March 18, 2019

Testing Causal Theories

Plan for Today:

(1) Correlation

  • correlation
    • technical details
  • random association
    • \(p\) values

Correlation

correlation:

degree of association or relationship between the observed values taken by two variables (\(X\) and \(Y\))

  • Many different ways of doing this (compare group means, regression) are all fundamentally about correlation.
  • correlations have a direction:
    • positive: implies that as \(X\) increases, \(Y\) increases
    • negative: \(X\) increases, \(Y\) decreases
  • correlations have strength (has nothing to do size of effect):
    • strong: \(X\) and \(Y\) almost always move together
    • weak: \(X\) and \(Y\) do not move together very much
  • There is also a technical definition of correlation (later)

Correlation

What is it?

(Pearson) correlation: also has specific mathematical definition (you don't need to know it):

\[r = \frac{\sum_{i}^n (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_i^n(x_i - \bar{x})^2}\sqrt{\sum_i^n (y_i - \bar{y})^2}}\]

This captures extent to which deviations from mean of \(X\) move with deviations from mean of \(Y\).

Correlation

What is it?

mathematically: correlation is the degree of linear association between \(X\) and \(Y\)

  • Takes values between \(-1\) and \(1\)
  • Values close to \(1\) or \(-1\) suggest high degree of linear association
  • Values close to \(0\) suggest low degree of linear association
  • Value of correlation does not tell us how much \(Y\) changes with \(X\)

Correlation

What is it?

negative correlation: (correlation \(< 0\)) values of \(X\) and \(Y\) move in opposite direction:

  • higher values of \(X\) appear with lower values of \(Y\)
  • lower values of \(X\) appear with higher values of \(Y\)

positive correlation: (correlation \(> 0\)) values of \(X\) and \(Y\) move in same direction:

  • higher values of \(X\) appear with higher values of \(Y\)
  • lower values of \(X\) appear with lower values of \(Y\)

Correlation