An ecological fallacy (also ecological ''inference'' fallacy or population fallacy) is a
formal fallacy in the interpretation of
statistic
A statistic (singular) or sample statistic is any quantity computed from values in a sample which is considered for a statistical purpose. Statistical purposes include estimating a population parameter, describing a sample, or evaluating a hypo ...
al data that occurs when
inferences about the nature of individuals are deduced from inferences about the group to which those individuals belong. "Ecological fallacy" is a term that is sometimes used to describe the
fallacy of division, which is not a statistical fallacy. The four common statistical ecological fallacies are: confusion between ecological correlations and individual correlations, confusion between group average and total average,
Simpson's paradox, and confusion between higher average and higher likelihood.
Examples
Mean and median
An example of ecological fallacy is the assumption that a population mean has a simple interpretation when considering likelihoods for an individual.
For instance, if the mean score of a group is larger than zero, this does not imply that a random individual of that group is more likely to have a positive score than a negative one (as long as there are more negative scores than positive scores an individual is more likely to have a negative score). Similarly, if a particular group of people is measured to have a lower mean IQ than the general population, it is an error to conclude that a randomly-selected member of the group is more likely than not to have a lower IQ than the mean IQ of the general population; it is also not necessarily the case that a randomly selected member of the group is more likely than not to have a lower IQ than a randomly-selected member of the 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 mean score is 51 points.
* Group B: 50% of people got 45 points and 50% got 55 points. The mean 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 – 0.8 × 0.5)
** A – 40, B – 55 (B wins, 40% probability – 0.8 × 0.5)
** A – 95, B – 45 (A wins, 10% probability – 0.2 × 0.5)
** A – 95, B – 55 (A wins, 10% probability – 0.2 × 0.5)
* Although Group A has a higher mean score, 80% of the time a random individual of A will score lower than a random individual of B.
Individual and aggregate correlations
Research dating back to
Émile Durkheim
David Émile Durkheim ( or ; 15 April 1858 – 15 November 1917) was a French sociologist. Durkheim formally established the academic discipline of sociology and is commonly cited as one of the principal architects of modern social science, al ...
suggests that predominantly Protestant localities have higher suicide rates than predominantly Catholic localities. According to Freedman,
[Freedman, D. A. (1999). Ecological Inference and the Ecological Fallacy. ''International Encyclopedia of the Social & Behavioral Sciences'', Technical Report No. 549. https://web.stanford.edu/class/ed260/freedman549.pdf] the idea that Durkheim's findings link, at an individual level, a person's religion to his or her suicide risk is an example of the ecological fallacy. A group-level relationship does not automatically characterize the relationship at the level of the individual.
Similarly, even if at the individual level, wealth is positively correlated to tendency to vote Republican, we observe that wealthier states tend to vote Democratic. For example, in 2004, the Republican candidate,
George W. Bush, won the fifteen poorest states, and the Democratic candidate,
John Kerry
John Forbes Kerry (born December 11, 1943) is an American attorney, politician and diplomat who currently serves as the first United States special presidential envoy for climate. A member of the Forbes family and the Democratic Party, he ...
, won 9 of the 11 wealthiest states. Yet 62% of voters with annual incomes over $200,000 voted for Bush, but only 36% of voters with annual incomes of $15,000 or less voted for Bush.
Aggregate-level correlation will differ from individual-level correlation if voting preferences are affected by the total wealth of the state even after controlling for individual wealth. It could be that the true driving factor in voting preference is self-perceived relative wealth; perhaps those who see themselves as better off than their neighbours are more likely to vote Republican. In this case, an individual would be more likely to vote Republican if she became wealthier, but she would be more likely to vote for a Democrat if her neighbor's wealth increased (resulting in a wealthier state).
However, the observed difference in voting habits based on state-level and individual-level wealth could also be explained by the common confusion between higher averages and higher likelihoods as discussed above. States may not be wealthier because they contain more wealthy people (i.e. more people with annual incomes over $200,000), but rather because they contain a small number of super-rich individuals; the ecological fallacy then results from incorrectly assuming that individuals in wealthier states are more likely to be wealthy.
Many examples of ecological fallacies can be found in studies of social networks, which often combine analysis and implications from different levels. This has been illustrated in an academic paper on networks of farmers in Sumatra.
Robinson's paradox
A 1950 paper by William S. Robinson computed the illiteracy rate and the proportion of the population born outside the US for each state and for the District of Columbia, as of the
1930 census
The United States census of 1930, conducted by the Census Bureau one month from April 1, 1930, determined the resident population of the United States to be 122,775,046, an increase of 13.7 percent over the 106,021,537 persons enumerated during ...
. He showed that these two figures were associated with a negative correlation of −0.53; in other words, the greater the proportion of immigrants in a state, the lower its average illiteracy (or, equivalently, the higher its average literacy). However, when individuals are considered, the correlation between illiteracy and nativity was +0.12 (immigrants were on average more illiterate than native citizens). Robinson showed that the negative correlation at the level of state populations was because immigrants tended to settle in states where the native population was more literate. He cautioned against deducing conclusions about individuals on the basis of population-level, or "ecological" data. In 2011, it was found that Robinson's calculations of the ecological correlations are based on the wrong state level data. The correlation of −0.53 mentioned above is in fact −0.46. Robinson's paper was seminal, but the term 'ecological fallacy' was not coined until 1958 by Selvin.
Formal problem
The correlation of aggregate quantities (or
ecological correlation
In statistics, an ecological correlation (also ''spatial correlation'') is a correlation between two variables that are group means, in contrast to a correlation between two variables that describe individuals. For example, one might study the corr ...
) is not equal to the correlation of individual quantities. Denote by ''X''
''i'', ''Y''
''i'' two quantities at the individual level. The formula for the covariance of the aggregate quantities in groups of size ''N'' is
:
The covariance of two aggregated variables depends not only on the covariance of two variables within the same individuals but also on covariances of the variables between different individuals. In other words, correlation of aggregate variables take into account cross sectional effects which are not relevant at the individual level.
The problem for correlations entails naturally a problem for regressions on aggregate variables: the correlation fallacy is therefore an important issue for a researcher who wants to measure causal impacts. Start with a regression model where the outcome
is impacted by
:
:
The regression model at the aggregate level is obtained by summing the individual equations:
:
:
Nothing prevents the regressors and the errors from being correlated at the aggregate level. Therefore, generally, running a regression on aggregate data does not estimate the same model than running a regression with individual data.
The aggregate model is correct if and only if
:
This means that, controlling for
,
does not determine
.
Choosing between aggregate and individual inference
There is nothing wrong in running regressions on aggregate data if one is interested in the aggregate model. For instance, for the governor of a state, it is correct to run regressions between police force on 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 police force in 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 (Lubinski & Humphreys, 1996). 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. For instance, in evaluating the impact of state policies, it is helpful to know that policy impacts vary less among the states than do the policies themselves, suggesting that the policy differences are not well translated into results, despite high ecological correlations (Rose, 1973).
Group 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.
Formally, denote