March 20, 2019

Correlation to Causation

Plan for Today:

(1) Recap

  • correlation
  • two problems:
    • confounding/bias
    • random association

(2) Solutions for Bias

  • design-based vs. adjustment-based
  • conditioning
    • what is it?
    • how does it work?

Correlation to Causation

Testing Causal Claims

  1. We start with a causal claim

  2. Turn claim into a causal theory
    • causal logic, independent \(X\)/dependent \(Y\) variables
  3. Turn causal logic and \(X\)/\(Y\) into hypotheses
    • expect that potential outcomes of \(Y\) change with \(X\)
  4. B/C of FPCI: imperfectly test hypotheses using correlation of observed values of \(X\) and \(Y\)

  5. Infer causality if:
    • assumptions about cases we compare let us ignore confounding/bias
    • correlation unlikely to have occured by chance

Problems with Correlation

Bias/Confounding

Equivalently:

  1. Observed cases with different values of \(X\) have different potential outcomes of \(Y\)
  2. Observed cases with different values of \(X\) have different values of \(W\), which is related to \(Y\)
  3. There is a "backdoor" causal path from \(X\) to \(Y\)

Random Error

  1. Correlation between \(X\) and \(Y\) is by chance and reflects no change in \(Y\) due to \(X\).

Solutions to Problems

Bias/Confounding

  1. adjustment-based solutions
  2. design-based solutions

Random Error

  1. Increase number of independent cases
  2. Take courses in statistics

Solutions to Bias

adjustment-based

  • Identify possible confounding variables (e.g. \(W, Z, V, U\))
  • Measure these variables
  • adjust correlation of \(X\) and \(Y\) by "conditioning" on confounding variables

design-based

  • Compare cases that, by assumption, are
    • similar in terms of confounding variables \(W\)/ potential outcomes of \(Y\)
    • exposed to \(X\) in a manner unrelated to \(W\)/potential outcomes of \(Y\)

Adjustment-Based Solutions

Intuition: Adjustment

Adjustment-based approaches start from the…

Comparative Method:

Sometimes called the "method of difference" (via John Stuart Mill), this assesses whether \(X\) causes \(Y\) by…

  • comparing two cases that are the same in all relevant respects, except for value of \(X\)
  • assess for these two cases whether \(Y\) changes when \(X\) changes (correlation)

Intuition: Adjustment

Question:

What causes the spread of cholera?

Causal claim:

Contaminated water causes cholera outbreaks

Obvious?

19th Century London saw repeated outbreaks of cholera, with mass death

  • Dominant view was that "miasmas" or bad air caused diseases like cholera

Intuition: Adjustment

John Snow, MD