Journal ArticleDOI
An Unbalanced Two‐way Model With Random Effects Having Unequal Variances
TLDR
In this article, the mixed model yijk, where xijkkm are known constants and βm's are unknown fixed effects, is transformed to a fixed effect model with equal variances so that least squares theory can be used to draw inferences about the β m's.Abstract:
Consider the model Yijk=μ + ai + bij + eijk (i=1, 2,…, t; j=1,2,…, Bi; k=1,2…,nij), where μ is a constant and a1,bij and eijk are distributed independently and normally with zero means and variances σ2adij and σ2, respectively, where it is assumed that the di's and dij's are known In this paper procedures for estimating the variance components (σ2, σ2a and σ2b) and for testing the hypothesis σ2b = 0 and σ2a = 0 are presented In the last section the mixed model yijk, where xijkkm are known constants and βm's are unknown fixed effects (m = 1, 2,…,p), is transformed to a fixed effect model with equal variances so that least squares theory can be used to draw inferences about the βm'sread more
Citations
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Journal ArticleDOI
Tests concerning a nested mixed model with heteroscedastic random effects
TL;DR: In this paper, the fixed and random effects for a two-fold nested mixed model are compared under the following conditions: (i) the nesting effect is fixed; (ii) the nested effect is random with an unknown variance component assumed to depend only on the levels of the nesting effects; (iii) all random effects, including the random errors, are independently distributed as normal variates.
Book ChapterDOI
Two-Way Nested Classification
Hardeo Sahai,Mario Miguel Ojeda +1 more
TL;DR: In this paper, the authors considered two-way nested classification, where replications are grouped into different sets arising from the levels of the nested factor for a given level of the main factor.
Book ChapterDOI
Two-Way Crossed Classification with Interaction
Hardeo Sahai,Mohammed I. Ageel +1 more
TL;DR: In this chapter, the model involving two factors with interaction terms, which is an important and somewhat restrictive implication of the simple additive model discussed in Chapter III, is studied.
Dissertation
Inference procedures for fixed effects in multivariate mixed models
Abstract: Except in special cases, exact tests of fixed effects in multivariate mixed models are not available. Available approximate methods either apply techniques developed for balanced multivariate mixed models or treat unknown variance components as known. The objectives of this study were to develop a test for fixed effects in unbalanced multivariate mixed models and to examine the properties of the test. The proposed test statistic is a generalization of the Lawley-Hotelling trace statistic. The sampling distribution of the test statistic was approximated by equating the first moment and a scalar function of the second moment of the two quadratic forms which make up the test statistic to the corresponding moments of two independent Wishart matrices. Four different scalar functions of the second moments were used to obtain four approximations to the sampling distribution. The approximations yield different critical values and, thus, give different tests. When the data are balanced, the test reduces to the usual Lawley-Hotelling test. Approximate simultaneous confidence intervals for linear functions of the fixed effects are given. A Monte-Carlo study of the size and power of the proposed test was conducted. The model used in the simulation was based on a data set comparing different breeds of beef cattle. Three covariance structures for the random effects and seven effect sizes for the fixed effects were simulated. For each data set, all. four versions of the proposed test were conducted. In addition, an exact test, obtained by using a balanced subset of the data was made. The Monte-Carlo based estimates of the size of the proposed test were within .003 of the nominal test size of .05. The estimated power curves of the proposed test were well above the estimated power curves of the exact test. The estimated power from the simulation agreed well with the computations based on the approximation to the non-null distribution. There appears to be no reason to prefer one method of approximating the distribution over another. Based on this simulation study, the proposed test is superior to an exact test which uses a subset of the data. INFERENCE PROCEDURES FOR FIXED EFFECTS IN MULTIVARIATE MIXED MODELS by Stephen Daniel Kachman A thesis submitted in partial,fulfillment of the requirements for the degree Doctor of Philosophy in
References
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Journal ArticleDOI
Estimation of variance and covariance components
TL;DR: The theory of variance component analysis has been discussed recently by Crump (1946, 1951) and by Eisenhart (1947), and most of the published works on estimating variance components deal with the one-way classification, with nested" classifications, and with factorial classifications having equal subclass numbers.
Journal ArticleDOI
Introduction to Matrices with Applications in Statistics
TL;DR: In this paper, the authors present an introduction to matrices with applications in statistics, and present a set of matrices that can be used in statistics applications in the field of computer vision.
Journal ArticleDOI
Transformations for Estimation of Linear Models with Nested-Error Structure
TL;DR: In this article, two linear models with error structure of the nested type are considered and transformations are presented by which uncorrelated errors with constant variances are obtained, where the transformed observations are differences between the original observations and multiples of averages of subsets of the observations.
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Bates and best quadratic unbiased estimators for variance components and heteroscedastie variances in linear models
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