Selective limitation of sibship size.

The belief that a morbid condition is attributable to their genes may lead parents to limit the size of the sibship in two ways. If the belief precedes the birth of an affected individual (A), the parents may decline to take the risk of producing another. We may then say that the criterion of termination is A = 1. If they have no such preconception, the birth of a second affected sib may lead them to the same conclusion with the same result. We may then say that the criterion of termination is A = 2. Such contingencies raise the question: does the decision to terminate the sibship affect the expected proportion of affected individuals in an otherwise random selection of fraternities? The issue so stated arises frequently in familial studies; and the answer is not so simple as it might appear to be. To clarify it, we shall initially postulate a fixed target value of s, the size the sibship would attain in the absence of any indication relevant to the criterion of termination. This assumption implies the existence of some fraternities which consist of s members, none of them affected. If p = (1q) be the probability that an individual of given parentage will be affected, the probability of this occurrence is qS; and p is also the expected proportion of affected sibs in a complete pool of s-fold fraternities chosen randomwise. We may hope to obtain a complete pool in this sense, if the criterion of ascertainment is the phenotype of one or both parents, as when the relevant morbid condition is hereditary in the sense formerly current in medical literature. The problem is then on all fours with the issue: what would happen if all parents terminated the sibship on the arrival of a boy ? When our concern is with so-called familial conditions, the method of ascertainment excludes fraternities containing no affected members. The expected proportion of affected sibs in a pool of a-fold fraternities, otherwise chosen at random, will then be greater than p, being in fact