QR: Q-matrix refinement method

In NPCDTools: The Nonparametric Classification Methods for Cognitive Diagnosis

Q-matrix refinement method

Description

The QR function refines a provisional Q-matrix by minimizing the residual sum of squares (RSS) between the observed and ideal item responses across all possible q-vectors, given the estimates of examinees' attribute profiles.

Usage

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

Arguments

Y	A N\times J matrix of binary responses (1=correct, 0=incorrect). Rows represent persons and columns represent items.
Q	A J\times K provisional Q-matrix to be refined. Rows represent items and columns represent attributes.
gate	A string, "AND" or "OR". "AND" is specified when a conjunctive relation between the attributes an examinee possesses and the attributes required by an item is assumed. "OR" is specified when a disjunctive relation between the attributes an examinee possesses and the attributes required by an item is assumed.
max.ite	The number of iterations to run until the RSS's of all items are stationary.

Details

This function implements the Q-matrix refinement (QR) method developed by Chiu (2013). The NPC method (Chiu & Douglas, 2013) is first used to classify examinees and the best q-vector for an item is identified by minimizing its RSS. Specifically, the RSS of item j for examinee i is defined as

RSS_{j} =\sum_{m=1}^{2^K} \sum_{i \in C_{m}} (Y_{ij} - \eta_{jm})^2,

where C_m for m = 1, \ldots, 2^K is the mth proficiency class, and N is the number of examinees. Chiu (2013) proved that the expected value of RSS_j corresponding to the correct q-vector is the minimum among the 2^K - 1 candidates.

Value

A list containing:

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

References

Chiu, C. Y. (2013). Statistical Refinement of the Q-matrix in Cognitive Diagnosis. Applied Psychological Measurement, 37(8), 598-618. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1177/0146621613488436")}

See Also

NPC

Examples

## Not run: 
## Generate data
library(GDINA)
N = 500
Q = sim30GDINA$simQ
J = nrow(Q)
K= ncol(Q)
gs = data.frame(guess = rep(0.2,J), slip = rep(0.2,J))
sim = simGDINA(N, Q, gs.parm = gs, model = "DINA")
Y = extract(sim,what = "dat")

## Randomly generate a misspecified Q with 20% of misspecifications
mis.Q = matrix(0, J, K)
while (any(rowSums(mis.Q)==0)==T){
  mis.q = sample(J*K, J*K*0.2) ## percentage of misspecified q
  ind = arrayInd(mis.q, dim(Q))
  mis.Q = Q
  mis.Q[ind] = 1-mis.Q[ind]
}

## Refine the misspecified Q-matrix
ref = QR(Y, mis.Q)
ref.Q = ref$modified.Q

## Compute the entry-wise and item-wise recovery rates
rr = RR(ref.Q, Q)
rr$entry.wise
rr$item.wise

## Compute the retention rate
retention.rate(ref.Q, mis.Q, Q)

## Compute the correction rate
correction.rate(ref.Q, mis.Q, Q)

## End(Not run)

NPCDTools documentation built on Sept. 1, 2026, 1:08 a.m.