Reversion to the Mean

Galton's Regression

• Problem set up: Like father like son?
  • Galton collected data on the heights of parantes and their adult children.
    • Made a scatter plot
    • Found the line of best fit.
  • He expected that the line would be a 45 degree line:
    • Intercept should be 0
    • Slope should be 1
  • That seems reasonable at first thought because it implies that our best guess about the height of a child would be their parents' height.
• However, the line of best fit is not a 45 degree line.
  • The slope was less than 1 (but positive)
  • Explained:
    • Tall parents tend to have children that are taller than average but shorter than them.
    • Short parents tend to have children that are shorter than average but taller than them.
  • Galton referred to this phenomenon as regression to mediocrity.
• Why regression to mediocrity?
  • Key: a person's height is determined by multiple things.
  • For simplified analysis, suppose that height is only influenced by:
    • Genes from parents
    • The temperature of the day they were born
  • Now, Height = Genes + Temperature. This equation implies that gene effect and temperature effect are independent of each other.
  • If we find someone extremely high, the only way they can be so is that they have high genes from their parents and were born on a super hot day.
    • However, their children are more likely to be born on a day with average temperature, so they will be shorter than their parents.
    • But since their children still inherit the high genes, they will be taller than average.
• Signal + Noise = Reversion to Mean:
  • Any outcome that is partly a function of Signal (gene) and noise (temperature).
  • Extreme observations probably arise from both extreme signals and extreme noise. ${\color{red}{\text{Estimate}}}= {\color{blue}{\text{Estimand}}} + {\color{purple}{\text{Bias}}} +{\color{green}{\text{Noise}}}$
    • We will ignore Bias for now.
    • In our setting, the estimate is the height, and the signal is the genes, which is the estimand. The noise is just the temperature. So, ${\color{red}{\text{Height}}}={\color{blue}{\text{Genes}}}+{\color{green}{\text{Temperature}}}$

Reversion to Mean Examples

• Golf: A player who scored higher in round 1 is more likely to score higher or lower in round 2?
  • Those players are likely to score lower in round 2, but still higher than average.
  • This is because their score = skills + luck.
• Medicine: Does vitamin C really help us get recovered from cold?
  • No!
  • Even though we feel better after taking vitamin C, it is just because we are getting better anyway.
• Where should we not expect reversion to mean?