Fun Challenge - weighted average of global rankings

[rm2dance]

***Fun Challenge -- weighted average of global rankings***

Goal
The goal is to create a weighted average of global rankings. Does not have to be perfectly accurate (if that was even possible).

Specific Example
To make this problem less abstract and easier, we are going to use a specific example -- Sweden. Sweden is ranked by the following numbers:

http://en.wikipedia.org/wiki/E-readiness#Economist_Intelligence_Unit_e-readiness_rankings
2 overall
8.67 index score

http://en.wikipedia.org/wiki/Global_Competitiveness_Report#2010-2011_rankings
2 overall
5.56 index score

On http://en.wikipedia.org/wiki/Global_Innovation_Index#Positively_ranked_countries
10 overall
1.56 index score

Expanded Goal
1 -- Now we don't want to use the "overall" rank because the "separation" between each overall rank are not "equal."

​

1a -- Doing this will lead to less accuracy in the combined ranking.

2 -- Instead, what we want to do is "factor" in the index score onto the overall rank.

​

2a -- The index score can be seen as a % or a proportion out of 100.

​

2b -- Using this new "factored number," a combined ranking can be made.

The Problem
How do you create a weighted average that is based off of (or at least accounts for) the index scores?

Stuff already known
The "standard" or "common" weighted average is basically the following method: ((a% * a) + (b% * b) + (c% * c))/3
So we can call the mentioned problem and goal a "factored average" instead of a "weighted average" as the terms may cause unnecessary confusion, unless there is already a phrase/label for specifically the mentioned problem and goal.

Next step -- Solution
There may be need for clarification, and I'll try my best to make something more specific, but I'm pretty sure a leading statistician can easily understand, and solve the problem, just as easily.

***Please do not use specific statistical terminologies unless you feel they are absolutely necessary. If a poster uses any terms outside of the language used in this post, feel free to explain them as soon as they are used (or link to a clear, concise explanation).

You should be able to explain the solution to a junior high school girl. If you cannot, then the solution is neither clear nor concise.

 

[Evo]

Bumping for OP.

 

[pmsrw3]

I don't understand what you want to do. What is the purpose of this so-called "factored average"? It seems pretty clear that you're asking for some sort of linear combination of the component rankings, but without any explanation of the purpose of this maneuver, there's no way to choose an optimum.

Furthermore, I don't understand why you're unhappy with what you call '"overall" rank'. What do you mean by saying the separation between overall ranks is not equal? Why is that bad? Isn't it desirable, in any ranking system, that separations not be equal? After all, if two things are more different from each other, we want the separation between them to be greater.

 

[rm2dance]

1 -- 1st half
If "there's no way to choose an optimum" without an "explanation of the purpose," then it's best to just relay the various options. Then I can decide on whatever is "optimum."

I don't understand what kind of "explanation" may be needed. It sounds like the "purpose" is already in the Expanded Goal.

2 -- 2nd half
As to why I'm "unhappy with" the overall rank (which is the correct terminology)[1][2], it's already clear -- "Doing this will lead to less accuracy in the combined ranking."

2.1 -- Not Relevant to the Goal (better to leave it aside)
I don't know how to respond to the rest as you seem to be inconsistent with your thought process and conflicting yourself: You first express the lack of clarity with the statement "the separation between each overall rank are not equal" by asking "What do you mean.." Given this presumed lack of clarity, you still go on to interpret it and say that it is "desirable, in any ranking system, that separations not be equal." This is an inconsistent thought process on your part. This makes no sense. If there was a "presumed lack of clarity," why would you still go on to interpret it? To me, this is a cognitive malfunction, but the more important thing is that this is simply not relevant to the goal/purpose.

2.2 -- Clarification Relevant to the Goal
So after thinking about the Expanded Goal, I need to clarify:

I agree that it is "desirable, in any ranking system, that separations not be equal" and that is exactly the problem with the overall rank "because the separation between each overall rank are not equal."

One reason why we don't want to use the overall rank is because if "two things are more different from each other, we want the separation between them to be [in appropriate proportion]." Because the index scores show that the separation between each rank is, in fact, not equal, we want the combined ranking to account for the index scores. Not "doing this will lead to less accuracy in the combined ranking."

So what I wrote may not have been as "explanatory" to some people but, at least to me, the Expanded Goal already relayed the problem, which lead to you answering your own question, "Why is that bad?" -- besides what was already stated in the main post and this post -- because it is "desirable, in any ranking system, that separations not be equal. After all, if two things are more different from each other, we want the separation between them to be [in appropriate proportion]."

3 -- Misc.
I have no idea if this is a "linear" combination or bilinear or multilinear, or what the difference is.

4 -- Best Solution
If after all this, you feel "there's [still] no way to choose an optimum," then it's easier to just relay the various options, and I'll figure out what is "optimum."

5 -- References
[1] see usage of overall rank -- http://en.wikipedia.org/wiki/Comparison_between_U.S._states_and_countries_by_GDP_(PPP )
[2] did you even check the link?? -- http://en.wikipedia.org/wiki/Global_Innovation_Index#Positively_ranked_countries

 

[pmsrw3]

Here is THE ANSWER.

Suppose there are n indexes to be combined. Let [itex]I_k[/itex] be the k^th index. Then the so-called "factored average" is

[tex]
\mbox{FA} = \frac{1}{n!} \sum_{\pi \in S_n}
\mbox{arcsinh}\left(
\sum_{k=1}^{n} I_{\pi(k)}^{\varphi^{k-1}}
\right)
[/tex]

[2] did you even check the link?? -- http://en.wikipedia.org/wiki/Global_...nked_countries

Yes! I did! It was VERY INFORMATIVE!