sample structure is identified used bootEGA and itemStability. If some variables have a replication less than 0.70 in their assigned dimension, then they are considered unstable and therefore not invariant. These variables are removed and this process is repeated until all items are considered stable (replication values greater than 0.70) or there are no variables left. If configural invariance cannot be established, then the last run of results are returned and metric invariance is not tested (because configural invariance is not met). Importantly, if any variables are removed, then configural invariance is not met for the original structure. Any removal would suggest only partial configural invariance is met.

Metric Invariance

The variables that remain after configural invariance are submitted to metric invariance. First, each group estimates a network and then network loadings (net.loads) are computed using the assigned community memberships (determined during configural invariance). Then, the difference between the assigned loadings of the groups is computed. This difference represents the empirical values. Second, the group memberships are permutated and networks are estimated based on the these permutated groups for iter times. Then, network loadings are computed and the difference between the assigned loadings of the group is computed, resulting in a null distribution. The empirical difference is then compared against the null distribution using a two-tailed p-value based on the number of null distribution differences that are greater and less than the empirical differences for each variable. Both uncorrected and false discovery rate corrected p-values are returned in the results. Uncorrected p-values are flagged for significance along with the direction of group differences.

Three or More Groups

When there are 3 or more groups, the function performs metric invariance testing by comparing all possible pairs of groups. Specifically:

Pairwise Comparisons: The function generates all possible unique group pairings and computes the differences in network loadings for each pair. The same community structure, derived from configural invariance or provided by the user, is used for all groups.
Permutation Testing: For each group pair, permutation tests are conducted to assess the statistical significance of the observed differences in loadings. p-values are calculated based on the proportion of permuted differences that are greater than or equal to the observed difference.
Result Compilation: The function compiles the results for each pair including both uncorrected (p) and FDR-corrected (Benjamini-Hochberg; p_BH) p-values, and the direction of differences. It returns a summary of the findings for all pairwise comparisons.

This approach allows for a detailed examination of metric invariance across multiple groups, ensuring that all potential differences are thoroughly assessed while maintaining the ability to identify specific group differences.

For more details, see Jamison, Golino, and Christensen (2023)

References

Original implementation
Jamison, L., Christensen, A. P., & Golino, H. F. (2024). Metric invariance in exploratory graph analysis via permutation testing. Methodology, 20(2), 144-186.

Author

Laura Jamison <lj5yn@virginia.edu>, Hudson F. Golino <hfg9s at virginia.edu>, and Alexander P. Christensen <alexpaulchristensen@gmail.com>,

Examples

# Load data
wmt <- wmt2[-1,7:24]

# Groups
groups <- rep(1:2, each = nrow(wmt) / 2)

if (FALSE) { # \dontrun{
# Measurement invariance
results <- invariance(wmt, groups, ncores = 2)

# Plot with uncorrected alpha = 0.05
plot(results, p_type = "p", p_value = 0.05)

# Plot with BH-corrected alpha = 0.10
plot(results, p_type = "p_BH", p_value = 0.10)} # }

Measurement Invariance of `EGA` Structure