How many steps does a rock take to get to its weight? Our measurement principles should be the same for properties of rocks as it is for the properties of people. What we say has to be consistent with physical measurement. We do not model how a rock got to its weight, even though we could model our process of locating the weight. If the thresholds on our ruler come out back to front, then our process of locating the person, or the rock has gone wrong.
The word "step" has no place in measurement. You can take steps over thresholds, but you can have steps without thresholds, and thresholds without steps, and measurement is not about modelling people taking steps, but modelling the responses to check where they are. The word "step" has a connotation of a local distance in the sense that the next step starts where the previous one finished and on movement. It is never used when we are trying to explain physical measurement.
The point at which the probability is 50% of a response being in one or other category, given that it is in one of them, is a threshold. No one goes anywhere in measurement - the person location is fixed. The Rasch model is a measurement model that models probabilistically where we reckon the person is, between thresholds, as in a ruler. It does not model how a person got there.
The threshold estimates are independent of the person distribution - so we can tell if there is something wrong with our instrument independently of the distribution. All those other statistics to check if the categories are "going up" are not distribution free and they do not answer the question unequivocally and they add to the confusion.
What I said over 20 years ago was simple and correct:
(i) The rating scale transition points are thresholds - they are points where there is 50% chance of being in either category, given that you are in one of them. Everyone understands that as a threshold;
(ii) If in the data, the thresholds are reversed as evidenced by the estimates, there is something wrong in the data. Nothing else. And the evidence on the thresholds is distribution free - very much consistent with Rasch philosophy.
When I say that there is "something wrong in the data", I, of course, mean in the sense of measurement. It is the same kind of wrong as when we have different item discriminations in the dichotomous case. In any particular set of data, it might be perfectly explainable why it is that some item really discriminates too much or too little, and that is a property of the data that we must acknowledge. Likewise when the thresholds in a multicategory item are reversed, they may be reversed for some good reason, and we should acknowledge it. However, the data demonstrate that the ordering is not working as intended.
If the discriminations at these Rasch thresholds are equal, then we have the integer scoring function - just as an integer score arises when the items are independent. Everything is the same as in the simple dichotomous case, except that the dichotomies are not independent. Being dependent does not turn them from dichotomies to steps. The dichotomous case with one dichotomy specializes from the general case. In the special case we have a threshold, not a step. No one goes from wrong to right, they are either wrong or right, and that is a big difference.
This is central - the category probability curves are about the implied distribution of responses of a single person to the item on replications. It is important to think of this as sample- and item-distribution free, that it is about the implied distribution of a single person with a single fixed location responding to a single item on repeated occasions with nothing but random error causing the variation - that is all there is in the model - together with the item parameters, which are thresholds. Those category characteristic curves pertain to only one person responding to the item, not the distribution of persons. Most important.
The Rasch model is not a model of how long any one spends in a category. If that is an issue in the data, then this is not the right model. So we should not try to force the model to do things it cannot do. This is model of where people are. The idea of time, of motion, is not relevant to this model, and it leads to misunderstanding. A rating scale is just like dichotomous items, except that there is a constraint on the dichotomies within the items - the constraint is that response must follow a Guttman pattern.
My suggestions are simple, and consistent: the transition points should be called "thresholds", and in the context of Thurstone or other thresholds, they should be called Rasch thresholds.
David Andrich, Murdoch University
Andrich D. 1978. A rating formulation for ordered response categories. Psychometrika, 43 561-573
Andrich D. 1979. A model for contingency tables having an ordered response classification. Biometrics, 35, 403-415.
Thresholds, Steps and Rating Scale Conceptualization.Andrich D.A. Rasch Measurement Transactions, 1998, 12:3 p. 648-9.
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