Simpson's Paradox — when the trend flips direction

Dr. R. Düsing · Osnabrück University
📋 Example — Treatment Dose & Recovery
Within every disease-severity group: higher dose → better recovery (the colored lines slope up). But more severe groups get a higher dose and recover worse. Ignore the groups, and the pooled line suddenly seems to say: higher dose → worse recovery. Same dataset, opposite message — that's Simpson's paradox. It arises from a confounding variable (here: severity).
① Sign Check
Within slope (grouped)
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mean of the group lines
Pooled slope (aggregated)
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regression ignoring groups
Pooled correlation r
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across all points
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② Scatterplot — Within vs. Pooled
Colored lines = within groups · red line = pooled (groups ignored)
Concepts
What is Simpson's paradox?
A relationship that points in one direction within every subgroup can reverse as soon as the groups are pooled together. The cause is almost always a confounding variable that relates to both the predictor and the outcome and is unevenly distributed across the groups.
Berkeley admissions (1973)
The most famous example: women were admitted less often overall — but in almost every single department equally or more often. Women simply applied more often to departments with lower admission rates. The department was the confounder.
Ecological fallacy
Inferring from the group level to the individual level (or vice versa) is not valid. Simpson's paradox is the extreme case: the aggregated trend states the opposite of the individual one. Which level is "correct" is decided by the causal question, not by statistics.
Connection to multilevel models
This is exactly where partial pooling comes in: a multilevel model separates the within- from the between-effect instead of blending both like pooled regression. Ignoring group structure (complete pooling) risks Simpson.
→ To the multilevel tool
Simpson's Paradox — Background
The phenomenon

A statistical relationship can be positive within every subgroup and still turn negative when the groups are combined (or vice versa). Named after Edward H. Simpson (1951), but already described by Pearson, Yule, and Kendall. It is not a calculation error — both trends are real, they just answer different questions.

Why does this happen?

The pooled (aggregated) slope blends two sources of covariance: the within-group variation and the between-group variation. If the groups are shifted along an opposing direction (confounding), the between-component can outvote the within-component:

b_pooled = [ Cov_within(x,y) + Cov_between(x,y) ] / [ Var_within(x) + Var_between(x) ] Cov_within = b_within · Var_within(x) Cov_between = b_between · Var_between(x)

If bwithin is positive but bbetween is strongly negative and the groups are far apart (large group offset → large Var_between), the second term dominates and bpooled turns negative. That's exactly when the sign flips.

The controls

Within slope bw: the true effect within each group.
Between trend bb: how the group centers are arranged (direction & strength of confounding).
Group offset: how far apart the groups are. At 0 they fully overlap → pooled = within, no paradox. The larger it is, the more the between trend dominates.

Three scenarios

Reversal: within positive, between negative → pooled negative (the classic paradox).
No confounding: group offset = 0 → within and pooled agree.
Amplification: within and between share the same sign → pooled overstates the effect (right direction, wrong magnitude). This too is a form of bias.

What to do?

The solution isn't "always break it down" or "always pool," but to clarify the causal structure (DAG): if the group variable is a confounder, you must adjust for it (look within groups). If it's a mediator on the causal path, adjusting would wrongly explain the effect away. A multilevel model (partial pooling) cleanly separates the within- and between-effect.

References

Simpson, E. H. (1951). The interpretation of interaction in contingency tables. JRSS B, 13(2), 238–241.
Bickel, P. J., Hammel, E. A. & O'Connell, J. W. (1975). Sex bias in graduate admissions: Data from Berkeley. Science, 187, 398–404.
Pearl, J. (2014). Comment: Understanding Simpson's paradox. The American Statistician, 68(1), 8–13.