Underlying Covariance Structure Research Articles

Empirical likelihood for the analysis of experimental designs

Empirical likelihood enables a nonparametric, likelihood-driven style of inference without restrictive assumptions routinely made in parametric models. We develop a framework for applying empirical likelihood to the analysis of experimental designs, addressing issues that arise from blocking and multiple hypothesis testing. In addition to popular designs such as balanced incomplete block designs, our approach allows for highly unbalanced, incomplete block designs. We derive an asymptotic multivariate chi-square distribution for a set of empirical likelihood test statistics and propose two single-step multiple testing procedures: asymptotic Monte Carlo and nonparametric bootstrap. Both procedures asymptotically control the generalised family-wise error rate and efficiently construct simultaneous confidence intervals for comparisons of interest without explicitly considering the underlying covariance structure. A simulation study demonstrates that the performance of the procedures is robust to violations of standard assumptions of linear mixed models. We also present an application to experiments on a pesticide.

ScreeNOT: Exact MSE-optimal singular value thresholding in correlated noise

We derive a formula for optimal hard thresholding of the singular value decomposition in the presence of correlated additive noise; although it nominally involves unobservables, we show how to apply it even where the noise covariance structure is not a priori known or is not independently estimable. The proposed method, which we call ScreeNOT, is a mathematically solid alternative to Cattell’s ever-popular but vague scree plot heuristic from 1966. ScreeNOT has a surprising oracle property: it typically achieves exactly, in large finite samples, the lowest possible MSE for matrix recovery, on each given problem instance, that is, the specific threshold it selects gives exactly the smallest achievable MSE loss among all possible threshold choices for that noisy data set and that unknown underlying true low rank model. The method is computationally efficient and robust against perturbations of the underlying covariance structure. Our results depend on the assumption that the singular values of the noise have a limiting empirical distribution of compact support; this property, which is standard in random matrix theory, is satisfied by many models exhibiting either cross-row correlation structure or cross-column correlation structure, and also by many situations with more general, interelement correlation structure. Simulations demonstrate the effectiveness of the method even at moderate matrix sizes. The paper is supplemented by ready-to-use software packages implementing the proposed algorithm: package ScreeNOT in Python (via PyPI) and R (via CRAN).

NetGAM: Using generalized additive models to improve the predictive power of ecological network analyses constructed using time-series data

Ecological network analyses are used to identify potential biotic interactions between microorganisms from species abundance data. These analyses are often carried out using time-series data; however, time-series networks have unique statistical challenges. Time-dependent species abundance data can lead to species co-occurrence patterns that are not a result of direct, biotic associations and may therefore result in inaccurate network predictions. Here, we describe a generalize additive model (GAM)-based data transformation that removes time-series signals from species abundance data prior to running network analyses. Validation of the transformation was carried out by generating mock, time-series datasets, with an underlying covariance structure, running network analyses on these datasets with and without our GAM transformation, and comparing the network outputs to the known covariance structure of the simulated data. The results revealed that seasonal abundance patterns substantially decreased the accuracy of the inferred networks. In addition, the GAM transformation increased the predictive power (F1 score) of inferred ecological networks on average and improved the ability of network inference methods to capture important features of network structure. This study underscores the importance of considering temporal features when carrying out network analyses and describes a simple, effective tool that can be used to improve results.

Theory for socio-demographic enrichment performance using the inverse discrete choice modelling approach

In light of the growing availability of big data sources and the essential role of socio-demographic information in travel behaviour and transport demand modelling more broadly, the enrichment of socio-demographic attributes for anonymous big datasets is a key issue that continues to be explored. The common shortcoming of existing socio-demographic enrichment approaches concerns their lack of consistent theory that can link their enrichment performance (i.e. the ability to correctly enrich the required attribute) to the underlying covariance structure in the anonymous big datasets. In other words, existing approaches are unable to indicate, prior to the enrichment, to what extent it will be successful. Instead, they require undertaking the enrichment itself to assess and validate it post factum, incurring the effort and cost of the activity. An alternative and arguably preferable way would be to have a prior indicator as to whether an enrichment is likely to be sufficiently effective for the desired application.Towards this end, this paper draws upon the Inverse Discrete Choice Modelling (IDCM) approach to demonstrate what is termed as the IDCM performance theory, which systematically and in a tractable manner links the socio-demographic enrichment performance of the IDCM approach to the structure of the underlying datasets. This is achieved by recalibration of the constant, a technique adopted from conventional discrete choice modelling practice, while also drawing upon information theory employed in the context of communication systems. The established IDCM performance theory is validated in two empirical applications where performance of the IDCM approach in enriching several socio-demographic attributes, given travel behaviour patterns, is successfully estimated. Additionally, the IDCM approach is found to perform comparably to commonly used methods in previous socio-demographic enrichment efforts. It is thus argued that the capability of the IDCM performance theory to predict and explain its enrichment performance under different data conditions can facilitate informed and transparent transferability of the IDCM framework in the socio-demographic enrichment for anonymous big datasets.

Multitrait transcriptome-wide association study (TWAS) tests.

Multitrait tests can improve power to detect associations between individual single-nucleotide polymorphisms (SNPs) and several related traits. Here, we develop methods for multi-SNP transcriptome-wide association (TWAS) tests to test the association between predicted gene expression levels and multiple phenotypes. We show that the correlation in TWAS test statistics for multiple phenotypes has the same form as multitrait statistics for the single-SNP setting. Thus, established methods for combining single-SNP test statistics across multiple traits can be extended directly to the TWAS setting. We performed an extensive evaluation across eight multitrait methods in simulations that varied gene-phenotype effect sizes in addition to the underlying covariance structure among the phenotypes. We found that all multitrait TWAS tests have well-calibrated Type I error (except ASSET, which can have a slightly elevated or depressed Type I error rate). Our results show that multitrait TWAS can improve statistical power compared with multiple single-trait TWAS followed by Bonferroni correction. To illustrate our approach to real data, we conducted a multitrait TWAS of four circulating lipid traits from the Global Lipids Genetics Consortium. We found that our multitrait Wald TWAS approach identified 506 genes associated with lipid levelscompared with 87 identified through Bonferroni-corrected single-trait TWAS. Overall, we find that our proposed multitrait TWAS framework outperforms single-trait approaches to identify new genetic associations, especially for functionally correlated phenotypes and phenotypes with overlapping genome-wide association studies samples, leading to insights into the genetic architecture of multiple phenotypes.

Ecomorphological divergence and habitat lability in the context of robust patterns of modularity in the cichlid feeding apparatus

BackgroundAdaptive radiations are characterized by extreme and/or iterative phenotypic divergence; however, such variation does not accumulate evenly across an organism. Instead, it is often partitioned into sub-units, or modules, which can differentially respond to selection. While it is recognized that changing the pattern of modularity or the strength of covariation (integration) can influence the range or rate of morphological evolution, the relationship between shape variation and covariation remains unclear. For example, it is possible that rapid phenotypic change requires concomitant changes to the underlying covariance structure. Alternatively, repeated shifts between phenotypic states may be facilitated by a conserved covariance structure. Distinguishing between these scenarios will contribute to a better understanding of the factors that shape biodiversity. Here, we explore these questions using a diverse Lake Malawi cichlid species complex, Tropheops, that appears to partition habitat by depth.ResultsWe construct a phylogeny of Tropheops populations and use 3D geometric morphometrics to assess the shape of four bones involved in feeding (mandible, pharyngeal jaw, maxilla, pre-maxilla) in populations that inhabit deep versus shallow habitats. We next test numerous modularity hypotheses to understand whether fish at different depths are characterized by conserved or divergent patterns of modularity. We further examine rates of morphological evolution and disparity between habitats and among modules. Finally, we raise a single Tropheops species in environments mimicking deep or shallow habitats to discover whether plasticity can replicate the pattern of morphology, disparity, or modularity observed in natural populations.ConclusionsOur data support the hypothesis that conserved patterns of modularity permit the evolution of divergent morphologies and may facilitate the repeated transitions between habitats. In addition, we find the lab-reared populations replicate many trends in the natural populations, which suggests that plasticity may be an important force in initiating depth transitions, priming the feeding apparatus for evolutionary change.

A Simple Scale-Invariant Two-Sample Test for High-dimensional Data

A new scale-invariant test for two-sample problems for high-dimensional data is proposed and studied. Under some regularity conditions and the null hypothesis, the proposed test statistic and a chi-square-type mixture are shown to have the same limiting distribution after they are normalized. The limiting distribution can be normal or non-normal, depending on the underlying covariance structure of the high-dimensional data. To approximate the null distribution of the proposed test, the well-known Welch-Satterthwaite chi-square approximation is applied. The resulting test is shown to be adaptive to the shape of the underlying null distribution in the sense that when the test statistic is asymptotically normally distributed under the null hypothesis, so is the approximation distribution, and when the approximation distribution is asymptotically non-normally distributed, so is the underlying null distribution of the test statistic. The asymptotic powers of the proposed test under some local alternatives are derived. Simulation studies and a real data application are used to demonstrate the good performance of the proposed test compared with several existing competitors in the literature.

Nonparametric estimation of the spatio-temporal covariance structure.

Spatio-temporal modeling is an active research problem with broad applications. In this problem, proper description and estimation of the data covariance structure plays an important role. In the literature, most available methods assume that the data covariance is stationary and follows a specific parametric form. In practice, however, such assumptions are hardly valid or difficult to verify. In this paper, we propose a new and flexible method for estimating the underlying covariance structure. Our proposed method does not require the covariance to be stationary or follow a parametric form. It can accommodate nonparametric space-time-varying mean structure of the observed data. Under some mild regularity conditions, it is shown that our estimated covariance structure converges to the true covariance structure. The proposed method is also justified numerically by a simulation study and an application to a hand, foot, and mouth disease data.

Modeling Dynamics of Structural Components of Forest Stands Based on Trivariate Stochastic Differential Equation

Research Highlights: Today’s approaches to modeling of forest stands are in most cases based on that the regression models and they are constructed as static sub-models describing individual stands variables. The disadvantages of this method; it is laborious because too many different equations need to be assessed and empirical choices of candidate equations make the results subjective; it does not relate to the stand variables dynamics against the age dimension (time); and does not consider the underlying covariance structure driving changes in the stand variables. In this study, the dynamical model defined by a fixed-and mixed effect parameters trivariate stochastic differential equation (SDE) is introduced and described how such a model can be used to model quadratic mean diameter, mean height, number of trees per hectare, self-thinning line, stand basal area, stand volume per hectare and much more. Background and Objectives: New developed marginal and conditional trivariate probability density functions, combining information generated from an age-dependent variance-covariance matrix of quadratic mean diameter, mean height and number of trees per hectare, improve stand growth prediction, and forecast (in forecast the future is completely unavailable and must only be estimated from historical patterns) accuracies. Materials and Methods: Fixed-and mixed effect parameters SDE models were harmonized to predict and forecast the dynamics of quadratic mean diameter, mean height, number of trees per hectare, basal area, stand volume per hectare, and their current and mean increments. The results and experience from applying the SDE concepts and techniques in an extensive whole stand growth and yield analysis are described using a Scots pine (Pinus sylvestris L.) experimental dataset in Lithuania. Results: The mixed effects scenario SDE model showed high accuracy, the percentage root mean square error values for quadratic mean diameter, mean height, number of trees per hectare, stand basal area and stand volume per hectare predictions (forecasts) were 3.37% (10.44%), 1.82% (2.07%), 1.76% (2.93%), 6.65% (10.41%) and 6.50% (8.93%), respectively. In the same way, the quadratic mean diameter, mean height, number of trees per hectare, stand basal area and stand volume per hectare prediction (forecast) relationships had high values of the coefficient of determination, R2, 0.998 (0.987), 0.997 (0.992), 0.997 (0.988), 0.968 (0.984) and 0.966 (0.980), respectively. Conclusions: The approach presented in this paper can be used for developing a new generation stand growth and yield models.

Methodological approaches to analyzing IVF data with multiple cycles.

Which methodological approaches are most appropriate for analyzing IVF data with multiple cycles in the context of a binary outcome? Both mixed effect models and generalized estimating equation (GEE) modeling approaches can account for multiple IVF cycles and may reduce bias over first-cycle only approaches, but CIs were narrowest with cluster-weighted generalized estimating equation models (CWGEE). There is a lack of consensus among investigators regarding how to best incorporate data from multiple cycles and whether to present odds or risks in the analysis of IVF data. Failure to account for correlated outcomes within individuals and informative cluster size may lead to invalid CIs and biased estimates. The Environment and Reproductive Health (EARTH) Study is an ongoing prospective cohort study of subfertile couples conducted at an academic medical center. This cohort was established in 2004 and follows couples seeking treatment for infertility throughout the course of their treatment and pregnancy. Women aged 18-46 years enrolled in the EARTH Study from 2004 to 2017 who initiated at least one IVF cycle were eligible. Cycle initiation was defined as beginning ovulation induction with the intent to progress through an IVF or ICSI cycle. This analysis included 442 women undergoing 642 cycles who met the study inclusion criteria. We compared the results and interpretations of log-binomial and logistic models restricting to the first cycle, as well as mixed effects models, unweighted GEE models, and CWGEE models including all cycles. This analysis was conducted for two distinct exposures: maternal age at cycle initiation, and maternal preconception urinary concentrations of di(2-ethylhexyl) phthalate (DEHP) metabolites (previously reported to be associated with a decreased probability of live birth). In general, the CIs were widest for mixed effects models and narrowest for CWGEE models. Further, in models evaluating the sum of urinary concentrations of DEHP metabolites (∑DEHP, available for 91% of women), the point estimates were surprisingly different between the first-cycle and multiple-cycle models. We observed significant associations between maternal age and live birth in all models. However, we observed no associations between ∑DEHP and live birth. This analysis was limited to an example dataset in which the true effect of any exposure is unknown. While this allows us to observe model performance in the context of real data, future analyses should be conducted within simulated datasets under various assumptions to further evaluate the appropriateness of each approach. In addition, we did not address differential loss to follow-up in our statistical approaches. The use of CWGEE models should be more widely considered in the analysis of IVF data with multiple cycles per woman. The CWGEE approach is computationally simple, addresses non-ignorable (informative) cluster size, and is robust against mis-specification of the underlying covariance structure. Among the methods compared in this analysis, CWGEE models generally yielded the narrowest CIs, possibly indicating the most precise estimates. We also stress the importance of estimating risks rather than odds in the analysis of IVF data. The project was funded by Grants (R01ES022955, R01ES009718, and P30ES000002) from the National Institutes of Health. None of the authors has any conflicts of interest to declare.

Asymptotics for empirical eigenvalue processes in high-dimensional linear factor models

When vector-valued observations are of high dimension N relative to the sample size T, it is common to employ a linear factor model in order to estimate the underlying covariance structure or to further understand the relationship between coordinates. Asymptotic analyses of such models often consider the case in which both N andT tend jointly to infinity. Within this framework, we derive weak convergence results for processes of partial sample estimates of the largest eigenvalues of the sample covariance matrix. It is shown that if the effect of the factors is sufficiently strong, then the processes associated with the largest eigenvalues have Gaussian limits under general conditions on the divergence rates of N andT, and the underlying observations. If the common factors are “weak”, then N must grow much more slowly in relation to T in order for the largest eigenvalue processes to have a Gaussian limit. We apply these results to develop general tests for structural stability of linear factor models that are based on measuring the fluctuations in the largest eigenvalues throughout the sample, which we investigate further by means of a Monte Carlo simulation study and an application to US treasury yield curve data.

Biomarkers associated with risk and recovery in cancer patients after hematopoietic stem cell transplantation

Behavioral factors affecting risk for morbidity and mortality following hematopoietic stem cell transplantation (HSCT) have been identified; however, the biological mediators have not been elucidated. Adults receiving allogeneic HSCT for a hematologic malignancy (N = 61) completed self-report measures (IDAS, PANAS, SPS, PSQI) of risk (depression, anxiety) and protective (sleep, social support) factors pre- HSCT. Blood samples were collected pre- and 100 days post-HSCT. Biomarkers relevant to immune recovery were determined via electrochemiluminescence assay. Exploratory factor analysis (EFA) identified the underlying covariance structure of the cytokines, chemokines and other analytes. Multiple regression analyses covarying for age, sex, and treatment regimen examined associations between behavioral measures and EFA-derived biomarker factors. With respect to risk, participants reporting more depressive symptoms had higher pre-transplant scores on a factor comprised of inflammatory cytokines and acute phase proteins ( β =.34,.29; p .05; betas in SD units). Regarding protective factors, participants with better sleep quality had lower pre- and post-HSCT scores on a factor comprised of monocytic chemokines ( β =.31,.37; p

Bayes Factor Covariance Testing in Item Response Models.

Two marginal one-parameter item response theory models are introduced, by integrating out the latent variable or random item parameter. It is shown that both marginal response models are multivariate (probit) models with a compound symmetry covariance structure. Several common hypotheses concerning the underlying covariance structure are evaluated using (fractional) Bayes factor tests. The support for a unidimensional factor (i.e., assumption of local independence) and differential item functioning are evaluated by testing the covariance components. The posterior distribution of common covariance components is obtained in closed form by transforming latent responses with an orthogonal (Helmert) matrix. This posterior distribution is defined as a shifted-inverse-gamma, thereby introducing a default prior and a balanced prior distribution. Based on that, an MCMC algorithm is described to estimate all model parameters and to compute (fractional) Bayes factor tests. Simulation studies are used to show that the (fractional) Bayes factor tests have good properties for testing the underlying covariance structure of binary response data. The method is illustrated with two real data studies.

Bayesian Cosmological inference beyond statistical isotropy

With advent of rich data sets, computationally challenge of inference in cosmology has relied on stochastic sampling method. First, I review the widely used MCMC approach used to infer cosmological parameters and present a adaptive improved implementation SCoPE developed by our group. Next, I present a general method for Bayesian inference of the underlying covariance structure of random fields on a sphere. We employ the Bipolar Spherical Harmonic (BipoSH) representation of general covariance structure on the sphere. We illustrate the efficacy of the method with a principled approach to assess violation of statistical isotropy (SI) in the sky maps of Cosmic Microwave Background (CMB) fluctuations. The general, principled, approach to a Bayesian inference of the covariance structure in a random field on a sphere presented here has huge potential for application to other many aspects of cosmology and astronomy, as well as, more distant areas of research like geosciences and climate modelling.

Improved power in crossover designs through linear combinations of baselines

In a crossover design in the absence of any carryover effect, including period-specific baselines as covariates in an analysis of covariance, is known to increase the precision of the estimated treatment effect. The extent of the efficiency gain is a function of the joint covariance structure of the baselines and post-treatment responses, as well as the metric used to incorporate the baselines into the analysis. Here, we show how the underlying covariance structure can be leveraged to find an optimal linear combination of baselines so as to minimize the theoretical variance of the analysis of covariance-based estimated treatment effect. Our work is relevant to complete designs with up to four periods, specifically the 2 × 2, 3 × 3, and 4 × 4. Given that the optimal linear combination of baselines is a function of the covariance structure, which in practice is unknown, we propose an adaptive method. Here, the covariance structure is chosen using information criterion to guide the choice of the linear combination of baselines. Evaluation of the proposed approach suggests that the type I error rate is maintained. Moreover, relative to previously published methods, sizeable gains in power are possible with this method. Results from a 2 × 2 trial exploring renal function, and a 3 × 3 trial with heart rate as the outcome, are used to illustrate the methods. Copyright © 2016 John Wiley & Sons, Ltd.

Regularization for high-dimensional covariance matrix

Abstract In many applications, high-dimensional problem may occur often for various reasons, for example, when the number of variables under consideration is much bigger than the sample size, i.e., p >> n. For highdimensional data, the underlying structures of certain covariance matrix estimates are usually blurred due to substantial random noises, which is an obstacle to draw statistical inferences. In this paper, we propose a method to identify the underlying covariance structure by regularizing a given/estimated covariance matrix so that the noises can be filtered. By choosing an optimal structure from a class of candidate structures for the covariance matrix, the regularization is made in terms of minimizing Frobenius-norm discrepancy. The candidate class considered here includes the structures of order-1 moving average, compound symmetry, order-1 autoregressive and order-1 autoregressive moving average. Very intensive simulation studies are conducted to assess the performance of the proposed regularization method for very high-dimensional covariance problem. The simulation studies also show that the sample covariance matrix, although performs very badly in covariance estimation for high-dimensional data, can be used to correctly identify the underlying structure of the covariance matrix. The approach is also applied to real data analysis, which shows that the proposed regularization method works well in practice.

Bayesian inference on the sphere beyond statistical isotropy

We present a general method for Bayesian inference of the underlying covariance structure of random fields on a sphere. We employ the Bipolar Spherical Harmonic (BipoSH) representation of general covariance structure on the sphere. We illustrate the efficacy of the method as a principled approach to assess violation of statistical isotropy (SI) in the sky maps of Cosmic Microwave Background (CMB) fluctuations. SI violation in observed CMB maps arise due to known physical effects such as Doppler boost and weak lensing; yet unknown theoretical possibilities like cosmic topology and subtle violations of the cosmological principle, as well as, expected observational artefacts of scanning the sky with a non-circular beam, masking, foreground residuals, anisotropic noise, etc. We explicitly demonstrate the recovery of the input SI violation signals with their full statistics in simulated CMB maps. Our formalism easily adapts to exploring parametric physical models with non-SI covariance, as we illustrate for the inference of the parameters of a Doppler boosted sky map. Our approach promises to provide a robust quantitative evaluation of the evidence for SI violation related anomalies in the CMB sky by estimating the BipoSH spectra along with their complete posterior.

Prediction of total fatty acid parameters and individual fatty acids in pork backfat using Raman spectroscopy and chemometrics: Understanding the cage of covariance between highly correlated fat parameters

This study investigates how Partial Least Squares regression models for predicting individual fatty acids (FAs) and total FA parameters depend on Raman spectral variation associated with the iodine value in pork backfat. The backfat was sampled from pigs, which were fed with different dietary fat sources and levels. Good correlations between the Raman spectra and the total FA composition parameters and most individual FAs were obtained (RCV2=0.78–0.90). However, the predictions of the individual FAs are indirect and to a high degree depend on co-variance with the total FA parameters. A new procedure was demonstrated for identifying and characterizing such indirect or non-targeted calibrations. This information is very useful when Raman spectroscopy or other vibrational spectroscopic techniques are used to predict non-targeted quality parameters such as individual FAs as they may lead to inaccurate predictions of future sample if the underlying covariance structure is changed e.g. by new dietary regimes or genotypes.

A Bayesian method for analyzing combinations of continuous, ordinal, and nominal categorical data with missing values

From a Bayesian perspective, we propose a general method for analyzing a combination of continuous, ordinal (including binary), and categorical/nominal multivariate measures with missing values. We assume multivariate normal linear regression models for multivariate continuous measures, multivariate probit models for correlated ordinal measures, and multivariate multinomial probit models for multivariate categorical/nominal measures. Then we assume a multivariate normal linear model on the continuous vector comprised of continuous variables and those underlying normal variables for ordinal variables from multivariate probit models and for categorical variables from multinomial probit models. We develop a Markov chain Monte Carlo (MCMC) algorithm to estimate unknown parameters including regression parameters, cut-points for ordinal data from the multivariate probit models, and the covariance matrix encompassing both continuous variables and the underlying normal latent variables. Combining the continuous variables and the normal latent variables allows us to model combinations of continuous, ordinal, and categorical multivariate data simultaneously. The framework incorporates flexible priors for the covariance matrix, provides a foundation for inference about the underlying covariance structure, and imputes missing data where needed. The method is illustrated through simulated examples and two real data applications.

Structured functional principal component analysis.

Motivated by modern observational studies, we introduce a class of functional models that expand nested and crossed designs. These models account for the natural inheritance of the correlation structures from sampling designs in studies where the fundamental unit is a function or image. Inference is based on functional quadratics and their relationship with the underlying covariance structure of the latent processes. A computationally fast and scalable estimation procedure is developed for high-dimensional data. Methods are used in applications including high-frequency accelerometer data for daily activity, pitch linguistic data for phonetic analysis, and EEG data for studying electrical brain activity during sleep.