QR: Refine the Q-matrix by Minimizing the RSS

In NPCDTools: The Nonparametric Classification Methods for Cognitive Diagnosis

Refine the Q-matrix by Minimizing the RSS

Description

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.

Usage

QR(Y, Q, gate = c("AND", "OR"), max.ite = 50)

Arguments

Y	A matrix of binary responses (1=correct, 0=incorrect). Rows represent persons and columns represent items.
Q	The Q-matrix of the test. Rows represent items and columns represent attributes.
gate	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.
max.ite	The number of iterations to run until all RSS of all items are stationary.

Value

A list containing:

initial.class	Initial classification
terminal.class	Terminal classification
modified.Q	The modified Q-matrix
modified.entries	The modified q-entries

The Q-Matrix Refinment (QR) Method

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.

References

Chiu, C. Y. (2013). Statistical Refinement of the Q-matrix in Cognitive Diagnosis. Applied Psychological Measurement, 37(8), 598-618.

NPCDTools documentation built on Sept. 23, 2024, 5:10 p.m.