Q.completeness: Check the completeness status of a binary Q-matrix

In NPCDTools: The Nonparametric Classification Methods for Cognitive Diagnosis

Check the completeness status of a binary Q-matrix

Description

Q.completeness is used to examine whether a given Q-matrix is complete when data conform to a specified CDM. A Q-matrix is said to be complete if it allows for the unique identification of all possible attribute profiles among examinees. So far, the function can only be used for a binary Q-matrix with binary responses.

Usage

Q.completeness(raw.Q, model = NULL)

Arguments

raw.Q	The Q-matrix that is to be checked, where J is the number of items and K is the number of attributes. It must be a binary (0/1) matrix or data frame that can be coerced to a matrix.
model	Character string specifying the cognitive diagnosis model. Valid options are "DINA", "DINO", or "General" (for general CDMs including G-DINA, LCDM, etc.). Case-insensitive. If not specified or invalid, the default is "General" with a warning.

Details

The conditions for one Q-matrix completeness are model-dependent: a Q-matrix may be complete for one CDM but incomplete for another. This function implements the theoretical work developed by Chiu et al. (2009) and Köhn and Chiu (2017).

For DINA and DINO models:

A Q-matrix is complete if and only if it contains all K single-attribute items (Chiu et al., 2009).

For More General CDMs:

The function implements a sequential procedure based on Theorems 3-4 and Propositions 1-2 in the work by Köhn and Chiu (2017).

1. If Q contains all K single-attribute items, it is complete (Proposition 1).

2. If Q has rank < K, it is incomplete (Theorem 3).

3. For full-rank Q-matrices without all single-attribute items, the function examines non-nested attribute pairs using indicator vectors to determine if distinct expected response patterns \mathbf{S}(\boldsymbol{\alpha}) can be guaranteed.

The theoretical framework establishes the sufficient conditions for Q completeness, which means completeness implies distinct expected item response patterns for all 2^K possible attribute profiles.

Value

A list of class "Qcompleteness" containing:

is_complete	Logical value indicating completeness: TRUE if complete, FALSE if incomplete, NA if uncertain.
status	Character string: "complete", "incomplete", or "uncertain".
message	Character string with detailed explanation of the result.
model	The CDM used for assessment.
K	Number of attributes in the Q-matrix.
J	Number of items in the Q-matrix.

The function also prints the status message to the console as a side effect.

References

Chiu, C.-Y., Douglas, J. A., & Li, X. (2009). Cluster analysis for cognitive diagnosis: Theory and applications. Psychometrika, 74(4), 633-665. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/s11336-009-9125-0")}

Köhn, H.-F., & Chiu, C.-Y. (2017). A procedure for assessing the completeness of the Q-matrices of cognitively diagnostic tests. Psychometrika, 82(1), 112-132. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/s11336-016-9536-7")}

Köhn, H.-F., & Chiu, C.-Y. (2018). How to build a complete Q-matrix for a #' cognitively diagnostic test. Journal of Classification, 35(2), 273-299. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/s00357-018-9255-0")}

Examples

## Not run: 
# Example 1: Complete Q-matrix for DINA model
# (contains all 3 single-attribute items)
Q1 <- matrix(c(1, 0, 0,
               0, 1, 0,
               0, 0, 1,
               1, 1, 0,
               1, 0, 1), ncol = 3, byrow = TRUE)
result1 <- Q.completeness(Q1, model = "DINA")
print(result1$is_complete)  # TRUE

# Example 2: Incomplete Q-matrix for DINA model
# (missing single-attribute items)
Q2 <- matrix(c(1, 1, 0,
               1, 0, 1,
               0, 1, 1), ncol = 3, byrow = TRUE)
result2 <- Q.completeness(Q2, model = "DINA")
print(result2$is_complete)  # FALSE

# Example 3: Check completeness for general CDM
Q3 <- matrix(c(1, 0, 0,
               0, 1, 0,
               0, 0, 1,
               1, 1, 0,
               1, 0, 1,
               0, 1, 1), ncol = 3, byrow = TRUE)
result3 <- Q.completeness(Q3, model = "General")

## End(Not run)

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