# Standard error of EAP ability estimation (dichotomous and polytomous IRT models) This command returns the estimated standard error of the 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. ```r eapSem(thEst, it, x, model = NULL, D = 1, priorDist = "norm", priorPar = c(0, 1), lower = -4, upper = 4, nqp = 33) ``` ## Arguments - `thEst`: numeric: the EAP ability estimate. - `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). ## Returns The estimated standard error of the EAP ability level. ## Details This command computes the standard error of the EAP (expected a posteriori) ability estimator (Bock and Mislevy, 1982). 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.mstR` 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. Note that in the current version, the EAP ability estimate must be specified through the `thEst` argument. ## 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. doi: 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 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/](http://www.jstatsoft.org/v48/i08/) ## Author(s) David Magis Department of Psychology, University of Liege, Belgium david.magis@uliege.be ## See Also `thetaEst`, `genPolyMatrix` ## Examples ```r ## Not run: ## Dichotomous models ## # Generation of an item bank under 3PL with 100 items m.3PL <- genDichoMatrix(100, model = "3PL") m.3PL <- as.matrix(m.3PL) # Creation of a response pattern (true ability level 0) set.seed(1) x <- genPattern(0, m.3PL) # EAP estimation, standard normal prior distribution th <- eapEst(m.3PL, x) c(th, eapSem(th, m.3PL, x)) # EAP estimation, uniform prior distribution upon range [-2,2] th <- eapEst(m.3PL, x, priorDist = "unif", priorPar = c(-2, 2)) c(th, eapSem(th, m.3PL, x, priorDist = "unif", priorPar=c(-2, 2))) # EAP estimation, Jeffreys' prior distribution th <- eapEst(m.3PL, x, priorDist = "Jeffreys") c(th, eapSem(th, m.3PL, x, priorDist = "Jeffreys")) ## 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 th <- eapEst(m.GRM, x, model = "GRM") c(th, eapSem(th, m.GRM, x, model = "GRM")) # EAP estimation, uniform prior distribution upon range [-2,2] th <- eapEst(m.GRM, x, model = "GRM", priorDist = "unif", priorPar = c(-2, 2)) c(th, eapSem(th, m.GRM, x, model = "GRM", priorDist = "unif", priorPar = c(-2, 2))) # EAP estimation, Jeffreys' prior distribution th <- eapEst(m.GRM, x, model = "GRM", priorDist = "Jeffreys") c(th, eapSem(th, m.GRM, x, model = "GRM", priorDist = "Jeffreys")) # Generation of a item bank under PCM with 20 items and at most 3 categories m.PCM <- genPolyMatrix(20, 3, "PCM") m.PCM <- as.matrix(m.PCM) # Creation of a response pattern (true ability level 0) set.seed(1) x <- genPattern(0, m.PCM, model = "PCM") # EAP estimation, standard normal prior distribution th <- eapEst(m.PCM, x, model = "PCM") c(th, eapSem(th, m.PCM, x, model = "PCM")) # EAP estimation, uniform prior distribution upon range [-2,2] th <- eapEst(m.PCM, x, model = "PCM", priorDist = "unif", priorPar = c(-2, 2)) c(th, eapSem(th, m.PCM, x, model = "PCM", priorDist = "unif", priorPar = c(-2, 2))) # EAP estimation, Jeffreys' prior distribution th <- eapEst(m.PCM, x, model = "PCM", priorDist = "Jeffreys") c(th, eapSem(th, m.PCM, x, model = "PCM", priorDist = "Jeffreys")) ## End(Not run) ```

eapSem function