The supplementary materials illustrate the use of the accompanying R package equaltestMI. The sample statistics from Table 1 of Lee and Al Otaiba (2015) are used as an example. The description of the data can be found in the original article and the results obtained from equaltestMI are discussed at length in the submitted article “Measurement Invariance Via Equivalence Testing”.
The R package equaltestMI is available on CRAN and can be downloaded for use on any R platform with version higher than 3.1.0. Users can pass different arguments to the main function eqMI.main() to examine measurement invariance using the conventional multiple-group approach or equivalence testing approach. Under the framework of equivalence testing, users obtain the minimum tolerable size (T-size) and adjusted cutoff values to evaluate the goodness-of-fit of each invariance test. The projection method is also available for testing the equality of latent means.
## load package
library(equaltestMI)## sample statistics where M1 and M2 are sample means, and Cov1 and Cov2 are sample covariance matrices;
## group 1 = boys ineligible for free-reduced lunches
## group 2 = boys eligible for free-reduced lunches
#setwd("C:/research/equaltestMI")
Group1 <- read.table('Group1.txt', header = TRUE)
Group2 <- read.table('Group2.txt', header = TRUE)
Group1 <- as.matrix(Group1)
Group2 <- as.matrix(Group2)
M1 <- Group1[1,]
M2 <- Group2[1,]
Cov1 <- Group1[2:7,]
Cov2 <- Group2[2:7,]## lavaan model syntax
model <- '
AlphabetKnowledge =~ Letter_Name+ Letter_Sound
PhonologicalAwareness =~ Blending + Elision
Spelling =~ Real_Words + Pseudo_Words
'## the results using equivalence testing and projection method
## full R output will be presented in Part 3
test <- eqMI.main(model = model,
sample.nobs = c(78, 174),
sample.mean = list(M1, M2),
sample.cov = list(Cov1, Cov2),
meanstructure = TRUE,
output = 'both',
quiet = FALSE,
equivalence.test = TRUE, adjRMSEA = TRUE,
projection = TRUE, bootstrap = FALSE)Sample means:
## Letter_Name Letter_Sound Blending Elision Real_Words
## 45.26 40.45 10.91 6.51 23.88
## Pseudo_Words
## 14.12Sample Covariance Matrix:
## Letter_Name Letter_Sound Blending Elision Real_Words
## Letter_Name 207.360 159.097 32.589 25.805 61.776
## Letter_Sound 159.097 280.228 42.888 36.748 76.123
## Blending 32.589 42.888 18.233 10.713 19.051
## Elision 25.805 36.748 10.713 20.070 20.372
## Real_Words 61.776 76.123 19.051 20.372 73.616
## Pseudo_Words 45.075 60.204 14.219 16.709 47.429
## Pseudo_Words
## Letter_Name 45.075
## Letter_Sound 60.204
## Blending 14.219
## Elision 16.709
## Real_Words 47.429
## Pseudo_Words 44.356Sample means:
## Letter_Name Letter_Sound Blending Elision Real_Words
## 41.32 34.88 9.08 4.45 19.24
## Pseudo_Words
## 11.07Sample Covariance Matrix:
## Letter_Name Letter_Sound Blending Elision Real_Words
## Letter_Name 295.840 232.200 38.996 20.174 67.593
## Letter_Sound 232.200 324.000 43.164 22.824 77.954
## Blending 38.996 43.164 19.010 9.260 23.428
## Elision 20.174 22.824 9.260 10.049 15.254
## Real_Words 67.593 77.954 23.428 15.254 64.320
## Pseudo_Words 57.771 60.458 16.272 11.042 38.411
## Pseudo_Words
## Letter_Name 57.771
## Letter_Sound 60.458
## Blending 16.272
## Elision 11.042
## Real_Words 38.411
## Pseudo_Words 38.688##
## ---------- Equality of Population Covariance Matrices under NHT ----------
##
## Chisq Df pvalue
## fit.pop.cov 48.85006 21 0.0005261173
##
## -------- Chi-Square and Chi-Square-Difference Test under NHT --------
##
## Chisq Df pvalue Chisq.diff Df.diff pvalue
## fit.pop.cov 48.850 21 0.001
## fit.configural.g1 4.408 6 0.622
## fit.configural.g2 10.641 6 0.100
## fit.combine.groups 15.049 12
## fit.metric 20.033 15 0.171 4.984 3 0.173
## fit.residuals 42.512 21 0.004 22.479 6 0.001
## fit.varfactor 54.175 27 0.001 11.663 6 0.070
## fit.scalar 23.732 18 0.164 3.699 3 0.296
## fit.strong.means 41.066 21 0.006 17.334 3 0.001
## fit.strict.residuals 45.968 24 0.004 22.237 6 0.001
## fit.strict.means 63.630 27 0.000 17.662 3 0.001
##
## -------- T-size epsilon, RMSEA, and Adjusted Cutoff Values under ET --------
##
## epsilon_t RMESA_t cut.01 cut.05 cut.08
## fit.pop.cov 0.209 0.141 0.076 0.097 0.121
## fit.configural.g1 0.028 0.097 0.116 0.133 0.157
## fit.configural.g2 0.071 0.154 0.116 0.133 0.157
## fit.metric 0.049 0.181 0.151 0.164 0.187
## fit.residuals 0.140 0.216 0.116 0.133 0.157
## fit.varfactor 0.078 0.161 0.116 0.133 0.157
## fit.scalar 0.040 0.163 0.151 0.164 0.187
## fit.strong.means 0.125 0.289 0.151 0.164 0.187
## fit.strict.residuals 0.138 0.215 0.116 0.133 0.157
## fit.strict.means 0.127 0.291 0.151 0.164 0.187
## cut.10 goodness-of-fit
## fit.pop.cov 0.139 poor
## fit.configural.g1 0.175 excellent
## fit.configural.g2 0.175 fair
## fit.metric 0.205 fair
## fit.residuals 0.175 poor
## fit.varfactor 0.175 mediocre
## fit.scalar 0.205 close
## fit.strong.means 0.205 poor
## fit.strict.residuals 0.175 poor
## fit.strict.means 0.205 poor
##
##
## ---------- Means of Latent and Specific Factors by the Projection Method and under NHT ----------
##
## Chisq Df pvalue
## fit.mvmean 22.388937 6 0.0010292258
## fit.common 19.433784 3 0.0002223613
## fit.specific 4.015386 3 0.2598074616
## Validity Index is 0.9885648
##
## ---------- Means of Latent and Specific Factors by the Projection Method and under ET ----------
##
## epsilon_t RMESA_t cut.01 cut.05 cut.08 cut.10
## fit.mvmean 0.139 0.215 0.116 0.133 0.157 0.175
## fit.common 0.137 0.302 0.151 0.164 0.187 0.205
## fit.specific 0.042 0.168 0.151 0.164 0.187 0.205
## goodness-of-fit
## fit.mvmean poor
## fit.common poor
## fit.specific fairtest1 <- eqMI.main(model = model,
sample.nobs = c(78, 174), sample.cov = list(Cov1, Cov2),
sample.mean = list(M1, M2), meanstructure = TRUE,
equivalence.test = FALSE, adjRMSEA = FALSE)test2 <- eqMI.main(model = model,
sample.nobs = c(78, 174), sample.cov = list(Cov1, Cov2),
sample.mean = list(M1, M2), meanstructure = TRUE,
equivalence.test = FALSE, adjRMSEA = FALSE,
projection = TRUE)test3 <- eqMI.main(model = model,
sample.nobs = c(78, 174), sample.cov = list(Cov1, Cov2),
sample.mean = list(M1, M2), meanstructure = TRUE,
equivalence.test = TRUE, adjRMSEA = FALSE)test4 <- eqMI.main(model = model,
sample.nobs = c(78, 174), sample.cov = list(Cov1, Cov2),
sample.mean = list(M1, M2), meanstructure = TRUE,
equivalence.test = TRUE, adjRMSEA = TRUE)test5 <- eqMI.main(model = model,
sample.nobs = c(78, 174), sample.cov = list(Cov1, Cov2),
sample.mean = list(M1, M2), meanstructure = TRUE,
equivalence.test = TRUE, adjRMSEA = TRUE,
projection = TRUE)test6 <- eqMI.main(model = model, structure = 'mean',
sample.nobs = c(78, 174), sample.cov = list(Cov1, Cov2),
sample.mean = list(M1, M2), meanstructure = TRUE,
equivalence.test = TRUE, adjRMSEA = TRUE,
projection = TRUE)test7 <- eqMI.main(model = model, data = literacy.dat,
group = "FRL", meanstructure = TRUE,
equivalence.test = TRUE, adjRMSEA = TRUE,
projection = TRUE)#test8 <- eqMI.main(model = model, data = literacy.dat,
# group = "FRL", meanstructure = TRUE,
# equivalence.test = TRUE, adjRMSEA = TRUE,
# projection = TRUE, bootstrap = TRUE)test9 <- eqMI.main(model = model, data = literacy.dat,
group = "FRL", meanstructure = TRUE,
equivalence.test = TRUE, adjRMSEA = TRUE,
projection = TRUE, bootstrap = FALSE,
quite = TRUE)