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If I demonstrated that there is no correlation between two random variables, does that mean that there is no cause and effect relation between them ?


marked as duplicate by Tim, Richard Hardy, gung, Silverfish, Christoph Hanck Apr 28 '16 at 10:23

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    $\begingroup$ See the discussion here : stats.stackexchange.com/questions/179511/… You can have a variable $X$, create a variable $Y=f(X)$, and still have that $X$ and $Y$ are not correlated. Because correlation is one measure of how variables are related, not THE measure. $\endgroup$ – Anthony Martin Apr 28 '16 at 6:58
  • $\begingroup$ It's easy to have perfect dependence without any linear correlation; that could certainly be causal. $\endgroup$ – Glen_b Apr 28 '16 at 7:32
  • $\begingroup$ correlation implies that there a cause.It could be mutual relationship - X And Y could depend on each other. To affirm a relationship (correlatio), we may use e.g. regression analysis or use alternative method such as Spearman rank correlation. The correlation coefficient is just one several options to ascertain the existence of cause and effect relationship. $\endgroup$ – Subhash C. Davar Apr 28 '16 at 8:07
  • $\begingroup$ @suchash Two things can be correlated without either causing the other. $\endgroup$ – Glen_b Apr 28 '16 at 16:04

Correlation, in the usual sense, measures the linear association between two variables. One variable can cause another without there being any correlation between the two. For example, you might have a perfectly sinusoidal relationship between a variable $x$ and $y$. When you obtain the correlation, it will be zero, but $x$ is still determining the value of $y$ through $f(x)=y=\sin(x)$. So if you show correlation is zero, that does not imply that a causal relationship does not exist.

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    $\begingroup$ Whether $\sin(x)$ is correlated with $x$ depends on what the distribution of the x-values is. Consider $x$ evenly and symmetrically spaced in the interval $(-0.1,0.1)$ (or perhaps uniformly distributed on it if we're talking about population correlation between random variables rather than sample correlation of a set of values)... $\endgroup$ – Glen_b Apr 28 '16 at 7:34
  • $\begingroup$ without correlation, how could one imagine a cause and therefore effect. $\endgroup$ – Subhash C. Davar Apr 28 '16 at 10:29
  • $\begingroup$ +1, @Glen_b. I should have been a bit more clear in my response. $\endgroup$ – StatsStudent Apr 28 '16 at 15:53
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    $\begingroup$ @subhash imagine a causal relationship (X causes Y) of the form $E(Y|X=x)=x^2/2$, with $x$ uniform on $(-1,1)$, What's the correlation? $\endgroup$ – Glen_b Apr 28 '16 at 16:07

Let $X \sim \mathcal{N}(0,1)$ follow the standard normal distribution. Let $Y = X^2$. Clearly there is an entirely deterministic relationship, but $\mathrm{Corr}(X, Y) = 0$.


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