Explain causal inference with an example

Question

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

Codemia · Accepted Answer

Causal inference is the process of determining whether a relationship between two variables is causal rather than merely correlational. To set up a mathematical framework for causal inference, we begin by defining two variables: the treatment variable (X) and the outcome variable (Y). We are interested in estimating the causal effect of X on Y, which can be expressed mathematically as:

$$ \text{Causal Effect} = E[Y | do(X)] - E[Y | do(X')] $$

Here, $E[Y | do(X)]$ represents the expected outcome when treatment X is applied, and $E[Y | do(X')]$ represents the expected outcome when treatment is not applied. This formula emphasizes the need for experimental or observational data to infer causation.

To determine the causal effect, we follow these steps:
1. **Define the Treatment and Outcome**: Identify what X (treatment) and Y (outcome) are.
2. **Collect Data**: Gather data through experiments (e.g., A/B testing) or observational studies.
3. **Control for Confounding Variables**: Ensure other variables that could affect Y are controlled or accounted for.
4. **Estimate the Causal Effect**: Use statistical methods such as regression or propensity score matching to estimate the causal effect.
5. **Conduct Hypothesis Testing**: Formulate null (H0) and alternative (H1) hypotheses to determine significance.

Let's consider a practical example: A company wants to know if a new feature in their app (X) increases user engagement (Y).

1. **Define Treatment and Outcome**: Here, X = new feature (1 = feature introduced, 0 = feature not introduced), Y = user engagement measured by session duration.
2. **Collect Data**: Conduct an A/B test with 1000 users, 500 with the feature and 500 without. 
   - Average session duration with feature: 10 minutes.
   - Average session duration without feature: 8 minutes.
3. **Estimate Causal Effect**: 
   - Causal Effect = E[Y | do(X=1)] - E[Y | do(X=0)] = 10 - 8 = 2 minutes.
4. **Hypothesis Testing**: 
   - H0: The new feature has no effect on engagement (mean difference = 0).
   - H1: The new feature increases engagement (mean difference > 0).
   - Calculate the p-value using a t-test.
   - Assume p-value = 0.03, which is less than 0.05, we reject H0.

The result indicates that introducing the new feature increases user engagement by an average of 2 minutes per session, and the statistical significance (p-value < 0.05) suggests that this effect is unlikely to be due to random chance. Thus, the company can make data-driven decisions about rolling out the feature more widely. However, we must consider assumptions such as random assignment and the absence of confounding variables; if violated, the causal inference may be compromised.

Causal inference connects to several related concepts:
- **Bayesian vs Frequentist Inference**: While frequentist methods focus on p-values and confidence intervals, Bayesian methods incorporate prior beliefs and update them with data.
- **Central Limit Theorem**: Ensures that the distribution of sample means approaches a normal distribution as the sample size increases, facilitating hypothesis testing.
- **Multiple Testing Correction**: In scenarios with multiple comparisons, techniques like Bonferroni correction or False Discovery Rate (FDR) control are essential to mitigate Type I errors.
- **Simpson's Paradox**: Highlights the importance of controlling for confounding variables, as a trend present in several groups can disappear or reverse when the groups are combined.

Explain causal inference with an example

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