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Correlation and dependence

correlationcorrelatedcorrelate
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.
However, in general, the presence of a correlation is not sufficient to infer the presence of a causal relationship (i.e., correlation does not imply causation).

Causality

causalcausationcause and effect
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. 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.
A mere observation of a correlation is not nearly adequate to establish causality.

Post hoc ergo propter hoc

post hoc, ergo propter hocpost hocpost hoc fallacy
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.
A logical fallacy of the questionable cause variety, it is subtly different from the fallacy cum hoc ergo propter hoc ("with this, therefore because of this"), in which two events occur simultaneously or the chronological ordering is insignificant or unknown.

Questionable cause

false causefallacy of false causenon causa pro causa
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.
Circular cause and consequence

Convergent cross mapping

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.
Convergent cross mapping (CCM) is a statistical test for a cause-and-effect relationship between two time series variables that, like the Granger causality test, seeks to resolve the problem that correlation does not imply causation.

Statistics

statisticalstatistical analysisstatistician
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.
(See Correlation does not imply causation.)

Epidemiology

epidemiologistepidemiologicalepidemiologists
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.
"Correlation does not imply causation" is a common theme for much of the epidemiological literature.

Statistical hypothesis testing

hypothesis testingstatistical teststatistical tests
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.
Layers of philosophical concerns. The probability of statistical significance is a function of decisions made by experimenters/analysts. If the decisions are based on convention they are termed arbitrary or mindless while those not so based may be termed subjective. To minimize type II errors, large samples are recommended. In psychology practically all null hypotheses are claimed to be false for sufficiently large samples so "...it is usually nonsensical to perform an experiment with the sole aim of rejecting the null hypothesis.". "Statistically significant findings are often misleading" in psychology. Statistical significance does not imply practical significance and correlation does not imply causation. Casting doubt on the null hypothesis is thus far from directly supporting the research hypothesis.

Coefficient of determination

R 2 R'' 2 explained
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.
A caution that applies to R 2, as to other statistical descriptions of correlation and association is that "correlation does not imply causation."

Coincidence

coincidencescoincidentalCoincidentally
There is no connection between A and B; the correlation is a coincidence.
To establish cause and effect (i.e., causality) is notoriously difficult, as is expressed by the commonly heard statement that "correlation does not imply causation."

Spurious relationship

spurious correlationspuriousmisleading results
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.
Correlation does not imply causation

Health effects of tobacco

health effectsdangers of smokinghealth risks associated with smoking
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.
Case-control studies were published in Germany in 1939 and 1943, and one in the Netherlands in 1948, but widespread attention was first drawn by five case-control studies published in 1950 by researchers from the US and UK. These studies were widely criticized as showing correlation, not causality.

Near-sightedness

myopiamyopicnearsighted
Young children who sleep with the light on are much more likely to develop myopia in later life.
Numerous studies have found correlations between myopia, on the one hand, and intelligence and academic achievement, on the other; it is not clear whether there is a causal relationship.

Redskins Rule

November 1, 1992November 2, 1952November 2, 1980
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).
Correlation does not imply causation

Mierscheid law

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.
Correlation does not imply causation

Ronald Fisher

FisherR.A. FisherR. A. Fisher
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.
Fisher publicly spoke out against the 1950 study showing that smoking tobacco causes lung cancer, arguing that correlation does not imply causation.

Variable (mathematics)

variablesvariableunknown
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.

Logical consequence

entailsentailmentfollows from
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.

Latin

Lat.Latin languagelat
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.

Hormone replacement therapy

menopausal hormone therapyhormone therapyestrogen replacement therapy
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.

Coronary artery disease

coronary heart diseaseischemic heart diseaseischaemic heart disease
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.

Randomized controlled trial

randomized controlled trialsrandomized clinical trialrandomized control trial
But randomized controlled trials showed that HRT caused a small but statistically significant increase in risk of CHD.

Statistical significance

statistically significantsignificantsignificantly
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. But randomized controlled trials showed that HRT caused a small but statistically significant increase in risk of CHD.

Social class

classsocial classesclasses
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.

NRS social grade

ABC1social gradeA
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.