The observed information function for item j is given by
$$OI_j= -\frac{\partial^2}{\partial \theta^2} \,\log L(\theta | x_j)$$
where \(\theta\) is the ability level, \(L\) is the likelihood function and \(x_j\) is the item response.
For dichotomous IRT models with success probability \(P_j(\theta)\), 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 \(P_j=P_j(\theta)\), \(Q_j=1-P_j\) 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 \(P_{jk}(\theta)=Pr(X_j=k | \theta)\) as the probability of answering response category \(k\) and set \(\tau_{jk}\) 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:
$$OI_j = \sum_{k=0}^{g_j} \tau_{jk}\,\left( \frac{{P_{jk}'(\theta)}^2}{{P_{jk}(\theta)}^2}-\frac{P_{jk}''(\theta)}{P_{jk}(\theta)}\right)$$
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
$$I_j=-E \left[\frac{\partial^2}{\partial \theta^2} \,\log L(\theta | x_j)\right] = \frac{{P_j'}^2}{P_j Q_j}$$
(van der Linden, 1998; Veerkamp, 1996).
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.