Correlation and Cause: Quick Revision

A brief note on Correlation and Cause. Read it before a test. Try each question at the end before you open its answer.

Why two variables move together

• Cause: $x$ causes $y$.
• Reverse cause: $y$ causes $x$.
• Lurking (confounding) variable: a third variable drives both.
• Mixed groups: groups that differ in both variables.
• Time or chance: two quantities that both grow over the years.
• Evidence of cause: a randomised experiment, or many agreeing studies with a sensible mechanism.

Simpson's paradox

• What: a comparison holds in every group but reverses when the groups are combined.
• Why: the groups are mixed in different proportions, and their rates differ a lot.
• What to do: compare within groups when the grouping affects the outcome.

Curved patterns

• Log scale: growth by an equal factor gives $\log_{10} y = a + bx$, a straight line.
• Growth factor: $10^{b}$ for each unit of $x$.
• Doubling time: $\dfrac{\log_{10} 2}{b}$.
• Square root: straightens a quantity that grows like a square.
• Predict: on the log scale, then undo with $y = 10^{a + bx}$.

A worked example

Two made-up bus companies record trips on time. Company P: $45$ of $50$ in dry weather, $90$ of $150$ in rain. Company Q: $136$ of $160$ dry, $22$ of $40$ in rain.

1. Dry. P has $90\%$ and Q has $85\%$.
2. Rain. P has $60\%$ and Q has $55\%$.
3. Overall. P has $\dfrac{135}{200} = 67.5\%$ and Q has $\dfrac{158}{200} = 79\%$. P wins in both kinds of weather but loses overall, because most of its trips were in rain.

Practice with new numbers

A program to try

The program finds the rates of the worked example in each kind of weather and overall.

Mistakes to avoid

• Reading cause into a correlation.
• Averaging rates of groups with different sizes.
• Forgetting to undo the logarithm in a prediction.
• Extrapolating growth beyond the data.

Check yourself

1. Towns with more swimming pools have more drownings. Name a lurking variable.

2. $20$ of $25$ members of one group and $15$ of $75$ of another say yes. What is the overall percentage?

3. A count follows $\log_{10} y = 2 + 0.5x$. Predict $y$ at $x = 2$.

4. Why does a randomised experiment protect against lurking variables?

Sources

• OpenStax, Introductory Statistics, Chapters 1, 3 and 12.