Correlation and Causation

Introduction

News stories often say "people who do X are more likely to Y." This describes a correlation, a pattern in which two variables change together. It is a start, but it does not by itself show that X causes Y.

The correlation coefficient

The correlation coefficient r ranges from -1 to +1.

• +1: a perfect rising straight-line relationship.
• 0: no linear relationship.
• -1: a perfect falling straight-line relationship.

Values like 0.2 are weak, 0.5 moderate and 0.8 strong, though what counts as strong depends on the field.

Example: hours studied (x) and test score out of 10 (y) for six students:

The means are 3.5 and 4.83. Calculating gives r ≈ 0.92, a strong positive correlation. Spreadsheets have a CORREL function.

Line of best fit

A regression line is the straight line that best matches the data, usually by minimising the squared vertical distances. It lets you predict y from x, for instance the expected score for a student who studied 7 hours, but only within the range of data, and with uncertainty. Predicting far outside the data (extrapolation) is risky.

Correlation is not causation

If A and B are correlated, there are several possible explanations:

1. A causes B.
2. B causes A (reverse causation).
3. A third factor causes both (a confounder). Ice cream sales and drowning rise together in summer. Hot weather drives both, and ice cream does not cause drowning.
4. Coincidence. With many variables, some will correlate by chance.
5. Selection or measurement effects.

Other cautions:

• Linear only: r measures straight-line patterns. A U-shaped relationship can have r near 0.
• Outliers can change r a lot.
• Aggregated data can hide different patterns in subgroups (Simpson's paradox).
• Small samples give unreliable estimates.

How we establish cause

• Randomised controlled trials: randomly assign people to groups, so other factors balance out on average.
• Natural experiments and careful observational studies with adjustments.
• Mechanisms: a plausible explanation of how.
• Consistency across many studies and settings.
• Timing: causes must come before effects.

Observational studies can suggest hypotheses, but are weaker evidence for cause than well-designed experiments.

Reading claims

Ask:

• What is the sample, and how large?
• Is it observational or experimental?
• What other factors might explain it?
• How big is the effect, not just whether it exists?
• Is the result replicated?
• Who funded it, and what is the headline leaving out?

Try it yourself

Find a news claim about a link between two things. List two alternative explanations other than direct cause. Use a spreadsheet to calculate CORREL for a small dataset of your own.

Summary

• The correlation coefficient r measures the strength and direction of a linear relationship, from -1 to +1.
• A line of best fit supports prediction within the range of the data.
• Correlation does not prove cause: consider reverse causation, confounders, chance and study design.