Decomposition of approaches of a general linear model with fixed parameters

The well-known ordinary least-squares estimators (OLSEs) and the best linear unbiased estimators (BLUEs) under linear regression models can be represented by certain closed-form formulas composed by the given matrices and their generalized inverses in the models. This paper provides a general algebraic approach to relationships between OLSEs and BLUEs of the whole and partial mean parameter vectors in a constrained general linear model (CGLM) with fixed parameters by using a variety of matrix analysis tools on generalized inverses of matrices and matrix rank formulas. In particular, it establishes a variety of necessary and sufficient conditions for OLSEs to be BLUEs under a CGLM, which include many reasonable statistical interpretations on the equalities of OLSEs and BLUEs of parameter space in the CGLM. The whole work shows how to effectively establish matrix equalities composed by matrices and their generalized inverses and how to use them when characterizing performances of estimators of parameter spaces in linear models under most general assumptions.

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On decompositions of estimators under a general linear model with partial parameter restrictions

Open Mathematics ◽

10.1515/math-2017-0109 ◽

2017 ◽

Vol 15 (1) ◽

pp. 1300-1322 ◽

Cited By ~ 6

Author(s):

Bo Jiang ◽

Yongge Tian ◽

Xuan Zhang

Keyword(s):

Linear Model ◽

General Linear Model ◽

Matrix Analysis ◽

General Linear ◽

Full Model ◽

Linear Regression Models ◽

Statistical Inferences ◽

Best Linear Unbiased ◽

Best Linear Unbiased Estimators ◽

The Given

Abstract A general linear model can be given in certain multiple partitioned forms, and there exist submodels associated with the given full model. In this situation, we can make statistical inferences from the full model and submodels, respectively. It has been realized that there do exist links between inference results obtained from the full model and its submodels, and thus it would be of interest to establish certain links among estimators of parameter spaces under these models. In this approach the methodology of additive matrix decompositions plays an important role to obtain satisfactory conclusions. In this paper, we consider the problem of establishing additive decompositions of estimators in the context of a general linear model with partial parameter restrictions. We will demonstrate how to decompose best linear unbiased estimators (BLUEs) under the constrained general linear model (CGLM) as the sums of estimators under submodels with parameter restrictions by using a variety of effective tools in matrix analysis. The derivation of our main results is based on heavy algebraic operations of the given matrices and their generalized inverses in the CGLM, while the whole contributions illustrate various skillful uses of state-of-the-art matrix analysis techniques in the statistical inference of linear regression models.

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On equality and proportionality of ordinary least squares, weighted least squares and best linear unbiased estimators in the general linear model

Statistics & Probability Letters ◽

10.1016/j.spl.2006.01.005 ◽

2006 ◽

Vol 76 (12) ◽

pp. 1265-1272 ◽

Cited By ~ 29

Author(s):

Yongge Tian ◽

Douglas P. Wiens

Keyword(s):

Least Squares ◽

Linear Model ◽

Weighted Least Squares ◽

General Linear Model ◽

Ordinary Least Squares ◽

General Linear ◽

Unbiased Estimators ◽

Best Linear Unbiased ◽

Best Linear Unbiased Estimators

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Least Squares Estimations for the General Linear Model Parameters with Epsilon Skew Normal Error Term

Iraqi Journal of Science ◽

10.24996/ijs.2020.61.3.20 ◽

2020 ◽

pp. 636-645

Author(s):

Hussain Karim Nashoor ◽

Ebtisam Karim Abdulah

Keyword(s):

Least Squares ◽

Linear Model ◽

Least Squares Method ◽

General Linear Model ◽

Ordinary Least Squares ◽

Skewed Distribution ◽

General Linear ◽

Average Error ◽

Model Parameters ◽

Skew Normal

Examination of skewness makes academics more aware of the importance of accurate statistical analysis. Undoubtedly, most phenomena contain a certain percentage of skewness which resulted to the appearance of what is -called "asymmetry" and, consequently, the importance of the skew normal family . The epsilon skew normal distribution ESN (μ, σ, ε) is one of the probability distributions which provide a more flexible model because the skewness parameter provides the possibility to fluctuate from normal to skewed distribution. Theoretically, the estimation of linear regression model parameters, with an average error value that is not zero, is considered a major challenge due to having difficulties, as no explicit formula to calculate these estimates can be obtained. Practically, values for these estimates can be obtained only by referring to numerical methods. This research paper is dedicated to estimate parameters of the Epsilon Skew Normal General Linear Model (ESNGLM) using an adaptive least squares method, as along with the employment of the ordinary least squares method for estimating parameters of the General Linear Model (GLM). In addition, the coefficient of determination was used as a criterion to compare the models’ preference. These methods were applied to real data represented by dollar exchange rates. The Matlab software was applied in this work and the results showed that the ESNGLM represents a satisfactory model.

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Linear models for diallel crosses: a review with R functions

Theoretical and Applied Genetics ◽

10.1007/s00122-020-03716-8 ◽

2020 ◽

Author(s):

Andrea Onofri ◽

Niccolò Terzaroli ◽

Luigi Russi

Keyword(s):

Linear Model ◽

Combining Ability ◽

Linear Models ◽

General Linear Model ◽

Ordinary Least Squares ◽

General Purpose ◽

General Linear ◽

Mixed Effect ◽

Typical Process ◽

R Functions

Abstract Key message A new R-software procedure for fixed/random Diallel models was developed. We eased the diallel schemes approach by considering them as specific cases with different parameterisations of a general linear model. Abstract Diallel experiments are based on a set of possible crosses between some homozygous (inbred) lines. For these experiments, six main diallel models are available in literature, to quantify genetic effects, such as general combining ability (GCA), specific combining ability (SCA), reciprocal (maternal) effects and heterosis. Those models tend to be presented as separate entities, to be fitted by using specialised software. In this manuscript, we reinforce the idea that diallel models should be better regarded as specific cases (different parameterisations) of a general linear model and might be fitted with general purpose software facilities, as used for all other types of linear models. We start from the estimation of fixed genetical effects within the R environment and try to bridge the gap between diallel models, linear models and ordinary least squares estimation (OLS). First, we review the main diallel models in literature. Second, we build a set of tools to enable geneticists, plant/animal breeders and students to fit diallel models by using the most widely known R functions for OLS fitting, i.e. the ‘lm()’ function and related methods. Here, we give three examples to show how diallel models can be built by using the typical process of GLMs and fitted, inspected and processed as all other types of linear models in R. Finally, we give a fourth example to show how our tools can be also used to fit random/mixed effect diallel models in the Bayesian framework.

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The use of MIXED models in the analysis of animal experiments with repeated measures data

Canadian Journal of Animal Science ◽

10.4141/a03-123 ◽

2004 ◽

Vol 84 (1) ◽

pp. 1-11 ◽

Cited By ~ 312

Author(s):

Z. Wang and L. A. Goonewardene

Keyword(s):

Least Squares ◽

Linear Model ◽

Repeated Measures ◽

Mixed Model ◽

Covariance Structure ◽

General Linear Model ◽

Ordinary Least Squares ◽

General Linear ◽

Repeated Measures Data ◽

Split Plot

The analysis of data containing repeated observations measured on animals (experimental unit) allocated to different treatments over time is a common design in animal science. Conventionally, repeated measures data were either analyzed as a univariate (split-plot in time) or a multivariate ANOVA (analysis of contrasts), both being handled by the General Linear Model procedure of SAS. In recent times, the mixed model has become more appealing for analyzing repeated data. The objective of this paper is to provide a background understanding of mixed model methodology in a repeated measures analysis and to use balanced steer data from a growth study to illustrate the use of PROC MIXED in the SAS system using five covariance structures. The split-plot in time approach assumes a constant variance and equal correlations (covariance) between repeated measures or compound symmetry, regardless of their proximity in time, and often these assumptions are not true. Recognizing this limitation, the analysis of contrasts was proposed. If there are missing measurements, or some of the data are measured at different times, such data were excluded resulting in inadequate data for a meaningful analysis. The mixed model uses the generalized least squares method, which is generally better than the ordinary least squares used by GLM, if the appropriate covariance structure is adopted. The presence of unequally spaced and/or missing data does not pose a problem for the mixed model. In the example analyzed, the first order ante dependence [ANTE(1)] covariance model had the lowest value for the Akaike and Schwarz’s Bayesian information criteria fit statistics and is therefore the model that provided the best fit to our data. Hence, F values, least square estimates and standard errors based on the ANTE (1) were considered the most appropriate from among the five models demonstrated. It is recommended that the mixed model be used for the analysis of repeated measures designs in animal studies. Key words: Repeated measures, General Linear Model, Mixed Model, split-plot, covariance structure

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Sensitivity analysis in linear models

Special Matrices ◽

10.1515/spma-2016-0021 ◽

2016 ◽

Vol 4 (1) ◽

Author(s):

Shuangzhe Liu ◽

Tiefeng Ma ◽

Yonghui Liu

Keyword(s):

Sensitivity Analysis ◽

Least Squares ◽

Linear Model ◽

Linear Models ◽

General Linear Model ◽

Ordinary Least Squares ◽

The Other ◽

Regression Coefficients ◽

General Linear ◽

Least Squares Estimators

AbstractIn this work, we consider the general linear model or its variants with the ordinary least squares, generalised least squares or restricted least squares estimators of the regression coefficients and variance. We propose a newly unified set of definitions for local sensitivity for both situations, one for the estimators of the regression coefficients, and the other for the estimators of the variance. Based on these definitions, we present the estimators’ sensitivity results.We include brief remarks on possible links of these definitions and sensitivity results to local influence and other existing results.

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