THRESHOLDS AND ICCs
What is an explanation behind reversed thresholds in RUMM2030?
The presence of reversed thresholds in a Rasch analysis raises the issue of the scoring of the categories with successive integers. If two successive thresholds are reversed, e.g., the threshold between x-1 and x is greater than the threshold between x and x+1, it means that the person on the boundary of the former has a greater ability than the person on the boundary of the latter. This has to be a problem of one kind or another.
If the person-item targeting is not an issue [see the next question for mis-targeted samples], then the distribution of the persons as a whole, or frequencies in the categories as a whole, is not the issue. The issue is that persons who should have responded in category x did not do so at the expected rate, not that there was not many people in category x.
Thus, if adjacent thresholds are disordered then it would be appropriate to experiment with the rescoring of all the categories for that item.
Rescoring should not be entertained lightly and the procedure should be undertaken only when the relative threshold locations are assessed carefully. The collapsing of categories [as rescoring is often referred to in the literature] is, in general, not compatible with the Rasch model when the data fit the model. Technically, the only time the category scoring function can be changed so that adjacent categories are collapsed is when discriminations at the thresholds are zero. Reversed thresholds may reveal problems in the discriminations at the thresholds (which are required to be the same in the Rasch model for ordered categories), and rescoring may solve the problem. Ideally, new categories would be defined from the collapsed categories and new data collected to confirm that the collapsing of adjacent categories has worked.
Andrich, D. (2000). Understanding resistance to the data-model relationship in Rasch's paradigm: A reflection for the next generation. Journal of Applied Measurement, 3, 325-357.
Andrich, D. (1995). Models for measurement, precision and the non-dichotimization of graded responses. Psychometrika, 60, (1) 7-26.
Andrich, D., de Jong, J.H.A.L. & Sheridan, B.E. (1997). Diagnostic opportunities with the Rasch model for ordered response categories. In J. Rost and R. Langeheine (Eds.), Applications of Latent Trait and Latent Class Models in the Social Sciences (pp. 59-70).
Jansen, P.G.W. & Roskam, E.E. (1986). Latent trait models and dichotomization of graded responses. Psychometrika, 51 (1) 69-91.
How does RUMM2030 calculate the slopes associated with the ICC displays?
The slope of an ICC is calculated at the location of the item simply as the first differential coefficient of the expected value curve as a function of the latent trait (beta).
The slopes of ICCs for dichotomous response items are identical.
In ordered category data:
What if the persons are mis-targetted with respect to the Items?
When the location of the persons is far away from the location of the items, this is considered bad targetting. Mis-targetting, as this problem is usually labelled, is really a problem with the distribution of the persons as a whole.
The best place to start in RUMM2030 is to examine the Person/Item Distribution display to see visually how well the persons are located with respect to the item location distribution.
The Person Separation Index is a pointer to the level of mis-targetting as this value will decrease towards zero as the mis-match between the person and item distributions becomes more pronounced.
Mis-targetting can also be inferred from the Category Frequencies display.
If there is no data present in the extreme categories, i.e. no persons are located there, it means that the estimates for these regions are unstable. The manifestation of this lack of data will almost certainly appear as disorder in the threshold estimates for categories within the region.
To address mis-targetted samples: