Estimate the Slope Parameter in Replicated Linear Structural Relationship Model

Replication of observation allows consistent estimation of slope parameter of a linear structural model when the ratio of variances is unknown or when some external information about parameters is not available. In this paper, we look at the way a linear structural relationship model work by replicating observations with two different estimation methods of slope parameter and different cases of existence of outliers. The maximum likelihood estimate (MLE) and a new nonparametric robust estimation method  are used to estimate the slope parameter in replicated linear structural relationship model (RLSRM). The simulation studies and the application of real data are used to investigate the performance of the estimated parameters.Â

Modifiers of mutation-selection balance: general approach and the evolution of mutation rates

SummaryA general approach is developed to estimate secondary selection at a modifier locus that influences some feature of a population under mutation-selection balance. The approach is based on the assumption that the properties of all available genotypes at this locus are similar. Then mutation-selection balance and weak associations between genotype distributions at selectable loci and the modifier locus are established rapidly. In contrast, changes of frequencies of the modifier genotypes are slow, and lead to only slow and small changes of the other features of the population. Thus, while these changes occur, the population remains in a state of quasi-equilibrium, where the mutation-selection balance and the associations between the selectable loci and the modifier locus are almost invariant. Selection at the modifier locus can be estimated by calculating quasiequilibrium values of these associations. This approach is developed for the situation where distributions of the number of mutations per genome within the individuals with a given modifier genotype are close to Gaussian. The results are used to study the evolution of the mutation rate. Because beneficial mutations are ignored, secondary selection at the modifier locus always diminishes the mutation rate. The coefficient of selection against an allele which increases the mutation rate by υ is approximately υδ2/[U(2−ρ)] = υŝ, where υ is the genomic deleterious mutation rate, δ is the selection differential of the number of mutations per individual in units of the standard deviation of the distribution of this number in the population, ρ is the ratio of variances of the number of mutations after and before selection, and ŝ is the selection coefficient against a mutant allele in the quasiequilibrium population. However, the decline of the mutation rate can be counterbalanced by the cost of fidelity, which can lead to an evolutionary equilibrium mutation rate.

Dynamics of unconditionally deleterious mutations: Gaussian approximation and soft selection

SummaryThis paper studies the influence of two opposite forces, unidirectional unconditionally deleterious mutations and directional selection against them, on an amphimictic population. Mutant alleles are assumed to be equally deleterious and rare, so that homozygous mutations can be ignored. Thus, a genotype is completely described by its value with respect to a quantitative trait x, the number of mutations it carries, while a population is described by its distribution p(x) with mean M[p] and variance V[p] = σ2[p]. When mutations are only slightly deleterious, so that M » 1, before selection p(x) is close to Gaussian with any mode of selection. I assume that selection is soft in the sense that the fitness of a genotype depends on the difference between its value of x and M, in units of σ. This leads to a simple system of equations connecting the values of M and V in successive generations. This system has a unique and stable equilibrium, where U is the genomic deleterious mutation rate, δ is the selection differential for x in units of σ, and p is the ratio of variances of p(x) after and before selection. Both δ and ρ are parameters of the mode of soft selection, and do not depend on M or V. In an equilibrium population, the selection coefficient against a mutant allele is ŝ = δ2[U(2–ρ)]−1. The mutation load can be tolerable only if the genome degradation rate υ = U/σ is below 2. Other features of mutation-selection equilibrium are also discussed.