Showing papers on "Random effects model published in 1969"

Iterative Estimation of Variance Components for Non-Orthogonal Data

Abstract: Eisenhart [1947] introduced the term 'mixed model' to describe models useful in experiments where some effects, such as block or animal effects, can be thought of as random effects and other effects, for example treatments, are regarded as fixed. The estimation of variance components from these experiments is fully understood for various incomplete block arrangements with a high degree of symmetry (Nelder [1968]). For a general non-orthogonal design, however, difficulties arise and no simple known method is optimal under all conditions. Lack of balance is very common in records on animals, especially those arising from studies of quantitative genetics. Experimenters are fortunate if families of animals are all of the same size; even if an experiment begins with reasonable symmetry, accidental losses may introduce non-orthogonality. Cunningham and Henderson [1968] have proposed a general method of estimation, using iterative calculations. An algebraic oversight has corrupted their formulae, giving an iterative process that will usually fail to converge to anything reasonable.

Exact first- and second-order moments of estimates of components of covariance

Abstract: SUMMARY This paper develops formulae for the first two moments of estimates of covariance for the general multivariate 'one-way' and 'two-way' models. The results are used to obtain the large sample dispersion matrix for estimated coefficients of two types of genetic selection indexes. These dispersion matrices provide the necessary extension of known results in balanced models to the unbalanced case. The problems of estimation associated with variance and covariance component analysis with unbalanced data have been of major concern to the statistical geneticist. This is primarily because most of the selection procedures applied to livestock require a knowledge of genetic variances and covariances which usually have to be estimated from the analyses of hierarchical models. Invariably, there is a marked lack of balance in the data thus rendering standard formulae for the variances of the estimates inapplicable. Serious consideration to these problems has been given by Henderson (1953), Searle (1956) and Hartley & Rao (1967). Searle gave particular attention to the one-way analysis of variance and covariance and used matrix methods to calculate the moments of the various estimators. Other work in this area concerns analysis of variance models of varying complexity; see Searle (1958, 1961), Mahamunulu (1963) and Blischke (1966). It is the purpose of the present paper to extend and complement existing results. With the advent of high speed computers, matrix operations can be handled with great speed and hence formulae for expectations and covariances of sums of squares and products can be left in a general computable form. Thus, explicit algebraic evaluation of each case is, in most cases, not only time-consuming but unnecessary. We consider here the general one-way and two-way analysis of covariance model with fixed and random effects. The number of variables included in the analysis is assumed to be arbitrary and this seems to lead to somewhat involved notation and algebra. However, general results are required in order to solve a number of practical problems. We give two examples from statistical genetics. In the theory of animal breeding interest centres around certain phenotypic and genetic parameters. Suppose that k characters of a particular breed of animal are relevant from the point of view of a selection programme. Then we let P and G be the phenotypic and additive genetic covariance matrices for the k characters and we consider two types of selection index which are based on these matrices.

Formal bayes estimation with application to a random effects analysis of variance model.

Abstract: : Techniques of formal Bayes estimation are discussed and applied to the estimation of the components of variance in the one way layout random effects ANOVA model. Invariant formal Bayes estimators are presented and proved to be admissible in certain classes of invariant rules and also minimax. To do this a general sufficient condition for admissible in certain multiparameter problems is given and applied to the ANOVA problem. Numerical calculations of mean squared errors of various estimators are also presented, and these results help justify the recommendation of the formal Bayes estimators. (Author)

Comparison of the sensitivities of similar independent and non-independent experiments

Abstract: SUMMARY The comparison of sensitivities is considered in detail for a random effects model in which the different methods of measurement are applied to the same experimental units. An exact distribution is obtained and tabulated and approximations to it considered. Other applications of the results are outlined. The problem of comparing two or more methods of measuring the effects of experimental treatments arises in various fields of research. Cochran (1943) discussed this problem in considerable detail under the assumption that analysis of variance techniques were applicable. He suggested that a comparison of the sensitivities of two scales of measurement, or two experimental techniques, should depend both on the magnitudes of treatment effects and the experimental errors associated with the measurement scales, or experimental techniques. A method of measurement for which the differences between treatment effects show up well, and which, at the same time, has a relatively small error variance, will be considered a sensitive method. In this paper we distinguish between three types of situation in experimentation: (i) Type 1. The different methods of measurement are applied to the same experimental units. This can be done whenever not more than one method affects the experimental units. The results of the methods will be correlated. (ii) Type 11. The different methods of measurement are applied to separate subsamples of the same experimental units. Here only the 'treatment effects' for the methods will be correlated. (iii) Type 111. The different methods of measurement are applied to independent experimental units. Schumann & Bradley (1957, 1959) considered the type III situation. Following Cochran's line of thought, they effected a comparison of the sensitivities of two experiments through a comparison of two variance ratios. Their work was restricted to independent experiments and they were mainly concerned with similar experiments in which the variance ratios compared had identical degrees of freedom. They calculated critical values of the ratios of two independent F's to be used in hypothesis tests to compare sensitivities. Schumann & Bradley assumed that analysis of variance techniques were applicable and considered both model I (fixed effects model) and model II (random effects model) of the analysis of variance. Dar (1962) derived a normal approximation to the Schumann & Bradley density function for the ratio of two F's. He showed that the logarithm of the ratio was approximately normally distributed for large degrees of freedom. He also extended the theory to obtain an approximate chi-square test for the comparison of the sensitivities of several