Multiple Group Estimation

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.

Usage

multipleGroup(
  data,
  model = 1,
  group,
  itemtype = NULL,
  invariance = "",
  method = "EM",
  dentype = "Gaussian",
  itemdesign = NULL,
  item.formula = NULL,
  ...
)

Arguments

data

a matrix or data.frame that consists of numerically ordered data, organized in the form of integers, with missing data coded as NA

model

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)

group

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

itemtype

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

invariance

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

method

a character object that is either 'EM', 'QMCEM', or 'MHRM' (default is 'EM'). See mirt for details

dentype

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

itemdesign

see mirt for details

item.formula

see mirt for details

...

additional arguments to be passed to the estimation engine. See mirt for details and examples

Value

function returns an object of class MultipleGroupClass (MultipleGroupClass-class).

Details

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).

References

Chalmers, R., P. (2012). mirt: A Multidimensional Item Response Theory Package for the R Environment. Journal of Statistical Software, 48(6), 1-29. doi:10.18637/jss.v048.i06

Magnus, B. E. and Garnier-Villarreal (2022). A multidimensional zero-inflated graded response model for ordinal symptom data. Psychological Methods, 27, 261-279.

Wall, M., M., Park, J., Y., and Moustaki I. (2015). IRT modeling in the presence of zero-inflation with application to psychiatric disorder severity. Applied Psychological Measurement 39: 583-597.

See also

mirt, DIF, extract.group, DRF

Author

Phil Chalmers rphilip.chalmers@gmail.com

Examples

if (FALSE) { # \dontrun{

# 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, though occasionally less stable
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)

#############
# Testing main effects in multiple independent grouping variables
set.seed(1234)
a <- matrix(abs(rnorm(15,1,.3)), ncol=1)
d <- matrix(rnorm(15,0,.7),ncol=1)
itemtype <- rep('2PL', nrow(a))
N <- 500

# generated data have interaction effect for latent means, as well as a
# main effect across D but no main effect across G
d11 <- simdata(a, d, N, itemtype, mu = 0)
d12 <- simdata(a, d, N, itemtype, mu = 0)
d13 <- simdata(a, d, N, itemtype, mu = 0)
d21 <- simdata(a, d, N, itemtype, mu = 1/2)
d22 <- simdata(a, d, N, itemtype, mu = 1/2)
d23 <- simdata(a, d, N, itemtype, mu = -1)
dat <- do.call(rbind, list(d11, d12, d13, d21, d22, d23))
group <- rep(c('G1.D1', 'G1.D2', 'G1.D3', 'G2.D1', 'G2.D2', 'G2.D3'), each=N)
table(group)

# in practice, group would be organized in a data.frame as follows
df <- data.frame(group)
dfw <- tidyr::separate_wider_delim(df, group, delim='.', names = c('G', 'D'))
head(dfw)

# for use with multipleGroup() combine into a single long group
group <- with(dfw, factor(G):factor(D))

# conditional information
itemstats(dat, group=group)

mod <- multipleGroup(dat, group = group, SE=TRUE,
                     invariance = c(colnames(dat), 'free_mean', 'free_var'))
coef(mod, simplify=TRUE)
sapply(coef(mod, simplify=TRUE), \(x) unname(x$means)) # mean estimates
wald(mod) # view parameter names for later testing

# test for main effect over G group (manually compute marginal mean)
wald(mod, "0 + MEAN_1.123 + MEAN_1.185 = MEAN_1.247 + MEAN_1.309 + MEAN_1.371")

# test for main effect over D group  (manually compute marginal means)
wald(mod, c("0 + MEAN_1.247 = MEAN_1.123 + MEAN_1.309",
            "0 + MEAN_1.247 = MEAN_1.185 + MEAN_1.371"))

# post-hoc tests (better practice would include p.adjust() )
wald(mod, "0 + MEAN_1.247 = MEAN_1.123 + MEAN_1.309") # D1 vs D2
wald(mod, "0 + MEAN_1.247 = MEAN_1.185 + MEAN_1.371") # D1 vs D3
wald(mod, "MEAN_1.123 + MEAN_1.309 = MEAN_1.185 + MEAN_1.371") # D2 vs D3


#############
# 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
if(interactive()) 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)

# Compare to single group
mod <- mirt(dat)
anova(mod, mod_mix2)

########################################
# Zero-inflated 2PL IRT model (Wall, Park, and Moustaki, 2015)

n <- 1000
nitems <- 20

a <- rep(2, nitems)
d <- rep(c(-2,-1,0,1,2), each=nitems/5)
zi_p <- 0.2 # Proportion of people in zero class

theta <- rnorm(n, 0, 1)
zeros <- matrix(0, n*zi_p, nitems)
nonzeros <- simdata(a, d, n*(1-zi_p), itemtype = '2PL',
                   Theta = as.matrix(theta[1:(n*(1-zi_p))]))
data <- rbind(nonzeros, zeros)

# define class with extreme theta but fixed item parameters
zi2PL <- "F = 1-20
          START [MIXTURE_1] = (GROUP, MEAN_1, -100), (GROUP, COV_11, .00001),
                              (1-20, a1, 1.0), (1-20, d, 0)
          FIXED [MIXTURE_1] = (GROUP, MEAN_1), (GROUP, COV_11),
                              (1-20, a1), (1-20, d)"

# define custom Theta integration grid that contains extreme theta + normal grid
technical <- list(customTheta = matrix(c(-100, seq(-6,6,length.out=61))))

# fit ZIM-IRT
zi2PL.fit <- multipleGroup(data, zi2PL, dentype = 'mixture-2', technical=technical)
coef(zi2PL.fit, simplify=TRUE)

# classification estimates
pi_hat <- fscores(zi2PL.fit, method = 'classify')
head(pi_hat)
tail(pi_hat)

# EAP estimates (not useful for zip class)
fs <- fscores(zi2PL.fit)
head(fs)
tail(fs)

########################################
# Zero-inflated graded response model (Magnus and Garnier-Villarreal, 2022)

n <- 1000
nitems <- 20

a <- matrix(rlnorm(20,.2,.3))

# for the graded model, ensure that there is enough space between the intercepts,
# otherwise closer categories will not be selected often (minimum distance of 0.3 here)
diffs <- t(apply(matrix(runif(20*4, .3, 1), 20), 1, cumsum))
diffs <- -(diffs - rowMeans(diffs))
d <- diffs + rnorm(20)

zi_p <- 0.2 # Proportion of people in zero/lowest category class

theta <- rnorm(n, 0, 1)
zeros <- matrix(0, n*zi_p, nitems)
nonzeros <- simdata(a, d, n*(1-zi_p), itemtype = 'graded',
                    Theta = as.matrix(theta[1:(n*(1-zi_p))]))
data <- rbind(nonzeros, zeros)

# intercepts will be labelled as d1 through d4
apply(data, 2, table)

# ignoring zero inflation (bad idea)
modGRM <- mirt(data)
coef(modGRM, simplify=TRUE)

# Define class with extreme theta but fixed item parameters
#   For GRM in zero-inflated class the intercept values are arbitrary
#   as the model forces the responses all into the first category (hence,
#   spacing arbitrarily set to 1)
ziGRM <- "F = 1-20
          START [MIXTURE_1] = (GROUP, MEAN_1, -100), (GROUP, COV_11, .00001),
                              (1-20, a1, 1.0),
                              (1-20, d1, 2), (1-20, d2, 1), (1-20, d3, 0), (1-20, d4, -1)
          FIXED [MIXTURE_1] = (GROUP, MEAN_1), (GROUP, COV_11),
                              (1-20, a1),
                              (1-20, d1), (1-20, d2), (1-20, d3), (1-20, d4)"

# define custom Theta integration grid that contains extreme theta + normal grid
technical <- list(customTheta = matrix(c(-100, seq(-6,6,length.out=61))))

# fit zero-inflated GRM
ziGRM.fit <- multipleGroup(data, ziGRM, dentype = 'mixture-2', technical=technical)
coef(ziGRM.fit, simplify=TRUE)

# classification estimates
pi_hat <- fscores(ziGRM.fit, method = 'classify')
head(pi_hat)
tail(pi_hat)

# EAP estimates (not useful for zip class)
fs <- fscores(ziGRM.fit)
head(fs)
tail(fs)

} # }