catR (version 2.0)

EPV: Expected Posterior Variance (EPV)

Description

This command returns the expected posterior variance (EPV) for a given item, as used for Minimum Expected Posterior Variance (MEPV) criterion.

Usage

EPV(itemBank, item, x, theta, it, priorDist="norm", 
 	priorPar=c(0,1), D=1, parInt=c(-4,4,33))

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.

theta

numeric: the provisional ability estimate.

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

priorDist

character: specifies the prior distribution. Possible values are "norm" (default) and "unif".

priorPar

numeric: vector of two components specifying the prior parameters (default is c(0,1)) of the prior ability distribution.

D

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

parInt

numeric: vector of three components, defining the sequence of ability values for computing the posterior variance. See Details.

Value

• The expected posterior variance for the selected item.

Details

The EPV can be used as a rule for selecting the next item in the CAT process (Choi and Swartz, 2009; Owen, 1975; van der Linden, 1998). 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 provisional response pattern. Set $\hat{\theta}_k$ as the provisional ability estimate (with the first k responses) and let j be the item of interest (not previously administered). Set also $P_j(\theta)$ as the probability of answering item j correctly for a given ability level $\theta$, and set $Q_j(\theta)=1-P_j(\theta)$. Finally, set $Var(\theta | x_1, ..., x_k, 0)$ and $Var(\theta | x_1, ..., x_k, 1)$ as the posterior variances of $\theta$, given the provisional response pattern (updated by response 0 and 1 respectively). Then, the EPV for item j equals $$EPV_j = P_j(\hat{\theta}_k)\,Var(\theta | x_1, ..., x_k, 1) + Q_j(\hat{\theta}_k)\,Var(\theta | x_1, ..., x_k, 0)$$. The posterior variance $Var(\theta | x_1, ..., x_k, x_j)$ (where x_j takes value 0 or 1) is computed as the squared standard error of the EAP estimate of ability, using the response pattern $(x_1, ..., x_k, x_j)$. This is done by a joint use of the eapEst and eapSem functions. The prior distribution is set up by the arguments priorDist and priorPar, with the by-default standard normal distribution. The range of integration is defined by the parInt argument, with by default, the sequence from -4 to 4 and of length 33 (or, by steps of 0.25). See the function eapEst for further details. 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. 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. Owen, R. J. (1975). A Bayesian sequential procedure for quantal response in the context of adaptive mental testing. Journal of the American Statistical Association, 70, 351-356. van der Linden, W. J. (1998). Bayesian item selection criteria for adaptive testing. Psychometrika, 63, 201-216.

See Also

nextItem, eapEst, eapSem

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)

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

 # With prior standard deviation 2
 EPV(bank, 1, x, 0, it, priorPar=c(0,2))