E. A. Thompson

Annealing Markov Chain Monte Carlo with Applications to Ancestral Inference

Abstract Markov chain Monte Carlo (MCMC; the Metropolis-Hastings algorithm) has been used for many statistical problems, including Bayesian inference, likelihood inference, and tests of significance. Though the method generally works well, doubts about convergence often remain. Here we propose MCMC methods distantly related to simulated annealing. Our samplers mix rapidly enough to be usable for problems in which other methods would require eons of computing time. They simulate realizations from a sequence of distributions, allowing the distribution being simulated to vary randomly over time. If the sequence of distributions is well chosen, then the sampler will mix well and produce accurate answers for all the distributions. Even when there is only one distribution of interest, these annealing-like samplers may be the only known way to get a rapidly mixing sampler. These methods are essential for attacking very hard problems, which arise in areas such as statistical genetics. We illustrate the methods wi...

Probability functions on complex pedigrees

The calculation of probabilities on pedigrees of arbitrary complexity is discussed for a basic model of transmission and penetrance (encompassing Mendelian inheritance, and certain environmental influences). The structure of pedigrees, and the types of loops occurring, is discussed. Some results in graph theory are obtained and, using these, a recurrence relation derived for certain probabilities. The recursive procedure enables the successive peeling off of certain members of the pedigree, and the condensation of the information on those individuals into a function on a subset of those remaining. The underlying theory is set out, and examples given of the utilization of the resulting algorithm. PEDIGREE; PROBABILITY; LOOPS; PEELING; GRAPH; OUSIOTYPE; GENETICS

The estimation of pairwise relationships

Relationships between the individuals of a population have been previously studied from the point of view of prediction. Edwards (1967) suggested that the problem of detailed population structure could also be studied from the point of view of inference. Even where inferences of practical applicability cannot be made, such an approach can increase understanding of the relation between genealogical and genetic structure. In this paper we consider a specific problem which provides an introduction to the ideas and methods of genealogical inference. This is the problem of estimating the pairwise relationship between two individuals on the basis of their phenotypes at several loci. There is no theoretical problem in the extension from pairwise to joint relationship.

Estimation of the Inbreeding Coefficient through Use of Genomic Data

Many linkage studies are performed in inbred populations, either small isolated populations or large populations with a long tradition of marriages between relatives. In such populations, there exist very complex genealogies with unknown loops. Therefore, the true inbreeding coefficient of an individual is often unknown. Good estimators of the inbreeding coefficient (f) are important, since it has been shown that underestimation of f may lead to false linkage conclusions. When an individual is genotyped for markers spanning the whole genome, it should be possible to use this genomic information to estimate that individuals f. To do so, we propose a maximum-likelihood method that takes marker dependencies into account through a hidden Markov model. This methodology also allows us to infer the full probability distribution of the identity-by-descent (IBD) status of the two alleles of an individual at each marker along the genome (posterior IBD probabilities) and provides a variance for the estimates. We simulate a full genome scan mimicking the true autosomal genome for (1) a first-cousin pedigree and (2) a quadruple-second-cousin pedigree. In both cases, we find that our method accurately estimates f for different marker maps. We also find that the proportion of genome IBD in an individual with a given genealogy is very variable. The approach is illustrated with data from a study of demyelinating autosomal recessive Charcot-Marie-Tooth disease.

Analysis of Parentage for Naturally Established Seedlings of Chamaelirium Luteum (Liliaceae)

The genealogical relationships among naturally established seedlings and flowering individuals were analyzed for a large population of the dioecious plant species Chamaelirium luteum. Genetic likelihoods based on 11 electrophoretic markers of all possible parent pairs within flowering seasons from 1974 to 1981 preceding observed establishment of seedlings from 1976 to 1982 were evaluated; and most—likely parents for 283 seedlings were thus identified, enabling a partial reconstruction of the genealogy of this population. This information, combined with the map location of each plant, was used to analyze realized gene—flow patterns. Intermate distances showed more nearby matings than would the kind of pollen dispersal profile that could be expected under random mating. Overall, seedlings tended to undergo establishment at locations between their maternal and paternal parents, indicating localized genetic adaptation along environmental gradients. Finally, a negative correlation was observed between inflorescence size (reproductive effort) and number of progeny observed (reproductive success). Techniques of genealogy reconstruction have previously been limited to human populations. As seen in the present study, such techniques also show great promise for revealing the evolutionary behavior of natural populations.

The relationship between single parent and parent pair genetic likelihoods in genealogy reconstruction

Abstract In the present paper, techniques of genealogy reconstruction based on genetic likelihoods of parent-offspring relationships are explored. Previous applications of such techniques have involved human populations, with emphasis placed on identification of parent pairs followed by reconstruction of families. In natural populations, this approach is neither practical nor necessarily a realistic representation of population structure. It is proposed that for natural populations emphasis should be placed first on locating the most likely mothers and fathers for a given individual, then seeking the most likely pair among that subset of genetically possible parents. Thus the genealogy is ultimately represented as a set of genotype triplets consisting of each individual coupled with its mother and father. Mathematical analyses show a strong positive correlation between single parent and parent pair likelihoods within triplets; this result is corroborated by statistical investigation of data from a natural plant population. Therefore the practice of constructing parent pairs using only likely single parents is justifiable on statistical grounds.

Human evolutionary trees

Preface 1. Inference and the evolutionary tree problem 2. The model 3. The likelihood approach 4. A likelihood solution 5. Further aspects of the problem and its likelihood solution 6. The Icelandic admixture problem Summary References References index Subject index.

MCMC segregation and linkage analysis

Our objective was to infer the genetic model for the quantitative traits using a variety of methods developed in our group. Only a single data set was analyzed in any one analysis, although some comparison between data sets was made. In addition, the simulated model was not known during the course of the analysis. Basic modeling and segregation analyses for the five quantitative traits was followed by several simple genome scans to indicate areas of interest. A Markov chain Monte Carlo (MCMC) multipoint quantitative trait locus (QTL) mapping approach was then used to estimate the posterior probabilities of linkage of QTL to each chromosome simultaneously with trait model parameters, and to further localize the genes. Comparisons between the nuclear family and pedigree data sets indicated a greater power for QTL detection and mapping with the pedigree data sets. Even with the pedigree data, however, precise localization of the QTL did not appear to be possible using single replicate data sets. Two of the three genes with effects on trait Q1 were detected by the MCMC method.

Bias in multipoint linkage analysis arising from map misspecification.

Multipoint linkage analysis methods are often used in human genetic studies. Although multipoint methods increase power for a linkage analysis and will become essential if use of diallelic markers becomes widespread, the methods in use assume an accurate meiotic marker map. Unfortunately, uncertainties in estimates of between‐marker meiotic distances are large. Also, sex‐averaged maps are generally used, but recombination rates differ in males and females. Both these types of map misspecification can lead to lod score bias, but such bias has not previously been systematically quantified. We examine multipoint lod score bias arising from these map misspecifications, in both the presence and absence of actual linkage. We define bias as the expected difference between the lod score computed under the misspecified map and that computed under the true map. With actual linkage, any map misspecification causes negative bias in lod scores, resulting in loss of power to detect linkage. In most cases, bias is modest, only reaching clearly detectable levels when both types of misspecification are substantial. In the absence of linkage, map misspecification can cause positive or negative bias: falsely assuming a 1:1 female:male ratio always causes positive bias; using too large a distance gives a positive bias; using too small a distance gives a negative bias. This bias can inflate the false‐positive rate, especially when the sample size is modest. We conclude that although current sex‐averaged maps are suitable for a first‐pass multipoint screen, the potential for bias from map misspecification should be evaluated in following up results from such an analysis. Genet. Epidemiol. 19:366–380, 2000.

Pedigree analysis for quantitative traits : variance components without matrix inversion

Recent developments in the animal breeding literature facilitate estimation of the variance components in quantitative genetic models. However, computation remains intensive, and many of the procedures are restricted to specialized designs and models, unsuited to data arising from studies of natural populations. We develop algorithms that allow maximum likelihood estimation of variance components for data on arbitrary pedigree structures. The proposed methods can be implemented on microcomputers, since no intensive matrix computations or manipulations are involved. Although parts of our procedures have been previously presented, we unify these into an overall scheme whose intuitive justification clarifies the approach. Two examples are analyzed: one of data on a natural population of Salivia lyrata and the other of simulated data on an extended pedigree.