Balgobin Nandram and Hongyan Xu
When there is a rare disease in a population, it is inefficient to take a random sample to estimate a parameter. Instead one takes a random sample of all nuclear families with the disease by ascertaining at least one sibling (proband) of each family. In these studies, if the ascertainment bias is ignored, an estimate of the proportion of siblings with the disease will be inflated. The problem arises in population genetics, and it is analogous to the well-known selection bias problem in survey sampling. For example, studies of the issue of whether a rare disease shows an autosomal recessive pattern of inheritance, where the Mendelian segregation ratios are of interest, have been investigated for several decades and corrections have been made for the ascertainment bias using maximum likelihood estimation. Here, we develop a Bayesian analysis to estimate the segregation ratio in nuclear families when there is an ascertainment bias. We consider the situation in which the proband probabilities are allowed to vary with the number of affected siblings, and we investigate the effect of familial correlation among siblings within the same family. We discuss an example on cystic fibrosis and a simulation study to assess the effect of the familial correlation.
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