Confounders and Simpson's Paradox

Цели урока

• Identify a confounder from a DAG and distinguish it from a mediator or collider
• Recognize Simpson's paradox in production metrics
• Apply back-door adjustment to estimate a causal effect
• Build a minimal adjustment set from a causal graph
• Explain why randomization eliminates confounding

Предварительные знания

• Basic causality and the do-operator
• Conditional distributions and conditional independence
• Linear regression

• Causal inference: foundations
• Bayes' theorem

The same Berkeley dataset in 1973 simultaneously proved and refuted gender discrimination in graduate admissions. The lawsuit collapsed not because of a new survey but because a statistician split the data by department. The case entered every textbook as the canonical confounding example and helped trigger the formal development of causal inference.

• **Instagram and TikTok recommenders**: models train on biased engagement data, Simpson's paradox shows up at scale - aggregate metrics rise while satisfaction drops
• **Clinical trials**: older patients receive more treatments and have more deaths - naive analysis makes treatments look lethal
• **Google search quality**: aggregate CTR rises while per-query CTR falls because the query mix shifts
• **RLHF in LLMs**: reward models trained on preference scores without controlling for prompt difficulty learn to confuse verbosity with quality

A paradox with three authors and one name

E.H. Simpson published 'The Interpretation of Interaction in Contingency Tables' in 1951, describing the paradox that now carries his name. Simpson himself noted the result was 'not new': the same phenomenon was discussed by Udny Yule in 1903 and by Karl Pearson around the same decade. The modern mechanistic framework, Directed Acyclic Graphs, was formalized only in the 1980s and 1990s by Judea Pearl, who won the Turing Award for this work in 2011.

What a confounder is

1968, UC Berkeley. Graduate school admitted 44% of male applicants and only 35% of female applicants. Newspapers ran with the discrimination story and the university was sued. Then a statistician broke the numbers down by department - and the direction reversed: in almost every single department, women were admitted at equal or higher rates than men. The very same dataset proved both discrimination and the absence of it. Both views are technically correct, and that contradiction is the heart of Simpson's Paradox.

A confounding variable is a third variable Z that influences both the supposed cause X and the outcome Y. Because of Z, the observed X-Y correlation no longer reflects the true causal effect. In the Berkeley case the confounder was choice of department: women applied disproportionately to humanities programs with low admission rates, men to engineering programs with high admission rates.

This DAG reads: Z causally drives both X and Y. Measuring the X-Y relationship without conditioning on Z yields a spurious correlation that has nothing to do with any direct causal path.

A confounder is not just any third variable. It must sit on a back-door path from X to Y. Mediators (on the causal pipeline) and colliders (common children) require completely different handling.

Confounding is the main technical reason correlation differs from causation. Without an explicit model of Z, any regression of Y on X measures a mixture of the causal effect and confounding bias.

Simpson's Paradox: the mechanism

Simpson's Paradox is a formal effect: the direction of a statistical association within each stratum of a confounder can be opposite to the direction of the aggregate association. Not a visual illusion, not a computational mistake - a clean arithmetic artifact of weighted averaging.

At the department level women are admitted at equal or higher rates in most cases. In the aggregate they lose by 9 percentage points. The cause: women apply heavily to departments C-F with low admission rates, men apply to A and B with high admission rates.

In production ML monitoring Simpson's paradox is especially treacherous. Overall accuracy can rise release after release while accuracy in every demographic stratum falls. Shifting traffic mix is enough to fool the team.

When comparing two policies or models, watch for very different population mixes on a key feature. If the mix is observational rather than experimentally fixed, Simpson is on the table.

Small-sample noise can also flip an aggregate effect. Simpson's paradox specifically requires structural distortion through a confounder, not random fluctuation.

Controlling for confounders

When confounders are observable and measured, the causal effect can be recovered. Three core techniques: stratification (analyze inside strata), matching (pair treatment and control with similar Z), regression adjustment (include Z as covariates).

Pearl's back-door adjustment formula. The left side is what would happen if X were forcibly set to x (intervention via the do-operator). The right side is computable from observed data, provided Z blocks every back-door path from X to Y.

Randomization (RCT) makes X independent of every Z by construction, so P(Z|X) = P(Z) and the naive difference of means equals the true causal effect. That is the structural reason A/B tests are the gold standard for causal inference.

If treatment assignment is not strictly random (for example, users self-select into a beta feature), conditioning on observed covariates will not save the analysis from unobserved confounders. IV, RDD, or DiD become necessary.

Regression adjustment of Y on X plus Z produces an unbiased effect estimate only under linearity and correct specification. In practice combine with propensity scores or use doubly robust methods.

DAGs: formalizing causal structure

A Directed Acyclic Graph is the language modern causal inference uses to record assumptions about dependence. Nodes are variables, arrows are direct causal links, no cycles allowed. A DAG immediately tells the analyst what to condition on and what to leave alone.

A back-door path from X to Y is any undirected path that starts with an arrow into X (that is, X<-...). If such a path exists and is not blocked by conditioning, it injects bias.

Multiple Z sets can satisfy the back-door criterion. The minimal one usually gives lower estimator variance. Tools such as dagitty and DoWhy find these sets automatically.

The graph is drawn by a human. A missed arrow or an unmodeled hidden confounder leaves every downstream effect biased. DAGs do not replace domain knowledge - they make it explicit.

Where this leads

Confounders are the central problem of causal inference. Every later method is a different way to defeat them.

• Randomized Controlled Trials — Randomization is the gold standard antidote to confounding - it breaks the back-door path between treatment and confounder
• Propensity Score Methods — Propensity score matching and IPW adjust for measured confounders when randomization is impossible

Key takeaways

• A confounder Z simultaneously drives the cause X and the outcome Y, producing spurious correlation
• Simpson's paradox is an arithmetic reversal of effect direction caused by aggregating across a confounder
• Back-door adjustment: average the conditional effect over marginal P(Z), not over P(Z|X)
• A DAG formalizes causal structure and tells the analyst what to control for
• Colliders are the opposite hazard: conditioning on one induces fake correlation
• Randomization makes X independent of every Z by construction - the radical antidote to confounding

Вопросы для размышления

• Which confounders may exist in the team's current production dataset and how to surface them?
• Could Simpson's paradox be lurking in the headline metric used to ship products?
• Among current model features, which are mediators and which are colliders?
• What needs to change in data collection to make the dominant confounder neutralizable?

Связанные уроки

• stat-20-causal — basic causal vocabulary and DAGs
• stat-40-causal-rct — RCTs neutralize confounders by design
• stat-42-causal-propensity — propensity score balances observed confounders
• stat-30-stats-ml — selection bias in ML training data
• prob-04-bayes