Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-23T12:15:53.742Z Has data issue: false hasContentIssue false

Application of the Galton-Watson process to the kin number problem

Published online by Cambridge University Press:  01 July 2016

W. A. O'n. Waugh*
Affiliation:
The University of Toronto
*
Postal Address: Department of Statistics, The University of Toronto, Sidney Smith Hall, Toronto, Ontario M5S 1A1, Canada.

Abstract

The kin number problem concerns the relationship between the distribution of the number of offspring of a randomly chosen individual in a population, and that of the number of relatives of various degrees of affinity of a randomly chosen individual (referred to as ‘Ego’). This problem is considered in terms of a population consisting of a large number of simultaneously developing Galton-Watson family trees. By time-reversal starting with the epoch at which Ego is sampled, it is shown that the number of offspring of Ego's parent (Ego plus siblings) and the number of offspring of Ego's grandparent (Ego's parent plus Ego's parent's siblings) have the same distribution, and the probability generating function (p.g.f.) is obtained in terms of the reproduction p.g.f. Further development of the method yields the joint p.g.f. of any number of generations prior, and subsequent, to that of Ego. Means and variances are obtained, and numerical examples are given based on data for the reproduction p.g.f. in two contrasting human populations. Various applications including demography and cancer research are discussed.

Type
Research Article
Copyright
Copyright © Applied Probability Trust 1981 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Research supported in part by an Operating Grant (No. A5304) from the Natural Sciences and Engineering Research Council Canada.

References

Adams, K. J. (1981) The role of children in the changing socioeconomic strategies of the Guyanese Caribs. Canad. J. Anthropol. Google Scholar
Burks, B. S. (1933) A statistical method for estimating the distribution of sizes of completed fraternities in a population represented by a random sampling of individuals. J. Amer. Statist. Assoc. 28, 388394.Google Scholar
Gokalp, C. (1979) Le réseau familial. Population 6, 10771094.Google Scholar
Goodman, L. A., Keyfitz, N. and Pullum, T. W. (1974) Family formation and the frequency of various kinship relationships. Theoret. Popn Biol. 5, 127.CrossRefGoogle ScholarPubMed
Goodman, L. A., Keyfitz, N. and Pullum, T. W. (1975) Addendum to ‘Family formation and the frequency of various kinship relationships’. Theoret. Popn Biol. 8, 376381.Google Scholar
Hwang, Tea-Yuan, and Wang, Nae-Sheng (1979) On best fractional linear generating function bounds. J. Appl. Prob. 16, 449453.CrossRefGoogle Scholar
Keyfitz, N. (1968) Introduction to the Mathematics of Populations. Addison-Wesley, Reading, Ma. Google Scholar
Keyfitz, N. (1977) Applied Mathematical Demography. Wiley, New York.Google Scholar
Le Bras, H. (1973) Parents, grandparents, bisaieux. Population 1, 938.Google Scholar
Lotka, A. J. (1939) Théorie analytique des associations biologiques, Vol. 2. Actualités scientifiques et industrielles 780, 123136, Hermann, Paris.Google Scholar
Lynch, H. T., Guirgis, H. D., Lynch, P. M., Lynch, J. F. and Harris, R. D. (1977a) Familial cancer syndromes, a survey. Cancer 39, 18671881.Google Scholar
Lynch, H. T., Guirgis, H. D., Lynch, P. M., Lynch, J. F. and Harris, R. D. (1977b) Role of heredity in multiple primary cancer. Cancer 40, 18491854.3.0.CO;2-U>CrossRefGoogle ScholarPubMed
McKusick, V. A. (1965) Some computer applications to problems in human genetics. Methods of Information in Medicine IV 4, 183189.Google Scholar
Morgan, W. K. C. (1979) Industrial carcinogens: the extent of the risk. Thorax 34, 431433.Google Scholar
Whittle, P. (1970) Probability. Penguin Books, Harmondsworth.Google Scholar