Unbalanced repeated-measures models with structured covariance matrices

The question of how to analyze unbalanced or incomplete repeated-measures data is a common problem facing analysts. We address this problem through maximum likelihood analysis using a general linear model for expected responses and arbitrary structural models for the within-subject covariances. Models that can be fit include standard univariate and multivariate models with incomplete data, random-effects models, and models with time-series and factor-analytic error structures. We describe Newton-Raphson and Fisher scoring algorithms for computing maximum likelihood estimates, and generalized EM algorithms for computing restricted and unrestricted maximum likelihood estimates. An example fitting several models to a set of growth data is included.     

Genomic runs of homozygosity record population history and consanguinity

The human genome is characterised by many runs of homozygous genotypes, where identical haplotypes were inherited from each parent. The length of each run is determined partly by the number of generations since the common ancestor: offspring of cousin marriages have long runs of homozygosity (ROH), while the numerous shorter tracts relate to shared ancestry tens and hundreds of generations ago. Human populations have experienced a wide range of demographic histories and hold diverse cultural attitudes to consanguinity. In a global population dataset, genome-wide analysis of long and shorter ROH allows categorisation of the mainly indigenous populations sampled here into four major groups in which the majority of the population are inferred to have: (a) recent parental relatedness (south and west Asians); (b) shared parental ancestry arising hundreds to thousands of years ago through long term isolation and restricted effective population size (N(e)), but little recent inbreeding (Oceanians); (c) both ancient and recent parental relatedness (Native Americans); and (d) only the background level of shared ancestry relating to continental N(e) (predominantly urban Europeans and East Asians; lowest of all in sub-Saharan African agriculturalists), and the occasional cryptically inbred individual. Moreover, individuals can be positioned along axes representing this demographic historic space. Long runs of homozygosity are therefore a globally widespread and under-appreciated characteristic of our genomes, which record past consanguinity and population isolation and provide a distinctive record of the demographic history of an individual's ancestors. Individual ROH measures will also allow quantification of the disease risk arising from polygenic recessive effects.     

Adaptive evolution in a spatially structured asexual population

We study the process of adaptation in a spatially structured asexual haploid population. The model assumes a local competition for replication, where each organism interacts only with its nearest neighbors. We observe that the substitution rate of beneficial mutations is smaller for a spatially structured population than that seen for populations without structure. The difference between structured and unstructured populations increases as the adaptive mutation rate increases. Furthermore, the substitution rate decreases as the number of neighbors for local competition is reduced. We have also studied the impact of structure on the distribution of adaptive mutations that fix during adaptation.     

The maintenance of sexual reproduction in a structured population

We present the results of a computer simulation model in which a sexual population produces an asexual mutant. We estimate the probability that the new asexual lineage will go extinct. We find that whenever the asexual lineage does not go extinct the sexual population is out-competed, and only asexual individuals remain after a sufficiently long period of time has elapsed. We call this type of outcome an asexual takeover. Our results suggest that, given repeated mutations to asexuality, asexual takeover is likely in an unstructured environment. However, if the environment is subdivided into demes that are connected by migration, then asexual takeover becomes less likely. The probability of asexual takeover declines towards zero as the number of demes increases and as the rate of migration decreases. The reason for this is that asexuality leads to a greater loss of fitness due to mutation and genetic drift, in comparison to what occurs under sexual reproduction. Population subdivision slows the spread of asexual lineages, which allows more time for the genetic degeneration caused by asexuality to take place.     

Travelling wave dynamics and self-organization in a spatio-temporally structured population

We analyse spatial population dynamics showing that periodic or period-like chaotic dynamics produce self-organization structures, such as travelling waves. We suggest that self-organized patterns are associated with spatial synchrony patterns that often depend on geographical distance between subpopulations. The population dynamics also show statistical spatial autocorrelation patterns. We contrast our theoretical simulations with empirical data on annual damages in young sapling stands caused by voles. We conclude, on the basis of the periodicity, synchrony, and spatial autocorrelation patterns, and our simulation results, that vole dynamics represent travelling waves in population dynamics. We suggest that because such synchrony patterns are frequently observed in natural populations, spatial self-organization may be more common in population dynamics than reported in the literature.     

Linkage disequilibrium and haplotype homozygosity in population samples genotyped at a high marker density

OBJECTIVE: Analyze the information contained in homozygous haplotypes detected with high density genotyping. METHODS: We analyze the genotypes of approximately 2,500 markers on chr 22 in 12 population samples, each including 200 individuals. We develop a measure of disequilibrium based on haplotype homozygosity and an algorithm to identify genomic segments characterized by non-random homozygosity (NRH), taking into account allele frequencies, missing data, genotyping error, and linkage disequilibrium. RESULTS: We show how our measure of linkage disequilibrium based on homozygosity leads to results comparable to those of R(2), as well as the importance of correcting for small sample variation when evaluating D'. We observe that the regions that harbor NRH segments tend to be consistent across populations, are gene rich, and are characterized by lower recombination. CONCLUSIONS: It is crucial to take into account LD patterns when interpreting long stretches of homozygous markers.     

Effective population size and population subdivision in demographically structured populations

A fast-timescale approximation is applied to the coalescent process in a single population, which is demographically structured by sex and/or age. This provides a general expression for the probability that a pair of alleles sampled from the population coalesce in the previous time interval. The effective population size is defined as the reciprocal of twice the product of generation time and the coalescence probability. Biologically explicit formulas for effective population size with discrete generations and separate sexes are derived for a variety of different modes of inheritance. The method is also applied to a nuclear gene in a population of partially self-fertilizing hermaphrodites. The effects of population subdivision on a demographically structured population are analyzed, using a matrix of net rates of movement of genes between different local populations. This involves weighting the migration probabilities of individuals of a given age/sex class by the contribution of this class to the leading left eigenvector of the matrix describing the movements of genes between age/sex classes. The effects of sex-specific migration and nonrandom distributions of offspring number on levels of genetic variability and among-population differentiation are described for different modes of inheritance in an island model. Data on DNA sequence variability in human and plant populations are discussed in the light of the results.     

Invasion and persistence of plant parasites in a spatially structured host population

Spatial structure is of central importance in the dynamics of plant-parasite interactions and is imposed by the growth habit and distribution of host plants and by parasite dispersal which is frequently restricted. To investigate the effects of spatial heterogeneity on the dynamics of plant parasites we introduce a simple model for epidemic development within a spatially structured host population. Here the host population is subdivided into a number of patches which are linked to allow for transmission from one patch to another with the connections defining the spatial structure of the host population. Three key parameters are identified that play a critical role in the ability of the parasite to invade and persist within the host population: the within-patch parasite basic reproductive number which characterises the infection dynamics at the local spatial scale; and the neighbourhood of interaction which describes which patches interact with which and the strength of coupling between patches within the neighbourhood which together characterise the spread of the parasite over larger spatial scales. Using both deterministic and stochastic formulations of the model, we investigate how the thresholds and probabilities of invasion and persistence are affected by these parameters, by demographic stochasticity and by differences in the initial level of infection.     

Impulsive culling of a structured population on two patches

We consider a model for a creature inhabiting two patches between which migration may occur. The creature is assumed to have a life cycle with two stages, namely juvenile and adult, giving rise to a delay differential system. The creature could represent an insect crop pest whilst the patches could represent neighbouring farms. Given that it is common to control crop pests by adult impulsive culling, we impose an adult impulsive culling regime on each patch. We find conditions on the regimes such that the pest will be eradicated on both patches simultaneously. The regime on one patch is assumed to be independent of the regime on the other patch to reflect the possibility that the patches represent farms with different owners where each owner has autonomy in their pest control decisions. In the special case where the birth functions on both patches are of an Allee type, we calculate explicit finite upper bounds for the number of culls needed on each patch to guarantee eradication. Simulations corroborate our theoretical results.     

Factors influencing the optimum sex ratio in a structured population

W. D. Hamilton (1967, Science 156, 477-488) calculated the optimum sex-ratio strategy for a population subdivided into local mating groups. He made three important assumptions: that the females founding each group responded precisely to the number of them initiating the group; that ail broods within a group matured synchronously; and that males were incapable of dispersing between groups. We have examined the effects of relaxing each of these assumptions and obtained the following results: (1) When broods mature asynchronously the optimum sex ratio is considerably more female biased than the Hamiltonian prediction. (2) Increasing male dispersal always decreases the optimum female bias to the sex ratio, but it is of particular interest that when moderate levels of dispersal are coupled with asynchrony of brood maturation then the optimum strategy is relatively insensitive to changes in foundress number. (3) When females cannot precisely determine the number of other foundresses initiating the group then the optimum strategy is almost exactly the strategy appropriate to a group of average size. These effects can be most easily understood in terms of local parental control (LPC) of the sex ratio. Through LPC a founding female can alter the mating success of her sons by altering the sex ratio of her brood. Asynchrony in the maturation of broods within a group increases the control that a founding female has over the mating success of her sons, whereas male dispersal reduces it. We have shown that the role of LPC and the role of inbreeding, which favors a female-biased sex ratio in haploidiploid species, are independent and that their effects can be combined into a single general formula r = (1-(r2/z2) E(alpha z/alpha r]/(1 + I). The concept of LPC can also be used to interpret two factors which have been proposed to select for the Hamiltonian sex ratios: local mate competition is LPC acting through sons; and sib mating is LPC acting through daughters.     

Complex transition to cooperative behavior in a structured population model

Cooperation plays an important role in the evolution of species and human societies. The understanding of the emergence and persistence of cooperation in those systems is a fascinating and fundamental question. Many mechanisms were extensively studied and proposed as supporting cooperation. The current work addresses the role of migration for the maintenance of cooperation in structured populations. This problem is investigated in an evolutionary perspective through the prisoner's dilemma game paradigm. It is found that migration and structure play an essential role in the evolution of the cooperative behavior. The possible outcomes of the model are extinction of the entire population, dominance of the cooperative strategy and coexistence between cooperators and defectors. The coexistence phase is obtained in the range of large migration rates. It is also verified the existence of a critical level of structuring beyond that cooperation is always likely. In resume, we conclude that the increase in the number of demes as well as in the migration rate favor the fixation of the cooperative behavior.     

Group beneficial norms can spread rapidly in a structured population

Group beneficial norms are common in human societies. The persistence of such norms is consistent with evolutionary game theory, but existing models do not provide a plausible explanation for why they are common. We show that when a model of imitation used to derive replicator dynamics in isolated populations is generalized to allow for population structure, group beneficial norms can spread rapidly under plausible conditions. We also show that this mechanism allows recombination of different group beneficial norms arising in different populations.     

The covariance of relatives derived from a random mating population.

Attention is drawn to errors common in the derivation of forms for the genotypic covariance of noninbred relatives from a Hardy-Weinberg population of diploids. A synthesis of Fisher's least-squares method of partitioning the genotypic variance and Malécot's probability method of expressing kinship, yields a general form. For one locus, the form is $\text{(P}{}_{\mathrm{ss}}\text{+ P}{}_{\mathrm{sd}}\text{+ P}{}_{\mathrm{ds}}\text{+ P}{}_{\mathrm{dd}}\text{)}\text{1}\text{2}\text{σ}{}_{\mathrm{a}}{}^2\text{+ (P}{}_{\mathrm{ss}}\text{P}{}_{\mathrm{dd}}\text{+ P}{}_{\mathrm{sd}}\text{P}{}_{\mathrm{ds}}\text{) σa}{}_{\mathrm{d}}{}^2$, where σ_a² is the additive genetic variance, α_d² is the variance of dominance deviations, p_ij is the probability that parental gamete i is identical by descent to parental gamete j, i = s, d indexes the parents of one relative, and j = s, d indexes those of the other. The form provides a framework for obtaining the covariance of relatives from an equilibrium population with linkage.     

Genetic variation versus recombination rate in a structured population of mice

The correlation between genetic variation and recombination rate was investigated in a structured mouse population. Nucleotide sequence data from 19 autosomal DNA loci from eight inbred strains of mouse (Mus musculus) sampled from three major subspecies were analyzed. The recombination rate was estimated from the comparison of genetic and physical map distances between markers flanking a 10-cM region of each locus. The strains were categorized into four groups (subpopulations) based on geography. By partitioning the genetic diversity into within-group and among-group variation, we detected a positive correlation between the recombination rate and nucleotide diversity within groups. The level of nucleotide differentiation among groups (G(ST)) showed a negative correlation with the rate of recombination. There was no significant correlation between recombination rate and nucleotide diversity when data from different subpopulations were pooled. No correlation was detected between recombination rate and nucleotide divergence of M. musculus and M. spicilegus. These patterns deviate from the strict neutral expectation under the constant nucleotide substitution rate, and they are likely to have been formed either by a hitchhiking effect of positively selected mutants or by background selection of deleterious mutants occurring in a subdivided population. Our series of comparisons show that because a real population always has some structure, incorporation of its information is important in detecting non-neutral evolution.     

Properties of components of covariance of inbred relatives and their estimates in a maize population

Summary Properties of three parameterizations, denoted as the C-model, D-model and Q-model, for covariances of inbred relatives under assumptions of no linkage or epistasis are explored and compared. Additive variance in an inbred population with inbreeding coefficient F, ² _AF =(1+F) ² _A where ² _A is additive variance in a panmictic population, if Q-model parameters Q _xx and Q _xy are both zero. Conditions sufficient for this to hold are presented in terms of gene frequencies and dominance contrasts (homozygotes vs. heterozygotes). Some other properties and potential uses of estimates of components in the models are also discussed. Estimates of components in the D-model and Q-model were calculated from a maize (Zea mays L.) study from which estimates of components in the C-model were previously published. Of particular interest were the covariance (Q _xy ) of effects of alleles at complete homozygosity with inbreeding depression effects, the covariance (D ₁) of additive effects at panmixia with inbreeding depression effects and the within-locus variance (D ₂, alias Q _xx ) of inbreeding depression effects. Estimates of Q _xy , D ₁, and D ₂ were small and nonsignificant in most cases. For ear height in the second year of the study, D ₂ appeared to be a major component. In some cases, results were obtained which had contradictory implications (negative D ₂ coupled with positive Q _xy or D ₁, and positive D ₂ coupled with negative ² _D ). A negative estimate of one or the other of ² _D or ² _A was obtained in one of the two within-year analyses for every character. Problems in getting realistic results were thought to be owing to excessive multicollinearity among the coefficients of the components in the expectations of the covariances of the kinds of relatives included in the study. Implications for future studies of this kind are discussed.Journal Article No. 87-3-14 of the Kentucky Agricultural Experiment Station published with the approval of the Director