Regression to the mean

Introduction to Regression in R

Richie Cotton

Data Evangelist

The concept

Response value = fitted value + residual
"The stuff you explained" + "the stuff you couldn't explain"
Residuals exist due to problems in the model and fundamental randomness
Extreme cases are often due to randomness
Regression to the mean means extreme cases don't persist over time

Pearson's father son dataset

1078 father/son pairs
Do tall fathers have tall sons?

father_height_cm	son_height_cm
165.2	151.8
160.7	160.6
165.0	160.9
167.0	159.5
155.3	163.3
...	...

¹ Adapted from https://www.rdocumentation.org/packages/UsingR/topics/father.son

Scatter plot

plt_son_vs_father <- ggplot(
  father_son, 
  aes(father_height_cm, son_height_cm)
) +
  geom_point() +
  geom_abline(color = "green", size = 1) +
  coord_fixed()

A scatter plot of sons' heights versus fathers' heights, with a line where the father and son would be the same height. As fathers get taller, so do the sons.

Adding a regression line

plt_son_vs_father +
  geom_smooth(method = "lm", se = FALSE)

The scatter plot of sons' heights versus fathers' heights, annotated with a linear trend line. The trend line is less steep than the line where fathers and sons would be the same height.

Running a regression

mdl_son_vs_father <- lm(
  son_height_cm ~ father_height_cm, 
  data = father_son
)

Call:
lm(formula = son_height_cm ~ father_height_cm, data = father_son)

Coefficients:
     (Intercept)  father_height_cm  
          86.072             0.514

Making predictions

really_tall_father <- tibble(
  father_height_cm = 190
)
predict(mdl_son_vs_father, really_tall_father)

183.7

really_short_father <- tibble(
  father_height_cm = 150
)
predict(mdl_son_vs_father, really_short_father)

163.2

Let's practice!

Introduction to Regression in R