Causal Inference Exam Prep: Solved Problems

📚 Quick Study Guide

• 🔬 Definition: Causal inference is the process of determining cause-and-effect relationships based on data. It goes beyond correlation to understand if one variable *causes* another.
• 🧪 Potential Outcomes Framework: This framework considers what would have happened to an individual under different treatments. We use notation like $Y_i(1)$ (outcome for individual $i$ under treatment) and $Y_i(0)$ (outcome for individual $i$ without treatment).
• 📈 Average Treatment Effect (ATE): ATE measures the average difference in outcomes between treated and untreated groups in the entire population. It's calculated as $ATE = E[Y(1) - Y(0)]$.
• 🧩 Conditional Average Treatment Effect (CATE): CATE is the average treatment effect for a specific subgroup, conditioned on certain characteristics. It's represented as $CATE = E[Y(1) - Y(0) | X = x]$.
• 🎲 Randomized Controlled Trials (RCTs): RCTs are the gold standard for causal inference because they randomize treatment assignment, minimizing bias.
• 📊 Observational Studies: In observational studies, treatment assignment is not randomized, so we need to control for confounding variables using techniques like regression, matching, or instrumental variables.
• 🐍 Confounding Variables: These are variables that are associated with both the treatment and the outcome, potentially distorting the causal relationship. Addressing confounders is key to unbiased causal inference.

Practice Quiz

1. Which of the following best describes the goal of causal inference?

   1. A. Identifying correlations between variables.
   2. B. Predicting future outcomes based on past data.
   3. C. Determining cause-and-effect relationships between variables.
   4. D. Summarizing data using descriptive statistics.

2. In the potential outcomes framework, what does $Y_i(1)$ represent?

   1. A. The observed outcome for individual $i$.
   2. B. The outcome for individual $i$ if they do not receive treatment.
   3. C. The outcome for individual $i$ if they receive treatment.
   4. D. The average outcome for all individuals.

3. What is the formula for the Average Treatment Effect (ATE)?

   1. A. $ATE = E[Y(1) + Y(0)]$
   2. B. $ATE = E[Y(1) - Y(0)]$
   3. C. $ATE = E[Y(0) - Y(1)]$
   4. D. $ATE = Y(1) - Y(0)$

4. What is a confounding variable?

   1. A. A variable that is caused by the treatment.
   2. B. A variable that is correlated with the outcome but not the treatment.
   3. C. A variable that is associated with both the treatment and the outcome.
   4. D. A variable that has no effect on the outcome.

5. Why are Randomized Controlled Trials (RCTs) considered the gold standard for causal inference?

   1. A. Because they are easy to conduct.
   2. B. Because they always produce statistically significant results.
   3. C. Because they randomize treatment assignment, minimizing bias.
   4. D. Because they are inexpensive.

6. What technique is used in observational studies to control for confounding variables?

   1. A. Randomization
   2. B. Blinding
   3. C. Regression, matching, or instrumental variables
   4. D. Placebo control

7. What does Conditional Average Treatment Effect (CATE) measure?

   1. A. The treatment effect on the entire population.
   2. B. The treatment effect only on individuals who received the treatment.
   3. C. The average treatment effect for a specific subgroup.
   4. D. The treatment effect under all possible conditions.