A classic example of a spurious correlation is the observed relationship between ice cream sales and crime rates. While data shows both increase concurrently, one does not cause the other. The association is more likely explained by a third, confounding variable: temperature. During warm weather, more people are outside, leading to increased social interactions and potential for crime. Simultaneously, warm weather drives up the demand for cool treats like ice cream. Therefore, the temperature independently influences both variables, creating a statistical correlation without a direct causal link.

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Correlation only indicates that usually two variable are related to each other, which might be due to a causal, neural, or reverse-causal relation. Chances are that two correlated variables are both caused by a third variable (confounder).

Correlation Does Not Imply Causation

A confounding variable is an external factor that influences the variables being studied, creating a misleading association between them. This variable can cause the systematic movement in both variables of interest, making it appear as though they have a direct correlational relationship when, in fact, the confounder is the true underlying cause of the observed connection.

Confounding Variables

Pearl, J., & Mackenzie, D. (2018). $\mathit{The \ Book \ of \ Why: \ The \ New \ Science \ of \ Cause \ and \ Effect.}$ Basic Books.

The Book of Why

Spielman, R. M., Jenkins, W. J., & Lovett, M. D. (2020, April 22). Psychology (2nd ed.). OpenStax. https://openstax.org/books/psychology-2e/pages/1-introduction

OpenStax Psychology (2nd ed.) Textbook

Event A directly causes event B, without any hidden contributing factors driving them both.

- A is the cause, B is the effect. 
- A correlation coefficient of 1 usually accompanies a causal relationship because it means that A occuring positively influences/ensures that B occurs. 

Causal Relationship

Events A and B have no impact on the occurence of the other. They do not cause the other.

Neutral Relationship

Instead of event A causing event B, event B causes event A. Sometimes called the 'cart before the horse bias'. 
Eg : If there is a patient at risk for a heart disease that reduces their intake of fatty/unhealthy foods, but then still suffers a heart attack. Their healthier lifestyle cannot be held as a cause for their heart attack, because the reason they shifted to a healthier lifestyle was because they were at high risk for a heart attack.

Reverse Causal relationship

When two heterogeneous data sets are aggregated into one, false correlations between variables can arise due to the differences in populations. 

For example, the correlation between skull length and breadth is statistically significant when measured on all humans, but when separated between male and female, there was no significant correlation.

Spurious Correlation: Aggregated Data

Inheritance of fur color in guinea pigs does not follow Mendelian laws. Sewell Wright postulated that this was because of 'developmental factors' (*d* ). He tried to find *d* by solving mathematical equations where *d* and hereditary factors *h* were the unknowns.

Wright showed that knowing *d* and *h* helped predict correlations, and drew a path diagram that allowed him to identify the causes after knowing correlations.

Reference : Pearl, J., & Mackenzie, D. (2019). The Book of Why. Penguin

S Wright's Guinea Pigs and the "First link between Causality and Probability"

Regression towards the mean is a phenomenon that occurs when resampling a single random variable. 

The phenomenon arises when the first sample of the random variable appears extreme, while the second sample appears much closer to the mean. This result occurs when the first sample is extremely above and below the mean.

Regression to the Mean

Proposed by Hans Reichenbach, the common cause principal argues against correlation doesn’t imply causation. It states that if there are two variables I and J and they are correlated, then I causes J or J causes I or there is another variable X that precedes and causes both I and J. 

This theory is not true because it doesn't account for how the data is selected and appeals to our human nature to look for casual explanations whenever there is a pattern. This is an example of collider bias.

Common Cause Principal 

Probability is a concept that is largely tied more toward association than it is causation as it in itself does not address the second and third rungs of the ladder of causation (Intervention and Counterfactual Reasoning, respectively). Therefore, to say that "X raises the probability of Y" is faulty as suggesting that it "raises" the probability implies causation, which is not an aspect of probabilistic expressions. Therefore, the proper way to discuss probability according to Judea Pearl would be to state that "If we see X, then the probability of Y increases." This delineates the association without jumping to causation and leaves the causal mechanism open to either interpretation or further research. 

Irreducibility of Causation to Probabilities 

Example of Spurious Correlation: Ice Cream Sales and Crime Rates

Claims of causation based on correlational data are frequently and mistakenly made, particularly in advertisements and news stories. These sources may present a statistical relationship in a way that implies a cause-and-effect link, often to promote a product or a particular narrative. This is sometimes done by maximizing and perhaps overstating the positive aspects of the findings, without acknowledging other possible explanations for the observed correlation.

Misinterpretation of Correlation as Causation in Media

There are two general approaches to removing confounds: 
• Randomize participants between conditions of the experiment 
• Experimentally control for potential confounds

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