12.3 GDINA discrimination index (GDI)

An essential component of these methods is the G-DINA discrimination index (GDI)

• GDI, denoted by ${\varsigma }_j^2\left(q\right)$ for item $j$, is the variance of success probabilities given a possible q-vector $q$ $${\varsigma }_j^2\left(q\right)=\sum _{l=1}^{2^{K_j^{\ast }}}p\left({\alpha }_{lj}^{\ast }|q\right){\left[P\right(Y=1|{\alpha }_{lj}^{\ast },q)-\overline{P}(Y=1|{\alpha }_{lj}^{\ast },q\left)\right]}^2,$$ where $q$ is a possible q-vector of item $j$ and $$\begin{array}{c}\overline{P}(Y=1|{\alpha }_{lj}^{\ast },q)=\sum _{l=1}^{2^{K_j^{\ast }}}p\left({\alpha }_{lj}^{\ast }\right)P(Y=1|{\alpha }_{lj}^{\ast },q).\end{array}$$
• Theoretically, or when the correct Q-matrix is used and models fit the data perfectly, the correct q-vector and overspecified q-vectors from the correct one produce the largest GDI
• In practice, however, overspecified q-vectors from the correct one have larger GDI than the correct q-vector due to random errors
• The q-vector with all 1s produced the largest GDI for each item in practice