Ascertainment corrections based on smaller family units

Ascertainment concerns the manner by which families are selected for genetic analysis and how to correct for it in likelihood models. Because such families are often neither drawn at random nor selected according to well-defined rules, the problem of ascertainment correction in the genetic analysis of family data has proved durable. This paper undertakes a systematic study of ascertainment corrections in terms of smaller distinct units, which will usually be sibships, nuclear families, or small pedigrees. Three principal results are presented. The first is that ascertainment corrections in likelihood models for family data can be made in terms of smaller units, without breaking up the pedigree. The second is that the appropriate correction for single ascertainment in a unit is the reciprocal of the sum of the marginal probabilities of all the persons relevant to its ascertainment, as if affected. The third result is a generalization of the single ascertainment-correction formula to k-plex ascertainment, in which each unit has k or more affecteds. The correction is the reciprocal of the sum of the joint probabilities of all distinct sets of k persons in the unit, as if they were all affected. In extended families, two additional ascertainment schemes will be considered and explicit formulas will be presented. One of these schemes is "uniform-proband-status ascertainment," in which nonmembers of a given unit have the same chance as members to become probands if they are affected; the other scheme is the "inverse law of ascertainment," in which the chance that nonmembers of a unit will become probands for that unit decreases with degree of relationship. Several specific recommendations are made for further study.