# OIi: Observed information function (dichotomous and polytomous models)

### Description

This command returns the observed information functions for a given ability value and a given matrix of item parameters, either under the 4PL model or any suitable polytomous IRT model.

### Usage

 1 2 OIi(th, it, x, model = NULL, D = 1)

### Arguments

 th numeric: the ability value. it numeric: a suitable matrix of item parameters. See Details. x numeric: a vector of item responses. 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.

### Details

The observed information function for item j is given by

OI_j= -\frac{\partial^2}{\partial θ^2} \,\log L(θ | x_j)

where θ is the ability level, L is the likelihood function and x_j is the item response.

For dichotomous IRT models with success probability P_j(θ), it takes the following form:

OI_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(θ), Q_j=1-P_j and P_j' and P_j'' are the first and second derivatives of P_j respectively.

For polytomous IRT models, set $X_j$ as the item response, taking values k \in \{0, 1, ..., g_j\}. Set P_{jk}(θ)=Pr(X_j=k | θ) as the probability of answering response category k and set τ_{jk} as the boolean factor equal to 1 if X_j=k and 0 otherwise. Then, the observed information function for item j takes the following form:

OI_j = ∑_{k=0}^{g_j} τ_{jk}\,≤ft( \frac{{P_{jk}'(θ)}^2}{{P_{jk}(θ)}^2}-\frac{P_{jk}''(θ)}{P_{jk}(θ)}\right)

with the same notations for the first and second derivatives as above.

Under the 2PL model, the observed information function is exactly equal to Fisher's information function

I_j=-E ≤ft[\frac{\partial^2}{\partial θ^2} \,\log L(θ | x_j)\right] = \frac{{P_j'}^2}{P_j Q_j}

(van der Linden, 1998; Veerkamp, 1996).

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.

The observed information function is used to compute some item selection criteria, such as the Maximum Expected Information (MEI). See nextItem and MEI for further details.

### Value

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

### Author(s)

David Magis
Department of Education, University of Liege, Belgium
david.magis@ulg.ac.be

### 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. 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

van der Linden, W. (1998). Bayesian item selection criteria for adaptive testing. Psychometrika, 63, 201-216. doi: 10.1007/BF02294775

Veerkamp, W. J. J. (1996). Statistical inference for adaptive testing. Internal report. Enschede, The Netherlands: University of Twente.