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I ran an experiment where I experimentally (randomly) manipulated $x$ to find out the effect on $y$. Both variables tend to have power law distributions, and this was indeed the case.

I ran some regressions and found out that $x$ is positively correlated with $y$ but $log(x)$ is uncorrelated with $log(y)$. Not only is it insignificant, the point estimate is actually negative. One can make a theoretical case for using either levels or logs.

What should I do for the purposes of an academic paper?

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    $\begingroup$ What would the purposes of the academic paper be? Regardless, one thing you should do in any case is to determine whether either correlation coefficient is sensitive to a small number of data values (as the power law distribution suggests might be the case). $\endgroup$ – whuber Jun 9 '18 at 20:31
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    $\begingroup$ Have you plotted $x$ against $y$, $\log x$ against $\log y$ and so forth? $\endgroup$ – Stephan Kolassa Jun 9 '18 at 21:10
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The correlation in the linear scale is probably being driven by few data points, and the influence of these data points is reduced in the log scale. Here's an example:

rm(list = ls())
x <- 1:5
y <- 5:1
x[6] <- 6
y[6] <- 15

oldpar <- par(mfrow = c(1, 2))
plot(y~x, pch = 20, main = "Natural Scale")
abline(coef(lm(y~x)), col = "red")
plot(log(y)~log(x), pch = 20, main = "Log Scale")
abline(coef(lm(log(y)~log(x))), col = "red")
par(oldpar)

enter image description here

So one thing you should do is to investigate those "outliers" and make sure they are not measurement errors, and that they are what you think they are.

However, what you should do depends heavily on the substantive knowledge of your field and the question you want to answer. The correlations are what they are, so if you are unsatisfied with them being different from of what you expected, this means you are probably trying to answer something deeper about the data generating process, so you need to define more precisely what you want to answer.

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