Linking of Fitted Unidimensional Item Response Models in TAM

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)
}