TSQE: Two-step Q-matrix estimation method

In NPCDTools: The Nonparametric Classification Methods for Cognitive Diagnosis

Two-step Q-matrix estimation method

Description

The function estimates the Q-matrix based on the response data using the two-step Q-matrix estimation method.

Usage

TSQE(
  Y,
  K,
  input.cor = c("tetrachoric", "pearson"),
  ref.method = c("QR", "GDI"),
  GDI.model = c("GDINA", "DINA", "ACDM", "RRUM"),
  cutoff = 0.8
)

Arguments

Y	A N \times J binary data matrix consisting of responses from N examinees to J items
K	The number of attributes in the Q-matrix
input.cor	The type of correlation used as input for the provisional attribute extraction (PAE) algorithm. It could be the tetrachoric or pearson correlation.
ref.method	The refinement method used to polish the provisional Q-matrix obtained from the PAE. Currently available methods include the Q-matrix refinement (QR) method and the G-DINA discrimination index (GDI).
GDI.model	The CDM used in the GDI algorithm to fit the data. Currently available models include the DINA model, the ACDM, the RRUM, and the G-DINA model.
cutoff	The cutoff used to dichotomize the entries in the provisional Q-matrix. The default is 0.8.

Value

The function returns the estimated Q-matrix.

Details

The TSQE method estimates a Q-matrix by integrating the provisional attribute extraction (PAE) algorithm with a Q-matrix refinement-and-validation method, such as the Q-Matrix Refinement (QR) method and the G-DINA Model Discrimination Index (GDI). Specifically, the PAE algorithm relies on classic exploratory factor analysis (EFA) combined with a unique stopping rule for identifying a provisional Q-matrix, and the resulting provisional Q-Matrix is "polished" by a refinement method to derive the finalized estimation of Q-matrix.

The PAE Algorithm starts with computing the inter-item tetrachoric correlation matrix. The reason for using tetrachoric correlation is that the examinee responses are binary, so it is more appropriate than the Pearson product moment correlation coefficient. See Köhn et al. (2025) for details. The next step is to use factor analysis on the item-correlation matrix, and treat the extracted factors as proxies for the latent attributes. The third step concerns the identification of specific attributes required for each item. The detailed algorithm is described below:

(1)

Initialize the item index as j = 1.

(2)

Let l_{jk} denote the loading of item j on factor k, where k = 1,2,...,K.

(3)

Arrange the loadings in descending order. Define a mapping function f(k) = t, where t is the order index. Hence, l_{j(1)} will indicate the maximum loading, while l_{j(K)} will indicate the minimum loading.

(4)

Define

p_j(t) = \frac{\sum_{h=1}^t l_{j(h)}^2}{\sum_{k=1}^K l_{jk}^2}

as the proportion of the communality of item j accounted for by the first t factors.

(5)

Define

K_j = \min \{ t \mid p_j(t) \geq \lambda \}

, where \lambda is the cut-off value for the desired proportion of item variance-accounted-for. Then, the ordered entries of the provisional q-vector of item j are obtained as

q_{j(t)}^* = \begin{cases} 1 & \text{if } t \leq K_j \\ 0 & \text{if } t > K_j \end{cases}

.

(6)

Identify q_j^* = (q_{j1}^*,q_{j2}^*,...,q_{jK}^*) by rearranging the ordered entries of the q-vector using the inverse function k = f^{-1}(t).

(7)

Set j = j + 1 and repeat (2) to (6) until j = J. Then denote the provisional Q-matrix as \mathbf{Q}^*.

The provisional Q-matrix \mathbf{Q}^* is then refined by using either the QR or GDI method.

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")}

de la Torre, J., & Chiu, C.-Y. (2016). A general method of empirical Q-matrix validation. Psychometrika, 81, 253-273. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/s11336-015-9467-8")}

Köhn, H. F., Chiu, C.-Y., Oluwalana, O., Kim, H. & Wang, J. (2025). A two-step Q-matrix estimation method, Applied Psychological Measurement, 49(1-2), 3-28. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1177/01466216241284418")}

See Also

QR

Examples

## Not run: 
library(GDINA)
N = 1000
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")

## Run TSQE method with QR
est.Q = TSQE(Y, K, input.cor = "tetrachoric", ref.method = "QR", cutoff = 0.8)

## If the recovery rate is to be computed, the columns of the estimated Q-matrix 
## should be permuted so that they align with those of the true Q-matrix. 
best.est.Q = bestQperm(est.Q, Q)

## Compute the recovery rate
RR(best.est.Q, Q)

## End(Not run)

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