12.8 Stepwise method
To address the both limitations, Ma & de la Torre (2020) proposed a stepwise procedure, which use the Wald test as the central component for determining the best q-vector for each item.
To show how Wald test can be used for q-vector validation, assume an item measures three attributes. We want to use Wald test to test whether Attribute 1 is required statistically, the null hypothesis (Attribute 1 is not necessary) would be \[\begin{equation} \begin{bmatrix} 1&-1&0&0&0&0&0&0\\ 0&0&1&0&-1&0&0&0\\ 0&0&0&1&0&-1&0&0\\ 0&0&0&0&0&0&-1&1 \end{bmatrix} \times \begin{bmatrix} {P}(000)\\ {P}(100)\\ {P}(010)\\ {P}(001)\\ {P}(110)\\ {P}(101)\\ {P}(011)\\ {P}(111) \end{bmatrix}=\mathbf{0}. \end{equation}\]