tam.linking {TAM}R Documentation

Linking of Fitted Unidimensional Item Response Models in TAM

Description

Performs linking of fitted unidimensional item response models in TAM according to the Stocking-Lord and the Haebara method (Kolen & Brennan, 2014; Gonzales & Wiberg, 2017). Several studies can either be linked by a chain of linkings of two studies (method="chain") or a joint linking approach (method="joint") comprising all pairwise linkings.

The linking of two studies is implemented in the tam_linking_2studies function.

Usage

tam.linking(tamobj_list, type="Hae", method="joint", pow_rob_hae=1, eps_rob_hae=1e-4,
   theta=NULL, wgt=NULL, wgt_sd=2, fix.slope=FALSE, elim_items=NULL,
   par_init=NULL, verbose=TRUE)

## S3 method for class 'tam.linking'
summary(object, file=NULL, ...)

## S3 method for class 'tam.linking'
print(x, ...)

tam_linking_2studies( B1, AXsi1, guess1, B2, AXsi2, guess2, theta, wgt, type,
    M1=0, SD1=1, M2=0, SD2=1, fix.slope=FALSE, pow_rob_hae=1)

## S3 method for class 'tam_linking_2studies'
summary(object, file=NULL, ...)

## S3 method for class 'tam_linking_2studies'
print(x, ...)

Arguments

tamobj_list

List of fitted objects in TAM

type

Type of linking method: "SL" (Stocking-Lord), "Hae" (Haebara) or "RobHae" (robust Haebara). See Details for more information. The default is the Haebara linking method.

method

Chain linking ("chain") or joint linking ("joint")

pow_rob_hae

Power for robust Heabara linking

eps_rob_hae

Value \varepsilon for numerical approximation of loss function |x|^p in robust Haebara linking

theta

Grid of \theta points. The default is seq(-6,6,len=101).

wgt

Weights defined for the theta grid. The default is
tam_normalize_vector( stats::dnorm( theta, sd=2 )).

wgt_sd

Standard deviation for \theta grid used for linking function

fix.slope

Logical indicating whether the slope transformation constant is fixed to 1.

elim_items

List of vectors refering to items which should be removed from linking (see Model 'lmod2' in Example 1)

par_init

Optional vector with initial parameter values

verbose

Logical indicating progress of linking computation

object

Object of class tam.linking or tam_linking_2studies.

x

Object of class tam.linking or tam_linking_2studies.

file

A file name in which the summary output will be written

...

Further arguments to be passed

B1

Array B for first study

AXsi1

Matrix A \xi for first study

guess1

Guessing parameter for first study

B2

Array B for second study

AXsi2

Matrix A \xi for second study

guess2

Guessing parameter for second study

M1

Mean of first study

SD1

Standard deviation of first study

M2

Mean of second study

SD2

Standard deviation of second study

Details

The Haebara linking is defined by minimizing the loss function

\sum_i \sum_k \int \left ( P_{ik} ( \theta ) - P_{ik}^\ast ( \theta ) \right )^2

A robustification of Haebara linking minimizes the loss function

\sum_i \sum_k \int \left ( P_{ik} ( \theta ) - P_{ik}^\ast ( \theta ) \right )^p

with a power p (defined in pow_rob_hae) smaller than 2. He, Cui and Osterlind (2015) consider p=1.

Value

List containing entries

parameters_list

List containing transformed item parameters

linking_list

List containing results of each linking in the linking chain

M_SD

Mean and standard deviation for each study after linking

trafo_items

Transformation constants for item parameters

trafo_persons

Transformation constants for person parameters

References

Battauz, M. (2015). equateIRT: An R package for IRT test equating. Journal of Statistical Software, 68(7), 1-22. doi:10.18637/jss.v068.i07

Gonzalez, J., & Wiberg, M. (2017). Applying test equating methods: Using R. New York, Springer. doi:10.1007/978-3-319-51824-4

He, Y., Cui, Z., & Osterlind, S. J. (2015). New robust scale transformation methods in the presence of outlying common items. Applied Psychological Measurement, 39(8), 613-626. doi:10.1177/0146621615587003

Kolen, M. J., & Brennan, R. L. (2014). Test equating, scaling, and linking: Methods and practices. New York, Springer. doi:10.1007/978-1-4939-0317-7

Weeks, J. P. (2010). plink: An R package for linking mixed-format tests using IRT-based methods. Journal of Statistical Software, 35(12), 1-33. doi:10.18637/jss.v035.i12

See Also

Linking or equating of item response models can be also conducted with plink (Weeks, 2010), equate, equateIRT (Battauz, 2015), equateMultiple, kequate and irteQ packages.

See also the sirt::linking.haberman, sirt::invariance.alignment and sirt::linking.haebara functions in the sirt package.

Examples

## Not run: 
#############################################################################
# EXAMPLE 1: Linking dichotomous data with the 2PL model
#############################################################################

data(data.ex16)
dat <- data.ex16
items <- colnames(dat)[-c(1,2)]

# fit grade 1
rdat1 <- TAM::tam_remove_missings( dat[ dat$grade==1, ], items=items )
mod1 <- TAM::tam.mml.2pl( resp=rdat1$resp[, rdat1$items], pid=rdat1$dat$idstud )
summary(mod1)

# fit grade 2
rdat2 <- TAM::tam_remove_missings( dat[ dat$grade==2, ], items=items )
mod2 <- TAM::tam.mml.2pl( resp=rdat2$resp[, rdat2$items], pid=rdat2$dat$idstud )
summary(mod2)

# fit grade 3
rdat3 <- TAM::tam_remove_missings( dat[ dat$grade==3, ], items=items )
mod3 <- TAM::tam.mml.2pl( resp=rdat3$resp[, rdat3$items], pid=rdat3$dat$idstud )
summary(mod3)

# define list of fitted models
tamobj_list <- list( mod1, mod2, mod3 )

#-- link item response models
lmod <- TAM::tam.linking( tamobj_list)
summary(lmod)

# estimate WLEs based on transformed item parameters
parm_list <- lmod$parameters_list

# WLE grade 1
arglist <- list( resp=mod1$resp, B=parm_list[[1]]$B, AXsi=parm_list[[1]]$AXsi )
wle1 <- TAM::tam.mml.wle(tamobj=arglist)

# WLE grade 2
arglist <- list( resp=mod2$resp, B=parm_list[[2]]$B, AXsi=parm_list[[2]]$AXsi )
wle2 <- TAM::tam.mml.wle(tamobj=arglist)

# WLE grade 3
arglist <- list( resp=mod3$resp, B=parm_list[[3]]$B, AXsi=parm_list[[3]]$AXsi )
wle3 <- TAM::tam.mml.wle(tamobj=arglist)

# compare result with chain linking
lmod1b <- TAM::tam.linking(tamobj_list)
summary(lmod1b)

#-- linking with some eliminated items

# remove three items from first group and two items from third group
elim_items <- list( c("A1", "E2","F1"), NULL,  c("F1","F2") )
lmod2 <- TAM::tam.linking(tamobj_list, elim_items=elim_items)
summary(lmod2)

#-- Robust Haebara linking with p=1
lmod3a <- TAM::tam.linking(tamobj_list, type="RobHae", pow_rob_hae=1)
summary(lmod3a)

#-- Robust Haeabara linking with initial parameters and prespecified epsilon value
par_init <- lmod3a$par
lmod3b <- TAM::tam.linking(tamobj_list, type="RobHae", pow_rob_hae=.1,
                eps_rob_hae=1e-3, par_init=par_init)
summary(lmod3b)

#############################################################################
# EXAMPLE 2: Linking polytomous data with the partial credit model
#############################################################################

data(data.ex17)
dat <- data.ex17

items <- colnames(dat)[-c(1,2)]

# fit grade 1
rdat1 <- TAM::tam_remove_missings( dat[ dat$grade==1, ], items=items )
mod1 <- TAM::tam.mml.2pl( resp=rdat1$resp[, rdat1$items], pid=rdat1$dat$idstud )
summary(mod1)

# fit grade 2
rdat2 <- TAM::tam_remove_missings( dat[ dat$grade==2, ], items=items )
mod2 <- TAM::tam.mml.2pl( resp=rdat2$resp[, rdat2$items], pid=rdat2$dat$idstud )
summary(mod2)

# fit grade 3
rdat3 <- TAM::tam_remove_missings( dat[ dat$grade==3, ], items=items )
mod3 <- TAM::tam.mml.2pl( resp=rdat3$resp[, rdat3$items], pid=rdat3$dat$idstud )
summary(mod3)

# list of fitted TAM models
tamobj_list <- list( mod1, mod2, mod3 )

#-- linking: fix slope because partial credit model is fitted
lmod <- TAM::tam.linking( tamobj_list, fix.slope=TRUE)
summary(lmod)

# WLEs can be estimated in the same way as in Example 1.

#############################################################################
# EXAMPLE 3: Linking dichotomous data with the multiple group 2PL models
#############################################################################

data(data.ex16)
dat <- data.ex16
items <- colnames(dat)[-c(1,2)]

# fit grade 1
rdat1 <- TAM::tam_remove_missings( dat[ dat$grade==1, ], items=items )
# create some grouping variable
group <- ( seq( 1, nrow( rdat1$dat ) ) %% 3 ) + 1
mod1 <- TAM::tam.mml.2pl( resp=rdat1$resp[, rdat1$items], pid=rdat1$dat$idstud, group=group)
summary(mod1)

# fit grade 2
rdat2 <- TAM::tam_remove_missings( dat[ dat$grade==2, ], items=items )
group <- 1*(rdat2$dat$dat$idstud > 500)
mod2 <- TAM::tam.mml.2pl( resp=rdat2$resp[, rdat2$items], pid=rdat2$dat$dat$idstud, group=group)
summary(mod2)

# fit grade 3
rdat3 <- TAM::tam_remove_missings( dat[ dat$grade==3, ], items=items )
mod3 <- TAM::tam.mml.2pl( resp=rdat3$resp[, rdat3$items], pid=rdat3$dat$idstud )
summary(mod3)

# define list of fitted models
tamobj_list <- list( mod1, mod2, mod3 )

#-- link item response models
lmod <- TAM::tam.linking( tamobj_list)

#############################################################################
# EXAMPLE 4: Linking simulated dichotomous data with two groups
#############################################################################

library(sirt)

#*** simulate data
N <- 3000  # number of persons
I <- 30    # number of items
b <- seq(-2,2, length=I)
# data for group 1
dat1 <- sirt::sim.raschtype( rnorm(N, mean=0, sd=1), b=b )
# data for group 2
dat2 <- sirt::sim.raschtype( rnorm(N, mean=1, sd=.6), b=b )

# fit group 1
mod1 <- TAM::tam.mml.2pl( resp=dat1 )
summary(mod1)

# fit group 2
mod2 <- TAM::tam.mml.2pl( resp=dat2 )
summary(mod2)

# define list of fitted models
tamobj_list <- list( mod1, mod2 )

#-- link item response models
lmod <- TAM::tam.linking( tamobj_list)
summary(lmod)

# estimate WLEs based on transformed item parameters
parm_list <- lmod$parameters_list

# WLE grade 1
arglist <- list( resp=mod1$resp, B=parm_list[[1]]$B, AXsi=parm_list[[1]]$AXsi )
wle1 <- TAM::tam.mml.wle(tamobj=arglist)

# WLE grade 2
arglist <- list( resp=mod2$resp, B=parm_list[[2]]$B, AXsi=parm_list[[2]]$AXsi )
wle2 <- TAM::tam.mml.wle(tamobj=arglist)
summary(wle1)
summary(wle2)

# estimation with linked and fixed item parameters for group 2
B <- parm_list[[2]]$B
xsi.fixed <- cbind( 1:I, -parm_list[[2]]$AXsi[,2] )
mod2f <- TAM::tam.mml( resp=dat2, B=B, xsi.fixed=xsi.fixed )
summary(mod2f)

## End(Not run)
[Package TAM version 4.2-21 Index]