We estimate memberships using the non-parametric classification method (weighted hamming), and comparisons of the residual sum of squares computed from the observed and the ideal item responses.
QR(Y, Q, gate = c("AND", "OR"), max.ite = 50)A list containing:
Initial classification
Terminal classification
The modified Q-matrix
The modified q-entries
A matrix of binary responses (1=correct, 0=incorrect). Rows represent persons and columns represent items.
The Q-matrix of the test. Rows represent items and columns represent attributes.
A string, "AND" or "OR". "AND": the examinee needs to possess all related attributes to answer an item correctly. "OR": the examinee needs to possess only one of the related attributes to answer an item correctly.
The number of iterations to run until all RSS of all items are stationary.
This function implements the Q-matrix refinement method developed by Chiu (2013), which is also based on the aforementioned nonparametric classification methods (Chiu & Douglas, 2013). This Q-matrix refinement method corrects potential misspecified entries of the Q-matrix through comparisons of the residual sum of squares computed from the observed and the ideal item responses.
The algorithm operates by minimizing the RSS. Recall that \(Y_{ij}\) is the observed response and \(\eta_{ij}\) is the ideal response. Then the RSS of item \(j\) for examinee \(i\) is defined as $$RSS_{ij} = (Y_{ij} - \eta_{ij})^2$$. The RSS of item \(j\) across all examinees is therefor $$RSS_{j} = \sum_{i=1}^{N} (Y_{ij} - \eta_{ij})^2 = \sum_{m=1}^{2^k} \sum_{i \in C_{m}} (Y_{ij} - \eta_{jm})^2$$ where \(C_m\) is the latent proficiency-class \(m\), and \(N\) is the number of examinees. Chiu(2013) proved that the expectation of \(RSS_j\) is minimized for the correct q-vector among the \(2^K - 1\) candidates. Please see the paper for the justification.
Chiu, C. Y. (2013). Statistical Refinement of the Q-matrix in Cognitive Diagnosis. Applied Psychological Measurement, 37(8), 598-618.