Residual Covariance Structure Research Articles

Fast Calculation of Gaussian Process Multiple-Fold Cross-Validation Residuals and their Covariances

We generalize fast Gaussian process leave-one-out formulas to multiple-fold cross-validation, highlighting in turn the covariance structure of cross-validation residuals in simple and universal kriging frameworks. We illustrate how resulting covariances affect model diagnostics. We further establish in the case of noiseless observations that correcting for covariances between residuals in cross-validation-based estimation of the scale parameter leads back to maximum likelihood estimation. Also, we highlight in broader settings how differences between pseudo-likelihood and likelihood methods boil down to accounting or not for residual covariances. The proposed fast calculation of cross-validation residuals is implemented and benchmarked against a naive implementation, all in R. Numerical experiments highlight the substantial speed-ups that our approach enables. However, as supported by a discussion on main drivers of computational costs and by a numerical benchmark, speed-ups steeply decline as the number of folds (say, all sharing the same size) decreases. An application to a contaminant localization test case illustrates that the way of grouping observations in folds may affect model assessment and parameter fitting compared to leave-one-out. Overall, our results enable fast multiple-fold cross-validation, have consequences in model diagnostics, and pave the way to future work on hyperparameter fitting as well as on goal-oriented fold design. Supplementary materials for this article are available online.

Modelling spatial and temporal correlation in multi-assessment perennial crop variety selection trials using a multivariate autoregressive model

Context Perennial crop variety selection trials are often conducted over several seasons or years. These field trials often exhibit spatial correlation between plots. When data from multiple assessment times are analysed, it is necessary to account for both spatial and temporal correlation. A current approach is to use linear mixed models with separable spatial and temporal residual covariance structures. A limitation of these separable models is that they assume the same spatial correlation structure for each assessment time, which may not hold in practice. Aims This study aims to provide more flexible methods for modelling the spatio-temporal correlation in multi-assessment perennial crop data, allowing for differing spatial parameters for each time, together with modelling genetic effects over time. Methods The paper investigates the suitability of two-directional invariant multivariate autoregressive (2DIMVAR1) models for analysis of multi-assessment perennial crop data. The analysis method is applied to persistence data from a pasture breeding trial. Key results The multivariate autoregressive spatio-temporal residual models are a significant improvement on separable residual models under different genetic models. The paper demonstrates how to fit the models in practice using the software ASReml-R. Conclusions A flexible modelling approach for multi-assessment perennial crop data is presented, allowing differing spatial correlation parameters for each time. The models allow investigation into genotype × time interactions, while optimally accounting for spatial and temporal correlation. Implications The models provide improvements on current approaches and hence will result in more accurate genetic predictions in multi-assessment perennial crop variety selection trials.

Serum Creatinine Patterns in Neonates Treated with Therapeutic Hypothermia for Neonatal Encephalopathy

Introduction: There is large variability in kidney function and injury in neonates with neonatal encephalopathy (NE) treated with therapeutic hypothermia (TH). Acute kidney injury (AKI) definitions that apply categorical approaches may lose valuable information about kidney function in individual patients. Centile serum creatinine (SCr) over postnatal age (PNA) may provide more valuable information in TH neonates. Methods: Data from seven TH neonates and one non-TH-treated, non-NE control cohorts were pooled in a retrospective study. SCr centiles over PNA, and AKI incidence (definition: SCr ↑≥0.3 mg/dL within 48 h, or ↑ ≥1.5 fold vs. the lowest prior SCr within 7 days) and mortality were calculated. Repeated measurement linear models were applied to SCr trends, modeling SCr on PNA, birth weight or gestational age (GA), using heterogeneous autoregressive residual covariance structure and maximum likelihood methods. Findings were compared to patterns in the control cohort. Results: Among 1,136 TH neonates, representing 4,724 SCr observations, SCr (10th–25th–50th–75th–90th–95th) PNA centiles (day 1–10) were generated. In TH neonates, the AKI incidence was 132/1,136 (11.6%), mortality 193/1,136 (17%). AKI neonates had a higher mortality (37.2–14.3%, p < 0.001). Median SCr patterns over PNA were significantly higher in nonsurvivors (p < 0.01) or AKI neonates (p < 0.001). In TH-treated neonates, PNA and GA or birth weight explained SCr variability. Patterns over PNA were significantly higher in TH neonates to controls (801 neonates, 2,779 SCr). Conclusions: SCr patterns in TH-treated NE neonates are specific. Knowing PNA-related patterns enable clinicians to better assess kidney function and tailor pharmacotherapy, fluids, or kidney supportive therapies.

Bayesian Mixture Model of Extended Redundancy Analysis.

Extended redundancy analysis (ERA), a generalized version of redundancy analysis (RA), has been proposed as a useful method for examining interrelationships among multiple sets of variables in multivariate linear regression models. As a limitation of the extant RA or ERA analyses, however, parameters are estimated by aggregating data across all observations even in a case where the study population could consist of several heterogeneous subpopulations. In this paper, we propose a Bayesian mixture extension of ERA to obtain both probabilistic classification of observations into a number of subpopulations and estimation of ERA models within each subpopulation. It specifically estimates the posterior probabilities of observations belonging to different subpopulations, subpopulation-specific residual covariance structures, component weights and regression coefficients in a unified manner. We conduct a simulation study to demonstrate the performance of the proposed method in terms of recovering parameters correctly. We also apply the approach to real data to demonstrate its empirical usefulness.

Discrepant Inferences When Modeling Associations Between Time-Varying Exposures and Linear Growth Trajectories in Infancy Using Length-For-Age Z Scores Versus Raw Length

Abstract Objectives To compare inferences from longitudinal models of the relationships between biomarkers of interest and linear growth outcomes in infancy using length-for-age z-scores (LAZ) based on age- and sex-specific growth standards, or raw length. Methods This was a secondary analysis of data from a study of the association between bone-related biomarkers and infant linear growth trajectories up to 1 year of life in a subset of infants (n = 820) enrolled in the Maternal Vitamin D for Infant Growth trial. The linear growth outcome (LAZ or raw length) was modelled as a function of the interaction between each biomarker and age using linear mixed effect models with restricted cubic splines. Models were specified to obtain the best fit and reconcile discrepancies in results from LAZ and length models. Inferences from marginal effects at birth, 3 months, 6 months, and 12 months were compared, for a total 4 effect estimates from each of 10 pairs of LAZ and length models, resulting in 40 pairs of estimates. The following biomarkers were included: fibroblast growth factor 21 (FGF21), fibroblast growth factor 23 (FGF23), N-terminal propeptide of C-type natriuretic peptide (NT-proCNP), osteocalcin, osteoprotegerin, receptor activator of nuclear activator kappa-b ligand (RANKL), 25-hydroxyvitamin D (25OHD), C-reactive protein (CRP), Interleukin 6 (IL6), and insulin-like growth factor-1 (IGF1). Biomarkers were time-varying, measured in cord blood and at 3 and 6 months of age. Results The best fitting model for LAZ had 3 knots with random slopes, and the best fitting model for raw length had 5 knots, random slopes, and an exponential residual covariance structure. Comparisons of the pairs of marginal estimates from the LAZ vs length models resulted in discrepant inferences for 25% of effect estimates (10/40). Results were consistently concordant only for FGF23, 25OHD, CRP, and IL6. Conclusions Length and LAZ represent the same biological construct but their use in longitudinal modelling may lead to different inferences about associations between time-varying exposures and infant growth, possibly due to residual confounding by age. These findings raise concerns about the reliability of studies of determinants or markers of infant linear growth based on longitudinal modelling of growth trajectories. Funding Sources Bill and Melinda Gates Foundation.

Fitting Nonlinear Mixed-effects Models With Alternative Residual Covariance Structures

Nonlinear mixed-effects models are models in which one or more coefficients of the growth model enter in a nonlinear manner, such as appearing in the exponent of the growth function. In their applications, the within-individual residuals are often assumed to be independent with constant variance across time, an assumption that implies that the assumed growth function fully accounts for the dependencies and patterns of variation in the data. Studies have shown that a poorly specified within-individual residual covariance structure of a linear mixed-effects model can impact the estimated covariance matrix of the random effects at the second-level, model fit and statistical inference. The consequences for nonlinear mixed-effects models are not, however, clearly understood. This is due in part to the differences in the estimation needs of the two types of models. Using empirical data examples, this work illustrates the impact of fitting alternative residual covariance structures in nonlinear mixed-effects models that do not entirely parallel the results from studies of strictly linear mixed-effects models and call for the need of researchers to consider alternative structures when fitting nonlinear mixed-effects models.

Impact of residual covariance structures on genomic prediction ability in multi-environment trials.

In plant breeding, one of the main purpose of multi-environment trial (MET) is to assess the intensity of genotype-by-environment (G×E) interactions in order to select high-performing lines of each environment. Most models to analyze such MET data consider only the additive genetic effects and the part of the non-additive genetic effects are confounded with the residual terms and this may lead to the non-negligible residual covariances between the same trait measured at multiple environments. In breeding programs it is also common to have the phenotype information from some environments available and values are missing in some other environments. In this study we focused on two problems: (1) to study the impact of different residual covariance structures on genomic prediction ability using different models to analyze MET data; (2) to compare the ability of different MET analysis models to predict the missing values in a single environment. Our results suggests that, it is important to consider the heterogeneous residual covariance structure for the MET analysis and multivariate mixed model seems to be especially suitable to predict the missing values in a single environment. We also present the prediction abilities based on Bayesian and frequentist approaches with different models using field data sets (maize and rice) having different levels of G×E interactions.

Maternal body composition in seedstock herds. 3. Multivariate analysis using factor analytic models and cluster analysis

Considerable information exists on genetic relationships of body composition and carcass quality of young and finished beef cattle. However, there is a dearth of information on genetic relationships of cow body composition over time and, also, relationships with young-animal body-composition measures. The aim of the present study is to understand genetic relationships among various cow body-composition traits of Angus cows over time, from yearling to weaning of a second calf at ~3.5 years. To determine genetic correlations among various composition traits over time, a multi-trait–multi-time analysis is required. For the Maternal Productivity Project, this necessitates modelling of five traits (namely weight and ultrasound measure for loin eye muscle area (EMA), rib fat, P8 rump fat and intramuscular fat) by five time combinations (recordings at yearling then pre-calving and weaning in first and second parity). The approach was based on including all 25 trait-by-time combinations in an analysis using factor analytic models to approximate the genetic covariance matrix. Various models for the residual covariance structure were investigated. The analyses yielded correlations that could be compared with those of past studies reported in the literature and, also, to a set of bivariate analyses. Clustering of the genetic multi-trait–multi-time correlation structure resulted in a separation of traits (weight and EMA, and the fat traits) and also of time effects into early (heifer = before first lactation) and late (cow = post-first lactation) measurements.

A Functional Estimate of Covariation

ABSTRACTThe analysis of functional data calls for a bivariate functional covariance function σ(s, t) that may be evaluated at any discrete set of points to define a variance-covariance matrix This article uses finite element methodology to construct a representation of a functional Choleski factor λ(w, s) to define σ(s, t) = ∫λ(w, s)λ(w, t) dw. An estimate of is especially important for applications and, where the eigenstructure of the covariance permits, this is readily available since the resulting is almost always positive definite. A simulation study compares the performance of estimates of and to those from the classic covariance matrix estimate and an estimate using glasso package in R. The method’s capability of constraining estimates of to be strongly band-structured resulted in superior estimates. The real data application is to the smoothing of the Fels female growth data where σ(s, t) estimates the residual covariance structure in the presence of sampling points varying from one case to another. Supplementary materials are available online.

Plasticity in Vegetative Growth over Contrasted Growing Sites of an F1 Olive Tree Progeny during Its Juvenile Phase.

Climatic changes impact fruit tree growth and severely limit their production. Investigating the tree ability to cope with environmental variations is thus necessary to adapt breeding and management strategies in order to ensure sustainable production. In this study, we assessed the genetic parameters and genotype by environment interaction (GxE) during the early tree growth. One hundred and twenty olive seedlings derived from the cross ‘Olivière’ x ‘Arbequina’ were examined across two sites with contrasted environments, accounting for ontogenetic trends over three years. Models including the year of growth, branching order, environment, genotype effects, and their interactions were built with variance function and covariance structure of residuals when necessary. After selection of a model, broad sense heritabilities were estimated. Despite strong environmental effect on most traits, no GxE was found. Moreover, the internal structure of traits co-variation was similar in both sites. Ontogenetic growth variation, related to (i) the overall tree form and (ii) the growth and branching habit at growth unit scale, was not altered by the environment. Finally, a moderate to strong genetic control was identified for traits at the whole tree scale and at internode scale. Among all studied traits, the maximal internode length exhibited the highest heritability (H2 = 0.74). Considering the determinant role of this trait in tree architecture and its stability across environments, this study consolidates its relevance for breeding.

Network Risk and Financial Crisis

A stock is prone to co-move with its related stocks is considered as network risky and more vulnerable to shocks transmitted through a network in a bad time. I propose a new econometric procedure to estimate network risk using a factor model. Specifically, I construct a new network risk factor by the Katz centrality based on the residual returns covariance structure of stocks. New risk metrics featuring network risk are developed. An empirical study shows that the component of the standard deviation, value-at-risk or expected shortfall attributing to the network risk better explains stocks performance than their conventional counterparts in the 2007-2008 financial crisis.

Do gamblers eat more salt? Testing a latent trait model of covariance in consumption.

A diverse class of stimuli, including certain foods, substances, media, and economic behaviours, may be described as ‘reward-oriented’ in that they provide immediate reinforcement with little initial investment. Neurophysiological and personality concepts, including dopaminergic dysfunction, reward sensitivity and rash impulsivity, each predict the existence of a latent behavioural trait that leads to increased consumption of all stimuli in this class. Whilst bivariate relationships (co-morbidities) are often reported in the literature, to our knowledge, a multivariate investigation of this possible trait has not been done. We surveyed 1,194 participants (550 male) on their typical weekly consumption of 11 types of reward-oriented stimuli, including fast food, salt, caffeine, television, gambling products, and illicit drugs. Confirmatory factor analysis was used to compare models in a 3×3 structure, based on the definition of a single latent factor (none, fixed loadings, or estimated loadings), and assumed residual covariance structure (none, a-priori / literature based, or post-hoc / data-driven). The inclusion of a single latent behavioural ‘consumption’ factor significantly improved model fit in all cases. Also confirming theoretical predictions, estimated factor loadings on reward-oriented indicators were uniformly positive, regardless of assumptions regarding residual covariances. Additionally, the latent trait was found to be negatively correlated with the non-reward-oriented indicators of fruit and vegetable consumption. The findings support the notion of a single behavioural trait leading to increased consumption of reward-oriented stimuli across multiple modalities. We discuss implications regarding the concentration of negative lifestyle-related health behaviours.

A semiparametric approach to simultaneous covariance estimation for bivariate sparse longitudinal data

Estimation of the covariance structure for irregular sparse longitudinal data has been studied by many authors in recent years but typically using fully parametric specifications. In addition, when data are collected from several groups over time, it is known that assuming the same or completely different covariance matrices over groups can lead to loss of efficiency and/or bias. Nonparametric approaches have been proposed for estimating the covariance matrix for regular univariate longitudinal data by sharing information across the groups under study. For the irregular case, with longitudinal measurements that are bivariate or multivariate, modeling becomes more difficult. In this article, to model bivariate sparse longitudinal data from several groups, we propose a flexible covariance structure via a novel matrix stick-breaking process for the residual covariance structure and a Dirichlet process mixture of normals for the random effects. Simulation studies are performed to investigate the effectiveness of the proposed approach over more traditional approaches. We also analyze a subset of Framingham Heart Study data to examine how the blood pressure trajectories and covariance structures differ for the patients from different BMI groups (high, medium, and low) at baseline.

Preliminary Analysis of AASHO Road Test Rigid Pavement Data Using Modern Regression Techniques

The normality assumptions with random errors and constant variance were often violated while analyzing multilevel pavement performance data using conventional regression techniques. Because of its hierarchical data structure, multilevel data are often analyzed using Linear Mixed-Effects (LME) models. The exploratory analysis, statistical modeling, and the examination of model-fit of LME models are more complicated than those of standard multiple regressions. A systematic modeling approach using visual-graphical techniques and LME models was proposed and demonstrated using the original AASHO road test rigid pavement data. The basic modeling approach includes: selecting a preliminary mean structure, selecting a random structure, selecting a residual covariance structure, model reduction, and examining the model fit. A goodness of fit plot indicates that the preliminary LME model provides better explanation to the data.

Selecting a linear mixed model for longitudinal data: Repeated measures analysis of variance, covariance pattern model, and growth curve approaches.

With increasing popularity, growth curve modeling is more and more often considered as the 1st choice for analyzing longitudinal data. Although the growth curve approach is often a good choice, other modeling strategies may more directly answer questions of interest. It is common to see researchers fit growth curve models without considering alterative modeling strategies. In this article we compare 3 approaches for analyzing longitudinal data: repeated measures analysis of variance, covariance pattern models, and growth curve models. As all are members of the general linear mixed model family, they represent somewhat different assumptions about the way individuals change. These assumptions result in different patterns of covariation among the residuals around the fixed effects. In this article, we first indicate the kinds of data that are appropriately modeled by each and use real data examples to demonstrate possible problems associated with the blanket selection of the growth curve model. We then present a simulation that indicates the utility of Akaike information criterion and Bayesian information criterion in the selection of a proper residual covariance structure. The results cast doubt on the popular practice of automatically using growth curve modeling for longitudinal data without comparing the fit of different models. Finally, we provide some practical advice for assessing mean changes in the presence of correlated data.

A hierarchical bayesian approach to multi-trait clinical quantitative trait locus modeling.

Recent advances in high-throughput genotyping and transcript profiling technologies have enabled the inexpensive production of genome-wide dense marker maps in tandem with huge amounts of expression profiles. These large-scale data encompass valuable information about the genetic architecture of important phenotypic traits. Comprehensive models that combine molecular markers and gene transcript levels are increasingly advocated as an effective approach to dissecting the genetic architecture of complex phenotypic traits. The simultaneous utilization of marker and gene expression data to explain the variation in clinical quantitative trait, known as clinical quantitative trait locus (cQTL) mapping, poses challenges that are both conceptual and computational. Nonetheless, the hierarchical Bayesian (HB) modeling approach, in combination with modern computational tools such as Markov chain Monte Carlo (MCMC) simulation techniques, provides much versatility for cQTL analysis. Sillanpää and Noykova (2008) developed a HB model for single-trait cQTL analysis in inbred line cross-data using molecular markers, gene expressions, and marker-gene expression pairs. However, clinical traits generally relate to one another through environmental correlations and/or pleiotropy. A multi-trait approach can improve on the power to detect genetic effects and on their estimation precision. A multi-trait model also provides a framework for examining a number of biologically interesting hypotheses. In this paper we extend the HB cQTL model for inbred line crosses proposed by Sillanpää and Noykova to a multi-trait setting. We illustrate the implementation of our new model with simulated data, and evaluate the multi-trait model performance with regard to its single-trait counterpart. The data simulation process was based on the multi-trait cQTL model, assuming three traits with uncorrelated and correlated cQTL residuals, with the simulated data under uncorrelated cQTL residuals serving as our test set for comparing the performances of the multi-trait and single-trait models. The simulated data under correlated cQTL residuals were essentially used to assess how well our new model can estimate the cQTL residual covariance structure. The model fitting to the data was carried out by MCMC simulation through OpenBUGS. The multi-trait model outperformed its single-trait counterpart in identifying cQTLs, with a consistently lower false discovery rate. Moreover, the covariance matrix of cQTL residuals was typically estimated to an appreciable degree of precision under the multi-trait cQTL model, making our new model a promising approach to addressing a wide range of issues facing the analysis of correlated clinical traits.

Bayesian functional mapping of dynamic quantitative traits

Without consideration of other linked QTLs responsible for dynamic trait, original functional mapping based on a single QTL model is not optimal for analyzing multiple dynamic trait loci. Despite that composite functional mapping incorporates the effects of genetic background outside the tested QTL in mapping model, the arbitrary choice of background markers also impact on the power of QTL detection. In this study, we proposed Bayesian functional mapping strategy that can simultaneously identify multiple QTL controlling developmental patterns of dynamic traits over the genome. Our proposed method fits the change of each QTL effect with the time by Legendre polynomial and takes the residual covariance structure into account using the first autoregressive equation. Also, Bayesian shrinkage estimation was employed to estimate the model parameters. Especially, we specify the gamma distribution as the prior for the first-order auto-regressive coefficient, which will guarantee the convergence of Bayesian sampling. Simulations showed that the proposed method could accurately estimate the QTL parameters and had a greater statistical power of QTL detection than the composite functional mapping. A real data analysis of leaf age growth in rice is used for the demonstration of our method. It shows that our Bayesian functional mapping can detect more QTLs as compared to composite functional mapping.

Linear Mixed-Effects Models for Preliminary Analysis of Flexible Pavement Serviceability Index Data

Multilevel data are common in many fields. Because of the hierarchical data structure, multilevel data are often analyzed with linear mixed-effects (LME) models. Exploratory analysis, statistical modeling, and examination of the model fit of LME models are more complex than those of standard multiple regressions. A systematic modeling approach that uses visual-graphical techniques and LME models is proposed and demonstrated with the original AASHO Road Test flexible pavement serviceability index data. The proposed approach includes exploring the growth patterns at both group and individual levels, identifying the important predictors and unusual subjects, choosing suitable statistical models, selecting a preliminary mean structure, selecting a random structure, selecting a residual covariance structure, model reduction, and examination of the model fit.

Application of multiple‐trait finite mixture model to test‐day records of milk yield and somatic cell score of Canadian Holsteins

Multiple-trait (MT) finite mixture random regression (MIX) model was applied using Bayesian methods to first lactation test-day (TD) milk yield and somatic cell score (SCS) of Canadian Holsteins, allowing for heterogeneity of distributions with respect to days in milk (DIM) in lactation. The assumption was that the associations between patterns of variation in these traits and mastitis would allow revealing the hidden structure in the data distribution because of unknown health status of cows. The MIX model assumed separate means and residual co-variance structures for two components in four intervals of lactation, in addition to fitting the fixed effect of herd-test-day, and fixed and random regressions with Legendre polynomials. Results indicated that the mixture model was superior to standard MT model, as supported by the Bayes factor. Approximately 20% of TD records were classified as originated from cows with a putative, sub-clinical form of mastitis. The proportion of records from mastitic cows was the largest at the beginning of lactation. The MIX model exhibited different distributions of data from healthy and infected cows in different parts of lactation. Records from sick cows were characterized by larger (smaller) means for SCS (milk) and larger variances. Residual, and daily genetic and environmental correlations between milk and SCS were smaller from the MIX model when compared with MT estimates. Heritabilities of both traits differed significantly among records from healthy, sick and MT model estimates. Both models fitted milk records from healthy cows relatively well. The ability of the MT model in handling SCS records, measured by model residuals, was low, but improved substantially, however, where the data were allowed to be separated into two components in the MIX parameterization. Correlations between estimated breeding values (EBV) for sires from both models were very high for cumulative milk yield (>0.99) and slightly lower (0.95 in the interval from 5 to 45 DIM) for daily SCS. EBV for SCS from MT and MIX models were weakly correlated with posterior probability of sub-clinical mastitis on the phenotypic scale.

Mapeo de loci de caracteres cuantitativos (QTL) usando un enfoque multivariado

AbstractF. Mora, A.I. Santos, and C.A. Scapim. 2008. Mappin g quantitative trait loci (QTLs) usin ga multivariate approach. Cien. Inv. Agr. 35(2):137-145. Statistical procedures for mappingquantitative trait loci (QTL) have been extensively studied because they are the essential for improving the accuracy of genetic analyses. The ob jective of the present study was to examineQTL using multivariate methods, considering the principle of simple interval mapping. A microsatellite marker data set from a F 2 population was simulated. It was assumed that theQTL control binomial and normal traits. In the normal case, fi ve QTL mapping models wereevaluated that had the following residual covariance structures: variance components (VC),compound s ymmetry (CS), unstructured (UN), banded main diagonal (UN1) and heterogeneoustoeplitz. Akaike’s information criterion (AIC) was used to select the appropriate structure. Inthe binar y case, the Generalized Estimating Equations (GEE) procedure was used. UN structureminimized the AIC value on the interval that indicated a higher probability of the QTL. In thebinomial case, a non-independent working correlation matrix (WCM) was fi tted (ρ = 0.47). Inboth cases, the additive effect of QTL was signi fi cant (p 0.05). Thus, QTL mapping using a multivariate approach ma y be a useful tool for breedingprograms that aim to improve quantitative traits that have phenotypic values that change over time.