More About Correlation

Ecological Fallacy

An ecological fallacy is an interpretation of statistical data where inferences about individuals are deduced from inferences about the group as a whole.

Learning Objectives

Discuss ecological fallacy in terms of aggregate versus individual inference and give specific examples of its occurrence.

Key Takeaways

Key Points

  • Ecological fallacy can refer to the following fallacy: the average for a group is approximated by the average in the total population divided by the group size.
  • A striking ecological fallacy is Simpson’s paradox.
  • Another example of ecological fallacy is when the average of a population is assumed to have an interpretation in terms of likelihood at the individual level.
  • Aggregate regressions lose individual level data but individual regressions add strong modeling assumptions.

Key Terms

  • Simpson’s paradox: That the association of two variables for one subset of a population may be similar to the association of those variables in another subset, but different from the association of the variables in the total population.
  • ecological correlation: A correlation between two variables that are group parameters, in contrast to a correlation between two variables that describe individuals.

Confusion Between Groups and Individuals

Ecological fallacy can refer to the following statistical fallacy: the correlation between individual variables is deduced from the correlation of the variables collected for the group to which those individuals belong. As an example, assume that at the individual level, being Protestant impacts negatively one’s tendency to commit suicide, but the probability that one’s neighbor commits suicide increases one’s tendency to become Protestant. Then, even if at the individual level there is negative correlation between suicidal tendencies and Protestantism, there can be a positive correlation at the aggregate level.

Choosing Between Aggregate and Individual Inference

Running regressions on aggregate data is not unacceptable if one is interested in the aggregate model. For instance, as a governor, it is correct to make inferences about the effect the size of a police force would have on the crime rate at the state level, if one is interested in the policy implication of a rise in police force. However, an ecological fallacy would happen if a city council deduces the impact of an increase in the police force on the crime rate at the city level from the correlation at the state level.

Choosing to run aggregate or individual regressions to understand aggregate impacts on some policy depends on the following trade off: aggregate regressions lose individual level data but individual regressions add strong modeling assumptions.

Some researchers suggest that the ecological correlation gives a better picture of the outcome of public policy actions, thus they recommend the ecological correlation over the individual level correlation for this purpose. Other researchers disagree, especially when the relationships among the levels are not clearly modeled. To prevent ecological fallacy, researchers with no individual data can model first what is occurring at the individual level, then model how the individual and group levels are related, and finally examine whether anything occurring at the group level adds to the understanding of the relationship.

Groups and Total Averages

Ecological fallacy can also refer to the following fallacy: the average for a group is approximated by the average in the total population divided by the group size. Suppose one knows the number of Protestants and the suicide rate in the USA, but one does not have data linking religion and suicide at the individual level. If one is interested in the suicide rate of Protestants, it is a mistake to estimate it by the total suicide rate divided by the number of Protestants.

Simpson’s Paradox

A striking ecological fallacy is Simpson’s paradox, diagramed in. Simpson’s paradox refers to the fact, when comparing two populations divided in groups of different sizes, the average of some variable in the first population can be higher in every group and yet lower in the total population.

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Simpson’s Paradox: Simpson’s paradox for continuous data: a positive trend appears for two separate groups (blue and red), a negative trend (black, dashed) appears when the data are combined.

Mean and Median

A third example of ecological fallacy is when the average of a population is assumed to have an interpretation in terms of likelihood at the individual level.

For instance, if the average score of group A is larger than zero, it does not mean that a random individual of group A is more likely to have a positive score. Similarly, if a particular group of people is measured to have a lower average IQ than the general population, it is an error to conclude that a randomly selected member of the group is more likely to have a lower IQ than the average general population. Mathematically, this comes from the fact that a distribution can have a positive mean but a negative median. This property is linked to the skewness of the distribution.

Consider the following numerical example:

Group A: 80% of people got 40 points and 20% of them got 95 points. The average score is 51 points.

Group B: 50% of people got 45 points and 50% got 55 points. The average score is 50 points.

If we pick two people at random from A and B, there are 4 possible outcomes:

  • A – 40, B – 45 (B wins, 40% probability)
  • A – 40, B – 55 (B wins, 40% probability)
  • A – 95, B – 45 (A wins, 10% probability)
  • A – 95, B – 55 (A wins, 10% probability)

Although Group A has a higher average score, 80% of the time a random individual of A will score lower than a random individual of B.

Correlation is Not Causation

The conventional dictum “correlation does not imply causation” means that correlation cannot be used to infer a causal relationship between variables.

Learning Objectives

Recognize that although correlation can indicate the existence of a causal relationship, it is not a sufficient condition to definitively establish such a relationship

Key Takeaways

Key Points

  • The assumption that correlation proves causation is considered a questionable cause logical fallacy, in that two events occurring together are taken to have a cause-and-effect relationship.
  • 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 cum hoc ergo propter hoc logical fallacy, one makes a premature conclusion about causality after observing only a correlation between two or more factors.

Key Terms

  • Granger causality test: A statistical hypothesis test for determining whether one time series is useful in forecasting another.
  • tautology: A statement that is true for all values of its variables.
  • convergent cross mapping: A statistical test that (like the Granger Causality test) tests whether one variable predicts another; unlike most other tests that establish a coefficient of correlation, but not a cause-and-effect relationship.

The conventional dictum that “correlation does not imply causation” means that correlation cannot be used to infer a causal relationship between the variables. This dictum does not imply that correlations cannot indicate the potential existence of causal relations. However, the causes underlying the correlation, if any, may be indirect and unknown, and high correlations also overlap with identity relations (tautology) where no causal process exists. Consequently, establishing a correlation between two variables is not a sufficient condition to establish a causal relationship (in either direction). Many statistical tests calculate correlation between variables. A few go further and calculate the likelihood of a true causal relationship. Examples include the Granger causality test and convergent cross mapping.

The assumption that correlation proves causation is considered a “questionable cause logical fallacy,” in that two events occurring together are taken to have 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. ” Consider the following:

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 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.

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 caused a decrease in coronary heart disease, but not to the degree suggested by the epidemiological studies, the assumption of causality would have been correct, although the logic behind the assumption would still have been flawed.

General Pattern

For any two correlated events A and B, the following relationships are possible:

  • A causes B;
  • B causes A;
  • A and B are consequences of a common cause, but do not cause each other;
  • There is no connection between A and B; the correlation is coincidental.

Less clear-cut correlations are also possible. For example, causality is not necessarily one-way; in a predator-prey relationship, predator numbers affect prey, but prey numbers (e.g., food supply) also affect predators.

The cum hoc ergo propter hoc logical fallacy can be expressed as follows:

  1. A occurs in correlation with B.
  2. Therefore, A causes B.

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:

  1. A may be the cause of B.
  2. B may be the cause of A.
  3. Some unknown third factor C may actually be the cause of both A and B.
  4. 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.
  5. The “relationship” is a coincidence 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 size helps to reduce the chance of a coincidence, unless there is a systematic error 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, a large effect size is observed, or a large part of the variance is explained.

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Greenhouse Effect: The greenhouse effect is a well-known cause-and-effect relationship. While well-established, this relationship is still susceptible to logical fallacy due to the complexity of the system.