Showing papers on "Random effects model published in 1970"

Estimation of Variance and Covariance Components in the Mixed Model

Abstract: A direct method of computing the coefficient of a variance or covariance component in the expectation of a mean square or mean product when the inverse of the coefficient matrix is available is presented for use with models containing only non-interacting sets of random effects in addition to error. These models may contain any number of sets of fixed effects including interactions and partial regressions for continuous variables. Shortcut computational procedures are presented for the estimation of components of variance and covariance when one set of random effects interacts with one or two sets of fixed main effects and subclass frequencies are unequal. A computational example is given for the two-way classification without interaction and one is available in mimeograph form from the author for the three-way classification with interactions.

Large Sample Variances of Maximum Likelihood Estimators of Variance Components Using Unbalanced Data

Abstract: Estimation of variance components from data that are unbalanced (having unequal numbers of observations in the subclasses) has been referred to by Hartley [1967] as involving 'algebraic heroics.' Obtaining sampling variances can be described similarly. There is little or no difficulty with balanced data because then, with customary normnality assumptions, the sums of squares of an analysis of variance have independent X2-distributions, variance component estimators are linear functionis of these, and their variances are readily derived. Furthermore, as Graybill and Hultquist [1961] point out, these estimators (which are unbiased) also have minimum variance properties. In contrast to the relatively manageable situation of balanced data we consider here the estimation of variance components from unbalanced data, of which balanced data are deemed to be simply a special case. In particular we deal with the sampling variances of large sample maximum likelihood (ML) estimators. Before doing so, a brief review of the present status of sampling variances of other estimators is in order. For random (and mixed) models Henderson [1953] developed three methods of estimating variance components from unbalanced data of any crossed and/or nested classifications, the techniques of which have been further discussed in Searle [1968]. Sampling variances of the resulting estimators have been developed extensively for only one of these methods, Method 1, that is analogous to the analysis of variance method for balanced data. Furthermore, in all cases, sampling variances have been considered only in the case of random effects models (Model II of Eisenhart [1947]), and only on the basis of having normality assumptions as part of such models. Within

Estimation of variance and covariance components in the mixed model

Abstract: A direct method of computing the coefficient of a variance or covariance component in the expectation of a mean square or mean product when the inverse of the coefficient matrix is available is presented for use with models containing only non-interacting sets of random effects in addition to error. These models may contain any number of sets of fixed effects including interactions and partial regressions for continuous variables. Shortcut computational procedures are presented for the estimation of components of variance and covariance when one set of random effects interacts with one or two sets of fixed main effects and subclass frequencies are unequal. A computational example is given for the two-way classification without interaction and one is available in mimeograph form from the author for the three-way classification with interactions.

A Comparison of Variances of Some Estimators in the Balanced Incomplete Block (BIB) Variance Components Model

Abstract: A way is described of obtaining the variances of Henderson's [1953] Methods 1 and 3 estimates in a BIB with random effects, when normality is assumed. We illustrate the method by an example, showing that an estimator indicated by the minimal sufficient but not complete set of statistics given by Weeks and Graybill [1961] has variance which can exceed that of another estimator, and so is not MV. We give tentative conclusions regarding the choice of estimator in practice.

An Application of Majorization to Comparison of Variances

Abstract: Empirical comparisons of variances of estimators of the population mean in a two-stage nested design, random effects model are given by Koch [3]. He uses the methods in [Z] to obtain a new estimator and then conjectures that its variance is intermediate with respect to better known estimators. His conjecture is proved by using theorems from [4]. All three estimators are compared with the expression that maximizes the likelihood function.