The Sisters' Riddle in Age-Parity-Structured Stable Population

Toru SUZUKI

N. Goldman (1978) developed a method to estimate growth rates from sibling data. She showed that the expected number of younger sister exceeds that of older when a population is growing, and the relation reverses in a decreasing population. A formula was derived to estimate r from ratio of the number of younger to older sisters.
K. Wachter (1980) evaluated Goldman's method and examined a fundamental question. When ego counts an older sister, that sister always counts ego as a younger sister. Because of this symmetry, the expected number of older sisters must be equal to younger sisters. Why the ratio of younger to older can fluctuate by Wachter.
Both Goldman and Wachter based on Lotka's age-structured stable population. However, the lack of parity structure in this model caused certain difficulties, and they had to set strong assumptions such as ignorance of death.
This article is an attempt to examine how the ratio of sisters is related with growth rate in age-parity-structured population model. Use of parity specific rates allows the expected number of sisters S that is free from Poison distribution.

In above, N is the Net Reproduction Rate, I is the last parity, ‚Œ_x,I-1 is parity specific survivorship function, and m_x,I-1 is parity specific fertility rate.
The age-parity model also enables us to obtain birth order distribution. If we consider all daughters of one mother cohort, the proportion of i-th daughter in this group is given as follows.

It is easy to calculate the expected number of older sisters from this distribution. The result is exactly a half of S, which means the number of older sisters is always same to younger sisters. This is one side of the Sisters' Riddle. The situation changes when we directly inspect one daughter cohort. Birth order distribution in a cohort apparently depends on the growth rate.

In this case, ratio of older and younger sisters changes with r. This is the other side of the Sisters' Riddle.
Deviation of O_i' from O_i results from growth in size of mother cohorts. Positive r causes a bias to young mothers. If younger mothers tend to bear babies of earlier birth order, which is a natural assumption, the number of younger sisters exceeds older sisters in a growing population.
Thus the relationship between ratio of sisters and growth rate depends on the fertility schedules by birth order. Though the condition, which affirms the anticipated relationship, is natural, it is not logically true as is in Lotka's model.

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