Syntax for measurement equivalence

Description

Automatically generates lavaan model syntax to specify a confirmatory factor analysis (CFA) model with equality constraints imposed on user-specified measurement (or structural) parameters. Optionally returns the fitted model (if data are provided) representing some chosen level of measurement equivalence/invariance across groups and/or repeated measures.

Usage

measEq.syntax(configural.model, ..., ID.fac = "std.lv",
  ID.cat = "Wu.Estabrook.2016", ID.thr = c(1L, 2L), group = NULL,
  group.equal = "", group.partial = "", longFacNames = list(),
  longIndNames = list(), long.equal = "", long.partial = "",
  auto = "all", warn = TRUE, debug = FALSE, return.fit = FALSE)

Arguments

Details

This function is a pedagogical and analytical tool to generate model syntax representing some level of measurement equivalence/invariance across any combination of multiple groups and/or repeated measures. Support is provided for confirmatory factor analysis (CFA) models with simple or complex structure (i.e., cross-loadings and correlated residuals are allowed). For any complexities that exceed the limits of automation, this function is intended to still be useful by providing a means to generate syntax that users can easily edit to accommodate their unique situations.

Limited support is provided for bifactor models and higher-order constructs. Because bifactor models have cross-loadings by definition, the option ID.fac = "effects.code" is unavailable. ID.fac = "UV" is recommended for bifactor models, but ID.fac = "UL" is available on the condition that each factor has a unique first indicator in the configural.model. In order to maintain generality, higher-order factors may include a mix of manifest and latent indicators, but they must therefore require ID.fac = "UL" to avoid complications with differentiating lower-order vs. higher-order (or mixed-level) factors. The keyword "loadings" in group.equal or long.equal constrains factor loadings of all manifest indicators (including loadings on higher-order factors that also have latent indicators), whereas the keyword "regressions" constrains factor loadings of latent indicators. Users can edit the model syntax manually to adjust constraints as necessary, or clever use of the group.partial or long.partial arguments could make it possible for users to still automated their model syntax. The keyword "intercepts" constrains the intercepts of all manifest indicators, and the keyword "means" constrains intercepts and means of all latent common factors, regardless of whether they are latent indicators of higher-order factors. To test equivalence of lower-order and higher-order intercepts/means in separate steps, the user can either manually edit their generated syntax or conscientiously exploit the group.partial or long.partial arguments as necessary.

ID.fac: If the configural.model fixes any (e.g., the first) factor loadings, the generated syntax object will retain those fixed values. This allows the user to retain additional constraints that might be necessary (e.g., if there are only 1 or 2 indicators). Some methods must be used in conjunction with other settings:

• ID.cat = "Millsap" requires ID.fac = "UL" and parameterization = "theta".

• ID.cat = "LISREL" requires parameterization = "theta".

• ID.fac = "effects.code" is unavailable when there are any cross-loadings.

ID.cat: Wu & Estabrook (2016) recommended constraining thresholds to equality first, and doing so should allow releasing any identification constraints no longer needed. For each ordered indicator, constraining one threshold to equality will allow the item's intercepts to be estimated in all but the first group or repeated measure. Constraining a second threshold (if applicable) will allow the item's (residual) variance to be estimated in all but the first group or repeated measure. For binary data, there is no independent test of threshold, intercept, or residual-variance equality. Equivalence of thresholds must also be assumed for three-category indicators. These guidelines provide the least restrictive assumptions and tests, and are therefore the default.

The default setting in Mplus is similar to Wu & Estabrook (2016), except that intercepts are always constrained to zero (so they are assumed to be invariant without testing them). Millsap & Tein (2004) recommended parameterization = "theta" and identified an item's residual variance in all but the first group (or occasion; Liu et al., 2017) by constraining its intercept to zero and one of its thresholds to equality. A second threshold for the reference indicator (so ID.fac = "UL") is used to identify the common-factor means in all but the first group/occasion. The LISREL software fixes the first threshold to zero and (if applicable) the second threshold to 1, and assumes any remaining thresholds to be equal across groups / repeated measures; thus, the intercepts are always identified, and residual variances (parameterization = "theta") are identified except for binary data, when they are all fixed to one.

Repeated Measures: If each repeatedly measured factor is measured by the same indicators (specified in the same order in the configural.model) on each occasion, without any cross-loadings, the user can let longIndNames be automatically generated. Generic names for the repeatedly measured indicators are created using the name of the repeatedly measured factors (i.e., names(longFacNames)) and the number of indicators. So the repeatedly measured first indicator ("ind") of a longitudinal construct called "factor" would be generated as "._factor_ind.1".

The same types of parameter can be specified for long.equal as for group.equal (see lavOptions for a list), except for "residual.covariances" or "lv.covariances". Instead, users can constrain autocovariances using keywords "resid.autocov" or "lv.autocov". Note that group.equal = "lv.covariances" or group.equal = "residual.covariances" will constrain any autocovariances across groups, along with any other covariances the user specified in the configural.model. Note also that autocovariances cannot be specified as exceptions in long.partial, so anything more complex than the auto argument automatically provides should instead be manually specified in the configural.model.

When users set orthogonal=TRUE in the configural.model (e.g., in bifactor models of repeatedly measured constructs), autocovariances of each repeatedly measured factor will still be freely estimated in the generated syntax.

Missing Data: If users wish to utilize the auxiliary function to automatically include auxiliary variables in conjunction with missing = "FIML", they should first generate the hypothesized-model syntax, then submit that syntax as the model to auxiliary(). If users utilized runMI to fit their configural.model to multiply imputed data, that model can also be passed to the configural.model argument, and if return.fit = TRUE, the generated model will be fitted to the multiple imputations.

Value

By default, an object of class measEq.syntax. If return.fit = TRUE, a fitted lavaan model, with the measEq.syntax object stored in the @external slot, accessible by fit@external$measEq.syntax.

Author(s)

Terrence D. Jorgensen (University of Amsterdam; TJorgensen314@gmail.com)

References

Little, T. D., Slegers, D. W., & Card, N. A. (2006). A non-arbitrary method of identifying and scaling latent variables in SEM and MACS models. Structural Equation Modeling, 13(1), 59–72. doi:10.1207/s15328007sem1301_3

Liu, Y., Millsap, R. E., West, S. G., Tein, J.-Y., Tanaka, R., & Grimm, K. J. (2017). Testing measurement invariance in longitudinal data with ordered-categorical measures. Psychological Methods, 22(3), 486–506. doi:10.1037/met0000075

Millsap, R. E., & Tein, J.-Y. (2004). Assessing factorial invariance in ordered-categorical measures. Multivariate Behavioral Research, 39(3), 479–515. doi:10.1207/S15327906MBR3903_4

Wu, H., & Estabrook, R. (2016). Identification of confirmatory factor analysis models of different levels of invariance for ordered categorical outcomes. Psychometrika, 81(4), 1014–1045. doi:10.1007/s11336-016-9506-0

See Also

compareFit

Examples

mod.cat <- ' FU1 =~ u1 + u2 + u3 + u4
             FU2 =~ u5 + u6 + u7 + u8 '
## the 2 factors are actually the same factor (FU) measured twice
longFacNames <- list(FU = c("FU1","FU2"))

## configural model: no constraints across groups or repeated measures
syntax.config <- measEq.syntax(configural.model = mod.cat, data = datCat,
                               ordered = paste0("u", 1:8),
                               parameterization = "theta",
                               ID.fac = "std.lv", ID.cat = "Wu.Estabrook.2016",
                               group = "g", longFacNames = longFacNames)
## print lavaan syntax to the Console
cat(as.character(syntax.config))
## print a summary of model features
summary(syntax.config)

## threshold invariance
syntax.thresh <- measEq.syntax(configural.model = mod.cat, data = datCat,
                               ordered = paste0("u", 1:8),
                               parameterization = "theta",
                               ID.fac = "std.lv", ID.cat = "Wu.Estabrook.2016",
                               group = "g", group.equal = "thresholds",
                               longFacNames = longFacNames,
                               long.equal = "thresholds")
## notice that constraining 4 thresholds allows intercepts and residual
## variances to be freely estimated in all but the first group & occasion
cat(as.character(syntax.thresh))
## print a summary of model features
summary(syntax.thresh)


## Fit a model to the data either in a subsequent step:
mod.config <- as.character(syntax.config)
fit.config <- cfa(mod.config, data = datCat, group = "g",
                  ordered = paste0("u", 1:8), parameterization = "theta")
## or in a single step:
fit.thresh <- measEq.syntax(configural.model = mod.cat, data = datCat,
                            ordered = paste0("u", 1:8),
                            parameterization = "theta",
                            ID.fac = "std.lv", ID.cat = "Wu.Estabrook.2016",
                            group = "g", group.equal = "thresholds",
                            longFacNames = longFacNames,
                            long.equal = "thresholds", return.fit = TRUE)

## compare their fit to test threshold invariance
anova(fit.config, fit.thresh)


## Not run: 

## compare several invariance models
test.seq <- c("thresholds","loadings","intercepts","means","residuals")
meq.list <- list()
for (i in 0:length(test.seq)) {
  if (i == 0L) {
    meq.label <- "configural"
    group.equal <- ""
    long.equal <- ""
  } else {
    meq.label <- test.seq[i]
    group.equal <- test.seq[1:i]
    long.equal <- test.seq[1:i]
  }
  meq.list[[meq.label]] <- measEq.syntax(configural.model = mod.cat,
                                         data = datCat,
                                         ordered = paste0("u", 1:8),
                                         parameterization = "theta",
                                         ID.fac = "std.lv",
                                         ID.cat = "Wu.Estabrook.2016",
                                         group = "g",
                                         group.equal = group.equal,
                                         longFacNames = longFacNames,
                                         long.equal = long.equal,
                                         return.fit = TRUE)
}

compareFit(meq.list)


## End(Not run)