ABSTRACT
The two main mechanisms for characterizing dichotomous responses of persons to items on a single dimension are the cumulative and the unfolding. In the former, the probability of a positive response is a monotonic function of the location of the relevant parameters on the continuum; in the latter, it is single-peaked. This chapter presents a unidimensional IRT model for unfolding, referred to as the hyperbolic cosine model (HCM). Table 21.1 shows a deterministic unfolding response pattern and, for purposes of contrast, the deterministic cumulative response patterns, for five items ordered on the continuum. It is evident that although the total score recovers the pattern of responses in the latter (Guttman, 1950), this is not the case with the unfolding patterns. In the former, if a person responds positively to an item, then the person will respond positively to all items to the left in the order; in the latter, the person responds positively only to items close to the person's location on the continuum, and negatively at greater distances from the person's location on both ends of the continuum. Deterministic Unfolding and Cumulative and Response Patterns to Five Statements
|
Statements |
Statements |
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|---|---|---|---|---|---|---|---|---|---|---|---|
|
1 |
2 |
3 |
4 |
5 |
Sum |
1 |
2 |
3 |
4 |
5 |
Sum |
|
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
|
1 |
0 |
0 |
0 |
0 |
1 |
1 |
0 |
0 |
0 |
0 |
1 |
|
1 |
1 |
0 |
0 |
0 |
2 |
1 |
1 |
0 |
0 |
0 |
2 |
|
0 |
1 |
1 |
0 |
0 |
2 |
1 |
1 |
1 |
0 |
0 |
3 |
|
0 |
0 |
1 |
1 |
0 |
2 |
1 |
1 |
1 |
1 |
0 |
4 |
|
0 |
0 |
0 |
0 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
5 |
|
Illustrative unfolding response patterns for five ordered statements |
Cumulative response patterns for five ordered statements |
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