Berkson's paradox and weighted distributions: An application to Alzheimer's disease

One reason for observing in practice a false positive or negative correlation between two random variables, which are either not correlated or correlated with a different direction, is the overrepresentation in the sample of individuals satisfying specific properties. In 1946, Berkson first illustrated the presence of a false correlation due to this last reason, which is known as Berkson's paradox and is one of the most famous paradox in probability and statistics. In this paper, the concept of weighted distributions is utilized to describe Berskon's paradox. Moreover, a proper procedure is suggested to make inference for the population given a biased sample which possesses all the characteristics of Berkson's paradox. A real data application for patients with dementia due to Alzheimer's disease demonstrates that the proposed method reveals characteristics of the population that are masked by the sampling procedure.