Family-size distribution and Ewens' equivalence theorem

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.