catR (version 2.2)

eapEst: EAP ability estimation under the 4PL model

Description

This command returns the EAP (expected a posteriori) ability estimate for a given matrix of item parameters of the 4PL model and a given response pattern.

Usage

eapEst(it, x, D=1, priorDist="norm", priorPar=c(0,1),
 	lower=-4, upper=4, nqp=33)

Arguments

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).

x

numeric: a vector of dichotomous item responses.

D

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

priorDist

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

priorPar

numeric: vector of two components specifying the prior parameters (default is c(0,1)). Ignored if priorDist="Jeffreys". See Details.

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).

Value

• The estimated EAP ability level.

Details

The EAP (expected a posteriori) ability estimator (Bock and Mislevy, 1982) is obtained by computing the average of the posterior distribution of ability, set as the prior distribution times the likelihood function. Three prior distributions are available: the normal distribution, the uniform distribution and Jeffreys' prior distribution (Jeffreys, 1939, 1946). The prior distribution is specified by the argument priorPar, with values "norm", "unif" and "Jeffreys", respectively. The argument priorPar determines either the prior mean and standard deviation of the normal prior distribution (if priorDist="norm"), or the range for defining the prior uniform distribution (if priorDist="unif"). This argument is ignored if priorDist="Jeffreys". The required integrals are approximated by numerical adaptive quadrature. This is achieved by using the integrate.xy function of the package sfsmisc. Arguments lower, upper and nqp define respectively the lower and upper bounds for numerical integration, and the number of quadrature points. By default, the numerical integration runs with 33 quadrature points on the range [-4; 4], that is, a sequence of values from -4 to 4 by steps of 0.25.

References

Bock, R. D., and Mislevy, R. J. (1982). Adaptive EAP estimation of ability in a microcomputer environment. Applied Psychological Measurement, 6, 431-444. 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. Jeffreys, H. (1939). Theory of probability. Oxford, UK: Oxford University Press. Jeffreys, H. (1946). An invariant form for the prior probability in estimation problems. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 186, 453-461. 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.

See Also

thetaEst, integrate.xy

Examples

# Loading the 'tcals' parameters 
 data(tcals)

 # Selecting item parameters only
 tcals <- as.matrix(tcals[,1:4])
 
 # Creation of a response pattern (tcals item parameters,
 # true ability level 0)
 set.seed(1)
 x <- rbinom(85, 1, Pi(0, tcals)$Pi)

 # EAP estimation, standard normal prior distribution
 eapEst(tcals, x)

 # EAP estimation, uniform prior distribution upon range [-2,2]
 eapEst(tcals, x, priorDist="unif", priorPar=c(-2,2))

 # EAP estimation, Jeffreys' prior distribution  
 eapEst(tcals, x, priorDist="Jeffreys")

 # Changing the integration settings
 eapEst(tcals, x, nqp=100)