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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 θk, the joint probability of a particular response pattern x = (x1,...,xn) 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:

The probability of the vector x given theta sub k and the item parameters equals the product over j from one to n and I from zero to m sub j minus one of uppercase p sub j i of theta k to the u sub j i power

where Pji (θk) is of the form appropriate to the type of item (dichotomous or polytomous), mj is equal to 2 for the dichotomously scored items, and uji is an indicator variable defined by

u sub ji equals one if response x sub j is in category i, otherwise u sub ji equals zero.

Last updated 29 April 2008 (PE)

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