catR (version 3.16)

eapEst: EAP ability estimation (dichotomous and polytomous IRT models)

Description

This command returns the EAP (expected a posteriori) ability estimate for a given response pattern and a given matrix of item parameters, either under the 4PL model or any suitable polytomous IRT model.

Usage

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

Arguments

it

numeric: a suitable matrix of item parameters. See Details.

x

numeric: a vector of item responses.

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.

D

numeric: the metric constant. Default is D=1 (for logistic metric); D=1.702 yields approximately the normal metric (Haley, 1952). Ignored if model is not NULL.

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, the latter being set as the prior distribution times the likelihood function.

Dichotomous IRT models are considered whenever model is set to NULL (default value). In this case, it 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 it 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.

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.catR function. 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

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.

Bock, R. D., and Mislevy, R. J. (1982). Adaptive EAP estimation of ability in a microcomputer environment. Applied Psychological Measurement, 6, 431-444. 10.1177/014662168200600405

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 Barrada, J. R. (2017). Computerized Adaptive Testing with R: Recent Updates of the Package catR. Journal of Statistical Software, Code Snippets, 76(1), 1-18. 10.18637/jss.v076.c01

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. 10.18637/jss.v048.i08

See Also

thetaEst, genPolyMatrix, integrate.catR

Examples

# NOT RUN {
## Dichotomous models ##

 # 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 <- genPattern(0, tcals)

 # 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)


## 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)

 # Creation of a response pattern (true ability level 0)
 set.seed(1)
 x <- genPattern(0, m.GRM, model = "GRM")

 # EAP estimation, standard normal prior distribution
 eapEst(m.GRM, x, model = "GRM")

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

 # EAP estimation, Jeffreys' prior distribution  
 eapEst(m.GRM, x, model = "GRM", priorDist = "Jeffreys")


 # Loading the cat_pav data
 data(cat_pav)
 cat_pav <- as.matrix(cat_pav)

 # Creation of a response pattern (true ability level 0)
 set.seed(1)
 x <- genPattern(0, cat_pav, model = "GPCM")

 # EAP estimation, standard normal prior distribution
 eapEst(cat_pav, x, model = "GPCM")

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

 # EAP estimation, Jeffreys' prior distribution  
 eapEst(cat_pav, x, model = "GPCM", priorDist = "Jeffreys")
 
# }