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:

$p_{1} = [P (Y_{1} = 1), \dots, P (Y_{j} = 1), \dots, P (Y_{J} = 1)]$

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

$p_{2} = [P (Y_{1} = 1, Y_{2} = 1), \dots, P (Y_{j} = 1, Y_{j^{'}} = 1), \dots, P (Y_{J - 1} = 1, Y_{J} = 1)]$ We can also find the model-predicted counterparts.