GV249 Seminar AT5: Causation
Lennard Metson
2024-11-12
Correlation \(\neq\) causation
📖 Key reading
Bueno de Mesquita & Fowler - Thinking Clearly with Data - Chapters 2 & 9
Causation
- Correlation \(\neq\) causation… but what does?
- Counterfactuals!
Causation
Definition
A is caused by B when A would not have happened if B had not happened.
- A causal effect is the comparison of the value of A in a world in which B happened versus a world in which B didn’t happen.
- These two values of A are called potential outcomes
Causation - Potential outcomes
- \(Y_{i}^{1}\)
- \(Y_{i}^{0}\)
😟 The fundamental problem of causal inference
⚠️ The problem
We cannot observe both potential outcomes for an individual.
- Once they are treated, we only observe their treated potential outcome.
- And when they are untreated, we only observe their untreated potential outcome.
- Causal inference is a field of statistics focused on getting around this problem.
😌 Solving the fundamental problem
- Causal inference methods try to fill in missing potential outcomes using information from treated and untreated units.
😌 Solving the fundamental problem
- The answer lies in comparing average potential outcomes across treated and untreated units.
\[ \text{mean}(Y_{i}^{1}-Y_{i}^{0}) = \text{mean}(Y_{i}^{1}) - \text{mean}(Y_{i}^{0}) \]
🧪 Treatment effects
🧪 ATE
- The ATE is the mean of the individual-level treatment effect (\(\tau_i\)) for all units (\(\text{mean}(Y_i^1 - Y_i^0)\)).
- This is mathematically equal to \(\text{mean}(Y_{i}^{1}) - \text{mean}(Y_{i}^{0})\)
🧪 ATT & ATU
ATT is the mean \(\tau_i\) for the subset of units who received treatment.
ATU is the mean \(\tau_i\) for the subset of units who did not receive treatment.
The ATT and ATU are different from the ATE when there is biased (i.e., non-random) selection into who receives treatment.
📺 Example: news and attitudes
📺 Set-up
- Imagine political attitudes range from 0-5, with 0 being the most left-wing and 5 being the most right-wing.
- We are interested in whether watching a left-wing news (IV) channel makes individuals more left wing (DV). This is a causal research question!
📺 Set-up
We have some variables:
- \(X\) = individual \(i\)’s political attitudes before watching left-leaning news (or not).
- \(D\) (IV) = whether \(i\) watches left-leaning news.
- \(Y\) (DV) = \(i\)’s political attitudes after watching left-leaning news (or not).
📺 The counter-factual world
- In the counter-factual world, we can see both potential outcomes.
- Which means we can calculate the individual causal effect for each individual.
- On the next slide we can see the full schedule of potential outcomes for 10 people.
📺 The counter-factual world
| \(i\) | \(X_i\) | \(D_i\) | \(Y_{i}^{0}\) | \(Y_{i}^{1}\) | \(\tau_i\) |
|---|---|---|---|---|---|
| \(i_1\) | 5 | 5 | 4 | ||
| \(i_2\) | 4 | 3 | 3 | ||
| \(i_3\) | 3 | 4 | 5 | ||
| \(i_4\) | 4 | 4 | 3 | ||
| \(i_5\) | 5 | 5 | 5 | ||
| \(i_6\) | 2 | 3 | 3 | ||
| \(i_7\) | 3 | 2 | 3 | ||
| \(i_8\) | 2 | 1 | 2 | ||
| \(i_9\) | 1 | 1 | 1 | ||
| \(i_{10}\) | 1 | 1 | 2 |
📺 The counter-factual world
| \(i\) | \(X_i\) | \(D_i\) | \(Y_{i}^{0}\) | \(Y_{i}^{1}\) | \(\tau_i\) |
|---|---|---|---|---|---|
| \(i_1\) | 5 | 5 | 4 | -1 | |
| \(i_2\) | 4 | 3 | 3 | 0 | |
| \(i_3\) | 3 | 4 | 5 | 1 | |
| \(i_4\) | 4 | 4 | 3 | -1 | |
| \(i_5\) | 5 | 5 | 5 | 0 | |
| \(i_6\) | 2 | 3 | 3 | 0 | |
| \(i_7\) | 3 | 2 | 3 | 1 | |
| \(i_8\) | 2 | 1 | 2 | 1 | |
| \(i_9\) | 1 | 1 | 1 | 0 | |
| \(i_{10}\) | 1 | 1 | 2 | 1 |
\(ATE = mean(\tau_i) = 0.2\)
\(mean(Y_{i}^{1}) = 3.1\)
\(mean(Y_{i}^{0}) = 2.9\)
→ \(ATE = 3.1-2.9 = 0.2\)
📺 The real-world (1)
- However, in the real-world, we only see one potential outcome.
- More right-wing people are less likely to choose to watch left-wing news.
- The correlation between pre-treatment variables and selecting into treatment is selection bias.
- Let’s see what happens when we compare potential outcomes letting individuals choose whether they watch left-wing news.
📺 The real-world (1)
| \(i\) | \(X_i\) | \(D_i\) | \(Y_{i}^{0}\) | \(Y_{i}^{1}\) |
|---|---|---|---|---|
| \(i_1\) | 5 | 0 | 5 | ? |
| \(i_2\) | 4 | 0 | 3 | ? |
| \(i_3\) | 3 | 0 | 4 | ? |
| \(i_4\) | 4 | 1 | ? | 3 |
| \(i_5\) | 5 | 0 | 5 | ? |
| \(i_6\) | 2 | 1 | ? | 3 |
| \(i_7\) | 3 | 1 | ? | 3 |
| \(i_8\) | 2 | 0 | 1 | ? |
| \(i_9\) | 1 | 1 | ? | 1 |
| \(i_{10}\) | 1 | 1 | ? | 2 |
- In the real world, we only see one potential outcome. This is defined by the value of \(D_i\).
- We refer to the observed outcome as \(Y_i\).
📺 The real-world (1)
| \(i\) | \(X_i\) | \(D_i\) | \(Y_i\) |
|---|---|---|---|
| \(i_1\) | 5 | 0 | 5 |
| \(i_2\) | 4 | 0 | 3 |
| \(i_3\) | 3 | 0 | 4 |
| \(i_4\) | 4 | 1 | 3 |
| \(i_5\) | 5 | 0 | 5 |
| \(i_6\) | 2 | 1 | 3 |
| \(i_7\) | 3 | 1 | 3 |
| \(i_8\) | 2 | 0 | 1 |
| \(i_9\) | 1 | 1 | 1 |
| \(i_{10}\) | 1 | 1 | 2 |
- In the real world, we only see one potential outcome. This is defined by the value of \(D_i\).
- We refer to the observed outcome as \(Y_i\).
📺 The real-world (1)
| \(i\) | \(X_i\) | \(D_i\) | \(Y_i\) |
|---|---|---|---|
| \(i_1\) | 5 | 0 | 5 |
| \(i_2\) | 4 | 0 | 3 |
| \(i_3\) | 3 | 0 | 4 |
| \(i_4\) | 4 | 1 | 3 |
| \(i_5\) | 5 | 0 | 5 |
| \(i_6\) | 2 | 1 | 3 |
| \(i_7\) | 3 | 1 | 3 |
| \(i_8\) | 2 | 0 | 1 |
| \(i_9\) | 1 | 1 | 1 |
| \(i_{10}\) | 1 | 1 | 2 |
\(mean(Y_i|D_i = 1)\) = 2.4
\(mean(Y_i|D_i = 0)\) = 3.6
Estimated ATE = \(2.4 - 3.6 = -1.2\)
📺 The real-world (1)
| \(i\) | \(X_i\) | \(D_i\) | \(Y_i\) |
|---|---|---|---|
| \(i_1\) | 5 | 0 | 5 |
| \(i_2\) | 4 | 0 | 3 |
| \(i_3\) | 3 | 0 | 4 |
| \(i_4\) | 4 | 1 | 3 |
| \(i_5\) | 5 | 0 | 5 |
| \(i_6\) | 2 | 1 | 3 |
| \(i_7\) | 3 | 1 | 3 |
| \(i_8\) | 2 | 0 | 1 |
| \(i_9\) | 1 | 1 | 1 |
| \(i_{10}\) | 1 | 1 | 2 |
\(mean(Y_i|D_i = 1)\) = 2.4
\(mean(Y_i|D_i = 0)\) = 3.6
Estimated ATE = \(2.4 - 3.6 = -1.2\)
What’s the problem?
📺 The real-world (2)
- Now imagine we can decide who watches left-wing news or not.
- We can randomly select individuals to reveal their treated (and untreated) potential outcomes.
- Because we select them randomly, there is no selection bias: individuals have the same probability of watching left-wing news regardless of their underlying attitudes.
📺 The real-world (2)
| \(i\) | \(X_i\) | \(D_i\) | \(Y_i\) |
|---|---|---|---|
| \(i_1\) | 5 | 0 | 5 |
| \(i_2\) | 4 | 1 | 3 |
| \(i_3\) | 3 | 1 | 5 |
| \(i_4\) | 4 | 0 | 4 |
| \(i_5\) | 5 | 1 | 5 |
| \(i_6\) | 2 | 0 | 3 |
| \(i_7\) | 3 | 0 | 2 |
| \(i_8\) | 2 | 0 | 1 |
| \(i_9\) | 1 | 1 | 1 |
| \(i_{10}\) | 1 | 1 | 2 |
📺 The real-world (2)
| \(i\) | \(X_i\) | \(D_i\) | \(Y_i\) |
|---|---|---|---|
| \(i_1\) | 5 | 0 | 5 |
| \(i_2\) | 4 | 1 | 3 |
| \(i_3\) | 3 | 1 | 5 |
| \(i_4\) | 4 | 0 | 4 |
| \(i_5\) | 5 | 1 | 5 |
| \(i_6\) | 2 | 0 | 3 |
| \(i_7\) | 3 | 0 | 2 |
| \(i_8\) | 2 | 0 | 1 |
| \(i_9\) | 1 | 1 | 1 |
| \(i_{10}\) | 1 | 1 | 2 |
\(mean(Y_i|D_i = 1)\) = 3.2
\(mean(Y_i|D_i = 0)\) = 3
Estimated ATE = \(3.2 - 3 = 0.2\) 1
📺 What does this tell us about causal inference?
- Sometimes, we can use information about the average outcomes for treated and untreated units to estimate average treatment effects.
- Where there is no selection bias, we can directly compare the mean outcomes of two groups.
- When there is selection bias, we need to use alternative strategies to “fill in” missing potential outcomes.
📺 Broockman and Kalla (2022) did this for real!
- On a sample of \(\approx\) 5,500 American voters, Broockman and Kalla (2022) randomized encouragement to watch CNN (rather than Fox News).
- They find evidence that actually, there is an effect, even when you randomise!
📊 Visualisation: extra resources
- Leland Wilkinson - Grammar of Graphics
- This is what the package
ggplot2is based on - hence “gg” - This blog has some detailed information about the grammar of graphics framework.
- This is what the package
- Claus Wilke - Fundamentals of Data Visualization.
- This chapter - on visualising uncertainty - links nicely to what we covered in AT4.
- Edward Tufte - The Visual Display of Quantitative Information
- Leans more heavily into graphic design
- Tufte’s 6 principle’s: this blog post dicusses them.
References
GV249 AT5 | 📨 email l.m.metson@lse.ac.uk 🤔 Question? 🙋 raise your hand or 🖥️ use the Moodle Forum.
Broockman, David, and Joshua Kalla. 2022. “Consuming Cross-Cutting Media Causes Learning and Moderates Attitudes: A Field Experiment with Fox News Viewers.” Preprint. Open Science Framework. https://doi.org/10.31219/osf.io/jrw26.