The EPV can be used as a rule for selecting the next item in the CAT process (Choi and Swartz, 2009; Owen, 1975; van der Linden, 1998). 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 \(\theta\) (thus \(t \in \{0, ..., g_j\}\)). Finally, set \(Var(\theta | x_1, ..., x_k, t)\) as the posterior variance of \(\theta\), given the provisional response pattern (updated by response \(t\)). Then, the EPV for item j equals
$$EPV_j = \sum_{t=0}^{g_j} P_{jt}(\hat{\theta}_k)\,Var(\theta | x_1, ..., x_k, t)$$.
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 \(Var(\theta | x_1, ..., x_k, 0)\) and \(Var(\theta | x_1, ..., x_k, 1)\) as the posterior variances of \(\theta\), given the provisional response pattern (updated by response 0 and 1 respectively). Then, the EPV for item j reduces to
$$EPV_j = P_j(\hat{\theta}_k)\,Var(\theta | x_1, ..., x_k, 1) + Q_j(\hat{\theta}_k)\,Var(\theta | x_1, ..., x_k, 0)$$.
The posterior variances \(Var(\theta | x_1, ..., x_k, x_j)\) (where \(x_j\) takes value 0 or 1 for dichotomous models, and 0, 1, ..., or \(g_j\) for polytomous models) is computed as the squared standard error of the EAP estimate of ability, using the response pattern \((x_1, ..., x_k, x_j)\). This is done by a joint use of the eapEst and eapSem functions.
The prior distribution is set up by the arguments priorDist and priorPar, with the by-default standard normal distribution. The range of integration is defined by the parInt argument, with by default, the sequence from -4 to 4 and of length 33 (or, by steps of 0.25). See the function
eapEst 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,].