Theory of genetic counseling for rare recessive traits with consanguinity

The theory of assessing genetic risk for rare autosomal recessive traits in the face of extensive consanguinity is explored. A general lemma is proved that justifies the partition of posterior probability and recursive incorporation of it, as a means of reducing the logic of analysis to steps of manageable complexity: the conditions are that either the components should be independent or that each component, as it is added, is conditioned on the occurrence of all earlier components. An example is given of a pedigree in which the tracing of a gene by the method of expected proportions is misleading, and another that points up the importance of the posterior information. Four classes of patterns are identified and appropriate methods devised for handling them: the single consanguineous loop, multiple mutually-external loops, nested loops, and multiple closures. The latter problem may be treated in general by the use of truncated generating functions, with or without the use of matrix methods.