Hodge S E
Am J Hum Genet. 1985 Jan;37(1):166-77.
Segregation analysis of a data set containing nuclear families of more than one sibship size is considered, and two different formulations of the likelihood are examined. One is the "separate-multinomials" formulation, which treats each family size as representing a separate multinomial distribution; the other is the "grand-multinomial" formulation, which treats the entire data set as representing one distribution. It is shown that these two formulations are equivalent, if and only if the population distribution of family sizes is completely unknown. However, if anything is known about the family-size distribution, the grand-multinomial formulation, although more cumbersome, makes more complete use of the data; moreover, it enables the use of one-child families in a segregation analysis. The relationship of this work to Ewens' equivalence theorem concerning "unconditional" and "conditional" likelihoods is discussed. The findings are illustrated with a simple example, and their practical relevance to real-life segregation analysis is discussed.
考虑对包含多个同胞规模的核心家庭的数据集进行分离分析，并研究似然性的两种不同表述。一种是“单独多项分布”表述，它将每个家庭规模视为代表一个单独的多项分布；另一种是“总体多项分布”表述，它将整个数据集视为代表一种分布。结果表明，当且仅当家庭规模的总体分布完全未知时，这两种表述才是等效的。然而，如果对家庭规模分布有所了解，总体多项分布表述虽然更繁琐，但能更充分地利用数据；此外，它使得在分离分析中能够使用独生子女家庭。讨论了这项工作与Ewens关于“无条件”和“有条件”似然性的等价定理之间的关系。通过一个简单的例子说明了这些发现，并讨论了它们在实际生活中的分离分析中的实际相关性。
