Berkson's paradox

Berkson's paradox also known as Berkson's bias or Berkson's fallacy is a result in conditional probability and statistics which is counterintuitive for some people, and hence a veridical paradox. It is a complicating factor arising in statistical tests of proportions. Specifically, it arises when there is an ascertainment bias inherent in a study design. The effect is related to the explaining away phenomenon in Bayesian networks.

It is often described in the fields of medical statistics or biostatistics, as in the original description of the problem by Joseph Berkson.

Statement

The result is that two independent events become conditionally dependent (negatively dependent) given that at least one of them occurs. Symbolically:

if 0 < P(A) < 1 and 0 < P(B) < 1
(in words: each event may or may not occur),
and P(A|B) = P(A)
(they are independent),
then P(A|B,C) < P(A|C) where C = A∪B (i.e. A or B), and the comma indicates 'and' ie P(A|B,C)=P(A|(B⋂C))
(in words: A is less likely to occur given that A or B occurs and given that B occurs, than if simply A or B occurs).

In words, given two independent events, if you only consider outcomes where at least one occurs, then they become negatively dependent.

Explanation

The cause is that the conditional probability of event A occurring, given that it or B occurs, is inflated: it is higher than the unconditional probability, because we have excluded cases where neither occur.

P(A|A∪B) > P(A)
conditional probability inflated relative to unconditional

One can see this in tabular form as follows: the gray regions are the outcomes where at least one event occurs (and ~A means "not A").

A ~A
B A & B ~A & B
~B A & ~B ~A & ~B

For instance, if one has a sample of 100, and both A and B occur independently half the time (So P(A) = P(B) = 1/2), one obtains:

A ~A
B 25 25
~B 25 25

So in 75 outcomes, either A or B occurs, of which 50 have A occurring, so

P(A|A∪B) = 50/75 = 2/3 > 1/2 = 50/100 = P(A).

Thus the probability of A is higher in the subset (of outcomes where it or B occurs), 2/3, than in the overall population, 1/2. On the other hand, the probability of A given both B and C (equivalently, given both B and (A or B), which is the same thing as simply given B since B itself implies A or B), is simply the unconditional probability of A since A is independent of B. In the numerical example, we have conditioned on being in the top row:

A ~A
B 25 25
~B 25 25

Here the probability of A is 25/50 = 1/2.

Berkson's paradox arises because the conditional probability of A given B within the three-cell subset equals the conditional probability in the overall population, but the unconditional probability within the subset is inflated relative to the unconditional probability in the overall population, hence, within the subset, the presence of B decreases the conditional probability of A (back to its overall unconditional probability):

P(A|B, A∪B) = P(A|B) = P(A)
P(A|A∪B) > P(A).

Examples

Berkson's original illustration involves a retrospective study examining a risk factor for a disease in a statistical sample from a hospital in-patient population. Because samples are taken from a hospital in-patient population, rather than from the general public, this can result in a spurious negative association between the disease and the risk factor. For example, if the risk factor is diabetes and the disease is cholecystitis, a hospital patient without diabetes is more likely to have cholecystitis than a member of the general population, since the patient must have had some non-diabetes (possibly cholecystitis-causing) reason to enter the hospital in the first place. That result will be obtained regardless of whether there is any association between diabetes and cholecystitis in the general population.

An example presented by Jordan Ellenberg: Suppose Alex will only date a man if his niceness plus his handsomeness exceeds some threshold. Then nicer men do not have to be as handsome to qualify for Alex's dating pool. So, among the men that Alex dates, Alex may observe that the nicer ones are less handsome on average (and vice versa), even if these traits are uncorrelated in the general population.

Note that this does not mean that men in the dating pool compare unfavorably with men in the population. On the contrary, Alex's selection criterion means that Alex has high standards. The average nice man that Alex dates is actually more handsome than the average man in the population (since even among nice men, the ugliest portion of the population is skipped). Berkson's negative correlation is an effect that arises within the dating pool: the rude men that Alex dates must have been even more handsome to qualify.

As a quantitative example, suppose a collector has 1000 postage stamps, of which 300 are pretty and 100 are rare, with 30 being both pretty and rare. 10% of all his stamps are rare and 10% of his pretty stamps are rare, so prettiness tells nothing about rarity. He puts the 370 stamps which are pretty or rare on display. Just over 27% of the stamps on display are rare (100/370), but still only 10% of the pretty stamps are rare (and 100% of the 70 not-pretty stamps on display are rare). If an observer only considers stamps on display, she will observe a spurious negative relationship between prettiness and rarity as a result of the selection bias (that is, not-prettiness strongly indicates rarity in the display, but not in the total collection).

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