catR (version 1.0)

thetaEst: Ability estimation under the 4PL model

Description

This command returns the ability estimate for a given matrix of item parameters of the 4PL model and a given response pattern. Available estimators are maximum likelihood, Bayes modal, expected a posteriori (EAP) and weighted likelihood.

Usage

thetaEst(it, x, D=1, method="BM", priorDist="norm", 
 	priorPar=c(0,1), range=c(-4,4), 
 	eapPar=c(-4,4,20))

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).

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)). Ignored if method is neither "BM" nor "EAP", or if priorDist="Jeffreys". See Details.

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".

eapPar

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, 20). Ignor

Value

• The estimated ability level.

Details

Four ability estimators are available: the maximum likelihood (ML) estimator (Lord, 1980), the Bayes modal (BM) estimator (Birnbaum, 1969), the expected a posteriori (EAP) estimator (Bock and Mislevy, 1982) and the weighted likelihood (WL) estimator (Warm, 1989). The selected estimator is specified by the method argument, with values "ML", "BM", "EAP" and "WL" respectively. For the BM and EAP estimators, 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 priorPar argument is ignored if method="ML" or method="WL". 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 eapPar argument sets the range and the number of quadrature points for numerical integration in the EAP process. By default, it takes the vector value (-4, 4, 20), that is, 20 quadrature points on the range [-4; 4]. See eapEst for further details. The range argument permits to limit the interval of investigation for the ML, BM and WL ability estimates (in particular, to avoid infinite ability estimates). The default range is [-4, 4].

References

Birnbaum, A. (1969). Statistical theory for logistic mental test models with a prior distribution of ability. Journal of Mathematical Psychology, 6, 258-276. 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. Lord, F.M. (1980). Applications of item response theory to practical testing problems. Hillsdale, NJ: Lawrence Erlbaum. Warm, T.A. (1989). Weighted likelihood estimation of ability in item response models. Psychometrika, 54, 427-450.

See Also

eapEst, semTheta

Examples

# Loading the 'tcals' parameters 
 data(tcals)
 tcals <- as.matrix(tcals)
 
 # Creation of a response pattern (tcals item parameters,
 # true ability level 0)
 set.seed(1)
 x <- rbinom(85, 1, Pi(0, tcals)$Pi)

 # ML estimation
 thetaEst(tcals, x, method="ML")

 # BM estimation, standard normal prior distribution
 thetaEst(tcals, x)

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

 # BM estimation, Jeffreys' prior distribution  
 thetaEst(tcals, x, method="BM", priorDist="Jeffreys")

 # EAP estimation, standard normal prior distribution
 thetaEst(tcals, x, method="EAP")

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

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

 # WL estimation
 thetaEst(tcals, x, method="WL")