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I've got two sets of measurements whose uncertainty has been quantified: $x_1..x_n$ and $y_1..y_n$. For each $x_i$ and $y_i$, I have not only a point estimate but also an interval that expresses my uncertainty in the accuracy of the measurement.

Now, I'd like to compute a correlation between $x$ and $y$. Is there a reasonable way of taking the intervals into account?

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I'm not sure whether this precisely matches your scenario, but I've played with correlation amidst estimable measurement error in the past and wrote up a description here. As I note there, the take-home message of my work is that if all you're interested in is whether the correlation is different from zero, it's ok to simply ignore the measurement error. However, if you are interested in getting an unbiased point estimate and confidence interval for a non-zero correlation, then you have to use the methods I describe.

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    $\begingroup$ Thanks, that sounds interesting. I'm "accepting" this answer, but I'd love to see additional thoughts on this topic in the future. $\endgroup$ – Jack Tanner Feb 13 '12 at 20:39

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