The maximum non-unit eigenvalue for the Wright-Fisher model of random genetic drift incorporating selection

The largest non-unit eigenvalue λ of the transition matrix for the Wright-Fisher Markov chain model of random genetic drift is found numerically with selective advantages of genotypes taken into account. Polynomials in the selection coefficients are fitted to λ in order to summarize the behaviour of λ with varying selection. Also found are the values of the selective advantages which give rise to an acceleration to the rate of fixation of alleles. These values are compared to results for the diffusion approximation to the Wright-Fisher model.     

Bayesian extrapolation of space-time trends in cancer registry data

We apply a full Bayesian model framework to a dataset on stomach cancer mortality in West Germany. The data are stratified by age group, year, and district. Using an age-period-cohort model with an additional spatial component, our goal is to investigate whether there is evidence for space-time interactions in these data. Furthermore, we will determine whether a period-space or a cohort-space interaction model is more appropriate to predict future mortality rates. The setup will be fully Bayesian based on a series of Gaussian Markov random field priors for each of the components. Statistical inference is based on efficient algorithms to block update Gaussian Markov random fields, which have recently been proposed in the literature.     

Identifying differentially expressed genes in cancer patients using a non-parameter Ising model

Identification of genes and pathways involved in diseases and physiological conditions is a major task in systems biology. In this study, we developed a novel non-parameter Ising model to integrate protein-protein interaction network and microarray data for identifying differentially expressed (DE) genes. We also proposed a simulated annealing algorithm to find the optimal configuration of the Ising model. The Ising model was applied to two breast cancer microarray data sets. The results showed that more cancer-related DE sub-networks and genes were identified by the Ising model than those by the Markov random field model. Furthermore, cross-validation experiments showed that DE genes identified by Ising model can improve classification performance compared with DE genes identified by Markov random field model.     

A Bayesian Approach to Detect Quantitative Trait Loci Using Markov Chain Monte Carlo

Markov chain Monte Carlo (MCMC) techniques are applied to simultaneously identify multiple quantitative trait loci (QTL) and the magnitude of their effects. Using a Bayesian approach a multi-locus model is fit to quantitative trait and molecular marker data, instead of fitting one locus at a time. The phenotypic trait is modeled as a linear function of the additive and dominance effects of the unknown QTL genotypes. Inference summaries for the locations of the QTL and their effects are derived from the corresponding marginal posterior densities obtained by integrating the likelihood, rather than by optimizing the joint likelihood surface. This is done using MCMC by treating the unknown QTL genotypes, and any missing marker genotypes, as augmented data and then by including these unknowns in the Markov chain cycle along with the unknown parameters. Parameter estimates are obtained as means of the corresponding marginal posterior densities. High posterior density regions of the marginal densities are obtained as confidence regions. We examine flowering time data from double haploid progeny of Brassica napus to illustrate the proposed method.     

Hierarchical Bayesian modeling of spatially correlated health service outcome and utilization rates

We present Bayesian hierarchical spatial models for spatially correlated small-area health service outcome and utilization rates, with a particular emphasis on the estimation of both measured and unmeasured or unknown covariate effects. This Bayesian hierarchical model framework enables simultaneous modeling of fixed covariate effects and random residual effects. The random effects are modeled via Bayesian prior specifications reflecting spatial heterogeneity globally and relative homogeneity among neighboring areas. The model inference is implemented using Markov chain Monte Carlo methods. Specifically, a hybrid Markov chain Monte Carlo algorithm (Neal, 1995, Bayesian Learning for Neural Networks; Gustafson, MacNab, and Wen, 2003, Statistics and Computing, to appear) is used for posterior sampling of the random effects. To illustrate relevant problems, methods, and techniques, we present an analysis of regional variation in intraventricular hemorrhage incidence rates among neonatal intensive care unit patients across Canada.     

Similarity selection and the evolution of sex: revisiting the red queen

For over 25 years, many evolutionary ecologists have believed that sexual reproduction occurs because it allows hosts to change genotypes each generation and thereby evade their coevolving parasites. However, recent influential theoretical analyses suggest that, though parasites can select for sex under some conditions, they often select against it. These models assume that encounters between hosts and parasites are completely random. Because of this assumption, the fitness of a host depends only on its own genotype (“genotypic selection”). If a host is even slightly more likely to encounter a parasite transmitted by its mother than expected by random chance, then the fitness of a host also depends on its genetic similarity to its mother (“similarity selection”). A population genetic model is presented here that includes both genotypic and similarity selection, allowing them to be directly compared in the same framework. It is shown that similarity selection is a much more potent force with respect to the evolution of sex than is genotypic selection. Consequently, similarity selection can drive the evolution of sex even if it is much weaker than genotypic selection with respect to fitness. Examination of explicit coevolutionary models reveals that even a small degree of mother–offspring parasite transmission can cause parasites to favor sex rather than oppose it. In contrast to previous predictions, the model shows that weakly virulent parasites are more likely to favor sex than are highly virulent ones. Parasites have figured prominently in discussions of the evolution of sex, but recent models suggest that parasites often select against sex rather than for it. With the inclusion of small and realistic exposure biases, parasites are much more likely to favor sex. Though parasites alone may not provide a complete explanation for sex, the results presented here expand the potential for parasites to contribute to the maintenance of sex rather than act against it.     

Wright-Fisher model of social insects with haploid males and diploid females

An inhomogeneous discrete Markov model is formulated for sexual random mating in finite populations of haploid male and diploid female individuals. This is a Wright-Fisher type of model for social insects. The generations are non-overlapping and of given finite sizes. Bottlenecks are included, allowing different sizes to change from generation to generation. Mutations and selection are included in this exact model for the stochastic process. Computations of the exact Markov model are presented, focussing on the sexually asymmetric genetic drift caused by haplodiploidy.     

Little loss of information due to unknown phase for fine-scale linkage-disequilibrium mapping with single-nucleotide-polymorphism genotype data

We present the results of a simulation study that indicate that true haplotypes at multiple, tightly linked loci often provide little extra information for linkage-disequilibrium fine mapping, compared with the information provided by corresponding genotypes, provided that an appropriate statistical analysis method is used. In contrast, a two-stage approach to analyzing genotype data, in which haplotypes are inferred and then analyzed as if they were true haplotypes, can lead to a substantial loss of information. The study uses our COLDMAP software for fine mapping, which implements a Markov chain-Monte Carlo algorithm that is based on the shattered coalescent model of genetic heterogeneity at a disease locus. We applied COLDMAP to 100 replicate data sets simulated under each of 18 disease models. Each data set consists of haplotype pairs (diplotypes) for 20 SNPs typed at equal 50-kb intervals in a 950-kb candidate region that includes a single disease locus located at random. The data sets were analyzed in three formats: (1). as true haplotypes; (2). as haplotypes inferred from genotypes using an expectation-maximization algorithm; and (3). as unphased genotypes. On average, true haplotypes gave a 6% gain in efficiency compared with the unphased genotypes, whereas inferring haplotypes from genotypes led to a 20% loss of efficiency, where efficiency is defined in terms of root mean integrated square error of the location of the disease locus. Furthermore, treating inferred haplotypes as if they were true haplotypes leads to considerable overconfidence in estimates, with nominal 50% credibility intervals achieving, on average, only 19% coverage. We conclude that (1). given appropriate statistical analyses, the costs of directly measuring haplotypes will rarely be justified by a gain in the efficiency of fine mapping and that (2). a two-stage approach of inferring haplotypes followed by a haplotype-based analysis can be very inefficient for fine mapping, compared with an analysis based directly on the genotypes.     

Exploring dependence with data on spatial lattices

Summary .  The application of Markov random field models to problems involving spatial data on lattice systems requires decisions regarding a number of important aspects of model structure. Existing exploratory techniques appropriate for spatial data do not provide direct guidance to an investigator about these decisions. We introduce an exploratory quantity that is directly tied to the structure of Markov random field models based on one-parameter exponential family conditional distributions. This exploratory diagnostic is shown to be a meaningful statistic that can inform decisions involved in modeling spatial structure with statistical dependence terms. In this article, we develop the diagnostic, illustrate its use in guiding modeling decisions with simulated examples, and reexamine a previously published application.     

A comparison of persistence-time estimation for discrete and continuous stochastic population models that include demographic and environmental variability

A discrete-time Markov chain model, a continuous-time Markov chain model, and a stochastic differential equation model are compared for a population experiencing demographic and environmental variability. It is assumed that the environment produces random changes in the per capita birth and death rates, which are independent from the inherent random (demographic) variations in the number of births and deaths for any time interval. An existence and uniqueness result is proved for the stochastic differential equation system. Similarities between the models are demonstrated analytically and computational results are provided to show that estimated persistence times for the three stochastic models are generally in good agreement when the models satisfy certain consistency conditions.     

Time-varying Markov regression random-effect model with Bayesian estimation procedures: Application to dynamics of functional recovery in patients with stroke

The rates of functional recovery after stroke tend to decrease with time. Time-varying Markov processes (TVMP) may be more biologically plausible than time-invariant Markov process for modeling such data. However, analysis of such stochastic processes, particularly tackling reversible transitions and the incorporation of random effects into models, can be analytically intractable. We make use of ordinary differential equations to solve continuous-time TVMP with reversible transitions. The proportional hazard form was used to assess the effects of an individual’s covariates on multi-state transitions with the incorporation of random effects that capture the residual variation after being explained by measured covariates under the concept of generalized linear model. We further built up Bayesian directed acyclic graphic model to obtain full joint posterior distribution. Markov chain Monte Carlo (MCMC) with Gibbs sampling was applied to estimate parameters based on posterior marginal distributions with multiple integrands. The proposed method was illustrated with empirical data from a study on the functional recovery after stroke.     

A conditional Markov model for clustered progressive multistate processes under incomplete observation

Clustered progressive chronic disease processes arise when interest lies in modeling damage in paired organ systems (e.g., kidneys, eyes), in diseases manifest in different organ systems, or in systemic conditions for which damage may occur in several locations of the body. Multistate Markov models have considerable appeal for modeling damage in such settings, particularly when patients are only under intermittent observation. Generalizations are necessary, however, to deal with the fact that processes within subjects may not be independent. We describe a conditional Markov model in which the clustering in processes within subjects is addressed by the use of multiplicative random effects for each transition intensity. The random effects for the different transition intensities may be correlated within subjects, but are assumed to be independent for different subjects. We apply the mixed Markov model to a motivating data set of patients with psoriatic arthritis, and characterize the progressive course of damage in joints of the hand. A generalization to accommodate a subpopulation of "stayers" and extensions which facilitate regression are indicated and illustrated.     

Random effects selection in linear mixed models

We address the important practical problem of how to select the random effects component in a linear mixed model. A hierarchical Bayesian model is used to identify any random effect with zero variance. The proposed approach reparameterizes the mixed model so that functions of the covariance parameters of the random effects distribution are incorporated as regression coefficients on standard normal latent variables. We allow random effects to effectively drop out of the model by choosing mixture priors with point mass at zero for the random effects variances. Due to the reparameterization, the model enjoys a conditionally linear structure that facilitates the use of normal conjugate priors. We demonstrate that posterior computation can proceed via a simple and efficient Markov chain Monte Carlo algorithm. The methods are illustrated using simulated data and real data from a study relating prenatal exposure to polychlorinated biphenyls and psychomotor development of children.     

On the dispersion index of a Markovian molecular clock

The number of nucleotide substitutions accumulated in a gene or in a lineage is an important random variable in the study of molecular evolution. Of particular interest is the ratio of the variance to the mean of that random variable, often known as the dispersion index. Because nucleotide substitution is most commonly modeled by a continuous-time four-state Markov chain, this paper provides a systematic method of computing the dispersion indices exhibited by a continuous-time four-state Markov chain. Using this method along with computer algebra and Monte Carlo simulation, this paper offers partially proven conjectures that were supported by thorough computer experiments. It is believed that the Tamura model, the equal-input model and the Takahata-Kimura model always exhibit dispersion indices less than 2. It is also believed that a general four-state model can be chosen to exhibit a dispersion index of any desired magnitude, although the chance of a randomly chosen such model exhibiting a dispersion index greater than 2 is as small as about 2%. Relevance of these findings to the neutral theory is discussed.     

A mixed model for two-state Markov processes under panel observation

Many chronic medical conditions can be meaningfully characterized in terms of a two-state stochastic process. Here we consider the problem in which subjects make transitions among two such states in continuous time but are only observed at discrete, irregularly spaced time points that are possibly unique to each subject. Data arising from such an observation scheme are called panel data, and methods for related analyses are typically based on Markov assumptions. The purpose of this article is to present a conditionally Markov model that accommodates subject-to-subject variation in the model parameters by the introduction of random effects. We focus on a particular random effects formulation that generates a closed-form expression for the marginal likelihood. The methodology is illustrated by application to a data set from a parasitic field infection survey.     

Utilizing Gaussian Markov Random Field Properties of Bayesian Animal Models

Summary In this article, we demonstrate how Gaussian Markov random field properties give large computational benefits and new opportunities for the Bayesian animal model. We make inference by computing the posteriors for important quantitative genetic variables. For the single‐trait animal model, a nonsampling‐based approximation is presented. For the multitrait model, we set up a robust and fast Markov chain Monte Carlo algorithm. The proposed methodology was used to analyze quantitative genetic properties of morphological traits of a wild house sparrow population. Results for single‐ and multitrait models were compared.     

Semi-Markov graph dynamics

In this paper, we outline a model of graph (or network) dynamics based on two ingredients. The first ingredient is a Markov chain on the space of possible graphs. The second ingredient is a semi-Markov counting process of renewal type. The model consists in subordinating the Markov chain to the semi-Markov counting process. In simple words, this means that the chain transitions occur at random time instants called epochs. The model is quite rich and its possible connections with algebraic geometry are briefly discussed. Moreover, for the sake of simplicity, we focus on the space of undirected graphs with a fixed number of nodes. However, in an example, we present an interbank market model where it is meaningful to use directed graphs or even weighted graphs.     

浙江东部常绿阔叶林次生演替的随机过程模型

 植物群落次生演替过程的有限序列在一定意义上构成一随机过程。浙江东部常绿阔叶林次生演替的随机过程系统可以近似地看成线性系统,因而可以用马尔可夫过程描述。本文以群落主要乔木优势种作为马尔可夫过程的状态变量。用“空间序列代替时间序列”的研究方法测得自然次生演替过程和干扰次生演替过程群落主要优势乔木种的更新概率,以此建立了马尔可夫过程的一步平稳转移概率矩阵。应用马尔可夫链模型对常绿阔叶林的自然和干扰次生演替过程进行了模拟。模型还给出了次生演替过程群落主要优势乔木种类的数量动态,为深入研究常绿阔叶林次生演替规律以及林业生产和管理等提供了依据。     

A Markov chain Monte Carlo strategy for sampling from the joint posterior distribution of pedigrees and population parameters under a Fisher-Wright model with partial selfing

A simple population genetic model is presented for a hermaphrodite annual species, allowing both selfing and outcrossing. Those male gametes (pollen) responsible for outcrossing are assumed to disperse much further than seeds. Under this model, the pedigree of a sample from a single locality is loop-free. A novel Markov chain Monte Carlo strategy is presented for sampling from the joint posterior distribution of the pedigree of such a sample and the parameters of the population genetic model (including the selfing rate) given the genotypes of the sampled individuals at unlinked marker loci. The computational costs of this Markov chain Monte Carlo strategy scale well with the number of individuals in the sample, and the number of marker loci, but increase exponentially with the age (time since colonisation from the source population) of the local population. Consequently, this strategy is particularly suited to situations where the sample has been collected from a population which is the result of a recent colonisation process.     

Invariant probabilities for systems in a random environment—With applications to the Brusselator

In this paper we are concerned with problems of the long-term behavior for nonlinear systems in random environment. The general model is assumed to be given by an ordinary differential equation with random parameters or random input. The disturbance process can be taken from a fairly general class of Markov processes having a bounded state space. In terms of the system’s dynamics we give sufficient conditions for the existence and uniqueness of invariant probabilities. Finally, we apply these results to the two-dimensional biochemical model which is known as the Brusselator. This work is part of a research project supported by the ‘Stiftung Volkswagenwerk’.