Correlation does not imply causation, as there could be many explanations for the correlation. But does causation imply correlation? Intuitively, I would think that the presence of causation means there is necessarily some correlation. But my intuition has not always served me well in statistics. Does causation imply correlation?
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As many of the answers above have stated, causation does not imply linear correlation. Since a lot of the correlation concepts come from fields that rely heavily on linear statistics, usually correlation is seen as equal to linear correlation. The wikipedia article is an alright source for this, I really like this image:
Look at some of the figures in the bottom row, for instance the parabola-ish shape in the 3rd example. This is kind of what happens in @StasK answer (with a little bit of noise added). Y can be fulled caused by X but if the numeric relationship is not linear and symmetric, you will still have a correlation of 0. The word you are looking for is mutual information: this is sort of the general non-linear version of correlation. In that case, your statement would be true: causation implies high mutual information. |
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The strict answer is "no, causation does not necessarily imply correlation". Consider $X\sim N(0,1)$ and $Y=X^2\sim\chi^2_1$. Causation does not get any stronger: $X$ determines $Y$. Yet, correlation between $X$ and $Y$ is 0. Proof: The (joint) moments of these variables are: $E[X]=0$; $E[Y]=E[X^2]=1$; $${\rm Cov}[X,Y]=E[ (X-0)(Y-1) ] = E[XY]-E[X]1 = E[X^3]-E[X]=0$$ using the property of the standard normal distribution that its odd moments are all equal to zero (can be easily derived from its moment-generating-function, say). Hence, the correlation is equal to zero. To address some of the comments: the only reason this argument works is because the distribution of $X$ is centered at zero, and is symmetric around 0. In fact, any other distribution with these properties that would have sufficient number of moments would have worked in place of $N(0,1)$, e.g., uniform on $(-10,10)$ or Laplace $\sim \exp(-|x|)$. An oversimplified argument is that for every positive value of $X$, there is an equally likely negative value of $X$ of the same magnitude, so when you square the $X$, you can't say that greater values of $X$ are associated with greater or smaller values of $Y$. However, if you take say $X\sim N(3,1)$, then $E[X]=3$, $E[Y]=E[X^2]=10$, $E[X^3]=36$, and ${\rm Cov}[X,Y]=E[XY]-E[X]E[Y]=36-30=6\neq0$. This makes perfect sense: for each value of $X$ below zero, there is a far more likely value of $-X$ which is above zero, so larger values of $X$ are associated with larger values of $Y$. (The latter has a non-central $\chi^2$ distribution; you can pull the variance from the Wikipedia page and compute the correlation if you are interested.) |
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Essentially, yes. Correlation does not imply causation because there could be other explanations for a correlation beyond cause. But in order for A to be a cause of B they must be associated in some way. Meaning there is a correlation between them - though that correlation does not necessarily need to be linear. As some of the commenters have suggested, it's likely more appropriate to use a term like 'dependence' or 'association' rather than correlation. Though as I've mentioned in the comments, I've seen "correlation does not mean causation" in response to analysis far beyond simple linear correlation, and so for the purposes of the saying, I've essentially extended "correlation" to any association between A and B. |
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Adding to @EpiGrad 's answer. I think, for a lot of people, "correlation" will imply "linear correlation". And the concept of nonlinear correlation might not be intuitive. So, I would say "no they don't have to be correlated but they do have to be related". We are agreeing on the substance, but disagreeing on the best way to get the substance across. One example of such a causation (at least people think it's causal) is that between the likelihood of answering your phone and income. It is known that people at both ends of the income spectrum are less likely to answer their phones than people in the middle. It is thought that the causal pattern is different for the poor (e.g. avoid bill collectors) and rich (e.g. avoid people asking for donations). |
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The cause and the effect will be correlated unless there is no variation at all in the incidence and magnitude of the cause and no variation at all in its causal force. The only other possibility would be if the cause is perfectly correlated with another causal variable with exactly the opposite effect. Basically, these are thought-experiment conditions. In the real world, causation will imply dependence in some form (although it might not be linear correlation). |
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