1. Correlation: Review
2. Problems with Correlation
- Random correlation
- Bias in correlation (confounding)
March 18, 2021
We solve the fundamental problem of causal inference by:
Correlation is the degree of association/relationship between the observed values of \(X\) (the independent variable) and \(Y\) (the dependent variable)
All empirical evidence for causal claims relies on correlation between the independent and dependent variables.
But, you’ve all heard this:
What do we need to assume to use correlation as evidence of causation?
random association: correlations between \(X\) and \(Y\) occur by chance and do not reflect causal relationship.
bias (spurious correlation, confounding): \(X\) and \(Y\) are correlated but the correlation does not result from a causal relationship between those variables
How do we know a correlation is systematic?
If you look at enough possible sets of variables, you might find a strong correlation
To see that random patterns can emerge, I use random number generators to
We can imagine these are the observed \(X\) and \(Y\) for \(5\) cases.
How easy is it to find a strong correlation?
Tries to get correlation \(> 0.9\): 1
Field of statistics investigates properties of chance events (stochastic processes):
This procedure works…
we know the chance processes that might affect this correlation
Tries to get correlation \(> 0.9\): 397
Tries to get correlation \(> 0.9\): 63963
Tries to get correlation \(> 0.45\): 76
An indication of how likely correlation we observe could have happened purely by chance.
higher degree of statistical significance indicates correlation is unlikely to have happened by chance
A numerical measure of statistical significance. Puts a number on how likely observed correlation would have occurred by chance, assuming a we know the chance procedure and the truth is a \(0\) correlation.
It is a probability, so is between \(0\) and \(1\).
Lower \(p\)-values indicate greater statistical significance
\(p < 0.05\) often used as threshold for “significant” result.
Be wary of “\(p\)-hacking”