Minimum Norm Quadratic Unbiased Estimation Research Articles

Additive‐genetic relationships of growth and morphometrics to meat yield traits in Macrobrachium nipponense (De Haan, 1849)

AbstractBased upon mixed linear model and incomplete diallel cross between different strains of Macrobrachium nipponense, the additive‐genetic correlations of growth and morphometrics with meat yield traits in M. nipponense were estimated using the minimum norm quadratic unbiased estimation procedure. Path analysis method was utilized to investigate the overall additive‐genetic determinations of growth and morphometric traits on the meat yield traits. Results indicated that growth and morphometrics were genetically positively correlated with abdominal meat weight (p < 0.01), but were not genetically correlated with abdominal meat percentage (p > 0.05) in both sexes. Growth and morphometrics were genetically positively correlated (p < 0.01). Abdominal meat weight was genetically positively correlated with abdominal meat percentage (p < 0.01). In terms of the overall additive‐genetic determination, body weight had the greatest impact on abdominal meat weight in both sexes. In contrast, abdominal meat weight, abdominal length, and body weight were most influential on abdominal meat percentage in both sexes. The genetic relationships of growth and morphometrics to meat yield traits were elucidated in M. nipponense for the first time, contributing to formulation of sound improvement plan for the meat yield traits in this commercially important prawn species.

Bayes estimates of variance components in mixed linear model

This paper proves that in mixed linear model, the analysis of variance estimation (ANOVAE), the minimum norm quadratic unbiased estimation (MINQUE), the spectral decomposition estimation (SDE) and the restricted maximum likelihood estimation (RMLE) of variance components are the same under some conditions. Based on this result, we construct a linear Bayes estimation (LBE) for the parameter vector consisting of variance components and establish its superiorities. Numerical computations and an illustration show that the LBE is comparable to Lindley's approximation, Tierney and Kadane's approximation and the usual Bayes estimation (UBE) obtained by the MCMC method and easy to use as well.

Calibrating error variance and scaling global covariance function of geoid gradients for optimal determinations of gravity anomaly and gravity gradient from altimetry

With the advancements of satellite altimeter technology and data volume, the accuracy and spatial resolution of altimeter-derived marine gravity fields have been steadily improved. Ideally, a best marine gravity field from multi-altimeter missions is one that combines all data by optimally calibrating their relative weights. To this end, we use the minimum norm quadratic unbiased estimator theory to calibrate error variances of geoid gradients (GGs) from the Cryosat-2 mission and the Jason-1/2 geodetic missions and then use a scaling factor to modify the global covariance functions into local covariance functions. The calibrated error variances and the scaling factors are used in the least-squares collocation to estimate the gridded north and east components of GGs, which are used to compute gravity anomaly and vertical gravity gradient (VGG) by the methods of inverse Vening-Meinesz (IVM) and numerical differentiations. The assessment of the altimeter-derived gravity anomalies in the northern part of the South China Sea using the shipborne gravity data shows an average gravity accuracy improvement of 9.5% by calibrated and scaled covariances of GGs compared to the initial variances. The method for VGG computations is confirmed by examining the extinct ridges in the Gulf of Mexico.

A New Algorithm for Fault Tolerance in Redundant Sensor Systems Based on Real-Time Variance Estimation

Low cost and small size integrated sensors bring a significant interest to redundant sensing systems such as sensor array systems. Redundancy may permit to increase both performance and dependability of individual sensors at the system level, however, the emergence of these redundant systems has generated the need for dynamic algorithms able to detect and act against the presence of errors in system elements. Then, a new adaptive algorithm for sensor array systems is presented in this work. This approach is generic and can be implemented for different types of sensor systems, besides it does not require any information about the inputs. The proposed real-time algorithm is based on the following: 1) estimation of sensor individual variances at each time step, 2) identification and removal of sensors affected by errors, and 3) reincorporation of sensors after recovery or replacement. Variance estimation is revisiting MInimum Norm Quadratic Unbiased Estimation (MINQUE) method to allow real-time operation and requires that the number of sensors is strictly greater than two times the number of signals to be measured. Consequently, the proposed method is able to support the presence of m-1 faulty devices, where m is the total number of sensors less two times the number of measured signals. Performance of the proposed algorithm is demonstrated through theoretical demonstrations, simulations, and experimental results.

Fast heritability estimation based on MINQUE and batch training.

Heritability, the proportion of phenotypic variance explained by genome-wide single nucleotide polymorphisms (SNPs) in unrelated individuals, is an important measure of the genetic contribution to human diseases and plays a critical role in studying the genetic architecture of human diseases. Linear mixed model (LMM) has been widely used for SNP heritability estimation, where variance component parameters are commonly estimated by using a restricted maximum likelihood (REML) method. REML is an iterative optimization algorithm, which is computationally intensive when applied to large-scale datasets (e.g. UK Biobank). To facilitate the heritability analysis of large-scale genetic datasets, we develop a fast approach, minimum norm quadratic unbiased estimator (MINQUE) with batch training, to estimate variance components from LMM (LMM.MNQ.BCH). In LMM.MNQ.BCH, the parameters are estimated by MINQUE, which has a closed-form solution for fast computation and has no convergence issue. Batch training has also been adopted in LMM.MNQ.BCH to accelerate the computation for large-scale genetic datasets. Through simulations and real data analysis, we demonstrate that LMM.MNQ.BCH is much faster than two existing approaches, GCTA and BOLT-REML.

45 Partitioning SNP Heritability in Related Individuals

Abstract Partitioning SNP heritability by many functional annotations has been a successful tool for understanding the genetic architecture of complex traits in human genetic studies. Similar analyses are being extended to animal research, as (imputed) whole-genome sequence data of many individuals and various functional annotations have become available in livestock animals. Though many approaches have been developed for heritability partition (e.g., LDSC and HE-reg), they are mostly based on approximations tailored to human populations and few can produce statistically efficient estimates for animal genomic studies where individuals are often related. To tackle this issue, we present a stochastic MINQUE (Minimum Norm Quadratic Unbiased Estimation) approach for partitioning SNP heritability, which we refer to as MPH. We provide a theoretical analysis comparing LDSC and HE-reg with REML and MPH and demonstrate what LDSC and HE-reg (and similar methods) take advantage of in their approximations: sparse relationships between individuals and relatively weak linkage disequilibrium. We also show that our method is mathematically equivalent to the MC-REML approach implemented in BOLT. MPH has three key features. First, it is comparable to genomic REML in terms of accuracy, while being at least one order of magnitude faster than GCTA and BOLT and using only ~1/4 of memory as much as GCTA, when applied to sequence data and many variance components (or functional annotation categories). Second, it can do weighted analyses if residual variances are unequal (such as DYD). Third, it works for many overlapping functional annotations. Using simulations based on a human pedigree and a dairy cattle pedigree, we illustrate the benefits of our method for partitioning SNP heritability in pedigree-based studies. We also demonstrate that it is feasible to efficiently partition SNP heritability for animal genomes with strong, long-span LD. MPH is freely available at https://jiang18.github.io/mph.

Nonlinear total least-squares variance component estimation for GM(1,1) model

The solution of the grey model (GM(1,1) model) generally involves equal-precision observations, and the (co)variance matrix is established from the prior information. However, the data are generally available with unequal-precision measurements in reality. To deal with the errors of all observations for GM(1,1) model with errors-in-variables (EIV) structure, we exploit the total least-squares (TLS) algorithm to estimate the parameters of GM(1,1) model in this paper. Ignoring that the effect of the improper prior stochastic model and the homologous observations may degrade the accuracy of parameter estimation, we further present a nonlinear total least-squares variance component estimation approach for GM(1,1) model, which resorts to the minimum norm quadratic unbiased estimation (MINQUE). The practical and simulative experiments indicate that the presented approach has significant merits in improving the predictive accuracy in comparison with control methods.

Stochastic modeling for VRS network-based GNSS RTK with residual interpolation uncertainty

Abstract The key concept of the virtual reference station (VRS) network-based technique is to use the observables of multiple reference stations to generate the network corrections in the form of a virtual reference station at a nearby user’s location. Regarding the expected positioning accuracy, the novice GNSS data processing strategies have been adopted in the server-side functional model for mitigating distance-dependent errors including atmospheric effects and orbital uncertainty in order to generate high-quality virtual reference stations. In addition, the realistic stochastic model also plays an important role to take account of the unmodelled error in the rover-side processing. The results of our previous study revealed that the minimum norm quadratic unbiased estimation (MINQUE) stochastic model procedure can improve baseline component accuracy and integer ambiguity reliability, however, it requires adequate epoch length in a solution to calculate the elements of the variance-covariance matrix. As a result, it may not be suitable for urban environment where the satellite signal interruptions take place frequently, therefore, the ambiguity resolution needs to be resolved within the limited epochs. In order to address this limitation, this study proposed the stochastic model based on using the residual interpolation uncertainty (RIU) as the weighting schemes. This indicator reflects the quality of network corrections for any satellite pair at a specific rover position and can be calculated on the epoch-by-epoch basis. The comparison results with the standard stochastic model indicated that the RIU-weight model produced slightly better positioning accuracy but increased significant level of the ambiguity resolution successful rate.

Investigation on Stochastic Model Refinement for Precise Underwater Positioning

More accurate and realistic stochastic model is required for high-precision underwater positioning. The common stochastic modeling procedure, assuming that the measurements are statistically independent in space and time and have same accuracy, is certainly not realistic. Any misspecification in the stochastic model may have a significant effect on underwater acoustic data processing. Taking the heteroscedastic space and time correlation into consideration, a novel stochastic modeling procedure based on an iterative minimum norm quadratic unbiased estimator method has been developed to improve underwater positioning accuracy. Testing results by experiment data show that the positioning accuracy can be superiorly improved from 1.226, 1.434, and 1.018 m based on a traditional stochastic model estimation scheme to 0.935, 0.336, and 0.190 m for three transponders when adopting the new method.

Genetic parameters for some growth traits of Nigerian local chickens

Genetic parameters were estimated for bodyweight (BWT), shank length (SHL), and wing length (WL) of Nigerian local chicken (NLC) from 4 to 20 weeks of age by fitting dyadic mixed model (dmm) equations which yield estimates of variance components equivalent to minimum norm quadratic unbiased estimator (MINQUE). Data obtained from 600 chicks, progenies of 300 hens and 30 cocks were used for the analysis. The heritability estimates range from 0.08 to 0.80 for BWT, 0.03 to 0.69 for SHL and 0.22 to 0.47 for WL. The genetic correlations among BWT, SHL and WL at different ages were high and positive and range from 0.18 to 0.96 with the exemption of SHL and WL at 16 weeks (- 0.06). The phenotypic correlations were positive and range from 0.10 to 0.91. The results imply that NLC could be improved on any of the studied traits through mass selection and that improvement in one trait will result to correlated improvement in the others.

Estimating Variance Components in Functional Linear Models With Applications to Genetic Heritability

Quantifying heritability is the first step in understanding the contribution of genetic variation to the risk architecture of complex human diseases and traits. Heritability can be estimated for univariate phenotypes from nonfamily data using linear mixed effects models. There is, however, no fully developed methodology for defining or estimating heritability from longitudinal studies. By examining longitudinal studies, researchers have the opportunity to better understand the genetic influence on the temporal development of diseases, which can be vital for populations with rapidly changing phenotypes such as children or the elderly. To define and estimate heritability for longitudinally measured phenotypes, we present a framework based on functional data analysis, FDA. While our procedures have important genetic consequences, they also represent a substantial development for FDA. In particular, we present a very general methodology for constructing optimal, unbiased estimates of variance components in functional linear models. Such a problem is challenging as likelihoods and densities do not readily generalize to infinite-dimensional settings. Our procedure can be viewed as a functional generalization of the minimum norm quadratic unbiased estimation procedure, MINQUE, presented by C. R. Rao, and is equivalent to residual maximum likelihood, REML, in univariate settings. We apply our methodology to the Childhood Asthma Management Program, CAMP, a 4-year longitudinal study examining the long term effects of daily asthma medications on children.

Comparison of REML and MINQUE for Estimated Variance Components and Predicted Random Effects

Linear mixed model (LMM) approaches have been widely applied in many areas of research data analysis because they offer great flexibility for different data structures and linear model systems. In this study, emphasis is placed on comparing the properties of two LMM approaches: restricted maximum likelihood (REML) and minimum norm quadratic unbiased estimation (MINQUE) with and without resampling techniques being included. Bias, testing power, Type I error, and computing time were compared between REML and MINQUE approaches with and without Jackknife technique based on 500 simulated data sets. Results showed that MINQUE and REML methods performed equally regarding bias, Type I error, and power. Jackknife-based MINQUE and REML greatly improved power compared to non-Jackknife based linear mixed model approaches. Results also showed that MINQUE is more time-saving compared to REML, especially with the use of resampling techniques and large data set analysis. Results from the actual cotton data analysis were in agreement with our simulated results. Therefore, Jackknife-based MINQUE approaches could be recommended to achieve desirable power with reduced time for a large data analysis and model simulations.

Application of variance components estimation to calibrate geoid error models.

The method of using Global Positioning System-leveling data to obtain orthometric heights has been well studied. A simple formulation for the weighted least squares problem has been presented in an earlier work. This formulation allows one directly employing the errors-in-variables models which completely descript the covariance matrices of the observables. However, an important question that what accuracy level can be achieved has not yet to be satisfactorily solved by this traditional formulation. One of the main reasons for this is the incorrectness of the stochastic models in the adjustment, which in turn allows improving the stochastic models of measurement noises. Therefore the issue of determining the stochastic modeling of observables in the combined adjustment with heterogeneous height types will be a main focus point in this paper. Firstly, the well-known method of variance component estimation is employed to calibrate the errors of heterogeneous height data in a combined least square adjustment of ellipsoidal, orthometric and gravimetric geoid. Specifically, the iterative algorithms of minimum norm quadratic unbiased estimation are used to estimate the variance components for each of heterogeneous observations. Secondly, two different statistical models are presented to illustrate the theory. The first method directly uses the errors-in-variables as a priori covariance matrices and the second method analyzes the biases of variance components and then proposes bias-corrected variance component estimators. Several numerical test results show the capability and effectiveness of the variance components estimation procedure in combined adjustment for calibrating geoid error model.

An investigation into the correlations among GNSS observations and their impact on height and zenith wet delay estimation for medium and long baselines

Most stochastic modelling techniques neglect the correlations among the raw un-differenced observations when forming the variance–covariance matrix of the Global Navigation Satellite System (GNSS) observations. Some methods were developed to model these correlations. One such method is the Minimum Norm Quadratic Unbiased Estimator (MINQUE). Studies have shown that MINQUE improves ambiguity resolution, and ultimately, the positioning solution in short baselines. However, its effect in cases of processing with longer baselines and on the estimation of zenith wet delay (ZWD) is somewhat unknown. In this paper, a comparison between the impact of neglecting the correlations among the observations using an elevation-angle-dependent model (EADM) and modelling the correlations using MINQUE on height determination and ZWD for medium and long baselines is carried out. The initial testing was carried out across two Australian GNSS stations with a medium-length baseline throughout a three-week campaign. The results showed that using MINQUE did not resolve the coordinate, height and wet delay components as accurately as the EADM. The results were further verified with two long-baseline campaigns whereby EADM was also able to provide better wet delay estimates. The coordinate results were, however, mixed. Overall, the study concluded that the inclusion of the correlations among the observations, in general, do not improve the resolution of the coordinate and wet delay estimates.

Evaluation of preharvest sprouting traits in a collection of spring wheat germplasm using genotype and genotype × environment interaction model

With 10 tablesAbstractPreharvest sprouting (PHS) can greatly affect the consistent production of high‐quality spring wheat (Triticum aestivum L.) in many regions worldwide including Western Canada. A worldwide collection of red‐ and white‐seeded spring wheat genotypes with different levels of sprouting response were characterized for three PHS traits: falling number, germination index and sprouting index in three different environments in Manitoba, Canada. The data sets were analysed by the genotype and genotype × environment interaction model. Variance components were estimated, genotypic and their interaction effects with environments were predicted by using one of mixed linear model approaches: minimum norm quadratic unbiased estimation approach. Genotypic variance expressed as proportion to the phenotypic variance was higher compared to genotype × environment (G × E) interaction effects for all three PHS traits, suggesting that these genotypes can be used to develop high level of PHS‐resistant cultivars regardless of environment. Strong correlations between PHS traits across environments suggest that all three traits are repeatable and reliable methods to determine PHS response in spring wheat depending on sample types (spike, grain or flour). Predicted genotypic effects, G × E interaction effects and the linear discriminant analysis revealed that white‐seeded genotypes ‘AUS1408’, ‘SC8019‐R1’ and Kanata, and red‐seeded genotypes ‘AC Domain’, ‘AC Majestic’ and ‘Red RL4137’ would be useful PHS‐resistant donors in spring wheat cultivar development programmes.

Prediction by Regression and kriging for Spatial Data with Application

This study deals with the prediction of the non-stationary spatial stochastic process. The prediction is done by two techniques which are regression technique (generalized least square estimation) and universal kriging technique. As it is familiar, that the non-stationary stochastic process has a trend (mean) as a linear or non-linear model. By this process we can find covariance function from knowing the variogram function and the latter is attributed to a spherical variogram model, also in order to estimate parameters of spherical model by minimum norm quadratic unbiased estimator which requires that the covariance function must be linear in the parameters, then changing the spherical model into an approximated linear model by Taylor series in the linear approximation The prediction in these two techniques is applied to real data which represent height levels of ground water of 47 wells with their regional coordinates in Sinjar district in Ninevah Governorate in Iraq. The results were so encouraging where we show the approximation between the predictive values and the real values as well as computing the variance of prediction in these two techniques. It is shown that the prediction variance of universal kringing is less than that of regression.

Estimation of the Variance Components in Various Covariance Matrix Structures

This article describes the procedure of obtaining variance component estimates with the Minimum Norm Quadratic Unbiased Estimator (MINQUE). Using the examples of simulated measurements, variance components are estimated in geodetic two-dimensional network models. The efficiency of the estimators was tested using simulated data in three different structures of variance component models. Additive, group and mixed models were analyzed using distance and direction measurements, and the efficiency of the estimator was analyzed in all three cases, focusing in particular on the impact of various ratios of the initial variance components and geodetic network characteristics on the variance components estimation.

Supervisor assessment of clinical and professional competence of medical trainees: a reliability study using workplace data and a focused analytical literature review

Even though rater-based judgements of clinical competence are widely used, they are context sensitive and vary between individuals and institutions. To deal adequately with rater-judgement unreliability, evaluating the reliability of workplace rater-based assessments in the local context is essential. Using such an approach, the primary intention of this study was to identify the trainee score variation around supervisor ratings, identify sampling number needs of workplace assessments for certification of competence and position the findings within the known literature. This reliability study of workplace-based supervisors' assessments of trainees has a rater-nested-within-trainee design. Score variation attributable to the trainee for each competency item assessed (variance component) were estimated by the minimum-norm quadratic unbiased estimator. Score variance was used to estimate the number needed for a reliability value of 0.80. The trainee score variance for each of 14 competency items varied between 2.3% for emergency skills to 35.6% for communication skills, with an average for all competency items of 20.3%; the "Overall rating" competency item trainee variance was 28.8%. These variance components translated into 169, 7, 17 and 28 assessments needed for a reliability of 0.80, respectively. Most variation in assessment scores was due to measurement error, ranging from 97.7% for emergency skills to 63.4% for communication skills. Similar results have been demonstrated in previously published studies. In summary, overall supervisors' workplace based assessments have poor reliability and are not suitable for use in certification processes in their current form. The marked variation in the supervisors' reliability in assessing different competencies indicates that supervisors may be able to assess some with acceptable reproducibility; in this case communication and possibly overall competence. However, any continued use of this format for assessment of trainee competencies necessitates the identification of what supervisors in different institutions can reliably assess rather than continuing to impose false expectations from unreliable assessments.

MULTI-PARENTAL MATING DESIGN ANALYSIS: MODEL EVALUATION AND APPLICATION IN SPRING WHEAT

Conventional quantitative genetics studies have mainly focused on bi-parental mating systems. However, genetic potential of selected individuals within a segregating population may be limited due to only two parents being used for each cross. Multiple-parental mating systems have been proposed that involve three or four diverse parents. This provides a higher potential of combining desirable genes. Due to complexity of the data structure of multi-parental mating systems, analysis of variance (ANOVA) methods are not applicable in analysis. The objective of this study is to validate and apply a mixed linear model approach, minimum norm quadratic unbiased estimation (MINQUE), to analyze a widely used additive-dominance (AD) genetic model. Various simulations were conducted to validate the use of this approach. Twelve different spring wheat genotypes were used to develop populations in the study. Phenotypic data containing parents and their F2 (second filial generation) on preharvest sprouting (PHS) resistance in spring wheat (Triticum aestivum L.) developed by multi-parental crosses were used as a demonstration. The simulation study showed that a modified AD model can be used to estimate variance components in an unbiased manner within this complex data structure. Actual data analysis revealed that both additive and dominance effects were responsible for PHS resistance. Several parents associated with desirable additive effects for PHS were identified. In addition, some crosses with desirable heterozygous dominance effects were also identified, which can be used for hybrid development. Results should help breeders to obtain useful genetic information by using the methods suggested in this study.

Distribution-free estimators of variance components for multivariate linear mixed models

Non-iterative, distribution-free, and unbiased estimators of variance components including minimum norm quadratic unbiased estimators and the method of moments estimators are derived for multivariate linear mixed models. A general inter-cluster variance matrix, a same-member only general inter-response variance matrix, and an uncorrelated intra-cluster error structure for each response are assumed. Some properties of the proposed estimators such as unbiasedness and existence are discussed, and related computational issues are addressed. A simulation study is conducted to compare the proposed estimators with Gaussian (restricted) maximum likelihood estimators in terms of bias and mean square error. An application of gene expression family study is presented to illustrate the proposed estimators.