catR (version 3.9)

GDI: Global-discrimination index (GDI) and posterior global-discrimination index (GDIP) for item selection

Description

This command returns the value of the global-discrimination index (GDI) and posterior global-discrimination index (GDIP) for a given item, an item bank and a set of previously administered items.

Usage

GDI(itemBank, item, x, it.given, model = NULL, lower = -4, upper = 4, nqp = 33,  	type = "GDI", priorDist="norm", priorPar = c(0, 1), D = 1, X = NULL,  	lik = NULL)

Arguments

itemBank

numeric: a suitable matrix of item parameters. See Details.

item

numeric: the item (referred to as its rank in the item bank) for which the GDI or GDIP must be computed.

x

numeric: a vector of item responses, coded as 0 or 1 only (for dichotomous items) or from 0 to the number of response categories minus one (for polytomous items).

it.given

numeric: a matrix with one row per item and four columns, with the values of the discrimination, the difficulty, the pseudo-guessing and the inattention parameters (in this order). The number of rows of it must be equal to the length of x.

model

now only NULL (default) for dichotomous models is allowed.

lower

numeric: the lower bound for numercal integration (default is -4).

upper

numeric: the upper bound for numercal integration (default is 4).

nqp

numeric: the number of quadrature points (default is 33).

type

character: the type of index to be computed. Possible values are "GDI" (default) and "GDIP". See Details.

priorDist

character: the prior ability distribution. Possible values are "norm" (default) for the normal distribution, and "unif" for the uniform distribution. Ignored if type is "GDI".

priorPar

numeric: a vector of two components with the prior parameters. If priorDist is "norm", then priorPar contains the mean and the standard deviation of the normal distribution. If priorDist is "unif", then priorPar contains the bounds of the uniform distribution. The default values are 0 and 1 respectively. Ignored if type is "GDI".

D

numeric: the metric constant. Default is D=1 (for logistic metric); D=1.702 yields approximately the normal metric (Haley, 1952).

X

either a vector of numeric values or NULL (default). See Details.

lik

either a vector of numeric values or NULL (default). See Details.

Value

The required GDI or GDIP value for the selected item.

Details

Global-discrimination index can be used as a rule for selecting the next item in the CAT process (Kaplan, de la Torre, and Barrada, 2015). This command serves as a subroutine for the nextItem function.

Dichotomous IRT models are considered whenever model is set to NULL (default value). In this case, itemBank must be a matrix with one row per item and four columns, with the values of the discrimination, the difficulty, the pseudo-guessing and the inattention parameters (in this order). These are the parameters of the four-parameter logistic (4PL) model (Barton and Lord, 1981).

Currently both GDI and GDIP are not implemented for polytomous IRT models.

The integrals within GDI and GDIP are approximated by the integrate.catR function. The range of integration is set up by the arguments lower, upper and nqp, giving respectively the lower bound, the upper bound and the number of quadrature points. The default range goes from -4 to 4 with length 33 (that is, by steps of 0.25).

To speed up the computation, both the range of integration of values $\theta$ and the values of the likelihood function $L(\theta)$ can be directly provided to the function through the arguments X and lik. If X is set to NULL (default), the sequence of ability values for integration is determined by the arguments lower, upper and nqp as explained above. If lik is NULL (default), it is also internally computed from an implementation of the likelihood function.

The provisional response pattern and the related item parameters are provided by the arguments x and it.given respectively. The target item (for which the KL information is computed) is given by its rank number in the item bank, through the item argument.

The argument type defines the type of KL information to be computed. The default value, "GDI", computes the GDI indexinformation, while the posterior GDI index is obtained with type="GDIP". For the latter, the priorDist and priorPar arguments fix the prior ability distribution. The normal distribution is set up by priorDist="norm" and then, priorPar contains the mean and the standard deviation of the normal distribution. If priorDist is "unif", then the uniform distribution is considered, and priorPar fixes the lower and upper bounds of that uniform distribution. By default, the standard normal prior distribution is assumed. These arguments are ignored whenever method is "GDI".

References

Barton, M.A., and Lord, F.M. (1981). An upper asymptote for the three-parameter logistic item-response model. Research Bulletin 81-20. Princeton, NJ: Educational Testing Service.

Haley, D.C. (1952). Estimation of the dosage mortality relationship when the dose is subject to error. Technical report no 15. Palo Alto, CA: Applied Mathematics and Statistics Laboratory, Stanford University.

Kaplan, M., de la Torre, J., and Barrada, J. R. (2015). New item selection methods for cognitive diagnosis computerized adaptive testing. Applied Psychological Measurement, 39, 167-188. doi: 10.1177/0146621614554650

Magis, D., and Raiche, G. (2012). Random Generation of Response Patterns under Computerized Adaptive Testing with the R Package catR. Journal of Statistical Software, 48 (8), 1-31. URL http://www.jstatsoft.org/v48/i08/

See Also

integrate.catR, nextItem, genPolyMatrix

Examples

## Dichotomous models ##

 # Loading the 'tcals' parameters 
 data(tcals)

 # Selecting item parameters only
 bank <- as.matrix(tcals[,1:4])
 
 # Selection of two arbitrary items (15 and 20) of the
 # 'tcals' data set
 it.given <- bank[c(15, 20),]

 # Creation of a response pattern
 x <- c(0, 1)

 # GDI for item 1
 GDI(bank, 1, x, it.given)

 # GDIP for item 1
 GDI(bank, 1, x, it.given, type = "GDIP")

 # GDIP for item 1, different integration range
 GDI(bank, 1, x, it.given, type = "GDIP", lower = -2, upper = 2, nqp = 20)

 # GDIP for item 1, uniform prior distribution on the range [-2,2]
 GDI(bank, 1, x, it.given, type = "GDIP", priorDist = "unif", 
    priorPar = c(-2, 2))

 # Computation of likelihood function beforehand
 L <- function(th, r, param) 
  prod(Pi(th, param)$Pi^r * (1 - Pi(th,param)$Pi)^(1 - r))
 xx <- seq(from = -4, to = 4, length = 33)
 y <- sapply(xx, L, x, it.given) 
 GDI(bank, 1, x, it.given, X = xx, lik = y)