Berkson's paradox
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Berkson's paradox or Berkson's fallacy is a result in conditional probability and statistics which is counter-intuitive for some people, and so has been described as a 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 result is that two independent events become conditionally dependent given that at least one of them occurs. Symbolically:
- if 0 < P(A) < 1 and 0 < P(B) < 1,
- and P(A|B) = P(A), i.e. they are independent,
- then P(A|B,C) < P(A|C) where C = A∪B (i.e. A or B).
It is often described in the fields of medical statistics or biostatistics, as in the original description of the problem by J Berkson.
A classic illustration involves a retrospective study examining a risk factor for a disease in a statistical sample from a hospital in-patient population. If a control group is also ascertained from the in-patient population, a difference in hospital admission rates for the case sample and control sample can result in a spurious association between the disease and the risk factor.
As another example, suppose I have 1000 postage stamps, of which 300 are pretty and 100 are rare, with 30 being both pretty and rare. 10% of all the stamps are rare and 10% of the pretty stamps are rare, so prettiness tells me nothing about rarity. I put the 370 stamps which are pretty or rare on display. Just over 27% of the stamps on display are rare, but still only 10% of the pretty stamps on display are rare. If I only consider stamps on display, I will observe a spurious negative relationship between prettiness and rarity as a result of my selection bias.
References
- Berkson, J. (1946) "Limitations of the application of fourfold tables to hospital data". Biometrics Bulletin, 2(3), 47-53.
Note on References
The reference Berkson (1946) cited above is frequently cited incorrectly in the literature as Berkson, J. (1949) Biological Bulletin 2, 47-53.
Biological Bulletin, established in the 19th century, does not publish statistical papers. The correct reference is to the biostatistical journal Biometrics Bulletin, established in 1945 which became Biometrics in 1947.
Acknowledgement and Attribution Regarding Sources of Content
Some of the initial content on this page may be incorporated in part from copyleft sources in the public domain including wikis such as Wikipedia and AskDrWiki. Drug information for patients came from the The National Library of Medicine. Infectious disease information may have come from the Centers for Disease Control (CDC). Differential Diagnoses are drawn from clinicians as well as an amalgamation of 3 sources: 1.The Disease Database; 2. Kahan, Scott, Smith, Ellen G. In A Page: Signs and Symptoms. Malden, Massachusetts: Blackwell Publishing, 2004:3; 3. Sailer, Christian, Wasner, Susanne. Differential Diagnosis Pocket. Hermosa Beach, CA: Borm Bruckmeir Publishing LLC, 2002:7 .
