Correlation and Causation

We must be very careful in interpreting correlation coefficients. Just because two variables are highly correlated does not mean that one causes the other. In statistical terms, we say that correlation does not imply causation. There are many good examples of correlation which are nonsensical when interpreted in terms of causation.

• Ice cream sales and the number of shark attacks on swimmers are correlated.
• Skirt lengths and stock prices are highly correlated (as stock prices go up, skirt lengths get shorter).
• The number of cavities in elementary school children and vocabulary size have a strong positive correlation.

Three relationships which can be taken (or mistaken) for causation are:

1. Causation: Changes in X cause changes in Y. For example, football weekends cause heavier traffic, more food sales, etc.
2. Common response: Both X and Y respond to changes in some unobserved variable. All three of our examples are examples of common response.
   • Ice cream sales and shark attacks both increase during summer.
   • Skirt lengths and stock prices are both controlled by the general attitude of the country, liberal or conservative.
   • The number of cavities and children's vocabulary are both related to a child's age.
3. Confounding: The effect of X on Y is hopelessly mixed up with the effects of other explanatory variables on y. For example, if we are studying the effects of Tylenol on reducing pain, and we give a group of pain-sufferers Tylenol and record how much their pain is reduced, we are confounding the effect of giving them Tylenol with giving them any pill. Many people report a reduction in pain by simply being given a sugar pill with no medication in it at all, this is called the placebo effect. To establish causation, a designed experiment must be run.