Function GNPC is used to estimate examinees' attribute profiles using
the general nonparametric classification (GNPC) method
(Chiu, Sun, & Bian, 2018; Chiu & Koehn, 2019). It can be
used with data conforming to any CDMs.
GNPC(
Y,
Q,
initial.dis = c("hamming", "whamming"),
initial.gate = c("AND", "OR", "Mix")
)The function returns a series of outputs, including
The estimates of examinees' attribute profiles
The estimates of examinees' class memberships
The weighted ideal responses
The weights used to compute the weighted ideal responses
A \(N \times J\) binary data matrix consisting of the responses from \(N\) examinees to \(J\) items.
A \(J \times K\) binary Q-matrix where the entry \(q_{jk}\) describing whether the \(k\)th attribute is required by the \(j\)th item.
The type of distance used in the AlphaNP to carry
out the initial attribute profiles for the GNPC method.
Allowable options are "hamming" and "whamming" representing
the Hamming and the weighted Hamming distances, respectively.
The type of relation between examinees' attribute profiles
and the items.
Allowable relations are "AND", "OR", and "Mix",
representing the conjunctive, disjunctive, and mixed relations, respectively
A weighted ideal response \(\eta^{(w)}\), defined as the convex combination of \(\eta^(c)\) and \(\eta^(d)\), is proposed. Suppose item j requires \(K_{j}^* \leq {K}\) attributes that, without loss of generality, have been permuted to the first \(K_{j}^*\) positions of the item attribute vector \(\boldsymbol{q_j}\). For each item j and \(\mathcal{C}_{l}\), the weighted ideal response \(\eta_{ij}^{(w)}\) is defined as the convex combination \(\eta_{ij}^{(w)} = w _{lj} \eta_{lj}^{(c)}+(1-w_{lj})\eta_{lj}^{(d)}\) where \(0\leq w_{lj}\leq 1\). The distance between the observed responses to item j and the weighted ideal responses \(w_{lj}^{(w)}\) of examinees in \(\mathcal{C}_{l}\) is defined as the sum of squared deviations: \(d_{lj} = \sum_{i \in \mathcal {C}_{l}} (y_{ij} - \eta_{lj}^{(w)})^2=\sum_{i \in \mathcal {C}_{l}}(y_{ij}-w_{lj}\eta_{lj}^{(c)}-(1-w_{lj})\eta_{lj}^{(d)})\) Thus, \(\widehat{w_{lj}}\) can be minimizing \(d_{lj}\): \(\widehat{w_{lj}}=\frac{\sum_{i \in \mathcal {C}_{l}}(y_{ij}-\eta_{lj}^{(d)})}{\left \| \mathcal{C}_{l} \right \|(\eta_{lj}^{(c)}-\eta_{lj}^{(d)})}\)
As a viable alternative to \(\boldsymbol{\eta^{(c)}}\) for obtaining initial estimates of the proficiency classes, Chiu et al. (2018) suggested to use an ideal response with fixed weights defined as \(\eta_{lj}^{(fw)}=\frac{\sum_{k=1}^{K}\alpha_{k}q_{jk}}{K}\eta_{lj}^{(c)}+(1-\frac{\sum_{k=1}^{K}\alpha_{k}q_{jk}}{K})\eta_{lj}^{(d)}\)