Factorial_Invariance: Factor Model Invariance

In EFA.dimensions: Exploratory Factor Analysis Functions for Assessing Dimensionality

Factor Model Invariance

Description

Assesses the equivalence of measurement and structural parameters of a CFA or ESEM factor model across multiple groups

Usage

Factorial_Invariance(data, 
                     group,
                     LV_model=NULL,
                     LV_keys=NULL,
                     LV_names=NULL, 
                     LV_resid_correls=NULL,
                     estimator = 'ML',
                     ordered = FALSE,
                     disp_options = list(models = c('Configural', 'Scalar'),
                                         coefs = c('std', 'LV_means')),
                     verbose=TRUE)

Arguments

data	A dataframe with the model variables and the group factor. Example: data = data_RSE_sex
group	The name of the group variable in data. Example: group = 'gender'
LV_model	(optional) lavaan package CFA model syntax. Example: model = ' pos_items =~ Q1 + Q2 + Q4 + Q6 + Q7 neg_items =~ Q3_R + Q5_R + Q8_R + Q9_R + Q10_R'
LV_keys	(optional) A vector or dataframe indicating which item (i.e., which variable in data) goes with which latent variable (which can be named or numbered). LV_keys is required when LV_model is not provided. If LV_keys is a dataframe, then the variable names must be in the first column and the factor memberships must be in the second column. Example of a vector entry: LV_keys = c(1, 1, 2, 1, 2, 1, 1, 2, 2, 2)
LV_names	(optional) A vector with the preferred names for the latent variables. Example: LV_names = c('pos', 'neg')
LV_resid_correls	(optional) A vector with the pairs of correlated error terms (if any). Example: LV_resid_correls = c('Q1 ~~ Q2', 'Q9 ~~ Q10')
estimator	The name of the lavaan estimator to be used in the analyses. The options for basic estimators (continuous data) are ML, GLS, WLS, DWLS, ULS, DLS, and PML. The options for robust estimators are MLM, MLMVS, MLMV, MLF, MLR, WLSM, WLSMVS, WLSMV, ULSM, ULSMVS, and ULSMV. Example: estimator = 'ML'
ordered	(optional) are the variables in data (if provided) ordered? For example, Likert scale responses are usually ordered. The possible values for ordered are TRUE, FALSE, or a vector with the names of the ordered variables.
disp_options	(optional) A list with display (print) options for model coefficients. The list elements are 'models' and 'coefs'. The possible entries for 'models' are: 'mod_Configural', 'mod_Metric', 'mod_Scalar', 'mod_Strict', 'mod_LV_vars', 'mod_LV_covars', & 'mod_LV_means'. The possible entries for 'coefs' are: 'raw', 'std', 'phi', & 'LV_means' See the Details section below for information on each element.
verbose	(optional) Should detailed results be displayed in console? TRUE (default) or FALSE

Details

This function assesses the equivalence of the measurement and structural parameters of a confirmatory factor analysis (CFA) model, or of an exploratory structural equation model (ESEM), across multiple groups. It is a wrapper function that uses functions from the lavaan (Rosseel, 2012) and semTools (Jorgensen et al., 2026) packages.

The measurement model pertains to the measurement characteristics of the indicators (observed measures) and thus consists of the factor loadings, intercepts, and residual variances. The evaluations of the equivalence of these parameters across groups are tests of measurement invariance (Brown, 2015).

Tests of the structural parameters involve evaluations of the latent variables themselves and focus on the factor variances, covariances, and latent means. These parameters are (estimated) features of the population from which a sample was drawn. The evaluations of the equivalence of these parameters across groups are tests of population heterogeneity: Does the dispersion, the interrelationships, and the levels of the factors vary across groups?

Brown (2015) stated that it is more prudent for model evaluation to work upward from the least restricted solution (equal form) to determine if further tests of measurement invariance and population heterogeneity are warranted. Here is the recommended sequence of multiple groups invariance evaluations that is used in the present function, with the function model names in bold:

• Configural invariance, or "equal form" invariance, tests whether the number of factors and the pattern of indicator factor loadings are identical across groups.

• Metric invariance, or "equal factor loadings" invariance, tests the equality of the factor loadings across groups.

• Scalar invariance, "equal intercepts" invariance, or "strong factorial" invariance, tests whether the indicator intercepts are equal across groups.

• Strict invariance, or "equal residual" invariance, tests whether the indicator residual variances are equal across groups.

• LV_vars tests whether the factor variances are equal across groups.

• LV_covars tests whether the factor covariances are equal across groups.

• LV_means tests whether the latent variable means are equal across groups.

Value

A list containing the following components:

lavaan_model_syntax	The lavaan CFA model syntax
fits_by_group	The model fit coefficients for each group separately
mod_Configural	The lavaan cfa output object for the configural invariance model
mod_Metric	The lavaan cfa output object for the metric invariance model
mod_Scalar	The lavaan cfa output object for the scalar invariance model
mod_Strict	The lavaan cfa output object for the strict invariance model
mod_LV_vars	The lavaan cfa output object for the model that restricts the latent variable variances to be equal
mod_LV_covars	The lavaan cfa output object for the model that restricts the latent variable covariances to be equal
mod_LV_means	The lavaan cfa output object for the model that restricts the latent variable means to be equal
model_fits	The invariance model fit coefficients
model_fit_diffs	Differences in the invariance model fit coefficients
chisq_diffs	Chi-Squared model differences tests
LV_scores	A list with scores on the latent variables for each of the models in disp_options

Author(s)

Brian P. O'Connor

References

Brown, T. A. (2015). Confirmatory Factor Analysis for Applied Research (2nd ed.). New York: The Guilford Press.

Jorgensen, T. D., Pornprasertmanit, S., Schoemann, A. M., & Rosseel, Y. (2026). semTools: Useful tools for structural equation modeling. R package version 0.5-8. Retrieved from https://CRAN.R-project.org/package=semTools

Milfont, T. L., & Fischer, R. (2015). Testing measurement invariance across groups: Applications in cross-cultural research. International Journal of Psychological Research, 3, 111130.

Rosseel, Y. (2012). lavaan: An R package for structural equation modeling. Journal of Statistical Software, 48, 136.

Stark, S., Chernyshenko, O. S., & Drasgow, F. (2006). Detecting differential item functioning with confirmatory factor analysis and item response theory: Toward a unified strategy. Journal of Applied Psychology, 91(6), 12921306.

Stein, J. A., Lee, J. W., & Jones, P. S. (2006). Assessing cross-cultural differences through use of multiple-group invariance analyses. Journal of Personality Assessment, 87(3), 249258.

Tan, T. (2024) Frequentist and Bayesian factorial invariance using R. Practical Assessment, Research, and Evaluation. 29(8), 1-34.

Examples

model_RSE <- '
   pos_items =~ Q1 + Q2 + Q4 + Q6 + Q7
   neg_items =~ Q3_R + Q5_R + Q8_R + Q9_R + Q10_R '

# using LV_model
Factorial_Invariance(data = data_RSE_sex, group = 'gender', LV_model = model_RSE)

# using LV_keys
Factorial_Invariance(data = data_RSE_sex, group = 'gender', 
                     LV_keys = c(1, 1, 2, 1, 2, 1, 1, 2, 2, 2), 
                     LV_names = c('pos', 'neg'))

# another way of using LV_keys
Factorial_Invariance(data = data_RSE_sex, group = 'gender', 
                     LV_keys = c( Q1 = 1, Q2 = 1, Q3_R = 2, Q4 = 1,   Q5_R = 2, 
                                  Q6 = 1, Q7 = 1, Q8_R = 2, Q9_R = 2, Q10_R = 2) )

# for an ESEM model
# first, use the ESEM function to obtain the esem_model_syntax (without the group variable, gender)
esem_output <- ESEM(data = subset(data_RSE_sex, select = -c(gender)), Nfactors = 2)

Factorial_Invariance(data = data_RSE_sex, group = 'gender',
                     LV_model = esem_output$esem_model_syntax)
        
         
# Run this command for additional Examples:  vignette("EXAMPLES_vignettes")

EFA.dimensions documentation built on Sept. 14, 2026, 9:08 a.m.