Package 'equateIRT'

IRT Equating Methods

Description

This package computes direct, chain and average (bisector) equating coefficients with standard errors using IRT methods for dichotomous items. The IRT models included are the three-parameter logistic model, the two-parameter logistic model, the one-parameter logistic model and the Rasch model. Test scoring can be performed by true score equating and observed score equating methods. DIF detection can be performed using a Wald-type test.

Details

Package:	equateIRT
Type:	Package
Version:	2.5.0
Date:	2024-07-30
License:	GPL-3

Direct equating coefficients and their standard errors between forms presenting common items can be computed using function direc. The equating methods implemented are "mean-mean", "mean-geometric mean", "mean-sigma", "Haebara" and "Stocking-Lord". Estimates of item parameters and their covariance matrix can be imported from the R packages ltm and mirt or from the IRT programs IRTPRO and flexMIRT using functions import.ltm, import.mirt, import.irtpro and import.flexmirt. Item parameter estimates from other software can be imported as well by the user. Data should be previously organized using function modIRT. Function alldirec computes all direct equating coefficients (with standard errors) between all pairs of a list of forms. Function chainec computes chain equating coefficients (and standard errors) given direct equating coefficients between forms directly linked. Average equating coefficients with standard errors can be calculated using function bisectorec, that implements the (weighted) bisector method. Once the equating coefficients are obtained, the computation of equated scores can be performed using function score, which implements true score equating and observed score equating. Standard errors of equated scores are also provided. Three simulated datasets are available for illustrative purposes. These datasets contain item parameter coefficients and their covariance matrix. In particular, est3pl concerns a three-parameter logistic model, est2pl regards a two-parameter logistic model, and estrasch refers to a Rasch model. The estimates included in est2pl are obtained from the dataset data2pl, also contained in the package. Function dif.test performs a Wald-type test for the detection of DIF (Battauz, 2018). The stability of the equating transformations can be assessed using function sd.test, which tests if the equating coefficients that link the same two forms are equal, and function id.test, which tests if the equating coefficients that link a form to itself through a chain of forms return the identity equating.

Author(s)

Michela Battauz

Maintainer: Michela Battauz <[email protected]>

References

Battauz, M. (2013). IRT Test Equating in Complex Linkage Plans. Psychometrika, 78, 464–480.

Battauz, M. (2015). equateIRT: An R Package for IRT Test Equating. Journal of Statistical Software, 68, 1–22.

Battauz, M. (2019). On Wald tests for differential item functioning detection. Statistical Methods and Applications. 28, 103–118.

Battauz, M. (2022). Testing for differences in chain equating. Statistica Neerlandica, 1–12.

Cai L. (2013). FlexMIRT version 2: Flexible Multilevel Multidimensional Item Analysis and Test Scoring [Computer Software]. Chapel Hill, NC: Vector Psychometric Group.

Cai, L., du Toit, S. H. C., Thissen, D. (2011). IRTPRO: Flexible, multidimensional, multiple categorical IRT modeling [Computer software]. Chicago: Scientific Software International.

Chalmers, R. P. (2012). mirt: A Multidimensional Item Response Theory Package for the R Environment. Journal of Statistical Software, 48, 1–29.

Holland, P.W. and Strawderman, W.E. (2011). How to average equating functions if you must. In A.A. von Davier (Ed.), Statistical models for test equating, scaling, and linking (pp. 89–107). New York: Springer.

Kolen, M.J. and Brennan, R.L. (2014). Test equating, scaling, and linking: methods and practices, 3rd ed., New York: Springer.

Ogasawara, H. (2000). Asymptotic standard errors of IRT equating coefficients using moments. Economic Review (Otaru University of Commerce), 51, 1–23.

Ogasawara, H. (2001a). Item response theory true score equatings and their standard errors. Journal of Educational and Behavioral Statistics, 26, 31–50.

Ogasawara, H. (2001b). Standard Errors of Item Response Theory Equating/Linking by Response Function Methods. Applied Psychological Measurement, 25, 53–67.

Ogasawara, H. (2003). Asymptotic standard errors of IRT observed-score equating methods. Psychometrika, 68, 193–211.

Rizopoulos, D. (2006). ltm: an R package for latent variable modelling and item response theory analyses. Journal of Statistical Software, 17, 1–25.

---

Direct Equating Coefficients Between All Pairs of a List of Forms

Description

Calculates direct equating coefficients and standard errors using IRT methods between all pairs of a list of forms.

Usage

alldirec(mods, method = "mean-mean", all = FALSE, quadrature = TRUE, nq = 30, 
	direction = "both")

Arguments

mods	an object of the class modIRT containing item parameter coefficients and their covariance matrix of the forms to be equated.
method	the equating method to be used. This should be one of "mean-mean", "mean-gmean", "mean-sigma", "Haebara" or "Stocking-Lord".
all	logical; if FALSE forms that do not have common items will not appear in the output.
quadrature	logical; if TRUE the Gauss-Hermite quadrature is used to approximate the integral in the function that is minimized in the Haebara and Stocking-Lord methods. If FALSE the integral is replaced with a sum over 40 equally spaced values ranging from -4 to 4 with an increment of 0.05 and weights equal to one for all values.
nq	number of quadrature points used for the Gauss-Hermite quadrature if quadrature is TRUE
direction	a character string specifying the direction used to equate pair of forms. If "back" newer forms are equated to older forms, if "forward" older forms are equated to newer forms, if "both" (the default) equating is performed in both directions. Form A is considered to be newer than Form B if it is listed after Form B in mods.

Value

An object of class eqclist consisting in a list with length equal to the number of pairs of forms equated. Each component of the list is an object of class eqc returned by function direc.

Author(s)

Michela Battauz

References

Battauz, M. (2015). equateIRT: An R Package for IRT Test Equating. Journal of Statistical Software, 68, 1–22.

Kolen, M.J. and Brennan, R.L. (2014). Test equating, scaling, and linking: methods and practices, 3rd ed., New York: Springer

Ogasawara, H. (2000). Asymptotic standard errors of IRT equating coefficients using moments. Economic Review (Otaru University of Commerce), 51, 1–23.

Ogasawara, H. (2001). Standard Errors of Item Response Theory Equating/Linking by Response Function Methods. Applied Psychological Measurement, 25, 53–67.

See Also

direc, eqc, itm, modIRT, summary.eqclist

Examples

# the following code can be used to start from item response data

## Not run: 
library("mirt")
data("data2pl")

m1 <- mirt(data2pl[[1]], SE = TRUE)
m2 <- mirt(data2pl[[2]], SE = TRUE)
m3 <- mirt(data2pl[[3]], SE = TRUE)
m4 <- mirt(data2pl[[4]], SE = TRUE)
m5 <- mirt(data2pl[[5]], SE = TRUE)

mlist<- list(m1,m2,m3,m4,m5)
test <- paste("test", 1:5, sep = "")
mod2pl <- modIRT(est.mods = mlist, names = test, display = FALSE)

direclist2pl <- alldirec(mods = mod2pl, method = "Haebara")
summary(direclist2pl)
summary(direclist2pl$test2.test1)

## End(Not run)

# ===========================================================================
# the following code uses item parameter estimates previously obtained

# three-parameter logistic model
# direct equating coefficients using the "Stocking-Lord" method
data(est3pl)
test <- paste("test", 1:5, sep = "")
mod3pl <- modIRT(coef = est3pl$coef, var = est3pl$var, names = test, display = FALSE)
direclist3pl <- alldirec(mods = mod3pl, method = "Stocking-Lord")
summary(direclist3pl)
summary(direclist3pl$test1.test2)

# two-parameter logistic model
# direct equating coefficients using the "Haebara" method
data(est2pl)
test <- paste("test", 1:5, sep = "")
mod2pl <- modIRT(coef = est2pl$coef, var = est2pl$var, names = test, display = FALSE)
direclist2pl <- alldirec(mods = mod2pl, method = "Haebara")
summary(direclist2pl)
summary(direclist2pl$test1.test5)

# Rasch model
# direct equating coefficients using the "mean-mean" method
data(estrasch)
test <- paste("test", 1:5, sep = "")
modrasch <- modIRT(coef = estrasch$coef, var = estrasch$var, names = test, 
	display = FALSE)
direclistrasch <- alldirec(mods = modrasch, method = "mean-mean", all = TRUE)
summary(direclistrasch)
summary(direclistrasch$test5.test4)

---

Bisector Equating Coefficients

Description

Calculates average equating coefficients using the bisector method and standard errors given a set of direct and chain equating coefficients.

Usage

bisectorec(ecall, mods = NULL, weighted = TRUE, unweighted = TRUE)

Arguments

ecall	list of objects of class eqc or ceqc returned by functions direc and chainec.
mods	an object of class modIRT containing item parameter coefficients and their covariance matrix of the forms to be equated. From version 2.0 it can be NULL
weighted	logical; if TRUE weighted bisector coefficients are computed.
unweighted	logical; if TRUE unweighted bisector coefficients are computed.

Value

An object of class meqc with components

coef	data frame containing link, path, coefficient A, coefficient B, standard error of coefficient A (seA), standard error of coefficient B (seB) and weights of direct, chain and bisector equating coefficients.
method	the equating method used.

Author(s)

Michela Battauz

References

Battauz, M. (2013). IRT Test Equating in Complex Linkage Plans. Psychometrika, 78, 464–480.

Battauz, M. (2015). equateIRT: An R Package for IRT Test Equating. Journal of Statistical Software, 68, 1–22.

Holland, P.W. and Strawderman, W.E. (2011). How to average equating functions if you must. In A.A. von Davier (Ed.), Statistical models for test equating, scaling, and linking (pp. 89–107). New York: Springer.

See Also

chainec, convert, direc, eqc, summary.meqc

Examples

# the following code can be used to start from item response data

## Not run: 
library("mirt")
data("data2pl")

m1 <- mirt(data2pl[[1]], SE = TRUE)
m2 <- mirt(data2pl[[2]], SE = TRUE)
m3 <- mirt(data2pl[[3]], SE = TRUE)
m4 <- mirt(data2pl[[4]], SE = TRUE)
m5 <- mirt(data2pl[[5]], SE = TRUE)

mlist<- list(m1,m2,m3,m4,m5)
test <- paste("test", 1:5, sep = "")
mod2pl <- modIRT(est.mods = mlist, names = test, display = FALSE)

direclist2pl <- alldirec(mods = mod2pl, method = "Haebara")

# compute chain equating coefficients for path 1,2,3,4
pth1 <- paste("test", 1:4, sep = "")
chainec1 <- chainec(direclist = direclist2pl, pths = pth1)
# compute chain equating coefficients for path 1,5,4
pth2 <- c(paste("test", c(1,5,4), sep = ""))
chainec2 <- chainec(direclist = direclist2pl, pths = pth2)
# create a list of objects of class ceqc
ecall <- c(chainec1, chainec2)
# compute bisector and weighted bisector coefficients
allec <- bisectorec(ecall = ecall, weighted = TRUE, unweighted = TRUE)
summary(allec)

## End(Not run)

# ===========================================================================
# the following code uses item parameter estimates previously obtained

# three-parameter logistic model
# direct equating coefficients using the "Stocking-Lord" method
data(est3pl)
test <- paste("test", 1:5, sep = "")
mod3pl <- modIRT(coef = est3pl$coef, var = est3pl$var, names = test, display = FALSE)
direclist3pl <- alldirec(mods = mod3pl, method = "Stocking-Lord")
# compute chain equating coefficients for path 1,2,3,4,5
pth3 <- paste("test", 1:5, sep = "")
chainec3 <- chainec(direclist = direclist3pl, pths = pth3)
# create a list of objects of class eqc or ceqc
ecall <- c(chainec3, direclist3pl["test1.test5"])
# compute bisector and weighted bisector coefficients
allec <- bisectorec(ecall = ecall, weighted = TRUE, unweighted = TRUE)
summary(allec)

# two-parameter logistic model
# direct equating coefficients using the "Haebara" method
data(est2pl)
test <- paste("test", 1:5, sep = "")
mod2pl <- modIRT(coef = est2pl$coef, var = est2pl$var, names = test, display = FALSE)
direclist2pl <- alldirec(mods = mod2pl, method = "Haebara")
# compute chain equating coefficients for path 1,2,3,4
pth1 <- paste("test", 1:4, sep = "")
chainec1 <- chainec(direclist = direclist2pl, pths = pth1)
# compute chain equating coefficients for path 1,5,4
pth2 <- c(paste("test", c(1,5,4), sep = ""))
chainec2 <- chainec(direclist = direclist2pl, pths = pth2)
# compute chain equating coefficients for path 1,2,3,4,5
pth3 <- paste("test", 1:5, sep = "")
chainec3 <- chainec(direclist = direclist2pl, pths = pth3)
# create a list of objects of class eqc or ceqc
ecall <- c(chainec1, chainec2, chainec3, direclist2pl["test1.test5"])
# compute bisector and weighted bisector coefficients
allec <- bisectorec(ecall = ecall, weighted = TRUE, unweighted = TRUE)
summary(allec)

# Rasch model
# direct equating coefficients using the "mean-mean" method
data(estrasch)
test <- paste("test", 1:5, sep = "")
modrasch <- modIRT(coef = estrasch$coef, var = estrasch$var, names = test, 
	display = FALSE)
direclistrasch <- alldirec(mods = modrasch, method = "mean-mean", all = TRUE)
# compute chain equating coefficients for path 1,2,3,4
pth1 <- paste("test", 1:4, sep = "")
chainec1 <- chainec(direclist = direclistrasch, pths = pth1)
# compute chain equating coefficients for path 1,5,4
pth2 <- c(paste("test", c(1,5,4), sep = ""))
chainec2 <- chainec(direclist = direclistrasch, pths = pth2)
# create a list of objects of class eqc or ceqc
ecall <- c(chainec1, chainec2)
# compute bisector and weighted bisector coefficients
allec <- bisectorec(ecall = ecall, weighted = TRUE, unweighted = TRUE)
summary(allec)

---

Chain Equating Coefficients

Description

Calculates chain (indirect) equating coefficients and standard errors using IRT methods.

Usage

chainec(r = NULL, direclist, f1 = NULL, f2 = NULL, pths = NULL)

Arguments

r	length of the chain, that is the number of forms used for equating including extremes. It should be at least 3. It does not need to be specified if argument pths is not NULL.
direclist	an object of the class eqclist return by function alldirec containing direct equating coefficients between pairs of forms.
f1	the name of the first form of the chain.
f2	the name of the last form of the chain.
pths	vector, matrix or data frame containing the path used for equating. The number of columns is equal to r and the number of rows is equal to the equatings that have to be performed. If NULL all the chain equating coefficients of length r will be computed.

Details

Equating coefficients perform the conversion from the scale of the first form to the scale of the last form of the path.

Value

An object of class ceqclist consisting in a list with length equal to the number of chain equating coefficients computed. Each component of the list is an object of class ceqc with components

tab1	item parameters of the first form.
tab2	item parameters of the last form.
tab	Data frame containing item names (Item), item parameters of the first form (e.g. test1), item parameters of the last form (e.g. test3), and item parameters of the first form converted in the scale of the last form (e.g. test1.as.test3).
varAll	covariance matrix of item parameters of all forms used in the chain.
partial	partial derivatives of equating coefficients A and B with respect to item parameters.
A	equating coefficient A.
B	equating coefficient B.
varAB	covariance matrix of the equating coefficients.
commonitem	list of length r-1 containing the names of common item parameters between adjacent forms.
ni	vector containing number of common items between pairs of adjacent forms.
forms	names of equated forms.
method	the equating method used.
itmp	number of item parameters of the IRT model.

Author(s)

Michela Battauz

References

Battauz, M. (2013). IRT Test Equating in Complex Linkage Plans. Psychometrika, 78, 464–480.

Battauz, M. (2015). equateIRT: An R Package for IRT Test Equating. Journal of Statistical Software, 68, 1–22.

Kolen, M.J. and Brennan, R.L. (2014). Test equating, scaling, and linking: methods and practices, 3rd ed., New York: Springer

See Also

alldirec, eqc, itm, summary.ceqc, summary.ceqclist

Examples

# the following code can be used to start from item response data

## Not run: 
library("mirt")
data("data2pl")

m1 <- mirt(data2pl[[1]], SE = TRUE)
m2 <- mirt(data2pl[[2]], SE = TRUE)
m3 <- mirt(data2pl[[3]], SE = TRUE)
m4 <- mirt(data2pl[[4]], SE = TRUE)
m5 <- mirt(data2pl[[5]], SE = TRUE)

mlist<- list(m1,m2,m3,m4,m5)
test <- paste("test", 1:5, sep = "")
mod2pl <- modIRT(est.mods = mlist, names = test, display = FALSE)

direclist2pl <- alldirec(mods = mod2pl, method = "Haebara")

# compute chain equating coefficients for path 1,2,3,4
pth1 <- paste("test", 1:4, sep = "")
chainec1 <- chainec(direclist = direclist2pl, pths = pth1)
summary(chainec1)

## End(Not run)

# ===========================================================================
# the following code uses item parameter estimates previously obtained

# three-parameter logistic model
# direct equating coefficients using the "Stocking-Lord" method
data(est3pl)
test<-paste("test", 1:5, sep = "")
mod3pl <- modIRT(coef = est3pl$coef, var = est3pl$var, names = test, display = FALSE)
direclist3pl <- alldirec(mods = mod3pl, method = "Stocking-Lord")
# compute all chain equating coefficients of length 4
chainec4 <- chainec(r = 4, direclist = direclist3pl)
summary(chainec4)
summary(chainec4$test1.test2.test3.test4)
# compute all chain equating coefficients of length 4 
# where the first form is test1
chainec4.1 <- chainec(r = 4, direclist = direclist3pl, f1 = "test1")
summary(chainec4.1)
# compute all chain equating coefficients of length 4 
# where the first form is test1 and the last form is test4
chainec4.14 <- chainec(r = 4, direclist = direclist3pl, f1 = "test1", f2 = "test4")
summary(chainec4.14)


# two-parameter logistic model
# direct equating coefficients using the "Haebara" method
data(est2pl)
test<-paste("test", 1:5, sep = "")
mod2pl <- modIRT(coef = est2pl$coef, var = est2pl$var, names = test, display = FALSE)
direclist2pl <- alldirec(mods = mod2pl, method = "Haebara")
# compute chain equating coefficients of a given path
pth <- paste("test", c(1,5,4), sep = "")
chainec154 <- chainec(direclist = direclist2pl, pths = pth)
summary(chainec154)


# Rasch model
# direct equating coefficients using the "mean-mean" method
data(estrasch)
test<-paste("test", 1:5, sep = "")
modrasch <- modIRT(coef = estrasch$coef, var = estrasch$var, names = test, 
	display = FALSE)
direclistrasch <- alldirec(mods = modrasch, method = "mean-mean", all = TRUE)
# compute chain equating coefficients of two given paths
pth1 <- paste("test", 1:3, sep = "")
pth2 <- paste("test", c(1,5,4), sep = "")
pths <- rbind(pth1, pth2)
chainec1 <- chainec(direclist = direclistrasch, pths = pths)
summary(chainec1)

---

Item Parameters Conversion

Description

Converts item and person parameters using equating coefficients.

Usage

convert(A, B, coef = NULL, person.par = NULL)

Arguments

A	equating coefficient A.
B	equating coefficient B.
coef	vector of item parameters return by function modIRT.
person.par	vector of person parameters estimates.

Details

Difficulty parameters $b$ are converted using transformation $b \cdot A+B$; discrimination parameters $a$ are converted using transformation $a/A$; guessing parameters c are not transformed.

Person parameters $\theta$ are converted using transformation $\theta \cdot A+B$.

Value

A list with components

coef	vector of item parameters transformed.
person.par	vector of person parameters transformed

Author(s)

Michela Battauz

References

Battauz, M. (2015). equateIRT: An R Package for IRT Test Equating. Journal of Statistical Software, 68, 1–22.

Kolen, M.J. and Brennan, R.L. (2014). Test equating, scaling, and linking: methods and practices, 3rd ed., New York: Springer

See Also

itm, modIRT

Examples

# conversion using direct coefficients
# three-parameter logistic model
# direct equating coefficients between forms 1 and 2 using the Stocking-Lord method
data(est3pl)
test <- paste("test", 1:5, sep = "")
mod3pl <- modIRT(coef = est3pl$coef, var = est3pl$var, names = test, display = FALSE)
l12 <- direc(mod1 = mod3pl[1], mod2 = mod3pl[2], method = "Stocking-Lord")
convert(A = l12$A, B = l12$B, coef = coef(mod3pl$test1))
# the conversion of item parameters is obtained also using
itm(l12)

# conversion using bisector coefficients
# two-parameter logistic model
# direct equating coefficients using the "Haebara" method
data(est2pl)
test <- paste("test", 1:5, sep = "")
mod2pl <- modIRT(coef = est2pl$coef, var = est2pl$var, names = test, display = FALSE)
direclist2pl <- alldirec(mods = mod2pl, method = "Haebara")
# compute chain equating coefficients for path 1,2,3,4
pth1 <- paste("test", 1:4, sep = "")
chainec1 <- chainec(direclist = direclist2pl, pths = pth1)
# compute chain equating coefficients for path 1,5,4
pth2 <- c(paste("test", c(1,5,4), sep = ""))
chainec2 <- chainec(direclist = direclist2pl, pths = pth2)
# create a list of objects of class ceqc
ecall <- c(chainec1, chainec2)
# compute bisector and weighted bisector coefficients
allec <- bisectorec(ecall = ecall, weighted = TRUE, unweighted = TRUE)
summary(allec)
eqc14<-eqc(allec, link = "test1.test4", path = "bisector")
convert(A = eqc14$A, B = eqc14$B, coef = coef(mod2pl$test1), person.par = seq(-3, 3, 0.5))

---

Simulated Data Sets

Description

Five simulated data sets from a two-parameter logistic model.

Usage

data(data2pl)

Format

A list of length 5, containing 5 data frames with 5000 dichotomous responses to 20 items.

Author(s)

Michela Battauz

See Also

est2pl, import.ltm

Examples

data(data2pl)
data2pl[[1]][1:3,]

---

Simulated Data Set with DIF

Description

Item responses and group membership.

Usage

data(dataDIF)

Format

A dataframe with 3000 dichotomous responses to 20 items (I01 to I20) and group membership (group).

Author(s)

Michela Battauz

See Also

dif.test

Examples

data(dataDIF)

---

DIF Test

Description

Performs a Wald-type test for Differential Item Functioning detection.

Usage

dif.test(est.mods = NULL, coef = NULL, var = NULL, names = NULL, 
  reference = NULL, method = "mean-mean",
  quadrature = TRUE, nq = 30, DIFtype = NULL, purification = FALSE, 
  signif.level = 0.05, trace = FALSE, maxiter = 30, anchor = NULL)

Arguments

est.mods	list of output objects from functions rasch, ltm, or tpm of the ltm package or from function mirt of the mirt package.
coef	list of matrices (one for each group) containing the item parameter estimates. Guessing, difficulty and discrimination parameters should strictly be given in this order and they are contained in different columns of the matrix. The names of the rows of each matrix should be the names of the items.
var	list of matrices (one for each group) containing the covariance matrix of item parameter estimates. They should be given in the same order of coefficients.
names	character vector containing the names of the groups. This should have the same length of coef and var. If NULL, the names of the groups will be "T1", "T2", ...
reference	reference group. Can be specified by name or number. The default is the first group.
method	the equating method to be used in function direc to convert the item parameters to the scale of the reference group. This should be one of "mean-mean", "mean-gmean", "mean-sigma", "Haebara" or "Stocking-Lord".
quadrature	logical; if TRUE the Gauss-Hermite quadrature is used in function direc to approximate the integral in the function that is minimized in the Haebara and Stocking-Lord methods. If FALSE the integral is replaced with a sum over 40 equally spaced values ranging from -4 to 4 with an increment of 0.05 and weights equal to one for all values.
nq	number of quadrature points used for the Gauss-Hermite quadrature if quadrature is TRUE.
DIFtype	character indicating which parameters to test for DIF. If NULL all parameters are tested for DIF. Use "beta1" for $\beta_1$, "beta2" for $\beta_2$, "beta3" for $\beta_3$, "beta12" for $\beta_1$ and $\beta_2$, "beta123" for $\beta_1$, $\beta_2$ and $\beta_3$. See details.
purification	logical. if TRUE the procedure described in Candell and Drasgow (1988) is applied.
signif.level	significance level to use in the purification process.
trace	logical. If TRUE tracing information is produced.
maxiter	The maximum number of iterations in the purification process.
anchor	Optional character vector containing the names of the items to use for equating. These should be items free of DIF.

Details

The parameterization of the IRT model is that commonly used for estimation. Under this parameterization, the three-parameter logistic model is as follows

$\pi_i = c_i + (1 - c_i) \frac{\exp(\beta_{1i} + \beta_{2i} z)}{1 + \exp(\beta_{1i} + \beta_{2i} z)},$

where $\pi_i$ denotes the conditional probability of responding correctly to the $i$th item given $z$, $c_i$ denotes the guessing parameter, $\beta_{1i}$ is the easiness parameter, $\beta_{2i}$ is the discrimination parameter, and $z$ denotes the latent ability. Furthermore, the guessing parameter is equal to

$c_i = \frac{\exp(\beta_{3i})}{1+\exp(\beta_{3i})}$

.

The test verifies whether the item parameters $\beta_{1i}, \beta_{2i}, \beta_{3i}$ are invariant across two or more groups as explained in Battauz (2018).

Value

An object of class dift with components

test	matrix containing the test statistic and the p-value for each item. "noGuess" is equal to 1 if the guessing parameter of a 3PL model was set to a fixed value.
eqmet	the equating method used.
DIFtype	character indicating which parameters were tested for DIF.
reference	the reference group.
focal	the focal groups.
names	names of the groups.
purification	logical. If TRUE thre purification procedure was applied.
signif.level	significance level used in the purification process
equatings	list containing the output of function direc.
coef_trasf	list containing the item parameters of each group transformed to the scale of the reference group.
var_trasf	list containing the covariance matrix of item parameters of each group transformed to the scale of the reference group.
items.dif	names of the items for which the null hypothesis of the test is rejected.
anchor	names of the items used as anchors.
niter	number of iterations.

Author(s)

Michela Battauz

References

Battauz, M. (2019). On Wald tests for differential item functioning detection. Statistical Methods and Applications. 28, 103-118.

Candell, G.L., Drasgow, F. (1988). An iterative procedure for linking metrics and assessing item bias in item response theory. Applid Psychological Measurement, 12, 253-260.

Examples

# load the data
data(dataDIF)
head(dataDIF)
# estimate a 2PL model for each group using the R package mirt
library(mirt)
data1 <- dataDIF[dataDIF$group == 1, 1:20]
data2 <- dataDIF[dataDIF$group == 2, 1:20]
data3 <- dataDIF[dataDIF$group == 3, 1:20]
mod1 <- mirt(data1, SE = TRUE)
mod2 <- mirt(data2, SE = TRUE)
mod3 <- mirt(data3, SE = TRUE)

# since package version 2.5.0 it is possible to skip function import.mirt
# as follows

# perform the test for DIF on two groups
res_diftest2 <- dif.test(est.mods = list(mod1, mod2))
res_diftest2

# perform the test for DIF on three groups
res_diftest3 <- dif.test(est.mods = list(mod1, mod2, mod3))
res_diftest3

# perform the test for DIF on three groups
# reference group: 2
# equating method: Haebara
# purification applied
res_diftest3 <- dif.test(est.mods = list(mod1, mod2, mod3), 
  reference = 2, method = "Haebara", purification = TRUE)
res_diftest3

---

Direct Equating Coefficients

Description

Calculates direct equating coefficients and standard errors using IRT methods.

Usage

direc(mods, which, mod1, mod2, method = "mean-mean", suff1 = ".1", suff2 = ".2",
   D = 1, quadrature = TRUE, nq = 30, items.select = NULL)

Arguments

mods	an object of the class modIRT containing item parameter coefficients and their covariance matrix of the forms to be equated.
which	which forms to equate. Can be specified by name or number.
mod1	deprecated; please use mods instead. An object of the class modIRT containing item parameter coefficients and their covariance matrix of the first form.
mod2	deprecated; please use mods instead. An object of the class modIRT containing item parameter coefficients and their covariance matrix of the second form.
method	the equating method to be used. This should be one of "mean-mean", "mean-gmean", "mean-sigma", "Haebara" or "Stocking-Lord".
suff1	suffix to identify the first form to be equated.
suff2	suffix to identify the second form to be equated.
D	constant D of the IRT model used to estimate item parameters. See below for more details.
quadrature	logical; if TRUE the Gauss-Hermite quadrature is used to approximate the integral in the function that is minimized in the Haebara and Stocking-Lord methods. If FALSE the integral is replaced with a sum over 40 equally spaced values ranging from -4 to 4 with an increment of 0.05 and weights equal to one for all values.
nq	number of quadrature points used for the Gauss-Hermite quadrature if quadrature is TRUE.
items.select	optional character vector including the names of the items to use for equating.

Details

Equating coefficients perform the conversion from the scale of the first form to the scale of the second form.

In the three-parameter logistic model the probability of a positive response on item $i$ is

$\pi_i = c_i + (1 - c_i) \frac{\exp[D a_i (\theta - b_i)]}{1 + \exp[D a_i (\theta - b_i)]}.$

where $a_i$ is the item discrimination parameter, $b_i$ is the item difficulty parameter, $c_i$ is the item guessing parameter and $\theta$ is the latent ability. The constant $D$ can be specified using argument D of the direc function. The two-parameter logistic model can be obtained by setting $c_i$ equal to zero, the one-parameter logistic model can be obtained by setting $c_i$ equal to zero and $a_i$ costant across items, while the Rasch model can be obtained by setting $c_i$ equal to zero and $a_i$ equal to 1.

The type of IRT model does not need to be specified as it is obtained from arguments mod1 and mod2.

Value

An object of class eqc with components

tab1	item parameters of the first form.
tab2	item parameters of the second form.
tab	Data frame containing item names (Item), item parameters of the first form (e.g. test1), item parameters of the second form (e.g. test2), and item parameters of the first form converted in the scale of the second form (e.g. test1.as.test2).
var12	covariance matrix of item parameters of the first and the second form (only items used for equating).
varFull	list of covariance matrices of the item parameters (one matrix for each form)
partial	partial derivatives of equating coefficients A and B with respect to item parameters.
A	equating coefficient A.
B	equating coefficient B.
varAB	covariance matrix of the equating coefficients.
commonitem	list of length 2 containing the names of common item parameters and the names of the common items selected for equating.
suffixes	suffixes used to identify the forms.
ni	number of common items.
nis	number of common items selected for equating.
forms	names of equated forms.
method	the equating method used.
itmp	number of item parameters of the IRT model.

Author(s)

Michela Battauz

References

Battauz, M. (2015). equateIRT: An R Package for IRT Test Equating. Journal of Statistical Software, 68, 1–22.

Kolen, M.J. and Brennan, R.L. (2014). Test equating, scaling, and linking: methods and practices, 3nd ed., New York: Springer

Ogasawara, H. (2000). Asymptotic standard errors of IRT equating coefficients using moments. Economic Review (Otaru University of Commerce), 51, 1–23.

Ogasawara, H. (2001). Standard Errors of Item Response Theory Equating/Linking by Response Function Methods. Applied Psychological Measurement, 25, 53–67.

See Also

eqc, itm, modIRT, summary.eqc

Examples

# the following code can be used to start from item response data

## Not run: 
library("mirt")
data("data2pl")

m1 <- mirt(data2pl[[1]], SE = TRUE)
m2 <- mirt(data2pl[[2]], SE = TRUE)
m3 <- mirt(data2pl[[3]], SE = TRUE)
m4 <- mirt(data2pl[[4]], SE = TRUE)
m5 <- mirt(data2pl[[5]], SE = TRUE)

mlist<- list(m1,m2,m3,m4,m5)
test <- paste("test", 1:5, sep = "")
mod2pl <- modIRT(est.mods = mlist, names = test, display = FALSE)

# two-parameter logistic model
# direct equating coefficients between forms 2 and 3 using the Haebara method
l23 <- direc(mods = mod2pl, which = c(2,3), method = "Haebara")
summary(l23)

## End(Not run)

# ===========================================================================
# the following code uses item parameter estimates previously obtained

# three-parameter logistic model
# direct equating coefficients between forms 1 and 2 using the Stocking-Lord method
data(est3pl)
test <- paste("test", 1:5, sep = "")
mod3pl <- modIRT(coef = est3pl$coef, var = est3pl$var, names = test, display = FALSE)
l12 <- direc(mods = mod3pl, which = c("test1", "test2"), method = "Stocking-Lord")
summary(l12)

# two-parameter logistic model
# direct equating coefficients between forms 1 and 5 using the Haebara method
data(est2pl)
test <- paste("test", 1:5, sep = "")
mod2pl <- modIRT(coef = est2pl$coef, var = est2pl$var, names = test, display = FALSE)
l15 <- direc(mods = mod2pl, which = c(1,5), method = "Haebara")
summary(l15)

# Rasch model
# direct equating coefficients between forms 5 and 4 using the mean-mean method
data(estrasch)
test <- paste("test", 1:5, sep = "")
modrasch <- modIRT(coef = estrasch$coef, var = estrasch$var, names = test, 
	display = FALSE)
l54 <- direc(mods = modrasch, which = c(5,4), method = "mean-mean")
summary(l54)

---

Extract Equating Coefficients

Description

eqc is a generic function which extracts the equating coefficients.

Usage

eqc(x, ...)

## S3 method for class 'eqc'
eqc(x, ...)

## S3 method for class 'eqclist'
eqc(x, link = NULL, ...)

## S3 method for class 'ceqc'
eqc(x, ...)

## S3 method for class 'ceqclist'
eqc(x, link = NULL, path = NULL, ...)

## S3 method for class 'meqc'
eqc(x, link = NULL, path = NULL, ...)

Arguments

x	object of the class eqc returned by function direc or of the class eqclist returned by function alldirec or of the class ceqc and ceqclist returned by function chainec or of the class meqc returned by function bisectorec.
link	a character string with the names of the two forms being linked separated by a dot (e.g. "test1.test3").
path	a character string with the names of the forms that constitute the path separated by a dot (e.g. "test1.test2.test3").
...	further arguments passed to or from other methods.

Value

A data frame containing the equating coefficients for every link and path.

Author(s)

Michela Battauz

See Also

alldirec, bisectorec, chainec, direc, modIRT

Examples

# the following code uses item parameter estimates previously obtained
# for examples that start from item response data see function modIRT

# two-parameter logistic model
data(est2pl)
test<-paste("test", 1:5, sep = "")
mod2pl <- modIRT(coef = est2pl$coef, var = est2pl$var, names = test, display = FALSE)
# direct equating coefficients between forms 1 and 2 using the Haebara method
l12 <- direc(mods = mod2pl, which = c(1,2), method = "Haebara")
# all direct equating coefficients using the Haebara method
direclist2pl <- alldirec(mods = mod2pl, method = "Haebara")
# compute all chain equating coefficients of length 3
chainec3 <- chainec(r = 3, direclist = direclist2pl)
# compute chain equating coefficients for path 1,2,3,4
pth1 <- paste("test", 1:4, sep = "")
chainec1 <- chainec(direclist = direclist2pl, pths = pth1)
# compute chain equating coefficients for path 1,5,4
pth2 <- c(paste("test", c(1,5,4), sep = ""))
chainec2 <- chainec(direclist = direclist2pl, pths = pth2)
# create a list of objects of class ceqc
ecall <- c(chainec1, chainec2)
# compute bisector and weighted bisector coefficients
allec <- bisectorec(ecall = ecall, weighted = TRUE, unweighted = TRUE)

# extract equating coefficients
eqc(l12)
eqc(direclist2pl)
eqc(direclist2pl, link = "test1.test2")
eqc(chainec3)
eqc(chainec3, link = "test1.test3")
eqc(allec)
eqc(allec, path = "bisector")

---

Item Parameter Estimates and Covariance Matrices of a Two-Parameter Logistic Model

Description

This dataset includes item parameter estimates and covariance matrices of a two-parameter logistic model applied to 5 simulated datasets with common items. The dichotomous item responses can be found in the dataset data2pl. See details for more information on the linkage plan.

Usage

data(est2pl)

Format

A list of length 2 with components:

coef

a list of length 5 containing the matrices of item parameter estimates. Each matrix presents 2 columns; the first column contains difficulty parameters and the second column contains discrimination parameters. See details for information on the parameterization used. Names of rows correspond to the names of the items.

var

a list of length 5 containing the covariance matrices of item parameter estimates.

Details

Every form is composed by 20 items and presents 10 items in common with adjacent forms. Furthermore, forms 1 and 5 present 10 common items. Use linkp to obtain a matrix with elements equal to the number of common items between different forms.

Item parameters are given under the parameterization used in the ltm package. Under this parameterization, the two-parameter logistic model is as follows

$\pi_i = \frac{\exp(\beta_{1i} + \beta_{2i} z)}{1 + \exp(\beta_{1i} + \beta_{2i} z)},$

where $\pi_i$ denotes the conditional probability of responding correctly to the $i$th item given $z$, $\beta_{1i}$ is the easiness parameter, $\beta_{2i}$ is the discrimination parameter, and $z$ denotes the latent ability.

Author(s)

Michela Battauz

See Also

data2pl, linkp, modIRT

Examples

data(est2pl)
est2pl$coef
est2pl$var
linkp(coef = est2pl$coef)

---

Item Parameter Estimates and Covariance Matrices of a Three-Parameter Logistic Model

Description

This dataset includes item parameter estimates and covariance matrices of a three-parameter logistic model applied to 5 simulated datasets with common items. See details for more information on the linkage plan.

Usage

data(est3pl)

Format

A list of length 2 with components:

coef

a list of length 5 containing the matrices of item parameter estimates. Each matrix presents 3 columns; the first column contains guessing parameters, the second column contains difficulty parameters and the third column contains discrimination parameters. See details for information on the parameterization used. Names of rows correspond to the names of the items.

var

a list of length 5 containing the covariance matrices of item parameter estimates.

Details

Every form is composed by 20 items and presents 10 items in common with adjacent forms. Furthermore, forms 1 and 5 present 10 common items. Use linkp to obtain a matrix with elements equal to the number of common items between different forms.

Item parameters are given under the parameterization used in the ltm package. Under this parameterization, the three-parameter logistic model is as follows

$\pi_i = c_i + (1 - c_i) \frac{\exp(\beta_{1i} + \beta_{2i} z)}{1 + \exp(\beta_{1i} + \beta_{2i} z)},$

where $\pi_i$ denotes the conditional probability of responding correctly to the $i$th item given $z$, $c_i$ denotes the guessing parameter, $\beta_{1i}$ is the easiness parameter, $\beta_{2i}$ is the discrimination parameter, and $z$ denotes the latent ability. Furthermore, the guessing parameters are given under this parameterization

$c_i = \frac{\exp(c_i^*)}{1+\exp(c_i^*)}.$

Author(s)

Michela Battauz

See Also

linkp, modIRT

Examples

data(est3pl)
est3pl$coef
est3pl$var
linkp(coef = est3pl$coef)

---

Item Parameter Estimates and Covariance Matrices of a Rasch Model

Description

This dataset includes item parameter estimates and covariance matrices of a Rasch model applied to 5 simulated datasets with common items. See details for more information on the linkage plan.

Usage

data(estrasch)

Format

A list of length 2 with components:

coef

a list of length 5 containing the matrices of item parameter estimates. Each matrix presents 2 columns; the first column contains difficulty parameters and the second column is equal to 1. See details for information on the parameterization used. Names of rows correspond to the names of the items.

var

a list of length 5 containing the covariance matrices of item parameter estimates.

Details

Every form is composed by 20 items and presents 10 items in common with adjacent forms. Furthermore, forms 1 and 5 present 10 common items. Use linkp to obtain a matrix with elements equal to the number of common items between different forms.

Item parameters are given under the parameterization used in the ltm package. Under this parameterization, the Rasch model is as follows

$\pi_i = \frac{\exp(\beta_{1i} + z)}{1 + \exp(\beta_{1i} + z)},$

where $\pi_i$ denotes the conditional probability of responding correctly to the $i$th item given $z$, $\beta_{1i}$ is the easiness parameter, and $z$ denotes the latent ability.

Author(s)

Michela Battauz

See Also

linkp, modIRT

Examples

data(estrasch)
estrasch$coef
estrasch$var
linkp(coef = estrasch$coef)

---

Identity Test

Description

Performs a statistical test to verify if the chain equating coeffients from one form to itself are A=1 and B=0.

Usage

id.test(chain)

Arguments

chain

an object of the class ceqc returned by function chainec.

Value

An object of class idtest with components

path
AB
statistic
df
p.value

Author(s)

Michela Battauz

References

Battauz, M. (2022). Testing for differences in chain equating. Statistica Neerlandica, 1-12.

See Also

chainec

Examples

# the following code uses item parameter estimates previously obtained
# for other examples see documentation in function chainec

# three-parameter logistic model
# direct equating coefficients using the "Haebara" method
data(est3pl)
test <- paste("test", 1:5, sep = "")
mod3pl <- modIRT(coef = est3pl$coef, var = est3pl$var, names = test, display = FALSE)
direclist3pl <- alldirec(mods = mod3pl, method = "Haebara")
# compute chain equating coefficients for path 1,2,3,4,5,1
pth3 <- paste("test", c(1:5,1), sep = "")
chainec_circle <- chainec(direclist = direclist3pl, pths = pth3)
summary(chainec_circle)
# test
id.test(chainec_circle)

---

Import Item Parameters Estimates and Covariance Matrices from IRT Software

Description

Import estimated item parameters and covariance matrix from the R packages ltm and mirt, and from external software IRTPRO and flexMIRT.

Usage

import.ltm(mod, display = TRUE, digits = 4) 

import.mirt(mod, display = TRUE, digits = 3) 

import.irtpro(fnamep, fnamev = NULL, fnameirt = NULL, display = TRUE, digits = 2) 

import.flexmirt(fnamep, fnamev = NULL, fnameirt = NULL, display = TRUE, digits = 2)

Arguments

mod	output object from functions rasch, ltm, or tpm of the ltm package or from function mirt of the mirt package.
display	logical; if TRUE coefficients and standard errors are printed.
digits	integer indicating the number of decimal places to be used if display is TRUE.
fnamep	name of the file containing the estimated item parameters. Typically, -prm.txt.
fnamev	name of the file containing the covariance matrix of the estimated item parameters. Typically, -cov.txt.
fnameirt	name of the file containing additional information to link item parameters with the covariance matrix. Typically, -irt.txt.

Details

Item parameters are imported with the parameterization used by the software to estimate the IRT model. The usual IRT parameterization can be obtained later by using function modIRT.

Since package version 2.5.0 this function can be skipped if the item parameters are estimated with packages ltm or mirt. See modIRT.

Value

A list with components

coef	item parameter estimates.
var	covariance matrix of item parameter estimates.

Author(s)

Michela Battauz

References

Battauz, M. (2015). equateIRT: An R Package for IRT Test Equating. Journal of Statistical Software, 68, 1–22.

Cai L. (2013). FlexMIRT version 2: Flexible Multilevel Multidimensional Item Analysis and Test Scoring [Computer Software]. Chapel Hill, NC: Vector Psychometric Group.

Cai, L., du Toit, S. H. C., Thissen, D. (2011). IRTPRO: Flexible, multidimensional, multiple categorical IRT modeling [Computer software]. Chicago: Scientific Software International.

Chalmers, R. P. (2012). mirt: A Multidimensional Item Response Theory Package for the R Environment. Journal of Statistical Software, 48, 1–29.

Rizopoulos, D. (2006). ltm: an R package for latent variable modelling and item response theory analyses. Journal of Statistical Software, 17, 1–25.

See Also

modIRT

Examples

#====================
# from package ltm
library(ltm)

# one-parameter logistic model
mod1pl <- rasch(LSAT)
est.mod1pl <- import.ltm(mod1pl)
est.mod1pl

# two-parameter logistic model
mod2pl <- ltm(LSAT ~ z1)
est.mod2pl <- import.ltm(mod2pl)
est.mod2pl

#====================
# from package mirt
library(mirt)

# one-parameter logistic model
data(LSAT, package = "ltm")
val <- mirt(LSAT, 1, SE = TRUE, pars = "values")
cnstr <- val[val$name == "a1",]$parnum
mod1pl.m <- mirt(LSAT, 1, SE = TRUE, SE.type = 'Richardson', constrain = list(cnstr))
est.mod1pl.m <- import.mirt(mod1pl.m, digits = 4)
est.mod1pl.m

# two-parameter logistic model
data(LSAT, package = "ltm")
mod2pl.m <- mirt(LSAT, 1, SE = TRUE, SE.type = 'Richardson')
est.mod2pl.m <- import.mirt(mod2pl.m, display = FALSE)
est.mod2pl.m

---

Extract Item Parameters

Description

itm is a generic function which extracts a data frame containing the item parameters of two forms being equated in the original scale and item parameters of the first form converted in the scale of the second form.

Usage

itm(x, ...)

## S3 method for class 'eqc'
itm(x, ...)

## S3 method for class 'eqclist'
itm(x, link, ...)

## S3 method for class 'ceqc'
itm(x, ...)

## S3 method for class 'ceqclist'
itm(x, path, ...)

## S3 method for class 'meqc'
itm(x, link, bistype = NULL, ...)

Arguments

x	object of the class eqc returned by function direc or of the class eqclist returned by function alldirec or of the class ceqc and ceqclist returned by function chainec or of the class meqc return by function bisectorec.
link	a character string with the names of the two forms being linked separated by a dot (e.g. "test1.test3").
path	a character string with the names of the forms that constitute the path separated by a dot (e.g. "test1.test2.test3").
bistype	bisector type. It should be weighted or unweighted. It could be NULL if only one type was computed by function bisectorec.
...	further arguments passed to or from other methods.

Value

A data frame containing item names (Item), item parameters of the first form (e.g. test1), item parameters of the last form (e.g. test3), and item parameters of the first form converted in the scale of the last form (e.g. test1.as.test3).

Author(s)

Michela Battauz

See Also

convert, alldirec, bisectorec, chainec, direc, modIRT

Examples

# the following code uses item parameter estimates previously obtained
# for examples that start from item response data see function modIRT

# two-parameter logistic model
data(est2pl)
test<-paste("test", 1:5, sep = "")
mod2pl <- modIRT(coef = est2pl$coef, var = est2pl$var, names = test, display = FALSE)
# direct equating coefficients between forms 1 and 2 using the Haebara method
l12 <- direc(mods = mod2pl, which = c(1,2), method = "Haebara")
# all direct equating coefficients using the Haebara method
direclist2pl <- alldirec(mods = mod2pl, method = "Haebara")
# compute all chain equating coefficients of length 3
chainec3 <- chainec(r = 3, direclist = direclist2pl)
# compute chain equating coefficients for path 1,2,3,4
pth1 <- paste("test", 1:4, sep = "")
chainec1 <- chainec(direclist = direclist2pl, pths = pth1)
# compute chain equating coefficients for path 1,5,4
pth2 <- c(paste("test", c(1,5,4), sep = ""))
chainec2 <- chainec(direclist = direclist2pl, pths = pth2)
# create a list of objects of class ceqc
ecall <- c(chainec1, chainec2)
# compute bisector and weighted bisector coefficients
allec <- bisectorec(ecall = ecall, weighted = TRUE, unweighted = TRUE)

# extract item parameters
itm(l12)
itm(direclist2pl, link = "test1.test2")
itm(chainec3, path = "test1.test2.test3")
itm(allec, bistype = "weighted")

---

Linkage Plan

Description

Calculates the number of common items between a list of forms.

Usage

linkp(mods = NULL, coef = NULL)

Arguments

mods	an object of the class modIRT containing item parameter coefficients and their covariance matrix of the forms to be equated.
coef	list of matrices (one for each form) containing item parameter estimates. The names of the rows of each matrix should be the names of the items.

Value

A matrix whose elements indicate the number of common items between the forms. On the diagonal of the matrix there are the number of items of each form.

Author(s)

Michela Battauz

See Also

est2pl, est3pl, estrasch

Examples

## Not run: 
library("mirt")
data("data2pl")

m1 <- mirt(data2pl[[1]], SE = TRUE)
m2 <- mirt(data2pl[[2]], SE = TRUE)
m3 <- mirt(data2pl[[3]], SE = TRUE)
m4 <- mirt(data2pl[[4]], SE = TRUE)
m5 <- mirt(data2pl[[5]], SE = TRUE)

mlist<- list(m1,m2,m3,m4,m5)
test <- paste("test", 1:5, sep = "")
mod2pl <- modIRT(est.mods = mlist, names = test, display = FALSE)
linkp(mod2pl)

## End(Not run)

# ===========================================================================
# the following code uses item parameter estimates previously obtained

data(est3pl)
linkp(coef = est3pl$coef)

---

Estimated Coefficients and Covariance Matrix of IRT Models

Description

Creates an object of the class modIRT containing estimated coefficients and covariance matrices of IRT models. Rasch, one-parameter logistic, two-parameter logistic and three-parameter logistic models are included.

Usage

modIRT(est.mods = NULL, coef = NULL, var = NULL, names = NULL, ltparam = TRUE, 
lparam = TRUE, display = TRUE, digits = 2)

Arguments

est.mods	list of output objects from functions rasch, ltm, or tpm of the ltm package or from function mirt of the mirt package.
coef	necessary only if est.mods is NULL. List of matrices (one for each form) containing item parameter estimates. Guessing, difficulty and discrimination parameters should strictly be given in this order and they are contained in different columns of the matrix. The names of the rows of each matrix should be the names of the items.
var	necessary only if est.mods is NULL. list of matrices (one for each form) containing the covariance matrix of item parameter estimates. They should be given in the same order of coefficients.
names	character vector containing the names of the forms. This should have the same length of est.mods, coef and var. If NULL, the names of the forms are assigned by function modIRT.
ltparam	logical; if TRUE the latent trait parameterization is used for difficulty parameters and the modIRT function performs a transformation of item parameters to return them in the usual IRT parameterization. Set to FALSE to avoid transformations. Not necessary if est.mods is not NULL. See below for more details.
lparam	logical; if TRUE the logistic parameterization is used for guessing parameters and the modIRT function performs a transformation of item parameters to return them in the usual IRT parameterization. Set to FALSE to avoid transformations. Not necessary if est.mods is not NULL. See below for more details.
display	logical; if TRUE coefficients and standard errors are printed.
digits	integer indicating the number of decimal places to be used if display is TRUE.

Details

Since package version 2.5.0 this function takes as input est.mods.

ltparam and lparam refers the the parameterization used by the software used to estimate item parameters. The R package ltm, and the programs IRTPRO and flexMIRT use these parameterizations. If ltparam is TRUE the latent trait parameterization is used. Under this parameterization, the three-parameter logistic model is as follows

$\pi_i = c_i + (1 - c_i) \frac{\exp(\beta_{1i} + \beta_{2i} z)}{1 + \exp(\beta_{1i} + \beta_{2i} z)},$

where $\pi_i$ denotes the conditional probability of responding correctly to the $i$th item given $z$, $c_i$ denotes the guessing parameter, $\beta_{1i}$ is the easiness parameter, $\beta_{2i}$ is the discrimination parameter, and $z$ denotes the latent ability. The two-parameter logistic model, the one-parameter logistic model and the Rasch model present the same formulation. The two-parameter logistic model can be obtained by setting $c_i$ equal to zero, the one-parameter logistic model can be obtained by setting $c_i$ equal to zero and $\beta_{2i}$ costant across items, while the Rasch model can be obtained by setting $c_i$ equal to zero and $\beta_{2i}$ equal to 1.

If lparam is TRUE the guessing parameters are given under this parameterization

$c_i = \frac{\exp(c_i^*)}{1+\exp(c_i^*)}.$

The modIRT function returns parameter estimates under the usual IRT parameterization, that is,

$\pi_i = c_i + (1 - c_i) \frac{\exp[D a_i (\theta - b_i)]}{1 + \exp[D a_i (\theta - b_i)]},$

where $D a_i = \beta_{2i}$, $b_i = -\beta_{1i}/\beta_{2i}$ and $\theta = z$.

If ltparam or lparam are TRUE, the covariance matrix is calculated using the delta method.

If item parameters are already given under the usual IRT parameterization, arguments ltparam and lparam should be set to FALSE.

Value

An object of class modIRT consisting in a list with length equal to the number of forms containing lists with components

coefficients	item parameter estimates.
var	covariance matrix of item parameter estimates.
itmp	number of item parameters of the IRT model. This is 1 for the Rasch model, 2 for the one-parameter logistic model with constant discriminations, 2 for the two-parameter logistic model and 3 for the three-parameter logistic model.

Author(s)

Michela Battauz

References

Battauz, M. (2015). equateIRT: An R Package for IRT Test Equating. Journal of Statistical Software, 68, 1–22.

Bartholomew, D., Knott, M. and Moustaki, I. (2011) Latent Variable Models and Factor Analysis: a Unified Approach, 3rd ed. Wiley.

Rizopoulos, D. (2006). ltm: an R package for latent variable modelling and item response theory analyses. Journal of Statistical Software, 17, 1–25.

See Also

direc, import.ltm

Examples

# the following code can be used to estimate the item parameters and then 
# extract the coefficients and their covariance matrices
# skipping functions import.mirt or import.ltm
## Not run: 
library("mirt")
data("data2pl")

m1 <- mirt(data2pl[[1]], SE = TRUE)
m2 <- mirt(data2pl[[2]], SE = TRUE)
m3 <- mirt(data2pl[[3]], SE = TRUE)
m4 <- mirt(data2pl[[4]], SE = TRUE)
m5 <- mirt(data2pl[[5]], SE = TRUE)

mlist<- list(m1,m2,m3,m4,m5)
test <- paste("test", 1:5, sep = "")
mod2pl <- modIRT(est.mods = mlist, names = test, display = FALSE)

## End(Not run)

# ===========================================================================
# the following code uses item parameter estimates previously obtained

# three-parameter logistic model
data(est3pl) # includes item parameter estimates
test <- paste("test", 1:5, sep = "")
mod3pl <- modIRT(coef = est3pl$coef, var = est3pl$var, names = test, display = FALSE)

# two-parameter logistic model
data(est2pl) # includes item parameter estimates
test <- paste("test", 1:5, sep = "")
mod2pl <- modIRT(coef = est2pl$coef, var = est2pl$var, names = test, display = FALSE)

# Rasch model
data(estrasch) # includes item parameter estimates
test <- paste("test", 1:5, sep = "")
modrasch <- modIRT(coef = estrasch$coef, var = estrasch$var, names = test, 
	display = FALSE)

# one-parameter logistic model imported from the R package ltm
library(ltm)
mod1pl <- rasch(LSAT)
summary(mod1pl)
est.mod1pl <- import.ltm(mod1pl)
mod1pl.ltm <- modIRT(coef = list(est.mod1pl$coef), var = list(est.mod1pl$var), digits = 4)

---

Scoring

Description

Relates number-correct scores on two forms.

Usage

score(obj, ...)

## S3 method for class 'eqc'
score(obj, method = "TSE", D = 1, scores = NULL, se = TRUE, nq = 30, 
  w = 0.5, theta = NULL, weights = NULL, anchor.type = "internal", ...)

## S3 method for class 'eqclist'
score(obj, link = NULL, method = "TSE", D = 1, scores = NULL, se = TRUE, nq = 30, 
  w = 0.5, theta = NULL, weights = NULL, anchor.type = "internal", ...)

## S3 method for class 'ceqc'
score(obj, method = "TSE", D = 1, scores = NULL, se = TRUE, nq = 30, 
  w = 0.5, theta = NULL, weights = NULL, anchor.type = "internal", ...)

## S3 method for class 'ceqclist'
score(obj, path = NULL, method = "TSE", D = 1, scores = NULL, se = TRUE, nq = 30, 
  w = 0.5, theta = NULL, weights = NULL, anchor.type = "internal", ...)

## S3 method for class 'meqc'
score(obj, link = NULL, method = "TSE", D = 1, scores = NULL, se = TRUE, 
  bistype = NULL, nq = 30, w = 0.5, theta = NULL, weights = NULL, 
  anchor.type = "internal", ...)

Arguments

obj	output object from functions direc, alldirec, chainec or bisectorec.
link	a character string with the names of the two forms being linked separated by a dot (e.g. "test1.test3"). Necessary if obj is output of functions alldirec or bisectorec.
path	a character string with the names of the forms that constitute the path separated by a dot (e.g. "test1.test2.test3"). Necessary if obj is output of function chainec.
method	the scoring method to be used. This should be one of "TSE" (the default) for true score equating or "OSE" for observed score equating.
D	constant D of the IRT model used to estimate item parameters.
scores	integer values to be converted.
se	logical; is TRUE standard errors of equated scores are computed.
bistype	bisector type. It should be specified when obj is an output object from function bisectorec. It should be weighted or unweighted. It can be NULL if only one type was computed by function bisectorec.
nq	number of quadrature points used to approximate integrals with observed score equating. Used only if arguments theta and weights are NULL.
w	synthetic weight for population 1. It should be a number between 0 and 1.
theta	vector of ability values used to approximate integrals with observed score equating.
weights	vector of weights used to approximate integrals with observed score equating.
anchor.type	The anchor type, which can be "internal" or "external". External anchors are not implemented for bisector equating coefficients. See details.
...	further arguments passed to or from other methods.

Details

If anchor.type is internal, the common items are used for computing the scores; if it is external, the common items are used for computing the equating coefficients but not for computing the scores. External anchors are not implemented for bisector equating coefficients because it is possible that all items are used as achors in one of the paths considered.

This function computes standard error of equated scores with the observed score equating method using $(\alpha_X',\alpha_{V1}',A,B)'$ for the description of $\beta_{X2,x}(At_m+B)$, and $(\alpha_Y',\alpha_{V2}',A,B)'$ for $\beta_{Y1,y}(t_m)$ (see Ogasawara, 2003).

Value

A data frame containing theta values (only for true score equating), scores in the form chosen as base, equated scores, and standard errors of equated scores.

Author(s)

Michela Battauz

References

Kolen, M.J. and Brennan, R.L. (2014). Test equating, scaling, and linking: methods and practices, 3nd ed., New York: Springer.

Ogasawara, H. (2001). Item response theory true score equatings and their standard errors. Journal of Educational and Behavioral Statistics, 26, 31–50.

Ogasawara, H. (2003). Asymptotic standard errors of IRT observed-score equating methods. Psychometrika, 68, 193–211.

See Also

alldirec, bisectorec, chainec, direc

Examples

# the following code uses item parameter estimates previously obtained
# for examples that start from item response data see function modIRT

# two-parameter logistic model
data(est2pl)
test<-paste("test", 1:5, sep = "")
mod2pl <- modIRT(coef = est2pl$coef, var = est2pl$var, names = test, display = FALSE)
# direct equating coefficients between forms 1 and 2 using the Haebara method
l12 <- direc(mods = mod2pl, which = c(1,2), method = "Haebara")
# scoring using direct equating coefficients and the true score equating method
score(l12)
# scoring using direct equating coefficients and the observed score equating method
score(l12, method = "OSE")
# specify only scores from 10 to 15
score(l12, method = "OSE", scores = 10:15)
# external anchors and true score equating method
score(l12, anchor.type = "external")
# external anchors and observed score equating method
score(l12, method = "OSE", anchor.type = "external")

# all direct equating coefficients using the Haebara method
direclist2pl <- alldirec(mods = mod2pl, method = "Haebara")
# scoring using direct equating coefficients and the true score equating method
score(direclist2pl, link = "test1.test2")

# compute chain equating coefficients for path 1,2,3,4
pth1 <- paste("test", 1:4, sep = "")
chainec1 <- chainec(direclist = direclist2pl, pths = pth1)
# compute chain equating coefficients for path 1,5,4
pth2 <- paste("test", c(1,5,4), sep = "")
chainec2 <- chainec(direclist = direclist2pl, pths = pth2)
# scoring using chain equating coefficients and the true score equating method for score 12
score(chainec1, path = "test1.test2.test3.test4", scores = 12)
score(chainec2, path = "test1.test5.test4", scores = 12)
# scoring using chain equating coefficients, true score equating and external anchors
score(chainec1, path = "test1.test2.test3.test4", anchor.type = "external")
# scoring using chain equating coefficients, observed score equating and external anchors
score(chainec1, path = "test1.test2.test3.test4", method = "OSE", anchor.type = "external")


# create a list of objects of class ceqc
ecall <- c(chainec1, chainec2)
# compute bisector and weighted bisector coefficients
allec <- bisectorec(ecall = ecall, weighted = TRUE, unweighted = TRUE)
# scoring using weighted bisector equating coefficients and the true score 
# equating method for score 12
score(allec, bistype = "weighted", link = "test1.test4", scores = 12)

---

Test on Chain Equating Coefficients

Description

Performs a statistical test to verify if the chain equating coeffients that link the same two forms are equal.

Usage

sd.test(ecall)

Arguments

ecall

list of objects of class eqc or ceqc returned by functions direc and chainec.

Value

An object of class idtest with components

link
paths
AB
statistic
df
p.value

Author(s)

Michela Battauz

References

Battauz, M. (2022). Testing for differences in chain equating. Statistica Neerlandica, 1-12.

See Also

chainec

Examples

# the following code uses item parameter estimates previously obtained
# for other examples see documentation in function chainec

# three-parameter logistic model
# direct equating coefficients using the "Haebara" method
data(est3pl)
test <- paste("test", 1:5, sep = "")
mod3pl <- modIRT(coef = est3pl$coef, var = est3pl$var, names = test, display = FALSE)
direclist3pl <- alldirec(mods = mod3pl, method = "Haebara")
# compute chain equating coefficients for path 1,2,3,4,5
pth3 <- paste("test", 1:5, sep = "")
chainec3 <- chainec(direclist = direclist3pl, pths = pth3)
# create a list of objects of class eqc or ceqc
ecall <- c(chainec3, direclist3pl["test1.test5"])
summary(chainec3)
summary(direclist3pl$test1.test5)
# test
sd.test(ecall)

---

Summarizing Estimated Chain Equating Coefficients

Description

summary method for class ceqc.

Usage

## S3 method for class 'ceqc'
summary(object, ...)

## S3 method for class 'summary.ceqc'
print(x, ...)

Arguments

object	an object of the class ceqc returned by function chainec.
x	an object of class summary.ceqc, a result of a call to summary.ceqc.
...	further arguments passed to or from other methods.

Value

The function summary.ceqc returns a list with components

forms	names of equated forms.
method	the equating method used.
coefficients	a $2 \times 2$ matrix with columns for the estimated coefficients A and B and standard errors.

Author(s)

Michela Battauz

See Also

chainec, modIRT

Examples

# the following code uses item parameter estimates previously obtained
# for examples that start from item response data see function modIRT

# two-parameter logistic model
# direct equating coefficients using the "Haebara" method
data(est2pl)
test<-paste("test", 1:5, sep = "")
mod2pl <- modIRT(coef = est2pl$coef, var = est2pl$var, names = test, display = FALSE)
direclist2pl <- alldirec(mods = mod2pl, method = "Haebara")
# compute all chain equating coefficients of length 3
chainec3 <- chainec(r = 3, direclist = direclist2pl)
summary(chainec3$test1.test5.test4)

---

Summarizing a List of Estimated Chain Equating Coefficients

Description

summary method for class ceqclist.

Usage

## S3 method for class 'ceqclist'
summary(object, path, ...)

## S3 method for class 'summary.ceqclist'
print(x, ...)

Arguments

object	an object of the class ceqclist returned by function chainec.
path	a vector of character strings with the names of the forms that constitute the path separated by a dot (e.g. "test1.test2.test3").
x	an object of class summary.ceqclist, a result of a call to summary.ceqclist.
...	further arguments passed to or from other methods.

Value

The function summary.ceqclist returns a list containing the output of function summary.ceqc for each path contained in object.

Author(s)

Michela Battauz

See Also

chainec, modIRT, summary.eqc

Examples

# the following code uses item parameter estimates previously obtained
# for examples that start from item response data see function modIRT

# two-parameter logistic model
# direct equating coefficients using the "Haebara" method
data(est2pl)
test<-paste("test", 1:5, sep = "")
mod2pl <- modIRT(coef = est2pl$coef, var = est2pl$var, names = test, display = FALSE)
direclist2pl <- alldirec(mods = mod2pl, method = "Haebara")
# compute all chain equating coefficients of length 3
chainec3 <- chainec(r = 3, direclist = direclist2pl)
summary(chainec3)	
summary(chainec3, path = "test1.test2.test3")	
summary(chainec3, path = c("test1.test2.test3","test1.test5.test4"))

---

Summarizing Estimated Equating Coefficients

Description

summary method for class eqc.

Usage

## S3 method for class 'eqc'
summary(object, ...)

## S3 method for class 'summary.eqc'
print(x, ...)

Arguments

object	an object of the class eqc returned by function direc.
x	an object of class summary.eqc, a result of a call to summary.eqc.
...	further arguments passed to or from other methods.

Value

The function summary.eqc returns a list with components

forms	names of equated forms.
method	the equating method used.
coefficients	a $2 \times 2$ matrix with columns for the estimated coefficients A and B and standard errors.

Author(s)

Michela Battauz

See Also

direc, modIRT

Examples

# the following code uses item parameter estimates previously obtained
# for examples that start from item response data see function modIRT

# three-parameter logistic model
# direct equating coefficients between forms 1 and 2 using the Stocking-Lord method
data(est3pl)
test <- paste("test", 1:5, sep = "")
mod3pl <- modIRT(coef = est3pl$coef, var = est3pl$var, names = test, display = FALSE)
l12 <- direc(mod1 = mod3pl[1], mod2 = mod3pl[2], method = "Stocking-Lord")
summary(l12)

---

Summarizing a List of Estimated Direct Equating Coefficients

Description

summary method for class eqclist.

Usage

## S3 method for class 'eqclist'
summary(object, link, ...)

## S3 method for class 'summary.eqclist'
print(x, ...)

Arguments

object	an object of the class eqclist returned by function alldirec.
link	a vector of character strings with the names of the two forms being linked separated by a dot (e.g. "test1.test2").
x	an object of class summary.eqclist, a result of a call to summary.eqclist.
...	further arguments passed to or from other methods.

Value

The function summary.eqclist returns a list containing the output of function summary.eqc for each link contained in object.

Author(s)

Michela Battauz

See Also

alldirec, modIRT, summary.eqc

Examples

# the following code uses item parameter estimates previously obtained
# for examples that start from item response data see function modIRT

# Rasch model
# direct equating coefficients using the "mean-mean" method
data(estrasch)
test<-paste("test", 1:5, sep = "")
modrasch <- modIRT(coef = estrasch$coef, var = estrasch$var, names = test, 
	display = FALSE)
direclistrasch <- alldirec(mods = modrasch, method = "mean-mean", all = TRUE)
summary(direclistrasch)
summary(direclistrasch, link = "test1.test2")
summary(direclistrasch, link = c("test1.test2", "test1.test5"))

---

Summarizing Bisector Equating Coefficients

Description

summary method for class meqc.

Usage

## S3 method for class 'meqc'
summary(object, ...)

## S3 method for class 'summary.meqc'
print(x, ...)

Arguments

object	an object of the class meqc returned by function bisectorec.
x	an object of class summary.meqc, a result of a call to summary.meqc.
...	further arguments passed to or from other methods.

Value

The function summary.meqc returns a list with components

link	character vector with names of equated forms.
method	the equating method used.
coefficients	list of data frames containing Path, Estimate and StdErr of direct, chain and bisector equating coefficients.

Author(s)

Michela Battauz

See Also

bisectorec, modIRT

Examples

# the following code uses item parameter estimates previously obtained
# for examples that start from item response data see function modIRT

# three-parameter logistic model
# direct equating coefficients using the "Stocking-Lord" method
data(est3pl)
test <- paste("test", 1:5, sep = "")
mod3pl <- modIRT(coef = est3pl$coef, var = est3pl$var, names = test, display = FALSE)
direclist3pl <- alldirec(mods = mod3pl, method = "Stocking-Lord")
# compute chain equating coefficients for path 1,2,3,4
pth1 <- paste("test", 1:4, sep = "")
pth1 <- data.frame(t(pth1), stringsAsFactors = FALSE)
chainec1 <- chainec(direclist = direclist3pl, pths = pth1)
# compute chain equating coefficients for path 1,5,4
pth2 <- c(paste("test", c(1,5,4), sep = ""))
pth2 <- data.frame(t(pth2), stringsAsFactors = FALSE)
chainec2 <- chainec(direclist = direclist3pl, pths = pth2)
# compute chain equating coefficients for path 1,2,3,4,5
pth3 <- paste("test", 1:5, sep = "")
pth3 <- data.frame(t(pth3), stringsAsFactors = FALSE)
chainec3 <- chainec(direclist = direclist3pl, pths = pth3)
# create a list of objects of class eqc or ceqc
ecall <- c(chainec1, chainec2, chainec3, direclist3pl["test1.test5"])
# compute bisector and weighted bisector coefficients
allec <- bisectorec(ecall = ecall, weighted = TRUE, unweighted = TRUE)
summary(allec)

---