statistics (non-parametric?), comparative rankings, advice

[overtone]

This has come up a few times in various threads, most recently for me in a thread on gun regulation.

 

Given two different orderings of a given set according to two different criteria, how does one obtain information about a partial correlation if any between the two criteria? The goal would be to use the compared orderings only, not any values or quantities of the criteria.

 

The specific situation was this: the 50 States of the US were ordered 1 - 50 by percentage of citizens possessing guns, and again 1 - 50 by homicide rate. The claim at hand was that gun ownership rate was positively correlated with homicide rate. What I was trying to sort out was an idea of how far apart one would expect the rankings of a given State to be by chance, and what it meant for the mean and median of the rank differences to be a given value different from that individual chance one.

 

A complication: There appeared to be reason to suspect some kind of bimodal distribution or correlation, both in data and in theory - in this case that low rates and high rates of each were correlated, but not middle rates. Is there a test for that situation?

[imatfaal]

Could you not use a simple linear correlation - ie plot data points rather than rank and see what the line of best fit, r etc are? Using only the rankings seems perverse if the data are available. I suppose if you only have the rankings and not the raw data from which they were formed then there are rank correlation tests such as Spearman Rank Correlation

 

http://en.wikipedia.org/wiki/Correlation#Rank_correlation_coefficients

 

http://en.wikipedia.org/wiki/Spearman%27s_rank_correlation_coefficient

 

Both Pearson and Spearman are tbomk distribution insensitive and non-parametric - but equally they can provide misleading answers to relatively simple data sets.