catR (version 1.2)

OIi: Observed information function (4PL)

Description

This command returns the observed information functions for a given matrix of item parameters of the 4PL model and a given ability value.

Usage

OIi(th, it, x, 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).

x

numeric: the item response (coded as 0 or 1). Can be either a single value or a vector of the same length of the number of items.

D

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

Value

• A vector with the observed item informations (one per item).

Details

The observed information function for item j is given by $$-\frac{\partial^2}{\partial \theta^2} \,\log L(\theta | x_j)$$ where $\theta$ is the ability level, $L$ is the likelihood function and $x_j$ is the item response. For dichotomous item response models with success probability $P_j(\theta)$, it takes the following form: $$-\frac{\partial^2}{\partial \theta^2} \,\log L(\theta | x_j) = \frac{P_j \,Q_j \,{P_j'}^2- (x_j-P_j) \,[P_j \,Q_j \,P_j'' + {P_j}^2 \,(P_j-Q_j]}{{P_j}^2 \,{Q_j}^2}$$ where $P_j=P_j(\theta)$, $Q_j=1-P_j$ and $P_j'$ and $P_j''$ are the first and second derivatives of $P_j$ respectively. Under the 2PL model, the observed information function is exactly equal to Fisher's information function $$-E \left[\frac{\partial^2}{\partial \theta^2} \,\log L(\theta | x_j)\right] = \frac{{P_j'}^2}{P_j Q_j}$$ (van der Linden, 1998; Veerkamp, 1996). The observed information function is used to compute some item selection criteria, such as the Maximum Expected Information (MEI). See nextItem for further details.

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. van der Linden, W. (1998). Bayesian item selection criteria for adaptive testing. Psychometrika, 63, 201-216. Veerkamp, W. J. J. (1996). Statistical inference for adaptive testing. Internal report. Enschede, The Netherlands: University of Twente.

See Also

createItemBank, nextItem

Examples

# Loading the 'tcals' parameters 
 data(tcals)
 tcals <- as.matrix(tcals)
 
 # Observed information functions 
 # (various th, x and D values)
 OIi(th=0, tcals, x=0)
 OIi(th=0, tcals, x=0, D=1.702)
 OIi(th=0, tcals, x=1)
 OIi(th=1, tcals, x=1)