Correlation does not imply causation

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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.[1][2] 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.


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 and convergent cross mapping.[clarification needed]


In logic, the technical use of the word "implies" means "is a sufficient circumstance for".[citation needed] 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."


General pattern[edit]

For any two correlated events, A and B, the different possible relationships include[citation needed]:

  • A causes B (direct causation);
  • B causes A (reverse causation);
  • 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);


Examples of illogically inferring causation from correlation[edit]

B causes A (reverse causation or reverse causality)[edit]

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


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.[4]

Example 3
Example 4

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 5

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 6

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.[citation needed]

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[5] or questionable cause[5]) 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 cause group 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
Therefore, sleeping with one's shoes on causes headache.


Example 2
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,[6] the study received much coverage at the time in the popular press.[7] 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.[8][9][10][11] In this case, the cause of both conditions is parental myopia, and the above-stated conclusion is false.

Example 3
Therefore, ice cream consumption causes drowning.


Example 4


Example 5
Hence, atmospheric CO2 causes obesity.

Richer populations tend to eat more food and produce more CO2.

Example 6


Bidirectional causation: A causes B, and B causes A[edit]

Causality is not necessarily one-way; in a predator-prey relationship, predator numbers affect prey numbers, but prey numbers, i.e. food supply, also affect predator numbers.

The relationship between A and B is coincidental[edit]

The two variables aren't related at all, but correlate by chance. The more things are examined, the more likely it is that two unrelated variables will appear to be related. For example:

  • The result of the last home game by the Washington Redskins prior to the presidential election predicted the outcome of every presidential election from 1936 to 2000 inclusive, despite the fact that the outcomes of football games had nothing to do with the outcome of the popular election. This streak was finally broken in 2004 (or 2012 using an alternative formulation of the original rule).
  • A collection of such coincidences[14] finds that for example, there is a 99.79% correlation for the period 1999-2009 between U.S. spending on science, space, and technology; and the number of suicides by suffocation, strangulation, and hanging.
  • The Mierscheid law, which correlates the Social Democratic Party of Germany's share of the popular vote with the size of crude steel production in Western Germany.
  • Alternating bald–hairy Russian leaders: A bald (or obviously balding) state leader of Russia has succeeded a non-bald ("hairy") one, and vice versa, for nearly 200 years.

Determining causation[edit]

In academia[edit]

The nature of causality is systematically investigated in several academic disciplines, including philosophy and physics.

In academia, there are a significant number of theories on causality; The Oxford Handbook of Causation (Beebee, Hitchcock & Menzies 2009) encompasses 770 pages. Among the more influential theories within philosophy are Aristotle's Four causes and Al-Ghazali's occasionalism.[15] David Hume argued that beliefs about causality are based on experience, and experience similarly based on the assumption that the future models the past, which in turn can only be based on experience – leading to circular logic. In conclusion, he asserted that causality is not based on actual reasoning: only correlation can actually be perceived.[16] Immanuel Kant, according to Beebee, Hitchcock & Menzies (2009), held that "a causal principle according to which every event has a cause, or follows according to a causal law, cannot be established through induction as a purely empirical claim, since it would then lack strict universality, or necessity".


From the point of view of thermodynamics, universal properties of causes as compared to effects have been identified through the Second law of thermodynamics, confirming the ancient, medieval and Cartesian[17] view that "the cause is greater than the effect" for the particular case of thermodynamic free energy. This, in turn, is challenged[dubious ] by popular interpretations of the concepts of nonlinear systems and the butterfly effect, in which small events cause large effects due to, respectively, unpredictability and an unlikely triggering of large amounts of potential energy.

Causality construed from counterfactual states[edit]




Causality predicted by an extrapolation of trends[edit]


Use of correlation as scientific evidence[edit]

Much of scientific evidence is based upon a correlation of variables[20] – 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.[20] 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.[21] 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,[22] as did biologist and statistician Ronald Fisher.[23][24][25][26][27][28][29]

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.[citation needed]

See also[edit]


  1. ^ Tufte 2006, p. 5
  2. ^ Aldrich, John (1995). "Correlations Genuine and Spurious in Pearson and Yule" (PDF). Statistical Science. 10 (4): 364–376. doi:10.1214/ss/1177009870. JSTOR 2246135. Archived from the original (PDF) on February 19, 2006. 
  3. ^ Lawlor DA, Davey Smith G, Ebrahim S (June 2004). "Commentary: the hormone replacement-coronary heart disease conundrum: is this the death of observational epidemiology?". Int J Epidemiol. 33 (3): 464–467. doi:10.1093/ije/dyh124. PMID 15166201. 
  4. ^ "Reinhart-Rogoff, Continued". 
  5. ^ a b Labossiere, M.C., Dr. LaBossiere's Philosophy Pages
  6. ^ Quinn, Graham E.; Shin, Chai H.; Maguire, Maureen G.; Stone, Richard A. (May 1999). "Myopia and ambient lighting at night". Nature. 399 (6732): 113–114. doi:10.1038/20094. PMID 10335839. 
  7. ^ CNN, May 13, 1999. Night-light may lead to nearsightedness
  9. ^ Zadnik, Karla; Jones, Lisa A.; Irvin, Brett C.; Kleinstein, Robert N.; Manny, Ruth E.; Shin, Julie A.; Mutti, Donald O. (2000). "Vision: Myopia and ambient night-time lighting". Nature. 404 (6774): 143–144. doi:10.1038/35004661. PMID 10724157. 
  10. ^ Gwiazda, J.; Ong, E.; Held, R.; Thorn, F. (2000). "Vision: Myopia and ambient night-time lighting". Nature. 404 (6774): 144–144. doi:10.1038/35004663. PMID 10724158. 
  11. ^ Stone, Richard A.; Maguire, Maureen G.; Quinn, Graham E. (2000). "Vision: reply: Myopia and ambient night-time lighting". Nature. 404 (6774): 144–144. doi:10.1038/35004665. 
  12. ^ Carducci, Bernardo J. (2009). The Psychology of Personality: Viewpoints, Research, and Applications (2nd ed.). John Wiley & Sons. ISBN 978-1-4051-3635-8. 
  14. ^ "Spurious Correlations". Retrieved 2017-10-08. 
  15. ^ a b Beebee, Hitchcock & Menzies 2009
  16. ^ Morris, William Edward (2001). "David Hume". The Stanford Encyclopedia of Philosophy. 
  17. ^ Lloyd, A.C. (1976). "The principle that the cause is greater than its effect". Phronesis. 21 (2): 146–156. doi:10.1163/156852876x00101. JSTOR 4181986. 
  19. ^ Pearl, Judea (2000). Causality: Models, Reasoning, and Inference. Cambridge University Press. ISBN 9780521773621. 
  21. ^ Nielsen, Michael (2012-01-23). "If correlation doesn't imply causation, then what does? | DDI". Retrieved 2017-10-08. 
  22. ^ "Evidence in Medicine: Correlation and Causation – Science-Based Medicine". 2009-11-18. Retrieved 2017-10-08. 
  23. ^ Silver, Nate (2015), The Signal and the Noise: Why So Many Predictions Fail – But Some Don't (2nd ed.), New York: Penguin Books, pp. 254–255 
  24. ^ Fisher, Ronald (July 6, 1957), "Dangers Of Cigarette-Smoking", The British Medical Journal, London: British Medical Association, 2: 43, doi:10.1136/bmj.2.5035.43, JSTOR 25383068 
  25. ^ Fisher, Ronald (August 3, 1957), "Dangers Of Cigarette-Smoking", The British Medical Journal, London: British Medical Association, 2: 297–298, doi:10.1136/bmj.2.5039.297-b, JSTOR 25383439 
  26. ^ Fisher, Ronald (1958), "Cigarettes, Cancer, and Statistics" (PDF), The Centennial Review of Arts & Science, East Lansing, Michigan: Michigan State University Press, 2: 151–166 
  27. ^ Fisher, Ronald (1958), "The Nature of Probability" (PDF), The Centennial Review of Arts & Science, East Lansing, Michigan: Michigan State University Press, 2: 261–274 
  28. ^ Fisher, Ronald (July 12, 1958), "Lung Cancer and Cigarettes" (PDF), Nature, London: Nature Publishing Group, 182: 108, doi:10.1038/182108a0 
  29. ^ Fisher, Ronald (August 30, 1958), "Cancer and Smoking" (PDF), Nature, London: Nature Publishing Group, 182: 596, doi:10.1038/182596a0 


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