irt.link {SNSequate}

IRT parameter linking methods

Description

The function implements parameter linking methods to transform IRT scales. Mean-mean, mean-sigma, Haebara, and Stocking and Lord methods are available (see details).

Usage

irt.link(parm, common, model, icc, D)

Arguments

parm

A 6 column matrix containing item parameter estimates from an IRT model. The first three columns contains the parameters for the form Y fit, and the last three those of form X. The order for item paramters in the matrix is discrimination, difficulty, and guessing. See details.

common

A vector indicating the position where common items are located

model

A character string indicating the underlying IRT model: "1PL", "2PL", "3PL".

icc

A character string indicating the type of icc used in the characteristic curve methods (see details). Available options are "logistic" and "cloglog".

D

A number indicating the value of the constant D (see details)

Details

The function implments various methods of IRT parameter linking (a.k.a, scale transformation methods). It calculates the linking constants A and B to tranform parameter estimates. When assuming a 1PL model, the matrix parm should contain a column of ones and a column of zeroes in the places where discrimination and guessing parameters are located, respectively.

The characteristic curve methods (Haebara and Stocking and Lord) rely on the item characteristic curve p_{ij}assumed for the probability of a correct answer

p_{ij}=P(Y_{ij}=1\mid\theta_i)=c_j+(1-c_j)\frac{\exp[Da_j(\theta_i-\beta_j)]}{1+\exp[Da_j(\theta_i-\beta_j)]}

Besides the traditional logistic model, the irt.link() function allows the use of an asymetric cloglog ICC. See the help for KB36.1PL data set, where some details on how to fit a 1PL model with cloglog link in lmer are given.

For more details on characteristic curve methods see Kolen and Brennan (2004).

Value

A list with the constants A and B calculated using the four different methods

Note

Currently, the cloglog ICC is only implmented for the 1PL model. A 1PL model with asymetric cloglog link can be fitted in R using the lmer() function in package lme4

Author(s)

Jorge Gonzalez [e-mail]

References

Gonzalez, J. (2014). SNSequate: Standard and Nonstandard Statistical Models and Methods for Test Equating. Journal of Statistical Software, 59(7), 1-30.

Kolen, M., and Brennan, R. (2004). Test Equating, Scaling and Linking. New York, NY: Springer-Verlag.

Estay, G. (2012). Characteristic Curves Scale Transformation Methods Using Asymmetric ICCs for IRT Equating. Unpublished MSc. Thesis. Pontificia Universidad Catolica de Chile

See Also

mea.eq, lin.eq, ker.eq

Examples

#### Example. KB, Table 6.6
data(KB36)
parm.x = KB36$KBformX_par
parm.y = KB36$KBformY_par	
comitems = seq(3,36,3)
parm = as.data.frame(cbind(parm.y, parm.x))

# Table 6.6 KB
irt.link(parm,comitems,model="3PL",icc="logistic",D=1.7)


# Same data but assuming a 1PL model. The parameter estimates are load from 
# the KB36.1PL data set. See the help for KB36.1PL data for details on how these
# estimates were obtained using \code{lmer()} (see also Table 6.13 in KB)
 
data(KB36.1PL)

#preparing the input data matrices for irt.link() function
b.log.y<-KB36.1PL$b.logistic[,2]
b.log.x<-KB36.1PL$b.logistic[,1]
b.clog.y<-KB36.1PL$b.cloglog[,2]
b.clog.x<-KB36.1PL$b.cloglog[,1]

parm2 = as.data.frame(cbind(1,b.log.y,0, 1,b.log.x, 0))
parm3 = as.data.frame(cbind(1,b.clog.y,0, 1,b.clog.x,0))

#vector indicating common items
comitems = seq(3,36,3)

#Calculating the B constant under the logistic-link model
irt.link(parm2,comitems,model="1PL",icc="logistic",D=1.7)

#Calculating the B constant under the cloglog-link model
irt.link(parm3,comitems,model="1PL",icc="cloglog",D=1.7)