Causal Inference
Correlation is easy to measure. Causation is hard to prove. How do we answer “does X cause Y?” — especially when we cannot run an experiment?
Correlation vs. Causation
Ice cream sales correlate with drowning deaths. Hot weather drives both. Correlation is symmetric — causation is directional and requires intervention.
Confounding Variables
A confounder influences both treatment and outcome, creating a spurious association. It is the primary obstacle to causal inference from observational data.
- Lighter carriers → lung cancer (confound: smoking)
- HRT users → less heart disease (confound: health status)
- Shoe size → reading ability (confound: age)
Simpson’s Paradox
A trend that appears within every subgroup reverses when data is combined. Not a mistake — it reveals a confounder you ignored.
The RCT
Randomized Controlled Trials eliminate confounding by design. Random assignment makes treated and control groups comparable on all variables — measured and unmeasured.
Average Treatment Effect
Each unit has two potential outcomes: Y(1) under treatment, Y(0) under control. We observe only one. The ATE averages their difference.
Propensity Scores
The probability of treatment given covariates. Matching on the propensity score collapses high-dimensional covariate adjustment to one number.
PS Methods
Once estimated, the propensity score can be used in three ways. All require the ignorability assumption: no unmeasured confounders.
- Matching — pair units with similar scores
- Stratification — bin by score, estimate within each bin
- IPW — weight by inverse of propensity score
Difference-in-Differences
Compare the change over time in a treated group to the change in a control group. Removes time-invariant confounding without requiring randomization.
Parallel Trends
DiD requires that, without treatment, both groups would have followed the same trend. Untestable directly — but pre-treatment parallel trends provide supporting evidence.
Instrumental Variables
An instrument Z shifts treatment “as if” at random. IV estimates the causal effect even when confounders are unmeasured — if the instrument is valid.
- Relevant — Z correlated with T
- Exclusion — Z affects Y only through T
- Exogenous — Z independent of unmeasured confounders
What you learned
Confounders create spurious associations. RCTs eliminate them by design. Propensity scores, DiD, and IV approximate experiments when randomization is impossible.