Genetic ascertainment with heterogeneous risk

Analysis of the bias of ascertainment is reformulated to deal with more general patterns commonly encountered in practice. The goal is to provide a unifying theory that will both replace the traditional, rather piecemeal, treatment of the problem and free it from certain restrictive assumptions. A compact algebraic method is furnished for analyzing the properties of the distributions by means of the probability generating function (PGF). The scope of the generalization is illustrated by applying it to the various classical patterns of bias of ascertainment. It is extended to other patterns in which the conditions of ascertainment, though more plausible, are also logically more complicated. It also accommodates cases hitherto inadequately dealt with, such as where the segregation ratios are heterogeneous (for example because of age-dependence); and cases where the ascertainment function is of arbitrary form and denies the authors such valuable, but demanding, assumptions as independence. Not only is the result unifying, but it leads to usable results in specific applications such as diseases that depend on age or birth order. While the commonest applications are in human genetics, there are many other issues (such as the use of batteries of tests) in which it is equally important.