A Note on Balanced Sampling and Robust Genetic Predictors

The concept of balanced sampling is applied to prediction in finite samples using model based inference procedures. Necessary and sufficient conditions are derived for a general linear model with arbitrary covariance structure to yield the expansion estimator as the best linear unbiased predictor for the mean. The analysis is extended to produce a robust estimator for the mean squared error under balanced sampling and the results are discussed in the context of statistical genetics where appropriate sampling produces simple efficient and robust genetic predictors free from unnecessary genetic assumptions.     

Evolutionary stability under the replicator and the gradient dynamics

We unite two general models for evolutionary change under the forces of selection, mutation and reproduction, a genetic model (replicator dynamics) and a cultural model (gradient dynamics). Under the assumption of normality, we find that the mean and variance dynamics are essentially identical under the two models and we relate these to the ESS and convergence stability conditions.     

Maximum likelihood estimation for incomplete repeated-measures experiments under an ARMA covariance structure

A stochastic model is presented for the analysis of incomplete repeated-measures experiments. The general linear model is used to relate the response measures to other variables which are thought to account for inherent variation; an autoregressive moving average (ARMA) time series representation is used to model disturbance terms. Maximum likelihood estimation procedures are considered, and the properties of these estimators are derived. It is concluded that while the assumptions underpinning the ARMA covariance models may be somewhat restrictive, they provide a useful inferential vehicle, particularly in the presence of missing values.     

Factor analysis models for structuring covariance matrices of additive genetic effects: a Bayesian implementation

Multivariate linear models are increasingly important in quantitative genetics. In high dimensional specifications, factor analysis (FA) may provide an avenue for structuring (co)variance matrices, thus reducing the number of parameters needed for describing (co)dispersion. We describe how FA can be used to model genetic effects in the context of a multivariate linear mixed model. An orthogonal common factor structure is used to model genetic effects under Gaussian assumption, so that the marginal likelihood is multivariate normal with a structured genetic (co)variance matrix. Under standard prior assumptions, all fully conditional distributions have closed form, and samples from the joint posterior distribution can be obtained via Gibbs sampling. The model and the algorithm developed for its Bayesian implementation were used to describe five repeated records of milk yield in dairy cattle, and a one common FA model was compared with a standard multiple trait model. The Bayesian Information Criterion favored the FA model.     

A PATH-ANALYTIC MODEL FOR THE MEASUREMENT OF SELECTION ON MORPHOLOGY

This paper describes a path model for the analysis of phenotypic selection upon continuous morphological characters. The path-analysis model assumes that selection occurs on unmeasured general size and shape allometry factors that summarize linear relations among sets of ontogenetically, phylogenetically, or functionally related traits. An unmeasured factor for general size is considered the only aspect of morphometric covariance matrices for which there is an a priori biological explanation. Consequently, selection coefficients are derived for each measured character by holding constant only a general size factor, rather than by using multiple regression to adjust for the full covariance matrix. Fitness is treated as an unmeasured factor with loadings, representing directional selection coefficients, computed as the covariances of the size-adjusted characters with the measured fitness indicator. The magnitudes and signs of the selection coefficients, combined with biological insight, may suggest hypotheses of selection on one or more shape allometry factors. Hypotheses of selection on general size and shape allometry factors are evaluated through cycles of measurement, analysis, and experimentation, designed to refine the path diagram depicting the covariances among the measured characters, the measured indicator of fitness, and unmeasured factors for morphology and fitness. The path-analysis and multiple-regression models were applied to data from remeasurement of Lande and Arnold's (1983) pentatomid bugs and to Bumpus's (1899) data on house sparrows. The path analysis suggested the hypothesis that variation in bug survivorship was an expression of directional selection on wing loading. Bumpus's data are consistent with a hypothesis of stabilizing selection on general size in females and directional selection for small wing size relative to body size in males.     

Genomic-assisted prediction of genetic value with semiparametric procedures

Semiparametric procedures for prediction of total genetic value for quantitative traits, which make use of phenotypic and genomic data simultaneously, are presented. The methods focus on the treatment of massive information provided by, e.g., single-nucleotide polymorphisms. It is argued that standard parametric methods for quantitative genetic analysis cannot handle the multiplicity of potential interactions arising in models with, e.g., hundreds of thousands of markers, and that most of the assumptions required for an orthogonal decomposition of variance are violated in artificial and natural populations. This makes nonparametric procedures attractive. Kernel regression and reproducing kernel Hilbert spaces regression procedures are embedded into standard mixed-effects linear models, retaining additive genetic effects under multivariate normality for operational reasons. Inferential procedures are presented, and some extensions are suggested. An example is presented, illustrating the potential of the methodology. Implementations can be carried out after modification of standard software developed by animal breeders for likelihood-based or Bayesian analysis.     

Methods for inferring phylogenies from nucleic acid sequence data by using maximum likelihood and linear invariants

Likelihood methods and methods using invariants are procedures for inferring the evolutionary relationships among species through statistical analysis of nucleic acid sequences. A likelihood-ratio test may be used to determine the feasibility of any tree for which the maximum likelihood can be computed. The method of linear invariants described by Cavender, which includes Lake's method of evolutionary parsimony as a special case, is essentially a form of the likelihood-ratio method. In the case of a small number of species (four or five), these methods may be used to find a confidence set for the correct tree. An exact version of Lake's asymptotic chi 2 test has been mentioned by Holmquist et al. Under very general assumptions, a one-sided exact test is appropriate, which greatly increases power.     

Analysis of Covariance Models for Data From Observational Field Studies

Abstract We outline the features of a general class of statistical models (i.e., analysis of covariance [ANCOVA] models) that has proven to be effective for the analysis of data from observational studies. In observational studies, treatments are assigned by Nature in a decidedly nonrandom manner; consequently, many of the crucial assumptions and safeguards of the classic experimental design either fail or are absent. Hence, inferences (causal or associative) are more difficult to justify. Typically, investigators can expect the primary factors of interest, which are usually called environmental exposures rather than treatments, to be involved in complex interactions with each other and with other factors, and these factors will be confounded with still other factors. We provide examples illustrating the application of ANCOVA models to adjust for confounding factors and complex interactions, thereby providing relatively clean estimates of association between exposure and response. We summarize information on available software and supporting literature for implementing ANCOVA models for the analysis of cross-sectional and longitudinal observational field data. We conclude with a brief discussion of critical model fitting issues, including proper specification of the functional form of continuous covariates and problems associated with overfitted models and misspecified models that lack important covariates.     

Symbolic modeling of epistasis

The workhorse of modern genetic analysis is the parametric linear model. The advantages of the linear modeling framework are many and include a mathematical understanding of the model fitting process and ease of interpretation. However, an important limitation is that linear models make assumptions about the nature of the data being modeled. This assumption may not be realistic for complex biological systems such as disease susceptibility where nonlinearities in the genotype to phenotype mapping relationship that result from epistasis, plastic reaction norms, locus heterogeneity, and phenocopy, for example, are the norm rather than the exception. We have previously developed a flexible modeling approach called symbolic discriminant analysis (SDA) that makes no assumptions about the patterns in the data. Rather, SDA lets the data dictate the size, shape, and complexity of a symbolic discriminant function that could include any set of mathematical functions from a list of candidates supplied by the user. Here, we outline a new five step process for symbolic model discovery that uses genetic programming (GP) for coarse-grained stochastic searching, experimental design for parameter optimization, graphical modeling for generating expert knowledge, and estimation of distribution algorithms for fine-grained stochastic searching. Finally, we introduce function mapping as a new method for interpreting symbolic discriminant functions. We show that function mapping when combined with measures of interaction information facilitates statistical interpretation by providing a graphical approach to decomposing complex models to highlight synergistic, redundant, and independent effects of polymorphisms and their composite functions. We illustrate this five step SDA modeling process with a real case-control dataset.     

Investigating the causes of changes in allele frequency during evolution

The use of models as teaching aids in the study of the causes of changes of gene frequency during evolution is now a well established practice in school and degree courses. The effects of migration, selection and genetic drift may all be investigated. This paper extends the range of procedures which may be used with sixth form and more advanced students by discussing possible migration models, by describing an improved selection model where selection is applied unconsciously and the selection coefficient can be calculated from the data and by comparing single and multiple generation models.

In each case there is a brief discussion of points to be extracted from class results.

An overall teaching programme is suggested which should enable the students to investigate all three factors while spending a minimum of time on repetitive manual procedures.

Finally, there is a brief discussion of the roles of migration, drift and selection in evolution.     

Dependence of epidemic and population velocities on basic parameters.

This paper describes the use of linear deterministic models for examining the spread of population processes, discussing their advantages and limitations. Their main advantages are that their assumptions are relatively transparent and that they are easy to analyze, yet they generally give the same velocity as more complex linear stochastic and nonlinear deterministic models. Their simplicity, especially if we use the elegant reproduction and dispersal kernel formulation of Diekmann and van den Bosch et al., allows us greater freedom to choose a biologically realistic model and greatly facilitates examination of the dependence of conclusions on model components and of how these are incorporated into the model and fitted from data. This is illustrated by consideration of a range of examples, including both diffusion and dispersal models and by discussion of their application to both epidemic and population dynamic problems. A general limitation on fitting models results from the poor accuracy of most ecological data, especially on dispersal distances. Confirmation of a model is thus rarely as convincing as those cases where we can clearly reject one. We also need to be aware that linear models provide only an upper bound for the velocity of more realistic nonlinear stochastic models and are almost wholly inadequate when it comes to modeling more complex aspects such as the transition to endemicity and endemic patterns. These limitations are, however, to a great extent shared by linear stochastic and nonlinear deterministic models.     

The genetics of phenotypic plasticity. V. Evolution of reaction norm shape

We present a general quantitative genetic model for the evolution of reaction norms. This model goes beyond previous models by simultaneously permitting any shaped reaction norm and allowing for the imposition of genetic constraints. Earlier models are shown to be special cases of our general model; we discuss in detail models involving just two macroenvironments, linear reaction norms, and quadratic reaction norms. The model predicts that, for the case of a temporally varying environment, a population will converge on (1) the genotype with the maximum mean geometric fitness over all environments, (2) a linear reaction norm whose slope is proportional to the covariance between the environment of development and the environment of selection, and (3) a linear reaction norm even if nonlinear reaction norms are possible. An examination of experimental studies finds some limited support for these predictions. We discuss the limitations of our model and the need for more realistic gametic models and additional data on the genetic and developmental bases of plasticity.     

Residual analysis for linear mixed models

Residuals are frequently used to evaluate the validity of the assumptions of statistical models and may also be employed as tools for model selection. For standard (normal) linear models, for example, residuals are used to verify homoscedasticity, linearity of effects, presence of outliers, normality and independence of the errors. Similar uses may be envisaged for three types of residuals that emerge from the fitting of linear mixed models. We review some of the residual analysis techniques that have been used in this context and propose a standardization of the conditional residual useful to identify outlying observations and clusters. We illustrate the procedures with a practical example.     

Robust Linear Models for Cis-eQTL Analysis

Expression Quantitative Trait Loci (eQTL) analysis enables characterisation of functional genetic variation influencing expression levels of individual genes. In outbread populations, including humans, eQTLs are commonly analysed using the conventional linear model, adjusting for relevant covariates, assuming an allelic dosage model and a Gaussian error term. However, gene expression data generally have noise that induces heavy-tailed errors relative to the Gaussian distribution and often include atypical observations, or outliers. Such departures from modelling assumptions can lead to an increased rate of type II errors (false negatives), and to some extent also type I errors (false positives). Careful model checking can reduce the risk of type-I errors but often not type II errors, since it is generally too time-consuming to carefully check all models with a non-significant effect in large-scale and genome-wide studies. Here we propose the application of a robust linear model for eQTL analysis to reduce adverse effects of deviations from the assumption of Gaussian residuals. We present results from a simulation study as well as results from the analysis of real eQTL data sets. Our findings suggest that in many situations robust models have the potential to provide more reliable eQTL results compared to conventional linear models, particularly in respect to reducing type II errors due to non-Gaussian noise. Post-genomic data, such as that generated in genome-wide eQTL studies, are often noisy and frequently contain atypical observations. Robust statistical models have the potential to provide more reliable results and increased statistical power under non-Gaussian conditions. The results presented here suggest that robust models should be considered routinely alongside other commonly used methodologies for eQTL analysis.     

Decay of heterogenous detritus: A general model

A new, general model of decay of organic matter assumes that the various constituents of the decaying mass have (1) different rates of decomposition and (2) different rates of transformation to more or less decomposable forms. Existing models of the decay process are special cases of the general model that share three assumptions: (a) no transformation of detrital components, (b) equal rates of decay for one or two homogeneous constituents of the decaying mass, and (6) a uniform distribution of each type of constituent at the beginning of decay. A solution of the general model that relaxes these three assumptions fits previously published data closely. The general model unites existing models, permits comparisons of their assumptions, and provides a theoretical framework for developing and evaluating particular models that embody assumptions more in harmony with empirical knowledge of decay mechanisms.     

Model choice in gene mapping: what and why

The choice of an appropriate genetic model describing the genetic architecture underlying a character of interest is an inherent part of the gene mapping studies of human and other living organisms. The genetic model specifies the statistical parameters for the number of genes, their positions, and the types and magnitudes of their contributions to the phenotype. There are many considerations involved in model formulation (choice) ranging from the assumptions concerning the data, the role of environment, and the number of oligogenes (or quantitative trait loci) influencing the trait behavior. There are several model selection procedures and criteria under specific sampling designs in the genetic literature. These approaches often have their origin in computer science or in general statistical theory. Our aim here is to give an overview of the most popular statistical criteria and to present principles behind them. Bayesian model averaging is suggested as a robust alternative for such methods.     

A review of some statistical methods for covariance analysis of categorical data

Three general methods for covariance analysis of categorical data are reviewed and applied to an example from a clinical trial in rheumatoid arthritis. The three methods considered are randomization-model nonparametric procedures, maximum likelihood logistic regression, and weighted least squares analysis of correlated marginal functions. A fourth heuristic approach, the unweighted linear model analysis, is an approximate procedure but it is easy to implement. The assumptions and statistical issues for each method are discussed so as to emphasize philosophical differences between their rationales. Attention is given to computational differences, but it is shown that the methods lead to similar results for analogous problems. It is argued that the essential differences between the methods lie in their underlying assumptions and the generality of the conclusions which may be drawn.