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NAEP Analysis and Scaling → Estimation of NAEP Score Scales → Checks for Violations of Assumptions → Comparisons of Empirical and Theoretical Item Response Functions → Calculation of Empirical Proportions for Empirical Item Response Functions

NAEP Technical DocumentationCalculation of Empirical Proportions for Empirical Item Response Functions

The theoretical curves are plots of the response functions based on the estimates of the item parameters. The empirical proportions are calculated from the posterior distributions of the s for each student who received the item. For dichotomous items, the sum of the values of the posterior distributions at a point on the θ scale for each student who answered an item correctly plus the sum of a fractional portion of the values of the posterior distribution at that point on the θ scale for each student who omitted the item is parallel in meaning to the number of students who actually answered the item correctly plus a fraction of the number of students who omitted the item. The sum of the values of the posterior distributions for all students receiving the item at each point on the θ scale is parallel in meaning to the empirical number of students at that point on the θ scale who received the item. The plotted values are sums of these individual posteriors at each point on the θ scale for those who got the item correct plus a fraction of the omitters divided by the sum of the posteriors of those administered the item, in the case of dichotomous items, and for those who scored in the category of interest over the sum for those who received the item, in the case of polytomous items.

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