Consistent ordered sampling distributions: characterization and convergence

1991 ◽  
Vol 23 (2) ◽  
pp. 229-258 ◽  
Author(s):  
Peter Donnelly ◽  
Paul Joyce
Keyword(s):  
Weak Convergence ◽  
Population Size ◽  
Wide Class ◽  
Unsolved Problem ◽  
Dirichlet Distribution ◽  
Allele Frequencies ◽  
Sampling Distributions ◽  
Population Distributions ◽  
Alternative Characterization ◽  
Central Result

This paper is concerned with models for sampling from populations in which there exists a total order on the collection of types, but only the relative ordering of types which actually appear in the sample is known. The need for consistency between different sample sizes limits the possible models to what are here called ‘consistent ordered sampling distributions'. We give conditions under which weak convergence of population distributions implies convergence of sampling distributions and conversely those under which population convergence may be inferred from convergence of sampling distributions. A central result exhibits a collection of ‘ordered sampling functions', none of which is continuous, which separates measures in a certain class. More generally, we characterize all consistent ordered sampling distributions, proving an analogue of de Finetti's theorem in this context. These results are applied to an unsolved problem in genetics where it is shown that equilibrium age-ordered population allele frequencies for a wide class of exchangeable reproductive models converge weakly, as the population size becomes large, to the so-called GEM distribution. This provides an alternative characterization which is more informative and often more convenient than Kingman's (1977) characterization in terms of the Poisson–Dirichlet distribution.

Consistent ordered sampling distributions: characterization and convergence

1991 ◽  
Vol 23 (02) ◽  
pp. 229-258 ◽  
Author(s):  
Peter Donnelly ◽  
Paul Joyce
Keyword(s):  
Weak Convergence ◽  
Population Size ◽  
Wide Class ◽  
Unsolved Problem ◽  
Dirichlet Distribution ◽  
Allele Frequencies ◽  
Sampling Distributions ◽  
Population Distributions ◽  
Alternative Characterization ◽  
Central Result

This paper is concerned with models for sampling from populations in which there exists a total order on the collection of types, but only the relative ordering of types which actually appear in the sample is known. The need for consistency between different sample sizes limits the possible models to what are here called ‘consistent ordered sampling distributions'. We give conditions under which weak convergence of population distributions implies convergence of sampling distributions and conversely those under which population convergence may be inferred from convergence of sampling distributions. A central result exhibits a collection of ‘ordered sampling functions', none of which is continuous, which separates measures in a certain class. More generally, we characterize all consistent ordered sampling distributions, proving an analogue of de Finetti's theorem in this context. These results are applied to an unsolved problem in genetics where it is shown that equilibrium age-ordered population allele frequencies for a wide class of exchangeable reproductive models converge weakly, as the population size becomes large, to the so-called GEM distribution. This provides an alternative characterization which is more informative and often more convenient than Kingman's (1977) characterization in terms of the Poisson–Dirichlet distribution.

POPULATION DYNAMICS OF SEX-DETERMINING ALLELES IN HONEY BEES AND SELF-INCOMPATIBILITY ALLELES IN PLANTS

Genetics ◽  
1979 ◽  
Vol 91 (3) ◽  
pp. 609-626 ◽  
Author(s):  
Shozo Yokoyama ◽  
Masatoshi Nei
Keyword(s):  
Population Dynamics ◽  
Population Size ◽  
Honey Bee ◽  
Honey Bees ◽  
Finite Population ◽  
Large Population ◽  
Allele Frequencies ◽  
Self Incompatibility ◽  
Overdominant Selection ◽  
Incompatibility Alleles

ABSTRACT Mathematical theories of the population dynamics of sex-determining alleles in honey bees are developed. It is shown that in an infinitely large population the equilibrium frequency of a sex allele is l/n, where n is the number of alleles in the population, and the asymptotic rate of approach to this equilibrium is 2/(3n) per generation. Formulae for the distribution of allele frequencies and the effective and actual numbers of alleles that can be maintained in a finite population are derived by taking into account the population size and mutation rate. It is shown that the allele frequencies in a finite population may deviate considerably from l/n. Using these results, available data on the number of sex alleles in honey bee populations are discussed. It is also shown that the number of self-incompatibility alleles in plants can be studied in a much simpler way by the method used in this paper. A brief discussion about general overdominant selection is presented.

Temporal changes in allele frequencies and low effective population size in greater prairie-chickens

2004 ◽  
Vol 13 (9) ◽  
pp. 2617-2630 ◽  
Author(s):  
JEFF A. JOHNSON ◽  
M. RENEE BELLINGER ◽  
JOHN E. TOEPFER ◽  
PETER DUNN
Keyword(s):  
Population Size ◽  
Effective Population Size ◽  
Temporal Changes ◽  
Allele Frequencies ◽  
Effective Population

scholarly journals Estimating the proportion of neutral mutants

1987 ◽  
Vol 50 (2) ◽  
pp. 155-163 ◽  
Author(s):  
G. A. Watterson
Keyword(s):  
Population Size ◽  
Estimation Method ◽  
Low Frequency ◽  
Practical Interest ◽  
Population Expansion ◽  
Allele Frequencies ◽  
New Method ◽  
Diploid Population ◽  
Deleterious Alleles ◽  
Deleterious Genes

SummaryKimura used the heterozygosity and the number of low-frequency alleles to estimate that about 14% of mutations are selectively neutral. The method is shown to be subject to biases and to disruption due to bottleneck effects. Let deleterious alleles have selective disadvantage, s, compared with neutral alleles and let Ne denote the effective diploid population size. The estimator, , of the proportion of neutral alleles is positively biased if (roughly) 4NeS < 25 or if 4Nes > 200. In the former case, one cannot adequately detect the different influences of deleterious and neutral alleles, whereas in the latter case, deleterious alleles will rarely appear in the sample. These difficulties cause the biases in , and are likely to cause similar biases for any estimation method based solely on allele frequencies. There is substantial sampling variability in in cases of practical interest, when data from 11 loci, or even as many as 31 loci, are pooled. If there has been a recent contraction in population size, will be positively biased, often yielding values greater than 1 or even being infinite. But after a recent expansion in population size, the heterozygosity will not have made as quick an increase and will be negatively biased. Population expansion alone can produce values close to those observed by Kimura, even if all alleles are neutral. In an appendix, a new method for simulating samples of neutral and deleterious genes is described.

Estimating the frequency of the oldest allele: a bayesian approach

1991 ◽  
Vol 23 (3) ◽  
pp. 456-475 ◽  
Author(s):  
Paul Joyce
Keyword(s):  
Stationary Distribution ◽  
Bayesian Approach ◽  
Dirichlet Distribution ◽  
Allele Frequencies ◽  
Posterior Distributions ◽  
Bayes Estimators ◽  
Infinite Alleles Model

In this paper we calculate posterior distributions associated with a version of the Poisson–Dirichlet distribution called the GEM. The GEM has been shown (by several authors) to be the limiting stationary distribution for allele frequencies listed in age order associated with the neutral infinite alleles model. In view of this result, we use our posterior distributions to calculate Bayes estimators for the frequency of the oldest allele given a sample.

The limiting properties of population distributions with particular application to manpower planning

1983 ◽  
Vol 20 (1) ◽  
pp. 19-30 ◽  
Author(s):  
Mark Woodward
Keyword(s):  
Population Size ◽  
Stable Distribution ◽  
Population Distribution ◽  
Manpower Planning ◽  
Expected Value ◽  
Practical Applications ◽  
Population Distributions ◽  
Number Of Classes

A model for predicting expected-value population distributions is developed, assuming that all movements are Markovian and time-homogeneous. Each individual is classified by the amount of time he has spent in the population and by which of a number of classes, of an unspecified nature, he inhabits. The limiting properties of the population distribution are derived, and, in particular, conditions for convergence to a stable distribution are given.Some discussion of the relevance of the theory to practical applications is given, primarily to manpower planning when recruitment occurs purely to maintain a specified overall population size.

Microsatellite loci over a thirty-three year period for a malaria parasite (Plasmodium mexicanum): bottleneck in effective population size and effect on allele frequencies

Parasitology ◽  
2012 ◽  
Vol 140 (1) ◽  
pp. 21-28 ◽  
Author(s):  
J. J. SCHALL ◽  
K. M. ST. DENIS
Keyword(s):  
Population Size ◽  
Effective Population Size ◽  
Malaria Parasite ◽  
Genetic Data ◽  
Parasite Prevalence ◽  
Allele Frequencies ◽  
The Other ◽  
Effective Population ◽  
Blood Smears ◽  
Parasite Plasmodium

SUMMARYChanges in population allele frequencies may be driven by several forces, including selection and drift, and are revealed only by sampling over many generations. Such studies, however, are rare for protist parasites. Microsatellite allele frequencies for 4 loci were followed in a population of Plasmodium mexicanum, a malaria parasite of lizards in California USA at 1 site from 1978 to 2010. Rapid turnover of the lizards indicates the parasite was studied for a minimum of 33 transmission cycles and possibly twice that number. Sample sizes ranged from 841 to 956 scored parasite clones per locus. DNA was extracted from frozen dried blood and blood removed from stained blood smears from the earliest years, and a verification study demonstrated DNA from the blood smears provided valid genetic data. Parasite prevalence and effective population size (Ne) dropped after 2000, remaining lower for the next decade. For 2 loci, allele frequencies appeared stable for the first 2 decades of the study, but changed more rapidly after the decline in prevalence. Allele frequencies changed more gradually for the other 2 loci. Genetic drift could account for changes in allele frequencies, especially after the drop in prevalence and Ne, but the force of selection could also have driven the observed patterns.

A DIRECT ASSESSMENT OF THE ROLE OF GENETIC DRIFT IN DETERMINING ALLELE FREQUENCY VARIATION IN POPULATIONS OF EUPHYDRYAS EDITHA

Genetics ◽  
1985 ◽  
Vol 110 (3) ◽  
pp. 495-511
Author(s):  
Laurence D Mueller ◽  
Bruce A Wilcox ◽  
Paul R Ehrlich ◽  
David G Heckel ◽  
Dennis D Murphy
Keyword(s):  
Population Size ◽  
Effective Population Size ◽  
Genetic Drift ◽  
Allele Frequencies ◽  
Frequency Variation ◽  
Effective Population ◽  
Variable Environment ◽  
Euphydryas Editha ◽  
Two Populations

ABSTRACT Estimates of allele frequencies at six polymorphic loci were collected over eight generations in two populations of Euphydryas editha. We have estimated, in addition, the effective population size for each generation for both populations with results from mark-recapture and other field data. The variation in allele frequencies generated by random genetic drift was then studied using computer simulations and our direct estimates of effective population size. Substantial differences between observed values and computer-generated expected values assuming drift alone were found for three loci (Got, Hk, Pgi) in one population. These observations are consistent with natural selection in a variable environment.

Weak convergence to the coalescent in neutral population models

1999 ◽  
Vol 36 (2) ◽  
pp. 446-460 ◽  
Author(s):  
M. Möhle
Keyword(s):  
Weak Convergence ◽  
Population Size ◽  
Large Class ◽  
Total Population ◽  
Population Models ◽  
Total Population Size ◽  
Hitting Probabilities ◽  
Time Scaling ◽  
Convergence Behaviour ◽  
Or Genes

For a large class of neutral population models the asymptotics of the ancestral structure of a sample of n individuals (or genes) is studied, if the total population size becomes large. Under certain conditions and under a well-known time-scaling, which can be expressed in terms of the coalescence probabilities, weak convergence in DE([0,∞)) to the coalescent holds. Further the convergence behaviour of the jump chain of the ancestral process is studied. The results are used to approximate probabilities which are of certain interest in applications, for example hitting probabilities.

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