item.invar: Between-Group and Longitudinal Measurement Invariance Evaluation

Description

This function is a wrapper function for evaluating configural, metric, scalar, and strict between-group or longitudinal (partial) measurement invariance using confirmatory factor analysis with continuous indicators by calling the cfa function in the R package lavaan. By default, the function evaluates configural, metric, and scalar measurement invariance by providing a table with model fit information (i.e., chi-square test, fit indices based on a proper null model, and information criteria) and model comparison (i.e., chi-square difference test, change in fit indices, and change in information criteria). Additionally, variance-covariance coverage of the data, descriptive statistics, parameter estimates, modification indices, and residual correlation matrix can be requested by specifying the argument print.

Usage

item.invar(..., data = NULL, model = NULL, rescov = NULL, rescov.long = TRUE,
           group = NULL, long = FALSE, cluster = NULL,
           invar = c("config", "metric", "scalar", "strict"),
           partial = NULL, ident = c("marker", "var", "effect"),
           estimator = c("ML", "MLM", "MLMV", "MLMVS", "MLF", "MLR",
                         "GLS", "WLS", "DWLS", "WLSM", "WLSMV",
                         "ULS", "ULSM", "ULSMV", "DLS", "PML"),
           missing = c("listwise", "pairwise", "fiml", "two.stage",
                       "robust.two.stage", "doubly.robust"), null.model = TRUE,
           print = c("all", "summary", "coverage", "descript", "fit", "est",
                     "modind", "resid"),
           print.fit = c("all", "standard", "scaled", "robust"),
           mod.minval = 6.63, resid.minval = 0.1, digits = 3, p.digits = 3,
           as.na = NULL, write = NULL, append = TRUE, check = TRUE, output = TRUE)

Value

Returns an object of class misty.object, which is a list with following entries:

call: function call
type: type of analysis
data: data frame including all variables used in the analysis, i.e., indicators for the factor, grouping variable and cluster variable
args: specification of function arguments
model: list with specified model for the configural, metric, scalar, and strict invariance model
model.fit: list with fitted lavaan object of the configural, metric, scalar, and strict invariance model
check: list with the results of the convergence and model identification check for the configural, metric, scalar, and strict invariance model
result: list with result tables, i.e., summary for the summary of the specification of the estimation method and missing data handling in lavaan, coverage for the variance-covariance coverage of the data, descript for descriptive statistics, fit for a list with model fit based on standard, scaled, and robust fit indices, est for a list with parameter estimates for the configural, metric, scalar, and strict invariance model, modind for the list with modification indices for the configural, metric, scalar, and strict invariance model, score for the list with result of the score tests for constrained parameters for the configural, metric, scalar, and strict invariance model, and resid for the list with residual correlation matrices and standardized residual means for the configural, metric, scalar, and strict invariance model

Arguments

...: a matrix or data frame. If model = NULL, confirmatory factor analysis based on a measurement model with one factor labeled f comprising all variables in the matrix or data frame specified in x for evaluating between-group measurement invariance for the grouping variable specified in the argument group is conducted. Longitudinal measurement invariance evaluation can only be conducted by specifying the model using the argument model. Note that the cluster variable is excluded from x when specifying cluster. If model is specified, the matrix or data frame needs to contain all variables used in the argument model and the cluster variable when specifying the name of the cluster variable in the argument cluster. Alternatively, an expression indicating the variable names in data e.g., item.invar(x1, x2, x2, data = dat). Note that the operators ., +, -, ~, :, ::, and ! can also be used to select variables, see 'Details' in the df.subset function.
data: a data frame when specifying one or more variables in the argument .... Note that the argument is NULL when specifying a vector, factor, matrix, array, data frame, or list for the argument ....
model: a character vector specifying a measurement model with one factor, or a list of character vectors for specifying a measurement model with more than one factor for evaluating between-group measurement invariance when long = FALSE or a list of character vectors for specifying a measurement model with one factor for each time of measurement for evaluating longitudinal measurement invariance when specifying long = TRUE. For example, model = c("x1", "x2", "x3", "x4") for specifying a measurement model with one factor labeled f comprising four indicators, or model = list(factor1 = c("x1", "x2", "x3", "x4"), factor2 = c("x5", "x6", "x7", "x8")) for specifying a measurement model with two latent factors labeled factor1 and factor2 each comprising four indicators for evaluating between-group measurement invariance, or model = list(time1 = c("ax1", "ax2", "ax3", "ax4"), time2 = c("bx1", "bx2", "bx3", "bx4"), time3 = c("cx1", "cx2", "cx3", "cx4")) for specifying a longitudinal measurement model with three time points comprising four indicators at each time point. This function cannot evaluate longitudinal measurement invariance for a measurement model with more than one factor. Note that the name of each list element is used to label factors, i.e., all list elements need to be named, otherwise factors are labeled with "f1", "f2", "f3" when long = FALSE and with "t1", "t2", "t3" when long = TRUE and so on.
rescov: a character vector or a list of character vectors for specifying residual covariances, e.g., rescov = c("x1", "x2") for specifying a residual covariance between items x1 and x2, or rescov = list(c("x1", "x2"), c("x3", "x4")) for specifying residual covariances between items x1 and x2, and items x3 and x4.
rescov.long: logical: if TRUE (default), residual covariances between parallel indicators are estimated across time when evaluating longitudinal measurement invariance (long = TRUE), i.e., residual variances of the same indicators that are measured at different time points are correlated across all possible time points. Note that residual covariances should be estimated even if the parameter estimates are statistically not significant since indicator-specific systematic variance is likely to correlate with itself over time (Little, 2013, p. 164).
group: either a character string indicating the variable name of the grouping variable in the matrix or data frame specified in x or a vector representing the groups for conducting multiple-group analysis to evaluate between-group measurement invariance.
long: logical: if TRUE, longitudinal measurement invariance evaluation is conducted. The longitudinal measurement model is specified by using the argument model. Note that this function can only evaluate either between-group or longitudinal measurement invariance, but not both at the same time.
cluster: either a character string indicating the variable name of the cluster variable in ... or data, or a vector representing the nested grouping structure (i.e., group or cluster variable) for computing cluster-robust standard errors. Note that cluster-robust standard errors are not available when treating indicators of the measurement model as ordered (ordinal).
invar: a character string indicating the level of measurement invariance to be evaluated, i.e., config to evaluate configural measurement invariance (i.e., same factor structure across groups or time), metric to evaluate configural and metric measurement invariance (i.e., equal factor loadings across groups or time), scalar (default) to evaluate configural, metric and scalar measurement invariance (i.e., equal intercepts or thresholds across groups or time), and strict to evaluate configural, metric, scalar, and strict measurement invariance (i.e., equal residual variances across groups or time).
partial: a character string or character vector containing the labels of the parameters which should be free in all groups or across time to specify a partial measurement invariance model. Note that the labels of the parameters need to match the labels shown in the output, i.e., "L" with a number for factor loadings, "T" with a number for intercepts, and "E" with a number for factor residual variances. The number attached to the "L", "T", or "E" label corresponds to the number of the indicator in the measurement model (e.g., "T3" for the intercept of the third indicator). When specifying the model using the argument model, however, the number for the factor loading is a combination of the number of the factor and the number of the indicator (e.g., "L23" is the third indicator of the second factor). Note that at least two invariant indicators are needed for a partial measurement invariance model. Otherwise there might be issues with model non-identification.
ident: a character string indicating the method used for identifying and scaling latent variables, i.e., "marker" for the marker variable method fixing the first factor loading of each latent variable to 1, "var" (default) for the fixed variance method fixing the variance of each latent variable to 1, or "effect" for the effects-coding method using equality constraints so that the average of the factor loading for each latent variable equals 1.
estimator: a character string indicating the estimator to be used (see 'Details' in the help page of the item.cfa() function). By default, "MLR" is used for CFA models with continuous indicators.
missing: a character string indicating how to deal with missing data, i.e., "listwise" for listwise deletion, "pairwise" for pairwise deletion, "fiml" for full information maximum likelihood method, two.stage for two-stage maximum likelihood method, robust.two.stage for robust two-stage maximum likelihood method, and doubly-robust for doubly-robust method (see 'Details' in the help page of theitem.cfa() function). By default, "fiml" is used for CFA models with continuous indicators which are estimated by using estimator = "MLR". However, argument missing switches to listwise when the data set is complete, i.e., it is not possible to use FIML in complete data. Note that the robust CFI, TLI, and RMSEA are different in complete data depending on whether FIML or listwise deletion was specified when estimating the model in lavaan.
null.model: logical: if TRUE (default), the proper null model for computing incremental fit indices (i.e., CFI and TLI) is used, i.e., means and variances of the indicators are constrained to be equal across group or time in the null model (Little, 2013, p. 112).
print: a character string or character vector indicating which results to show on the console, i.e. "all" for all results, "summary" for a summary of the specification of the estimation method and missing data handling in lavaan, "coverage" for the variance-covariance coverage of the data, "descript" for descriptive statistics, "fit" for model fit and model comparison, "est" for parameter estimates, "modind" for modification indices, and "resid" for the residual correlation matrix and standardized residual means. By default, a summary of the specification, model fit, and parameter estimates are printed. Note that parameter estimates, modification indices, and residual correlation matrix is only provided for the model investigating the level of measurement invariance specified in the argument "invar".
print.fit: a character string or character vector indicating which version of the CFI, TLI, and RMSEA to show on the console when using a robust estimation method involving a scaling correction factor, i.e., "all" for all versions of the CFI, TLI, and RMSEA, "standard" (default when estimator is one of "ML", "MLF", "GLS", "WLS", "DWLS", "ULS", "PML") for fit indices without any non-normality correction, "scaled" for population-corrected robust fit indices with ad hoc non-normality correction, and robust (default when estimator is one of "MLM", "MLMV", "MLMVS", "MLR", "WLSM", "WLSMV", "ULSM", "ULSMV", "DLS") for sample-corrected robust fit indices based on formula provided by Li and Bentler (2006) and Brosseau-Liard and Savalei (2014).
mod.minval: numeric value to filter modification indices and only show modifications with a modification index value equal or higher than this minimum value. By default, modification indices equal or higher 6.63 are printed. Note that a modification index value of 6.63 is equivalent to a significance level of \(\alpha = .01\).
resid.minval: numeric value indicating the minimum absolute residual correlation coefficients and standardized means to highlight in boldface. By default, absolute residual correlation coefficients and standardized means equal or higher 0.1 are highlighted. Note that highlighting can be disabled by setting the minimum value to 1.
digits: an integer value indicating the number of decimal places to be used for displaying results. Note that information criteria and chi-square test statistic are printed with digits minus 1 decimal places.
p.digits: an integer value indicating the number of decimal places to be used for displaying p-values, covariance coverage (i.e., p.digits - 1), and residual correlation coefficients.
as.na: a numeric vector indicating user-defined missing values, i.e., these values are converted to NA before conducting the analysis. Note that as.na() function is only applied to x but not to group or cluster.
write: a character string naming a file for writing the output into either a text file with file extension ".txt" (e.g., "Output.txt") or Excel file with file extension ".xlsx" (e.g., "Output.xlsx"). If the file name does not contain any file extension, an Excel file will be written.
append: logical: if TRUE (default), output will be appended to an existing text file with extension .txt specified in write, if FALSE existing text file will be overwritten.
check: logical: if TRUE (default), argument specification is checked and convergence and model identification checks are conducted for all estimated models.
output: logical: if TRUE (default), output is shown.

Author

Takuya Yanagida takuya.yanagida@univie.ac.at

References

Brosseau-Liard, P. E., & Savalei, V. (2014) Adjusting incremental fit indices for nonnormality. Multivariate Behavioral Research, 49, 460-470. https://doi.org/10.1080/00273171.2014.933697

Li, L., & Bentler, P. M. (2006). Robust statistical tests for evaluating the hypothesis of close fit of misspecified mean and covariance structural models. UCLA Statistics Preprint #506. University of California.

Little, T. D. (2013). Longitudinal structural equation modeling. Guilford Press.

Rosseel, Y. (2012). lavaan: An R Package for Structural Equation Modeling. Journal of Statistical Software, 48, 1-36. https://doi.org/10.18637/jss.v048.i02

Examples

Run this code

if (FALSE) {
# Load data set "HolzingerSwineford1939" in the lavaan package
data("HolzingerSwineford1939", package = "lavaan")

#-------------------------------------------------------------------------------
# Between-Group Measurement Invariance Evaluation

#..................
# Measurement model with one factor

# Example 1a: Specification of the grouping variable in 'x'
item.invar(HolzingerSwineford1939[, c("x1", "x2", "x3", "x4", "sex")], group = "sex")

# Example 1b: Specification of the grouping variable in 'group'
item.invar(HolzingerSwineford1939[, c("x1", "x2", "x3", "x4")],
           group = HolzingerSwineford1939$sex)

# Example 1c: Alternative specification using the 'data' argument
item.invar(x1:x4, data = HolzingerSwineford1939, group = "sex")

# Example 1d: Alternative specification using the argument 'model'
item.invar(HolzingerSwineford1939, model = c("x1", "x2", "x3", "x4"), group = "sex")

# Example 1e: Alternative specification using the 'data' and 'model' argument
item.invar(., data = HolzingerSwineford1939, model = c("x1", "x2", "x3", "x4"), group = "sex")

#..................
# Measurement model with two factors

item.invar(HolzingerSwineford1939,
           model = list(c("x1", "x2", "x3", "x4"),
                        c("x5", "x6", "x7", "x8")), group = "sex")

#..................
# Configural, metric, scalar, and strict measurement invariance

# Example 2: Evaluate configural, metric, scalar, and strict measurement invariance
item.invar(HolzingerSwineford1939, model = c("x1", "x2", "x3", "x4"),
           group = "sex", invar = "strict")

#..................
# Partial measurement invariance

# Example 3: Free second factor loading (L2) and third intercept (T3)
item.invar(HolzingerSwineford1939, model = c("x1", "x2", "x3", "x4"),
           group = "sex", partial = c("L2", "T3"), print = c("fit", "est"))

#..................
# Residual covariances

# Example 4a: One residual covariance
item.invar(HolzingerSwineford1939, model = c("x1", "x2", "x3", "x4"),
           rescov = c("x3", "x4"), group = "sex")

# Example 4b: Two residual covariances
item.invar(HolzingerSwineford1939, model = c("x1", "x2", "x3", "x4"),
           rescov = list(c("x1", "x2"), c("x3", "x4")), group = "sex")

#..................
# Scaled test statistic and cluster-robust standard errors

# Example 5a: Specify cluster variable using a variable name in 'x'
item.invar(HolzingerSwineford1939, model = c("x1", "x2", "x3", "x4"),
           group = "sex", cluster = "agemo")

# Example 5b: Specify vector of the cluster variable in the argument 'cluster'
item.invar(HolzingerSwineford1939, model = c("x1", "x2", "x3", "x4"),
           group = "sex", cluster = HolzingerSwineford1939$agemo)

#..................
# Default Null model

# Example 6: Specify default null model for computing incremental fit indices
item.invar(HolzingerSwineford1939, model = c("x1", "x2", "x3", "x4"),
           group = "sex", null.model = FALSE)

#..................
# Print argument

# Example 7a: Request all results
item.invar(HolzingerSwineford1939, model = c("x1", "x2", "x3", "x4"),
           group = "sex", print = "all")

# Example 7b: Request fit indices with ad hoc non-normality correction
item.invar(HolzingerSwineford1939, model = c("x1", "x2", "x3", "x4"),
           group = "sex", print.fit = "scaled")

# Example 7c: Request modification indices with value equal or higher than 10
# and highlight residual correlations equal or higher than 0.3
item.invar(HolzingerSwineford1939, model = c("x1", "x2", "x3", "x4"),
           group = "sex", print = c("modind", "resid"),
           mod.minval = 10, resid.minval = 0.3)

#..................
# Model syntax and lavaan summary of the estimated model

# Example 8
mod <- item.invar(HolzingerSwineford1939, model = c("x1", "x2", "x3", "x4"),
                  group = "sex", output = FALSE)

# lavaan model syntax scalar invariance model
cat(mod$model$scalar)

# lavaan summary of the scalar invariance model
lavaan::summary(mod$model.fit$scalar, standardized = TRUE, fit.measures = TRUE)

#-------------------------------------------------------------------------------
# Longitudinal Measurement Invariance Evaluation

# Example 9: Two time points with three indicators at each time point
item.invar(HolzingerSwineford1939,
           model = list(c("x1", "x2", "x3"),
                        c("x5", "x6", "x7")), long = TRUE)

#------------------------------------------------
# Write Results

# Example 10a: Write results into a text file
item.invar(HolzingerSwineford1939, model = c("x1", "x2", "x3", "x4"),
            group = "sex", print = "all", write = "Invariance.txt", output = FALSE)

# Example 10b: Write results into an Excel file
item.invar(HolzingerSwineford1939, model = c("x1", "x2", "x3", "x4"),
            group = "sex", print = "all", write = "Invariance.xlsx", output = FALSE)

result <- item.invar(HolzingerSwineford1939, model = c("x1", "x2", "x3", "x4"),
                     group = "sex", print = "all", output = FALSE)
write.result(result, "Invariance.xlsx")
}

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