LordChi2: Lord's chi-square DIF statistic
In difR: Collection of Methods to Detect Dichotomous Differential Item Functioning (DIF)

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

Calculates the Lord's chi-square statistics for DIF detection.

1 2	LordChi2(mR, mF)

`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.

This command computes the Lord's chi-square statistic (Lord, 1980) in the specific framework of differential item functioning. It forms the basic command of difLord and is specifically designed for this call.

The matrices mR and mF must have the same format as the output of the command itemParEst with one the possible models (1PL, 2PL, 3PL or constrained 3PL). The number of columns therefore equals two, five, nine or six, respectively. 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.

A vector with the values of the Lord's chi-square DIF statistics.

WARNING: the previous versions of LordChi2 were holding an error: under the 3PL model, the covariance matrices Sig_1 and Sig_2 were wrongly computed as the variance of the pseudo-guessing parameters were replaced by the parameter estimates. This has been fixed from version 4.0 of difR. Many thanks to J. Patrick Meyer (Curry School of Education, University of Virginia) for having discovered this mistake.

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.

Lord, F. (1980). Applications of item response theory to practical testing problems. Hillsdale, NJ: Lawrence Erlbaum Associates.

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

itemParEst, itemRescale, difLord, 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)
 LordChi2(mR, mF)

 # Pre-estimation of the item parameters (2PL model)
 mR <- itemParEst(data.ref, model = "2PL")
 mF <- itemParEst(data.focal, model = "2PL")
 mF <- itemRescale(mR, mF)
 LordChi2(mR, mF)

 # 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)
 LordChi2(mR, mF)
 
## End(Not run)