Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-23T05:30:00.619Z Has data issue: false hasContentIssue false
  • English
  • Français

Estimating and testing selection: the two-alleles, genic selection diffusion model

Published online by Cambridge University Press:  01 July 2016

G. A. Watterson
G. A. Watterson*
Affiliation:
Monash University
*
∗Postal address: Department of Mathematics, Monash University, Clayton, Victoria 3168 Australia.
Get access Rights & Permissions [Opens in a new window]

Abstract

The estimation, and testing for the presence, of a selective advantage of one allele over another is considered. It is assumed that a population's allele frequencies are known from some initial time until fixation of one or other allele occurs. The statistics needed to perform the estimation and testing are the heterozygosity of the population summed over all generations, and the observation of which allele fixes. It is shown that certain asymptotic probability distributions arise which are similar to those proved by Brown and Hewitt for statistical inference from diffusion processes, but their results assumed that the diffusion had a stationary density which is not the case for alleles which fix.

The genetic diffusion may be transformed to Brownian motion with constant drift, and the inference questions concerning selection can be transformed to questions about the first exit of a Brownian motion from an interval. It is thus possible to construct significance tests, and to calculate the power of those tests, for detecting selection.

Keywords

INFERENCE FROM DIFFUSIONSBROWNIAN MOTIONPOWER OF HETEROZYGOSITY TESTNEUTRAL ALLELES HYPOTHESIS
Type
Research Article
Information
Advances in Applied Probability , Volume 11 , Issue 1 , March 1979 , pp. 14 - 30
Copyright
Copyright © Applied Probability Trust 1979 

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.)

References

Breiman, L. (1968) Probability. Addison-Wesley, Reading, Mass. Google Scholar
Brown, B. M. (1974) A sequential procedure for diffusion processes. In Studies in Probability and Statistics. Papers in Honour of E. J. G. Pitman, ed. Williams, E. J.. Jerusalem Academic Press.Google Scholar
Brown, B. M. and Hewitt, J. I. (1975a) Asymptotic likelihood theory for diffusion processes. J. Appl. Prob. 12, 228–238.Google Scholar
Brown, B. M. and Hewitt, J. I. (1975b) Inference for the diffusion branching process. J. Appl. Prob. 12, 588–594.Google Scholar
Crow, J. and Kumura, M. (1970) An Introduction to Population Genetics Theory. Harper and Row, New York.Google Scholar
Curtiss, J. H. (1942) A note on the theory of moment generating functions. Ann. Math. Statist. 13, 430–433.CrossRefGoogle Scholar
Dynkin, E. B. (1965) Markov Processes, Vol. 2. Springer-Verlag, Berlin.Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. 2, 2nd edn. Wiley, New York.Google Scholar
Holland, C. J. (1976) On a formula in diffusion processes in population genetics. Proc. Amer. Math. Soc. 54, 316–318.Google Scholar
Kozakiewicz, W. (1947) On the convergence of sequences of moment generating functions Ann. Math. Statist. 18, 61–69.Google Scholar
Maruyama, T. (1977) Stochastic Problems in Population Genetics. Lecture Notes in Biomathematics, 17, Springer-Verlag, Berlin.CrossRefGoogle Scholar
Maruyama, T. and Kimura, M. (1971) Some methods of treating continuous stochastic processes in population genetics. Japan. J. Genetics 46, 407–410.CrossRefGoogle Scholar
Nagylaki, T. (1974) The moments of stochastic integrals and the distribution of sojourn times. Proc. Natn. Acad. Sci. U.S.A. 71, 746–749.Google Scholar
Pakes, A. G. (1974) Some limit theorems for Markov chains with applications to branching processes. In Studies in Probability and Statistics. Paper in Honour of E. J. G. Pitman, ed. Williams, E. J.. Jerusalem Academic Press.Google Scholar
Watterson, G. A. (1978) The homozygosity test of neutrality. Genetics 88, 405–417.Google Scholar