Correlations are oft interpreted as evidence for causation; this is oft falsified; do causal graphs explain why this is so common?
created: 24 Jun 2014; modified: 18 Apr 2019; status: in progress; confidence: log; importance: 10
It is widely understood that statistical correlation between two variables ≠ causation. But despite this admonition, people are routinely overconfident in claiming correlations to support particular causal interpretations and are surprised by the results of randomized experiments, suggesting that they are biased & systematically underestimating the prevalence of confounds/common-causation. I speculate that in realistic causal networks or DAGs, the number of possible correlations grows faster than the number of possible causal relationships. So confounds really are that common, and since people do not think in DAGs, the imbalance also explains overconfidence.
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.
“Hubris is the greatest danger that accompanies formal data analysis…Let me lay down a few basics, none of which is easy for all to accept… 1. The data may not contain the answer. The combination of some data and an aching desire for an answer does not ensure that a reasonable answer can be extracted from a given body of data.”
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 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 plan1; wearing XXXL clothing clearly doesn’t cause heart attacks, although one might wonder if diet soda causes obesity; the more firemen are around, the worse fires are; judging by grades of tutored vs non-tutored students, tutors would seem to be harmful rather than helpful; black skin does not cause sickle cell anemia nor, to borrow an example from Pearson2, 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 (or should we blame organic food sales?); height & vocabulary or foot size & math skills may correlate strongly (in children); national chocolate consumption correlates with Nobel prizes3, as do borrowing from commercial banks & buying luxury cars & serial killers/mass-murderers/traffic fatalities4; moderate alcohol consumption predicts increased lifespan and earnings; the role of storks in delivering babies may have been underestimated; children and people with high self-esteem have higher grades & lower crime rates etc, so “we all know in our gut that it’s true” that raising people’s self-esteem “empowers us to live responsibly and that inoculates us against the lures of crime, violence, substance abuse, teen pregnancy, child abuse, chronic welfare dependency and educational failure” - unless perhaps high self-esteem is caused by high grades & success, boosting self-esteem has no experimental benefits, and may backfire?
Now, the correlation could be bogus 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. Those last can be generated ad nauseam: Shaun Gallagher’s Correlated (also a book) surveys users & compares against all previous surveys with 1k+ correlations. Tyler Vigen’s “spurious correlations” catalogues 35k+ correlations, many with r>0.9, based primarily on US Census & CDC data. Google Correlate “finds Google search query patterns which correspond with real-world trends” based on geography or user-provided data, which offers endless fun (“Facebook”/“tapeworm in humans”, r=0.8721; “Superfreakonomic”/“Windows 7 advisor”, r=0.9751; Irish electricity prices/“Stanford webmail”, r=0.83; “heart attack”/“pink lace dress”, r=0.88; US states’ parasite loads/“booty models”, r=0.92; US states’ family ties/“how to swim”; metronidazole/“Is Lil’ Wayne gay?”, r=0.89; Clojure/“prnhub”, r=0.9784; “accident”/“itchy bumps”, r=0.87; “migraine headaches”/“sciences”, r=0.77; “Irritable Bowel Syndrome”/“font download”, r=0.94; interest-rate-index/“pill identification”, r=0.98; “advertising”/“medical research”, r=0.99; Barack Obama 2012 vote-share/“Top Chef”, r=0.88; “losing weight”/“houses for rent”, r=0.97; “Bieber”/tonsillitis, r=0.95; “paternity test”/“food for dogs”, r=0.83; “breast enlargement”/“reverse telephone search”, r=0.95; “theory of evolution” / “the Sumerians” or “Hector of Troy” or “Jim Crow laws”; “gwern”/“Danny Brown lyrics”, r=0.92; “weed”/“new Family Guy episodes”, r=0.8; a drawing of a bell curve matches “MySpace” while a penis matches “STD symptoms in men” r=0.95, not to mention Kurt Vonnegut stories). (And on less secular themes, do churches cause obesity & do Welsh rugby victories predict papal deaths?) Financial data-mining offers some fun examples; there’s the Super Bowl/stock-market one which worked well for several decades; and it’s not very elegant, but a 3-variable model (Bangladeshi butter, American cheese, joint sheep population) reaches R2=0.99 on 20 years of the S&P 500
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.
One of the constant problems I face in my reading is that I constantly want to know about causal relationships but I only have correlational data, and as we all know, that is an unreliable guide at best.
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 as being “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 reveals one’s unsophisticated middlebrow thinking or denialism. So it’s unsurprising that one so often runs into researchers for whom indeed correlation=causation; 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 slide 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 observations, 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…?
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 under 10%. (For example, Fraker & Maynard 1987 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 are so bad?
We’d expect that the a priori odds are good, by the principle of indifference: ! After all, you can divvy up the possibilities as:
A causes B
B causes A
both A and B are caused by a C
(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. And we can draw on all sorts of knowledge to do better5
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 1960-1969 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”6 that these correlations were more than just some coefficients in a predictive model - that they were causal? To quote Rutter 2007, 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. 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. 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.
“…we think so much reversal is based on ‘We think something should work, and so we’re going to adopt it before we know that it actually does work,’ and one of the reasons for this is because that’s how medical education is structured. We learn the biochemistry, the physiology, the pathophysiology as the very first things in medical school. And over the first two years we kind of get convinced that everything works mechanistically the way we think it does.”
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 (Walker 2014). 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 & Moffa 2012, 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.
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 & Hestenes 1985 & 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 e.g. 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”, Pearl 2014 [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.
Statistical folklore asserts that“everything is correlated”: in any real-world dataset, most or all measured variables will have non-zero correlations, even between variables which appear to be completely independent of each other, and that these correlations are not merely sampling error flukes but will appear in large-scale datasets to arbitrarily designated levels of statistical-significance or posterior probability.
This raises serious questions for null-hypothesis statistical-significance testing, as it implies the null hypothesis of 0 will always be rejected with sufficient data, meaning that a failure to reject only implies insufficient data, and provides no actual test or confirmation of a theory. Even a directional prediction is minimally confirmative since there is a 50% chance of picking the right direction at random.
It also has implications for conceptualizations of theories & causal models, interpretations of structural models, and other statistical principles such as the“sparsity principle”.
“Everything Is Correlated”.
Although this may have been suggested:
I used to read a magazine called Milo that covered a bunch of different strength sports. I ended my subscription after reading an article in which an entirely serious author wrote about how he noticed that shortly after he started hearing birds singing in the morning, plants started to grow. His conclusion was that birdsong made plants grow. If I remember correctly, he then concluded that it was the vibrations in the birdsong that made the plants grow, therefore vibrations were good for strength, therefore you could make your muscles grow through being exposed to certain types of vibrations, i.e. birdsong. It was my favorite article of all time, just for the way the guy started out so absurdly wrong and just kept digging.
I used to read old weight training books. In one of them the author proudly recalled how his secretary had asked him for advice on how to lose weight. This guy went around studying all the secretaries and noticed that the thin ones sat more upright compared to the fat ones. He then recommended to his secretary that she sit more upright, and if she did this she would lose weight. What I loved most about that whole story was that the guy was so proud of his analysis and conclusion that he made it an entire chapter of his book, and that no one in the entire publishing chain from the writer to the editor to the proofreader to the librarian who put the book on the shelves noticed any problems with any of it.
Slate provides a nice example from Pearson’s The Grammar of Science (pg407):
All causation as we have defined it is correlation, but the converse is not necessarily true, i.e. where we find correlation we cannot always predict causation. In a mixed African population of Kaffirs and Europeans, the former may be more subject to smallpox, yet it would be useless to assert darkness of skin (and not absence of vaccination) as a cause.
I should mention this one is not quite as silly as it sounds as there is experimental evidence for cocoa improving cognitive function↩︎
The same authors offer up a number of country-level correlation such as “Linguistic Diversity/Traffic accidents”, alcohol consumption/morphological complexity, and acacia trees vs tonality, which feed into their paper “Constructing knowledge: nomothetic approaches to language evolution” on the dangers of naive approaches to cross-country comparisons due to the high intercorrelation of cultural traits. More sophisticated approaches might be better; they derive a fairly-plausible looking graph of the relationships between variables.↩︎
like temporal order or biological plausibility - for example, in medicine you can generally rule out some of the relationships this way: if you find a correlation between taking supertetrohydracyline™ and pancreas cancer remission, it seems unlikely that #2 curing pancreas cancer causes a desire to take supertetrohydracyline™ so the causal relationship is probably either #1 supertetrohydracyline™ cures cancer or #3 a common cause like ‘doctors prescribe supertetrohydracyline™ to patients who are getting better’.↩︎
I borrow this phrase from the paper “Looking to the 21st century: have we learned from our mistakes, or are we doomed to compound them?”, Shapiro 2004:
In 1968, when I attended a course in epidemiology 101, Dick Monson was fond of pointing out that when it comes to relative risk estimates, epidemiologists are not intellectually superior to apes. Like them, we can count only three numbers: 1, 2 and BIG (I am indebted to Allen Mitchell for Figure 7). In adequately designed studies we can be reasonably confident about BIG relative risks, sometimes; we can be only guardedly confident about relative risk estimates of the order of 2.0, occasionally; we can hardly ever be confident about estimates of less than 2.0, and when estimates are much below 2.0, we are quite simply out of business. Epidemiologists have only primitive tools, which for small relative risks are too crude to enable us to distinguish between bias, confounding and causation.
…To illustrate that point, I have to allude to a problem that is usually avoided because to mention it in public is considered impolite: I refer to bias (unconscious, to be sure, but bias all the same) on the part of the investigator. And in order not to obscure the issue by considering studies of questionable quality, I have chosen the example of putatively causal (or preventive) associations published by the Nurses Health Study (NHS). For that study, the investigators have repeatedly claimed that their methods are almost perfect. Over the years, the NHS investigators have published a torrent of papers and Figure 8 gives an entirely fictitious but nonetheless valid distribution of the relative risk estimates derived from them (for relative risk estimates of less than unity, assume the inverse values). The overwhelming majority of the estimates have been less than 2 and mostly less than 1.5, and the great majority have been interpreted as causal (or preventive). Well, perhaps they are and perhaps they are not: we cannot tell. But, perhaps as a matter of quasi-religious faith, the investigators have to believe that the small risk increments they have observed can be interpreted and that they can be interpreted as causal (or preventive). Otherwise they can hardly justify their own existence. They have no choice but to ignore Feinstein’s dictum [Several years ago, Alvan Feinstein made the point that if some scientific fallacy is demonstrated and if it cannot be rebutted, a convenient way around the problem is simply to pretend that it does not exist and to ignore it.]