×

zbMATH — the first resource for mathematics

Approximate Ewens formulae for symmetric overdominance selection. (English) Zbl 1010.62100
From the paper: Ewens distributions [see W.J. Ewens, Mathematical population genetics. (1979; Zbl 0422.92011)] arise naturally in a number of sampling problems in the biological and physical sciences. In population genetics, Ewens distributions have been used to describe samples at genetic loci which follow the “infinitely-many neutral alleles” model. The basic features of this model are: (i) alleles (distinct versions of a gene) are equivalent with respect to natural selection, and (ii) mutation, which occurs in any gene copy with probability \(u\) each generation, transforms the copy into a completely new allele.
We derive a family of approximate sampling distributions for the symmetric overdominance model of population genetics. The distributions are selective versions of the Ewens Sampling Formula, which gives sample likelihoods under a model of neutral evolution. We draw on basic results for the general selection model of S.N. Ethier and T.G. Kurtz [see Lect. Notes Biomath. 70, 72-86 (1987; Zbl 0644.92011)] and use mathematical tools well-suited for calculating expectations of symmetric functions of Poisson-Dirichlet atoms. We conclude by briefly examining a Human Leukocyte Antigen data set, in light of a distribution conditional on the number of sample atoms.

MSC:
62P10 Applications of statistics to biology and medical sciences; meta analysis
92D15 Problems related to evolution
92D10 Genetics and epigenetics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] ALDOUS, D. J. (1985). Exchangeability and related topics. In Ecole d’Été de Probabilités de SaintFlour XIII. Lecture Notes in Math. 1117. Springer, Berlin. · Zbl 0562.60042
[2] BILLINGSLEY, P. (1986). Probability and Measure. Wiley, New York. · Zbl 0649.60001
[3] CAVALLI-SFORZA, L. L., MENOZZI, P. and PIAZZA, A. (1994). The History and Geography of Human Genes. Princeton Univ. Press. · Zbl 0506.62093
[4] ETHIER, S. N. and KURTZ, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York. · Zbl 0592.60049
[5] ETHIER, S. N. and KURTZ, T. G. (1987). The infinitely-many-alleles model with selection as a measure-valued diffusion. Lecture Notes in Biomathematics 70 72-86. Springer, Berlin. · Zbl 0644.92011
[6] ETHIER, S. N. and KURTZ, T. G. (1994). Convergence to Fleming-Viot processes in the weak atomic topology. Stochastic Process. Appl. 54 1-27. · Zbl 0817.60029
[7] EWENS, W. J. (1972). The sampling theory of selectively neutral alleles. Theoret. Popululation Biol. 3 87-112. · Zbl 0245.92009
[8] EWENS, W. J. (1979). Mathematical Population Genetics. Springer, Berlin. · Zbl 0422.92011
[9] EWENS, W. J. (1990). Population genetics theory the past and the future. In Mathematical and Statistical Developments of Evolutionary Theory (S. Lessard, ed.) 177-227. Kluwer, Dordrecht. · Zbl 0718.92010
[10] FUERST, P. A., CHAKRABORTY, R. and NEI, M. (1977). Statistical studies on protein poly morphism in natural populations I: Distribution of single-locus heterozy gosity. Genetics 86 455-483.
[11] GEy ER, C. J. and THOMPSON, E. A. (1992). Constrained Monte Carlo maximum likelihood for dependent data. J. Roy. Statist. Soc. Ser. B 54 657-699. JSTOR:
[12] GRIFFITHS, R. C. (1983). Allele frequencies with genic selection. J. Math. Biol. 17 1-10. · Zbl 0549.92012
[13] JOHNSON, N. L., KOTZ, S. and BALAKRISHNAN, N. (1997). Discrete Multivariate Distributions. Wiley, New York. · Zbl 0868.62048
[14] JOy CE, P. (1994). Likelihood ratios for the infinite alleles model. J. Appl. Probab. 31 595-605. JSTOR: · Zbl 0812.62102
[15] JOy CE, P. (1995). Robustness of the Ewens Sampling Formula. J. Appl. Prob. 32 609-622. JSTOR: · Zbl 0836.62022
[16] JOy CE, P. and TAVARÉ, S. (1995). The distribution of rare alleles. J. Math. Biol. 33 602-618. · Zbl 0830.92014
[17] KIMURA, M. and CROW, J. F. (1964). The number of alleles that can be maintained in a finite population. Genetics 49 725-738.
[18] KINGMAN, J. F. C. (1975). Random discrete distributions. J. Roy. Statist. Soc. Ser. B 37 1-22. JSTOR: · Zbl 0331.62019
[19] KINGMAN, J. F. C. (1977). The population structure associated with the Ewens Sampling Formula. Theoret. Population Biol. 11 274-283. · Zbl 0421.92011
[20] PARHAM, P. and OHTA, T. (1996). Population biology of antigen presentation by MHC class I molecules. Science 272 67-74.
[21] PENTTINEN, A. (1984). Modelling interaction in spatial point patterns: parameter estimation by the maximum likelihood method. Jy väsky lä Studies in Computer Science, Economics and Statistics 7.
[22] PITMAN, J. (1995). Exchangeable and partially exchangeable random partitions. Probab. Theory Related Fields 102 145-158. · Zbl 0821.60047
[23] PITMAN, J. (1996). Some developments of the Blackwell-MacQueen Urn Scheme. In Statistics, Probability and Game Theory: Papers in Honor of David Blackwell (T. S. Ferguson, L. S. Shapley and J. B. MacQueen, eds.). IMS, Hay ward, CA.
[24] RIPLEY, B. D. (1987). Stochastic Simulation. Wiley, New York. · Zbl 0613.65006
[25] SATTA, Y., O’HUIGIN, C., TAKAHATA, N. and KLEIN, J. (1994). Intensity of natural selection at the Major Histocompatibility Complex loci. Proceedings of the National Academy of Sciences of the United States of America 91 7184-7188.
[26] STUART, A. and ORD, J. K. (1994). Kendall’s Advanced Theory of Statistics 1: Distribution Theory. Arnold, London.
[27] TSUJI, K., AIZAWA, M. and SASAZUKI, T. (eds.) (1992). HLA 1991: Proceedings of the Eleventh International Histocompatibility Workshop and Conference 1. Oxford Science Publications, Oxford.
[28] WATTERSON, G. A. (1974). The sampling theory of selectively neutral alleles. Adv. Appl. Probab. 6 463-488. JSTOR: · Zbl 0289.62020
[29] WATTERSON, G. A. (1976). The stationary distribution of the infinitely-many neutral alleles diffusion model. J. Appl. Probab. 13 639-651. JSTOR: · Zbl 0356.92012
[30] WATTERSON, G. A. (1977). Heterosis or neutrality? Genetics 85 789-814.
[31] WATTERSON, G. A. (1978). The homozy gosity test of neutrality. Genetics 88 405-417.
[32] WEBER, J. L. and WONG, C. (1993). Mutation of human short tandem repeats. Human Molecular Genetics 2 1123-1128.
[33] DAVIS, CA 95616 E-MAIL: mngrote@ucdavis.edu DEPARTMENT OF STATISTICS 367 EVANS HALL #3860 UNIVERSITY OF CALIFORNIA BERKELEY, CA 94720-3860 E-MAIL: terry@stat.berkeley.edu
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.