In statistics, many statistical tests calculate correlations between variables and when two variables are found to be correlated, it is tempting to assume that this shows that one variable causes the other. That “correlation proves causation” is considered a questionable cause logical fallacy when two events occurring together are taken to have established a cause-and-effect relationship. This fallacy is also known as cum hoc ergo propter hoc, Latin for “with this, therefore because of this”, and “false cause”. A similar fallacy, that an event that followed another was necessarily a consequence of the first event, is the post hoc ergo propter hoc (Latin for “after this, therefore because of this.”) fallacy.
For example, in a widely studied case, numerous epidemiological studies showed that women taking combined hormone replacement therapy (HRT) also had a lower-than-average incidence of coronary heart disease (CHD), leading doctors to propose that HRT was protective against CHD. But randomized controlled trials showed that HRT caused a small but statistically significant increase in risk of CHD. Re-analysis of the data from the epidemiological studies showed that women undertaking HRT were more likely to be from higher socio-economic groups (ABC1), with better-than-average diet and exercise regimens. The use of HRT and decreased incidence of coronary heart disease were coincident effects of a common cause (i.e. the benefits associated with a higher socioeconomic status), rather than a direct cause and effect, as had been supposed.
As with any logical fallacy, identifying that the reasoning behind an argument is flawed does not imply that the resulting conclusion is false. In the instance above, if the trials had found that hormone replacement therapy does in fact have a negative incidence on the likelihood of coronary heart disease the assumption of causality would have been correct, although the logic behind the assumption would still have been flawed. Indeed, a few go further, using correlation as a basis for testing a hypothesis to try to establish a true causal relationship; examples are the Granger causality test, convergent cross mapping, and Liang-Kleeman information flow.
In logic, the technical use of the word “implies” means “is a sufficient circumstance for”. This is the meaning intended by statisticians when they say causation is not certain. Indeed, p implies q has the technical meaning of the material conditional: if p then q symbolized as p → q. That is “if circumstance p is true, then q follows.” In this sense, it is always correct to say “Correlation does not imply causation.”
However, in casual use, the word “implies” loosely means suggests rather than requires. Where there is causation, there is a likely correlation. Indeed, correlation is often used when inferring causation; the important point is that correlation is not sufficient.
For any two correlated events, A and B, the different possible relationships include
A causes B (direct causation);
B causes A (reverse causation);
A and B are consequences of a common cause, but do not cause each other;
A and B both cause C, which is (explicitly or implicitly) conditioned on;
A causes B and B causes A (bidirectional or cyclic causation);
A causes C which causes B (indirect causation);
There is no connection between A and B; the correlation is a coincidence.
Thus there can be no conclusion made regarding the existence or the direction of a cause-and-effect relationship only from the fact that A and B are correlated. Determining whether there is an actual cause-and-effect relationship requires further investigation, even when the relationship between A and B is statistically significant, a large effect size is observed, or a large part of the variance is explained.
Reverse causation or reverse causality or wrong direction is an informal fallacy of questionable cause where cause and effect are reversed. The cause is said to be the effect and vice versa.
- Example 1
- The faster windmills are observed to rotate, the more wind is observed to be.
- Therefore wind is caused by the rotation of windmills. (Or, simply put: windmills, as their name indicates, are machines used to produce wind.)
In this example, the correlation (simultaneity) between windmill activity and wind velocity does not imply that wind is caused by windmills. It is rather the other way around, as suggested by the fact that wind doesn’t need windmills to exist, while windmills need wind to rotate. Wind can be observed in places where there are no windmills or non-rotating windmills—and there are good reasons to believe that wind existed before the invention of windmills.
- Example 2
- When a country’s debt rises above 90% of GDP, growth slows.
- Therefore, high debt causes slow growth.
This argument by Carmen Reinhart and Kenneth Rogoff was refuted by Paul Krugman on the basis that they got the causality backwards: in actuality, slow growth causes debt to increase.
- Example 3
In other cases it may simply be unclear which is the cause and which is the effect. For example:
- Children that watch a lot of TV are the most violent. Clearly, TV makes children more violent.
This could easily be the other way round; that is, violent children like watching more TV than less violent ones.
- Example 4
A correlation between recreational drug use and psychiatric disorders might be either way around: perhaps the drugs cause the disorders, or perhaps people use drugs to self medicate for preexisting conditions. Gateway drug theory may argue that marijuana usage leads to usage of harder drugs, but hard drug usage may lead to marijuana usage (see also confusion of the inverse). Indeed, in the social sciences where controlled experiments often cannot be used to discern the direction of causation, this fallacy can fuel long-standing scientific arguments. One such example can be found in education economics, between the screening/signaling and human capital models: it could either be that having innate ability enables one to complete an education, or that completing an education builds one’s ability.
- Example 5
A historical example of this is that Europeans in the Middle Ages believed that lice were beneficial to your health, since there would rarely be any lice on sick people. The reasoning was that the people got sick because the lice left. The real reason however is that lice are extremely sensitive to body temperature. A small increase of body temperature, such as in a fever, will make the lice look for another host. The medical thermometer had not yet been invented, so this increase in temperature was rarely noticed. Noticeable symptoms came later, giving the impression that the lice left before the person got sick.
In other cases, two phenomena can each be a partial cause of the other; consider poverty and lack of education, or procrastination and poor self-esteem. One making an argument based on these two phenomena must however be careful to avoid the fallacy of circular cause and consequence. Poverty is a cause of lack of education, but it is not the sole cause, and vice versa.
The third-cause fallacy (also known as ignoring a common cause or questionable cause) is a logical fallacy where a spurious relationship is confused for causation. It asserts that X causes Y when, in reality, X and Y are both caused by Z. It is a variation on the post hoc ergo propter hoc fallacy and a member of the questionable causegroup of fallacies.
All of these examples deal with a lurking variable, which is simply a hidden third variable that affects both causes of the correlation. A difficulty often also arises where the third factor, though fundamentally different from A and B, is so closely related to A and/or B as to be confused with them or very difficult to scientifically disentangle from them (see Example 4).
- Example 1
- Sleeping with one’s shoes on is strongly correlated with waking up with a headache.
- Therefore, sleeping with one’s shoes on causes headache.
The above example commits the correlation-implies-causation fallacy, as it prematurely concludes that sleeping with one’s shoes on causes headache. A more plausible explanation is that both are caused by a third factor, in this case going to bed drunk, which thereby gives rise to a correlation. So the conclusion is false.
- Example 2
- Young children who sleep with the light on are much more likely to develop myopia in later life.
- Therefore, sleeping with the light on causes myopia.
This is a scientific example that resulted from a study at the University of Pennsylvania Medical Center. Published in the May 13, 1999 issue of Nature, the study received much coverage at the time in the popular press. However, a later study at Ohio State University did not find that infants sleeping with the light on caused the development of myopia. It did find a strong link between parental myopia and the development of child myopia, also noting that myopic parents were more likely to leave a light on in their children’s bedroom. In this case, the cause of both conditions is parental myopia, and the above-stated conclusion is false.
- Example 3
- As ice cream sales increase, the rate of drowning deaths increases sharply.
- Therefore, ice cream consumption causes drowning.
This example fails to recognize the importance of time of year and temperature to ice cream sales. Ice cream is sold during the hot summer months at a much greater rate than during colder times, and it is during these hot summer months that people are more likely to engage in activities involving water, such as swimming. The increased drowning deaths are simply caused by more exposure to water-based activities, not ice cream. The stated conclusion is false.
- Example 4
- A hypothetical study shows a relationship between test anxiety scores and shyness scores, with a statistical r value (strength of correlation) of +.59.
- Therefore, it may be simply concluded that shyness, in some part, causally influences test anxiety.
However, as encountered in many psychological studies, another variable, a “self-consciousness score”, is discovered that has a sharper correlation (+.73) with shyness. This suggests a possible “third variable” problem, however, when three such closely related measures are found, it further suggests that each may have bidirectional tendencies (see “bidirectional variable”, above), being a cluster of correlated values each influencing one another to some extent. Therefore, the simple conclusion above may be false.
- Example 5
- Since the 1950s, both the atmospheric CO2 level and obesity levels have increased sharply.
- Hence, atmospheric CO2 causes obesity.
Richer populations tend to eat more food and produce more CO2.
- Example 6
- HDL (“good”) cholesterol is negatively correlated with incidence of heart attack.
- Therefore, taking medication to raise HDL decreases the chance of having a heart attack.
Further research has called this conclusion into question. Instead, it may be that other underlying factors, like genes, diet and exercise, affect both HDL levels and the likelihood of having a heart attack; it is possible that medicines may affect the directly measurable factor, HDL levels, without affecting the chance of heart attack.
Much of scientific evidence is based upon a correlation of variables – they are observed to occur together. Scientists are careful to point out that correlation does not necessarily mean causation. The assumption that A causes B simply because A correlates with B is often not accepted as a legitimate form of argument.
However, sometimes people commit the opposite fallacy – dismissing correlation entirely. This would dismiss a large swath of important scientific evidence. Since it may be difficult or ethically impossible to run controlled double-blind studies, correlational evidence from several different angles may be useful for prediction despite failing to provide evidence for causation. For example, social workers might be interested in knowing how child abuse relates to academic performance. Although it would be unethical to perform an experiment in which children are randomly assigned to receive or not receive abuse, researchers can look at existing groups using a non-experimental correlational design. If in fact a negative correlation exists between abuse and academic performance, researchers could potentially use this knowledge of a statistical correlation to make predictions about children outside the study who experience abuse, even though the study failed to provide causal evidence that abuse decreases academic performance. The combination of limited available methodologies with the dismissing correlation fallacy has on occasion been used to counter a scientific finding. For example, the tobacco industry has historically relied on a dismissal of correlational evidence to reject a link between tobacco and lung cancer, as did biologist and statistician Ronald Fisher.
Correlation is a valuable type of scientific evidence in fields such as medicine, psychology, and sociology. But first correlations must be confirmed as real, and then every possible causative relationship must be systematically explored. In the end correlation alone cannot be used as evidence for a cause-and-effect relationship between a treatment and benefit, a risk factor and a disease, or a social or economic factor and various outcomes. It is one of the most abused types of evidence, because it is easy and even tempting to come to premature conclusions based upon the preliminary appearance of a correlation.
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What Everyone Should Know about Statistical Correlation