irt.link: IRT parameter linking methods
In jagonzalb/SNSequate: Standard and Nonstandard Statistical Models and Methods for Test Equating

Description Usage Arguments Details Value Note Author(s) References See Also Examples

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

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

`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)
`...`	Further arguments currently not used.

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|theta_i)=cj+(1-cj){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).

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

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

Jorge Gonzalez B. jgonzale@mat.puc.cl

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

mea.eq, lin.eq, ker.eq

#### 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)