difGenLord {difR}R Documentation

Generalized Lord's chi-square DIF method

Description

Performs DIF detection among multiple groups using generalized Lord's chi-square method.

Usage

 difGenLord(Data, group, focal.names, model, c=NULL, engine="ltm", 
 irtParam=NULL, nrFocal=2, same.scale=TRUE, alpha=0.05, 
 purify=FALSE, nrIter=10)
 ## S3 method for class 'GenLord':
 print(x, ...)
 ## S3 method for class 'GenLord':
 plot(x, pch=8, number=TRUE, col="red", ...)
 

Arguments

Data numeric: either the data matrix only, or the data matrix plus the vector of group membership. See Details.
group numeric or character: either the vector of group membership or the column indicator (within Data) of group membership. See Details.
focal.names numeric or character vector indicating the levels of group which correspond to the focal groups.
model character: the IRT model to be fitted (either "1PL", "2PL" or "3PL").
c optional numeric value or vector giving the values of the constrained pseudo-guessing parameters. See Details.
engine character: the engine for estimating the 1PL model, either "ltm" (default) or "lme4".
irtParam matrix with 2J rows (where J is the number of items) and at most 9 columns containing item parameters estimates. See Details.
nrFocal numeric: the number of focal groups (default is 2).
same.scale logical: are the item parameters of the irtParam matrix on the same scale? (default is TRUE). See Details.
alpha numeric: significance level (default is 0.05).
purify logical: should the method be used iteratively to purify the set of anchor items? (default is FALSE).
nrIter numeric: the maximal number of iterations in the item purification process. Default is 10.
x the result from a GenLord class object.
pch, col type of usual pch and col graphical options.
number logical: should the item number identification be printed (default is TRUE).
... other generic parameters for the plot or the print functions.

Details

The generalized Lord's chi-square method (Kim, Cohen and Park, 1995), also referred to as Qj statistic, allows for detecting uniform or non-uniform differential item functioning among multiple groups by setting an appropriate item response model. The input can be of two kinds: either by displaying the full data, the group membership, the focal groups and the model, or by giving the item parameter estimates (with the option irtParam). Both can be supplied, but in this case only the parameters in irtParam are used for computing generalized Lord's chi-square statistic.

The Data is a matrix whose rows correspond to the subjects and columns to the items. Missing values are not allowed. In addition, Data can hold the vector of group membership. If so, group indicates the column of Data which corresponds to the group membership, either by specifying its name or by giving the column number. Otherwise, group must be a vector of same length as nrow(Data).

The vector of group membership must hold only at least three different values, either as numeric or character. The focal groups are defined by the values of the argument focal.names.

If the model is not the 1PL model, or if engine is equal to "ltm", the selected IRT model is fitted using marginal maximum likelihood by means of the functions from the ltm package (Rizopoulos, 2006). Otherwise, the 1PL model is fitted as a generalized linear mixed model, by means of the glmer function of the lme4 package (Bates and Maechler, 2009). The 3PL model can be fitted either unconstrained (by setting c to NULL) or by fixing the pseudo-guessing values. In the latter case, the argument c is either a numeric vector of same length of the number of items, with one value per item pseudo-guessing parameter, or a single value which is duplicated for all the items. If c is different from NULL then the 3PL model is always fitted (whatever the value of model).

The irtParam matrix has a number of rows equal to the number of groups (reference and focal ones) times the number of items J. The first J rows refer to the item parameter estimates in the reference group, while the next sets of J rows correspond to the same items in each of the focal groups. The number of columns depends on the selected IRT model: 2 for the 1PL model, 5 for the 2PL model, 6 for the constrained 3PL model and 9 for the unconstrained 3PL model. The columns of irtParam have to follow the same structure as the output of itemParEst command (the latter can actually be used to create the irtParam matrix). The number of focal groups has to be specified with argument nrFocal (default value is 2).

In addition to the matrix of parameter estimates, one has to specify whether items in the focal groups were rescaled to those of the reference group. If not, rescaling is performed by equal means anchoring (Cook and Eignor, 1991). Argument same.scale is used for this choice (default option is TRUE and assumes therefore that the parameters are already placed on a same scale).

The threshold (or cut-score) for classifying items as DIF is computed as the quantile of the chi-square distribution with lower-tail probability of one minus alpha and p degrees of freedom. The value of p is the product of the number of focal groups by the number of item parameters to be tested (1 for the 1PL model, 2 for the 2PL model or the constrained 3PL model, and 3 for the unconstrained 3PL model).

Item purification can be performed by setting purify to TRUE. In this case, the purification occurs in the equal means anchoring process: items detected as DIF are iteratively removed from the set of items used for equal means anchoring, and the procedure is repeated until either the same items are identified twice as functioning differently, or when nrIter iterations have been performed. In the latter case a warning message is printed. See Candell and Drasgow (1988) for further details.

Value

A list of class "GenLord" with the following arguments:

genLordChi the values of the generalized Lord's chi-square statistics.
alpha the value of alpha argument.
thr the threshold (cut-score) for DIF detection.
df the degrees of freedom of the asymptotic null distribution of the statistics.
DIFitems either the column indicators of the items which were detected as DIF items, or "No DIF item detected".
purification the value of purify option.
nrPur the number of iterations in the item purification process. Returned only if purify is TRUE.
difPur a binary matrix with one row per iteration in the item purification process and one column per item. Zeros and ones in the i-th row refer to items which were classified respectively as non-DIF and DIF items at the (i-1)-th step. The first row corresponds to the initial classification of the items. Returned only if purify is TRUE.
convergence logical indicating whether the iterative item purification process stopped before the maximal number nrIterof allowed iterations. Returned only if purify is TRUE.
model the value of model argument.
c The value of the c argument.
engine The value of the engine argument.
itemParInit the matrix of initial parameter estimates,with the same format as irtParam either provided by the user (through irtParam) or estimated from the data (and displayed without rescaling).
itemParFinal the matrix of final parameter estimates, with the same format as irtParam, obtained after item purification. Returned only if purify is TRUE.
estPar a logical value indicating whether the item parameters were estimated (TRUE) or provided by the user (FALSE).
names the names of the items.

Author(s)

Sebastien Beland
Centre sur les Applications des Modeles de Reponses aux Items (CAMRI)
Universite du Quebec a Montreal
sebastien.beland.1@hotmail.com
David Magis
Research Group of Quantitative Psychology and Individual Differences
Katholieke Universiteit Leuven
David.Magis@psy.kuleuven.be, http://ppw.kuleuven.be/okp/home/
Gilles Raiche
Centre sur les Applications des Modeles de Reponses aux Items (CAMRI)
Universite du Quebec a Montreal
raiche.gilles@uqam.ca, http://www.er.uqam.ca/nobel/r17165/

References

Bates, D. and Maechler, M. (2009). lme4: Linear mixed-effects models using S4 classes. R package version 0.999375-31. http://CRAN.R-project.org/package=lme4

Candell, G.L. and Drasgow, F. (1988). An iterative procedure for linking metrics and assessing item bias in item response theory. Applied Psychological Measurement, 12, 253-260.

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

Kim, S.-H., Cohen, A.S. and Park, T.-H. (1995). Detection of differential item functioning in multiple groups. Journal of Educational Measurement, 32, 261-276.

Rizopoulos, D. (2006). ltm: An R package for latent variable modelling and item response theory analyses. Journal of Statistical Software, 17, 1-25. URL: http://www.jstatsoft.org/v17/i05/

See Also

itemParEst

Examples

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

# Creating four groups according to gender ("Man" or "Woman") and trait 
# anger score ("Low" or "High")
group<-rep("WomanLow",nrow(verbal))
group[Anger>20 & Gender==0]<-"WomanHigh"
group[Anger<=20 & Gender==1]<-"ManLow"
group[Anger>20 & Gender==1]<-"ManHigh"

# New data set
Verbal<-cbind(verbal[,1:24],group)

# Reference group: "WomanLow"
names<-c("WomanHigh","ManLow","ManHigh")

# Three equivalent settings of the data matrix and the group membership
# 1PL model, "ltm" engine (remove #)

# difGenLord(Verbal, group=25, focal.names=names, model="1PL")
# difGenLord(Verbal, group="group", focal.name=names, model="1PL")
difGenLord(Verbal[,1:24], group=Verbal[,25], focal.names=names, 
model="1PL")

# 1PL model, "lme4" engine (remove #)

# difGenLord(Verbal, group="group", focal.name=names, model="1PL", 
# engine="lme4")

# With item purification (remove #)

# difGenLord(Verbal, group=25, focal.names=names, model="1PL", purify=TRUE)

# Splitting the data into the four subsets according to "group"
data0<-data1<-data2<-data3<-NULL
for (i in 1:nrow(verbal)){
if (group[i]=="WomanLow") data0<-rbind(data0,as.numeric(verbal[i,1:24]))
if (group[i]=="WomanHigh") data1<-rbind(data1,as.numeric(verbal[i,1:24]))
if (group[i]=="ManLow") data2<-rbind(data2,as.numeric(verbal[i,1:24]))
if (group[i]=="ManHigh") data3<-rbind(data3,as.numeric(verbal[i,1:24]))
}

# Estimation of the item parameters (1PL model)
m0.1PL<-itemParEst(data0, model="1PL")
m1.1PL<-itemParEst(data1, model="1PL")
m2.1PL<-itemParEst(data2, model="1PL")
m3.1PL<-itemParEst(data3, model="1PL")

# Merging the item parameters WITHOUT rescaling
irt.noscale<-rbind(m0.1PL,m1.1PL,m2.1PL,m3.1PL)
rownames(irt.noscale)<-rep(colnames(verbal[,1:24]),4)

# merging the item parameters WITH rescaling
irt.scale<-rbind(m0.1PL, itemRescale(m0.1PL,m1.1PL),
itemRescale(m0.1PL,m2.1PL) ,itemRescale(m0.1PL,m3.1PL))
rownames(irt.scale)<-rep(colnames(verbal[,1:24]),4)

# Equivalent calculations
difGenLord(irtParam=irt.noscale, nrFocal=3, same.scale=FALSE)
difGenLord(irtParam=irt.scale, nrFocal=3, same.scale=TRUE)

# With item purification
difGenLord(irtParam=irt.noscale, nrFocal=3, same.scale=FALSE, purify=TRUE)

[Package difR version 1.1 Index]