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To model the item characteristics of a large-scale assessment, item difficulty parameters obtained from the Rasch model in the empirical study were used as the population parameters when generating item responses in the simulation study.

For a dichotomous item in the Rasch model, the item information is P i  × (1 − P i ), where P i is the probability of correctly solving item i. Accordingly, the maximum value of each item information curve in the dichotomous Rasch model is 0.25 at P i  = .50.50

In the factor model, the item variance is decomposed in to a part due to (explained by) the common factor(s) and a residual part.

Based on the model, the item parameters were calculated.

Given overall fit to the model, the item was retained.

Under the conventions of the Rasch model, the item thresholds are scaled so that their arithmetic mean is zero.

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In the Rasch model, the items included in an instrument must be defined as only a single construct (unidimensionality) [ 5].

(See Table  1; initial analysis) To improve the fit of the scale to the Rasch model the items with disordered thresholds were rescored to reorder them.

In the model the items were matched to find out which of their combination would best reproduce the mean value of the summary score.

The conditioning model is the item response model multiplied by the population model (Adams & Wu [2007]; Embretson & Reise [2000]).

A man wearing a Eurovision-branded straitjacket then came into view, modelling the item by turning around as Mede observed: "You know what they say – crazy is the new black".

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