The variability among data may have quite complex structure and the splitting of variation into its individual components is important in many areas of applications. In particular, the study of variation 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. Sometimes these models are called components of variance models since the interest in the model lies in investigating the variances of the effects of factor levels which are assumed to be randomly selected from a large population of levels. These variances are also known as variance components since the total variance can be expressed as the sum of these variances and the error variance. Knowledge of the variance components, i.e., the information on the relative importance and magnitude of the sources of variability in the data is of critical importance in many problems since the isolation of the total variance into the individual components of variance would result in the physical elimination of the sources of variability in question.

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A bibliography on variance components an introduction and an update: 1984-2002

Let y represent an n × 1 observable random vector that follows the general mixed linear model , where β is a p×l vector of unknown parameterss i is a qi ×1 unobservable random vector whose distribution is (multivariate normal with mean vector 0 and variance-covariance matrix , and e is an n × 1 unobservable random vector whose distribution is . The problem considered is that of computing restricted maximum likelihood (REML) estimates of . In general, closed-form expressions for the REML estimates do not exist, in which case the estimate must be computed by an iterative numerical method. Before applying an iterative algorithm, it may be advisable to reparameterize, to “linearize” the likelihood equations, or to eliminate one or more of the likelihood equations (by absorption). “Linearized” versions of two common algorithms, the method of successive approximations (MSA) and the Newton-Raphson (NR) algorithm, are proposed. Numerical results suggest that these algorithms improve on the MSA and the NR algorith...

Some new algorithms for computing restricted maximum likelihood estimates of variance components

MANOVA in the multivariate components of variance model

There is an increased interest in estimating the (co)variance components of additive animal models with direct and competition effects (AMC). However, some attempts to estimate the dispersion parameters in different animal species faced problems of convergence or inaccurate estimates when pen effects entered the model. We argue that the problem relates to lack of identifiability of the (co)variance components in some AMC. The check for identifiability of the dispersion parameters in mixed models with linear (co)variance structure requires that all the eigenvalues of the restricted maximum likelyhood information matrix (I(theta)) be positive. We show, by way of simple numerical examples, that the singularity of I(theta) is due to confounding between fixed pen effects and the additive competition effects (SBVs). It is also observed that setting pen effects as random does not always remedy the collinearity with SBVs. 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. Examples are presented in which the ICs are related to time. The distribution of families of full and half sibs across pens also plays a role in the identifiability and asymptotic variances of the (co)variance components.

On identifiability of (co)variance components in animal models with competition effects

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 paper. The paper examines when such a test statistic can be computed and what are its distributional properties with respect to the actual mixed model.

https://www.tandfonline.com/doi/pdf/10.1080/03610928808829673

When can random effects be treated as fixed effects for computing a test statistic for a linear hypothesis

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.

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K. G. Brown

Papers

On Analysis of Variance in the Mixed Model