scholarly journals A comparison of two methods for predicting changes in the distribution of gene frequency when selection is applied repeatedly to a finite population

1969 ◽  
Vol 13 (2) ◽  
pp. 117-126 ◽  
Author(s):  
Derek J. Pike
Keyword(s):  
Iterative Procedure ◽  
Gene Frequency ◽  
Finite Population ◽  
Transition Matrices ◽  
Single Cycle ◽  
Gene Frequencies ◽  
Mean And Variance ◽  
The Mean ◽  
The One ◽  
Selection Of

Robertson (1960) used probability transition matrices to estimate changes in gene frequency when sampling and selection are applied to a finite population. Curnow & Baker (1968) used Kojima's (1961) approximate formulae for the mean and variance of the change in gene frequency from a single cycle of selection applied to a finite population to develop an iterative procedure for studying the effects of repeated cycles of selection and regeneration. To do this they assumed a beta distribution for the unfixed gene frequencies at each generation.These two methods are discussed and a result used in Kojima's paper is proved. A number of sets of calculations are carried out using both methods and the results are compared to assess the accuracy of Curnow & Baker's method in relation to Robertson's approach.It is found that the one real fault in the Curnow-Baker method is its tendency to fix too high a proportion of the genes, particularly when the initial gene frequency is near to a fixation point. This fault is largely overcome when more individuals are selected. For selection of eight or more individuals the Curnow-Baker method is very accurate and appreciably faster than the transition matrix method.

scholarly journals On sojourn times at particular gene frequencies

1975 ◽  
Vol 25 (2) ◽  
pp. 89-94 ◽  
Author(s):  
Edward Pollak ◽  
Barry C. Arnold
Keyword(s):  
Markov Chains ◽  
Gene Frequency ◽  
Overlapping Generations ◽  
Finite Population ◽  
Diffusion Method ◽  
Gene Frequencies ◽  
Sojourn Times ◽  
The Mean ◽  
Number Of Visits ◽  
Finite Markov Chains

SUMMARYThe distribution of visits to a particular gene frequency in a finite population of size N with non-overlapping generations is derived. It is shown, by using well-known results from the theory of finite Markov chains, that all such distributions are geometric, with parameters dependent only on the set of bij's, where bij is the mean number of visits to frequency j/2N, given initial frequency i/2N. The variance of such a distribution does not agree with the value suggested by the diffusion method. An improved approximation is derived.

scholarly journals Quantifying superspreading for COVID-19 using Poisson mixture distributions

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Cécile Kremer ◽  
Andrea Torneri ◽  
Sien Boesmans ◽  
Hanne Meuwissen ◽  
Selina Verdonschot ◽  
...  
Keyword(s):  
Disease Transmission ◽  
Negative Binomial ◽  
Mixture Distributions ◽  
Infectious Individual ◽  
Transmission Process ◽  
Mean And Variance ◽  
The Mean ◽  
The One ◽  
Variance Estimates ◽  
Selection Of

AbstractThe number of secondary cases, i.e. the number of new infections generated by an infectious individual, is an important parameter for the control of infectious diseases. When individual variation in disease transmission is present, like for COVID-19, the distribution of the number of secondary cases is skewed and often modeled using a negative binomial distribution. However, this may not always be the best distribution to describe the underlying transmission process. We propose the use of three other offspring distributions to quantify heterogeneity in transmission, and we assess the possible bias in estimates of the mean and variance of this distribution when the data generating distribution is different from the one used for inference. We also analyze COVID-19 data from Hong Kong, India, and Rwanda, and quantify the proportion of cases responsible for 80% of transmission, $$p_{80\%}$$ p 80 % , while acknowledging the variation arising from the assumed offspring distribution. In a simulation study, we find that variance estimates may be biased when there is a substantial amount of heterogeneity, and that selection of the most accurate distribution from a set of distributions is important. In addition we find that the number of secondary cases for two of the three COVID-19 datasets is better described by a Poisson-lognormal distribution.

Expected consequences of the segregation of a major gene in a sheep population in relation to observations on the ovulation rate of a flock of Cambridge sheep

1990 ◽  
Vol 51 (2) ◽  
pp. 277-282 ◽  
Author(s):  
J. B. Owen ◽  
C. J. Whitaker ◽  
R. F. E. Axford ◽  
I. Ap Dewi
Keyword(s):  
Gene Frequency ◽  
Major Gene ◽  
Ovulation Rate ◽  
Phenotypic Effect ◽  
Data Set ◽  
Sheep Population ◽  
Mean Values ◽  
Population Mean ◽  
Mean And Variance ◽  
The Mean

ABSTRACTA simple model was derived relating the phenotypic effect (g) of a major gene to observed values of the population mean and variance for a trait, at specified values of the major gene frequency and at specified basal values of the population mean and variance (in the absence of the major gene). This model was applied to a total of 549 observed values of ovulation rate in ewes of the Cambridge breed at Bangor under a range of assumptions. The mean values of ovulation rate were 2·44 for 243 ewes of 1 year of age and 37·54 for 306 ewes of 2 and 3 years of age with a coefficient of variation for both age sets of 0·50.The results indicate a minimum value for g, in this data set, of 1·07 for 1 year old and 1·72 for 2 and 3 year old ewes. The results are also consistent with a frequency value in the region of 0·3 to 0·4, with the absence of dominance and with a reasonable concordance with Hardy-Weinburg equilibrium. The results also indicate that the value of g varies according to the background phenotype since it is lower for younger as compared with older ewes.

scholarly journals Effects of dominance and size of population on response to mass selection

1961 ◽  
Vol 2 (2) ◽  
pp. 177-188 ◽  
Author(s):  
Ken-Ichi Kojima
Keyword(s):  
Gene Frequency ◽  
Finite Size ◽  
Prediction Equation ◽  
Small Population ◽  
Mass Selection ◽  
Gene Frequencies ◽  
Random Drift ◽  
Expected Gain ◽  
The Mean ◽  
Frequency Changes

A theory of mass selection in a small population was developed, and the mean change in gene frequencies, the variance of gene frequency changes and the expected gain in the mean phenotypic value of an offspring population were formulated in terms of a generalized selection differential and the additive and dominance effects of genes.The magnitude of the variance of changes in gene frequency was compared with the magnitude of the variance expected from the genetic random drift in a population with the same gene frequency and of the same size in absence of selection. The former was found to be usually smaller than the latter when the gene frequency ranged from intermediate to high and when selection was directed for a high performance.The usual prediction equation for gain from selection in an infinite population was compared with the expected gain formula derived for a small population. The size of the population did not cause a serious difference between the two expected gains when there was no dominance effect of genes. Dominance alone could cause the usual prediction to be slightly more biased. The joint effects of the finite size of population and dominance gene action could amount to a considerable bias in the usual prediction equation. Such a bias can be, in the main, accounted for by the inbreeding depression.

the ‘Area Under the Curve’ or AUC. The AUC is taken as a measure of exposure of the drug to the subject. The peak or maximum concen-tration is referred to as Cmax and is an important safety measure. For regulatory approval of bioequivalence it is necessary to show from the trial results that the mean values of AUC and Cmax for T and R are not significantly different. The AUC is calculated by adding up the ar-eas of the regions identified by the vertical lines under the plot in Figure 7.1 using an arithmetic technique such as the trapezoidal rule (see, for example, Welling, 1986, 145–149, Rowland and Tozer, 1995, 469–471). Experience (e.g., FDA Guidance, 1992, 1997, 1999b, 2001) has dictated that AUC and Cmax need to be transformed to the natural logarithmic scale prior to analysis if the usual assumptions of normally distributed errors are to be made. Each of AUC and Cmax is analyzed separately and there is no adjustment to significance levels to allow for multiple testing (Hauck et al., 1995). We will refer to the derived variates as log(AUC) and log(Cmax), respectively. In bioequivalence trials there should be a wash-out period of at least five half-lives of the drugs between the active treatment periods. If this is the case, and there are no detectable pre-dose drug concentrations, there is no need to assume that carry-over effects are present and so it is not necessary to test for a differential carry-over effect (FDA Guidance, 2001). The model that is fitted to the data will be the one used in Section 5.3 of Chapter 5, which contains terms for subjects, periods and treatments. Following common practice we will also fit a sequence or group effect and consider subjects as a random effect nested within sequence. An example of fitting this model will be given in the next section. In the following sections we will consider three forms of bioequivalence: average (ABE), population (PBE) and individual (IBE). To simplify the following discussion we will refer only to log(AUC); the discussion for log(Cmax) is identical. To show that T and R are average bioequivalent it is only necessary to show that the mean log(AUC) for T is not significantly different from the mean log(AUC) for R. In other words we need to show that, ‘on average’, in the population of intended patients, the two drugs are bioequivalent. This measure does not take into account the variability of T and R. It is possible for one drug to be much more variable than the other, yet be similar in terms of mean log(AUC). It was for this reason that PBE was introduced. As we will see in Section 7.5, the measure of PBE that has been recommended by the regulators is a mixture of the mean and variance of the log(AUC) values (FDA Guidance, 1997, 1999a,b, 2000, 2001). Of course, two drugs could be similar in mean and variance over the

2003 ◽  
pp. 350-350
Keyword(s):  
Multiple Testing ◽  
Random Effect ◽  
Area Under The Curve ◽  
Mean Values ◽  
Fda Guidance ◽  
Significance Levels ◽  
Mean And Variance ◽  
The Mean ◽  
Carry Over ◽  
The One

scholarly journals The average number and the variance of generations at particular gene frequency in the course of fixation of a mutant gene in a finite population

1972 ◽  
Vol 19 (2) ◽  
pp. 109-113 ◽  
Author(s):  
Takeo Maruyama
Keyword(s):  
Computer Simulations ◽  
Diffusion Approximation ◽  
Gene Frequency ◽  
Finite Population ◽  
Mutant Gene ◽  
Gene Frequencies ◽  
Explicit Formulas ◽  
Two Generations

SUMMARYIn the case of an allele which is going to become fixed in a population, the average number of generations for which the population assumes particular gene frequencies is investigated, using the diffusion approximation. Explicit formulas were obtained and they were checked by computer simulations. As a particular case, it is shown that if a new mutant that is selectively neutral is eventually fixed in a population of size N, it spends two generations on average at each of the intermediate frequencies (1/2N, 2/2N, …, (2N−1)/2N), and the variance at each frequency is four generations.

scholarly journals Models of long-term artificial selection in finite population with recurrent mutation

1986 ◽  
Vol 48 (2) ◽  
pp. 125-131 ◽  
Author(s):  
William G. Hill ◽  
Jonathan Rasbash
Keyword(s):  
Population Size ◽  
First Generation ◽  
Finite Population ◽  
Effective Population ◽  
Term Selection ◽  
Cumulative Response ◽  
Mean And Variance ◽  
The Mean ◽  
Selection For

SummaryThe effects of mutation on mean and variance of response to selection for quantitative traits are investigated. The mutants are assumed to be unlinked, to be additive, and to have their effects symmetrically distributed about zero, with absolute values of effects having a gamma distribution. It is shown that the ratio of expected cumulative response to generation t from mutants, , and expected response over one generation from one generation of mutants, , is a function of t/N, where t is generations and N is effective population size. Similarly, , is a function of t/N, where is the increment in genetic variance from one generation of mutants. The mean and standard deviation of response from mutations relative to that from initial variation in the population, in the first generation, are functions of . Evaluation of these formulae for a range of parameters quantifies the important role that population size can play in response to long-term selection.

scholarly journals The effect of repeated cycles of selection and regeneration in populations of finite size

1968 ◽  
Vol 11 (1) ◽  
pp. 105-112 ◽  
Author(s):  
R. N. Curnow ◽  
L. H. Baker
Keyword(s):  
Gene Frequency ◽  
Random Mating ◽  
Finite Size ◽  
Transition Matrices ◽  
Gene Frequencies ◽  
Multiple Alleles ◽  
Expected Gain ◽  
Approximate Form ◽  
Breeding Programmes ◽  
Selection Intensities

Kojima's (1961) approximate formulae for the mean and variance of the change in gene frequency from a single cycle of selection applied to a finite population are used to develop an iterative method for studying the effects of repeated cycles of selection and random mating. This is done by assuming a particular, but flexible and probably realistic, approximate form for the distribution of gene frequencies at each generation.The method gives for each generation the first two moments of the gene frequency distribution, the expected gain from selection, the probabilities of fixation and also the variability of gain. The variability of gain is of considerable importance in evolution, selection experiments and in plant and animal breeding programmes.Kojima's (1961) formulae have been extended to allow for differentiation between males and females. Hence different selection intensities and population sizes for the two sexes can be studied. Selfing with selection is considered separately. Extensions to cover simple examples of multiple alleles, linkage and epistasis are possible. Reference is made to previous work using transition matrices.

scholarly journals THE EFFECT OF INTRAGENIC RECOMBINATION ON THE NUMBER OF ALLELES IN A FINITE POPULATION

Genetics ◽  
1978 ◽  
Vol 88 (4) ◽  
pp. 829-844 ◽  
Author(s):  
Curtis Strobeck ◽  
Kenneth Morgan
Keyword(s):  
Finite Population ◽  
Natural Populations ◽  
Intragenic Recombination ◽  
Infinite Allele Model ◽  
Site Model ◽  
Significant Process ◽  
Mean And Variance ◽  
The Mean ◽  
Allele Model ◽  
Size Data

ABSTRACT A two-site infinite allele model is constructed to study the effect of intragenic recombination on the number of neutral alleles and the distribution of their frequencies in a finite population. The results of theory and Monte Carlo simulation of the two-site model demonstrate that intragenic recombination significantly increases the mean and variance of the number of alleles when the rates of mutation and recombination are as large as the reciprocal of the population size. Data from natural populations indicate that this may be a significant process in generating variation and determining its distribution.

scholarly journals THE ISLAND MODEL WITH STOCHASTIC MIGRATION

Genetics ◽  
1979 ◽  
Vol 91 (1) ◽  
pp. 163-176
Author(s):  
Thomas Nagylaki
Keyword(s):  
Migration Rate ◽  
Gene Frequency ◽  
Finite Population ◽  
Island Model ◽  
Diploid Population ◽  
Absolute Value ◽  
Evolutionary Forces ◽  
Population Selection ◽  
Mean And Variance ◽  
Normally Distributed

ABSTRACT The island model with stochastically variable migration rate and immigrant gene frequency is investigated. It is supposed that the migration rate and the immigrant gene frequency are independent of each other in each generation, and each of them is independently and identically distributed in every generation. The treatment is confined to a single diallelic locus without mutation. If the diploid population is infinite, selection is absent and the immigrant gene frequency is fixed, then the gene frequency on the island converges to the immigrant frequency, and the logarithm of the absolute value of its deviation from it is asymptotically normally distributed. Assuming only neutrality, the evolution of the exact mean and variance of the gene frequency are derived for an island with finite population. Selection is included in the diffusion approximation: if all evolutionary forces have comparable roles, the gene frequency will be normally distributed at all times. All results in the paper are completely explicit.

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