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 ?
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5$\begingroup$ 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 MartinCommented Apr 28, 2016 at 6:58
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$\begingroup$ It's easy to have perfect dependence without any linear correlation; that could certainly be causal. $\endgroup$– Glen_bCommented Apr 28, 2016 at 7:32
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$\begingroup$ 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. $\endgroup$– user10619Commented Apr 28, 2016 at 8:07
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$\begingroup$ @suchash Two things can be correlated without either causing the other. $\endgroup$– Glen_bCommented Apr 28, 2016 at 16:04
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2 Answers
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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.
answered Apr 28, 2016 at 7:12
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3$\begingroup$ 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_bCommented Apr 28, 2016 at 7:34
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$\begingroup$ without correlation, how could one imagine a cause and therefore effect. $\endgroup$– user10619Commented Apr 28, 2016 at 10:29
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$\begingroup$ +1, @Glen_b. I should have been a bit more clear in my response. $\endgroup$ Commented Apr 28, 2016 at 15:53
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2$\begingroup$ @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_bCommented Apr 28, 2016 at 16:07
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Let $X \sim \mathcal{N}(0,1)$ follow the standard normal distribution. Let $Y = X^2$. Clearly there is an entirely deterministic relationship, but $\mathrm{Corr}(X, Y) = 0$.
answered Apr 28, 2016 at 9:22