Murphy E A, Trojak J E
Am J Med Genet. 1987 Mar;26(3):667-81. doi: 10.1002/ajmg.1320260322.
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.
探讨了宾果 - 伽马模型（即竞争、独立、多次打击系统中最短等待时间）的遗传结构与表型结构之间的广泛关系。推导了有限算法以封闭形式计算联合分布和边缘分布；失效时间的分布、密度和危险函数；每个竞争系统因失效而死亡的各自总概率；以及原始矩和中心矩。该算法是生成函数的计算机对应物。竞争系统的数量及其各自的阶数和转移参数可以随意选择。经典的高尔顿 - 费舍尔理论不适用：均值和方差既不是可加的，其效应也不是齐次的；相反，平均生存时间较短的系统或多或少掩盖了平均生存时间较长的系统的影响。因此，即使任何一个组分等位基因的均值之间存在巨大差异，在表型上也可能几乎完全被掩盖；即使最大的遗传协方差在实际中可能仍然完全未被识别，并且遗传力估计接近零。比例特定死亡率是一个较不随意的指标，并且自然是可加的，但是，尽管它是基础参数的单调函数，但它既不是线性的也不是齐次的。
