fixedCalib: Fixed-item calibration method

In mirt: Multidimensional Item Response Theory

Fixed-item calibration method

Description

Implements the set of fixed-item calibration methods described by Kim (2006). The initial calibrated model must be fitted via mirt, is currently limited to unidimensional models only, and should only be utilized when the new set of responses are obtained from a population with similar distributional characteristics in the latent traits. For more flexible calibration of items, including a fixed-item calibration variant involving anchor items for equating, see multipleGroup.

Usage

fixedCalib(
  data,
  model = 1,
  old_mod,
  PAU = "MWU",
  NEMC = "MEM",
  technical = list(),
  ...
)

Arguments

data	new data to be used for calibration. Note that to be consistent with the mod object, observed responses/NA placeholders must be included to link the item names used in the original mod definition (i.e., extract.mirt(mod, what = 'itemnames'))
model	type of model to fit for the complete dataset (not that for the fixed items in old_mod the factor loadings/constraints specified by the potential mirt.model specification is not relevant)
old_mod	a model of class SingleGroupClass fitted using mirt
PAU	prior ability update (PAU) approach. Supports none ("NWU"), one ("OWU"), and many ("MWU")
NEMC	number of EM cycles (NEMC) to use for the to-be-estimated parameters. Supports one ("OEM") and many ("MEM")
technical	list of technical estimation arguments (see mirt for details)
...	additional arguments to pass to mirt

References

Kim, S. (2006). A comparative study of IRT fixed parameter calibration methods. Journal of Educational Measurement, 4(43), 355-381.

See Also

mirt, multipleGroup

Examples

# single factor
set.seed(12345)
J <- 50
a <- matrix(abs(rnorm(J,1,.3)), ncol=1)
d <- matrix(rnorm(J,0,.7),ncol=1)
itemtype <- rep('2PL', nrow(a))

# calibration data theta ~ N(0,1)
N <- 3000
dataset1 <- simdata(a, d, N = N, itemtype=itemtype)

# new data (again, theta ~ N(0,1))
dataset2 <- simdata(a, d, N = 1000, itemtype=itemtype)

# last 40% of experimental items not given to calibration group
#     (unobserved; hence removed)
dataset1 <- dataset1[,-c(J:(J*.6))]
head(dataset1)

#--------------------------------------

# calibrated model from dataset1 only
mod <- mirt(dataset1, model = 1)
coef(mod, simplify=TRUE)

# No Prior Weights Updating and One EM Cycle (NWU-OEM)
NWU_OEM <- fixedCalib(dataset2, model=1, old_mod=mod, PAU='NWU', NEMC='OEM')
coef(NWU_OEM, simplify=TRUE)
data.frame(coef(NWU_OEM, simplify=TRUE)$items[,c('a1','d')],
           pop_a1=a, pop_d=d)
plot(NWU_OEM, type = 'empiricalhist')

# No Prior Weights Updating and Multiple EM Cycles (NWU-MEM)
NWU_MEM <- fixedCalib(dataset2, model = 1, old_mod = mod, PAU = 'NWU')
coef(NWU_MEM, simplify=TRUE)
data.frame(coef(NWU_MEM, simplify=TRUE)$items[,c('a1','d')],
           pop_a1=a, pop_d=d)
plot(NWU_MEM, type = 'empiricalhist')

# One Prior Weights Updating and One EM Cycle (OWU-OEM)
OWU_OEM <- fixedCalib(dataset2, model=1, old_mod=mod, PAU='OWU', NEMC="OEM")
coef(OWU_OEM, simplify=TRUE)
data.frame(coef(OWU_OEM, simplify=TRUE)$items[,c('a1','d')], pop_a1=a, pop_d=d)
plot(OWU_OEM, type = 'empiricalhist')

# One Prior Weights Updating and Multiple EM Cycles (OWU-MEM)
OWU_MEM <- fixedCalib(dataset2, model = 1, old_mod = mod, PAU = 'OWU')
coef(OWU_MEM, simplify=TRUE)
data.frame(coef(OWU_MEM, simplify=TRUE)$items[,c('a1','d')],
           pop_a1=a, pop_d=d)
plot(OWU_MEM, type = 'empiricalhist')

# Multiple Prior Weights Updating and Multiple EM Cycles (MWU-MEM)
MWU_MEM <- fixedCalib(dataset2, model = 1, old_mod = mod)
coef(MWU_MEM, simplify=TRUE)
data.frame(coef(MWU_MEM, simplify=TRUE)$items[,c('a1','d')],
           pop_a1=a, pop_d=d)
plot(MWU_MEM, type = 'empiricalhist')

# factor scores distribution check
fs <- fscores(MWU_MEM)
hist(fs)
c(mean_calib=mean(fs[1:N, ]), sd_calib=sd(fs[1:N, ]))
c(mean_exper=mean(fs[-c(1:N), ]), sd_exper=sd(fs[-c(1:N), ]))


############################
## Item length constraint example for each participant in the experimental
## items group. In this example, all participants were forced to have a test
## length of J=30, though the item pool had J=50 total items.

# new experimental data (relatively extreme, theta ~ N(.5,1.5))
dataset2 <- simdata(a, d, N = 1000, itemtype=itemtype,
    mu=.5, sigma=matrix(1.5))

# Add missing values to each participant in new dataset where individuals
# were randomly administered 10 experimental items, subject to the constraint
# that each participant received a test with J=30 items.
dataset2 <- t(apply(dataset2, 1, function(x){
   NA_precalib <- sample(1:30, 10)
   NA_experimental <- sample(31:50, 10)
   x[c(NA_precalib, NA_experimental)] <- NA
   x
}))
head(dataset2)

# check that all individuals had 30 items
all(rowSums(!is.na(dataset2)) == 30)

# Multiple Prior Weights Updating and Multiple EM Cycles (MWU-MEM)
MWU_MEM <- fixedCalib(dataset2, model = 1, old_mod = mod)
coef(MWU_MEM, simplify=TRUE)
data.frame(coef(MWU_MEM, simplify=TRUE)$items[,c('a1','d')],
           pop_a1=a, pop_d=d)
plot(MWU_MEM, type = 'empiricalhist')

## factor scores check
fs <- fscores(MWU_MEM)
hist(fs)
c(mean_calib=mean(fs[1:N, ]), sd_calib=sd(fs[1:N, ]))

## shrinkage, but generally different from calibrated sample
c(mean_exper=mean(fs[-c(1:N), ]), sd_exper=sd(fs[-c(1:N), ]))

mirt documentation built on Sept. 10, 2025, 1:07 a.m.