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 determine whether a change in one variable (the treatment) directly causes a change in another variable (the outcome). To set up the problem, let us denote:

- **Treatment (T)**: A binary variable where T = 1 indicates the presence of a treatment (e.g., a new software feature) and T = 0 indicates its absence.
- **Outcome (Y)**: A continuous variable representing the effect we are measuring (e.g., user engagement score).
- **Confounders (C)**: Additional variables that could influence both T and Y, such as user demographics or previous engagement levels.

Our goal is to estimate the causal effect of the treatment on the outcome, which can be expressed mathematically as:

$$ \text{Causal Effect} = E[Y | T=1] - E[Y | T=0] $$

Where $ E[Y | T=1] $ is the expected outcome with treatment and $ E[Y | T=0] $ is the expected outcome without treatment.

To infer causality, we can use a randomized controlled trial (RCT) to minimize confounding factors. The step-by-step approach is as follows:

1. **Randomization**: Randomly assign subjects to either the treatment group (T=1) or the control group (T=0) to ensure that both groups are statistically similar regarding confounders.
2. **Data Collection**: Measure the outcome (Y) for both groups after the treatment is applied.
3. **Estimation of Effects**: Calculate the average outcomes for both groups and compute the causal effect as defined previously.
4. **Statistical Testing**: Use a t-test to determine if the difference in means is statistically significant, assessing p-values to avoid common pitfalls like p-hacking.

Suppose we conduct an experiment where we have 100 users, and we randomly assign 50 to receive a new software feature (T=1) and 50 to not receive it (T=0). After a month, we collect the following engagement scores:

- Treatment Group (T=1): [8, 7, 9, 10, 6, 8, 9, 10, 7, 9, 8, 9, 10, 7, 8, 9, 10, 8, 9, 8, 7, 10, 9, 8, 8, 9, 7, 10, 9, 8, 9, 9, 8, 10, 9, 8, 7, 9, 10, 8, 9, 7, 10, 8, 9, 9, 8, 10, 9, 8]
- Control Group (T=0): [5, 6, 5, 6, 7, 5, 6, 5, 6, 6, 7, 5, 6, 7, 5, 6, 5, 7, 6, 5, 6, 5, 6, 6, 5, 7, 6, 5, 6, 7, 5, 6, 5, 5, 6, 7, 5, 6, 5, 6, 5, 6, 7, 6, 5, 6, 6, 7, 5, 6]

First, we calculate the means:
- $ E[Y | T=1] = 8.6 $ 
- $ E[Y | T=0] = 6.0 $

Then, the causal effect is:
$$ \text{Causal Effect} = 8.6 - 6.0 = 2.6 $$

Next, we perform a t-test to check if this difference is statistically significant.

The causal effect of 2.6 indicates that the new software feature leads to an increase of 2.6 points in the engagement score on average. This suggests that implementing the feature could meaningfully enhance user engagement. However, we must caution that this interpretation assumes that randomization was successful in eliminating confounders. If any confounding biases remain, the true causal effect may differ.

This discussion of causal inference connects to several related concepts:
- **Bayesian vs Frequentist Inference**: While we used frequentist methods (t-test) here, Bayesian approaches could also be applied by estimating the posterior distribution of the causal effect given the observed data.
- **Multiple Testing Correction**: If we were to test multiple features simultaneously, we would need to consider corrections like the Bonferroni correction to control for Type I error rates.
- **Simpson's Paradox**: This example is a reminder of the importance of controlling for confounders; failing to do so could lead to Simpson's Paradox, where the association between treatment and outcome reverses when controlling for confounders.

Explain causal inference with an example

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