Correlation and Causation
What are correlation and causation and how are they different?
Two or more variables considered to be related, in a statistical context, if their values change so that as the value of one variable increases or decreases so does the value of the other variable (although it may be in the opposite direction).
For example, for the two variables "hours worked" and "income earned" there is a relationship between the two if the increase in hours worked is associated with an increase in income earned. If we consider the two variables "price" and "purchasing power", as the price of goods increases a person's ability to buy these goods decreases (assuming a constant income).
Correlation is a statistical measure (expressed as a number) that describes the size and direction of a relationship between two or more variables. A correlation between variables, however, does not automatically mean that the change in one variable is the cause of the change in the values of the other variable.
Causation indicates that one event is the result of the occurrence of the other event; i.e. there is a causal relationship between the two events. This is also referred to as cause and effect.
Theoretically, the difference between the two types of relationships are easy to identify — an action or occurrence can cause another (e.g. smoking causes an increase in the risk of developing lung cancer), or it can correlate with another (e.g. smoking is correlated with alcoholism, but it does not cause alcoholism). In practice, however, it remains difficult to clearly establish cause and effect, compared with establishing correlation.
Why are correlation and causation important?
The objective of much research or scientific analysis is to identify the extent to which one variable relates to another variable. For example:
- Is there a relationship between a person's education level and their health?
- Is pet ownership associated with living longer?
- Did a company's marketing campaign increase their product sales?
These and other questions are exploring whether a correlation exists between the two variables, and if there is a correlation then this may guide further research into investigating whether one action causes the other. By understanding correlation and causality, it allows for policies and programs that aim to bring about a desired outcome to be better targeted.
How is correlation measured?
For two variables, a statistical correlation is measured by the use of a Correlation Coefficient, represented by the symbol (r), which is a single number that describes the degree of relationship between two variables.
The coefficient's numerical value ranges from +1.0 to –1.0, which provides an indication of the strength and direction of the relationship.
If the correlation coefficient has a negative value (below 0) it indicates a negative relationship between the variables. This means that the variables move in opposite directions (ie when one increases the other decreases, or when one decreases the other increases).
If the correlation coefficient has a positive value (above 0) it indicates a positive relationship between the variables meaning that both variables move in tandem, i.e. as one variable decreases the other also decreases, or when one variable increases the other also increases.
Where the correlation coefficient is 0 this indicates there is no relationship between the variables (one variable can remain constant while the other increases or decreases).
While the correlation coefficient is a useful measure, it has its limitations:
Correlation coefficients are usually associated with measuring a linear relationship.
For example, if you compare hours worked and income earned for a tradesperson who charges an hourly rate for their work, there is a linear (or straight line) relationship since with each additional hour worked the income will increase by a consistent amount.
If, however, the tradesperson charges based on an initial call out fee and an hourly fee which progressively decreases the longer the job goes for, the relationship between hours worked and income would be non-linear, where the correlation coefficient may be closer to 0.
Care is needed when interpreting the value of 'r'. It is possible to find correlations between many variables, however the relationships can be due to other factors and have nothing to do with the two variables being considered.
For example, sales of ice creams and the sales of sunscreen can increase and decrease across a year in a systematic manner, but it would be a relationship that would be due to the effects of the season (ie hotter weather sees an increase in people wearing sunscreen as well as eating ice cream) rather than due to any direct relationship between sales of sunscreen and ice cream.
The correlation coefficient should not be used to say anything about cause and effect relationship. By examining the value of 'r', we may conclude that two variables are related, but that 'r' value does not tell us if one variable was the cause of the change in the other.
How can causation be established?
Causality is the area of statistics that is commonly misunderstood and misused by people in the mistaken belief that because the data shows a correlation that there is necessarily an underlying causal relationship
The use of a controlled study is the most effective way of establishing causality between variables. In a controlled study, the sample or population is split in two, with both groups being comparable in almost every way. The two groups then receive different treatments, and the outcomes of each group are assessed.
For example, in medical research, one group may receive a placebo while the other group is given a new type of medication. If the two groups have noticeably different outcomes, the different experiences may have caused the different outcomes.
Due to ethical reasons, there are limits to the use of controlled studies; it would not be appropriate to use two comparable groups and have one of them undergo a harmful activity while the other does not. To overcome this situation, observational studies are often used to investigate correlation and causation for the population of interest. The studies can look at the groups' behaviours and outcomes and observe any changes over time.
The objective of these studies is to provide statistical information to add to the other sources of information that would be required for the process of establishing whether or not causality exists between two variables.
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Simpson’s paradox, also called Yule-Simpson effect, in statistics, an effect that occurs when the marginal association between two categorical variables is qualitatively different from the partial association between the same two variables after controlling for one or more other variables. Simpson’s paradox is important for three critical reasons. First, people often expect statistical relationships to be immutable. They often are not. The relationship between two variables might increase, decrease, or even change direction depending on the set of variables being controlled. Second, Simpson’s paradox is not simply an obscure phenomenon of interest only to a small group of statisticians. Simpson’s paradox is actually one of a large class of association paradoxes. Third, Simpson’s paradox reminds researchers that causal inferences, particularly in nonexperimental studies, can be hazardous. Uncontrolled and even unobserved variables that would eliminate or reverse the association observed between two variables might exist.
Understanding Simpson’s paradox is easiest in the context of a simple example. Suppose that a university is concerned about sex bias during the admission process to graduate school. To study this, applicants to the university’s graduate programs are classified based on sex and admissions outcome. These data would seem to be consistent with the existence of a sex bias because men (40 percent were admitted) were more likely to be admitted to graduate school than women (25 percent were admitted).
To identify the source of the difference in admission rates for men and women, the university subdivides applicants based on whether they applied to a department in the natural sciences or to one in the social sciences and then conducts the analysis again. Surprisingly, the university finds that the direction of the relationship between sex and outcome has reversed. In natural science departments, women (80 percent were admitted) were more likely to be admitted to graduate school than men (46 percent were admitted); similarly, in social science departments, women (20 percent were admitted) were more likely to be admitted to graduate school than men (4 percent were admitted).
Although the reversal in association that is observed in Simpson’s paradox might seem bewildering, it is actually straightforward. In this example, it occurred because both sex and admissions were related to a third variable, namely, the department. First, women were more likely to apply to social science departments, whereas men were more likely to apply to natural science departments. Second, the acceptance rate in social science departments was much less than that in natural science departments. Because women were more likely than men to apply to programs with low acceptance rates, when department was ignored (i.e., when the data were aggregated over the entire university), it seemed that women were less likely than men to be admitted to graduate school, whereas the reverse was actually true. Although hypothetical examples such as this one are simple to construct, numerous real-life examples can be found easily in the social science and statistics literatures.
A-B-C, 1-2-3… If you consider that counting numbers is like reciting the alphabet, test how fluent you are in the language of mathematics in this quiz.
Consider three random variables X, Y, and Z. Define a 2 × 2 × K cross-classification table by assuming that X and Y can be coded either 0 or 1, and Z can be assigned values from 1 to K.
The marginal association between X and Y is assessed by collapsing across or aggregating over the levels of Z. The partial association between X and Y controlling for Z is the association between X and Y at each level of Z or after adjusting for the levels of Z. Simpson’s paradox is said to have occurred when the pattern of marginal association and the pattern of partial association differ.
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Various indices exist for assessing the association between two variables. For categorical variables, the odds ratio and the relative risk ratio are the two most common measures of association. Simpson’s paradox is the name applied to differences in the association between two categorical variables, regardless of how that association is measured.
Association paradoxes, of which Simpson’s paradox is a special case, can occur between continuous (a variable that can take any value) or categorical variables (a variable that can take only certain values). For example, the best-known measure of association between two continuous variables is the correlation coefficient. It is well known that the marginal correlation between two variables can have one sign, whereas the partial correlation between the same two variables after controlling for one or more additional variables has the opposite sign.
Two Oregon settlers flipped a coin to decide whose hometown would be used to name their village. Had the man from Portland, Maine, not won, Oregon’s biggest city would now be named Boston.
See All Good FactsReversal paradoxes, in which the marginal and partial associations between two variables have different signs, such as Simpson’s paradox, are the most dramatic of the association paradoxes. A weaker form of association paradox occurs when the marginal and partial associations have the same sign, but the magnitude of the marginal association falls outside of the range of values of the partial associations computed at individual levels of the variable(s) being controlled. These have been termed amalgamation or aggregation paradoxes.