The bingo model of survivorship. III. Genetic principles

The broad relationships are explored between the genetic and the phenotypic structures of the bingo-gamma model (ie, the shortest waiting time among competing, independent, multiple-hit systems). Finite algorithms are derived to compute in closed form the joint and marginal distributions; the distribution, density, and hazard functions of time to failure; the respective total probabilities of dying from failure of each competing system; and the raw and central moments. The algorithm is the computer counterpart of a generating function. The number of competing systems and their individual orders and transition parameters may be chosen at will. Classical Galton-Fisher theory does not apply: neither means nor variances are additive nor are their effects homogeneous; rather, those systems with shorter mean survival more or less mask the impact of those with longer means. Thus even huge differences among means for alleles of any one component may be almost totally concealed phenotypically; even the maximal genetic covariation may in practice remain totally unrecognized and the heritability estimated close to zero. The proportional specific mortality is a less capricious index and is naturally additive, but, though a monotonic function of the underlying parameters, it is neither linear nor homogeneous.