RajuZ: Raju's area DIF statistic
In difR: Collection of Methods to Detect Dichotomous Differential Item Functioning (DIF)

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

Calculates the Raju's statistics for DIF detection.

1 2	RajuZ(mR, mF, signed = FALSE)

`mR`	numeric: the matrix of item parameter estimates (one row per item) for the reference group. See Details.
`mF`	numeric: the matrix of item parameter estimates (one row per item) for the focal group. See Details.
`signed`	logical: should the signed area be computed, or the unsigned (i.e. in absolute value) ara? Default is `FALSE`, i.e. the unsigned area. See Details.

This command computes the Raju's area statistic (Raju, 1988, 1990) in the specific framework of differential item functioning. It forms the basic command of difRaju and is specifically designed for this call.

The matrices mR and mF must have the same format as the output of the command itemParEst and one the possible models (1PL, 2PL or constrained 3PL). The number of columns therefore equals two, five or six, respectively. Note that the unconstrained 3PL model cannot be used in this method: all pseudo-guessing parameters must be equal in both groups of subjects. Moreover, item parameters of the focal must be on the same scale of that of the reference group. If not, make use of e.g. equal means anchoring (Cook and Eignor, 1991) and itemRescale to transform them adequately.

By default, the unsigned area, given by Equation (57) in Raju (1990), is computed. It makes use of Equations (14), (15), (23) and (46) for the numerator, and Equations (17), (33) to (39), and (52) for the denominator of the Z statistic. However, the signed area, given by Equation (56) in Raju (1990), can be used instead. In this case, Equations (14), (21) and (44) are used for the numerator, and Equations (17), (25) and (48) for the denominator. The choice of the type of area is fixed by the logical signed argument, with default value FALSE.

A list with two components:

`res`	a matrix with one row per item and three columns, holding respectively Raju's area between the two item characteristic curves, its standard error and the Raju DIF statistic (the latter being the ratio of the first two columns).
`signed`	the value of the `signed` argument.

Sebastien Beland
Collectif pour le Developpement et les Applications en Mesure et Evaluation (Cdame)
Universite du Quebec a Montreal
sebastien.beland.1@hotmail.com, http://www.cdame.uqam.ca/
David Magis
Department of Psychology, University of Liege
Research Group of Quantitative Psychology and Individual Differences, KU Leuven
David.Magis@uliege.be, http://ppw.kuleuven.be/okp/home/
Gilles Raiche
Collectif pour le Developpement et les Applications en Mesure et Evaluation (Cdame)
Universite du Quebec a Montreal
raiche.gilles@uqam.ca, http://www.cdame.uqam.ca/

Cook, L. L. and Eignor, D. R. (1991). An NCME instructional module on IRT equating methods. Educational Measurement: Issues and Practice, 10, 37-45.

Magis, D., Beland, S., Tuerlinckx, F. and De Boeck, P. (2010). A general framework and an R package for the detection of dichotomous differential item functioning. Behavior Research Methods, 42, 847-862. doi: 10.3758/BRM.42.3.847

Raju, N.S. (1988). The area between two item characteristic curves. Psychometrika, 53, 495-502. doi: 10.1007/BF02294403

Raju, N. S. (1990). Determining the significance of estimated signed and unsigned areas between two item response functions. Applied Psychological Measurement, 14, 197-207. doi: 10.1177/014662169001400208

itemParEst, itemRescale, difRaju, dichoDif

## Not run: 

 # Loading of the verbal data
 data(verbal)
 attach(verbal)

 # Splitting the data into reference and focal groups
 nF <- sum(Gender)
 nR <- nrow(verbal)-nF
 data.ref <- verbal[,1:24][order(Gender),][1:nR,]
 data.focal <- verbal[,1:24][order(Gender),][(nR+1):(nR+nF),]

 # Pre-estimation of the item parameters (1PL model)
 mR <- itemParEst(data.ref,model = "1PL")
 mF <- itemParEst(data.focal,model = "1PL")
 mF <- itemRescale(mR, mF)

 # Signed and unsigned Raju statistics
 RajuZ(mR, mF)
 RajuZ(mR, mF, signed = TRUE)

 # Pre-estimation of the item parameters (2PL model)
 mR <- itemParEst(data.ref, model = "2PL")
 mF <- itemParEst(data.focal, model = "2PL")
 mF <- itemRescale(mR, mF)

 # Signed and unsigned Raju statistics
 RajuZ(mR, mF)
 RajuZ(mR, mF, signed = TRUE)
 
 # Pre-estimation of the item parameters (constrained 3PL model)
 mR <- itemParEst(data.ref, model = "3PL", c = 0.05)
 mF <- itemParEst(data.focal, model = "3PL", c =0 .05)
 mF <- itemRescale(mR, mF)
 
 # Signed and unsigned Raju statistics
 RajuZ(mR, mF)
 RajuZ(mR, mF, signed = TRUE)
 
## End(Not run)