Linking of Fitted Unidimensional Item Response Models in <span class="pkg">TAM</span>

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.

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 tam.linking
summary(object, file=NULL, ...)

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

# S3 method for 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

Examples

if (FALSE) {
#############################################################################
# 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)
}

Linking of Fitted Unidimensional Item Response Models in TAM

Arguments

Details

Value

References

See also

Examples