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NAEP Analysis and Scaling → Estimation of NAEP Score Scales → Assumptions of the Item Scaling Models → Local Independence

NAEP Technical DocumentationLocal Independence

A basic assumption of Item Response Theory (IRT) is the conditional independence of the responses by an individual to a set of items, given the individual's scale score. That is, conditional on the individual's θ_k, the joint probability of a particular response pattern x = (x₁,...,x_n) across a set of n items is simply the product of terms based on the equations for the two-parameter and three-parameter IRT models and for the generalized partial credit model:

where P_ji (θ_k) is of the form appropriate to the type of item (dichotomous or polytomous), m_j is equal to 2 for the dichotomously scored items, and u_ji is an indicator variable defined by

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