A hyperbolic cosine latent trait model for unfolding polytomous responses: Reconciling Thurstone and Likert methodologies

In the 1920s, Thurstone articulated a theory for the measurement of social variables that involved two distinct steps: first constructing and operationalizing a linear continuum by locating statements according to their affective values; second measuring persons. The first step involved the pair-comparison design and the cumulative response mechanism, the second step involved the direct-response design (of the form Agree or Disagree) and the unfolding response mechanism. In the 1930s, Likert proposed a procedure that obviated the need for the first step and apparently permitted measuring persons from their responses to statements that were similar to those used by Thurstone, but which required a response that indicated degrees of intensity of the form Strongly Agree, Agree, Undecided, Disagree or Strongly Disagree. Furthermore, and in contrast to Thurstone, he implicitly used the cumulative mechanism, scoring the successive categories with successive integers and simply summing them to obtain a measurement for each person. The two procedures were considered to be distinct and alternative, and in general, this is still the perception with a number of matters still not reconciled between the two procedures. By resolving the Disagree response in the unfolding mechanism into its two constituent components, this paper presents an unfolding model for direct responses from first principles, and then generalises it to provide an unfolding model for polytomous responses of the Likert style. This model permits an understanding of those matters still not reconciled between the Thurstone and Likert approaches: First, Likert's success in using Thurstone-like statements with the cumulative mechanism rather than the unfolding mechanism; second, the gap found between two clusters of locations when statements from a Likert-style questionnaire are scaled using the Thurstone procedure; and third, the consistent problem with the middle category of Undecided in Likert-style response formats.