Inference for nonparanormal partial correlation via regularized rank-based nodewise regression

Partial correlation is a common tool in studying conditional dependence for Gaussian distributed data. However, partial correlation being zero may not be equivalent to conditional independence under non-Gaussian distributions. In this paper, we propose a statistical inference procedure for partial correlations under the high-dimensional nonparanormal (NPN) model where the observed data are normally distributed after certain monotone transformations. The NPN partial correlation is the partial correlation of the normal transformed data under the NPN model, which is a more general measure of conditional dependence. We estimate the NPN partial correlations by regularized nodewise regression based on the empirical ranks of the original data. A multiple testing procedure is proposed to identify the nonzero NPN partial correlations. The proposed method can be carried out by a simple coordinate descent algorithm for lasso optimization. It is easy-to-implement and computationally more efficient compared to the existing methods for estimating NPN graphical models. Theoretical results are developed to show the asymptotic normality of the proposed estimator and to justify the proposed multiple testing procedure. Numerical simulations and a case study on brain imaging data demonstrate the utility of the proposed procedure and evaluate its performance compared to the existing methods. Data used in preparation of this article were obtained from the Alzheimer's Disease Neuroimaging Initiative (ADNI) database.     

Realized Sampling Variances of Estimates of Genetic Parameters and the Difference between Genetic and Phenotypic Correlations

A data set of 1572 heritability estimates and 1015 pairs of genetic and phenotypic correlation estimates, constructed from a survey of published beef cattle genetic parameter estimates, provided a rare opportunity to study realized sampling variances of genetic parameter estimates. The distribution of both heritability estimates and genetic correlation estimates, when plotted against estimated accuracy, was consistent with random error variance being some three times the sampling variance predicted from standard formulae. This result was consistent with the observation that the variance of estimates of heritabilities and genetic correlations between populations were about four times the predicted sampling variance, suggesting few real differences in genetic parameters between populations. Except where there was a strong biological or statistical expectation of a difference, there was little evidence for differences between genetic and phenotypic correlations for most trait combinations or for differences in genetic correlations between populations. These results suggest that, even for controlled populations, estimating genetic parameters specific to a given population is less useful than commonly believed. A serendipitous discovery was that, in the standard formula for theoretical standard error of a genetic correlation estimate, the heritabilities refer to the estimated values and not, as seems generally assumed, the true population values.     

Spatial and Space-Time Correlations in Systems of Subpopulations with Genetic Drift and Migration

The geographic distribution of genetic variation is an important theoretical and experimental component of population genetics. Previous characterizations of genetic structure of populations have used measures of spatial variance and spatial correlations. Yet a full understanding of the causes and consequences of spatial structure requires complete characterization of the underlying space-time system. This paper examines important interactions between processes and spatial structure in systems of subpopulations with migration and drift, by analyzing correlations of gene frequencies over space and time. We develop methods for studying important features of the complete set of space-time correlations of gene frequencies for the first time in population genetics. These methods also provide a new alternative for studying the purely spatial correlations and the variance, for models with general spatial dimensionalities and migration patterns. These results are obtained by employing theorems, previously unused in population genetics, for space-time autoregressive (STAR) stochastic spatial time series. We include results on systems with subpopulation interactions that have time delay lags (temporal orders) greater than one. We use the space-time correlation structure to develop novel estimators for migration rates that are based on space-time data (samples collected over space and time) rather than on purely spatial data, for real systems. We examine the space-time and spatial correlations for some specific stepping stone migration models. One focus is on the effects of anisotropic migration rates. Partial space-time correlation coefficients can be used for identifying migration patterns. Using STAR models, the spatial, space-time, and partial space-time correlations together provide a framework with an unprecedented level of detail for characterizing, predicting and contrasting space-time theoretical distributions of gene frequencies, and for identifying features such as the pattern of migration and estimating migration rates in experimental studies of genetic variation over space and time.     

A method for assessing patterns of familial resemblance in complex human pedigrees, with an application to the nevus-count data in Utah kindreds.

An analytic method is described for estimating phenotypic correlations between pairs of members of specific relationships in pedigrees. In estimating correlations, this new method allows simultaneous adjustment for available covariates such as age, gender, environmental factors, and variables reflecting ascertainment mode, through mean- and variance-regression models. The estimated correlations and regression coefficients corresponding to covariates are consistent and asymptotically normally distributed. Differing from a full-likelihood approach, this new method does not require the assumption of a particular joint distribution of phenotypes from a pedigree, such as the multivariate normal distribution, but instead only requires correct specification of mean- and variance-regression models. Within this framework, missing data, if they are missing completely at random, can be ignored without biasing estimates. The method is illustrated by an application using nevus-count data from 28 Utah kinships. The results from the analysis are that covariate-adjusted nevus counts are correlated between parents and children (correlation .22; P less than .001) and between siblings (correlation .32; P less than .001), while the correlation of -.04 between husband and wife is not significantly different (P = .31) from 0. This result is consistent with a genetic etiology of nevus count.     

Analysis of functional abilities for elderly Danish twins using GEE models.

In this work we present a new method for genetic analysis of twin data which is based on generalized estimating equations and allows for analysis of various response types (e.g., continuous, binary, counts) combined with estimation of residual correlations. The new approach allows for control of covariates of any kind (e.g., continuous, counts) by modeling the dependence of mean and variance on background variables. The proposed method was applied to identify the covariates that have a significant influence on elderly people's functional abilities, and find the estimates for the correlation coefficients of residuals for MZ and DZ twins in a sample of 2401 Danish twin 75 years of age or older. The bootstrap method was used to obtain standard errors for correlation coefficients. It was shown, that the chosen covariates have similar effects on MZ and DZ twins, and that the residual correlation in MZ twins is significantly higher than in DZ twins, which indicates that genetic factors play an etiological role in the determination of physical status of elderly people, controlled for 10 background variables.     

Using clonal replicates to explore genetic variation in a perennial plant species

Summary An experimental design is presented for estimating genetic parameters using a family structure with clonally replicated individuals. This experimental design provides a technique to quantify genetic variation in a population, with partial separation of additive, dominance and epistatic gene action. Our method is offered as an alternative to techniques for estimating epistatic gene action that require several generations and/or inbreeding. Such methods are not particularly useful for long-lived perennials with long generation cycles. An example of the analysis is given with a forest tree species, Populus deltoides Bartr., and parameter estimates are presented for traits measured over 8 years.     

Bootstrap methods for estimating spatial synchrony of fluctuating populations

We describe and examine methods for estimating spatial correlations used in population ecology. We base our analyses on a hypothetical example of a species that has been censured at 30 different locations for 20 years. We assume that the population fluctuations can be described by a simple linear model on logarithmic scale. Stochastic simulations is utilized to check how seven different ways of resampling perform when the goal is to find nominal 95% confidence intervals for the spatial correlation in growth rates at given distances. It turns out that resampling of locations performs badly, with true coverage level as low as 30–40%, especially for small correlations at long distances. Resampling of timepoints performs much better, with coverage varying from 80 to 90%, depending on the strength of density regulation and whether the spatial correlation is estimated for the response variable or for the error terms in the model. Assuming that the underlying model is known, the best results are achieved for parametric bootstrapping, a result that strongly emphasize the importance of defining and estimating a proper population model when studying spatial processes.     

Genetic parameters for somatic cell score according to udder infection status in Valle del Belice dairy sheep and impact of imperfect diagnosis of infection

Background

Somatic cell score (SCS) has been promoted as a selection criterion to improve mastitis resistance. However, SCS from healthy and infected animals may be considered as separate traits. Moreover, imperfect sensitivity and specificity could influence animals'' classification and impact on estimated variance components. This study was aimed at: (1) estimating the heritability of bacteria negative SCS, bacteria positive SCS, and infection status, (2) estimating phenotypic and genetic correlations between bacteria negative and bacteria positive SCS, and the genetic correlation between bacteria negative SCS and infection status, and (3) evaluating the impact of imperfect diagnosis of infection on variance component estimates.

Methods

Data on SCS and udder infection status for 1,120 ewes were collected from four Valle del Belice flocks. The pedigree file included 1,603 animals. The SCS dataset was split according to whether animals were infected or not at the time of sampling. A repeatability test-day animal model was used to estimate genetic parameters for SCS traits and the heritability of infection status. The genetic correlation between bacteria negative SCS and infection status was estimated using an MCMC threshold model, implemented by Gibbs Sampling.

Results

The heritability was 0.10 for bacteria negative SCS, 0.03 for bacteria positive SCS, and 0.09 for infection status, on the liability scale. The genetic correlation between bacteria negative and bacteria positive SCS was 0.62, suggesting that they may be genetically different traits. The genetic correlation between bacteria negative SCS and infection status was 0.51. We demonstrate that imperfect diagnosis of infection leads to underestimation of differences between bacteria negative and bacteria positive SCS, and we derive formulae to predict impacts on estimated genetic parameters.

Conclusions

The results suggest that bacteria negative and bacteria positive SCS are genetically different traits. A positive genetic correlation between bacteria negative SCS and liability to infection was found, suggesting that the approach of selecting animals for decreased SCS should help to reduce mastitis prevalence. However, the results show that imperfect diagnosis of infection has an impact on estimated genetic parameters, which may reduce the efficiency of selection strategies aiming at distinguishing between bacteria negative and bacteria positive SCS.     

On the relationship between plant species diversity and genetic diversity of Plantago lanceolata (Plantaginaceae) within and between grassland communities

Aims and Methods The relationship between genetic diversity and species diversity and the underlying mechanisms are of both fundamental and applied interest. We used amplified fragment length polymorphism (AFLP) and vegetation records to investigate the association between genetic diversity of Plantago lanceolata and plant species diversity using 15 grassland communities in central Germany. We used correlation and partial correlation analyses to examine whether relationships between genetic and species diversity were direct or mediated by environmental differences between habitats.Important findings Both within- and between-population genetic diversity of P. lanceolata were significantly positively correlated with plant species diversity within and between sites. Simple and partial correlations revealed that the positive correlations indirectly resulted from the effects of abiotic habitat characteristics on plant species diversity and, via abundance, on genetic diversity of P. lanceolata. Thus, they did not reflect a direct causal relationship between plant species diversity and genetic diversity of P. lanceolata, as would have been expected based on the hypothesis of a positive relationship between plant species diversity and niche diversity.     

Measuring the costs of reproduction

The measurement of costs of reproduction is of interest because such costs are generally assumed by life history theory. There is some controversy concerning how to measure costs: common methods include experimental manipulations of life history, such as preventing some individuals from reproducing, or estimates of genetic correlations. These two methods often yield similar results, suggesting that one can serve as a substitute for the other. There are now experiments which demonstrate that there are different mechanisms underlying the response to an experimental manipulation versus a genetic correlation, so the two methods are not equivalent in estimating costs.     

Heritabilities and genetic correlations for reproductive traits in an F2 reciprocal cross chicken population

Studies estimating genetic parameters for reproductive traits in chickens can be useful for understanding and improvement of their genetic architecture. A total of 1276 observations of fertility (FERT), hatchability of fertile eggs (HFE) and hatchability of total eggs (HTE) were used to estimate the genetic and phenotypic parameters of 467 females from an F2 population generated by reciprocal crossing between a broiler line and a layer line, which were developed through a poultry genetics breeding program, maintained by Embrapa Swine and Poultry, Concordia, Santa Catarina, Brazil. Estimates of heritability and genetic and phenotypic correlations were obtained using restricted maximum likelihood calculations under the two-trait animal model, including the fixed effect of group (hatching of birds from the same genetic group) and the random additive genetic and residual effects. The mean percentages for FERT, HFE and HTE were 87.91 ± 19.77, 80.07 ± 26.81 and 70.67 ± 28.55%, respectively. The highest heritability estimate (h(2)) was 0.28 ± 0.04 for HTE. Genetic correlations for FERT with HFE (0.43 ± 0.17), HFE with HTE (0.98 ± 0.02) and FERT with HTE (0.69 ± 0.10) were positive and significant. Individuals with high breeding value for HTE would have high breeding values for HFE and FERT because of the high genetic association between them. These results suggest that HTE should be included as a selection criterion in genetic breeding programs to improve the reproductive performance of chickens, because HTE had the highest heritability estimate and high genetic correlation with FERT and HFE, and it is the easiest to measure.     

Shortest path analysis using partial correlations for classifying gene functions from gene expression data

MOTIVATION: Gaussian graphical models (GGMs) are a popular tool for representing gene association structures. We propose using estimated partial correlations from these models to attach lengths to the edges of the GGM, where the length of an edge is inversely related to the partial correlation between the gene pair. Graphical lasso is used to fit the GGMs and obtain partial correlations. The shortest paths between pairs of genes are found. Where terminal genes have the same biological function intermediate genes on the path are classified as having the same function. We validate the method using genes of known function using the Rosetta Compendium of yeast (Saccharomyces Cerevisiae) gene expression profiles. We also compare our results with those obtained using a graph constructed using correlations. RESULTS: Using a partial correlation graph, we are able to classify approximately twice as many genes to the same level of accuracy as when using a correlation graph. More importantly when both methods are tuned to classify a similar number of genes, the partial correlation approach can increase the accuracy of the classifications.     

Estimation of genetic parameters for BW and body measurements in Brahman cattle

Body weight and body measurements are commonly used to represent growth and measured at several growth stages in beef cattle. Those economically important traits should be genetically improved. To achieve breeding programs, genetic parameters are prerequisite, as they are needed for designing and predicting outcomes of breeding programs, as well as estimating of breeding values. (Co)variance components were estimated for BW and body measurements on Brahman cattle born between 1990 and 2016 from 17 research herds across Thailand. The traits measured were BW, heart girth (GR), hip height (HH) and body length (BL) and were measured at birth, 200 days, 400 days and 600 days of age. The number of records varied between traits from 18 890 for birth BW to 876 for GR at 600 days. Estimation of variance components was performed using restricted maximum likelihood using univariate and multivariate animal models. Pre-weaning traits were influenced by genetic and/or permanent environmental effects of the dam, except for BL. Heritability estimates from birth to 600 days of age ranged from 0.28±0.01 to 0.50±0.06 for BW, 0.27±0.01 to 0.43±0.09 for GR, 0.28±0.01 to 0.58±0.08 for HH and 0.34±0.01 to 0.51±0.08 for BL using univariate analysis. Heritability estimates for the traits studied increased with age. A similar trend was observed for the phenotypic and genetic correlations between subsequent BW and body measurements. A positive correlation was observed between different traits measured at a similar age, ranging from 0.22±0.01 to 0.72±0.01 for the phenotypic correlation and 0.25±0.04 to 0.97±0.11 for the genetic correlation. Also, a positive correlation was observed for similar traits across different age classes ranging from 0.07±0.03 to 0.76±0.02 for the phenotypic correlation and 0.24±0.11 to 0.92±0.05 for the genetic correlation. Therefore, all correlations between body measurements at the same age and across age classes were positive. The results show the potential improvement of growth traits in Brahman cattle, and those traits can be improved simultaneously under the same breeding program.     

Genetic parameters of milk fatty acid profile in sheep: comparison between gas chromatographic measurements and Fourier-transform IR spectroscopy predictions

Fatty acid (FA) composition is a key component of sheep milk nutritional quality. However, breeding for FA composition in dairy sheep is hampered by the logistic and phenotyping costs. This study was aimed at estimating genetic parameters for sheep milk FA and to test the feasibility of their routine measurement by using Fourier-transform IR (FTIR) spectroscopy. Milk FA composition of 989 Sarda ewes farmed in 48 flocks was measured by gas chromatography (FA_GC). Moreover, FTIR spectrum was collected for each sample, and it was used to predict FA composition (FA_FTIR) by partial least squares regression: 700 ewes were used for estimating model parameters, whereas the remaining 289 animals were used to validate the model. One hundred replicates were performed by randomly assigning animals to estimation and validation data set, respectively. Variance components for both measured and predicted traits were estimated with an animal model that included the fixed effects of parity, days in milking interval, lambing month, province, altitude of flock location, the random effects of flock-test-date and animal genetic additive. Genetic correlations among FA_GC, and between corresponding FA_GC and FA_FTIR were estimated by bivariate analysis. Coefficients of determination between FA_GC and FA_FTIR ranged from moderate (about 0.50 for odd- and branched-chain FA) to high (about 0.90 for de novo FA) values. Low-to-moderate heritabilities were observed for individual FA (ranging from 0.01 to 0.47). The largest value was observed for GC measured C16:0. Low–to-moderate heritabilities were estimated for FA groups. In most of cases, heritabilites were slightly larger for FA_GC than FA_FTIR. Estimates of genetic correlations among FA_GC showed a large variability in magnitude and sign. The genetic correlation between FA_FTIR and FA_GC was higher than 60% for all investigated traits. Results of the present study confirm the existence of genetic variability of the FA composition in sheep and suggest the feasibility of using FA_FTIR as proxies for these traits in large scale breeding programs.     

The genetic variance but not the genetic covariance of life‐history traits changes towards the north in a time‐constrained insect

Seasonal time constraints are usually stronger at higher than lower latitudes and can exert strong selection on life‐history traits and the correlations among these traits. To predict the response of life‐history traits to environmental change along a latitudinal gradient, information must be obtained about genetic variance in traits and also genetic correlation between traits, that is the genetic variance‐covariance matrix, G . Here, we estimated G for key life‐history traits in an obligate univoltine damselfly that faces seasonal time constraints. We exposed populations to simulated native temperatures and photoperiods and common garden environmental conditions in a laboratory set‐up. Despite differences in genetic variance in these traits between populations (lower variance at northern latitudes), there was no evidence for latitude‐specific covariance of the life‐history traits. At simulated native conditions, all populations showed strong genetic and phenotypic correlations between traits that shaped growth and development. The variance–covariance matrix changed considerably when populations were exposed to common garden conditions compared with the simulated natural conditions, showing the importance of environmentally induced changes in multivariate genetic structure. Our results highlight the importance of estimating variance–covariance matrixes in environments that mimic selection pressures and not only trait variances or mean trait values in common garden conditions for understanding the trait evolution across populations and environments.     

Sampling variance of the correlation coefficients estimated from analyses of variance and covariance

Summary A generalized sampling variance of correlation coefficients is derived for phenotypic, genetic and en vironmental correlations estimated from nested analyses of variance and covarianee for the equal number case. A numerical example is presented to estimate the sampling variance for the genetic correlation coefficient based on the relationship among full sibs using unequal subclass numbers.Journal Paper No. 3472 of the Purdue University Agricultural Experiment Station. This research was supported, in part, by NIH Biometry Training Grant, GM-00024.     

A Regression Modeling Approach for Describing Patterns of HIV Genetic Variation

We introduce a novel approach for describing patterns of HIV genetic variation using regression modeling techniques. Parameters are defined for describing genetic variation within and between viral populations by generalizing Simpson's index of diversity. Regression models are specified for these variation parameters and the generalized estimating equation framework is used for estimating both the regression parameters and their corresponding variances. Conditions are described under which the usual asymptotic approximations to the distribution of the estimators are met. This approach provides a formal statistical framework for testing hypotheses regarding the changing patterns of HIV genetic variation over time within an infected patient. The application of these methods for testing biologically relevant hypotheses concerning HIV genetic variation is demonstrated in an example using sequence data from a subset of patients from the Multicenter AIDS Cohort Study.     

Genetic correlations do not constrain the evolution of sexual dimorphism in the moss Ceratodon purpureus

The trajectory of phenotypic evolution is constrained in the short term by genetic correlations among traits. However, the extent to which genetic correlations impose a lasting constraint is generally unknown. Here, I examine the genetic architecture of life-history variation in male and female gametophytes from two populations of the moss Ceratodon purpureus, focusing on genetic correlations within and between the sexes. A significant negative correlation between allocation to vegetative and reproductive tissue was evident in males of both populations, but not females. All traits showed between-sex correlations of significantly less than one, indicating additive genetic variance for sexual dimorphism. The degree of dimorphism for traits was significantly negatively associated with the strength of the between-sex correlation. The structure of genetic correlations among life-history traits was more divergent between the two populations in females than in males. Collectively, these results suggest that genetic correlations do not impose a lasting constraint on the evolution of life-history variation in the species.     

Making Large-Scale Networks from fMRI Data

Pairwise correlations are currently a popular way to estimate a large-scale network (> 1000 nodes) from functional magnetic resonance imaging data. However, this approach generally results in a poor representation of the true underlying network. The reason is that pairwise correlations cannot distinguish between direct and indirect connectivity. As a result, pairwise correlation networks can lead to fallacious conclusions; for example, one may conclude that a network is a small-world when it is not. In a simulation study and an application to resting-state fMRI data, we compare the performance of pairwise correlations in large-scale networks (2000 nodes) against three other methods that are designed to filter out indirect connections. Recovery methods are evaluated in four simulated network topologies (small world or not, scale-free or not) in scenarios where the number of observations is very small compared to the number of nodes. Simulations clearly show that pairwise correlation networks are fragmented into separate unconnected components with excessive connectedness within components. This often leads to erroneous estimates of network metrics, like small-world structures or low betweenness centrality, and produces too many low-degree nodes. We conclude that using partial correlations, informed by a sparseness penalty, results in more accurate networks and corresponding metrics than pairwise correlation networks. However, even with these methods, the presence of hubs in the generating network can be problematic if the number of observations is too small. Additionally, we show for resting-state fMRI that partial correlations are more robust than correlations to different parcellation sets and to different lengths of time-series.     

Experimental evolution of phenotypic plasticity: how predictive are cross-environment genetic correlations?

Genetic correlations are often predictive of correlated responses of one trait to selection on another trait. There are examples, however, in which genetic correlations are not predictive of correlated responses. We examine how well a cross-environment genetic correlation predicts correlated responses to selection and the evolution of phenotypic plasticity in the seed beetle Stator limbatus. This beetle exhibits adaptive plasticity in egg size by laying large eggs on a resistant host and small eggs on a high-quality host. From a half-sib analysis, the cross-environment genetic correlation estimate was large and positive (rA=0.99). However, an artificial-selection experiment on egg size found that the realized genetic correlations were positive but asymmetrical; that is, they depended on both the host on which selection was imposed and the direction of selection. The half-sib estimate poorly predicted the evolution of egg size plasticity; plasticity evolved when selection was imposed on one host but did not evolve when selection was imposed on the other host. We use a simple two-locus additive genetic model to explore the conditions that can generate the observed realized genetic correlation and the observed pattern of plasticity evolution. Our model and experimental results indicate that the ability of genetic correlations to predict correlated responses to selection depends on the underlying genetic architecture producing the genetic correlation.