Permutation Randomization Tests of Measurement Equivalence and Differential Item Functioning (DIF)

Description

The function permuteMeasEq provides tests of hypotheses involving measurement equivalence, in one of two frameworks:

(1) For multiple-group CFA models, provide a pair of nested lavaan objects, the less constrained of which (uncon) freely estimates a set of measurement parameters (e.g., factor loadings, intercepts, or thresholds; specified in param) in all groups, and the more constrained of which (con) constrains those measurement parameters to equality across groups. Group assignment is repeatedly permuted and the models are fit to each permutation, in order to produce an empirical distribution under the null hypothesis of no group differences, both for (a) changes in user-specified fit measures (see AFIs and moreAFIs) and for (b) the maximum modification index among the user-specified equality constraints. Configural invariance can also be tested by providing that fitted lavaan object to con and leaving uncon = NULL, in which case param must be NULL as well.

(2) In MIMIC models, one or a set of continuous and/or discrete covariates can be permuted, and a constrained model is fit to each permutation in order to provide a distribution of any fit measures (namely, the maximum modification index among fixed parameters in param) under the null hypothesis of measurement equivalence across levels of those covariates.

In either framework, modification indices for equality constraints or fixed parameters specified in param are calculated from the constrained model (con) using the function lavTestScore.

Usage

permuteMeasEq(nPermute, modelType = c("mgcfa","mimic"),
              con, uncon = NULL, null = NULL,
              param = NULL, freeParam = NULL, covariates = NULL,
              AFIs = NULL, moreAFIs = NULL,
              maxSparse = 10L, maxNonconv = 10L, showProgress = TRUE,
              warn = -1L, datafun, extra,
              parallelType = c("none", "multicore", "snow"),
              ncpus = NULL, cl = NULL, iseed = 12345L)

Arguments

Details

For multiple-group CFA models, the multiparameter omnibus null hypothesis of measurement equivalence/invariance is that there are no group differences in any measurement parameters (of a particular type). This can be tested using the anova method on nested lavaan objects, as seen in the output of measurementInvariance, or by inspecting the change in alternative fit indices (AFIs) such as the CFI. The permutation randomization method employed by permuteMeasEq generates an empirical distribution of any AFIs under the null hypothesis, so the user is not restricted to using fixed cutoffs proposed by Cheung & Rensvold (2002), Chen (2007), or Meade, Johnson, & Braddy (2008).

If the multiparameter omnibus null hypothesis is rejected, partial invariance can still be established by freeing invalid equality constraints, as long as equality constraints are valid for at least two indicators per factor. Modification indices can be calculated from the constrained model (con), but multiple testing leads to inflation of Type I error rates. The permutation randomization method employed by permuteMeasEq creates a distribution of the maximum modification index if the null hypothesis is true, which allows the user to control the familywise Type I error rate in a manner similar to Tukey's q (studentized range) distribution for the Honestly Significant Difference (HSD) post hoc test.

For MIMIC models, DIF can be tested by comparing modification indices of regression paths to the permutation distribution of the maximum modification index, which controls the familywise Type I error rate. The MIMIC approach could also be applied with multiple-group models, but the grouping variable would not be permuted; rather, the covariates would be permuted separately within each group to preserve between-group differences. So whether parameters are constrained or unconstrained across groups, the MIMIC approach is only for testing null hypotheses about the effects of covariates on indicators, controlling for common factors.

In either framework, lavaan's group.label argument is used to preserve the order of groups seen in con when permuting the data.

Value

The permuteMeasEq object representing the results of testing measurement equivalence (the multiparameter omnibus test) and DIF (modification indices), as well as diagnostics and any extra output.

Author(s)

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

References

Chen, F. F. (2007). Sensitivity of goodness of fit indexes to lack of measurement invariance. Structural Equation Modeling, 14(3), 464-504. doi:10.1080/10705510701301834

Cheung, G. W., & Rensvold, R. B. (2002). Evaluating goodness-of-fit indexes for testing measurement invariance. Structural Equation Modeling, 9(2), 233-255. doi:10.1207/S15328007SEM0902_5

Meade, A. W., Johnson, E. C., & Braddy, P. W. (2008). Power and sensitivity of alternative fit indices in tests of measurement invariance. Journal of Applied Psychology, 93(3), 568-592. doi:10.1037/0021-9010.93.3.568

Widamin, K. F., & Thompson, J. S. (2003). On specifying the null model for incremental fit indices in structural equation modeling. Psychological Methods, 8(1), 16-37. doi:10.1037/1082-989X.8.1.16

See Also

TukeyHSD, lavTestScore, measurementInvariance, measurementInvarianceCat

Examples

## Not run: 

########################
## Multiple-Group CFA ##
########################

## create 3-group data in lavaan example(cfa) data
HS <- lavaan::HolzingerSwineford1939
HS$ageGroup <- ifelse(HS$ageyr < 13, "preteen",
                      ifelse(HS$ageyr > 13, "teen", "thirteen"))

## specify and fit an appropriate null model for incremental fit indices
mod.null <- c(paste0("x", 1:9, " ~ c(T", 1:9, ", T", 1:9, ", T", 1:9, ")*1"),
              paste0("x", 1:9, " ~~ c(L", 1:9, ", L", 1:9, ", L", 1:9, ")*x", 1:9))
fit.null <- cfa(mod.null, data = HS, group = "ageGroup")

## fit target model with varying levels of measurement equivalence
mod.config <- '
visual  =~ x1 + x2 + x3
textual =~ x4 + x5 + x6
speed   =~ x7 + x8 + x9
'
miout <- measurementInvariance(mod.config, data = HS, std.lv = TRUE,
                               group = "ageGroup")

(fit.config <- miout[["fit.configural"]])
(fit.metric <- miout[["fit.loadings"]])
(fit.scalar <- miout[["fit.intercepts"]])


####################### Permutation Method

## fit indices of interest for multiparameter omnibus test
myAFIs <- c("chisq","cfi","rmsea","mfi","aic")
moreAFIs <- c("gammaHat","adjGammaHat")

## Use only 20 permutations for a demo.  In practice,
## use > 1000 to reduce sampling variability of estimated p values

## test configural invariance
set.seed(12345)
out.config <- permuteMeasEq(nPermute = 20, con = fit.config)
out.config

## test metric equivalence
set.seed(12345) # same permutations
out.metric <- permuteMeasEq(nPermute = 20, uncon = fit.config, con = fit.metric,
                            param = "loadings", AFIs = myAFIs,
                            moreAFIs = moreAFIs, null = fit.null)
summary(out.metric, nd = 4)

## test scalar equivalence
set.seed(12345) # same permutations
out.scalar <- permuteMeasEq(nPermute = 20, uncon = fit.metric, con = fit.scalar,
                            param = "intercepts", AFIs = myAFIs,
                            moreAFIs = moreAFIs, null = fit.null)
summary(out.scalar)

## Not much to see without significant DIF.
## Try using an absurdly high alpha level for illustration.
outsum <- summary(out.scalar, alpha = .50)

## notice that the returned object is the table of DIF tests
outsum

## visualize permutation distribution
hist(out.config, AFI = "chisq")
hist(out.metric, AFI = "chisq", nd = 2, alpha = .01,
     legendArgs = list(x = "topright"))
hist(out.scalar, AFI = "cfi", printLegend = FALSE)


####################### Extra Output

## function to calculate expected change of Group-2 and -3 latent means if
## each intercept constraint were released
extra <- function(con) {
  output <- list()
  output["x1.vis2"] <- lavTestScore(con, release = 19:20, univariate = FALSE,
                                    epc = TRUE, warn = FALSE)$epc$epc[70]
  output["x1.vis3"] <- lavTestScore(con, release = 19:20, univariate = FALSE,
                                    epc = TRUE, warn = FALSE)$epc$epc[106]
  output["x2.vis2"] <- lavTestScore(con, release = 21:22, univariate = FALSE,
                                    epc = TRUE, warn = FALSE)$epc$epc[70]
  output["x2.vis3"] <- lavTestScore(con, release = 21:22, univariate = FALSE,
                                    epc = TRUE, warn = FALSE)$epc$epc[106]
  output["x3.vis2"] <- lavTestScore(con, release = 23:24, univariate = FALSE,
                                    epc = TRUE, warn = FALSE)$epc$epc[70]
  output["x3.vis3"] <- lavTestScore(con, release = 23:24, univariate = FALSE,
                                    epc = TRUE, warn = FALSE)$epc$epc[106]
  output["x4.txt2"] <- lavTestScore(con, release = 25:26, univariate = FALSE,
                                    epc = TRUE, warn = FALSE)$epc$epc[71]
  output["x4.txt3"] <- lavTestScore(con, release = 25:26, univariate = FALSE,
                                    epc = TRUE, warn = FALSE)$epc$epc[107]
  output["x5.txt2"] <- lavTestScore(con, release = 27:28, univariate = FALSE,
                                    epc = TRUE, warn = FALSE)$epc$epc[71]
  output["x5.txt3"] <- lavTestScore(con, release = 27:28, univariate = FALSE,
                                    epc = TRUE, warn = FALSE)$epc$epc[107]
  output["x6.txt2"] <- lavTestScore(con, release = 29:30, univariate = FALSE,
                                    epc = TRUE, warn = FALSE)$epc$epc[71]
  output["x6.txt3"] <- lavTestScore(con, release = 29:30, univariate = FALSE,
                                    epc = TRUE, warn = FALSE)$epc$epc[107]
  output["x7.spd2"] <- lavTestScore(con, release = 31:32, univariate = FALSE,
                                    epc = TRUE, warn = FALSE)$epc$epc[72]
  output["x7.spd3"] <- lavTestScore(con, release = 31:32, univariate = FALSE,
                                    epc = TRUE, warn = FALSE)$epc$epc[108]
  output["x8.spd2"] <- lavTestScore(con, release = 33:34, univariate = FALSE,
                                    epc = TRUE, warn = FALSE)$epc$epc[72]
  output["x8.spd3"] <- lavTestScore(con, release = 33:34, univariate = FALSE,
                                    epc = TRUE, warn = FALSE)$epc$epc[108]
  output["x9.spd2"] <- lavTestScore(con, release = 35:36, univariate = FALSE,
                                    epc = TRUE, warn = FALSE)$epc$epc[72]
  output["x9.spd3"] <- lavTestScore(con, release = 35:36, univariate = FALSE,
                                    epc = TRUE, warn = FALSE)$epc$epc[108]
  output
}

## observed EPC
extra(fit.scalar)

## permutation results, including extra output
set.seed(12345) # same permutations
out.scalar <- permuteMeasEq(nPermute = 20, uncon = fit.metric, con = fit.scalar,
                            param = "intercepts", AFIs = myAFIs,
                            moreAFIs = moreAFIs, null = fit.null, extra = extra)
## summarize extra output
summary(out.scalar, extra = TRUE)


###########
## MIMIC ##
###########

## Specify Restricted Factor Analysis (RFA) model, equivalent to MIMIC, but
## the factor covaries with the covariate instead of being regressed on it.
## The covariate defines a single-indicator construct, and the
## double-mean-centered products of the indicators define a latent
## interaction between the factor and the covariate.
mod.mimic <- '
visual  =~ x1 + x2 + x3
age =~ ageyr
age.by.vis =~ x1.ageyr + x2.ageyr + x3.ageyr

x1 ~~ x1.ageyr
x2 ~~ x2.ageyr
x3 ~~ x3.ageyr
'

HS.orth <- indProd(var1 = paste0("x", 1:3), var2 = "ageyr", match = FALSE,
                         data = HS[ , c("ageyr", paste0("x", 1:3))] )
fit.mimic <- cfa(mod.mimic, data = HS.orth, meanstructure = TRUE)
summary(fit.mimic, stand = TRUE)

## Whereas MIMIC models specify direct effects of the covariate on an indicator,
## DIF can be tested in RFA models by specifying free loadings of an indicator
## on the covariate's construct (uniform DIF, scalar invariance) and the
## interaction construct (nonuniform DIF, metric invariance).
param <- as.list(paste0("age + age.by.vis =~ x", 1:3))
names(param) <- paste0("x", 1:3)
# param <- as.list(paste0("x", 1:3, " ~ age + age.by.vis")) # equivalent

## test both parameters simultaneously for each indicator
do.call(rbind, lapply(param, function(x) lavTestScore(fit.mimic, add = x)$test))
## or test each parameter individually
lavTestScore(fit.mimic, add = as.character(param))


####################### Permutation Method

## function to recalculate the interaction terms after permuting the covariate
datafun <- function(data) {
  d <- data[, !names(data) %in% paste0("x", 1:3, ".ageyr")]
  indProd(var1 = paste0("x", 1:3), var2 = "ageyr", match = FALSE, data = d)
}

set.seed(12345)
perm.mimic <- permuteMeasEq(nPermute = 20, modelType = "mimic", con = fit.mimic,
                            param = param, covariates = "ageyr",
                            datafun = datafun)
summary(perm.mimic)


## End(Not run)