Explain causal inference with an example

Question

Explain causal inference in simple terms and provide a concrete example.

Codemia · Accepted Answer

Causal inference aims to identify a cause-and-effect relationship between variables. In this context, we want to determine whether an increase in Snapchat advertising spend causes an increase in user engagement. We can represent this mathematically using a causal model, where:
- Let $ A $ represent the amount spent on advertising.
- Let $ E $ represent user engagement, measured by metrics like daily active users (DAU) or time spent on the app.

The primary question is: Does changing $ A $ lead to a change in $ E $? This can be framed in terms of potential outcomes: If we increase $ A $ by a certain amount, what is the expected change in $ E $?

To analyze the causal relationship, we can use a regression analysis framework. The process involves the following steps:
1. **Data Collection**: Gather historical data on advertising spend and user engagement metrics.
2. **Model Specification**: Choose a regression model, e.g., $ E = eta_0 + eta_1 A + \epsilon $, where $ \epsilon $ represents error terms.
3. **Estimation**: Use ordinary least squares (OLS) to estimate $ eta_0 $ and $ eta_1 $.
4. **Statistical Testing**: Conduct hypothesis testing to evaluate if $ eta_1 $ is significantly different from zero, indicating a causal effect.
5. **Assumptions Check**: Verify the assumptions of the regression model, including linearity, independence, and homoscedasticity.

Suppose we have the following data for two months:
| Month | Advertising Spend (\$) | User Engagement (DAU) |
|-------|------------------------|-----------------------|
| 1     | 10,000                 | 200,000               |
| 2     | 20,000                 | 250,000               |

We can fit a linear regression model:
- $ E = eta_0 + eta_1 A $

Using software, we find:
- $ eta_0 = 150,000 $
- $ eta_1 = 5 $ (indicating that for every \$1,000 increase in advertising spend, DAU increases by 5,000)

Testing the hypothesis $ H_0: eta_1 = 0 $, we find $ p < 0.05 $, so we reject $ H_0 $. This suggests a significant positive effect of advertising spend on user engagement.

The result demonstrates that increased advertising spend correlates with higher user engagement on Snapchat. Specifically, for every \$1,000 spent on advertising, we can expect an increase of approximately 5,000 daily active users. This information is crucial for Snapchat's marketing strategies, as it quantifies the impact of advertising investments and helps in budget allocation.

Understanding causal inference also opens discussions on advanced topics such as:
- **Bayesian Methods**: Utilizing prior distributions to inform the causal model, which can be particularly useful when data is scarce or noisy.
- **Simpson's Paradox**: Recognizing situations where the correlation between variables can reverse when controlling for a third variable, stressing the importance of careful analysis.
- **Experimental Design**: Discussing randomized controlled trials (RCTs) where advertising spend is randomized to directly observe causal effects, minimizing confounding variables.

Explain causal inference with an example

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