catR (version 3.16)

Ii: Item information functions, first and second derivatives (dichotomous and polytomous models)

Description

This command returns the Fisher information functions for a given ability value and a given matrix of item parameters under either the 4PL model or any suitable polytomous model. Numerical values of the first and second derivatives of the item information functions are also returned.

Usage

Ii(th, it, model = NULL, D = 1)

Arguments

th

numeric: the ability value.

it

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

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.

Value

A list with three arguments:

Ii

the vector with item informations (one value per item).

dIi

the vector with first derivatives of the item information functions (one value per item).

d2Ii

the vector with second derivatives of the item information functions (one value per item).

Details

The first and second derivatives are computed algebraically, either from the four-parameter logistic (4PL) model (Barton and Lord, 1981) or from the corresponding polytomous model. These derivatives are necessary for both the estimation of ability and the computation of related standard errors.

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.

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.

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.

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

Pi, thetaEst, genPolyMatrix

Examples

# NOT RUN {
## Dichotomous models ##
 
 # Loading the 'tcals' parameters 
 data(tcals)

 # Selecting item parameters only
 tcals <- as.matrix(tcals[,1:4])
 
 # Item information functions and derivatives
 # (various th and D values)
 Ii(th = 0, tcals)
 Ii(th = 0, tcals, D = 1.702)
 Ii(th = 1, tcals)

## 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)

 # Computation of item information and derivatives for ability level 0
 Ii(0, m.GRM, model = "GRM")

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

 # Computation of item information and derivatives for ability level 1
 Ii(1, cat_pav, model = "GPCM")
 
# }