linking.haebara: Haebara Linking of the 2PL Model for Multiple Studies

Description

The function linking.haebara is a generalization of Haebara linking of the 2PL model to multiple groups (or multiple studies; see Battauz, 2017, for a similar approach). The optimization estimates transformation parameters for means and standard deviations of the groups and joint item parameters. The function allows two different distance functions dist="L2" and dist="L1" where the latter is a robustified version of Haebara linking (see Details; He, Cui, & Osterlind, 2015; He & Cui, 2020; Hu, Rogers, & Vukmirovic, 2008).

Usage

linking.haebara(itempars, dist="L2", theta=seq(-4,4, length=61),
        optimizer="optim", center=FALSE, eps=1e-3, par_init=NULL, use_rcpp=TRUE,
        pow=2, use_der=TRUE, ...)
# S3 method for linking.haebara
summary(object, digits=3, file=NULL, ...)

Value

A list with following entries

pars: Estimated means and standard deviations (transformation parameters)
item: Estimated joint item parameters
a.orig: Original $a_{it}$ parameters
b.orig: Original $b_{it}$ parameters
a.resid: Residual $a_{it}$ parameters (DIF parameters)
b.resid: Residual $b_{it}$ parameters (DIF parameters)
res_optim: Value of optimization routine

Arguments

itempars: A data frame with four or five columns. The first four columns contain in the order: study name, item name, $a$ parameter, $b$ parameter. The fifth column is an optional weight for every item and every study. See linking.haberman for a function which uses the same argument.
dist: Distance function. Options are "L2" for squared loss and "L1" for absolute value loss.
theta: Grid of theta points for 2PL item response functions
optimizer: Name of the optimizer chosen for alignment. Options are "optim" (using stats::optim) or "nlminb" (using stats::nlminb).
center: Logical indicating whether means and standard deviations should be centered after estimation
eps: Small value for smooth approximation of the absolute value function
par_init: Optional vector of initial parameter estimates
use_rcpp: Logical indicating whether Rcpp is used for computation
pow: Power for method dist="Lq"
use_der: Logical indicating whether analytical derivative should be used
object: Object of class linking.haabara.
digits: Number of digits after decimals for rounding in summary.
file: Optional file name if summary should be sunk into a file.
...: Further arguments to be passed

Details

For $t=1,\ldots,T$ studies, item difficulties $b_{it}$ and item slopes $a_{it}$ are available. The 2PL item response functions are given by $$ logit P(X_{pi}=1| \theta_p )=a_i ( \theta_p - b_i ) $$

Haebara linking compares the observed item response functions $P_{it}$ based on the equation for the logits $a_{it}(\theta - b_{it})$ and the expected item response functions $P_{it}^\ast$ based on the equation for the logits $a_i^\ast \sigma_t ( \theta - ( b_i - \mu_t)/\sigma_t )$ where the joint item parameters $a_i$ and $b_i$ and means $\mu_t$ and standard deviations $\sigma_t$ are estimated.

Two loss functions are implemented. The quadratic loss of Haebara linking (dist="L2") minimizes $$f_{opt, L2}=\sum_t \sum_i \int ( P_{it} (\theta ) - P_{it}^\ast (\theta ) )^2 w(\theta)$$ was originally proposed by Haebara. A robustified version (dist="L1") uses the optimization function (He et al., 2015) $$f_{opt, L1}=\sum_t \sum_i \int | P_{it} (\theta ) - P_{it}^\ast (\theta ) | w(\theta)$$ As a further generalization, the follwing distance function (dist="Lp") can be minimized: $$f_{opt, Lp}=\sum_t \sum_i \int | P_{it} (\theta ) - P_{it}^\ast (\theta ) |^p w(\theta)$$

References

Battauz, M. (2017). Multiple equating of separate IRT calibrations. Psychometrika, 82, 610-636. tools:::Rd_expr_doi("10.1007/s11336-016-9517-x")

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. tools:::Rd_expr_doi("10.1177/0146621615587003")

He, Y., & Cui, Z. (2020). Evaluating robust scale transformation methods with multiple outlying common items under IRT true score equating. Applied Psychological Measurement, 44(4), 296-310. tools:::Rd_expr_doi("10.1177/0146621619886050")

Hu, H., Rogers, W. T., & Vukmirovic, Z. (2008). Investigation of IRT-based equating methods in the presence of outlier common items. Applied Psychological Measurement, 32(4), 311-333. tools:::Rd_expr_doi("10.1177/0146621606292215")

Examples

Run this code

if (FALSE) {
#############################################################################
# EXAMPLE 1: Robust linking methods in the presence of outliers
#############################################################################

#** simulate data
I <- 10
a <- seq(.9, 1.1, len=I)
b <- seq(-2, 2, len=I)

#- define item parameters
item_names <- paste0("I",100+1:I)
# th=SIG*TH+MU=> logit(p)=a*(SIG*TH+MU-b)=a*SIG*(TH-(-MU)/SIG-b/SIG)
d1 <- data.frame( study="S1", item=item_names, a=a, b=b )
mu <- .5; sigma <- 1.3
d2 <- data.frame( study="S2", item=item_names, a=a*sigma, b=(b-mu)/sigma )
mu <- -.3; sigma <- .7
d3 <- data.frame( study="S3", item=item_names, a=a*sigma, b=(b-mu)/sigma )

#- define DIF effect
# dif <- 0  # no DIF effects
dif <- 1
d2[4,"a"] <- d2[4,"a"] * (1-.8*dif)
d3[5,"b"] <- d3[5,"b"] - 2*dif
itempars <- rbind(d1, d2, d3)

#* Haebara linking non-robust
mod1 <- sirt::linking.haebara( itempars, dist="L2", control=list(trace=2) )
summary(mod1)

#* Haebara linking robust
mod2 <- sirt::linking.haebara( itempars, dist="L1", control=list(trace=2) )
summary(mod2)

#* using initial parameter estimates
par_init <- mod1$res_optim$par
mod2b <- sirt::linking.haebara( itempars, dist="L1", par_init=par_init)
summary(mod2b)

#* power p=.25
mod2c <- sirt::linking.haebara( itempars, dist="Lp", pow=.25, par_init=par_init)
summary(mod2c)

#* Haberman linking non-robust
mod3 <- sirt::linking.haberman(itempars)
summary(mod3)

#* Haberman linking robust
mod4 <- sirt::linking.haberman(itempars, estimation="BSQ", a_trim=.25, b_trim=.5)
summary(mod4)

#* compare transformation parameters (means and standard deviations)
mod1$pars
mod2$pars
mod3$transf.personpars
mod4$transf.personpars
}

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