I intend to aggregate two ordinal (or if easier an ordinal and a quasi-cardinal) variables – Can you suggest a sound approach?
The context of the question is the analysis of the design of decision support tools and respective user requirements.
The ordinal variable represents for a distinct user group its preference ranking of six pre-defined tool properties a to f, e.g. the group favors property a over b over c … over f (from 1st to 6th rank position).
Next, a variable judges the quality of different tools with regard to each a, b, c, d, e and f property – like school marks from 1 "very weak" to 5 "excellent" quality. This second variable is actually ordinal, but might be interpreted (like marks) as quasi-cardinal (e.g. assuming that 3 is not only better than 1 and worse than 5, but that 3 is actually some-what half-way between the extremes).
In an aggregation step to assess overall tool quality, marks and ranking information on user requirements are to be linked. Here is the point, I kindly request your view on / suggestion for appropriate approaches: Assuming both variables to be quasi-cardinal would allow, e.g., to multiply the rank position and the mark and compare the sums for the individual tools; but is there a way to aggregate two ordinal variables and then be able to compare the aggregation results?
The attached file gives examlary data to better illustrate the problem.
Thank you for your advice!
To share with those interested in the question an interim result: I found some answer on how to aggregate ordinal information for distance-based consensus in ordinal ranking models in a paper by Cook et al (1996), which is linked below. The authors revisit the concerns of aggregating sets of ordinal rankings of n alternatives in models using average ranks. Here, the ordinal rank position numbers are interpreted directly as the values of being ranked at those levels. The authors show - for deriving consensus based on a general framework for associating value or worth with ordinal ranks - that ℓp-distance models using the general framework are equivalent to the conventional ordinal ℓ1 or ℓ2 models for any p ⩾ 1. - I'll appreciate any further suggestions.
http://www.nps.navy.mil/orfacpag/resumePages/papers/Kresspa/AGeneral.pdf
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