It takes two

Heavy drinkers get more cancers of the mouth and throat, and the relationship is beautifully behaved: the more someone drinks, the higher the risk, in a smooth curve you could draw with a confident hand. It looks like exactly what it appears to be — alcohol, doing damage, dose by dose.

But heavy drinkers also tend to smoke. So that tidy “alcohol” curve is compatible with more than one story. Maybe alcohol scars the tissue directly. Maybe alcohol does nothing at all, and the drinkers' extra cancers come entirely from the cigarettes that keep the drinks company. Or — the possibility this post is about — maybe the damage is done by the two together: the alcohol stripping away the defences that let the smoke's carcinogens in, so that neither is very dangerous alone but the pair is ruinous.

Here is the unsettling part. That third story, where the real cause is the combination, produces the same smooth, graded “alcohol causes cancer” curve as the first. You can get a perfect dose-response for a substance that, on its own, does nothing — as long as it usually travels with its partner. The curve is real as a description of the data and misleading as a guide to the mechanism, and staring harder at the same numbers will never tell the two apart.

Three ways to draw one curve

Hold three structures apart and the rest of the post follows.

The first is the one everyone has heard of: confounding, an innocent marker. Some third thing — a certain kind of life — makes people both drink and smoke, and the smoking does the damage. Alcohol is just a flag flying over the real culprit. Cut it and nothing changes, because the smoker keeps smoking. Under this story alcohol genuinely is not a cause.

The second is the one this post is about: combination, or synergy. The harm is done by co-consumption. Here, cutting the drinking does lower risk — because you have removed half of every harmful pairing. Alcohol is a cause: not a lone sufficient cause, but a component of one, in the epidemiologist Kenneth Rothman's picture of causes as pies made of slices. No single slice is “the” cause; the pie needs all its slices to be complete; and every slice, alcohol included, is correctly called a cause of the outcome.

The third is the plain one: direct. Alcohol harms on its own and the cigarettes are beside the point.

One intuition is worth killing straight away, because it is both tempting and wrong: “it's really the combination” does not mean “alcohol is innocent.” In the sense that actually matters for deciding what to do — if you change it, the outcome changes — alcohol under the combination story is still a cause. It simply isn't a cause that works alone.

A curve from a cause that does nothing alone

Watch the illusion assemble itself out of almost nothing. Picture a person's life as a long stream of occasions. On some fraction of them they drink; call that rate d. On some fraction they smoke; call that rate s. Now suppose — to see the effect in its purest form — that the only thing which does any damage is an occasion where they do both, so that harm piles up in proportion to the rate at which drinking and smoking coincide.

harm accrues in proportion to d × s — the rate of drinking-and-smoking together

Now do what an epidemiologist does: ignore the pairing, measure each person's total drinking (which tracks d), and plot it against cancer. Because harm rises with d × s, then at any fixed level of smoking, more drinking means more harm. Across a population that smokes to varying degrees, those individual lines average into a single smooth curve that climbs with drinking — even though drinking, in this world, is biologically inert on its own. The whole thing falls straight out of the multiplication. Do the same for smoking and you get an equally convincing “smoking” curve. Two persuasive single-variable dose-responses, and the real cause is neither variable but their product.

One subtlety is worth holding onto: what does the damage is the running count of paired occasions racked up over the years, not the single question “did they ever once drink and smoke together.” “Ever paired” would sort people into a crude in-or-out group; the graded curve comes from the accumulating total. Which sets up the strangest part of all.

Why a personal switch looks like a gentle slope

Suppose that inside any one person the rule is not gradual at all but a threshold: nothing happens until their lifetime tally of paired occasions crosses a personal line — a level of accumulated damage the body can no longer repair — and past it, risk jumps. Each individual is a step function: flat, then a cliff.

Average a crowd of step functions whose cliffs sit at different places, and the sum is smooth. The population curve you measure is essentially the distribution of individual thresholds laid on its side — the fraction of people whose personal line you have crossed by a given dose. A field of a thousand switches, each flipping at its own point, reads from a distance as a single gentle dimmer.

The shape of the population curve carries almost no information about the rule inside any one person.

This is the old idea behind probit analysis, and its consequence is sharp. A graded individual response and an all-or-nothing threshold response can produce the very same population S-curve; you need only choose the spread of thresholds to match. At the level of aggregate data they are the same picture — observationally equivalent, indistinguishable in principle. That is not a puzzle cleverness can solve. It is a ceiling.

The fix depends on the shape

This would be a mathematical curiosity if it did not change what you should do — and it does. Suppose the combination story is the true one. Does telling people to drink less help? It depends entirely on which drinks they give up. Cut drinking at random and you thin out the paired occasions in proportion, and risk falls. But cut only the solo drinks — the lunchtime glass with no cigarette in sight — while keeping every drink that comes with a smoke, and you have removed most of the alcohol and none of the harm.

The realistic version is worse than merely useless. The paired occasions are often the social, salient, hard-to-surrender ones — the drink with the cigarette out on the terrace — so a well-meaning “just drink less” message can trim exactly the innocent occasions and spare the dangerous ones. Then the study that follows finds no benefit, and the null gets written up as “so it wasn't the alcohol after all.” Same correlation; opposite lessons; and under the combination story the genuinely useful advice is almost free — don't pair them — while “give up drinking entirely” is either wasted effort or a sledgehammer for a job a scalpel would do.

Can you settle it by re-running the numbers?

No — and the examples in this post are themselves the proof. If several different causal structures all reproduce the same aggregate curve, then the aggregate curve cannot tell you which structure you are in. The numbers under-determine the mechanism, full stop. Every escape route involves changing what you measure or what you do, not what arithmetic you run on the data you already have.

The most powerful move is finer-grained data. Stop recording each person's yearly totals and start recording co-consumption at the level of the occasion. Then the pure-combination hypothesis makes a bet it can lose: once you know someone's count of paired occasions, their total drinking and total smoking should add nothing to the prediction — the pairing count “screens off” the marginals. (Reaching instead for a “drinking × smoking” interaction term built from the totals is the obvious move, and it is a lossy stand-in: two people with identical totals can have completely different pairing habits, and the totals cannot tell them apart.) A natural experiment does similar work: find the heavy drinkers who have essentially never smoked. Pure combination predicts their risk should be flat; if it isn't, pure combination is dead and only some mixed story survives. And the clean, if hypothetical, design is intervention: hold each person's total drinking and total smoking fixed, and randomise only whether they happen together. A lower-risk “never pair” arm would isolate the combination from every confounder at once — which is exactly the point, because it shows what the question demanded all along.

The rogues' gallery

The structure is no thought experiment; medicine is full of it, arranged along a gradient from “one ingredient truly does nothing alone” to “both bite, and together they maul.”

At the pure end sits favism. Fava beans are ordinary food to almost everyone — but in a person born with a deficiency of the enzyme G6PD, a plate of them can trigger a sudden, dangerous breakdown of red blood cells. Measure fava-bean intake across a population where that gene variant runs at, say, one in six — as it does in parts of the Mediterranean, the Middle East and Africa — and you would see a respectable “beans are dangerous” dose-response that is almost pure aggregation artifact: the bean does essentially nothing to the other five in six. It is the cleanest match to the toy model above, with the twist that the hidden second factor is not another habit but a gene.

A step along the gradient is the textbook case of asbestos and smoking. Each raises the risk of lung cancer on its own, but together they do far more than add. In the classic figures from Selikoff and Hammond's studies of insulation workers, against someone who never smoked and never met asbestos, asbestos alone multiplied lung-cancer risk perhaps fivefold and smoking alone perhaps tenfold — while the two together pushed it toward fiftyfold, roughly the product rather than the sum. (Modern pooled estimates are gentler, and whether the interaction is exactly multiplicative is still argued — itself a small monument to how hard the aggregate data make the mechanism to pin down.) In a population of smokers, much of what looks like “the asbestos curve” is really the interaction; a non-smoker's asbestos curve is far shallower than the headline.

The most elegant demonstration of the aggregation trap is aflatoxin and hepatitis B. Aflatoxin, a mould toxin on badly stored grain and nuts, traces a dose-response for liver cancer across exposed populations. But split people by whether they carry chronic hepatitis B and the effect all but collapses into the infected group. In a landmark Shanghai cohort, aflatoxin alone raised risk around three-and-a-half-fold and hepatitis B alone around sevenfold — but the two together multiplied to roughly sixtyfold. The smooth “aflatoxin” curve for everyone was largely the steep response of the hepatitis-B-positive minority, diluted across a mostly-uninfected majority. One curve; two utterly different slopes hiding inside it.

And the historical cautionary tale — a slightly different mechanism, the very same moral: pellagra. In the American South a century ago, the disfiguring disease tracked corn consumption so tightly that the medical mainstream went hunting for a toxin or a germ in the corn. Joseph Goldberger showed the near-opposite: pellagra was a deficiency — of what we now call niacin — and corn-heavy diets simply lacked it. Corn drew a convincing, causal-looking curve while being, in the sense that mattered, innocent; the culprit was what the corn-eaters were not eating. Not a combination but an absence, and still a single dietary variable painting a persuasive dose-response for a reason that had nothing to do with the food being harmful.

What the curve can't tell you

In every one of these, the single-variable curve was real as data and false as mechanism, and in every one the resolution came from the same move — not re-analysing the correlation but measuring or splitting on the second factor: the G6PD gene, the hepatitis-B serology, the smoking history, the missing vitamin. The correlation had already told you everything it was ever going to.

That is the sharper cousin of “correlation isn't causation,” and it is worth stating plainly because it is less comforting. We like to imagine that a clean, strong, dose-dependent association is the safe kind — the kind that has earned the right to be believed. But a combination cause delivers exactly that clean, strong, dose-dependent curve and hides its mechanism inside it. The smoothness you find so reassuring is, at least sometimes, just a thousand switches averaged from a distance. When the mechanism matters — because the right thing to do depends on it — the honest response to a beautiful curve is not to admire it, and not to re-run it, but to go and measure the thing it isn't showing you.

None of which lets alcohol, or tobacco, off the hook. They earn their standing as causes the hard way — the associations hold up across populations with very different drinking and smoking cultures, and there are concrete mechanisms that need no partner in the other hand. The combination structure is not a claim that either is harmless; it is a rival explanation that a single correlation, however lovely, has no power to dismiss. Which is the whole point. The curve is not the cause. It is a shadow the cause happens to cast — and more than one thing can throw the same shadow.