Item difficulty: definition

As modeled in Winsteps, the difficulty (challenge, easiness, etc.) of an item (task, prompt, etc.) is the point on the latent variable (unidimensional continuum) at which the highest and lowest categories have equal probability of being observed. This is usually near the center of the middle category of an odd number of categories, or close to the transition between adjacent central categories of an even number of categories.

 

For a dichotomous item, this is the point at which each category has a 50% probability of being observed.

 

For a Rasch-Andrich rating-scale item, this definition implies that the sum of the rating-scale-structure measures sum to zero relative to the item difficulty, i.e., the sum of the Rasch-Andrich thresholds is zero, i.e., sum(Fj) = 0.

 

For a Masters partial-credit item, this definition implies that the item difficulty is the average of the difficulties of the Rasch-Masters thresholds for the item, i.e., Di = average of the delta (Dij) values, so that re parameterizing, Dij = Di + Fj, then sum(Fij) = 0 for each item i.