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Let's say as a business owner (or marketing or anyone who understands a scatter plot) is shown a scatter plot of two variables: number of advertisements vs number of product sales per month for the past 5 years (or another time-scale so that you have more samples. I just made this one up).

Now he/she see's the scatter plot and is told that the correlation coefficient (corr) is:

  1. 1 or
  2. 0.5 or
  3. 0.11 or
  4. 0 or
  5. -0.75 or
  6. -1

Basically any valid value for corr

Question: What does this even mean to a decision maker or any consumer of the scatter plot? What decisions can one take just based on this?

I.e.: What is the use of seeing correlation between any two variables and what can one do with that information in isolation? Is it only to see what to and not to consider for inclusion in regression analysis or is there a more practical use?

Just curious, I've always worked with this technique, but I've been told that correlation by itself is not of much use - so what "IS" the use?

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A few thoughts:

  • The old canard about correlation not being causation is only half the story. Correlation may not be causation, but some form of association between the two variables is a necessary step along the path to showing causation, and correlation can help show that.
  • It helps point out trends. Show it to a business owner, and they may say "Yeah, that makes sense, you see Widget X and Widget Y both end up being used by a particular group of people, even though they're not really related. Or they might say "that's...odd", at which point you prompted further investigation.
  • Look at it this way. Correlation is a tool. A hammer, by itself, isn't all that useful. It certainly won't build a house all by itself. But have you ever tried to build a house without a hammer?
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    $\begingroup$ In your first bullet you say that correlation is a necessary condition for causation - that's not true. If there's a non-monotonic relationship between two variables then they can be uncorrelated - this does not preclude causation. $\endgroup$ – Macro Nov 7 '11 at 2:10
  • $\begingroup$ @Macro - true, and edited $\endgroup$ – Fomite Nov 7 '11 at 2:15
  • $\begingroup$ @Macro True, but in practice you can apply a function to your variable to make the relation to be tested monotonic. if you don't know this function, then... you don't know much about what you are looking for $\endgroup$ – RockScience Nov 7 '11 at 9:27
  • $\begingroup$ @EpiGrad: Assume the X-Y correlation graph of two variables look like a happy smiley (or any other shape per se). The correlation coefficient would actually be quite small, but there certainly would be some inter-relationship, right? How/what should one do in such a case? $\endgroup$ – PhD Nov 7 '11 at 19:24
  • $\begingroup$ @Nupul A somewhat more complicated exploration of X-Y beyond linearity. $\endgroup$ – Fomite Nov 7 '11 at 21:13
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Look at it from a gambling perspective. Let's say we know that on average people who wear workboots to work will have 1.5 injuries on the job, and people wear loafers will have .05 injuries on average. Or, maybe the chances of an injury for a person wearing workboots is .85, and the chances of injury to a person wearing loafers is .1.

If I randomly select a person from the population, and tell you the person is wearing workboots, and offer you an even money bet on whether or not they had a workplace injury last year, would you take the bet? Well, you'd take the bet if you were to be able to bet on the side that they had an injury.. 85% of the time you'll win, and you're getting even money.

The point is, knowing that piece of information gives us information about whether or not they are likely to experience an injury at work.. The shoes have nothing to do with it, in fact, the workboots prevent injury.. But the confounding variable here is the type of job that goes along with the workboots.. And maybe other things like the person possibly being more reckless.

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The phrase "correlation doesn't imply causation" gets overplayed. (As Cohen wrote, "it's an awfully big hint".) We beat this phrase into students because of a bias intrinsic to the human mind. When you hear 'the crime rate is correlated with the poverty rate', or something like that, you cannot help but think this means that poverty causes the crime. It is natural for people to assume this, because that's the way the mind works. We use the phrase over and over in the hopes of counteracting that. However, once you've absorbed the idea, the phrase loses most of it's value, and it's time to move on to a more sophisticated understanding.

When there is a correlation between two variables, there are two possibilities: it's all a coincidence, or there's some causal pattern at work. Calling a pattern in the world a coincidence is a terrible explanatory framework and should probably be your last resort. That leaves causality. The problem is that we don't know the nature of that causal pattern. It could well be that poverty causes crime, but it could also be that crime causes poverty (e.g., people don't want to live in a high-crime area, so they move out and property values fall, etc.). It could also be that there is some third variable or group of variables that cause both crime and poverty, but that there is, in fact, no direct causal link between crime and poverty (known as the 'common cause' model). This is especially pernicious, because, in a statistical model, all other sources of variation are collapsed into the dependent variable's error term. As a result, the independent variable is correlated with (caused by) the error term, leading to the problem of endogeneity. These problems are very difficult, and should not be taken lightly. Nonetheless, even in this scenario, it is important to recognize that there is real causality at work.

In short, when you see a correlation, you should think that there probably is some sort of causality at play somewhere, but that you don't know the nature of that causal pattern.

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I thought I was knowledgeable about these things, but it was only last month that I looked up "imply" in the dictionary and found it had two strikingly different meanings. 1. Suggest and 2. Necessitate. (!) Correlation seldom necessitates causation, but it certainly can suggest it. As @EpiGrad points out, it is a necessary though not sufficient condition for establishing causation.

As time goes on one hopefully finds a middle ground between seeing correlation as the end-all and as completely useless. And one takes into account subject-/domain-/content-specific knowledge in interpreting correlational results. Few people would question there being at least some causal link when seeing the advertising-sales results you describe. But it's always good to stay open to other possibilities, other variables that could at least partly explain the relationship observed. Readings about confounding variables, validity, and the like pay off with big dividends. For example, Cook and Campbell's old classic Quasi-Experimentation has a good section on validity and threats to validity.

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  • $\begingroup$ As I pointed out to @EpiGrad, correlation is not a necessary condition for causation. There is a widespread conception in data analysis that a relationship between variables always refers to a monotonic relationship, which is tacitly assumed by suggesting that correlation is a necessary condition for causation. $\endgroup$ – Macro Nov 7 '11 at 2:27
  • $\begingroup$ Fair enough. Let's say "statistical association" is necessary, then. $\endgroup$ – rolando2 Nov 7 '11 at 22:44
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A correlation coefficient, as other measures of association, is useful if you want to know how much knowing the value of X is informative about the value of Y. This is different from knowing whether if you were to set X to a particular value, what value of Y you would get (which is the essence of a counterfactual interpretation of causation).

Nevertheless, in many contexts (e.g. prediction) inferences based on correlation would be valuable in their own right. Yellow teeth are correlated with lung cancer (as they are both probabilistically caused by cancer). There is no causation between the two: whitening teeth would not cure lung cancer. But if you need a quick screening test for who is likely to have lung cancer, checking for yellow teeth might be a good first step.

It is a different question whether the correlation coefficient is the best available measure of association, but I think the question is more about what is the value of knowing non-causal association.

Btw, not only is correlation not sufficient demonstration of causation, but it is not necessary neither. Two variables can be causally related yet exhibit no correlation in any particular dataset (e.g. due to selection bias or confounders).

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correlation by itself is not of much use - so what "IS" the use?

Let me disagree with this phrase, correlation let to know the level of association between 2 variables. Then, it is useful when trying to explain relation between such variables. On the other hand, (as Macro wrote) correlation is not a necessary condition for causation, however, is enough to explain the level of association. Furthermore, you can test the independence of the variables, but correlation can give you another useful information, the coefficient of determination.

Nevertheless, analyst must know the domain to be able to explain the kind of relation.

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  • $\begingroup$ I'm not sure what you mean by this: Furthermore, you can test the independence of the variables, but correlation can give you another useful information, the coefficient of determination $\endgroup$ – PhD Nov 7 '11 at 19:07
  • $\begingroup$ What I mean was: "you can test the independence of the variables" but anyway, even when not testing the independence, the information of correlation and the coef. of determination are "useful" to understand and to explain the kind of relation between the variables. $\endgroup$ – Jose Zubcoff Nov 9 '11 at 10:45
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I think data collection and study design may also play a role in answer this question. You won't design a study and collect a set of data completely irrelevant to each other, even in the observational studies.Therefore "the correlation does not imply causation" may be justified. Even though it is not causal relationship, there might be an association related.

However if you are talking about two dataset completely irrelevant, but you still wanna use correlation to explain the association and causation, then it might be inappropriate. For example, if two dataset all have downward trends, say ice cream sales and number of marriages, the correlation coefficient might be very high. But is it necessary to mean an association?

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