Berkson's Paradox
2025 Jun 18
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Berkson's Paradox
Berkson's paradox occurs when sampling bias makes two independent
variables appear correlated.
- In particular the sampling bias happens because you condition on a
"collider variable" (an output variable which is influenced by both the
independent variables). This causes the two independent variable to
appear correlated.
- In Berkson's paradox, conditioning creates false correlation.
Example 1: college admissions
A mid-tier college admits students based on basketball ability OR
math scores. What's the probability a student has good math scores,
given they're good at basketball?
It's counter-intuitive to see that the two might be negatively
correlated.
- for a low-tier college, the students are not qualifying in
either.
- for a high-tier college, the students have to be good at both.
- but for a mid-tier college, the students are probably good at one
things - either maths or basketball.
So if you find a basketball player at a mid-tier college, they're
likely weak in math. Two independent variables appear correlated due to
sampling bias.
Here the collider variable is "admitted to college", while the
independent variables are basketball ability and maths score.
Example 2: who wrote this bad
book?
Consider 4 categories of author:
- A&R (Adept & Resilient): 40 books, 75%
good.
- A&NR (Adept & Not Resilient): 5 books, 80%
good
- NA&R (Not Adept & Resilient): 40 books, 20%
good
- NA&NR (Not Adept & Not Resilient): 5 books,
20% good
Suppose you find a bad book, for each author, what's
the probability each author type wrote it?
$$ \begin{align} P(A,R|Bad) &=
P(Bad|A,R)*P(A,R)/P(Bad) \\ &= 0.25*0.44/0.52 \\ &= 0.21
\end{align} $$ $$ \begin{align}
P(A,NR|Bad) &= P(Bad|A,NR)*P(A,NR)/P(Bad) \\ &= 0.20*0.0556/0.52
\\ &= 0.021 \end{align} $$ $$
\begin{align} P(NA,R|Bad) &= P(Bad|NA,R)*P(NA,R)/P(Bad) \\ &=
0.8*0.44/0.52 \\ &= 0.67 \end{align} $$ $$ \begin{align} P(NA,NR|Bad) &=
P(Bad|NA,NR)*P(NA,NR)/P(Bad) \\ &= 0.8*0.0556/0.52 \\ &= 0.0856
\end{align} $$
As expected, the not-adept, resilient author stands out. What might
be surprising is that the adept, resilient author has a high posterior
probability too. Because of their resiliency, the A&R author also
churns out quite a few bad books.
Here the collider variable is "wrote a bad book", while the
independent variables are "adeptness" and "resilience".
Berkson's Paradox
2025 Jun 18 See all postsBerkson's paradox occurs when sampling bias makes two independent variables appear correlated.
Example 1: college admissions
A mid-tier college admits students based on basketball ability OR math scores. What's the probability a student has good math scores, given they're good at basketball?
It's counter-intuitive to see that the two might be negatively correlated.
So if you find a basketball player at a mid-tier college, they're likely weak in math. Two independent variables appear correlated due to sampling bias.
Here the collider variable is "admitted to college", while the independent variables are basketball ability and maths score.
Example 2: who wrote this bad book?
Consider 4 categories of author:
Suppose you find a bad book, for each author, what's the probability each author type wrote it?
$$ \begin{align} P(A,R|Bad) &= P(Bad|A,R)*P(A,R)/P(Bad) \\ &= 0.25*0.44/0.52 \\ &= 0.21 \end{align} $$ $$ \begin{align} P(A,NR|Bad) &= P(Bad|A,NR)*P(A,NR)/P(Bad) \\ &= 0.20*0.0556/0.52 \\ &= 0.021 \end{align} $$ $$ \begin{align} P(NA,R|Bad) &= P(Bad|NA,R)*P(NA,R)/P(Bad) \\ &= 0.8*0.44/0.52 \\ &= 0.67 \end{align} $$ $$ \begin{align} P(NA,NR|Bad) &= P(Bad|NA,NR)*P(NA,NR)/P(Bad) \\ &= 0.8*0.0556/0.52 \\ &= 0.0856 \end{align} $$
As expected, the not-adept, resilient author stands out. What might be surprising is that the adept, resilient author has a high posterior probability too. Because of their resiliency, the A&R author also churns out quite a few bad books.
Here the collider variable is "wrote a bad book", while the independent variables are "adeptness" and "resilience".