Variance-covariance Structure Research Articles

Special Issue on INFORMS 2015 Annual Meeting

Special Issue on INFORMS 2015 Annual Meeting

Multilevel models for cost-effectiveness analyses that use cluster randomised trial data: An approach to model choice

Multilevel models provide a flexible modelling framework for cost-effectiveness analyses that use cluster randomised trial data. However, there is a lack of guidance on how to choose the most appropriate multilevel models. This paper illustrates an approach for deciding what level of model complexity is warranted; in particular how best to accommodate complex variance-covariance structures, right-skewed costs and missing data. Our proposed models differ according to whether or not they allow individual-level variances and correlations to differ across treatment arms or clusters and by the assumed cost distribution (Normal, Gamma, Inverse Gaussian). The models are fitted by Markov chain Monte Carlo methods. Our approach to model choice is based on four main criteria: the characteristics of the data, model pre-specification informed by the previous literature, diagnostic plots and assessment of model appropriateness. This is illustrated by re-analysing a previous cost-effectiveness analysis that uses data from a cluster randomised trial. We find that the most useful criterion for model choice was the deviance information criterion, which distinguishes amongst models with alternative variance-covariance structures, as well as between those with different cost distributions. This strategy for model choice can help cost-effectiveness analyses provide reliable inferences for policy-making when using cluster trials, including those with missing data.

A Modified Jonckheere Test Statistic for Ordered Alternatives in Repeated Measures Design

In this article, a new test based on Jonckheere test [1] for randomized blocks which have dependent observations within block is presented. A weighted sum for each block statistic rather than the unweighted sum proposed by Jonckheereis included. For Jonckheere type statistics, the main assumption is independency of observations within block. In the case of repeated measures design, the assumption of independence is violated. The weighted Jonckheere type statistic for the situation of dependence for different variance-covariance structure and the situation based on ordered alternative hypothesis structure of each block on the design is used. Also, the proposed statistic is compared to the existing test based on Jonckheere in terms of type I error rates by performing Monte Carlo simulation. For the strong correlations, circular bootstrap version of the proposed Jonckheere test provides lower rates of type I error.

A Genomic Bayesian Multi-trait and Multi-environment Model.

When information on multiple genotypes evaluated in multiple environments is recorded, a multi-environment single trait model for assessing genotype × environment interaction (G × E) is usually employed. Comprehensive models that simultaneously take into account the correlated traits and trait × genotype × environment interaction (T × G × E) are lacking. In this research, we propose a Bayesian model for analyzing multiple traits and multiple environments for whole-genome prediction (WGP) model. For this model, we used Half- priors on each standard deviation term and uniform priors on each correlation of the covariance matrix. These priors were not informative and led to posterior inferences that were insensitive to the choice of hyper-parameters. We also developed a computationally efficient Markov Chain Monte Carlo (MCMC) under the above priors, which allowed us to obtain all required full conditional distributions of the parameters leading to an exact Gibbs sampling for the posterior distribution. We used two real data sets to implement and evaluate the proposed Bayesian method and found that when the correlation between traits was high (>0.5), the proposed model (with unstructured variance–covariance) improved prediction accuracy compared to the model with diagonal and standard variance–covariance structures. The R-software package Bayesian Multi-Trait and Multi-Environment (BMTME) offers optimized C++ routines to efficiently perform the analyses.

Statistical inference for stochastic processes: Two-sample hypothesis tests

In this paper, we present inferential procedures to compare the means of two samples of functional data. The proposed tests are based on a suitable generalization of Mahalanobis distance to the Hilbert space of square integrable functions defined on a compact interval. The only conditions required concern the moments and the independence of the functional data, while the distribution of the processes generating the data is not needed to be specified. Test procedures are proposed for both the cases of known and unknown variance–covariance structures, and asymptotic properties of test statistics are deeply studied. A simulation study and a real case data analysis are also presented.

Fitting Unstructured Finite Mixture Models in Longitudinal Design: A Recommendation for Model Selection and Estimation of the Number of Classes

In longitudinal design, investigating interindividual differences of intraindividual changes enables researchers to better understand the potential variety of development and growth. Although latent growth curve mixture models have been widely used, unstructured finite mixture models (uFMMs) are also useful as a preliminary tool and are expected to be more robust in identifying classes under the influence of possible model misspecifications, which are very common in actual practice. In this study, large-scale simulations were performed in which various normal uFMMs and nonnormal uFMMs were fit to evaluate their utility and the performance of each model selection procedure for estimating the number of classes in longitudinal designs. Results show that normal uFMMs assuming invariance of variance–covariance structures among classes perform better on average. Among model selection procedures, the Calinski–Harabasz statistic, which has a nonparametric nature, performed better on average than information criteria, including the Bayesian information criterion.

Disjoint factor analysis with cross-loadings

Disjoint factor analysis (DFA) is a new latent factor model that we propose here to identify factors that relate to disjoint subsets of variables, thus simplifying the loading matrix structure. Similarly to exploratory factor analysis (EFA), the DFA does not hypothesize prior information on the number of factors and on the relevant relations between variables and factors. In DFA the population variance–covariance structure is hypothesized block diagonal after the proper permutation of variables and estimated by Maximum Likelihood, using an Coordinate Descent type algorithm. Inference on parameters on the number of factors and to confirm the hypothesized simple structure are provided. Properties such as scale equivariance, uniqueness, optimal simplification of loadings are satisfied by DFA. Relevant cross-loadings are also estimated in case they are detected from the best DFA solution. DFA has also the option to constrain a variable to load on a pre-specified factor so that the researcher can assume, a priori, some relations between variables and loadings. A simulation study shows performances of DFA and an application to optimally identify the dimensions of well-being is used to illustrate characteristics of the new methodology. A final discussion concludes the paper.

A test of separate hypotheses for comparing linear mixed models with non nested fixed effects

ABSTRACTAs researchers increasingly rely on linear mixed models to characterize longitudinal data, there is a need for improved techniques for selecting among this class of models which requires specification of both fixed and random effects via a mean model and variance-covariance structure. The process is further complicated when fixed and/or random effects are non nested between models. This paper explores the development of a hypothesis test to compare non nested linear mixed models based on extensions of the work begun by Sir David Cox. We assess the robustness of this approach for comparing models containing correlated measures of body fat for predicting longitudinal cardiometabolic risk.

Multivariate Bayesian variable selection exploiting dependence structure among outcomes: Application to air pollution effects on DNA methylation.

The analysis of multiple outcomes is becoming increasingly common in modern biomedical studies. It is well-known that joint statistical models for multiple outcomes are more flexible and more powerful than fitting a separate model for each outcome; they yield more powerful tests of exposure or treatment effects by taking into account the dependence among outcomes and pooling evidence across outcomes. It is, however, unlikely that all outcomes are related to the same subset of covariates. Therefore, there is interest in identifying exposures or treatments associated with particular outcomes, which we term outcome-specific variable selection. In this work, we propose a variable selection approach for multivariate normal responses that incorporates not only information on the mean model, but also information on the variance-covariance structure of the outcomes. The approach effectively leverages evidence from all correlated outcomes to estimate the effect of a particular covariate on a given outcome. To implement this strategy, we develop a Bayesian method that builds a multivariate prior for the variable selection indicators based on the variance-covariance of the outcomes. We show via simulation that the proposed variable selection strategy can boost power to detect subtle effects without increasing the probability of false discoveries. We apply the approach to the Normative Aging Study (NAS) epigenetic data and identify a subset of five genes in the asthma pathway for which gene-specific DNA methylations are associated with exposures to either black carbon, a marker of traffic pollution, or sulfate, a marker of particles generated by power plants.

Testing the equivalence of modern human cranial covariance structure: Implications for bioarchaeological applications.

Estimation of the variance-covariance (V/CV) structure of fragmentary bioarchaeological populations requires the use of proxy extant V/CV parameters. However, it is currently unclear whether extant human populations exhibit equivalent V/CV structures. Random skewers (RS) and hierarchical analyses of common principal components (CPC) were applied to a modern human cranial dataset. Cranial V/CV similarity was assessed globally for samples of individual populations (jackknifed method) and for pairwise population sample contrasts. The results were examined in light of potential explanatory factors for covariance difference, such as geographic region, among-group distance, and sample size. RS analyses showed that population samples exhibited highly correlated multivariate responses to selection, and that differences in RS results were primarily a consequence of differences in sample size. The CPC method yielded mixed results, depending upon the statistical criterion used to evaluate the hierarchy. The hypothesis-testing (step-up) approach was deemed problematic due to sensitivity to low statistical power and elevated Type I errors. In contrast, the model-fitting (lowest AIC) approach suggested that V/CV matrices were proportional and/or shared a large number of CPCs. Pairwise population sample CPC results were correlated with cranial distance, suggesting that population history explains some of the variability in V/CV structure among groups. The results indicate that patterns of covariance in human craniometric samples are broadly similar but not identical. These findings have important implications for choosing extant covariance matrices to use as proxy V/CV parameters in evolutionary analyses of past populations.

A Multivariate Loss Function Approach to Robust Design of Systems with Multiple Performance Characteristics

The Taguchi robust design method traditionally deals with single‐characteristic problems. Various methods have been developed for extending the Taguchi single‐characteristic robust design method to the case of multi‐characteristic robust design problems. However, most of those methods have shortcomings in that they do not properly consider the variance–covariance structures among performance characteristics and/or do not preserve the original properties of the Taguchi signal‐to‐noise ratio for single‐characteristic robust design problems. To overcome these shortcomings, this paper develops a multivariate loss function approach to multi‐characteristic robust design problems with an appropriately defined signal‐to‐noise ratio. Its performance is evaluated using simulated examples, and the results indicate that it generally outperforms existing representative methods for correlated as well as uncorrelated experimental data. Copyright © 2016 John Wiley & Sons, Ltd.

A multi-environment trials diallel analysis provides insights on the inheritance of fumonisin contamination resistance in tropical maize

In maize, the fungi that cause Fusarium ear rot result not only in decreased grain yield and quality, but also grain contamination by fumonisin. This study investigated the inheritance of fumonisin contamination resistance (FCR) in tropical maize, based on a multi-environment trials diallel analysis via mixed models. For this purpose, based on 13 inbred lines, single-cross hybrids were created and assessed in three environments. A mixed model diallel joint analysis across environments was performed, considering the existence of environment-specific variances and correlations between pairs of environments for general combining ability (GCA) and specific combining ability (SCA) effects, and additive genomic relationship between inbred lines for the prediction of GCA and SCA. For all environments, the SCA variance had a higher magnitude than the GCA variance, indicating a predominance of the dominance effects underlying FCR in tropical maize. Moreover, the proportion of the variance among single-cross hybrids that was due to GCA varied from 16 to 22 % across environments, suggesting that SCA is important to predict the hybrids performance. Through modeling variance–covariance structures for GCA and SCA, it was possible to observe that the GCA effects were stable, whereas the SCA effects were specific for each environment. Therefore, these results suggest that the selection of the best parents for the development of new inbred lines can be carried out through the average performance across the evaluated environments. Due to the importance of SCA effects and their complex interaction with environments, the selection of superior hybrids should be performed into specific environments.

Genome-Assisted Prediction of Quantitative Traits Using the R Package sommer.

Most traits of agronomic importance are quantitative in nature, and genetic markers have been used for decades to dissect such traits. Recently, genomic selection has earned attention as next generation sequencing technologies became feasible for major and minor crops. Mixed models have become a key tool for fitting genomic selection models, but most current genomic selection software can only include a single variance component other than the error, making hybrid prediction using additive, dominance and epistatic effects unfeasible for species displaying heterotic effects. Moreover, Likelihood-based software for fitting mixed models with multiple random effects that allows the user to specify the variance-covariance structure of random effects has not been fully exploited. A new open-source R package called sommer is presented to facilitate the use of mixed models for genomic selection and hybrid prediction purposes using more than one variance component and allowing specification of covariance structures. The use of sommer for genomic prediction is demonstrated through several examples using maize and wheat genotypic and phenotypic data. At its core, the program contains three algorithms for estimating variance components: Average information (AI), Expectation-Maximization (EM) and Efficient Mixed Model Association (EMMA). Kernels for calculating the additive, dominance and epistatic relationship matrices are included, along with other useful functions for genomic analysis. Results from sommer were comparable to other software, but the analysis was faster than Bayesian counterparts in the magnitude of hours to days. In addition, ability to deal with missing data, combined with greater flexibility and speed than other REML-based software was achieved by putting together some of the most efficient algorithms to fit models in a gentle environment such as R.

Yield response of winter wheat cultivars to environments modeled by different variance-covariance structures in linear mixed models

The main objectives of multi-environmental trials (METs) are to assess cultivar adaptation patterns under different environmental conditions and to investigate genotype by environment (G×E) interactions. Linear mixed models (LMMs) with more complex variance-covariance structures have become recognized and widely used for analyzing METs data. Best practice in METs analysis is to carry out a comparison of competing models with different variance-covariance structures. Improperly chosen variance-covariance structures may lead to biased estimation of means resulting in incorrect conclusions. In this work we focused on adaptive response of cultivars on the environments modeled by the LMMs with different variance-covariance structures. We identified possible limitations of inference when using an inadequate variance-covariance structure. In the presented study we used the dataset on grain yield for 63 winter wheat cultivars, evaluated across 18 locations, during three growing seasons (2008/2009-2010/2011) from the Polish Post-registration Variety Testing System. For the evaluation of variance-covariance structures and the description of cultivars adaptation to environments, we calculated adjusted means for the combination of cultivar and location in models with different variance-covariance structures. We concluded that in order to fully describe cultivars adaptive patterns modelers should use the unrestricted variance-covariance structure. The restricted compound symmetry structure may interfere with proper interpretation of cultivars adaptive patterns. We found, that the factor-analytic structure is also a good tool to describe cultivars reaction on environments, and it can be successfully used in METs data after determining the optimal component number for each dataset.

Estimation of genotypic correlation and heritability of apricot traits, using restricted maximum likelihood in repeated measures data

In estimation of genetic parameters in perennial tree species on the basis of analysis of variance (ANOVA), heterogeneity of years and genotype × environment interaction for data sets during the juvenility to maturity life period is ignored. Therefore, a linear mixed model based on restricted maximum likelihood (REML) approximation for modeling of covariance structure of longitudinal data can improve our ability to analyze repeated measures data. In the present research, a modeling of variance-covariance structure by mixed model based on the REML approach has been used for characteristics of 26 apricot genotypes recorded during three years. Fitting unstructured covariance (UN) models for all traits indicated a great heterogeneity of variances among repeated years and the trends of response variables in the genotypes (except for RWC) was due to imperfect correlation of subjects measured in different years. Based on the same structure, positive correlations were estimated among fruit set, potassium content, and yield of pistil in repetitive years, and most traits showed high heritability estimation. To our knowledge, this is the first report in plant that genotypic correlation and heritability and their standard errors are estimated in a repeated measures data over years using REML approximation.

419. Efficacy of Rambo for Sarcoma Therapy

Soft Tissue Sarcoma (STS), a cancer which forms in the connective tissue, requires aggressive treatment. Metastatic soft tissue sarcomas are associated with a median prognosis of about twelve months accompanied frequently with many signs and symptoms leading to a poor quality of life. These tumors frequently have angiogenic properties. Angiogenesis is responsible for the formation of new blood vessels and is associated with local progression and increased metastatic disease. Strategies that work to block angiogenesis are currently being investigated as a treatment for STS. Oncolytic virus therapy is a novel treatment strategy that utilizes modified viruses to specifically replicate and lyse cancer cells. The combination of angiogenesis inhibitors with oncolytic virus treatment has not been tested for STS. Here we tested the effect of combining oncolytic viral therapy with antiangiogenic gene therapy. RAMBO is a HSV-1 derived oncolytic virus that codes for the production of Vstat120, a gene with potent antiangiogenic effects. We have tested a panel of sarcoma cells and shown effective infection and lysis of these cells by RAMBO and HSVQ-1 in vitro. In vivo, we compared the antitumor efficacy of RAMBO and HSVQ-1 against sarcoma tumors grown in mice. Briefly, mice were implanted with 150-300mm3 A673 sarcoma tumors were injected with a PBS, HSVQ-1, or RAMBO (5.5×106 pfu). Mice were followed for tumor size over a period of time. Tumor volumes over time were logarithm (base 2) transformed for variance stabilization. The linear mixed model was used to compare the three groups while accounting for the variance-covariance structure due to repeated measures at different days from the same mouse. Statistically significant improvement in antitumor efficacy was observed in RAMBO treated tumors, compared to HSVQ-1 and control. H&E staining of tumor tissue reveals large areas of necrosis in RAMBO treated tumors. RAMBO and its inherent antiangiogenic gene therapy demonstrated significant efficacy in these sarcoma models and further investigations are underway with combination therapeutic strategies.

Long-Term Experiments with cropping systems: Case studies on data analysis

Data analysis for Long-Term Experiments (LTEs) with cropping systems requires some careful thinking, especially for the most complex designs, characterised by rotations with different durations and/or a different number of test-crops per rotation cycle. This paper takes an example-based approach, built upon a number of datasets, covering the main types of LTEs, with increasing levels of complexity. A procedure is outlined to build statistical models for data analysis that is useful for all LTEs characterised by the simultaneous presence of all rotation phases in all years, together with within-year replication. This procedure is based on the assumption that correct analyses can be performed separately for each year. The use of mixed models and REML estimation is advocated for model fitting with all LTEs, due to the fact that most designs are non-orthogonal, as plots may not produce data for the test-crop under study in all years. Mixed models are also useful to account for the autocorrelation of residuals over time and hints are given for the selection of an appropriate variance-covariance structure. For all our examples, variances were not constant across years and compound symmetry correlation structures with variance heterogeneity of years proved to be the best compromise between parsimony and statistical accuracy. Methods are outlined to test for the need of other more complex correlation structures and examples are also given on how to test for fixed effects, model fertility trends and assess the long-term stability of cropping systems.

The Detection of Metabolite-Mediated Gene Module Co-Expression Using Multivariate Linear Models.

Investigating whether metabolites regulate the co-expression of a predefined gene module is one of the relevant questions posed in the integrative analysis of metabolomic and transcriptomic data. This article concerns the integrative analysis of the two high-dimensional datasets by means of multivariate models and statistical tests for the dependence between metabolites and the co-expression of a gene module. The general linear model (GLM) for correlated data that we propose models the dependence between adjusted gene expression values through a block-diagonal variance-covariance structure formed by metabolic-subset specific general variance-covariance blocks. Performance of statistical tests for the inference of conditional co-expression are evaluated through a simulation study. The proposed methodology is applied to the gene expression data of the previously characterized lipid-leukocyte module. Our results show that the GLM approach improves on a previous approach by being less prone to the detection of spurious conditional co-expression.

Generating Correlated and/or Overdispersed Count Data: ASASImplementation

Analysis of longitudinal count data has, for long, been done using a generalized linear mixed model (GLMM), in its Poisson-normal version, to account for correlation by specifying normal random effects. Univariate counts are often handled with the negativebinomial (NEGBIN) model taking into account overdispersion by use of gamma random effects. Inherently though, longitudinal count data commonly exhibit both features of correlation and overdispersion simultaneously, necessitating analysis methodology that can account for both. The introduction of the combined model (CM) by Molenberghs, Verbeke, and Demetrio (2007) and Molenberghs, Verbeke, Demetrio, and Vieira (2010) serves this purpose, not only for count data but for the general exponential family of distributions. Here, a Poisson model is specified as the parent distribution of the data with a normally distributed random effect at the subject or cluster level and/or a gamma distribution at observation level. The GLMM and NEGBIN model are special cases. Data can be simulated from (1) the general CM, with random effects, or, (2) its marginal version directly. This paper discusses an implementation of (1) in SAS software (SAS Inc. 2011). One needs to reflect on the mean of both the combined (hierarchical) and marginal models in order to generate correlated and/or overdispersed counts. A pre-specification of the desired marginal mean (in terms of covariates and marginal parameters), a marginal variance-covariance structure and the hierarchical mean (in terms of covariates and regression parameters) is required. The implied hierarchical parameters, the variance-covariance matrix of the random effects, and the variance-covariance matrix of the overdispersion part are then derived from which correlated Poisson data are generated. Sample calls of the SAS macro are presented as well as output.

Nonresolvable Row–Column Designs with an Even Distribution of Treatment Replications

When generating experimental designs for field trials laid out on a rectangular grid of plots, it is useful to allow for blocking in both rows and columns. When the design is nonresolvable, randomized classical row–column designs may occasionally involve clustered placement of several replications of a treatment. In our experience, this feature prevents the more frequent use of these useful designs in practice. Practitioners often prefer a more even distribution of treatment replications. In this paper we illustrate how spatial variance–covariance structures can be used to achieve a more even distribution of treatment replications across the field and how such designs compare with classical row–column designs in terms of efficiency factors. We consider both equally and unequally replicated designs, including partially replicated designs. Supplementary materials accompanying this paper appear online.