12.3 GDINA discrimination index (GDI)

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

• GDI, denoted by \(\varsigma_{j}^2(\mathbf{q})\) for item \(j\), is the variance of success probabilities given a possible q-vector \(q\) \[ \varsigma_{j}^2(\mathbf{q})=\sum_{l=1}^{2^{K_{j}^*}}p({\alpha}_{lj}^*|\mathbf{q})\left[P(Y=1|{\alpha}_{lj}^*,\mathbf{q})-\bar{P}(Y=1|{\alpha}_{lj}^*,\mathbf{q})\right]^2, \] where \(\mathbf{q}\) is a possible q-vector of item \(j\) and

\[ \bar{P}(Y=1|{\alpha}_{lj}^*,\mathbf{q})=\sum_{l=1}^{2^{K_{j}^*}}p({\alpha}_{lj}^*)P(Y=1|{\alpha}_{lj}^*,\mathbf{q}). \]

• 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