Analysis of Variance for Random Models, Volume 2: Unbalanced by Hardeo Sahai

By Hardeo Sahai

Systematic therapy of the widely hired crossed and nested category versions utilized in research of variance designs with an in depth and thorough dialogue of sure random results versions no longer regularly present in texts on the introductory or intermediate point. it's also numerical examples to research information from a large choice of disciplines in addition to any labored examples containing computing device outputs from common software program programs resembling SAS, SPSS, and BMDP for every numerical instance.

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Extra info for Analysis of Variance for Random Models, Volume 2: Unbalanced Data: Theory, Methods, Applications, and Data Analysis

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R. L. Anderson and P. P. Crump (1967), Comparisons of designs and estimation procedures for estimating parameters in a two-stage nested process, Technometrics, 9, 499–516. T. R. Bainbridge (1963), Staggered nested designs for estimating variance components, in American Society for Quality Control Annual Conference Transactions, American Society for Quality Control, Milwaukee, 93–103. R. B. Bapat and T. E. S. Raghavan (1997), Nonnegative Matrices and Applications, Cambridge University Press, Cambridge, UK.

0 ⎨⎢ 0 .. · ·· = tr ⎢ ⎣ ⎪ ⎪ . ⎩ 0 .. X2 [I − X1 (X1 X1 )− X1 ]X2 ⎡ ⎤⎫ .. ⎬ E(β1 β1 ) . E(β1 β2 ) ⎦ ×⎣ . ⎭ E(β2 β1 ) .. E(β2 β2 ) ⎤ ⎥ ⎥ ⎦ + σe2 [rank(X) − rank(X1 )] = tr{X2 [I − X1 (X1 X1 )− X1 ]X2 E(β2 β2 )} + σe2 [rank(X) − rank(X1 )]. 12) is a function only of E(β2 β2 ) and σe2 and has been derived without any assumption on the form of E(ββ ). 12) states that if the vector β is partitioned as (β1 , β2 ), where β1 represents all the fixed effects and β2 represents all the random effects, then E{R(β2 |β1 )} contains only σe2 and the variance components associated with the random effects; it contains no terms due to the fixed effects.

NA . 6) where n(Ai , θj ) is the number of observations in the ith level of the factor A and the j th level of the factor θ. 6) is generally applicable to any T in any random model. 7) θ=A and for Tµ , the correction factor for the mean, it is equal to ⎧ ⎫ P ⎨ Nθ ⎬ σ2 θ + σe2 . 8) j =1 Thus the term N µ2 occurs in the expectation of every T . But since sums of squares (SSs) involve only differences between T s, expectations of SSs do not contain N µ2 , and their coefficients of σe2 are equal to their corresponding degrees of freedom.

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