reg_to_mean.R

set.seed(432)

# imagine these are scores before a program:
before <- rnorm(50, mean = 0, sd = 10)

# imagine these are scores after a program
after <- before + rnorm(50, mean = 5, sd = 20)

# the scores are correlated, but not perfectly correlated
# correlation = 0.32... the program helps, 
# on average, it helps with +5 points, but there is
# luck involved in the "after" assessment
cor(before, after)

# calculate z-scores for the before scores, to see how far from the mean
# calculate z-scores for the after scores, to see how extreme
before_zscore <- (before - mean(before))/sd(before)
after_zscore <- (after - mean(after))/sd(after)

# plot the zscored observations (before vs. after)
plot(x = before_zscore, y = after_zscore, 
     pch = 16, col = "blue",
     xlim = c(-3, 3), ylim = c(-3, 3))

# run a linear regression (fit a line) to the points in the plot
# there are technical reasons why this line will NEVER have a 
# slope of greater than 1 (or less than -1) -- basically b/c
# the slope of a regression line drawn through zscored (scaled) 
# observations is always the simple correlation and correlation is
# never less than -1 or greater than +1.
reg1 <- lm(after_zscore ~ before_zscore)
abline(reg1, col = "red", lwd = 4)

# you can see that the slope of the line is 0.32, the correlation above,
# it's the number in the first column, second row
summary(reg1)

# just to show you that the correlation is still 0.32
cor(before_zscore, after_zscore)

# and this is why whenever you have correlated before/after data you have 
# regression to the mean--because, for any given value of 'before' (call that
# value "b", e.g., b = 1), the expected value of the corresponding 'after'
# value (i.e., the values of 'after' along the regression line)
# will be less than or equal or |b| (absolute value of b). 
# The values of 'after' along the regression line will be
# equal to absolute value of b when correlation is perfect and the slope of
# the regression line is exactly 1. Otherwise, the slope will be shallower
# less steep than 1, and the 'after' value will be smaller than |b|.