How to aggregate variables.pdf

How to aggregate ordinal–ordinal or quasi-cardinal–ordinal variables? I intend to aggregate two ordinal (or if easier an ordinal and a quasi-cardinal) variable – 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 i its preference regarding the ranking of six pre-defined tool properties a to f, e.g. group i favors property a over b over c … over f (i.e. f is rank position 1, e = 2nd, d = 3rd … a = 6th [and best]). Tab. 1: Requirements a-f regarding tools of user groups i Tool Requirement criterion a

User i User ii User iii Ranking 1 ≺ 2 ≺ 3 ≺ 4 ≺ 5 ≺ 6  quasi-cardinal interpretation 1 6 2

User i User ii User iii Ranking α ≺ β ≺ γ ≺ δ ≺ ε ≺ ζ  ordinal interpretaiton α ζ β

b

6

3

4

ζ

γ

δ

c

3

4

6

γ

δ

ζ

d

2

5

5

β

ε

ε

e

5

1

1

ε

α

α

f

4

2

3

δ

β

γ

Next, a variable judges the quality of different tools I 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 (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). Tab. 2: Judging the tools Tool Requirement criterion a

Tool I Tool II Tool III Marking „1“ weak to „5“ excellent  quasi-cardinal interpretation 4 2 2

Tool I Tool II Tool III Ranking A ≺ B ≺ C ≺ D ≺ E  ordinal interpretaiton D B B

b

3

2

5

C

B

E

c

2

5

1

B

E

A

d

3

3

3

C

C

C

e

1

3

4

A

C

D

f

1

3

2

A

C

B

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 (e.g. User i for Tool I = 1*4+6*3+3*2+2*3+5*1+4*1 = 43) and compare the sums for the individual tools; but is there a way to aggregate two ordinal variables (like User i for Tool I = α*D + ζ*C + γ*B + … = __??) and then be able to compare the aggregation results?