Variance Components In Mixed Models Research Articles

Estimability of variance components when all model matrices commute

This paper deals with estimability of variance components in mixed models when all model matrices commute. In this situation, it is well known that the best linear unbiased estimators of fixed effects are the ordinary least squares estimators. If, in addition, the family of possible variance–covariance matrices forms an orthogonal block structure, then there are the same number of variance components as strata, and the variance components are all estimable if and only if there are non-zero residual degrees of freedom in each stratum.We investigate the case where the family of possible variance–covariance matrices, while still commutative, no longer forms an orthogonal block structure. Now the variance components may or may not all be estimable, but there is no clear link with residual degrees of freedom. Whether or not they are all estimable, there may or may not be uniformly best unbiased quadratic estimators of those that are estimable. Examples are given to demonstrate all four possibilities.

Restricted likelihood ratio tests for linearity in scalar-on-function regression

We propose a procedure for testing the linearity of a scalar-on-function regression relationship. To do so, we use the functional generalized additive model (FGAM), a recently developed extension of the functional linear model. For a functional covariate X(t), the FGAM models the mean response as the integral with respect to t of F{X(t),t} where F is an unknown bivariate function. The FGAM can be viewed as the natural functional extension of generalized additive models. We show how the functional linear model can be represented as a simple mixed model nested within the FGAM. Using this representation, we then consider restricted likelihood ratio tests for zero variance components in mixed models to test the null hypothesis that the functional linear model holds. The methods are general and can also be applied to testing for interactions in a multivariate additive model or for testing for no effect in the functional linear model. The performance of the proposed tests is assessed on simulated data and in an application to measuring diesel truck emissions, where strong evidence of nonlinearities in the relationship between the functional predictor and the response are found.

A note on genetic variance components in mixed models.

Burton et al. ([1999] Genet. Epidemiol. 17:118-140) proposed a series of generalized linear mixed models for pedigree data that account for residual correlation between related individuals. These models may be fitted using Markov chain Monte Carlo methods, but the posterior mean for small variance components can exhibit marked positive bias. Burton et al. ([1999] Genet. Epidemiol. 17:118-140) suggested that this problem could be overcome by allowing the variance components to take negative values. We examine this idea in depth, and show that it can be interpreted as a computational device for locating the posterior mode without necessarily implying that the original random effects structure is incorrect. We illustrate the application of this technique to mixed models for familial data.

Métodos de estimação de componentes de variância em modelos mistos desbalanceados

A estimação dos componentes de variância em modelos mistos tem merecido, nas últimas décadas, atenção especial dos pesquisadores. Sendo assim, vários métodos de estimação têm sido propostos, e diante de tal disponibilidade, escolher um deles pode não ser uma questão trivial. Nesse contexto, buscando fornecer aos usuários dos métodos de estimação de componentes de variância em modelos mistos desbalanceados, subsídios para uma opção tão confortável quanto possível do método mais adequado para a obtenção das estimativas, em suas pesquisas, o presente artigo apresenta, com ilustrações numéricas, os principais métodos de estimação. Ademais, apresenta, aos usuários não iniciados, conceitos introdutórios necessários para a utilização do PROC MIXED do Sistema Estatístico SAS, na obtenção das estimativas dos componentes de variância através dos métodos ali disponíveis, considerando a ocorrência de diferentes matrizes de variâncias e covariâncias.

ANOVA estimates of variance components for quasi-balanced mixed models

This article examines a simple and intuitively appealing way to estimate variance components in mixed models: perform as many ordinary least-squares regressions as there are variance components, using dummy variables for all but one random effect in each regression. By equating these residual sums of squares to their expectations, unbiased estimates of all of the variance components can be obtained. This approach is a special case of Method 3 of Henderson (Biometrics 9 (1953) 226–252). A quasi-balanced mixed model is defined here to be a model in which the indicator matrix for one of the random effects is essentially arbitrary, but the model obtained by omitting this random effect is nested and balanced. For quasi-balanced normal models the exact distributions of residual sums of squares are derived; they lead to the exact distributions of Henderson Method 3 estimates. An example involving strength data for a composite material is discussed in detail.

Generalized Confidence Intervals

Abstract The definition of a confidence interval is generalized so that problems such as constructing exact confidence regions for the difference in two normal means can be tackled without the assumption of equal variances. Under certain conditions, the extended definition is shown to preserve a repeated sampling property that a practitioner expects from exact confidence intervals. The proposed procedure is also applied to the problem of constructing confidence intervals for the difference in two exponential means and for variance components in mixed models. A repeated sampling property of generalized p values is also given. With this characterization one can carry out fixed level tests of parameters of continuous distributions on the basis of generalized p values. Finally, Pratt's paradox is revisited, and a procedure that resolves the paradox is given.

Improved estimation of variance components in mixed models

Taking Albert's (1976) formulation of a mixed model ANOVA, we consider improved estimation of the variance components for balanced designs under squared error loss. Two approaches are presented. One extends the ideas of Stein (1964), The other is developed from the fact that variance components can be expressed as linear combinations of chi-square scale parameters. Encouraging simulation results are presented.