Hello! Welcome to the second lesson in our module on the shift from correlation to causation.
In our last lesson, we contrasted the classical Humean view of causation as mere regularity with the modern interventionist account. We concluded that causation isn't about passive observation of patterns but about active manipulation and control. A cause is a "handle" for changing an effect.
Today, we will see a dramatic and famous illustration of why this shift is not just a philosophical nuance but a practical necessity. Our learning outcome is to use Simpson's paradox to demonstrate the insufficiency of purely statistical reasoning for causal claims. We will explore a scenario where purely statistical analysis leads to a direct contradiction, forcing us to introduce a causal perspective to make a rational decision.
1. The Paradoxical Data
Let's dive straight into a numerical example. Imagine you are a public health official tasked with deciding which of two treatments, A or B, to approve for a new disease. You have observational data from hospitals about patient outcomes.
The following video presents this scenario with clear, hypothetical data. It will walk you through the data first at an aggregate level and then broken down by patient subgroups.
1.2 - Motivating Example: Simpson’s Paradox
This video from Brady Neal's Causal Inference course provides a perfect introduction to Simpson's Paradox. It sets up a medical decision problem and presents the contradictory data that defines the paradox.
Please watch from the beginning to 03:17. Pay close attention to the mortality rates for Treatment A and Treatment B in the overall population, and then within the 'mild' and 'severe' condition subgroups.
As you just saw, we have a perplexing situation:
- Overall Population: Treatment A appears better (16% mortality vs. 19% for Treatment B).
- Subgroups: For patients with a 'mild' condition, Treatment B is better (10% vs. 15%). For patients with a 'severe' condition, Treatment B is also better (20% vs. 30%).
So, Treatment B is superior for both men and women, yet Treatment A is superior for the population as a whole. This reversal of association is the essence of Simpson's Paradox. As a decision-maker, which treatment do you choose? Which statistic do you trust?
2. The Statistical Explanation: A Matter of Weighting
Before we address the decision, let's understand how this is mathematically possible. It's not a trick; it's a consequence of how the groups are weighted.
The video continues by explaining the arithmetic behind the paradox.
1.2 - Motivating Example: Simpson’s Paradox
Now, let's look at the mathematical reason for the reversal. This next segment explains how unequal distributions of patients across the subgroups create the paradox.
Please watch from 03:17 to 05:28. The key idea is understanding how the overall mortality rate for each treatment is a weighted average of the subgroup rates.
The explanation is that the two treatments were not distributed evenly.
- Treatment A was given mostly to patients with a mild condition (who have a low mortality rate to begin with).
- Treatment B, being scarce, was reserved mostly for patients with a severe condition (who have a high mortality rate to begin with).
Therefore, the aggregate result for Treatment B looks worse because it was disproportionately used on the sickest patients. This statistical explanation clarifies how the numbers work, but it still doesn't tell us which treatment is causally more effective.
3. The Causal Resolution: It Depends on the Story
The key insight, and the reason this paradox is so important for our course, is that the statistical data alone is insufficient to answer the question, "Which treatment should we choose?". The correct answer depends on the causal story that generated the data.
Let's watch the final part of the video, which presents two different causal scenarios for the exact same data.
1.2 - Motivating Example: Simpson’s Paradox
Here is the crucial part of the lesson. The video will now present two different causal models that could explain the data we saw. Notice how a different model leads to a different conclusion, even though the numbers are identical.
Please watch from 05:28 to the end (11:42). Focus on the difference between the two causal graphs presented and why they lead to opposite recommendations.
This is the central point: the decision hinges on the causal structure.
Scenario 1: Condition Causes Treatment (Confounding)
- Story: Doctors choose the treatment based on the patient's condition. They save the scarce Treatment B for severe cases.
- Causal Graph:
Condition -> Treatment - Conclusion: We should trust the subgroup data. The patient's pre-existing condition is a common cause (a "confounder") of both the treatment choice and the outcome. To see the true effect of the treatment, we must "control for" this confounder by looking within each condition group. In this case, Treatment B is better.
Scenario 2: Treatment Causes Condition (Mediation)
- Story: Treatment B is scarce and has a long waiting list. While waiting, a patient's condition can worsen from mild to severe.
- Causal Graph:
Treatment -> Condition - Conclusion: We should trust the aggregate data. Here, the treatment itself is causing a change in the patient's condition, which then affects the outcome. The patient's condition is a mediator of the causal effect. If we were to control for it, we would be blocking part of the causal pathway and getting a misleading picture of the treatment's total effect. The harm caused by waiting for Treatment B (worsening condition) is part of its overall causal impact. In this case, Treatment A is better.
This demonstrates the core learning outcome: purely statistical reasoning, which only looks at the numbers, is insufficient. It can't distinguish between these two scenarios. To make a causal claim and a sound decision, we must bring in causal knowledge about the world.
4. From Intuition to Formalism
This idea is the foundation of modern causal inference, pioneered by Judea Pearl. He argues that the "paradox" only seems paradoxical because our causal intuition rightly rebels against the idea that a treatment could be good for men, good for women, but bad for people overall.
Let's hear Pearl himself describe the paradox as a conflict between two different kinds of logic.
Keynote: Judea Pearl - The New Science of Cause and Effect
In this clip, Judea Pearl presents Simpson's Paradox using a different example. He frames it as a clash between the logic of correlation and the logic of causation, which your intuition understands.
Please watch from 26:21 to 34:31. Notice how he emphasizes that your intuition 'rebels' against the paradoxical conclusion, and that this rebellion is a clue that a different, causal logic is at play.
Pearl's work provides a formal language to resolve this. The key is to distinguish between merely seeing a correlation and asking what happens when we do something (intervene). This connects directly to our last lesson on interventionism.
The following text from the Stanford Encyclopedia of Philosophy explains how Pearl's framework formalizes the solution to the paradox.
Simpson's Paradox - Stanford Encyclopedia of Philosophy
This article explains how graphical causal models provide a formal solution. It introduces key concepts like 'confounding' and the 'back-door criterion' that allow us to determine when and what to condition on to find a true causal effect.
Please read section 3.4, 'Confounding and Pearl’s Analysis of the Paradox'. Focus on how the causal graph helps us decide whether to partition the data (as in our Scenario 1) or not (as in Scenario 2).
The article formalizes what we saw in the video. When the third variable (Condition) is a confounder (a common cause of Treatment and Outcome), the back-door criterion tells us we must condition on it to identify the causal effect. When it's a mediator (part of the causal chain from Treatment to Outcome), we must not condition on it if we want the total causal effect.
The purely statistical statement (the probability of recovery given that we observe a treatment) is different from the causal statement (the probability of recovery if we intervene to give a treatment). Simpson's paradox arises from the fact that these two quantities can be, and often are, different. Your work as a statistician has focused on calculating quantities like ; the shift to causal inference is about developing the tools to estimate quantities like .
Conclusion
Today we've seen how a seemingly simple statistical observation can conceal deep causal complexities. Simpson's paradox is more than just a statistical curiosity; it's a fundamental lesson in epistemology.
Key Takeaways:
- Simpson's Paradox: A statistical trend observed in an aggregate population can reverse direction when that population is partitioned into subgroups.
- Statistical vs. Causal Explanation: The reversal can be explained mathematically by unequal weighting, but this doesn't resolve the paradox from a decision-making standpoint.
- The Primacy of Causal Models: Deciding whether to trust the aggregate or subgroup data is impossible without a causal model of the data-generating process. The right choice depends on whether the partitioning variable is a confounder or a mediator.
- The Insufficiency of Statistical Reasoning: The paradox vividly demonstrates that causal questions cannot be answered by statistical methods alone. We need to supplement data with causal assumptions, often expressed in graphical models.
This lesson has firmly established the need to distinguish between observing and intervening. In our next lesson, we will formalize this distinction and others by exploring Judea Pearl's Ladder of Causation. This three-level hierarchy—seeing, doing, and imagining—provides a powerful framework for understanding different kinds of causal questions and the evidence needed to answer them.