Correlation does not imply causation
Encyclopedia
"Correlation does not imply causation" (related to "ignoring a common cause" and questionable cause
) is a phrase used in science
and statistics
to emphasize that correlation
between two variables does not automatically imply that one causes
the other (though correlation is necessary for linear causation in the absence of any third and countervailing causative variable, and can indicate possible causes or areas for further investigation; in other words, correlation can be a hint).
The opposite belief, correlation proves causation, is a logical fallacy by which two events that occur together are claimed to have a cause-and-effect relationship. The fallacy is also known as cum hoc ergo propter hoc (Latin
for "with this, therefore because of this") and false cause. By contrast, the fallacy post hoc ergo propter hoc
requires that one event occur before the other and so may be considered a type of cum hoc fallacy.
In a widely-studied example, numerous epidemiological studies showed that women who were 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 cause and effect as had been supposed.
, the technical use of the word "implies" means "to be a sufficient circumstance". This is the meaning intended by statisticians when they say causation is not certain. Indeed, p implies q has the technical meaning of logical implication: if p then q symbolized as p → q. That is "if circumstance p is true, then q necessarily follows." In this sense, it is always correct to say "Correlation does not imply causation".
However, in casual use, the word "imply" loosely means suggests rather than requires. The idea that correlation and causation are connected is certainly true; where there is causation, there is likely to be correlation. Indeed, correlation is used when inferring causation; the important point is that such inferences are not always correct because there are other possibilities, as explained later in this article.
Edward Tufte
, in a criticism of the brevity of Microsoft PowerPoint
presentations, deprecates the use of "is" to relate correlation and causation (as in "Correlation is not causation"), citing its inaccuracy as incomplete. While it is not the case that correlation is causation, simply stating their nonequivalence omits information about their relationship. Tufte suggests that the shortest true statement that can be made about causality and correlation is one of the following:
In this type of logical fallacy, one makes a premature conclusion about causality
after observing only a correlation
between two or more factors. Generally, if one factor (A) is observed to only be correlated with another factor (B), it is sometimes taken for granted that A is causing B even when no evidence supports it. This is a logical fallacy because there are at least five possibilities:
In other words, 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
.
In this example, the correlation between the number of firemen at a scene and the size of the fire does not imply that the firemen cause the fire. Firemen are sent according to the severity of the fire and if there is a large fire, a greater number of firemen are sent; therefore it is rather that fire causes firemen to arrive at the scene. So the above conclusion is false.
The ideal gas law
,
, describes the direct relationship between pressure and temperature (along with other factors) to show that there is a direct correlation between the two properties. For a fixed volume and mass of gas, an increase in temperature will cause an increase in pressure; likewise, increased pressure will cause an increase in temperature. This demonstrates bidirectional causation. The conclusion that pressure causes temperature is true but is not logically guaranteed by the premise.
, which is simply a hidden third variable that affects both causes of the correlation; for example, the fact that it is summer in Example 3. 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
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
This is a recent 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 The Ohio State University did not find that infant
s 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
The aforementioned example fails to recognize the importance of time in relationship to ice cream sales. Ice cream is sold during the summer
months at a much greater rate, and it is during the 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
However, as encountered in many psychological studies, another variable, a "self-consciousness score," is discovered which 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
As car sales increase, carbon dioxide levels increase as well as obesity as people do less walking and biking.
Example 6
Recent research calls this conclusion into question. Instead, it may be that an underlying factor, genes, affects 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.
This example is used satirically
by the parody religion
Pastafarianism
to illustrate the logical fallacy of assuming that correlation equals causation.
argued that causality is 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.
Intuitively, causation seems to require not just a correlation, but a counterfactual dependence. Suppose that a student performed poorly on a test and guesses that the cause was his not studying. To prove this, one thinks of the counterfactual – the same student writing the same test under the same circumstances but having studied the night before. If one could rewind history, and change only one small thing (making the student study for the exam), then causation could be observed (by comparing version 1 to version 2). Because one cannot rewind history and replay events after making small controlled changes, causation can only be inferred, never exactly known. This is referred to as the Fundamental Problem of Causal Inference – it is impossible to directly observe causal effects.
A major goal of scientific experiments and statistical methods is to approximate as best as possible the counterfactual state of the world. For example, one could run an experiment on identical twins
who were known to consistently get the same grades on their tests. One twin is sent to study for six hours while the other is sent to the amusement park. If their test scores suddenly diverged by a large degree, this would be strong evidence that studying (or going to the amusement park) had a causal effect on test scores. In this case, correlation between studying and test scores would almost certainly imply causation.
Well-designed experimental studies
replace equality of individuals as in the previous example by equality of groups. This is achieved by randomization of the subjects to two or more groups. Although not a perfect system, the likeliness of being equal in all aspects rises with the number of subjects placed randomly in the treatment/placebo
groups. From the significance of the difference of the effect of the treatment vs. the placebo, one can conclude the likeliness of the treatment having a causal effect on the disease. This likeliness can be quantified in statistical terms by the P-value
.
When experimental studies are impossible and only pre-existing data are available, as is usually the case for example in economics, regression analysis
can be used. Factors other than the potential causative variable of interest are controlled for by including them as regressors in addition to the regressor representing the variable of interest. False inferences of causation due to reverse causation (or wrong estimates of the magnitude of causation due the presence of bidirectional causation) can be avoided by using explanators (regressors) that are necessarily exogenous
, such as physical explanators like rainfall amount (as a determinant of, say, futures prices), lagged variables whose values were determined before the dependent variable's value was determined, instrumental variables for the explanators (chosen based on their known exogeneity), etc. See Causality#Economics. Spurious correlation due to mutual influence from a third, common causative, variable, is harder to avoid: the model must be specified such that there is a theoretical reason to believe that no such underlying causative variable has been omitted from the model; in particular, underlying time trends of both the dependent variable and the independent (potentially causative) variable must be controlled for by including time as another independent variable.
Questionable cause
Fallacies of questionable cause, also known as causal fallacies, non causa pro causa or false cause, are informal fallacies where a cause is incorrectly identified...
) is a phrase used in science
Science
Science is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe...
and statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
to emphasize that correlation
Correlation
In statistics, dependence refers to any statistical relationship between two random variables or two sets of data. Correlation refers to any of a broad class of statistical relationships involving dependence....
between two variables does not automatically imply that one causes
Causality
Causality is the relationship between an event and a second event , where the second event is understood as a consequence of the first....
the other (though correlation is necessary for linear causation in the absence of any third and countervailing causative variable, and can indicate possible causes or areas for further investigation; in other words, correlation can be a hint).
The opposite belief, correlation proves causation, is a logical fallacy by which two events that occur together are claimed to have a cause-and-effect relationship. The fallacy is also known as cum hoc ergo propter hoc (Latin
Latin
Latin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...
for "with this, therefore because of this") and false cause. By contrast, the fallacy post hoc ergo propter hoc
Post hoc ergo propter hoc
Post hoc ergo propter hoc, Latin for "after this, therefore because of this," is a logical fallacy that states, "Since that event followed this one, that event must have been caused by this one." It is often shortened to simply post hoc and is also sometimes referred to as false cause,...
requires that one event occur before the other and so may be considered a type of cum hoc fallacy.
In a widely-studied example, numerous epidemiological studies showed that women who were taking combined hormone replacement therapy
Hormone replacement therapy (menopause)
Hormone replacement therapy is a system of medical treatment for surgically menopausal, perimenopausal and to a lesser extent postmenopausal women...
(HRT) also had a lower-than-average incidence of coronary heart disease
Coronary heart disease
Coronary artery disease is the end result of the accumulation of atheromatous plaques within the walls of the coronary arteries that supply the myocardium with oxygen and nutrients. It is sometimes also called 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
NRS social grade
The NRS social grades are a system of demographic classification used in the United Kingdom. They were originally developed by the National Readership Survey in order to classify readers, but are now used by many other organisations for wider applications and have become a standard for market...
), 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 cause and effect as had been supposed.
Usage
In logicLogic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
, the technical use of the word "implies" means "to be a sufficient circumstance". This is the meaning intended by statisticians when they say causation is not certain. Indeed, p implies q has the technical meaning of logical implication: if p then q symbolized as p → q. That is "if circumstance p is true, then q necessarily follows." In this sense, it is always correct to say "Correlation does not imply causation".
However, in casual use, the word "imply" loosely means suggests rather than requires. The idea that correlation and causation are connected is certainly true; where there is causation, there is likely to be correlation. Indeed, correlation is used when inferring causation; the important point is that such inferences are not always correct because there are other possibilities, as explained later in this article.
Edward Tufte
Edward Tufte
Edward Rolf Tufte is an American statistician and professor emeritus of political science, statistics, and computer science at Yale University. He is noted for his writings on information design and as a pioneer in the field of data visualization....
, in a criticism of the brevity of Microsoft PowerPoint
Microsoft PowerPoint
Microsoft PowerPoint, usually just called PowerPoint, is a non-free commercial presentation program developed by Microsoft. It is part of the Microsoft Office suite, and runs on Microsoft Windows and Apple's Mac OS X operating system...
presentations, deprecates the use of "is" to relate correlation and causation (as in "Correlation is not causation"), citing its inaccuracy as incomplete. While it is not the case that correlation is causation, simply stating their nonequivalence omits information about their relationship. Tufte suggests that the shortest true statement that can be made about causality and correlation is one of the following:
- "Empirically observed covariation is a necessary but not sufficient condition for causality."
- "Correlation is not causation but it sure is a hint."
General pattern
The cum hoc ergo propter hoc logical fallacy can be expressed as follows:- A occurs in correlation with B.
- Therefore, A causes B.
In this type of logical fallacy, one makes a premature conclusion about causality
Causality
Causality is the relationship between an event and a second event , where the second event is understood as a consequence of the first....
after observing only a correlation
Correlation
In statistics, dependence refers to any statistical relationship between two random variables or two sets of data. Correlation refers to any of a broad class of statistical relationships involving dependence....
between two or more factors. Generally, if one factor (A) is observed to only be correlated with another factor (B), it is sometimes taken for granted that A is causing B even when no evidence supports it. This is a logical fallacy because there are at least five possibilities:
- A may be the cause of B.
- B may be the cause of A.
- some unknown third factor C may actually be the cause of both A and B.
- there may be a combination of the above three relationships. For example, B may be the cause of A at the same time as A is the cause of B (contradicting that the only relationship between A and B is that A causes B). This describes a self-reinforcing system.
- the "relationship" is a coincidenceCoincidenceA coincidence is an event notable for its occurring in conjunction with other conditions, e.g. another event. As such, a coincidence occurs when something uncanny, accidental and unexpected happens under conditions named, but not under a defined relationship...
or so complex or indirect that it is more effectively called a coincidence (i.e. two events occurring at the same time that have no direct relationship to each other besides the fact that they are occurring at the same time). A larger sample sizeSample sizeSample size determination is the act of choosing the number of observations to include in a statistical sample. The sample size is an important feature of any empirical study in which the goal is to make inferences about a population from a sample...
helps to reduce the chance of a coincidence, unless there is a systematic errorSystematic errorSystematic errors are biases in measurement which lead to the situation where the mean of many separate measurements differs significantly from the actual value of the measured attribute. All measurements are prone to systematic errors, often of several different types...
in the experiment.
In other words, 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
Statistical significance
In statistics, a result is called statistically significant if it is unlikely to have occurred by chance. The phrase test of significance was coined by Ronald Fisher....
, a large effect size
Effect size
In statistics, an effect size is a measure of the strength of the relationship between two variables in a statistical population, or a sample-based estimate of that quantity...
is observed, or a large part of the variance is explained
Coefficient of determination
In statistics, the coefficient of determination R2 is used in the context of statistical models whose main purpose is the prediction of future outcomes on the basis of other related information. It is the proportion of variability in a data set that is accounted for by the statistical model...
.
B causes A (reverse causation)
- The more firemen fighting a fire, the bigger the fire is observed to be.
- Therefore firemen cause fire.
In this example, the correlation between the number of firemen at a scene and the size of the fire does not imply that the firemen cause the fire. Firemen are sent according to the severity of the fire and if there is a large fire, a greater number of firemen are sent; therefore it is rather that fire causes firemen to arrive at the scene. So the above conclusion is false.
A causes B and B causes A (bidirectional causation)
- Increased pressure is associated with increased temperature.
- Therefore pressure causes temperature.
The ideal gas law
Ideal gas law
The ideal gas law is the equation of state of a hypothetical ideal gas. It is a good approximation to the behavior of many gases under many conditions, although it has several limitations. It was first stated by Émile Clapeyron in 1834 as a combination of Boyle's law and Charles's law...
,
, describes the direct relationship between pressure and temperature (along with other factors) to show that there is a direct correlation between the two properties. For a fixed volume and mass of gas, an increase in temperature will cause an increase in pressure; likewise, increased pressure will cause an increase in temperature. This demonstrates bidirectional causation. The conclusion that pressure causes temperature is true but is not logically guaranteed by the premise.Third factor C (the common-causal variable) causes both A and B
All these examples deal with a lurking variableLurking variable
In statistics, a confounding variable is an extraneous variable in a statistical model that correlates with both the dependent variable and the independent variable...
, which is simply a hidden third variable that affects both causes of the correlation; for example, the fact that it is summer in Example 3. 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 headacheHeadacheA headache or cephalalgia is pain anywhere in the region of the head or neck. It can be a symptom of a number of different conditions of the head and neck. The brain tissue itself is not sensitive to pain because it lacks pain receptors. Rather, the pain is caused by disturbance of the...
. - 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
Drunkenness
Alcohol intoxication is a physiological state that occurs when a person has a high level of ethanol in his or her blood....
, 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 myopiaMyopiaMyopia , "shortsightedness" ) is a refractive defect of the eye in which collimated light produces image focus in front of the retina under conditions of accommodation. In simpler terms, myopia is a condition of the eye where the light that comes in does not directly focus on the retina but in...
in later life. - Therefore, sleeping with the light on causes myopia.
This is a recent scientific example that resulted from a study at the University of Pennsylvania
University of Pennsylvania
The University of Pennsylvania is a private, Ivy League university located in Philadelphia, Pennsylvania, United States. Penn is the fourth-oldest institution of higher education in the United States,Penn is the fourth-oldest using the founding dates claimed by each institution...
Medical Center
Penn Presbyterian Medical Center
Penn Presbyterian Medical Center is a hospital located in the University City section of West Philadelphia. It is located between Market Street and Powelton Avenue, and N. 38th Street and N. Sloan Street....
. Published in the May 13, 1999 issue of Nature
Nature (journal)
Nature, first published on 4 November 1869, is ranked the world's most cited interdisciplinary scientific journal by the Science Edition of the 2010 Journal Citation Reports...
, the study received much coverage at the time in the popular press. However, a later study at The Ohio State University did not find that infant
Infant
A newborn or baby is the very young offspring of a human or other mammal. A newborn is an infant who is within hours, days, or up to a few weeks from birth. In medical contexts, newborn or neonate refers to an infant in the first 28 days after birth...
s 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 creamIce creamIce cream is a frozen dessert usually made from dairy products, such as milk and cream, and often combined with fruits or other ingredients and flavours. Most varieties contain sugar, although some are made with other sweeteners...
sales increase, the rate of drowningDrowningDrowning is death from asphyxia due to suffocation caused by water entering the lungs and preventing the absorption of oxygen leading to cerebral hypoxia....
deaths increases sharply. - Therefore, ice cream causes drowning.
The aforementioned example fails to recognize the importance of time in relationship to ice cream sales. Ice cream is sold during the summer
Summer
Summer is the warmest of the four temperate seasons, between spring and autumn. At the summer solstice, the days are longest and the nights are shortest, with day-length decreasing as the season progresses after the solstice...
months at a much greater rate, and it is during the 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 which 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 CO2Carbon dioxideCarbon dioxide is a naturally occurring chemical compound composed of two oxygen atoms covalently bonded to a single carbon atom...
level and obesityObesityObesity is a medical condition in which excess body fat has accumulated to the extent that it may have an adverse effect on health, leading to reduced life expectancy and/or increased health problems...
levels have increased sharply. - Hence, atmospheric CO2 causes obesity.
As car sales increase, carbon dioxide levels increase as well as obesity as people do less walking and biking.
Example 6
- HDLHDLHDL may refer to one of the following:* Hardware description language* High-density lipoprotein, so-called "good cholesterol".* Hong Kong Disneyland, in China.* Les Hurlements d'Léo, an alternative rock band from France.* GE HDL diesel engine...
("good") cholesterolCholesterolCholesterol is a complex isoprenoid. Specifically, it is a waxy steroid of fat that is produced in the liver or intestines. It is used to produce hormones and cell membranes and is transported in the blood plasma of all mammals. It is an essential structural component of mammalian cell membranes...
is negatively correlated with incidence of heart attack. - Therefore, taking medication to raise HDL will decrease the chance of having a heart attack.
Recent research calls this conclusion into question. Instead, it may be that an underlying factor, genes, affects 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.
Coincidence
- With a decrease in the number of pirates, there has been an increase in global warmingGlobal warmingGlobal warming refers to the rising average temperature of Earth's atmosphere and oceans and its projected continuation. In the last 100 years, Earth's average surface temperature increased by about with about two thirds of the increase occurring over just the last three decades...
over the same period. - Therefore, global warming is caused by a lack of pirates.
This example is used satirically
Satire
Satire is primarily a literary genre or form, although in practice it can also be found in the graphic and performing arts. In satire, vices, follies, abuses, and shortcomings are held up to ridicule, ideally with the intent of shaming individuals, and society itself, into improvement...
by the parody religion
Parody religion
A parody religion or mock religion is a parody of a religion, sect or cult. A parody religion can be a parody of several religions, sects, gurus and cults at the same time. Or, it can be a parody of no particular religion, instead parodying the concept of religious belief...
Pastafarianism
Flying Spaghetti Monster
The Flying Spaghetti Monster is the deity of the parody religion the Church of the Flying Spaghetti Monster or Pastafarianism...
to illustrate the logical fallacy of assuming that correlation equals causation.
Determining causation
David HumeDavid Hume
David Hume was a Scottish philosopher, historian, economist, and essayist, known especially for his philosophical empiricism and skepticism. He was one of the most important figures in the history of Western philosophy and the Scottish Enlightenment...
argued that causality is 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
Problem of induction
The problem of induction is the philosophical question of whether inductive reasoning leads to knowledge. That is, what is the justification for either:...
: only correlation can actually be perceived.
Intuitively, causation seems to require not just a correlation, but a counterfactual dependence. Suppose that a student performed poorly on a test and guesses that the cause was his not studying. To prove this, one thinks of the counterfactual – the same student writing the same test under the same circumstances but having studied the night before. If one could rewind history, and change only one small thing (making the student study for the exam), then causation could be observed (by comparing version 1 to version 2). Because one cannot rewind history and replay events after making small controlled changes, causation can only be inferred, never exactly known. This is referred to as the Fundamental Problem of Causal Inference – it is impossible to directly observe causal effects.
A major goal of scientific experiments and statistical methods is to approximate as best as possible the counterfactual state of the world. For example, one could run an experiment on identical twins
Twin study
Twin studies help disentangle the relative importance of environmental and genetic influences on individual traits and behaviors. Twin research is considered a key tool in behavioral genetics and related fields...
who were known to consistently get the same grades on their tests. One twin is sent to study for six hours while the other is sent to the amusement park. If their test scores suddenly diverged by a large degree, this would be strong evidence that studying (or going to the amusement park) had a causal effect on test scores. In this case, correlation between studying and test scores would almost certainly imply causation.
Well-designed experimental studies
Experiment
An experiment is a methodical procedure carried out with the goal of verifying, falsifying, or establishing the validity of a hypothesis. Experiments vary greatly in their goal and scale, but always rely on repeatable procedure and logical analysis of the results...
replace equality of individuals as in the previous example by equality of groups. This is achieved by randomization of the subjects to two or more groups. Although not a perfect system, the likeliness of being equal in all aspects rises with the number of subjects placed randomly in the treatment/placebo
Placebo
A placebo is a simulated or otherwise medically ineffectual treatment for a disease or other medical condition intended to deceive the recipient...
groups. From the significance of the difference of the effect of the treatment vs. the placebo, one can conclude the likeliness of the treatment having a causal effect on the disease. This likeliness can be quantified in statistical terms by the P-value
P-value
In statistical significance testing, the p-value is the probability of obtaining a test statistic at least as extreme as the one that was actually observed, assuming that the null hypothesis is true. One often "rejects the null hypothesis" when the p-value is less than the significance level α ,...
.
When experimental studies are impossible and only pre-existing data are available, as is usually the case for example in economics, regression analysis
Regression analysis
In statistics, regression analysis includes many techniques for modeling and analyzing several variables, when the focus is on the relationship between a dependent variable and one or more independent variables...
can be used. Factors other than the potential causative variable of interest are controlled for by including them as regressors in addition to the regressor representing the variable of interest. False inferences of causation due to reverse causation (or wrong estimates of the magnitude of causation due the presence of bidirectional causation) can be avoided by using explanators (regressors) that are necessarily exogenous
Exogenous
Exogenous refers to an action or object coming from outside a system. It is the opposite of endogenous, something generated from within the system....
, such as physical explanators like rainfall amount (as a determinant of, say, futures prices), lagged variables whose values were determined before the dependent variable's value was determined, instrumental variables for the explanators (chosen based on their known exogeneity), etc. See Causality#Economics. Spurious correlation due to mutual influence from a third, common causative, variable, is harder to avoid: the model must be specified such that there is a theoretical reason to believe that no such underlying causative variable has been omitted from the model; in particular, underlying time trends of both the dependent variable and the independent (potentially causative) variable must be controlled for by including time as another independent variable.
See also
- Affirming the consequentAffirming the consequentAffirming the consequent, sometimes called converse error, is a formal fallacy, committed by reasoning in the form:#If P, then Q.#Q.#Therefore, P....
- CausalityCausalityCausality is the relationship between an event and a second event , where the second event is understood as a consequence of the first....
- Chain reactionChain reactionA chain reaction is a sequence of reactions where a reactive product or by-product causes additional reactions to take place. In a chain reaction, positive feedback leads to a self-amplifying chain of events....
- Confirmation biasConfirmation biasConfirmation bias is a tendency for people to favor information that confirms their preconceptions or hypotheses regardless of whether the information is true.David Perkins, a geneticist, coined the term "myside bias" referring to a preference for "my" side of an issue...
- ConfoundingConfoundingIn statistics, a confounding variable is an extraneous variable in a statistical model that correlates with both the dependent variable and the independent variable...
- Design of experimentsDesign of experimentsIn general usage, design of experiments or experimental design is the design of any information-gathering exercises where variation is present, whether under the full control of the experimenter or not. However, in statistics, these terms are usually used for controlled experiments...
- Domino effectDomino effectThe domino effect is a chain reaction that occurs when a small change causes a similar change nearby, which then will cause another similar change, and so on in linear sequence. The term is best known as a mechanical effect, and is used as an analogy to a falling row of dominoes...
- Ecological fallacyEcological fallacyAn ecological fallacy is a logical fallacy in the interpretation of statistical data in an ecological study, whereby inferences about the nature of specific individuals are based solely upon aggregate statistics collected for the group to which those individuals belong...
- Mierscheid LawMierscheid LawThe Mierscheid law is hypothesis, published in the German magazine Vorwärts on 14 July 1983 and attributed to the fictitious politician Jakob Maria Mierscheid. It forecasts the Social Democratic Party of Germany 's share of the popular vote based on the size of crude steel production in Western...
- Normally distributed and uncorrelated does not imply independentNormally distributed and uncorrelated does not imply independentIn probability theory, two random variables being uncorrelated does not imply their independence. In some contexts, uncorrelatedness implies at least pairwise independence ....
- Observational studyObservational studyIn epidemiology and statistics, an observational study draws inferences about the possible effect of a treatment on subjects, where the assignment of subjects into a treated group versus a control group is outside the control of the investigator...
- Post hoc ergo propter hocPost hoc ergo propter hocPost hoc ergo propter hoc, Latin for "after this, therefore because of this," is a logical fallacy that states, "Since that event followed this one, that event must have been caused by this one." It is often shortened to simply post hoc and is also sometimes referred to as false cause,...
(coincidental correlation) - Spurious relationshipSpurious relationshipIn statistics, a spurious relationship is a mathematical relationship in which two events or variables have no direct causal connection, yet it may be wrongly inferred that they do, due to either coincidence or the presence of a certain third, unseen factor In statistics, a spurious relationship...
- Third-cause fallacyThird-cause fallacyThe third cause fallacy is a logical fallacy that 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....
External links
- "The Art and Science of cause and effect": a slide show and tutorial lecture by Judea Pearl
- Causal inference in statistics: An overview, by Judea Pearl (September 2009)