The MEI (van der Linden, 1998; van der Linden and Pashley, 2000) can be used as a rule for selecting the next item in the CAT process (see also Choi and Swartz, 2009), both with dichotomous and polytomous IRT models. This command serves as a subroutine for the nextItem function.
Dichotomous IRT models are considered whenever model is set to NULL (default value). In this case, itemBank 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 itemBank 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.
Under polytomous IRT models, let k be the number of administered items, and set \(x_1, ..., x_k\) as the provisional response pattern (where each response \(x_l\) takes values in \(\{0, 1, ..., g_l\}\)). Set \(\hat{\theta}_k\) as the provisional ability estimate (with the first k responses) and let j be the item of interest (not previously administered). Set also \(P_{jt}(\theta)\) as the probability of answering response category t to item j for a given ability level \(\theta\). Finally, set
\(\hat{\theta}_{k+1}^t\) as the ability estimates computed under the condition that the response to item j is t (with \(t=0, ..., g_j\)). Then, the EI for item j equals
$$EI_j = \sum_{t=0}^{g_j} P_{jt}(\hat{\theta}_k)\,I_j(\hat{\theta}_{k+1}^t)$$
where \(I_j(\theta)\) is the information function for item j.
In case of dichotomous IRT models, all \(g_l\) values reduce to 1, so that item responses \(x_l\) equal either 0 or 1. Set simply \(P_j(\theta)\) as the probability of answering item j correctly for a given ability level \(\theta\), and set \(Q_j(\theta)=1-P_j(\theta)\). Finally, set \(\hat{\theta}_{k+1}^0\) and \(\hat{\theta}_{k+1}^1\) as the ability estimates computed under the condition that the response to item j is 0 or 1 respectively (that is, if the response pattern is updated by 0 or 1 for item j). Then, the EI for item j reduces to
$$EI_j = P_j(\hat{\theta}_k)\,I_j(\hat{\theta}_{k+1}^1) + Q_j(\hat{\theta}_k)\,I_j(\hat{\theta}_{k+1}^0)$$.
Two types of information functions are available. The first one is the observed information function, defined as
$$OI_j(\theta) = -\frac{\partial^2}{\partial \theta^2} \,\log L(\theta | x_j)$$
(van der Linden, 1998), where \(L(\theta | x_j)\) is the likelihood related to item j. The second one is Fisher information function:
$$I_j(\theta) = -E\,\left[\frac{\partial^2}{\partial\,\theta^2} \,\log L(\theta | x_j))\right].$$
Under the 1PL and the 2PL models, these functions are identical (Veerkamp, 1996). See also OIi.
The observed and Fisher information functions are specified by the infoType argument, with respective values "observed" and "Fisher". By default, the observed information function is considered (Choi and Swartz, 2009; van der Linden, 1998).
The estimator of provisional ability is defined by means of the arguments method, priorDist, priorPar, D, range and parInt of the thetaEst function. See the corresponding help file for further details.
The provisional response pattern and the related item parameters are provided by the arguments x and it.given respectively. The target item (for which the maximum information computed) is given by its number in the item bank, through the
item argument.
Note that the provisional response pattern x can also be set to NULL (which is useful when the number nrItems of starting items is set to zero). In this case, it.given must be a matrix with zero rows, such as e.g., itemBank[NULL,].