Advancing prediction in linear mixed models: a case study ongreenhouse gas emissions

In this paper, we consider mixed linear models, possibly with singular covariance matrices, by supplementing a particular fixed effects model with appropriate stochastic restrictions. We show that all representations of the best linear unbiased estimator (BLUE) and best linear unbiased predictor (BLUP) can be obtained through the augmented model including stochastic restrictions. Using this approach, we consider two mixed linear models, \({\fancyscript {M}_1}\) and \({\fancyscript {M}_2}\) , say, which have different covariance matrices. We give necessary and sufficient conditions that the BLUP and/or BLUE under the the model \({\fancyscript {M}_{1}}\) continue to be BLUP and/or BLUE also under the model \({\fancyscript {M}_{2}}\) .

Ordinary least squares estimator (OLSE), best linear unbiased estimator (BLUE), and best linear unbiased predictor (BLUP) in the general linear model with new observations are generalized to the general multivariate linear model. The fundamental equations of BLUE and BLUP in the multivariate linear model are derived by two methods, including the vectorization method and projection method. By using the matrix rank method, some new results of linear BLUE-sufficiency, linear BLUP-sufficiency, and the equality of OLSE, BLUE, and BLUP are given in the multivariate linear model.

The number of parameters mushrooms in a linear mixed effects (LME) model in the case of multivariate repeated measures data. Computation of these parameters is a real problem with the increase in the number of response variables or with the increase in the number of time points. The problem becomes more intricate and involved with the addition of additional random effects. A multivariate analysis is not possible in a small sample setting. We propose a method to estimate these many parameters in bits and pieces from baby models, by taking a subset of response variables at a time, and finally using these bits and pieces at the end to get the parameter estimates for the mother model, with all variables taken together. Applying this method one can calculate the fixed effects, the best linear unbiased predictions (BLUPs) for the random effects in the model, and also the BLUPs at each time of observation for each response variable, to monitor the effectiveness of the treatment for each subject. The proposed method is illustrated with an example of multiple response variables measured over multiple time points arising from a clinical trial in osteoporosis.

ABSTRACTIn this paper, we proposed a dynamic stress–strength model for coherent system. It is supposed that the system consists of n components with initial random strength and each component is subjected to random stresses. The stresses, applied repeatedly at random cycle times, will cause the degradation of strength. In addition, the number of cycles in an interval is assumed to follow a Poisson distribution. In the case of the strength and stress random variables following exponential distributions, the expression for the reliability of the proposed dynamic stress–strength model is derived based on survival signature. The reliability is estimated by using the best linear unbiased estimation (BLUE). Considering the Type-II censored failure times, the best linear unbiased predictors (BLUP) for the unobserved coherent system failure times are developed based on the observed failure times. Monte Carlo simulations are performed to compare the BLUE of parameters with different values and compute the BLUP. A real data set is also analysed for an illustration of the findings.

In field experiments with large numbers of treatments, inference can be affected by 1) local variation, and 2) method of analysis .

Pharmacokinetic-Pharmacodynamic Modeling and Simulation

Inference regarding the inclusion or exclusion of random effects in linear mixed models is challenging because the variance components are located on the boundary of their parameter space under the usual null hypothesis. As a result, the asymptotic null distribution of the Wald, score, and likelihood ratio tests will not have the typical χ(2) distribution. Although it has been proved that the correct asymptotic distribution is a mixture of χ(2) distributions, the appropriate mixture distribution is rather cumbersome and nonintuitive when the null and alternative hypotheses differ by more than one random effect. As alternatives, we present two permutation tests, one that is based on the best linear unbiased predictors and one that is based on the restricted likelihood ratio test statistic. Both methods involve weighted residuals, with the weights determined by the among- and within-subject variance components. The null permutation distributions of our statistics are computed by permuting the residuals both within and among subjects and are valid both asymptotically and in small samples. We examine the size and power of our tests via simulation under a variety of settings and apply our test to a published data set of chronic myelogenous leukemia patients.

This study investigates the effectiveness of high-throughput phenotyping (HTP) using RGB images from unmanned aerial vehicles (UAVs) to assess vegetation indices (VIs) in different soybean pure lines. The VIs were accessed at various stages of crop development and correlated with agronomic performance traits. The field research was conducted in the experimental area of the Mato Grosso do Sul Foundation, Brazil, with 60 soybean pure lines. RGB images were captured at multiple stages of development (28, 37, 49, 70, 86, 105, 115, and 120 days after sowing). We used a linear mixed effects model, with restricted maximum likelihood (REML)/best linear unbiased prediction (BLUP) methods, to estimate variance components and genetic correlations, and to predict genotypic values. Significant genetic differences were identified among genotypes for all agronomic traits evaluated (p< 0.001), with high accuracy and heritability for plant height, maturity at R8, and 100-seed weight. There was a significant genotype flight data interaction impact on VI expression, emphasizing the importance of timing data collection to enhance HTP with VIs in agronomic performance evaluation. In the early stages, the indices varied depending on the environment. On the other hand, the indices showed higher correlations with the traits of plant height and maturity at the R8 stage, at 105, 115, and 120 days after sowing. HTP with VIs based on RGB images from UAVs has proven to be more effective in the early and final stages of soybean development, providing essential information for the selection of superior genotypes. This study highlights the importance of the temporal approach in HTP, optimizing the selection of soybean genotypes and refining agricultural management strategies.

ABSTRACTThe Dirichlet process has been used extensively in Bayesian non parametric modeling, and has proven to be very useful. In particular, mixed models with Dirichlet process random effects have been used in modeling many types of data and can often outperform their normal random effect counterparts. Here we examine the linear mixed model with Dirichlet process random effects from a classical view, and derive the best linear unbiased estimator (BLUE) of the fixed effects. We are also able to calculate the resulting covariance matrix and find that the covariance is directly related to the precision parameter of the Dirichlet process, giving a new interpretation of this parameter. We also characterize the relationship between the BLUE and the ordinary least-squares (OLS) estimator and show how confidence intervals can be approximated.

The screening of poplar varieties that demonstrate tolerance to low nitrogen (N) represents a promising strategy for improving nitrogen-use efficiency in trees. Such an approach could reduce reliance on N fertilizers while mitigating environmental pollution associated with their cultivation. In this study, a total of 87 poplar varieties were evaluated in a controlled greenhouse pot experiment. Under both low-nitrogen (LN) and normal-nitrogen (NN) conditions, 18 traits spanning four categories—growth performance, leaf morphology, chlorophyll fluorescence, and N isotope parameters were measured. For 13 of these traits (growth, leaf morphology, chlorophyll fluorescence), genetic variation and parameters, including genotypic values, were analyzed using best linear unbiased prediction (BLUP) within a linear mixed model (LMM). LN tolerance of tested poplar varieties was comprehensively assessed with three MGIDI strategies by integrating means, BLUPs, and low-nitrogen tolerance coefficient (LNindex) to rank poplar varieties. The results exhibited highly significant differences across all traits between LN and NN experiments, as well as among varieties. LN stress markedly inhibited growth, altered leaf morphology, and reduced chlorophyll fluorescence parameters in young poplar plants. Among the selection strategies, the MGIDI_LNindex approach demonstrated the highest selection differential percent (SD% = 10.5–35.23%). Using a selection intensity (SI) of 20%, we systematically identified 17 superior genotypes across all three strategies. In a thorough, comprehensive MGIDI-based evaluation, these varieties exhibited exceptional adaptability and stability under LN stress. The selected genotypes represent valuable genetic resources for developing improved poplar cultivars with enhanced low-nitrogen tolerance.

Assume that a pair of general Linear Random-effects Models (LRMs) are given with a correlated covariance matrix for their error terms. This paper presents an algebraic approach to the statistical analysis and inference of the two correlated LRMs using some state-of-the-art formulas in linear algebra and matrix theory. It is shown first that the best linear unbiased predictors (BLUPs) of all unknown parameters under LRMs can be determined by certain linear matrix equations, and thus the BLUPs under the two LRMs can be obtained in exact algebraic expressions. We also discuss algebraical and statistical properties of the BLUPs, as well as some additive decompositions of the BLUPs. In particular, we present necessary and sufficient conditions for the separated and simultaneous BLUPs to be equivalent. The whole work provides direct access to a very simple algebraic treatment of predictors/estimators under two LRMs with correlated covariance matrices.

The problem considered is that of predicting the value of an unobservable random variable w from the value of an observable random vector y. This problem is considered under each of four states of knowledge about the joint distribution of w and y, ranging from complete knowledge to “no” knowledge. Point predictors, estimators of the mean squared error of prediction, and interval predictors are presented for each case. Both frequentist and Bayesian approaches are discussed and relationships between the two are pointed out. Specifics are given for the prediction of a linear combination of the fixed and random effects in a mixed linear model. The results are illustrated by applying them to some animal breeding data.KeywordsRoot Mean Square ErrorMean Square ErrorMixed Linear ModelJoint DistributionConditional DistributionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Previous chapters discussed linear mixed effects models and autoregressive linear mixed effects models for analysis of longitudinal data. This chapter discusses multivariate extensions of these models. In longitudinal clinical studies, multivariate responses are often collected at each measurement time point from each subject. When two response variables, such as an efficacy measurement and a safety measurement are obviously correlated, there are advantages in analyzing the bivariate responses jointly. Parathyroid hormone (PTH) and serum calcium (Ca) measurements in the treatment of secondary hyperparathyroidism in chronic hemodialysis patients provide an example in which joint bivariate responses are of interest. We introduce multivariate longitudinal data and explain bivariate autoregressive linear mixed effects models in which the current responses are regressed on the previous responses of both variables, fixed effects, and random effects. The dependent bivariate responses approach equilibria, and the equilibria are modeled using fixed and random effects. These type of profiles are observed in long-term clinical studies. We also explain bivariate linear mixed effects models.

A central point of disagreement, in certain long-standing discussions about a particular whaling dataset in the Scientific Committee of the International Whaling Commission, has directly involved model selection issues for linear mixed effect models. The biological question under discussion is associated with a clearly defined parameter of primary interest, a focus parameter, which makes model selection with the Focused Information Criterion (FIC) more appropriate than other selection methods. Since the existing FIC methodology has not covered the case of linear mixed effects models, this article sets up the required framework and develops the necessary formulae for the relevant FIC. Our new criterion requires the asymptotic distribution of estimators derived for a given candidate linear mixed model but with behaviour examined under a wider linear mixed model. These results, needed here to build our FIC, also have independent interest.

BackgroundVariance component (VC) models are commonly used for Quantitative Trait Loci (QTL) mapping in outbred populations. Here, the QTL effect is given as a random effect and a critical part of the model is the relationship between the phenotypic values and the random effect. In the traditional VC model, each individual has a unique QTL effect and the relationship between these random effects is given as a covariance structure (known as the identity-by-descent (IBD) matrix).ResultsWe present an alternative notation of the variance component model, where the elements of the random effect are independent base generation allele effects and sampling term effects. The relationship between the phenotypic vales and the random effect is given by an incidence matrix, which results in a novel, but statistically equivalent, version of the traditional VC model. A general algorithm to estimate this incidence matrix is presented. Since the model is given in terms of base generation allele effects and sampling term effects, these effects can be estimated separately using best linear unbiased prediction (BLUP). From simulated data, we showed that biallelic QTL effects could be accurately clustered using the BLUP obtained from our model notation when markers are fully informative, and that the accuracy increased with the size of the QTL effect. We also developed a measure indicating whether a base generation marker homozygote is a QTL heterozygote or not, by comparing the variances of the sampling term BLUP and the base generation allele BLUP. A ratio greater than one gives strong support for a QTL heterozygote.ConclusionWe developed a simple presentation of the VC QTL model for identification of base generation allele effects in QTL linkage analysis. The base generation allele effects and sampling term effects were separated in our model notation. This clarifies the assumptions of the model and should also enhance the development of genome scan methods.