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

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 <michela.battauz@uniud.it>

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

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

Value

An object of class meqc with components

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

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

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

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

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

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 ith 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

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

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

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

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 ith 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 ith 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 ith 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

Value

An object of class idtest with components

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

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

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

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

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

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 ith 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

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

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

Value

An object of class idtest with components

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

Value

The function summary.ceqc returns a list with components

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

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

Value

The function summary.eqc returns a list with components

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

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

Value

The function summary.meqc returns a list with components

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)