On Analysis of Variance in the Mixed Model
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In this article, conditions for an ANOVA and the number of terms in it both can depend on which effects in the model are fixed and which are random, which is not taken into account by those procedures for partitioning a sum of squares which distinguish between random and fixed effects only in the calculation of expected mean squares.Abstract:
An analysis of variance (ANOVA) is defined to be a partition of the total sum of squares into independent terms which, when suitably scaled, are chi-squared variables. A partition of less than the total sum of squares, but with these properties, will often suffice and is referred to as a partial ANOVA. Conditions for an ANOVA, and for partial ANOVAs selected to contain only specific parameters, are given. Implications for estimation of variance components from an ANOVA are also discussed. These results are largely an extension of work by Graybill and Hultquist (1961). With unbalanced data, conditions for an ANOVA and the number of terms in it both can depend on which effects in the model are fixed and which are random. This is not taken into account by those procedures for partitioning a sum of squares which distinguish between random and fixed effects only in the calculation of expected mean squares. Several examples are given.read more
Citations
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A bibliography on variance components an introduction and an update: 1984-2002
Hardeo Sahai,Anwer Khurshid +1 more
TL;DR: In particular, the study of variance through a class of linear models known as random and mixed models is a central topic in statistics with wide ramifications in both theory and applications as discussed by the authors.
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Some new algorithms for computing restricted maximum likelihood estimates of variance components
TL;DR: “Linearized” versions of two common algorithms, the method of successive approximations (MSA) and the Newton-Raphson (NR) algorithm, are proposed and numerical results suggest that these algorithms improve on the MSA and the NR algorith...
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MANOVA in the multivariate components of variance model
TL;DR: In this article, conditions are obtained for the multivariate components of variance model to admit a multivariate analysis of variance (MANOVA), defined as a partition of the sum of squares and sum of products matrix into independent Wishart matrices.
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On identifiability of (co)variance components in animal models with competition effects
TL;DR: It is observed that setting pen effects as random does not always remedy the collinearity with SBVs, and an alternative AMC is presented in which the incidence matrix of the SBVs can be written as a function of the 'intensity of competition' (IC) among animals in the same pen.
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When can random effects be treated as fixed effects for computing a test statistic for a linear hypothesis
TL;DR: The idea of treating the random effects as fixed for constructing a test for a linear hypothesis (of fixed effects) in a mixed linear model is considered in this article, where the authors examine when such a test statistic can be computed and what are its distributional properties with respect to the actual mixed model.
References
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Journal ArticleDOI
The Analysis of Variance
TL;DR: In this paper, the basic theory of analysis of variance by considering several different mathematical models is examined, including fixed-effects models with independent observations of equal variance and other models with different observations of variance.
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XV.—The Correlation between Relatives on the Supposition of Mendelian Inheritance.
TL;DR: In this paper, it was shown that the variance of a human measurement from its mean follows the Normal Law of Errors, and that the variability may be measured by the standard deviation corresponding to the square root of the mean square error.
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Maximum Likelihood Approaches to Variance Component Estimation and to Related Problems
TL;DR: In this paper, the authors proposed a restricted maximum likelihood (reml) approach which takes into account the loss in degrees of freedom resulting from estimating fixed effects, and developed a satisfactory asymptotic theory for estimators of variance components.
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Asymptotically Efficient Estimation of Covariance Matrices with Linear Structure
TL;DR: In this article, it was shown that the solution of the generalized least squares equations is asymptotically efficient if consistent estimates of the covariance matrix are used to obtain the coefficients of the linear equations.