linking.haberman: Linking in the 2PL/Generalized Partial Credit Model
In sirt: Supplementary Item Response Theory Models
View source: R/linking.haberman.R
linking.haberman R Documentation
Linking in the 2PL/Generalized Partial Credit Model
Description
This function does the linking of several studies which are calibrated
using the 2PL or the generalized item response model according to
Haberman (2009). This method is a generalization of log-mean-mean
linking from one study to several studies. The default a_log=TRUE
logarithmizes item slopes for linking while otherwise an additive regression
model is assumed for the original item loadings (see Details; Battauz, 2017)
Usage
linking.haberman(itempars, personpars, estimation="OLS", a_trim=Inf, b_trim=Inf,
lts_prop=.5, a_log=TRUE, conv=1e-05, maxiter=1000, progress=TRUE,
adjust_main_effects=TRUE, vcov=TRUE)
## S3 method for class 'linking.haberman'
summary(object, digits=3, file=NULL, ...)
linking.haberman.lq(itempars, pow=2, eps=1e-3, a_log=TRUE, use_nu=FALSE,
est_pow=FALSE, lower_pow=.1, upper_pow=3)
## S3 method for class 'linking.haberman.lq'
summary(object, digits=3, file=NULL, ...)
## prepare 'itempars' argument for linking.haberman()
linking_haberman_itempars_prepare(b, a=NULL, wgt=NULL)
## conversion of different parameterizations of item parameters
linking_haberman_itempars_convert(itempars=NULL, lambda=NULL, nu=NULL, a=NULL, b=NULL)
## L0 polish precedure minimizing number of interactions in two-way table
L0_polish(x, tol, conv=0.01, maxiter=30, type=1, verbose=TRUE)
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.
personpars
A list with vectors (e.g. EAPs or WLEs) or data frames
(e.g. plausible values) containing person parameters which
should be transformed.
If a data frame in each list entry has se or SE
(standard error) in a column name, then the corresponding
column is only multiplied by A_t.
If a column is labeled as pid (person ID),
then it is left untransformed.
estimation
Estimation method. Can be "OLS" (ordinary least squares),
"BSQ" (bisquare weighted regression), "HUB" (regression using Huber
weights), "MED" (median regression), "LTS" (trimmed least squares),
"L1" (median polish), "L0" (minimizing number of interactions)
a_trim
Trimming parameter for item slopes a_{it} in
bisquare regression (see Details).
b_trim
Trimming parameter for item slopes b_{it} in
bisquare regression (see Details).
lts_prop
Proportion of retained observations in "LTS"
regression estimation
a_log
Logical indicating whether item slopes should be logarithmized
for linking.
conv
Convergence criterion.
maxiter
Maximum number of iterations.
progress
An optional logical indicating whether computational progress
should be displayed.
adjust_main_effects
Logical indicating whether all elements in the vector
of main effects should be simultaneously adjusted
vcov
Optional indicating whether covariance matrix for linking errors
should be computed
pow
Power q
eps
Epsilon value used in differentiable approximating function
use_nu
Logical indicating whether item intercepts instead of
item difficulties are used in linking
est_pow
Logical indicating whether power values should be estimated
lower_pow
Lower bound for estimated power
upper_pow
Upper bound for estimated power
lambda
Matrix containing item loadings
nu
Matrix containing item intercepts
object
Object of class linking.haberman.
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
b
Matrix of item intercepts (items times studies)
a
Matrix of item slopes
wgt
Matrix of weights
x
Matrix
tol
Tolerance value
type
Can be 1 (using Tukey's median polish) or
2 (alternating median regression).
verbose
Logical indicating whether iteration progress should be displayed
Details
For t=1,\ldots,T studies, item difficulties b_{it} and
item slopes a_{it} are available. For dichotomous responses, these
parameters are defined by the 2PL response equation
logit P(X_{pi}=1| \theta_p )=a_i ( \theta_p - b_i )
while for polytomous responses the generalized partial credit model holds
log \frac{P(X_{pi}=k| \theta_p )}{P(X_{pi}=k-1| \theta_p )}
=a_i ( \theta_p - b_i + d_{ik} )
The parameters \{ a_{it}, b_{it} \} of all items and studies are
linearly transformed using equations a_{it} \approx a_i / A_t
(if a_log=TRUE) or a_{it} \approx a_i + A_t
(if a_log=FALSE) and
b_{it} \cdot A_t \approx B_t + b_i. For identification reasons,
we define A_1=1 and B_1=0.
The optimization function (which is a least squares criterion;
see Haberman, 2009) seeks the transformation parameters A_t and
B_t with an alternating least squares
method (estimation="OLS"). Note that every item i and every study t can
be weighted (specified in the fifth column of itempars).
Alternatively, a robust regression method based on bisquare weighting (Fox, 2015)
can be employed for linking using the argument estimation="BSQ".
For example, in the case of item loadings, bisquare weighting is applied to
residuals e_{it}=a_{it} - a_i - A_t (where logarithmized or non-logarithmized
item loadings are employed) forming weights
w_{it}=[ 1 - ( e_{it} / k )^2 ]^2 for e_{it} <k and 0 for e_{it} \ge k
where k is the trimming constant which can be estimated or fixed
during estimation using arguments a_trim or b_trim. Items in studies with
large residuals
(i.e., presence differential item functioning) are effectively set to zero in the
linking procedure. Alternatively, Huber weights (estimation="HUB") downweight
large residuals by applying w_{it}=k / | e_{it} | for residuals
|e_{it}|>k. The method estimation="LTS" employs trimmed least squares
where the proportion
of data retained is specified in lts_prop with default set to .50.
The method estimation="MED" estimates item parameters and linking constants
based on alternating median regression. A similar approach is the median polish
procedure of Tukey (Tukey, 1977, p. 362ff.; Maronna, Martin & Yohai, 2006, p. 104;
see also stats::medpolish) implemented in
estimation="L1" which aims to minimize \sum_{i,t} | e_{it} |.
For a pre-specified tolerance value t (in a_trim or b_trim),
the approach estimation="L0" minimizes the number of interactions
(i.e., DIF effects) in the e_{it} effects. In more detail, it minimizes
\sum_{i,t} \# \{ | e_{it} | > t \} which is computationally conducted
by repeatedly applying the median polish procedure in which one cell is
omitted (Davies, 2012; Terbeck & Davies, 1998).
Effect sizes of invariance are calculated as R-squared measures
of explained item slopes and intercepts after linking
in comparison to item parameters across groups
(Asparouhov & Muthen, 2014).
The function linking.haberman.lq uses the loss function \rho(x)=|x|^q.
The originally proposed Haberman linking can be obtained with pow=2 (q=2).
The powers can also be estimated (argument est_pow=TRUE).
Value
A list with following entries
transf.pars
Data frame with transformation parameters
A_t and B_t
transf.personpars
Data frame with linear transformation functions
for person parameters
joint.itempars
Estimated joint item parameters a_i and b_i
a.trans
Transformed a_{it} parameters
b.trans
Transformed b_{it} 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)
personpars
Transformed person parameters
es.invariance
Effect size measures of invariance,
separately for item slopes and intercepts.
In the rows, R^2 and \sqrt{1-R^2} are reported.
es.robust
Effect size measures of invariance based on
robust estimation (if used).
selitems
Indices of items which are present in more than one
study.
References
Asparouhov, T., & Muthen, B. (2014). Multiple-group factor analysis alignment.
Structural Equation Modeling, 21(4), 1-14.
\Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/10705511.2014.919210")}
Battauz, M. (2017). Multiple equating of separate IRT calibrations.
Psychometrika, 82(3), 610-636.
\Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/s11336-016-9517-x")}
Davies, P. L. (2012). Interactions in the analysis of variance.
Journal of the American Statistical Association, 107(500), 1502-1509.
\Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/01621459.2012.726895")}
Fox, J. (2015). Applied regression analysis and generalized linear models.
Thousand Oaks: Sage.
Haberman, S. J. (2009). Linking parameter estimates derived
from an item response model through separate calibrations.
ETS Research Report ETS RR-09-40. Princeton, ETS.
\Sexpr[results=rd]{tools:::Rd_expr_doi("10.1002/j.2333-8504.2009.tb02197.x")}
Kolen, M. J., & Brennan, R. L. (2014). Test equating, scaling, and linking:
Methods and practices. New York: Springer.
\Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/978-1-4939-0317-7")}
Magis, D., & De Boeck, P. (2012). A robust outlier approach to prevent type I error
inflation in differential item functioning.
Educational and Psychological Measurement, 72(2), 291-311.
\Sexpr[results=rd]{tools:::Rd_expr_doi("10.1177/0013164411416975")}
Maronna, R. A., Martin, R. D., & Yohai, V. J. (2006). Robust statistics.
West Sussex: Wiley. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1002/0470010940")}
Terbeck, W., & Davies, P. L. (1998). Interactions and outliers in the two-way
analysis of variance. Annals of Statistics, 26(4), 1279-1305.
doi: 10.1214/aos/1024691243
Tukey, J. W. (1977). Exploratory data analysis. Addison-Wesley.
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.
\Sexpr[results=rd]{tools:::Rd_expr_doi("10.18637/jss.v035.i12")}
See Also
See the plink package (Weeks, 2010) for a diversity of linking methods.
Mean-mean linking, Stocking-Lord and Haebara linking (see Kolen & Brennan, 2014,
for an overview) in the generalized logistic item response model can be conducted with
equating.rasch. See also TAM::tam.linking
in the TAM package. Haebara linking and a robustified version of it can be
found in linking.haebara.
The invariance alignment method employs an optimization function based on
pairwise loss functions of item parameters (Asparouhov & Muthen, 2014),
see invariance.alignment.
Examples
#############################################################################
# EXAMPLE 1: Item parameters data.pars1.rasch and data.pars1.2pl
#############################################################################
# Model 1: Linking three studies calibrated by the Rasch model
data(data.pars1.rasch)
mod1 <- sirt::linking.haberman( itempars=data.pars1.rasch )
summary(mod1)
# Model 1b: Linking these studies but weigh these studies by
# proportion weights 3 : 0.5 : 1 (see below).
# All weights are the same for each item but they could also
# be item specific.
itempars <- data.pars1.rasch
itempars$wgt <- 1
itempars[ itempars$study=="study1","wgt"] <- 3
itempars[ itempars$study=="study2","wgt"] <- .5
mod1b <- sirt::linking.haberman( itempars=itempars )
summary(mod1b)
# Model 2: Linking three studies calibrated by the 2PL model
data(data.pars1.2pl)
mod2 <- sirt::linking.haberman( itempars=data.pars1.2pl )
summary(mod2)
# additive model instead of logarithmic model for item slopes
mod2b <- sirt::linking.haberman( itempars=data.pars1.2pl, a_log=FALSE )
summary(mod2b)
## Not run:
#############################################################################
# EXAMPLE 2: Linking longitudinal data
#############################################################################
data(data.long)
#******
# Model 1: Scaling with the 1PL model
# scaling at T1
dat1 <- data.long[, grep("T1", colnames(data.long) ) ]
resT1 <- sirt::rasch.mml2( dat1 )
itempartable1 <- data.frame( "study"="T1", resT1$item[, c("item", "a", "b" ) ] )
# scaling at T2
dat2 <- data.long[, grep("T2", colnames(data.long) ) ]
resT2 <- sirt::rasch.mml2( dat2 )
summary(resT2)
itempartable2 <- data.frame( "study"="T2", resT2$item[, c("item", "a", "b" ) ] )
itempartable <- rbind( itempartable1, itempartable2 )
itempartable[,2] <- substring( itempartable[,2], 1, 2 )
# estimate linking parameters
mod1 <- sirt::linking.haberman( itempars=itempartable )
#******
# Model 2: Scaling with the 2PL model
# scaling at T1
dat1 <- data.long[, grep("T1", colnames(data.long) ) ]
resT1 <- sirt::rasch.mml2( dat1, est.a=1:6)
itempartable1 <- data.frame( "study"="T1", resT1$item[, c("item", "a", "b" ) ] )
# scaling at T2
dat2 <- data.long[, grep("T2", colnames(data.long) ) ]
resT2 <- sirt::rasch.mml2( dat2, est.a=1:6)
summary(resT2)
itempartable2 <- data.frame( "study"="T2", resT2$item[, c("item", "a", "b" ) ] )
itempartable <- rbind( itempartable1, itempartable2 )
itempartable[,2] <- substring( itempartable[,2], 1, 2 )
# estimate linking parameters
mod2 <- sirt::linking.haberman( itempars=itempartable )
#############################################################################
# EXAMPLE 3: 2 Studies - 1PL and 2PL linking
#############################################################################
set.seed(789)
I <- 20 # number of items
N <- 2000 # number of persons
# define item parameters
b <- seq( -1.5, 1.5, length=I )
# simulate data
dat1 <- sirt::sim.raschtype( stats::rnorm( N, mean=0,sd=1 ), b=b )
dat2 <- sirt::sim.raschtype( stats::rnorm( N, mean=0.5,sd=1.50 ), b=b )
#*** Model 1: 1PL
# 1PL Study 1
mod1 <- sirt::rasch.mml2( dat1, est.a=rep(1,I) )
summary(mod1)
# 1PL Study 2
mod2 <- sirt::rasch.mml2( dat2, est.a=rep(1,I) )
summary(mod2)
# collect item parameters
dfr1 <- data.frame( "study1", mod1$item$item, mod1$item$a, mod1$item$b )
dfr2 <- data.frame( "study2", mod2$item$item, mod2$item$a, mod2$item$b )
colnames(dfr2) <- colnames(dfr1) <- c("study", "item", "a", "b" )
itempars <- rbind( dfr1, dfr2 )
# Haberman linking
linkhab1 <- sirt::linking.haberman(itempars=itempars)
## Transformation parameters (Haberman linking)
## study At Bt
## 1 study1 1.000 0.000
## 2 study2 1.465 -0.512
##
## Linear transformation for item parameters a and b
## study A_a A_b B_b
## 1 study1 1.000 1.000 0.000
## 2 study2 0.682 1.465 -0.512
##
## Linear transformation for person parameters theta
## study A_theta B_theta
## 1 study1 1.000 0.000
## 2 study2 1.465 0.512
##
## R-Squared Measures of Invariance
## slopes intercepts
## R2 1 0.9979
## sqrtU2 0 0.0456
#*** Model 2: 2PL
# 2PL Study 1
mod1 <- sirt::rasch.mml2( dat1, est.a=1:I )
summary(mod1)
# 2PL Study 2
mod2 <- sirt::rasch.mml2( dat2, est.a=1:I )
summary(mod2)
# collect item parameters
dfr1 <- data.frame( "study1", mod1$item$item, mod1$item$a, mod1$item$b )
dfr2 <- data.frame( "study2", mod2$item$item, mod2$item$a, mod2$item$b )
colnames(dfr2) <- colnames(dfr1) <- c("study", "item", "a", "b" )
itempars <- rbind( dfr1, dfr2 )
# Haberman linking
linkhab2 <- sirt::linking.haberman(itempars=itempars)
## Transformation parameters (Haberman linking)
## study At Bt
## 1 study1 1.000 0.000
## 2 study2 1.468 -0.515
##
## Linear transformation for item parameters a and b
## study A_a A_b B_b
## 1 study1 1.000 1.000 0.000
## 2 study2 0.681 1.468 -0.515
##
## Linear transformation for person parameters theta
## study A_theta B_theta
## 1 study1 1.000 0.000
## 2 study2 1.468 0.515
##
## R-Squared Measures of Invariance
## slopes intercepts
## R2 0.9984 0.9980
## sqrtU2 0.0397 0.0443
#############################################################################
# EXAMPLE 4: 3 Studies - 1PL and 2PL linking
#############################################################################
set.seed(789)
I <- 20 # number of items
N <- 1500 # number of persons
# define item parameters
b <- seq( -1.5, 1.5, length=I )
# simulate data
dat1 <- sirt::sim.raschtype( stats::rnorm( N, mean=0, sd=1), b=b )
dat2 <- sirt::sim.raschtype( stats::rnorm( N, mean=0.5, sd=1.50), b=b )
dat3 <- sirt::sim.raschtype( stats::rnorm( N, mean=-0.2, sd=0.8), b=b )
# set some items to non-administered
dat3 <- dat3[, -c(1,4) ]
dat2 <- dat2[, -c(1,2,3) ]
#*** Model 1: 1PL in sirt
# 1PL Study 1
mod1 <- sirt::rasch.mml2( dat1, est.a=rep(1,ncol(dat1)) )
summary(mod1)
# 1PL Study 2
mod2 <- sirt::rasch.mml2( dat2, est.a=rep(1,ncol(dat2)) )
summary(mod2)
# 1PL Study 3
mod3 <- sirt::rasch.mml2( dat3, est.a=rep(1,ncol(dat3)) )
summary(mod3)
# collect item parameters
dfr1 <- data.frame( "study1", mod1$item$item, mod1$item$a, mod1$item$b )
dfr2 <- data.frame( "study2", mod2$item$item, mod2$item$a, mod2$item$b )
dfr3 <- data.frame( "study3", mod3$item$item, mod3$item$a, mod3$item$b )
colnames(dfr3) <- colnames(dfr2) <- colnames(dfr1) <- c("study", "item", "a", "b" )
itempars <- rbind( dfr1, dfr2, dfr3 )
# use person parameters
personpars <- list( mod1$person[, c("EAP","SE.EAP") ], mod2$person[, c("EAP","SE.EAP") ],
mod3$person[, c("EAP","SE.EAP") ] )
# Haberman linking
linkhab1 <- sirt::linking.haberman(itempars=itempars, personpars=personpars)
# compare item parameters
round( cbind( linkhab1$joint.itempars[,-1], linkhab1$b.trans )[1:5,], 3 )
## aj bj study1 study2 study3
## I0001 0.998 -1.427 -1.427 NA NA
## I0002 0.998 -1.290 -1.324 NA -1.256
## I0003 0.998 -1.140 -1.068 NA -1.212
## I0004 0.998 -0.986 -1.003 -0.969 NA
## I0005 0.998 -0.869 -0.809 -0.872 -0.926
# summary of person parameters of second study
round( psych::describe( linkhab1$personpars[[2]] ), 2 )
## var n mean sd median trimmed mad min max range skew kurtosis
## EAP 1 1500 0.45 1.36 0.41 0.47 1.52 -2.61 3.25 5.86 -0.08 -0.62
## SE.EAP 2 1500 0.57 0.09 0.53 0.56 0.04 0.49 0.84 0.35 1.47 1.56
## se
## EAP 0.04
## SE.EAP 0.00
#*** Model 2: 2PL in TAM
library(TAM)
# 2PL Study 1
mod1 <- TAM::tam.mml.2pl( resp=dat1, irtmodel="2PL" )
pvmod1 <- TAM::tam.pv(mod1, ntheta=300, normal.approx=TRUE) # draw plausible values
summary(mod1)
# 2PL Study 2
mod2 <- TAM::tam.mml.2pl( resp=dat2, irtmodel="2PL" )
pvmod2 <- TAM::tam.pv(mod2, ntheta=300, normal.approx=TRUE)
summary(mod2)
# 2PL Study 3
mod3 <- TAM::tam.mml.2pl( resp=dat3, irtmodel="2PL" )
pvmod3 <- TAM::tam.pv(mod3, ntheta=300, normal.approx=TRUE)
summary(mod3)
# collect item parameters
#!! Note that in TAM the parametrization is a*theta - b while linking.haberman
#!! needs the parametrization a*(theta-b)
dfr1 <- data.frame( "study1", mod1$item$item, mod1$B[,2,1], mod1$xsi$xsi / mod1$B[,2,1] )
dfr2 <- data.frame( "study2", mod2$item$item, mod2$B[,2,1], mod2$xsi$xsi / mod2$B[,2,1] )
dfr3 <- data.frame( "study3", mod3$item$item, mod3$B[,2,1], mod3$xsi$xsi / mod3$B[,2,1] )
colnames(dfr3) <- colnames(dfr2) <- colnames(dfr1) <- c("study", "item", "a", "b" )
itempars <- rbind( dfr1, dfr2, dfr3 )
# define list containing person parameters
personpars <- list( pvmod1$pv[,-1], pvmod2$pv[,-1], pvmod3$pv[,-1] )
# Haberman linking
linkhab2 <- sirt::linking.haberman(itempars=itempars,personpars=personpars)
## Linear transformation for person parameters theta
## study A_theta B_theta
## 1 study1 1.000 0.000
## 2 study2 1.485 0.465
## 3 study3 0.786 -0.192
# extract transformed person parameters
personpars.trans <- linkhab2$personpars
#############################################################################
# EXAMPLE 5: Linking with simulated item parameters containing outliers
#############################################################################
# simulate some parameters
I <- 38
set.seed(18785)
b <- stats::rnorm( I, mean=.3, sd=1.4 )
# simulate DIF effects plus some outliers
bdif <- stats::rnorm(I,mean=.4,sd=.09)+( stats::runif(I)>.9 )* rep( 1*c(-1,1)+.4, each=I/2 )
# create item parameter table
itempars <- data.frame( "study"=paste0("study",rep(1:2, each=I)),
"item"=paste0( "I", 100 + rep(1:I,2) ), "a"=1,
"b"=c( b, b + bdif ) )
#*** Model 1: Haberman linking with least squares regression
mod1 <- sirt::linking.haberman( itempars=itempars )
summary(mod1)
#*** Model 2: Haberman linking with robust bisquare regression with fixed trimming value
mod2 <- sirt::linking.haberman( itempars=itempars, estimation="BSQ", b_trim=.4)
summary(mod2)
#*** Model 2: Haberman linking with robust bisquare regression with estimated trimming value
mod3 <- sirt::linking.haberman( itempars=itempars, estimation="BSQ")
summary(mod3)
## see also Example 3 of ?sirt::robust.linking
#############################################################################
# EXAMPLE 6: Toy example of Magis and De Boeck (2012)
#############################################################################
# define item parameters from Magis & De Boeck (20212, p. 293)
b1 <- c(1,1,1,1)
b2 <- c(1,1,1,2)
itempars <- data.frame(study=rep(1:2, each=4), item=rep(1:4,2), a=1, b=c(b1,b2) )
#- Least squares regression
mod1 <- sirt::linking.haberman( itempars=itempars, estimation="OLS")
summary(mod1)
#- Bisquare regression with estimated and fixed trimming factors
mod2 <- sirt::linking.haberman( itempars=itempars, estimation="BSQ")
mod2a <- sirt::linking.haberman( itempars=itempars, estimation="BSQ", b_trim=.4)
mod2b <- sirt::linking.haberman( itempars=itempars, estimation="BSQ", b_trim=1.2)
summary(mod2)
summary(mod2a)
summary(mod2b)
#- Least squares trimmed regression
mod3 <- sirt::linking.haberman( itempars=itempars, estimation="LTS")
summary(mod3)
#- median regression
mod4 <- sirt::linking.haberman( itempars=itempars, estimation="MED")
summary(mod4)
#############################################################################
# EXAMPLE 7: Simulated example with directional DIF
#############################################################################
set.seed(98)
I <- 8
mu <- c(-.5, 0, .5)
b <- sample(seq(-1.5,1.5, len=I))
sd_dif <- 0.001
pars <- outer(b, mu, "+") + stats::rnorm(I*3, sd=sd_dif)
ind <- c(1,2); pars[ind,1] <- pars[ind,1] + c(.5,.5)
ind <- c(3,4); pars[ind,2] <- pars[ind,2] + (-1)*c(.6,.6)
ind <- c(5,6); pars[ind,3] <- pars[ind,3] + (-1)*c(1,1)
# median polish (=stats::medpolish())
tmod1 <- sirt:::L1_polish(x=pars)
# L0 polish with tolerance criterion of .3
tmod2 <- sirt::L0_polish(x=pars, tol=.3)
#- prepare itempars input
itempars <- sirt::linking_haberman_itempars_prepare(b=pars)
#- compare different estimation functions for Haberman linking
mod01 <- sirt::linking.haberman(itempars, estimation="L1")
mod02 <- sirt::linking.haberman(itempars, estimation="L0", b_trim=.3)
mod1 <- sirt::linking.haberman(itempars, estimation="OLS")
mod2 <- sirt::linking.haberman(itempars, estimation="BSQ")
mod2a <- sirt::linking.haberman(itempars, estimation="BSQ", b_trim=.4)
mod3 <- sirt::linking.haberman(itempars, estimation="MED")
mod4 <- sirt::linking.haberman(itempars, estimation="LTS")
mod5 <- sirt::linking.haberman(itempars, estimation="HUB")
mod01$transf.pars
mod02$transf.pars
mod1$transf.pars
mod2$transf.pars
mod2a$transf.pars
mod3$transf.pars
mod4$transf.pars
mod5$transf.pars
#############################################################################
# EXAMPLE 8: Many studies and directional DIF
#############################################################################
## dataset 2
set.seed(98)
I <- 10 # number of items
S <- 7 # number of studies
mu <- round( seq(0, 1, len=S))
b <- sample(seq(-1.5,1.5, len=I))
sd_dif <- 0.001
pars0 <- pars <- outer(b, mu, "+") + stats::rnorm(I*S, sd=sd_dif)
# select n_dif items at random per group and set it to dif or -dif
n_dif <- 2
dif <- .6
for (ss in 1:S){
ind <- sample( 1:I, n_dif )
pars[ind,ss] <- pars[ind,ss] + dif*sign( runif(1) - .5 )
}
# check DIF
pars - pars0
#* estimate models
itempars <- sirt::linking_haberman_itempars_prepare(b=pars)
mod0 <- sirt::linking.haberman(itempars, estimation="L0", b_trim=.2)
mod1 <- sirt::linking.haberman(itempars, estimation="OLS")
mod2 <- sirt::linking.haberman(itempars, estimation="BSQ")
mod2a <- sirt::linking.haberman(itempars, estimation="BSQ", b_trim=.4)
mod3 <- sirt::linking.haberman(itempars, estimation="MED")
mod3a <- sirt::linking.haberman(itempars, estimation="L1")
mod4 <- sirt::linking.haberman(itempars, estimation="LTS")
mod5 <- sirt::linking.haberman(itempars, estimation="HUB")
mod0$transf.pars
mod1$transf.pars
mod2$transf.pars
mod2a$transf.pars
mod3$transf.pars
mod3a$transf.pars
mod4$transf.pars
mod5$transf.pars
#* compare results with Haebara linking
mod11 <- sirt::linking.haebara(itempars, dist="L2")
mod12 <- sirt::linking.haebara(itempars, dist="L1")
summary(mod11)
summary(mod12)
## End(Not run)
sirt documentation built on May 29, 2024, 8:43 a.m.
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| linking.haberman | R Documentation |
Linking in the 2PL/Generalized Partial Credit Model
Description
This function does the linking of several studies which are calibrated
using the 2PL or the generalized item response model according to
Haberman (2009). This method is a generalization of log-mean-mean
linking from one study to several studies. The default a_log=TRUE
logarithmizes item slopes for linking while otherwise an additive regression
model is assumed for the original item loadings (see Details; Battauz, 2017)
Usage
linking.haberman(itempars, personpars, estimation="OLS", a_trim=Inf, b_trim=Inf,
lts_prop=.5, a_log=TRUE, conv=1e-05, maxiter=1000, progress=TRUE,
adjust_main_effects=TRUE, vcov=TRUE)
## S3 method for class 'linking.haberman'
summary(object, digits=3, file=NULL, ...)
linking.haberman.lq(itempars, pow=2, eps=1e-3, a_log=TRUE, use_nu=FALSE,
est_pow=FALSE, lower_pow=.1, upper_pow=3)
## S3 method for class 'linking.haberman.lq'
summary(object, digits=3, file=NULL, ...)
## prepare 'itempars' argument for linking.haberman()
linking_haberman_itempars_prepare(b, a=NULL, wgt=NULL)
## conversion of different parameterizations of item parameters
linking_haberman_itempars_convert(itempars=NULL, lambda=NULL, nu=NULL, a=NULL, b=NULL)
## L0 polish precedure minimizing number of interactions in two-way table
L0_polish(x, tol, conv=0.01, maxiter=30, type=1, verbose=TRUE)
Arguments
itempars |
A data frame with four or five columns. The first four columns contain
in the order: study name, item name, |
personpars |
A list with vectors (e.g. EAPs or WLEs) or data frames
(e.g. plausible values) containing person parameters which
should be transformed.
If a data frame in each list entry has |
estimation |
Estimation method. Can be |
a_trim |
Trimming parameter for item slopes |
b_trim |
Trimming parameter for item slopes |
lts_prop |
Proportion of retained observations in |
a_log |
Logical indicating whether item slopes should be logarithmized for linking. |
conv |
Convergence criterion. |
maxiter |
Maximum number of iterations. |
progress |
An optional logical indicating whether computational progress should be displayed. |
adjust_main_effects |
Logical indicating whether all elements in the vector of main effects should be simultaneously adjusted |
vcov |
Optional indicating whether covariance matrix for linking errors should be computed |
pow |
Power |
eps |
Epsilon value used in differentiable approximating function |
use_nu |
Logical indicating whether item intercepts instead of item difficulties are used in linking |
est_pow |
Logical indicating whether power values should be estimated |
lower_pow |
Lower bound for estimated power |
upper_pow |
Upper bound for estimated power |
lambda |
Matrix containing item loadings |
nu |
Matrix containing item intercepts |
object |
Object of class |
digits |
Number of digits after decimals for rounding in |
file |
Optional file name if |
... |
Further arguments to be passed |
b |
Matrix of item intercepts (items |
a |
Matrix of item slopes |
wgt |
Matrix of weights |
x |
Matrix |
tol |
Tolerance value |
type |
Can be |
verbose |
Logical indicating whether iteration progress should be displayed |
Details
For t=1,\ldots,T studies, item difficulties b_{it} and
item slopes a_{it} are available. For dichotomous responses, these
parameters are defined by the 2PL response equation
logit P(X_{pi}=1| \theta_p )=a_i ( \theta_p - b_i )
while for polytomous responses the generalized partial credit model holds
log \frac{P(X_{pi}=k| \theta_p )}{P(X_{pi}=k-1| \theta_p )}
=a_i ( \theta_p - b_i + d_{ik} )
The parameters \{ a_{it}, b_{it} \} of all items and studies are
linearly transformed using equations a_{it} \approx a_i / A_t
(if a_log=TRUE) or a_{it} \approx a_i + A_t
(if a_log=FALSE) and
b_{it} \cdot A_t \approx B_t + b_i. For identification reasons,
we define A_1=1 and B_1=0.
The optimization function (which is a least squares criterion;
see Haberman, 2009) seeks the transformation parameters A_t and
B_t with an alternating least squares
method (estimation="OLS"). Note that every item i and every study t can
be weighted (specified in the fifth column of itempars).
Alternatively, a robust regression method based on bisquare weighting (Fox, 2015)
can be employed for linking using the argument estimation="BSQ".
For example, in the case of item loadings, bisquare weighting is applied to
residuals e_{it}=a_{it} - a_i - A_t (where logarithmized or non-logarithmized
item loadings are employed) forming weights
w_{it}=[ 1 - ( e_{it} / k )^2 ]^2 for e_{it} <k and 0 for e_{it} \ge k
where k is the trimming constant which can be estimated or fixed
during estimation using arguments a_trim or b_trim. Items in studies with
large residuals
(i.e., presence differential item functioning) are effectively set to zero in the
linking procedure. Alternatively, Huber weights (estimation="HUB") downweight
large residuals by applying w_{it}=k / | e_{it} | for residuals
|e_{it}|>k. The method estimation="LTS" employs trimmed least squares
where the proportion
of data retained is specified in lts_prop with default set to .50.
The method estimation="MED" estimates item parameters and linking constants
based on alternating median regression. A similar approach is the median polish
procedure of Tukey (Tukey, 1977, p. 362ff.; Maronna, Martin & Yohai, 2006, p. 104;
see also stats::medpolish) implemented in
estimation="L1" which aims to minimize \sum_{i,t} | e_{it} |.
For a pre-specified tolerance value t (in a_trim or b_trim),
the approach estimation="L0" minimizes the number of interactions
(i.e., DIF effects) in the e_{it} effects. In more detail, it minimizes
\sum_{i,t} \# \{ | e_{it} | > t \} which is computationally conducted
by repeatedly applying the median polish procedure in which one cell is
omitted (Davies, 2012; Terbeck & Davies, 1998).
Effect sizes of invariance are calculated as R-squared measures of explained item slopes and intercepts after linking in comparison to item parameters across groups (Asparouhov & Muthen, 2014).
The function linking.haberman.lq uses the loss function \rho(x)=|x|^q.
The originally proposed Haberman linking can be obtained with pow=2 (q=2).
The powers can also be estimated (argument est_pow=TRUE).
Value
A list with following entries
transf.pars |
Data frame with transformation parameters
|
transf.personpars |
Data frame with linear transformation functions for person parameters |
joint.itempars |
Estimated joint item parameters |
a.trans |
Transformed |
b.trans |
Transformed |
a.orig |
Original |
b.orig |
Original |
a.resid |
Residual |
b.resid |
Residual |
personpars |
Transformed person parameters |
es.invariance |
Effect size measures of invariance,
separately for item slopes and intercepts.
In the rows, |
es.robust |
Effect size measures of invariance based on robust estimation (if used). |
selitems |
Indices of items which are present in more than one study. |
References
Asparouhov, T., & Muthen, B. (2014). Multiple-group factor analysis alignment. Structural Equation Modeling, 21(4), 1-14. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/10705511.2014.919210")}
Battauz, M. (2017). Multiple equating of separate IRT calibrations. Psychometrika, 82(3), 610-636. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/s11336-016-9517-x")}
Davies, P. L. (2012). Interactions in the analysis of variance. Journal of the American Statistical Association, 107(500), 1502-1509. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/01621459.2012.726895")}
Fox, J. (2015). Applied regression analysis and generalized linear models. Thousand Oaks: Sage.
Haberman, S. J. (2009). Linking parameter estimates derived from an item response model through separate calibrations. ETS Research Report ETS RR-09-40. Princeton, ETS. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1002/j.2333-8504.2009.tb02197.x")}
Kolen, M. J., & Brennan, R. L. (2014). Test equating, scaling, and linking: Methods and practices. New York: Springer. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/978-1-4939-0317-7")}
Magis, D., & De Boeck, P. (2012). A robust outlier approach to prevent type I error inflation in differential item functioning. Educational and Psychological Measurement, 72(2), 291-311. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1177/0013164411416975")}
Maronna, R. A., Martin, R. D., & Yohai, V. J. (2006). Robust statistics. West Sussex: Wiley. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1002/0470010940")}
Terbeck, W., & Davies, P. L. (1998). Interactions and outliers in the two-way analysis of variance. Annals of Statistics, 26(4), 1279-1305. doi: 10.1214/aos/1024691243
Tukey, J. W. (1977). Exploratory data analysis. Addison-Wesley.
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.
\Sexpr[results=rd]{tools:::Rd_expr_doi("10.18637/jss.v035.i12")}
See Also
See the plink package (Weeks, 2010) for a diversity of linking methods.
Mean-mean linking, Stocking-Lord and Haebara linking (see Kolen & Brennan, 2014,
for an overview) in the generalized logistic item response model can be conducted with
equating.rasch. See also TAM::tam.linking
in the TAM package. Haebara linking and a robustified version of it can be
found in linking.haebara.
The invariance alignment method employs an optimization function based on
pairwise loss functions of item parameters (Asparouhov & Muthen, 2014),
see invariance.alignment.
Examples
#############################################################################
# EXAMPLE 1: Item parameters data.pars1.rasch and data.pars1.2pl
#############################################################################
# Model 1: Linking three studies calibrated by the Rasch model
data(data.pars1.rasch)
mod1 <- sirt::linking.haberman( itempars=data.pars1.rasch )
summary(mod1)
# Model 1b: Linking these studies but weigh these studies by
# proportion weights 3 : 0.5 : 1 (see below).
# All weights are the same for each item but they could also
# be item specific.
itempars <- data.pars1.rasch
itempars$wgt <- 1
itempars[ itempars$study=="study1","wgt"] <- 3
itempars[ itempars$study=="study2","wgt"] <- .5
mod1b <- sirt::linking.haberman( itempars=itempars )
summary(mod1b)
# Model 2: Linking three studies calibrated by the 2PL model
data(data.pars1.2pl)
mod2 <- sirt::linking.haberman( itempars=data.pars1.2pl )
summary(mod2)
# additive model instead of logarithmic model for item slopes
mod2b <- sirt::linking.haberman( itempars=data.pars1.2pl, a_log=FALSE )
summary(mod2b)
## Not run:
#############################################################################
# EXAMPLE 2: Linking longitudinal data
#############################################################################
data(data.long)
#******
# Model 1: Scaling with the 1PL model
# scaling at T1
dat1 <- data.long[, grep("T1", colnames(data.long) ) ]
resT1 <- sirt::rasch.mml2( dat1 )
itempartable1 <- data.frame( "study"="T1", resT1$item[, c("item", "a", "b" ) ] )
# scaling at T2
dat2 <- data.long[, grep("T2", colnames(data.long) ) ]
resT2 <- sirt::rasch.mml2( dat2 )
summary(resT2)
itempartable2 <- data.frame( "study"="T2", resT2$item[, c("item", "a", "b" ) ] )
itempartable <- rbind( itempartable1, itempartable2 )
itempartable[,2] <- substring( itempartable[,2], 1, 2 )
# estimate linking parameters
mod1 <- sirt::linking.haberman( itempars=itempartable )
#******
# Model 2: Scaling with the 2PL model
# scaling at T1
dat1 <- data.long[, grep("T1", colnames(data.long) ) ]
resT1 <- sirt::rasch.mml2( dat1, est.a=1:6)
itempartable1 <- data.frame( "study"="T1", resT1$item[, c("item", "a", "b" ) ] )
# scaling at T2
dat2 <- data.long[, grep("T2", colnames(data.long) ) ]
resT2 <- sirt::rasch.mml2( dat2, est.a=1:6)
summary(resT2)
itempartable2 <- data.frame( "study"="T2", resT2$item[, c("item", "a", "b" ) ] )
itempartable <- rbind( itempartable1, itempartable2 )
itempartable[,2] <- substring( itempartable[,2], 1, 2 )
# estimate linking parameters
mod2 <- sirt::linking.haberman( itempars=itempartable )
#############################################################################
# EXAMPLE 3: 2 Studies - 1PL and 2PL linking
#############################################################################
set.seed(789)
I <- 20 # number of items
N <- 2000 # number of persons
# define item parameters
b <- seq( -1.5, 1.5, length=I )
# simulate data
dat1 <- sirt::sim.raschtype( stats::rnorm( N, mean=0,sd=1 ), b=b )
dat2 <- sirt::sim.raschtype( stats::rnorm( N, mean=0.5,sd=1.50 ), b=b )
#*** Model 1: 1PL
# 1PL Study 1
mod1 <- sirt::rasch.mml2( dat1, est.a=rep(1,I) )
summary(mod1)
# 1PL Study 2
mod2 <- sirt::rasch.mml2( dat2, est.a=rep(1,I) )
summary(mod2)
# collect item parameters
dfr1 <- data.frame( "study1", mod1$item$item, mod1$item$a, mod1$item$b )
dfr2 <- data.frame( "study2", mod2$item$item, mod2$item$a, mod2$item$b )
colnames(dfr2) <- colnames(dfr1) <- c("study", "item", "a", "b" )
itempars <- rbind( dfr1, dfr2 )
# Haberman linking
linkhab1 <- sirt::linking.haberman(itempars=itempars)
## Transformation parameters (Haberman linking)
## study At Bt
## 1 study1 1.000 0.000
## 2 study2 1.465 -0.512
##
## Linear transformation for item parameters a and b
## study A_a A_b B_b
## 1 study1 1.000 1.000 0.000
## 2 study2 0.682 1.465 -0.512
##
## Linear transformation for person parameters theta
## study A_theta B_theta
## 1 study1 1.000 0.000
## 2 study2 1.465 0.512
##
## R-Squared Measures of Invariance
## slopes intercepts
## R2 1 0.9979
## sqrtU2 0 0.0456
#*** Model 2: 2PL
# 2PL Study 1
mod1 <- sirt::rasch.mml2( dat1, est.a=1:I )
summary(mod1)
# 2PL Study 2
mod2 <- sirt::rasch.mml2( dat2, est.a=1:I )
summary(mod2)
# collect item parameters
dfr1 <- data.frame( "study1", mod1$item$item, mod1$item$a, mod1$item$b )
dfr2 <- data.frame( "study2", mod2$item$item, mod2$item$a, mod2$item$b )
colnames(dfr2) <- colnames(dfr1) <- c("study", "item", "a", "b" )
itempars <- rbind( dfr1, dfr2 )
# Haberman linking
linkhab2 <- sirt::linking.haberman(itempars=itempars)
## Transformation parameters (Haberman linking)
## study At Bt
## 1 study1 1.000 0.000
## 2 study2 1.468 -0.515
##
## Linear transformation for item parameters a and b
## study A_a A_b B_b
## 1 study1 1.000 1.000 0.000
## 2 study2 0.681 1.468 -0.515
##
## Linear transformation for person parameters theta
## study A_theta B_theta
## 1 study1 1.000 0.000
## 2 study2 1.468 0.515
##
## R-Squared Measures of Invariance
## slopes intercepts
## R2 0.9984 0.9980
## sqrtU2 0.0397 0.0443
#############################################################################
# EXAMPLE 4: 3 Studies - 1PL and 2PL linking
#############################################################################
set.seed(789)
I <- 20 # number of items
N <- 1500 # number of persons
# define item parameters
b <- seq( -1.5, 1.5, length=I )
# simulate data
dat1 <- sirt::sim.raschtype( stats::rnorm( N, mean=0, sd=1), b=b )
dat2 <- sirt::sim.raschtype( stats::rnorm( N, mean=0.5, sd=1.50), b=b )
dat3 <- sirt::sim.raschtype( stats::rnorm( N, mean=-0.2, sd=0.8), b=b )
# set some items to non-administered
dat3 <- dat3[, -c(1,4) ]
dat2 <- dat2[, -c(1,2,3) ]
#*** Model 1: 1PL in sirt
# 1PL Study 1
mod1 <- sirt::rasch.mml2( dat1, est.a=rep(1,ncol(dat1)) )
summary(mod1)
# 1PL Study 2
mod2 <- sirt::rasch.mml2( dat2, est.a=rep(1,ncol(dat2)) )
summary(mod2)
# 1PL Study 3
mod3 <- sirt::rasch.mml2( dat3, est.a=rep(1,ncol(dat3)) )
summary(mod3)
# collect item parameters
dfr1 <- data.frame( "study1", mod1$item$item, mod1$item$a, mod1$item$b )
dfr2 <- data.frame( "study2", mod2$item$item, mod2$item$a, mod2$item$b )
dfr3 <- data.frame( "study3", mod3$item$item, mod3$item$a, mod3$item$b )
colnames(dfr3) <- colnames(dfr2) <- colnames(dfr1) <- c("study", "item", "a", "b" )
itempars <- rbind( dfr1, dfr2, dfr3 )
# use person parameters
personpars <- list( mod1$person[, c("EAP","SE.EAP") ], mod2$person[, c("EAP","SE.EAP") ],
mod3$person[, c("EAP","SE.EAP") ] )
# Haberman linking
linkhab1 <- sirt::linking.haberman(itempars=itempars, personpars=personpars)
# compare item parameters
round( cbind( linkhab1$joint.itempars[,-1], linkhab1$b.trans )[1:5,], 3 )
## aj bj study1 study2 study3
## I0001 0.998 -1.427 -1.427 NA NA
## I0002 0.998 -1.290 -1.324 NA -1.256
## I0003 0.998 -1.140 -1.068 NA -1.212
## I0004 0.998 -0.986 -1.003 -0.969 NA
## I0005 0.998 -0.869 -0.809 -0.872 -0.926
# summary of person parameters of second study
round( psych::describe( linkhab1$personpars[[2]] ), 2 )
## var n mean sd median trimmed mad min max range skew kurtosis
## EAP 1 1500 0.45 1.36 0.41 0.47 1.52 -2.61 3.25 5.86 -0.08 -0.62
## SE.EAP 2 1500 0.57 0.09 0.53 0.56 0.04 0.49 0.84 0.35 1.47 1.56
## se
## EAP 0.04
## SE.EAP 0.00
#*** Model 2: 2PL in TAM
library(TAM)
# 2PL Study 1
mod1 <- TAM::tam.mml.2pl( resp=dat1, irtmodel="2PL" )
pvmod1 <- TAM::tam.pv(mod1, ntheta=300, normal.approx=TRUE) # draw plausible values
summary(mod1)
# 2PL Study 2
mod2 <- TAM::tam.mml.2pl( resp=dat2, irtmodel="2PL" )
pvmod2 <- TAM::tam.pv(mod2, ntheta=300, normal.approx=TRUE)
summary(mod2)
# 2PL Study 3
mod3 <- TAM::tam.mml.2pl( resp=dat3, irtmodel="2PL" )
pvmod3 <- TAM::tam.pv(mod3, ntheta=300, normal.approx=TRUE)
summary(mod3)
# collect item parameters
#!! Note that in TAM the parametrization is a*theta - b while linking.haberman
#!! needs the parametrization a*(theta-b)
dfr1 <- data.frame( "study1", mod1$item$item, mod1$B[,2,1], mod1$xsi$xsi / mod1$B[,2,1] )
dfr2 <- data.frame( "study2", mod2$item$item, mod2$B[,2,1], mod2$xsi$xsi / mod2$B[,2,1] )
dfr3 <- data.frame( "study3", mod3$item$item, mod3$B[,2,1], mod3$xsi$xsi / mod3$B[,2,1] )
colnames(dfr3) <- colnames(dfr2) <- colnames(dfr1) <- c("study", "item", "a", "b" )
itempars <- rbind( dfr1, dfr2, dfr3 )
# define list containing person parameters
personpars <- list( pvmod1$pv[,-1], pvmod2$pv[,-1], pvmod3$pv[,-1] )
# Haberman linking
linkhab2 <- sirt::linking.haberman(itempars=itempars,personpars=personpars)
## Linear transformation for person parameters theta
## study A_theta B_theta
## 1 study1 1.000 0.000
## 2 study2 1.485 0.465
## 3 study3 0.786 -0.192
# extract transformed person parameters
personpars.trans <- linkhab2$personpars
#############################################################################
# EXAMPLE 5: Linking with simulated item parameters containing outliers
#############################################################################
# simulate some parameters
I <- 38
set.seed(18785)
b <- stats::rnorm( I, mean=.3, sd=1.4 )
# simulate DIF effects plus some outliers
bdif <- stats::rnorm(I,mean=.4,sd=.09)+( stats::runif(I)>.9 )* rep( 1*c(-1,1)+.4, each=I/2 )
# create item parameter table
itempars <- data.frame( "study"=paste0("study",rep(1:2, each=I)),
"item"=paste0( "I", 100 + rep(1:I,2) ), "a"=1,
"b"=c( b, b + bdif ) )
#*** Model 1: Haberman linking with least squares regression
mod1 <- sirt::linking.haberman( itempars=itempars )
summary(mod1)
#*** Model 2: Haberman linking with robust bisquare regression with fixed trimming value
mod2 <- sirt::linking.haberman( itempars=itempars, estimation="BSQ", b_trim=.4)
summary(mod2)
#*** Model 2: Haberman linking with robust bisquare regression with estimated trimming value
mod3 <- sirt::linking.haberman( itempars=itempars, estimation="BSQ")
summary(mod3)
## see also Example 3 of ?sirt::robust.linking
#############################################################################
# EXAMPLE 6: Toy example of Magis and De Boeck (2012)
#############################################################################
# define item parameters from Magis & De Boeck (20212, p. 293)
b1 <- c(1,1,1,1)
b2 <- c(1,1,1,2)
itempars <- data.frame(study=rep(1:2, each=4), item=rep(1:4,2), a=1, b=c(b1,b2) )
#- Least squares regression
mod1 <- sirt::linking.haberman( itempars=itempars, estimation="OLS")
summary(mod1)
#- Bisquare regression with estimated and fixed trimming factors
mod2 <- sirt::linking.haberman( itempars=itempars, estimation="BSQ")
mod2a <- sirt::linking.haberman( itempars=itempars, estimation="BSQ", b_trim=.4)
mod2b <- sirt::linking.haberman( itempars=itempars, estimation="BSQ", b_trim=1.2)
summary(mod2)
summary(mod2a)
summary(mod2b)
#- Least squares trimmed regression
mod3 <- sirt::linking.haberman( itempars=itempars, estimation="LTS")
summary(mod3)
#- median regression
mod4 <- sirt::linking.haberman( itempars=itempars, estimation="MED")
summary(mod4)
#############################################################################
# EXAMPLE 7: Simulated example with directional DIF
#############################################################################
set.seed(98)
I <- 8
mu <- c(-.5, 0, .5)
b <- sample(seq(-1.5,1.5, len=I))
sd_dif <- 0.001
pars <- outer(b, mu, "+") + stats::rnorm(I*3, sd=sd_dif)
ind <- c(1,2); pars[ind,1] <- pars[ind,1] + c(.5,.5)
ind <- c(3,4); pars[ind,2] <- pars[ind,2] + (-1)*c(.6,.6)
ind <- c(5,6); pars[ind,3] <- pars[ind,3] + (-1)*c(1,1)
# median polish (=stats::medpolish())
tmod1 <- sirt:::L1_polish(x=pars)
# L0 polish with tolerance criterion of .3
tmod2 <- sirt::L0_polish(x=pars, tol=.3)
#- prepare itempars input
itempars <- sirt::linking_haberman_itempars_prepare(b=pars)
#- compare different estimation functions for Haberman linking
mod01 <- sirt::linking.haberman(itempars, estimation="L1")
mod02 <- sirt::linking.haberman(itempars, estimation="L0", b_trim=.3)
mod1 <- sirt::linking.haberman(itempars, estimation="OLS")
mod2 <- sirt::linking.haberman(itempars, estimation="BSQ")
mod2a <- sirt::linking.haberman(itempars, estimation="BSQ", b_trim=.4)
mod3 <- sirt::linking.haberman(itempars, estimation="MED")
mod4 <- sirt::linking.haberman(itempars, estimation="LTS")
mod5 <- sirt::linking.haberman(itempars, estimation="HUB")
mod01$transf.pars
mod02$transf.pars
mod1$transf.pars
mod2$transf.pars
mod2a$transf.pars
mod3$transf.pars
mod4$transf.pars
mod5$transf.pars
#############################################################################
# EXAMPLE 8: Many studies and directional DIF
#############################################################################
## dataset 2
set.seed(98)
I <- 10 # number of items
S <- 7 # number of studies
mu <- round( seq(0, 1, len=S))
b <- sample(seq(-1.5,1.5, len=I))
sd_dif <- 0.001
pars0 <- pars <- outer(b, mu, "+") + stats::rnorm(I*S, sd=sd_dif)
# select n_dif items at random per group and set it to dif or -dif
n_dif <- 2
dif <- .6
for (ss in 1:S){
ind <- sample( 1:I, n_dif )
pars[ind,ss] <- pars[ind,ss] + dif*sign( runif(1) - .5 )
}
# check DIF
pars - pars0
#* estimate models
itempars <- sirt::linking_haberman_itempars_prepare(b=pars)
mod0 <- sirt::linking.haberman(itempars, estimation="L0", b_trim=.2)
mod1 <- sirt::linking.haberman(itempars, estimation="OLS")
mod2 <- sirt::linking.haberman(itempars, estimation="BSQ")
mod2a <- sirt::linking.haberman(itempars, estimation="BSQ", b_trim=.4)
mod3 <- sirt::linking.haberman(itempars, estimation="MED")
mod3a <- sirt::linking.haberman(itempars, estimation="L1")
mod4 <- sirt::linking.haberman(itempars, estimation="LTS")
mod5 <- sirt::linking.haberman(itempars, estimation="HUB")
mod0$transf.pars
mod1$transf.pars
mod2$transf.pars
mod2a$transf.pars
mod3$transf.pars
mod3a$transf.pars
mod4$transf.pars
mod5$transf.pars
#* compare results with Haebara linking
mod11 <- sirt::linking.haebara(itempars, dist="L2")
mod12 <- sirt::linking.haebara(itempars, dist="L1")
summary(mod11)
summary(mod12)
## End(Not run)
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