multipleGroup performs a full-information
maximum-likelihood multiple group analysis for any combination of dichotomous and polytomous
data under the item response theory paradigm using either Cai's (2010)
Metropolis-Hastings Robbins-Monro (MHRM) algorithm or with an EM algorithm approach. This
function may be used for detecting differential item functioning (DIF), thought the
DIF function may provide a more convenient approach. If the grouping
variable is not specified then the dentype input can be modified to fit
mixture models to estimate any latent group components.
multipleGroup(
data,
model = 1,
group,
itemtype = NULL,
invariance = "",
method = "EM",
dentype = "Gaussian",
...
)function returns an object of class MultipleGroupClass
(MultipleGroupClass-class).
a matrix or data.frame that consists of
numerically ordered data, with missing data coded as NA
string to be passed to, or a model object returned from, mirt.model
declaring how the global model is to be estimated (useful to apply constraints here)
a character or factor vector indicating group membership. If a character
vector is supplied this will be automatically transformed into a factor variable.
As well, the first level of the (factorized) grouping variable will be treated as the "reference" group
can be same type of input as is documented in mirt, however may also be a
ngroups by nitems matrix specifying the type of IRT models for each group, respectively.
Rows of this input correspond to the levels of the group input. For mixture models the rows correspond
to the respective mixture grouping variables to be constructed, and the IRT models should be within these
mixtures
a character vector containing the following possible options:
'free_mean' or 'free_means'freely estimate all latent means in all focal groups (reference group constrained to a vector of 0's)
'free_var', 'free_vars', 'free_variance', or 'free_variances'freely estimate all latent variances in focal groups (reference group variances all constrained to 1)
'slopes'to constrain all the slopes to be equal across all groups
'intercepts'to constrain all the intercepts to be equal across all groups, note for nominal models this also includes the category specific slope parameters
Additionally, specifying specific item name bundles (from colnames(data)) will
constrain all freely estimated parameters in each item to be equal across groups. This is
useful for selecting 'anchor' items for vertical and horizontal scaling, and for detecting
differential item functioning (DIF) across groups
a character object that is either 'EM', 'QMCEM', or 'MHRM'
(default is 'EM'). See mirt for details
type of density form to use for the latent trait parameters. Current options include
all of the methods described in mirt, as well as
'mixture-#' estimates mixtures of Gaussian distributions,
where the # placeholder represents the number of potential grouping variables
(e.g., 'mixture-3' will estimate 3 underlying classes). Each class is
assigned the group name MIXTURE_#, where # is the class number.
Note that internally the mixture coefficients are stored as log values where
the first mixture group coefficient is fixed at 0
additional arguments to be passed to the estimation engine. See mirt
for details and examples
Phil Chalmers rphilip.chalmers@gmail.com
By default the estimation in multipleGroup assumes that the models are maximally
independent, and therefore could initially be performed by sub-setting the data and running
identical models with mirt and aggregating the results (e.g., log-likelihood).
However, constrains may be automatically imposed across groups by invoking various
invariance keywords. Users may also supply a list of parameter equality constraints
to by constrain argument, of define equality constraints using the
mirt.model syntax (recommended).
Chalmers, R., P. (2012). mirt: A Multidimensional Item Response Theory Package for the R Environment. Journal of Statistical Software, 48(6), 1-29. tools:::Rd_expr_doi("10.18637/jss.v048.i06")
mirt, DIF, extract.group, DRF
if (FALSE) {
#single factor
set.seed(12345)
a <- matrix(abs(rnorm(15,1,.3)), ncol=1)
d <- matrix(rnorm(15,0,.7),ncol=1)
itemtype <- rep('2PL', nrow(a))
N <- 1000
dataset1 <- simdata(a, d, N, itemtype)
dataset2 <- simdata(a, d, N, itemtype, mu = .1, sigma = matrix(1.5))
dat <- rbind(dataset1, dataset2)
group <- c(rep('D1', N), rep('D2', N))
# marginal information
itemstats(dat)
# conditional information
itemstats(dat, group=group)
mod_configural <- multipleGroup(dat, 1, group = group) #completely separate analyses
#limited information fit statistics
M2(mod_configural)
mod_metric <- multipleGroup(dat, 1, group = group, invariance=c('slopes')) #equal slopes
#equal intercepts, free variance and means
mod_scalar2 <- multipleGroup(dat, 1, group = group,
invariance=c('slopes', 'intercepts', 'free_var','free_means'))
mod_scalar1 <- multipleGroup(dat, 1, group = group, #fixed means
invariance=c('slopes', 'intercepts', 'free_var'))
mod_fullconstrain <- multipleGroup(dat, 1, group = group,
invariance=c('slopes', 'intercepts'))
extract.mirt(mod_fullconstrain, 'time') #time of estimation components
#optionally use Newton-Raphson for (generally) faster convergence in the M-step's
mod_fullconstrain <- multipleGroup(dat, 1, group = group, optimizer = 'NR',
invariance=c('slopes', 'intercepts'))
extract.mirt(mod_fullconstrain, 'time') #time of estimation components
summary(mod_scalar2)
coef(mod_scalar2, simplify=TRUE)
residuals(mod_scalar2)
plot(mod_configural)
plot(mod_configural, type = 'info')
plot(mod_configural, type = 'trace')
plot(mod_configural, type = 'trace', which.items = 1:4)
itemplot(mod_configural, 2)
itemplot(mod_configural, 2, type = 'RE')
anova(mod_metric, mod_configural) #equal slopes only
anova(mod_scalar2, mod_metric) #equal intercepts, free variance and mean
anova(mod_scalar1, mod_scalar2) #fix mean
anova(mod_fullconstrain, mod_scalar1) #fix variance
# compared all at once (in order of most constrained to least)
anova(mod_fullconstrain, mod_scalar2, mod_configural)
#test whether first 6 slopes should be equal across groups
values <- multipleGroup(dat, 1, group = group, pars = 'values')
values
constrain <- list(c(1, 63), c(5,67), c(9,71), c(13,75), c(17,79), c(21,83))
equalslopes <- multipleGroup(dat, 1, group = group, constrain = constrain)
anova(equalslopes, mod_configural)
#same as above, but using mirt.model syntax
newmodel <- '
F = 1-15
CONSTRAINB = (1-6, a1)'
equalslopes <- multipleGroup(dat, newmodel, group = group)
coef(equalslopes, simplify=TRUE)
############
# vertical scaling (i.e., equating when groups answer items others do not)
dat2 <- dat
dat2[group == 'D1', 1:2] <- dat2[group != 'D1', 14:15] <- NA
head(dat2)
tail(dat2)
# items with missing responses need to be constrained across groups for identification
nms <- colnames(dat2)
mod <- multipleGroup(dat2, 1, group, invariance = nms[c(1:2, 14:15)])
# this will throw an error without proper constraints (SEs cannot be computed either)
# mod <- multipleGroup(dat2, 1, group)
# model still does not have anchors, therefore need to add a few (here use items 3-5)
mod_anchor <- multipleGroup(dat2, 1, group,
invariance = c(nms[c(1:5, 14:15)], 'free_means', 'free_var'))
coef(mod_anchor, simplify=TRUE)
# check if identified by computing information matrix
mod_anchor <- multipleGroup(dat2, 1, group, pars = mod2values(mod_anchor), TOL=NaN, SE=TRUE,
invariance = c(nms[c(1:5, 14:15)], 'free_means', 'free_var'))
mod_anchor
coef(mod_anchor)
coef(mod_anchor, printSE=TRUE)
#############
#DIF test for each item (using all other items as anchors)
itemnames <- colnames(dat)
refmodel <- multipleGroup(dat, 1, group = group, SE=TRUE,
invariance=c('free_means', 'free_var', itemnames))
#loop over items (in practice, run in parallel to increase speed). May be better to use ?DIF
estmodels <- vector('list', ncol(dat))
for(i in 1:ncol(dat))
estmodels[[i]] <- multipleGroup(dat, 1, group = group, verbose = FALSE,
invariance=c('free_means', 'free_var', itemnames[-i]))
anova(refmodel, estmodels[[1]])
(anovas <- lapply(estmodels, function(x, refmodel) anova(refmodel, x),
refmodel=refmodel))
#family-wise error control
p <- do.call(rbind, lapply(anovas, function(x) x[2, 'p']))
p.adjust(p, method = 'BH')
#same as above, except only test if slopes vary (1 df)
#constrain all intercepts
estmodels <- vector('list', ncol(dat))
for(i in 1:ncol(dat))
estmodels[[i]] <- multipleGroup(dat, 1, group = group, verbose = FALSE,
invariance=c('free_means', 'free_var', 'intercepts',
itemnames[-i]))
(anovas <- lapply(estmodels, function(x, refmodel) anova(refmodel, x),
refmodel=refmodel))
#quickly test with Wald test using DIF()
mod_configural2 <- multipleGroup(dat, 1, group = group, SE=TRUE)
DIF(mod_configural2, which.par = c('a1', 'd'), Wald=TRUE, p.adjust = 'fdr')
#############
# Three group model where the latent variable parameters are constrained to
# be equal in the focal groups
set.seed(12345)
a <- matrix(abs(rnorm(15,1,.3)), ncol=1)
d <- matrix(rnorm(15,0,.7),ncol=1)
itemtype <- rep('2PL', nrow(a))
N <- 1000
dataset1 <- simdata(a, d, N, itemtype)
dataset2 <- simdata(a, d, N, itemtype, mu = .1, sigma = matrix(1.5))
dataset3 <- simdata(a, d, N, itemtype, mu = .1, sigma = matrix(1.5))
dat <- rbind(dataset1, dataset2, dataset3)
group <- rep(c('D1', 'D2', 'D3'), each=N)
# marginal information
itemstats(dat)
# conditional information
itemstats(dat, group=group)
model <- 'F1 = 1-15
FREE[D2, D3] = (GROUP, MEAN_1), (GROUP, COV_11)
CONSTRAINB[D2,D3] = (GROUP, MEAN_1), (GROUP, COV_11)'
mod <- multipleGroup(dat, model, group = group, invariance = colnames(dat))
coef(mod, simplify=TRUE)
#############
#multiple factors
a <- matrix(c(abs(rnorm(5,1,.3)), rep(0,15),abs(rnorm(5,1,.3)),
rep(0,15),abs(rnorm(5,1,.3))), 15, 3)
d <- matrix(rnorm(15,0,.7),ncol=1)
mu <- c(-.4, -.7, .1)
sigma <- matrix(c(1.21,.297,1.232,.297,.81,.252,1.232,.252,1.96),3,3)
itemtype <- rep('2PL', nrow(a))
N <- 1000
dataset1 <- simdata(a, d, N, itemtype)
dataset2 <- simdata(a, d, N, itemtype, mu = mu, sigma = sigma)
dat <- rbind(dataset1, dataset2)
group <- c(rep('D1', N), rep('D2', N))
#group models
model <- '
F1 = 1-5
F2 = 6-10
F3 = 11-15'
#define mirt cluster to use parallel architecture
mirtCluster()
#EM approach (not as accurate with 3 factors, but generally good for quick model comparisons)
mod_configural <- multipleGroup(dat, model, group = group) #completely separate analyses
mod_metric <- multipleGroup(dat, model, group = group, invariance=c('slopes')) #equal slopes
mod_fullconstrain <- multipleGroup(dat, model, group = group, #equal means, slopes, intercepts
invariance=c('slopes', 'intercepts'))
anova(mod_metric, mod_configural)
anova(mod_fullconstrain, mod_metric)
#same as above, but with MHRM (generally more accurate with 3+ factors, but slower)
mod_configural <- multipleGroup(dat, model, group = group, method = 'MHRM')
mod_metric <- multipleGroup(dat, model, group = group, invariance=c('slopes'), method = 'MHRM')
mod_fullconstrain <- multipleGroup(dat, model, group = group, method = 'MHRM',
invariance=c('slopes', 'intercepts'))
anova(mod_metric, mod_configural)
anova(mod_fullconstrain, mod_metric)
############
#polytomous item example
set.seed(12345)
a <- matrix(abs(rnorm(15,1,.3)), ncol=1)
d <- matrix(rnorm(15,0,.7),ncol=1)
d <- cbind(d, d-1, d-2)
itemtype <- rep('graded', nrow(a))
N <- 1000
dataset1 <- simdata(a, d, N, itemtype)
dataset2 <- simdata(a, d, N, itemtype, mu = .1, sigma = matrix(1.5))
dat <- rbind(dataset1, dataset2)
group <- c(rep('D1', N), rep('D2', N))
model <- 'F1 = 1-15'
mod_configural <- multipleGroup(dat, model, group = group)
plot(mod_configural)
plot(mod_configural, type = 'SE')
itemplot(mod_configural, 1)
itemplot(mod_configural, 1, type = 'info')
plot(mod_configural, type = 'trace') # messy, score function typically better
plot(mod_configural, type = 'itemscore')
fs <- fscores(mod_configural, full.scores = FALSE)
head(fs[["D1"]])
fscores(mod_configural, method = 'EAPsum', full.scores = FALSE)
# constrain slopes within each group to be equal (but not across groups)
model2 <- 'F1 = 1-15
CONSTRAIN = (1-15, a1)'
mod_configural2 <- multipleGroup(dat, model2, group = group)
plot(mod_configural2, type = 'SE')
plot(mod_configural2, type = 'RE')
itemplot(mod_configural2, 10)
############
## empirical histogram example (normal and bimodal groups)
set.seed(1234)
a <- matrix(rlnorm(50, .2, .2))
d <- matrix(rnorm(50))
ThetaNormal <- matrix(rnorm(2000))
ThetaBimodal <- scale(matrix(c(rnorm(1000, -2), rnorm(1000,2)))) #bimodal
Theta <- rbind(ThetaNormal, ThetaBimodal)
dat <- simdata(a, d, 4000, itemtype = '2PL', Theta=Theta)
group <- rep(c('G1', 'G2'), each=2000)
EH <- multipleGroup(dat, 1, group=group, dentype="empiricalhist", invariance = colnames(dat))
coef(EH, simplify=TRUE)
plot(EH, type = 'empiricalhist', npts = 60)
#DIF test for item 1
EH1 <- multipleGroup(dat, 1, group=group, dentype="empiricalhist", invariance = colnames(dat)[-1])
anova(EH, EH1)
#--------------------------------
# Mixture model (no prior group variable specified)
set.seed(12345)
nitems <- 20
a1 <- matrix(.75, ncol=1, nrow=nitems)
a2 <- matrix(1.25, ncol=1, nrow=nitems)
d1 <- matrix(rnorm(nitems,0,1),ncol=1)
d2 <- matrix(rnorm(nitems,0,1),ncol=1)
itemtype <- rep('2PL', nrow(a1))
N1 <- 500
N2 <- N1*2 # second class twice as large
dataset1 <- simdata(a1, d1, N1, itemtype)
dataset2 <- simdata(a2, d2, N2, itemtype)
dat <- rbind(dataset1, dataset2)
# group <- c(rep('D1', N1), rep('D2', N2))
# Mixture Rasch model (Rost, 1990)
models <- 'F1 = 1-20
CONSTRAIN = (1-20, a1)'
mod_mix <- multipleGroup(dat, models, dentype = 'mixture-2', GenRandomPars = TRUE)
coef(mod_mix, simplify=TRUE)
summary(mod_mix)
plot(mod_mix)
plot(mod_mix, type = 'trace')
itemplot(mod_mix, 1, type = 'info')
head(fscores(mod_mix)) # theta estimates
head(fscores(mod_mix, method = 'classify')) # classification probability
itemfit(mod_mix)
# Mixture 2PL model
mod_mix2 <- multipleGroup(dat, 1, dentype = 'mixture-2', GenRandomPars = TRUE)
anova(mod_mix, mod_mix2)
coef(mod_mix2, simplify=TRUE)
itemfit(mod_mix2)
# Zero-inflated 2PL IRT model
model <- "F = 1-20
START [MIXTURE_1] = (GROUP, MEAN_1, -100), (GROUP, COV_11, .00001),
(1-20, a1, 1.0), (1-20, d, 0.0)
FIXED [MIXTURE_1] = (GROUP, MEAN_1), (GROUP, COV_11),
(1-20, a1), (1-20, d)"
zip <- multipleGroup(dat, model, dentype = 'mixture-2')
coef(zip, simplify=TRUE)
}
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