Is the $Q$ Matrix Generically Identifiable?

Checks if the $Q$ matrix is generically identified.

is_q_generic(x)

Arguments

x	A `q_matrix()` or `base::matrix()` to test.

Value

A logical vector

Details

If $\mathbf{Q}$ is in the generically identifiable set $\mathcal{Q}$, then it must satisfy the following conditions:

(G1) The true $\boldsymbol{Q}$ matrix takes the form of $\boldsymbol{Q}^\top=\left[{\boldsymbol{Q}_1,\boldsymbol{Q}_2, (\boldsymbol{Q}^\ast)^\top}\right]^\top$ after row swapping, where $\boldsymbol{Q}^\ast$ is a $(J-2K)\times K$ binary matrix and $\boldsymbol{Q}_1,\boldsymbol{Q}_2 \in \mathcal{Q}_g$ with $${\mathbb {Q}}_g =\left\{ \boldsymbol{Q} \in \{0,1\}^{J\times K}: \boldsymbol{Q} = \begin{bmatrix} * &{} 1 &{} *&{} \dots &{} *&{}\dots &{} *\\ * &{}*&{} 1 &{} \dots &{} *&{}\dots &{} *\\ \vdots &{} \vdots &{} &{} \ddots &{} &{} &{} \vdots \\ * &{} * &{} *&{} \dots &{} 1&{}\dots &{} *\\ \end{bmatrix}\right\}$$ where * can be either 0 or 1.
(G2) For any $k = 1, 2, \ldots, K$, there exists a $j_k > 2K$, such that $q_{j_k, k} = 1$.

References

Gu, Yuqi, and Gongjun Xu. "Sufficient and Necessary Conditions for the Identifiability of the $Q$-matrix." arXiv preprint arXiv:1810.03819 (2018).

The function implemented is a translation of the publicly available MATLAB code from https://github.com/yuqigu/Identify_Q into C++.

Author

James Joseph Balamuta and Steven Andrew Culpepper

Examples

## Check if Q Matrix is Generically Identified ---
# Create a generically identified Q matrix
q3_generic = rbind(diag(3),
                   c(1, 1, 0),
                   c(1, 0, 1),
                   c(0, 1, 1),
                   c(1, 1, 1))

# Check if Q matrix is generic
is_q_generic(q3_generic)
#> [1] TRUE

# (Extra) Check if Q matrix is strict
is_q_strict(q3_generic)
#> [1] FALSE