(Maximum) Expected Information (MEI)

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Description

This command returns the expected information (EI) for a given item (under dichotomous and polytomous IRT models), as used for Maximum Expected Information (MEI) criterion.

Usage

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MEI(itemBank, item, x, theta, it.given, model = NULL, method = "BM", 
 	priorDist = "norm", priorPar = c(0, 1), D = 1, range = c(-4, 4), 
 	parInt = c(-4, 4, 33), infoType = "observed")
 

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 expected information 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).

theta

numeric: the provisional ability estimate.

it.given

numeric: a suitable matrix of item parameters for previously administered items. The number of rows of it.given must be equal to the length of x.

model

either NULL (default) for dichotomous models, or any suitable acronym for polytomous models. Possible values are "GRM", "MGRM", "PCM", "GPCM", "RSM" and "NRM". See Details.

method

character: the ability estimator. Possible values are "BM" (default), "ML" and "WL". See Details.

priorDist

character: specifies the prior distribution. Possible values are "norm" (default), "unif" and "Jeffreys". Ignored if method is neither "BM" nor "EAP". See Details.

priorPar

numeric: vector of two components specifying the prior parameters (default is c(0,1)) of the prior ability distribution. Ignored if method is neither "BM" nor "EAP", or if priorDist="Jeffreys". See Details.

D

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

range

numeric: vector of two components specifying the range wherein the ability estimate must be looked for (default is c(-4,4)). Ignored if method=="EAP".

parInt

numeric: vector of three components, holding respectively the values of the arguments lower, upper and nqp of the eapEst command. Default vector is (-4, 4, 33). Ignored if method is not "EAP".

infoType

character: the type of information function to be used. Possible values are "observed" (default) for observed information function, and "Fisher" for Fisher information function.

Details

The MEI (van der Linden, 1998; van der Linden and Pashley, 2000) can be used as a rule for selecting the next item in the CAT process (see also Choi and Swartz, 2009), both with dichotomous and polytomous IRT models. 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).

Polytomous IRT models are specified by their respective acronym: "GRM" for Graded Response Model, "MGRM" for Modified Graded Response Model, "PCM" for Partical Credit Model, "GPCM" for Generalized Partial Credit Model, "RSM" for Rating Scale Model and "NRM" for Nominal Response Model. The itemBank still holds one row per item, end the number of columns and their content depends on the model. See genPolyMatrix for further information and illustrative examples of suitable polytomous item banks.

Under polytomous IRT models, let k be the number of administered items, and set x_1, ..., x_k as the provisional response pattern (where each response x_l takes values in \{0, 1, ..., g_l\}). Set \hat{θ}_k as the provisional ability estimate (with the first k responses) and let j be the item of interest (not previously administered). Set also P_{jt}(θ) as the probability of answering response category t to item j for a given ability level θ. Finally, set \hat{θ}_{k+1}^t as the ability estimates computed under the condition that the response to item j is t (with t=0, ..., g_j). Then, the EI for item j equals

EI_j = ∑_{t=0}^{g_j} P_{jt}(\hat{θ}_k)\,I_j(\hat{θ}_{k+1}^t)

where I_j(θ) is the information function for item j.

In case of dichotomous IRT models, all g_l values reduce to 1, so that item responses x_l equal either 0 or 1. Set simply P_j(θ) as the probability of answering item j correctly for a given ability level θ, and set Q_j(θ)=1-P_j(θ). Finally, set \hat{θ}_{k+1}^0 and \hat{θ}_{k+1}^1 as the ability estimates computed under the condition that the response to item j is 0 or 1 respectively (that is, if the response pattern is updated by 0 or 1 for item j). Then, the EI for item j reduces to

EI_j = P_j(\hat{θ}_k)\,I_j(\hat{θ}_{k+1}^1) + Q_j(\hat{θ}_k)\,I_j(\hat{θ}_{k+1}^0)

.

Two types of information functions are available. The first one is the observed information function, defined as

OI_j(θ) = -\frac{\partial^2}{\partial θ^2} \,\log L(θ | x_j)

(van der Linden, 1998), where L(θ | x_j) is the likelihood related to item j. The second one is Fisher information function:

I_j(θ) = -E\,≤ft[\frac{\partial^2}{\partial\,θ^2} \,\log L(θ | x_j))\right].

Under the 1PL and the 2PL models, these functions are identical (Veerkamp, 1996). See also OIi.

The observed and Fisher information functions are specified by the infoType argument, with respective values "observed" and "Fisher". By default, the observed information function is considered (Choi and Swartz, 2009; van der Linden, 1998).

The estimator of provisional ability is defined by means of the arguments method, priorDist, priorPar, D, range and parInt of the thetaEst function. See the corresponding help file for further details.

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 maximum information computed) is given by its number in the item bank, through the item argument.

Note that the provisional response pattern x can also be set to NULL (which is useful when the number nrItems of starting items is set to zero). In this case, it.given must be a matrix with zero rows, such as e.g., itemBank[NULL,].

Value

The required expected information for the selected item.

Author(s)

David Magis
Department of Education, University of Liege, Belgium
david.magis@ulg.ac.be

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.

Choi, S. W., and Swartz, R. J. (2009). Comparison of CAT item selection criteria for polytomous items. Applied Psychological Measurement, 32, 419-440. doi: 10.1177/0146621608327801

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.

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/

van der Linden, W. J. (1998). Bayesian item selection criteria for adaptive testing. Psychometrika, 63, 201-216. doi: 10.1007/BF02294775

van der Linden, W. J., and Pashley, P. J. (2000). Item selection and ability estimation 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. (1996). Statistical inference for adaptive testing. Internal report. Enschede, The Netherlands: University of Twente.

See Also

Ii, OIi, nextItem, thetaEst, genPolyMatrix

Examples

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## 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)

 # MEI for item 1, provisional ability level 0
 MEI(bank, 1, x, 0, it.given)

 # With Fisher information instead
 MEI(bank, 1, x, 0, it.given, infoType = "Fisher")

 # With WL estimator instead
 MEI(bank, 1, x, 0, it.given, method = "WL")


## Polytomous models ##

 # Generation of an item bank under GRM with 100 items and at most 4 categories
 m.GRM <- genPolyMatrix(100, 4, "GRM")
 m.GRM <- as.matrix(m.GRM)

 # Selection of two arbitrary items (15 and 20) 
it.given <- m.GRM[c(15, 20),]

 # Generation of a response pattern (true ability level 0)
 x <- genPattern(0, it.given, model = "GRM")

 # EPV for item 1, provisional ability level 0
 MEI(m.GRM, 1, x, 0, it.given, model = "GRM")

 # With WL method
 MEI(m.GRM, 1, x, 0, it.given, model = "GRM", method = "WL")

 # With Fisher information
 MEI(m.GRM, 1, x, 0, it.given, model = "GRM", infoType = "Fisher")

 # Generation of an item bank under PCM with 20 items and 4 categories
 m.PCM <- genPolyMatrix(20, 4, "PCM", same.nrCat = TRUE)
 m.PCM <- as.matrix(m.PCM)

 # Selection of two arbitrary items (15 and 20) 
 it.given <- m.PCM[c(15, 20),]

 # Generation of a response pattern (true ability level 0)
 x <- genPattern(0, it.given, model = "PCM")

 # EPV for item 1, provisional ability level 0
 MEI(m.PCM, 1, x, 0, it.given, model = "PCM")

 # With WL method
 MEI(m.PCM, 1, x, 0, it.given, model = "PCM", method = "WL")

 # With Fisher information
 MEI(m.PCM, 1, x, 0, it.given, model = "PCM", infoType = "Fisher") 
 

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