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I'm comparing a set of methods. Each method generates a list of observations, then I compute the correlation between the observations of the methods. Is it correct to say that I'm checking the correlation between the methods? if so, why is it correct?

Or is it more correct to say that what I'm doing is checking the correlation of the results of the methods?!

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If you are using method to describe a transformation that makes the observations correlated then I would say yes. It is OK, but confusing to say that. I would go with checking the correlation of the results.

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