The essence of single ascertainment.

We propose a fundamental new definition of single ascertainment, namely, that single ascertainment is any ascertainment scheme in which P[pedigree is ascertained/true structure of pedigree] alpha p(theta), where p(theta) is some function of genetic parameters theta but is not a function of pedigree structure. Stated in words: Under single ascertainment, all pedigrees have equal (or proportional with respect to genetic parameters) probabilities of being ascertained, independent of pedigree size or structure. This new definition of single ascertainment allows us to show several results: (1) The correct likelihood consists of the probability of the data conditioned on the observed pedigree, divided by the function p(theta), whether sampling is "proband-independent" or "proband-dependent." (2) More-familiar definitions of single ascertainment all represent special cases of our definition. (3) When p(theta) represents the prevalence of the trait being studied, our definition corresponds to "classical" single ascertainment, i.e., ascertainment through a single "proband." However, the concept of p(theta) can also be generalized to represent the population frequency of configurations of affected relatives (such as affected sib pairs); we call this "generalized single ascertainment."