I'd like to compare two distributions of measurements for the same entities, to determine the degree to which the two distributions measure the same underlying phenomenon. For example, I'd like to compare countries in the Human Development Index to the same countries in the OECD Better Life Index to determine the degree to which the two indices measure the same thing.
I know that if I were comparing samples to see whether they came from the same population, I'd use, for example, the Kolmogorov–Smirnov test. In my case, though, I have additional information: the entity (country) in each distribution (index). Which test should I use that accounts for this additional information?