Nonparametric Modeling of Longitudinal Covariance Structure in Functional Mapping of Quantitative Trait Loci

ArticleinBiometrics 65(4):1068-77 · March 2009with 27 Reads 
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Abstract
Estimation of the covariance structure of longitudinal processes is a fundamental prerequisite for the practical deployment of functional mapping designed to study the genetic regulation and network of quantitative variation in dynamic complex traits. We present a nonparametric approach for estimating the covariance structure of a quantitative trait measured repeatedly at a series of time points. Specifically, we adopt Huang et al.'s (2006, Biometrika 93, 85-98) approach of invoking the modified Cholesky decomposition and converting the problem into modeling a sequence of regressions of responses. A regularized covariance estimator is obtained using a normal penalized likelihood with an L(2) penalty. This approach, embedded within a mixture likelihood framework, leads to enhanced accuracy, precision, and flexibility of functional mapping while preserving its biological relevance. Simulation studies are performed to reveal the statistical properties and advantages of the proposed method. A real example from a mouse genome project is analyzed to illustrate the utilization of the methodology. The new method will provide a useful tool for genome-wide scanning for the existence and distribution of quantitative trait loci underlying a dynamic trait important to agriculture, biology, and health sciences.

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    Precise identification of biological samples remains the most important proof in the forensic science. Illegal logging has become the urgent issue in Poland during the last decades, and conventional methods of investigation turn out to be often insufficient. Recently, the DNA-based markers (SSR and cytoplasmic genes) can remarkably help in the forensic botany performed by the Forest Service Guards and the Police investigation in illegal logging of timber. The identification method relies on comparison of the piece of evidence (i.e., stolen wood fragments) with the piece of reference (e.g., tree parts remained in the forest). We present the usefulness of the DNA neutral markers (i.e., microsatellite loci) and cytoplasmic genes in forensic botany based on several case studies of illegal wood identification in Poland, concerning the most economically important coniferous tree species such as Pinus sylvestris L., Picea abies (L.) Karst., Abies alba Mill., and Larix decidua (L.). Thanks to the DNA profiles established on the basis of minimum 4 microsatellite nuclear DNA loci, and at least one cytoplasmic organelle (mitochondrial or chloroplast) DNA marker, the determination of the DNA profiles provided fast and reliable comparison between material of evidence (also wood and needles) and material of reference (first of all tree stumps) in the forest. These data strongly supported the decision taken by several District Courts in Poland, as far as the identification of wood samples was proved with a high probability (approximately 98–99 %). The aim of the below publication is to present Polish case study on DNA use to fight illegal logging which became very successful among foresters.
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    Cells with the same genotype growing under the same conditions can show different phenotypes, which is known as "population heterogeneity". The heterogeneity of hematopoietic progenitor cells has an effect on their differentiation potential and lineage choices. However, the genetic mechanisms governing population heterogeneity remain unclear. Here, we present a statistical model for mapping the quantitative trait locus (QTL) that affects hematopoietic cell heterogeneity. This strategy, termed systems mapping, integrates a system of differential equations into the framework for systems mapping, allowing hypotheses regarding the interplay between genetic actions and cell heterogeneity to be tested. A simulation approach based on cell heterogeneity dynamics has been designed to test the statistical properties of the model. This model not only considers the traditional QTLs, but also indicates the methylated QTLs that can illustrate non-genetic individual differences. It has significant implications for probing the molecular, genetic and epigenetic mechanisms of hematopoietic progenitor cell heterogeneity.
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    Phenotypic traits, such as seed development, are a consequence of complex biochemical interactions among genes, proteins and metabolites, but the underlying mechanisms that operate in a coordinated and sequential manner remain elusive. Here, we address this issue by developing a computational algorithm to monitor proteome changes during the course of trait development. The algorithm is built within the mixture-model framework in which each mixture component is modeled by a specific group of proteins that display a similar temporal pattern of expression in trait development. A nonparametric approach based on Legendre orthogonal polynomials was used to fit dynamic changes of protein expression, increasing the power and flexibility of protein clustering. By analyzing a dataset of proteomic dynamics during early embryogenesis of the Chinese fir, the algorithm has successfully identified several distinct types of proteins that coordinate with each other to determine seed development in this forest tree commercially and environmentally important to China. The algorithm will find its immediate applications for the characterization of mechanistic underpinnings for any other biological processes in which protein abundance plays a key role.
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    In this paper, penalized regression using the L1 norm on the estimated parameters is proposed for chemometric calibration. The algorithm is of the lasso type, introduced by Tibshirani in 1996 as a linear regression method with bound on the absolute length of the parameters, but a modification is suggested to cope with the singular design matrix most often seen in chemometric calibration. Furthermore, the proposed algorithm may be generalized to all convex norms like ∑|βj| where  ≥ 1, i.e. a method that continuously varies from ridge regression to the lasso. The lasso is applied both directly as a calibration method and as a method to select important variables/wavelengths. It is demonstrated that the lasso algorithm, in general, leads to parameter estimates of which some are zero while others are quite large (compared to e.g. the traditional PLS or RR estimates). By using several benchmark data sets, it is shown that both the direct lasso method and the regression where the lasso acts as a wavelength selection method most often outperform the PLS and RR methods. Copyright © 2001 John Wiley & Sons, Ltd.
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    Genetic interactions or epistasis may play an important role in the genetic etiology of drug response. With the availability of large-scale, high-density single nucleotide polymorphism markers, a great challenge is how to associate haplotype structures and complex drug response through its underlying pharmacodynamic mechanisms. We have derived a general statistical model for detecting an interactive network of DNA sequence variants that encode pharmacodynamic processes based on the haplotype map constructed by single nucleotide polymorphisms. The model was validated by a pharmacogenetic study for two predominant beta-adrenergic receptor (betaAR) subtypes expressed in the heart, beta1AR and beta2AR. Haplotypes from these two receptors trigger significant interaction effects on the response of heart rate to different dose levels of dobutamine. This model will have implications for pharmacogenetic and pharmacogenomic research and drug discovery. A computer program written in Matlab can be downloaded from the webpage of statistical genetics group at the University of Florida. Supplementary data are available at Bioinformatics online.
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    Graphical models are frequently used to explore networks, such as genetic networks, among a set of variables. This is usually carried out via exploring the sparsity of the precision matrix of the variables under consideration. Penalized likelihood methods are often used in such explorations. Yet, positive-definiteness constraints of precision matrices make the optimization problem challenging. We introduce nonconcave penalties and the adaptive LASSO penalty to attenuate the bias problem in the network estimation. Through the local linear approximation to the nonconcave penalty functions, the problem of precision matrix estimation is recast as a sequence of penalized likelihood problems with a weighted $L_1$ penalty and solved using the efficient algorithm of Friedman et al. [Biostatistics 9 (2008) 432--441]. Our estimation schemes are applied to two real datasets. Simulation experiments and asymptotic theory are used to justify our proposed methods. Comment: Published in at http://dx.doi.org/10.1214/08-AOAS215 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)
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    SUMMARY Two major reasons for the popularity of the EM algorithm are that its maximum step involves only complete-data maximum likelihood estimation, which is often computationally simple, and that its convergence is stable, with each iteration increasing the likelihood. When the associated complete-data maximum likelihood estimation itself is complicated, EM is less attractive because the M-step is computationally unattractive. In many cases, however, complete-data maximum likelihood estimation is relatively simple when conditional on some function of the parameters being estimated. We introduce a class of generalized EM algorithms, which we call the ECM algorithm, for Expectation/Conditional Maximization (CM), that takes advantage of the simplicity of complete-data conditional maximum likelihood estimation by replacing a complicated M-step of EM with several computationally simpler CM-steps. We show that the ECM algorithm shares all the appealing convergence properties of EM, such as always increasing the likelihood, and present several illustrative examples.
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    The positive-definiteness constraint is the most awkward stumbling block in modelling the covariance matrix. Pourahmadi's (1999) unconstrained parameterisation models covariance using covariates in a similar manner to mean modelling in generalised linear models. The new covariance parameters have statistical interpretation as the regression coefficients and logarithms of prediction error variances corresponding to regressing a response on its predecessors. In this paper, the maximum likelihood estimators of the parameters of a generalised linear model for the covariance matrix, their consistency and their asymptotic normality are studied when the observations are normally distributed. These results along with the likelihood ratio test and penalised likelihood criteria such as BIC for model and variable selection are illustrated using a real dataset.
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    We provide unconstrained parameterisation for and model a covariance using covariates. The Cholesky decomposition of the inverse of a covariance matrix is used to associate a unique unit lower triangular and a unique diagonal matrix with each covariance matrix. The entries of the lower triangular and the log of the diagonal matrix are unconstrained and have meaning as regression coefficients and prediction variances when regressing a measurement on its predecessors. An extended generalised linear model is introduced for joint modelling of the vectors of predictors for the mean and covariance subsuming the joint modelling strategy for mean and variance heterogeneity, Gabriel's antedependence models, Dempster's covariance selection models and the class of graphical models. The likelihood function and maximum likelihood estimators of the covariance and the mean parameters are studied when the observations are normally distributed. Applications to modelling nonstationary dependence structures and multivariate data are discussed and illustrated using real data. A graphical method, similar to that based on the correlogram in time series, is developed and used to identify parametric models for nonstationary covariances.
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    Functional mapping of dynamic traits measured in a longitudinal study was originally derived within the maximum likelihood (ML) context and implemented with the EM algorithm. Although ML-based functional mapping possesses many favorable statistical properties in parameter estimation, it may be computationally intractable for analyzing longitudinal data with high dimensions and high measurement errors. In this article, we derive a general functional mapping framework for quantitative trait locus mapping of dynamic traits within the Bayesian paradigm. Markov chain Monte Carlo techniques were implemented for functional mapping to estimate biologically and statistically sensible parameters that model the structures of time-dependent genetic effects and covariance matrix. The Bayesian approach is useful to handle difficulties in constructing confidence intervals as well as the identifiability problem, enhancing the statistical inference of functional mapping. We have undertaken simulation studies to investigate the statistical behavior of Bayesian-based functional mapping and used a real example with F2 mice to validate the utilization and usefulness of the model.
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    Strain intercross experiments provide a powerful means for mapping genes affecting complex quantitative traits. We report on the genetic variability of the intercross of the Large (LG/J) and Small (SM/J) inbred mouse strains as a guide to gene mapping studies. Ten SM/J males were crossed to 10 LG/J females, after which animals were randomly mated to produce F1, F2, and F3 intercross generations. The 1632 F3 animals from 200 full-sib families were used to estimate heritabilities and genetic correlations of the traits measured. A subset of families was cross-fostered at birth to allow measurement of the importance of post-natal maternal effects. Data was collected on weekly body weight from one to 10 weeks and on organ weights, body weight, reproductive fat pad weight, and tail length at necropsy in the intercross generations. There was no heterosis for age-specific weights or necropsy traits, except that one-week weight was the highest in the F2 generation, indicating heterosis for maternal effect in the F1 mothers. We found moderate to high heritability for most age-specific weights and necropsy traits. Maternal effects were significant for age-specific weights from one to four weeks but disappeared completely at ten-week weight. Maternal effects for necropsy traits were low and not statistically significant. Age-specific weights showed a typical correlation pattern, with correlation declining as the difference in ages increased. Among necropsy traits, reproductive fat pad and body weights were very highly genetically correlated. Most other genetic correlations were low to moderate. The intercross between SM/J and LG/J inbred mouse strains provides a valuable resource for mapping quantitative trait loci for body size, composition, and morphology
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    In the past two decades a parametric multivariate regression modelling approach for analyzing growth curve data has achieved prominence. The approach, which has several advantages over classical analysis-of-variance and general multivariate approaches, consists of postulating, fitting, evaluating, and comparing parametric models for the data's mean structure and covariance structure. This article provides an overview of the approach, using unified terminology and notation. Well-established models and some developed more recently are described, with emphasis given to those models that allow for nonstationarity and for measurement times that differ across subjects and are unequally spaced. Graphical diagnostics that can assist with model postulation and evaluation are discussed, as are more formal methods for fitting and comparing models. Three examples serve to illustrate the methodology and to reveal the relative strengths and weaknesses of the various parametric models.
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    The genetic architecture of growth traits plays a central role in shaping the growth, development, and evolution of organisms. While a limited number of models have been devised to estimate genetic effects on complex phenotypes, no model has been available to examine how gene actions and interactions alter the ontogenetic development of an organism and transform the altered ontogeny into descendants. In this article, we present a novel statistical model for mapping quantitative trait loci (QTL) determining the developmental process of complex traits. Our model is constructed within the traditional maximum-likelihood framework implemented with the EM algorithm. We employ biologically meaningful growth curve equations to model time-specific expected genetic values and the AR(1) model to structure the residual variance-covariance matrix among different time points. Because of a reduced number of parameters being estimated and the incorporation of biological principles, the new model displays increased statistical power to detect QTL exerting an effect on the shape of ontogenetic growth and development. The model allows for the tests of a number of biological hypotheses regarding the role of epistasis in determining biological growth, form, and shape and for the resolution of developmental problems at the interface with evolution. Using our newly developed model, we have successfully detected significant additive additive epistatic effects on stem height growth trajectories in a forest tree.
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    The detection of genes that control quantitative characters is a problem of great interest to the genetic mapping community. Methods for locating these quantitative trait loci (QTL) relative to maps of genetic markers are now widely used. This paper addresses an issue common to all QTL mapping methods, that of determining an appropriate threshold value for declaring significant QTL effects. An empirical method is described, based on the concept of a permutation test, for estimating threshold values that are tailored to the experimental data at hand. The method is demonstrated using two real data sets derived from F(2) and recombinant inbred plant populations. An example using simulated data from a backcross design illustrates the effect of marker density on threshold values.
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    Adequate separation of effects of possible multiple linked quantitative trait loci (QTLs) on mapping QTLs is the key to increasing the precision of QTL mapping. A new method of QTL mapping is proposed and analyzed in this paper by combining interval mapping with multiple regression. The basis of the proposed method is an interval test in which the test statistic on a marker interval is made to be unaffected by QTLs located outside a defined interval. This is achieved by fitting other genetic markers in the statistical model as a control when performing interval mapping. Compared with the current QTL mapping method (i.e., the interval mapping method which uses a pair or two pairs of markers for mapping QTLs), this method has several advantages. (1) By confining the test to one region at a time, it reduces a multiple dimensional search problem (for multiple QTLs) to a one dimensional search problem. (2) By conditioning linked markers in the test, the sensitivity of the test statistic to the position of individual QTLs is increased, and the precision of QTL mapping can be improved. (3) By selectively and simultaneously using other markers in the analysis, the efficiency of QTL mapping can be also improved. The behavior of the test statistic under the null hypothesis and appropriate critical value of the test statistic for an overall test in a genome are discussed and analyzed. A simulation study of QTL mapping is also presented which illustrates the utility, properties, advantages and disadvantages of the method.
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    The problem of detecting minor quantitative trait loci (QTL) responsible for genetic variation not explained by major QTL is of importance in the complete dissection of quantitative characters. Two extensions of the permutation-based method for estimating empirical threshold values are presented. These methods, the conditional empirical threshold (CET) and the residual empirical threshold (RET), yield critical values that can be used to construct tests for the presence of minor QTL effects while accounting for effects of known major QTL. The CET provides a completely nonparametric test through conditioning on markers linked to major QTL. It allows for general nonadditive interactions among QTL, but its practical application is restricted to regions of the genome that are unlinked to the major QTL. The RET assumes a structural model for the effect of major QTL, and a threshold is constructed using residuals from this structural model. The search space for minor QTL is unrestricted, and RET-based tests may be more powerful than the CET-based test when the structural model is approximately true.
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    Body size is an archetypal quantitative trait with variation due to the segregation of many gene loci, each of relatively minor effect, and the environment. We examine the effects of quantitative trait loci (QTLs) on age-specific body weights and growth in the F2 intercross of the LG/J and SM/J strains of inbred mice. Weekly weights (1-10 wk) and 75 microsatellite genotypes were obtained for 535 mice. Interval mapping was used to locate and measure the genotypic effects of QTLs on body weight and growth. QTL effects were detected on 16 of the 19 autosomes with several chromosomes carrying more than one QTL. The number of QTLs for age-specific weights varied from seven at 1 week to 17 at 10 wk. The QTLs were each of relatively minor, subequal effect. QTLs affecting early and late growth were generally distinct, mapping to different chromosomal locations indicating separate genetic and physiological systems for early and later murine growth.
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    A new statistical method for mapping quantitative trait loci (QTL), called multiple interval mapping (MIM), is presented. It uses multiple marker intervals simultaneously to fit multiple putative QTL directly in the model for mapping QTL. The MIM model is based on Cockerham's model for interpreting genetic parameters and the method of maximum likelihood for estimating genetic parameters. With the MIM approach, the precision and power of QTL mapping could be improved. Also, epistasis between QTL, genotypic values of individuals, and heritabilities of quantitative traits can be readily estimated and analyzed. Using the MIM model, a stepwise selection procedure with likelihood ratio test statistic as a criterion is proposed to identify QTL. This MIM method was applied to a mapping data set of radiata pine on three traits: brown cone number, tree diameter, and branch quality scores. Based on the MIM result, seven, six, and five QTL were detected for the three traits, respectively. The detected QTL individually contributed from approximately 1 to 27% of the total genetic variation. Significant epistasis between four pairs of QTL in two traits was detected, and the four pairs of QTL contributed approximately 10.38 and 14.14% of the total genetic variation. The asymptotic variances of QTL positions and effects were also provided to construct the confidence intervals. The estimated heritabilities were 0.5606, 0.5226, and 0. 3630 for the three traits, respectively. With the estimated QTL effects and positions, the best strategy of marker-assisted selection for trait improvement for a specific purpose and requirement can be explored. The MIM FORTRAN program is available on the worldwide web (http://www.stat.sinica.edu.tw/chkao/).
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    Over 20 years ago, D. S. Falconer and others launched an important avenue of research into the quantitative of body size growth in mice. This study continues in that tradition by locating quantitative trait loci (QTLs) responsible for murine growth, such as age-specific weights and growth periods, and examining the genetic architecture for body weight. We identified a large number of potential QTLs in an earlier F2 intercross (Intercross I) of the SM/J and LG/J inbred mouse strains. Many of these QTLs are replicated in a second F2 intercross (Intercross II) between the same two strains. These replicated regions provide candidate regions for future fine-mapping studies. We also examined body size and growth QTLs using the combined data set from these two intercrosses, resulting in 96 microsatellite markers being scored for 1045 individuals. An examination of the genetic architecture for age-specific weight and growth periods resulted in locating 20 separate QTLs, which were mainly additive in nature, although dominance was found to affect early growth and body size. QTLs affecting early and late growth were generally distinct, mapping to separate chromosome locations. This QTL pattern indicates largely separate genetic and physiological systems for early and later murine growth, as Falconer suggested. We also found sex-specific QTLs for body size with implications for the evolution of sexual dimorphism.
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    Several equations have been proposed to describe ontogenetic growth trajectories for organisms justified primarily on the goodness of fit rather than on any biological mechanism. Here, we derive a general quantitative model based on fundamental principles for the allocation of metabolic energy between maintenance of existing tissue and the production of new biomass. We thus predict the parameters governing growth curves from basic cellular properties and derive a single parameterless universal curve that describes the growth of many diverse species. The model provides the basis for deriving allometric relationships for growth rates and the timing of life history events.
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    Most organisms display remarkable differences in morphological, anatomical, and developmental features between the two sexes. It has been recognized that these sex-dependent differences are controlled by an array of specific genetic factors, mediated through various environmental stimuli. In this paper, we present a unifying statistical model for mapping quantitative trait loci (QTL) that are responsible for sexual differences in growth trajectories during ontogenetic development. This model is derived within the maximum likelihood context, incorporated by sex-stimulated differentiation in growth form that is described by mathematical functions. A typical structural model is implemented to approximate time-dependent covariance matrices for longitudinal traits. This model allows for a number of biologically meaningful hypothesis tests regarding the effects of QTL on overall growth trajectories or particular stages of development. It is particularly powerful to test whether and how the genetic effects of QTL are expressed differently in different sexual backgrounds. Our model has been employed to map QTL affecting body mass growth trajectories in both male and female mice of an F2 population derived from the large (LG/J) and small (SM/J) mouse strains. We detected four growth QTL on chromosomes 6, 7, 11, and 15, two of which trigger different effects on growth curves between the two sexes. All the four QTL display significant genotype-sex interaction effects on the timing of maximal growth rate in the ontogenetic growth of mice. The implications of our model for studying the genetic architecture of growth trajectories and its extensions to some more general situations are discussed.
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    The incorporation of developmental control mechanisms of growth has proven to be a powerful tool in mapping quantitative trait loci (QTL) underlying growth trajectories. A theoretical framework for implementing a QTL mapping strategy with growth laws has been established. This framework can be generalized to an arbitrary number of time points, where growth is measured, and becomes computationally more tractable, when the assumption of variance stationarity is made. In practice, however, this assumption is likely to be violated for age-specific growth traits due to a scale effect. In this article, we present a new statistical model for mapping growth QTL, which also addresses the problem of variance stationarity, by using a transform-both-sides (TBS) model advocated by Carroll and Ruppert (1984, Journal of the American Statistical Association 79, 321-328). The TBS-based model for mapping growth QTL cannot only maintain the original biological properties of a growth model, but also can increase the accuracy and precision of parameter estimation and the power to detect a QTL responsible for growth differentiation. Using the TBS-based model, we successfully map a QTL governing growth trajectories to a linkage group in an example of forest trees. The statistical and biological properties of the estimates of this growth QTL position and effect are investigated using Monte Carlo simulation studies. The implications of our model for understanding the genetic architecture of growth are discussed.
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    Are there specific genes that control the pathogenesis of HIV infection? This question, which is of fundamental importance in designing personalized strategies of gene therapy to control HIV infection, can be examined by genetic mapping approaches. In this article, we present a new statistical model for unravelling the genetic mechanisms for the dynamic change of HIV that causes AIDS by marker-based linkage disequilibrium (LD) analyses. This new model is the extension of our functional mapping theory to integrate viral load trajectories within a genetic mapping framework. Earlier studies of HIV dynamics have led to various mathematical functions for modelling the kinetic curves of plasma virions and CD4 lymphocytes in HIV patients. Through incorporating these functions into the LD-based mapping procedure, we can identify and map individual quantitative trait loci (or QTL) responsible for viral pathogenesis. We derive a closed-form solution for estimating QTL allele frequency and marker-QTL linkage disequilibrium in the context of EM algorithm and implement the simplex algorithm to estimate the mathematical parameters describing the curve shapes of HIV pathogenesis. We performed different simulation scenarios based on currently used clinical designs in AIDS/HIV research to illustrate the utility and power of our model for genetic mapping of HIV dynamics. The implications of our model for genetic and genomic research into AIDS pathogenesis are discussed.
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    Understanding the genetic control of growth is fundamental to agricultural, evolutionary and biomedical genetic research. In this article, we present a statistical model for mapping quantitative trait loci (QTL) that are responsible for genetic differences in growth trajectories during ontogenetic development. This model is derived within the maximum likelihood context, implemented with the expectation-maximization algorithm. We incorporate mathematical aspects of growth processes to model the mean vector and structured antedependence models to approximate time-dependent covariance matrices for longitudinal traits. Our model has been employed to map QTL that affect body mass growth trajectories in both male and female mice of an F2 population derived from the Large and Small mouse strains. The results from this model are compared with those from the autoregressive-based functional mapping approach. Based on results from computer simulation studies, we suggest that these two models are alternative to one another and should be used simultaneously for the same dataset.
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    A general growth model derived from basic cellular properties can be used to describe the dynamic process of cancer growth with mathematical equations. It has been recognized that cancer growth is under genetic control, with a multitude of interacting genes each segregating in a Mendelian fashion and displaying environmental sensitivity. In this article, we integrate the mathematical aspects of the pervasive growth model into a statistical framework for the identification of quantitative trait nucleotides that underlie cancer growth. This integrative framework is constructed with a single nucleotide polymorphism-based haplotype blocking analysis. Simulation studies have been performed to demonstrate the usefulness of the model. The proposed model provides a generic platform model for testing and detecting specific DNA sequence variants that regulates the timing of cancer emergence, growth and differentiation.
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    Many biological processes, from cellular metabolism to population dynamics, are characterized by particular allometric scaling relationships between rate and size (power laws). A statistical model for mapping specific quantitative trait loci (QTLs) that are responsible for allometric scaling laws has been developed. We present an improved model for allometric mapping of QTLs based on a more general allometry equation. This improved model includes two steps: (1) use model II regression analysis to estimate the parameters underlying universal allometric scaling laws, and (2) substitute the estimated allometric parameters in the mixture-based mapping model to obtain the estimation of QTL position and effects. This model has been validated by a real example for a mouse F2 progeny, in which two QTLs were detected on different chromosomes that determine the allometric relationship between growth rate and body weight.
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    Genes that control circadian rhythms in organisms have been recognized, but have been difficult to detect because circadian behavior comprises periodically dynamic traits and is sensitive to environmental changes. We present a statistical model for mapping and characterizing specific genes or quantitative trait loci (QTL) that affect variations in rhythmic responses. This model integrates a system of differential equations into the framework for functional mapping, allowing hypotheses about the interplay between genetic actions and periodic rhythms to be tested. A simulation approach based on sustained circadian oscillations of the clock proteins and their mRNAs has been designed to test the statistical properties of the model. The model has significant implications for probing the molecular genetic mechanism of rhythmic oscillations through the detection of the clock QTL throughout the genome.
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    Whether and how thermal reaction norm is under genetic control is fundamental to understand the mechanistic basis of adaptation to novel thermal environments. However, the genetic study of thermal reaction norm is difficult because it is often expressed as a continuous function or curve. Here we derive a statistical model for dissecting thermal performance curves into individual quantitative trait loci (QTL) with the aid of a genetic linkage map. The model is constructed within the maximum likelihood context and implemented with the EM algorithm. It integrates the biological principle of responses to temperature into a framework for genetic mapping through rigorous mathematical functions established to describe the pattern and shape of thermal reaction norms. The biological advantages of the model lie in the decomposition of the genetic causes for thermal reaction norm into its biologically interpretable modes, such as hotter-colder, faster-slower and generalist-specialist, as well as the formulation of a series of hypotheses at the interface between genetic actions/interactions and temperature-dependent sensitivity. The model is also meritorious in statistics because the precision of parameter estimation and power of QTLdetection can be increased by modeling the mean-covariance structure with a small set of parameters. The results from simulation studies suggest that the model displays favorable statistical properties and can be robust in practical genetic applications. The model provides a conceptual platform for testing many ecologically relevant hypotheses regarding organismic adaptation within the Eco-Devo paradigm.
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    Functional mapping has emerged as a powerful tool for mapping quantitative trait loci (QTL) that control developmental patterns of complex dynamic traits. Original functional mapping has been constructed within the context of simple interval mapping, without consideration of separate multiple linked QTL for a dynamic trait. In this article, we present a statistical framework for mapping QTL that affect dynamic traits by capitalizing on the strengths of functional mapping and composite interval mapping. Within this so-called composite functional-mapping framework, functional mapping models the time-dependent genetic effects of a QTL tested within a marker interval using a biologically meaningful parametric function, whereas composite interval mapping models the time-dependent genetic effects of the markers outside the test interval to control the genome background using a flexible nonparametric approach based on Legendre polynomials. Such a semiparametric framework was formulated by a maximum-likelihood model and implemented with the EM algorithm, allowing for the estimation and the test of the mathematical parameters that define the QTL effects and the regression coefficients of the Legendre polynomials that describe the marker effects. Simulation studies were performed to investigate the statistical behavior of composite functional mapping and compare its advantage in separating multiple linked QTL as compared to functional mapping. We used the new mapping approach to analyze a genetic mapping example in rice, leading to the identification of multiple QTL, some of which are linked on the same chromosome, that control the developmental trajectory of leaf age.
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    We propose a nonparametric method for identifying parsimony and for producing a statistically efficient estimator of a large covariance matrix. We reparameterise a covariance matrix through the modified Cholesky decomposition of its inverse or the one-step-ahead predictive representation of the vector of responses and reduce the nonintuitive task of modelling covariance matrices to the familiar task of model selection and estimation for a sequence of regression models. The Cholesky factor containing these regression coefficients is likely to have many off-diagonal elements that are zero or close to zero. Penalised normal likelihoods in this situation with L-sub-1 and L-sub-2 penalities are shown to be closely related to Tibshirani's (1996) LASSO approach and to ridge regression. Adding either penalty to the likelihood helps to produce more stable estimators by introducing shrinkage to the elements in the Cholesky factor, while, because of its singularity, the L-sub-1 penalty will set some elements to zero and produce interpretable models. An algorithm is developed for computing the estimator and selecting the tuning parameter. The proposed maximum penalised likelihood estimator is illustrated using simulation and a real dataset involving estimation of a 102 × 102 covariance matrix. Copyright 2006, Oxford University Press.
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    Estimation of an unstructured covariance matrix is difficult because of its positive-definiteness constraint. This obstacle is removed by regressing each variable on its predecessors, so that estimation of a covariance matrix is shown to be equivalent to that of estimating a sequence of varying-coefficient and varying-order regression models. Our framework is similar to the use of increasing-order autoregressive models in approximating the covariance matrix or the spectrum of a stationary time series. As an illustration, we adopt Fan & Zhang's (2000) two-step estimation of functional linear models and propose nonparametric estimators of covariance matrices which are guaranteed to be positive definite. For parsimony a suitable order for the sequence of (auto)regression models is found using penalised likelihood criteria like AIC and BIC. Some asymptotic results for the local polynomial estimators of components of a covariance matrix are established. Two longitudinal datasets are analysed to illustrate the methodology. A simulation study reveals the advantage of the nonparametric covariance estimator over the sample covariance matrix for large covariance matrices. Copyright Biometrika Trust 2003, Oxford University Press.
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    This article proposes a data-driven method to identify parsimony in the covariance matrix of longitudinal data and to exploit any such parsimony to produce a statistically efficient estimator of the covariance matrix. The approach parameterizes the covariance matrix through the Cholesky decomposition of its inverse. For longitudinal data, this is a one-step-ahead predictive representation, and the Cholesky factor is likely to have off diagonal elements that are zero or close to zero. A hierarchical Bayesian model is used to identify any such zeros in the Cholesky factor, similar to approaches that have been successful in Bayesian variable selection. The model is estimated using a Markov chain Monte Carlo sampling scheme that is computationally efficient and can be applied to covariance matrices of high dimension. It is demonstrated through simulations that the proposed method compares favorably in terms of statistical efficiency with a highly regarded competing approach. The estimator is applied to three real examples in which the dimension of the covariance matrix is large relative to the sample size. The first two examples are from biometry and electricity demand modeling and are longitudinal. The third example is from finance and highlights the potential of our method for estimating cross-sectional covariance matrices.