catR (version 2.2)

MWI: Maximum likelihood weighted information (MLWI) and maximum posterior weighted information (MPWI)

Description

This command returns the maximum likelihood (MLWI) or the maximum posterior (MPWI) weighted information for a given item and an item bank.

Usage

MWI(itemBank, item, x, it, lower=-4, upper=4, nqp=33, 
 	type="MLWI", priorDist="norm", priorPar=c(0,1))

Arguments

itemBank

an item bank, i.e. a list of class itBank, typically an output of the function createItemBank.

item

numeric: the item (referred to as its rank in the item bank) for which the maximum information must be computed.

x

binary: a vector of item responses, coded as 0 or 1 only.

it

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

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 information to be computed. Possible values are "MLWI" (default) and "MPWI". 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 not "MPWI".

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

Value

• The required maximum information for the selected item.

Details

Both the MLWI (Veerkamp and Berger, 1997) and the MPWI (van der Linden, 1998; van der Linden and Pashley, 2000) can be used as rules for selecting the next item in the CAT process (see also Choi and Swartz, 2009). This command serves as a subroutine for the nextItem function. Let k be the number of administered items, and set $x_1, ..., x_k$ as the binary responses to the first k administered items. Set also $I_j(\theta)$ as the information function of item j evaluated at $\theta$, and set $L(\theta | x_1, ..., x_k)$ as the likelihood function evaluated at $\theta$, given the provisional response pattern. Then, the MLWI for item j is given by $$MLWI_j = \int I_j(\theta) L(\theta | x_1, ..., x_k) d\theta$$ and the MPWI by $$MPWI_j = \int I_j(\theta) \pi(\theta) L(\theta | x_1, ..., x_k) d\theta$$ where $\pi(\theta)$ is the prior distribution of the ability level. These integrals are approximated by the integrate.xy function from the package sfsmisc. 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). The argument type defines the type of information to be computed. The default value, "MLWI", computes the MLWI value, while the MPWI value is obtained with type="MPWI". 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. This argument is ignored whenever method is not "MPWI". The item bank is provided through the argument itemBank. The provisional response pattern and the related item parameters are provided by the arguments x and it respectively. The target item (for which the maximum information computed) is given by its number in the item bank, through the item argument.

References

Choi, S. W., and Swartz, R. J. (2009). Comparison of CAT item selection criteria for polytomous items. Applied PScyhological Measurement, 32, 419-440. Magis, D., and Raiche, G. (in press). Random generation of response patterns under computerized adaptive testing with the R package catR. Journal of Statistical Software. van der Linden, W. J. (1998). Bayesian item selection criteria for adaptive testing. Psychometrika, 63, 201-216. van der Linden, W. J., and Pashley, P. J. (2000). Item selection and ability estimlation in adaptive testing. In W. J. van der Linden and C. A. W. Glas (Eds.), Computerized adaptive testing. Theory and practice (pp. 1-25). Boston, MA: Kluwer. Veerkamp, W. J. J., and Berger, M. P. F. (1997). Some new item selection criteria for adaptive testing. Journal of Educational and Behavioral Statistics, 22, 203-226.

See Also

Ii, nextItem, integrate.xy

Examples

# Loading the 'tcals' parameters 
 data(tcals)

 # Selecting item parameters only
 tcals <- as.matrix(tcals[,1:4])
 
 # Item bank creation with 'tcals' item parameters
 bank <- createItemBank(tcals)

 # Selection of two arbitrary items (15 and 20) of the
 # 'tcals' data set
 it <- bank$itemPar[c(15,20),]

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

 # MLWI for item 1
 MWI(bank, 1, x, it)

 # MPWI for item 1
 MWI(bank, 1, x, it, type="MPWI")

 # MLWI for item 1, different integration range
 MWI(bank, 1, x, it, lower=-2, upper=2, nqp=20)

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