catR (version 2.6)

Ii: Item information functions, first and second derivatives (4PL)

Description

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

Usage

Ii(th, it, D=1)

Arguments

th

numeric: the ability value.

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).

D

numeric: the metric constant. Default is D=1 (for logistic metric); D=1.702 yields approximately the normal metric (Haley, 1952).

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 from the four-parameter logistic (4PL) model (Barton and Lord, 1981). These derivatives are necessary for both the estimation of ability and the computation of related standard errors.

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 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/

See Also

Pi, thetaEst

Examples

# 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)