catR (version 2.2)

Pi: Item response probabilities, first, second and third derivatives (4PL)

Description

This command returns the item reponse probabilities for a given matrix of item parameters of the 4PL model and a given ability value. Numerical values of the first, second and third derivatives of the response probabilities are also returned.

Usage

Pi(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 four arguments:
• Pi
  the vector with response probabilities (one value per item)
• dPi
  the vector with first derivatives of the response probabilities (one value per item)
• d2Pi
  the vector with second derivatives of the response probabilities (one value per item)
• d3Pi
  the vector with third derivatives of the response probabilities (one value per item)

Details

The first, second and third 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. (in press). Random generation of response patterns under computerized adaptive testing with the R package catR. Journal of Statistical Software.

See Also

Ii, thetaEst

Examples

# Loading the 'tcals' parameters 
 data(tcals)
 
 # Selecting item parameters only
 tcals <- as.matrix(tcals[,1:4])
 
 # Response probabilities and derivatives (various th and D values)
 Pi(th=0, tcals)
 Pi(th=0, tcals, D=1.702)
 Pi(th=1, tcals)