Why Correlation Usually ≠ Causation

I have noticed I seem to be unusually willing to bite the correlation ≠ causation bullet, and I think it’s due to an idea I had some time ago about the nature of reality. It ties into the cumulative evidence from meta-analysis and replication initiatives and the history of randomized experiments about the Replication crisis, correlations’ ability to predict randomized experiments in practice, behavioral genetics results from the genome sequencing era, and countless failed social engineering attempts.

Most scientifically-inclined people are reasonably aware that one of the major divides in research is that correlation ≠ causation: that having discovered some relationship between various data X and Y (not necessarily Pearson’s r, but any sort of mathematical or statistical relationship, whether it be a humble r or an opaque deep neural network’s predictions), we do not know how Y would change if we manipulated X. Y might increase, decrease, do something complicated, or remain implacably the same.

This might be because the correlation is not a real one, and is spurious, in the sense that it would disappear if we gathered more data, and was an illusory correlation due to biases; or it could be an artifact of our mathematical procedures as in “spurious correlations”; or it is a Type I error, a correlation thrown up by the standard statistical problems we all know about, such as too-small n, false positives from sampling error (A & B just happened to sync together due to randomness), data-mining/multiple testing, p-hacking, data snooping, selection bias, publication bias, misconduct, inappropriate statistical tests, etc. The Replication Crisis has seriously shaken faith in the published research literature in many fields, and it’s clear that many correlations are over-estimated in strength by severalfold, or the sign is in fact the opposite direction.

I’ve read about those problems at length, and despite knowing about all that, there still seems to be a problem: I don’t think those issues explain away all the correlations which turn out to be confounds—correlation too often ≠ causation.

But let’s say we get past that and we have established beyond a reasonable doubt that some X and Y really do correlate. We still have not solved the problem.

This point can be made by listing examples of correlations where we intuitively know changing X should have no effect on Y, and it’s a spurious relationship: the number of churches in a town may correlate with the number of bars, but we know that’s because both are related to how many people are in it; the number of pirates may inversely correlate with global temperatures (but we know pirates don’t control global warming and it’s more likely something like economic development leads to suppression of piracy but also CO₂ emissions); sales of ice cream may correlate with snake bites or violent crime or death from heat-strokes (but of course snakes don’t care about sabotaging ice cream sales); thin people may have better posture than fat people, but sitting upright does not seem like a plausible weight loss plan; wearing XXXL clothing clearly doesn’t cause heart attacks, although one might wonder if diet soda causes obesity; black skin does not cause sickle cell anemia nor, to borrow an example from Pearson, would black skin cause smallpox or malaria; more recently, part of the psychology behind linking vaccines with autism is that many vaccines are administered to children at the same time autism would start becoming apparent… In these cases, we can see what the correlation, which is surely true (in the sense that we can go out and observe it any time we like), doesn’t work like we might think: there is some third variable which causes both X and Y, or it turns out we’ve gotten it backwards.

In fact, if we go out and look at large datasets, we will find that two variables being correlated is nothing special—because “everything is correlated”. As Paul Meehl noted, the correlations can seem completely arbitrary, yet are firmly established by extremely large n (eg. n = 57,000 & n = 50,000 in his 2 examples).

We can also establish this by simply looking at the results of randomized experiments which take a correlation and nail down what a manipulation does.

To measure this directly you need a clear set of correlations which are proposed to be causal, randomized experiments to establish what the true causal relationship is in each case, and both categories need to be sharply delineated in advance to avoid issues of cherrypicking and retroactively confirming a correlation. Then you’d be able to say something like ‘11 out of the 100 proposed A → B causal relationships panned out’, and start with a prior of 11% that in your case, A → B.

This sort of dataset is pretty rare, although the few examples I’ve found tend to indicate that our prior should be low. (For example, Fraker & Maynard1987 analyze a government jobs program and got data on randomized participants & others, permitting comparison of randomized inference to standard regression approaches; they find roughly that 0⁄12 estimates—many statistically-significant—were reasonably similar to the causal effect for one job program & 4⁄12 for another job program, with the regression estimates for the former heavily biased.) Not great. Why are our best analyses & guesses at causal relationships so bad?

We’d expect that the a priori odds are good, by the principle of indifference: 1⁄3! After all, you can divvy up the possibilities as:

1. A causes B (A → B)

2. B causes A (A ← B)

3. both A and B are caused by C (A ← C → B)

   (Possibly in a complex way like Berkson’s paradox or conditioning on unmentioned variables, like a phone-based survey inadvertently generating conclusions valid only for the phone-using part of the population, causing amusing pseudo-correlations.)

If it’s either #1 or #2, we’re good and we’ve found a causal relationship; it’s only outcome #3 which leaves us baffled & frustrated. If we were guessing at random, you’d expect us to still be right at least 33% of the time. (As in the joke about the lottery—you’ll either win or lose, and you don’t know which, so it’s 50-50, and you like dem odds!) And we can draw on all sorts of knowledge to do better.

I think a lot of people tend to put a lot of weight on any observed correlation because of this intuition that a causal relationship is normal & probable because, well, “how else could this correlation happen if there’s no causal connection between A & B‽” And fair enough—there’s no grand cosmic conspiracy arranging matters to fool us by always putting in place a C factor to cause scenario #3, right? If you question people, of course they know correlation doesn’t necessarily mean causation—everyone knows that—since there’s always a chance of a lurking confound, and it would be great if you had a randomized experiment to draw on; but you think with the data you have, not the data you wish you had, and can’t let the perfect be the enemy of the better. So when someone finds a correlation between A and B, it’s no surprise that suddenly their language & attitude change and they seem to place great confidence in their favored causal relationship even if they piously acknowledge “Yes, correlation is not causation, but… obviously hanging out with fat people will make you fat / parents influence their kids a lot eg. smoking encourages their kids to smoke / when we gave babies a new drug, fewer went blind / female-named hurricanes increase death tolls due to sexistly underestimating women / Kaposi’s sarcoma correlates so highly with AIDS that it must be another consequence of HIV (actually caused by HHV-8 which is transmitted simultaneously with HIV) / vitamin and anti-oxidant use (among many other lifestyle choices) will save lives / LSD & marijuana use associates with and thus surely causes schizophrenia and other forms of insanity (despite increases in marijuana use not followed by any schizophrenia increases) / hormone replacement therapy correlates with mortality reduction in women so it definitely helps and doesn’t hurt” etc.

Besides the intuitiveness of correlation=causation, we are also desperate and want to believe: correlative data is so rich and so plentiful, and experimental data so rare. If it is not usually the case that correlation=causation, then what exactly are we going to do for decisions and beliefs, and what exactly have we spent all our time to obtain? When I look at some dataset with a number of variables and I run a multiple regression and can report that variables A, B, and C are all statistically-significant and of large effect-size when regressed on D, all I have really done is learned something along the lines of “in a hypothetical dataset generated in the exact same way, if I somehow was lacking data on D, I could make a better prediction in a narrow mathematical sense of no importance (squared error) based on A/B/C”. I have not learned whether A/B/C cause D, or whether I could predict values of D in the future, or anything about how I could intervene and manipulate any of A-D, or anything like that—rather, I have learned a small point about prediction. To take a real example: when I learn that moderate alcohol consumption means the actuarial prediction of lifespan for drinkers should be increased slightly, why on earth would I care about this at all unless it was causal? When epidemiologists emerge from a huge survey reporting triumphantly that steaks but not egg consumption slightly predicts decreased lifespan, why would anyone care aside from perhaps life insurance companies? Have you ever been abducted by space aliens and ordered as part of an inscrutable alien blood-sport to take a set of data about Midwest Americans born 1960^–₉1969_57ya with dietary predictors you must combine linearly to create predictors of heart attacks under a squared error loss function to outpredict your fellow abductees from across the galaxy? Probably not. Why would anyone give them grant money for this, why would they spend their time on this, why would they read each others’ papers unless they had a “quasi-religious faith” that these correlations were more than just some coefficients in a predictive model—that they were causal? To quote Rutter2007, most discussions of correlations fall into two equally problematic camps:

…all behavioral scientists are taught that statistically-significant correlations do not necessarily mean any kind of causative effect. Nevertheless, the literature is full of studies with findings that are exclusively based on correlational evidence. Researchers tend to fall into one of two camps with respect to how they react to the problem.

1. First, there are those who are careful to use language that avoids any direct claim for causation, and yet, in the discussion section of their papers, they imply that the findings do indeed mean causation.

2. Second, there are those that completely accept the inability to make a causal inference on the basis of simple correlation or association and, instead, take refuge in the claim that they are studying only associations and not causation. This second, “pure” approach sounds safer, but it is disingenuous because it is difficult to see why anyone would be interested in statistical associations or correlations if the findings were not in some way relevant to an understanding of causative mechanisms.

So, correlations tend to not be causation because it’s almost always #3, a shared cause. This commonness is contrary to our expectations, based on a simple & unobjectionable observation that of the 3 possible relationships, 2 are causal; and so we often reason as though correlation were strong evidence for causation. This leaves us with a paradox: experimental results seem to contradict intuition. To resolve the paradox, I need to offer a clear account of why shared causes/confounds are so common, and hopefully motivate a different set of intuitions.

Here’s where Bayes nets & causal networks (seen previously on LW & Michael Nielsen) come up. Even simulating the simplest possible model of linear regression, adding covariates barely increase the probability of correctly inferring direction of causality, and the effect sizes remain badly imprecise (Walker2014). And when networks are inferred on real-world data, they look gnarly: tons of nodes, tons of arrows pointing all over the place. Daphne Koller early on in her Probabilistic Graphical Models course shows an example from a medical setting where the network has like 600 nodes and you can’t understand it at all. When you look at a biological causal network like metabolism:

You start to appreciate how everything might be correlated with everything, but (usually) not cause each other.

This is not too surprising if you step back and think about it: life is complicated, we have limited resources, and everything has a lot of moving parts. (How many discrete parts does an airplane have? Or your car? Or a single cell? Or think about a chess player analyzing a position: ‘if my bishop goes there, then the other pawn can go here, which opens up a move there or here, but of course, they could also do that or try an en passant in which case I’ll be down in material but up on initiative in the center, which causes an overall shift in tempo…’) Fortunately, these networks are still simple compared to what they could be, since most nodes aren’t directly connected to each other, which tamps down on the combinatorial explosion of possible networks. (How many different causal networks are possible if you have 600 nodes to play with? The exact answer is complicated but it’s much larger than 2⁶⁰⁰—so very large!)

One interesting thing I managed to learn from PGM (before concluding it was too hard for me and I should try it later) was that in a Bayes net even if two nodes were not in a simple direct correlation relationship A → B, you could still learn a lot about A from setting B to a value, even if the two nodes were ‘way across the network’ from each other. You could trace the influence flowing up and down the pathways to some surprisingly distant places if there weren’t any blockers.

The bigger the network, the more possible combinations of nodes to look for a pairwise correlation between them (eg. If there are 10 nodes/variables and you are looking at bivariate correlations, then you have 10 choose 2 = 45 possible comparisons, and with 20, 190, and 40, 780. 40 variables is not that much for many real-world problems.) A lot of these combos will yield some sort of correlation. But does the number of causal relationships go up as fast? I don’t think so (although I can’t prove it).

If not, then as causal networks get bigger, the number of genuine correlations will explode but the number of genuine causal relationships will increase slower, and so the fraction of correlations which are also causal will collapse.

(Or more concretely: suppose you generated a randomly connected causal network with x nodes and y arrows perhaps using the algorithm in Kuipers & Moffa2012, where each arrow has some random noise in it; count how many pairs of nodes are in a causal relationship; now, n times initialize the root nodes to random values and generate a possible state of the network & storing the values for each node; count how many pairwise correlations there are between all the nodes using the n samples (using an appropriate significance test & alpha if one wants); divide # of causal relationships by # of correlations, store; return to the beginning and resume with x+1 nodes and y+1 arrows… As one graphs each value of x against its respective estimated fraction, does the fraction head toward 0 as x increases? My thesis is it does. Or, since there must be at least as many causal relationships in a graph as there are arrows, you could simply use that as an upper bound on the fraction.)

It turns out, we weren’t supposed to be reasoning ‘there are 3 categories of possible relationships, so we start with 33%’, but rather: ‘there is only one explanation “A causes B”, only one explanation “B causes A”, but there are many explanations of the form “C₁ causes A and B”, “C₂ causes A and B”, “C₃ causes A and B”…’, and the more nodes in a field’s true causal networks (psychology or biology vs physics, say), the bigger this last category will be.

The real world is the largest of causal networks, so it is unsurprising that most correlations are not causal, even after we clamp down our data collection to narrow domains. Hence, our prior for “A causes B” is not 50% (it’s either true or false) nor is it 33% (either A causes B, B causes A, or mutual cause C) but something much smaller: the number of causal relationships divided by the number of pairwise correlations for a graph, which ratio can be roughly estimated on a field-by-field basis by looking at existing work or directly for a particular problem (perhaps one could derive the fraction based on the properties of the smallest inferrable graph that fits large datasets in that field). And since the larger a correlation relative to the usual correlations for a field, the more likely the two nodes are to be close in the causal network and hence more likely to be joined causally, one could even give causality estimates based on the size of a correlation (eg. an r = 0.9 leaves less room for confounding than an r of 0.1, but how much will depend on the causal network).

This is exactly what we see. How do you treat cancer? Thousands of treatments get tried before one works. How do you deal with poverty? Most programs are not even wrong. Or how do you fix societal woes in general? Most attempts fail miserably and the higher-quality your studies, the worse attempts look (leading to Rossi’s Metallic Rules). This even explains why ‘everything correlates with everything’ and Andrew Gelman’s dictum about how coefficients are never zero: the reason large datasets find most of their variables to have non-zero correlations (often reaching statistical-significance) is because the data is being drawn from large complicated causal networks in which almost everything really is correlated with everything else.

And thus I was enlightened.

To go on a tangent: why is it so important that we hammer this in?

The unreliability is bad enough, but I’m also worried that the knowledge correlation ≠ causation, one of the core ideas of the scientific method and fundamental to fields like modern medicine, is going underappreciated and is being abandoned by meta-contrarians due to its inconvenience. Pointing it out is “nothing helpful” or “meaningless”, and justified skepticism is actually just “a dumb-ass thing to say”, a “statistical cliché that closes threads and ends debates, the freshman platitude turned final shutdown” often used by “party poopers” “Internet blowhards” to serve an “agenda” & is sometimes “a dog whistle”; in practice, such people seem to go well beyond the XKCD comic and proceed to take any correlations they like as strong evidence for causation, and any disagreement is an unsophisticated middlebrow dismissal & denialism.

So it’s unsurprising that one so often runs into researchers for whom indeed correlation=causation (we certainly wouldn’t want to be freshmen or Internet blowhards, would we?). It is common to use causal language and make recommendations (Prasad et al 2013), but even if they don’t, you can be sure to see them confidently talking causally to other researchers or journalists or officials. (I’ve noticed this sort of constant motte-and-bailey slide from vague mentions of how results are correlative tucked away at the end of the paper to freely dispensing advice for policymakers about how their research proves X should be implemented is particularly common in medicine, sociology, and education.)

Bandying phrases with meta-contrarians won’t help much here; I agree with them that correlation ought to be some evidence for causation. eg if I suspect that A → B, and I collect data and establish beyond doubt that A&B correlates r = 0.7, surely this observation, which is consistent with my theory, should boost my confidence in my theory, just as an observation like r = 0.0001 would trouble me greatly. But how much…?

As it is, it seems we fall readily into intellectual traps of our own making. When you believe every correlation adds support to your causal theory, you just get more and more wrong as you collect more data.

Now assuming the foregoing to be right (which I’m not sure about; in particular, I’m dubious that correlations in causal nets really do increase much faster than causal relations do), what’s the psychology of this? I see a few major ways that people might be incorrectly reasoning when they overestimate the evidence given by a correlation:

• they might be aware of the imbalance between correlations and causation, but underestimate how much more common correlation becomes compared to causation.

  This could be shown by giving causal diagrams and seeing how elicited probability changes with the size of the diagrams: if the probability is constant, then the subjects would seem to be considering the relationship in isolation and ignoring the context.

  It might be remediable by showing a network and jarring people out of a simplistic comparison approach.

• they might not be reasoning in a causal-net framework at all, but starting from the naive 33% base-rate you get when you treat all 3 kinds of causal relationships equally.

  This could be shown by eliciting estimates and seeing whether the estimates tend to look like base rates of 33% and modifications thereof.

  Sterner measures might be needed: could we draw causal nets with not just arrows showing influence but also another kind of arrow showing correlations? For example, the arrows could be drawn in black, inverse correlations drawn in red, and regular correlations drawn in green. The picture would be rather messy, but simply by comparing how few black arrows there are to how many green and red ones, it might visually make the case that correlation is much more common than causation.

• alternately, they may really be reasoning causally and suffer from a truly deep & persistent cognitive illusion that when people say ‘correlation’ it’s really a kind of causation and don’t understand the technical meaning of ‘correlation’ in the first place (which is not as unlikely as it may sound, given examples like David Hestenes’s demonstration of the persistence of Aristotelian folk-physics in physics students as all they had learned was guessing passwords; on the test used, see eg. Halloun & Hestenes1985 & Hestenes et al 1992); in which cause it’s not surprising that if they think they’ve been told a relationship is ‘causation’, then they’ll think the relationship is causation. Ilya remarks:

  Pearl has this hypothesis that a lot of probabilistic fallacies/paradoxes/biases are due to the fact that causal and not probabilistic relationships are what our brain natively thinks about. So eg. Simpson’s paradox is surprising because we intuitively think of a conditional distribution (where conditioning can change anything!) as a kind of “interventional distribution” (no Simpson’s type reversal under interventions: “Understanding Simpson’s Paradox”, Pearl2014 [see also Pearl’s comments on Nielsen’s blog)).

  This hypothesis would claim that people who haven’t looked into the math just interpret statements about conditional probabilities as about “interventional probabilities” (or whatever their intuitive analogue of a causal thing is).

  This might be testable by trying to identify simple examples where the two approaches diverge, similar to Hestenes’s quiz for diagnosing belief in folk-physics.