Description Usage Arguments Details Value Author(s) References See Also Examples

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.

1 2 3 |

`thEst` |
numeric: the EAP ability estimate. |

`it` |
numeric: a suitable matrix of item parameters. See |

`x` |
numeric: a vector of item responses. |

`model` |
either |

`D` |
numeric: the metric constant. Default is |

`priorDist` |
character: specifies the prior distribution. Possible values are |

`priorPar` |
numeric: vector of two components specifying the prior parameters (default is |

`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). |

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.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.

Note that in the current version, the EAP ability estimate must be specified through the `thEst`

argument.

The estimated standard error of the EAP ability level.

David Magis

[email protected]

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 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. doi: 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. doi: 10.18637/jss.v048.i08

`thetaEst`

, `genPolyMatrix`

, `integrate.catR`

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 | ```
## 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
th <- eapEst(tcals, x)
c(th, eapSem(th, tcals, x))
# EAP estimation, uniform prior distribution upon range [-2,2]
th <- eapEst(tcals, x, priorDist = "unif", priorPar = c(-2, 2))
c(th, eapSem(th, tcals, x, priorDist = "unif", priorPar=c(-2, 2)))
# EAP estimation, Jeffreys' prior distribution
th <- eapEst(tcals, x, priorDist = "Jeffreys")
c(th, eapSem(th, tcals, 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"))
# 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
th <- eapEst(cat_pav, x, model = "GPCM")
c(th, eapSem(th, cat_pav, x, model = "GPCM"))
# EAP estimation, uniform prior distribution upon range [-2,2]
th <- eapEst(cat_pav, x, model = "GPCM", priorDist = "unif", priorPar = c(-2, 2))
c(th, eapSem(th, cat_pav, x, model = "GPCM", priorDist = "unif", priorPar = c(-2, 2)))
# EAP estimation, Jeffreys' prior distribution
th <- eapEst(cat_pav, x, model = "GPCM", priorDist = "Jeffreys")
c(th, eapSem(th, cat_pav, x, model = "GPCM", priorDist = "Jeffreys"))
``` |

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