7.9 Limited information statistics

Limited-information statistics focus on marginal proportions or probabilities. We consider \(M_2\) statistic, which only focuses on the first two marginals.

The first marginal proportions are the proportions of students who answer each item correctly:

\[ \mathbf{p}_1=\big[P(Y_{1}=1),\ldots,P(Y_{j}=1),\ldots,P(Y_{J}=1)\big] \]

The second marginal proportions are the proportions of students who answer a pair of items correctly:

\[ \mathbf{p}_2=\big[P(Y_{1}=1,Y_{2}=1),\ldots,P(Y_{j}=1,Y_{j'}=1),\ldots,P(Y_{J-1}=1,Y_{J}=1)\big] \] We can also find the model-predicted counterparts.