Cum hoc ergo propter hoc

The front row

A professor looks at her course records and finds that students who sit in the first two rows get noticeably higher grades than students who sit at the back. She announces that from now on, seats will be assigned so that struggling students sit at the front.

The clear-cut case. Students aren’t assigned to rows at random; they choose them, and the students who choose the front are likely to differ from the others in ways that affect grades on their own, such as how motivated they are or how much they care about the course. Those differences could produce the whole correlation. Sitting at the front might help a little too, but the records can’t say how much, if at all. Moving a struggling student forward changes their seat, not their motivation.

The busy café

A café manager reviews a year of records and finds that the hours with the most baristas on shift are also the hours with the longest lines. “Adding staff doesn’t shorten the line. If anything it makes it worse.”

Here the correlation is real and the causal direction is backwards. The café schedules more baristas because those are the busy hours: the lines cause the staffing, not the other way round. Within any given hour, more baristas very likely would shorten the line. This is reverse causation, and it’s easy to miss because the data don’t record which came first; both are measured over the same hour.

The restaurant guide

A food writer goes through a city guide that lists only restaurants worth visiting, and notices that among them, the places with plain, shabby dining rooms tend to have the best food. “It makes sense. When a restaurant doesn’t spend money on decor, it spends it in the kitchen.”

The pattern in the guide may be exactly as described, and it may appear even if decor and food quality are unrelated across all restaurants. A place gets into the guide by having great food or a great room. So a listed restaurant with a shabby room is likely to be there for its food, and a listed restaurant with mediocre food is likely to be there for its room. Selecting on something that both factors influence creates a correlation between them. This is the respectable-looking case: the data are real and the story is plausible, but the correlation was manufactured by the selection. The structure is described under Collider bias.

The bare argument has one premise, and its conclusion goes beyond it. Douglas Walton named this pattern the argument from correlation to cause. Martijn Demollin compares it with a fuller scheme from the textbook authors Leo Groarke and Christopher Tindale, which adds the premises the bare version leaves out:

Each added premise answers one alternative explanation. Cum hoc is the bare version offered as if it were the full one. Demollin’s own version of the scheme also requires the causal link itself to be plausible. He notes one further possibility, a correlation produced by conditioning on a common effect (as in the restaurant guide), but leaves it outside his scheme.

Each variant is a different explanation the argument overlooks:

• Coincidence. Compare enough things and some will be correlated by chance. Searching many comparisons and reporting the ones that match is the problem behind P-hacking and the Texas sharpshooter fallacy.
• Reverse causation. B causes A, as with the café. Austin Bradford Hill’s version: does a diet lead to a disease, or do the early stages of the disease lead to that diet?
• A common cause. A third factor produces both A and B, as with the front row. When this distorts a research finding it’s called Confounding.
• Selection. The cases were chosen, or chose themselves, on something that both A and B affect, as with the restaurant guide. See Collider bias and Selection bias.

Two neighboring errors start from a different mistake. A correlation that exists only in perception, not in the data, is an illusory correlation. And a correlation measured across groups (countries, towns, schools) assumed to hold for the individuals within them is the ecological fallacy, whether or not a cause is claimed.

The family. Cum hoc and post hoc ergo propter hoc (“after this, therefore because of this”) are siblings: post hoc rests on order in time, cum hoc on occurring together. Demollin groups them as “post hoc and cum hoc fallacies, which are also called fallacies of questionable cause”. Bradley Dowden’s Internet Encyclopedia of Philosophy entry lists both as kinds of false cause, describing cum hoc as a false cause that “doesn’t depend on time order (as does the post hoc fallacy)”. Patrick Hurley and Lori Watson’s textbook doesn’t use the name cum hoc, but its non causa pro causa variety of false cause covers the same ground: mistakes “based on something other than mere temporal succession”, including taking an effect for its cause and taking a coincidence for a connection. None of these sources makes one a kind of the other, so this site doesn’t either. The family name itself is used inconsistently: non causa pro causa is Hurley and Watson’s name for this sibling, but older logic books, and Dowden, use it for the whole family, as the post hoc page explains. The general principle, and what does establish causes, is on the correlation vs. causation page.

Reasoning from a correlation to a cause isn’t fallacious in itself. Walton, as Demollin quotes him, wrote that “quite often, arguments fitting this argumentation scheme are presumptively correct”, while warning that they are often weak “because other factors are overlooked”. It holds up when:

• The alternatives have been addressed. Chance, reverse causation, common causes and selection have each been considered and made unlikely, with evidence rather than assumption.
• Background knowledge rules some out. Demollin’s example: when the speed of a windmill’s sails is correlated with the strength of the wind, reverse causation can be dismissed because windmills don’t make wind. Not every alternative needs a study to dismiss it.
• The variation in A was assigned at random. In a randomized experiment, whoever flips the coin decides who gets A, so nothing that affects B could also have decided who got A. As Miguel Hernán and James Robins put it, “in ideal randomized experiments, association is causation.”
• The conclusion is stated as a hypothesis. “This correlation suggests A may affect B; here’s what would test it” claims no more than the evidence shows.
• No causal claim is being made. Using a correlation to predict B from A is legitimate even when A doesn’t cause B (see the look-alike below).

The test: what else, besides A causing B, would make these two go together, and what rules it out?

The A/B test

An online bookstore shows a randomly chosen half of its visitors a green “Buy now” button and the other half a gray one. Over a month and 400,000 visitors, 3.3% of those who see the green button buy something, against 3.0% of those who see the gray one. The team concludes the green button increases purchases and switches every visitor to it.

The conclusion rests on a correlation between button color and buying, which is exactly the shape of cum hoc. The difference is how the correlation arose. Because a random process decided who saw which button, the visitors who saw green can’t systematically differ from those who saw gray in motivation, budget or taste; reverse causation is impossible, since buying couldn’t have chosen the color; and with 200,000 visitors in each group, a gap of that size would be very unlikely to arise by chance. That’s the assigned at random condition, with chance also addressed.

The line at the food truck

Two food trucks are parked side by side in an unfamiliar town. One has a line of fifteen people and the other has none. A visitor picks the one with the line: “Where people line up, the food is usually good.”

The visitor is reasoning from a correlation between long lines and good food, and she may be picking up a partly reversed relationship: good food causes long lines, not the other way round. But she isn’t claiming a cause, and she isn’t trying to make the food better by making the line longer. She is using the line as a sign of something it tends to go with. Predictions from correlations are reasonable even when the causal story is unknown or runs the other way, as long as nothing is done on the assumption that changing A would change B. That’s the no causal claim condition.

A correlation invites a story, and the story that comes to mind first is usually the direct one: the thing we noticed first causes the thing we noticed second. The alternatives take more work to imagine, and some, like selection effects, are genuinely counterintuitive.

Everyday language also blurs the step. Findings reported as “linked to”, “associated with” or “tied to” are easy to hear as “causes”, and the causal reading is often more interesting and more useful to act on. Demollin analyzes a popular science article about a study that correlated smartphone Facebook use with the volume of a brain region. The study’s authors had explicitly said they hadn’t determined the nature of the correlation, but the article presented it as the discovery of negative side effects. He describes the article as committing cum hoc ergo propter hoc, and suggests that the pressure to simplify and to make findings sound significant pushes science communication toward it.

Demollin proposes critical questions for the argument from correlation to cause; the first three below follow his, and the last two add alternatives from the rest of this page:

• Is the correlation real? Is it significant, and could it be chance? Chance is a bigger worry when many comparisons were made.
• Could a third factor explain it? Name the most plausible one and ask whether it was accounted for.
• Could it run the other way? Would B plausibly lead to A?
• How were the cases selected? Were they filtered on something both A and B influence?
• Does the causal link make sense? Is there a plausible route from A to B?

And on the other side: pointing out that an argument goes from correlation to cause shows it’s incomplete, not that A doesn’t cause B. The useful response names a specific alternative explanation; a bare “correlation isn’t causation” doesn’t. The correlation vs. causation page discusses how the slogan can be misused to wave away good evidence.