Causal Inference

1. Definition

• Causal Inference is the process of determining whether a cause-and-effect relationship exists between variables, not just correlation.
• Example:
  • Correlation: Ice cream sales and drowning incidents rise together (but both are caused by hot weather).
  • Causal inference: A new drug causes lower blood pressure.

The key goal is to separate causation from mere association.

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2. Why It’s Important

• Businesses: Did the marketing campaign cause more sales, or would sales have increased anyway?
• Medicine: Did the new treatment cause patient improvement, or was it natural recovery?
• Policy: Did raising the minimum wage cause unemployment changes, or were other factors involved?

Without causal inference, decisions may be based on spurious relationships.

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3. Main Approaches to Causal Inference

A) Randomized Controlled Trials (RCTs)

• The gold standard.
• Randomly assign subjects to treatment and control → balances out confounding variables.
• Example: Give one group the new drug, the other a placebo, compare outcomes.

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B) Observational Studies (Non-Randomized)

When randomization is not possible (e.g., ethics, cost), statisticians use:

1. Regression Analysis
   • Controls for confounding variables statistically.
   • Example: Regression of wages on education, controlling for age and gender.
2. Matching Methods
   • Propensity Score Matching (PSM): Match treated units with similar untreated units based on observed covariates.
3. Difference-in-Differences (DiD)
   • Compare before/after changes between treatment and control groups.
   • Example: A state raises taxes → compare economic outcomes vs. a state that didn’t.
4. Instrumental Variables (IV)
   • Use a variable that influences treatment but not directly the outcome.
   • Example: Distance to a hospital as an instrument for receiving medical treatment.
5. Regression Discontinuity Design (RDD)
   • If treatment is assigned based on a cutoff (e.g., test score ≥ 80 gets scholarship), compare those just above vs. just below cutoff.

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C) Causal Graphs & Counterfactual Framework

1. Causal Diagrams (Directed Acyclic Graphs, DAGs)
   • Visualize relationships between variables, confounders, mediators.
2. Counterfactuals (Rubin Causal Model)
   • The “what if” framework:
     • Treatment outcome ($Y_1$​) vs. Non-treatment outcome ($Y_0$​).
     • Causal Effect = $E[Y_1 – Y_0]$.
   • Problem: We never observe both outcomes for the same person → need methods like RCT, DiD, etc.

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4. Key Challenges

• Confounding Variables: Third variables that affect both cause & effect.
• Selection Bias: When groups differ systematically (e.g., motivated students take extra tutoring).
• Reverse Causality: Outcome may also influence the cause (e.g., health affects income, but income also affects health).
• Unobserved Variables: Cannot be controlled if not measured.

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5. Examples

Example 1 – Medicine

• Research question: Does Drug A lower blood pressure?
• RCT: Randomly give patients Drug A or placebo.
• Observed significant drop in blood pressure in Drug A group → causal effect established.

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Example 2 – Business

• A company launches an online ad campaign.
• Sales rise, but was it because of the ads or because of seasonal demand?
• Use difference-in-differences: compare sales trends in a region with ads vs. without ads.

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Example 3 – Policy

• Does free school lunch improve test scores?
• Use regression discontinuity: compare students just above vs. just below the income threshold for free lunch.

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6. Causal Inference vs Correlation

• Correlation: Variables move together (but may be coincidental or driven by third factors).
• Causation: A change in one variable directly produces a change in another.
• Causal inference is the toolkit for making that distinction.

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In short:
Causal inference is about rigorously identifying whether one thing actually causes another, using tools like experiments (RCTs), observational methods (DiD, IV, PSM), and counterfactual reasoning.