Read e-book online Analysis of Variance for Random Models, Volume 2: Unbalanced PDF

By Hardeo Sahai

ISBN-10: 0817632298

ISBN-13: 9780817632298

Systematic therapy of the widely hired crossed and nested class types utilized in research of variance designs with an in depth and thorough dialogue of definite random results versions no longer usually present in texts on the introductory or intermediate point. it's also numerical examples to investigate information from a wide selection of disciplines in addition to any labored examples containing machine outputs from general software program programs akin to SAS, SPSS, and BMDP for every numerical instance.

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Read or Download Analysis of Variance for Random Models, Volume 2: Unbalanced Data: Theory, Methods, Applications, and Data Analysis PDF

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

Example text

2) and P Var(Y ) = Xθ Var(βθ )Xθ + σe2 IN . θ=A Now, let y. (Ai ) and n(Ai ) denote the total value and the number of observations in the ith level of the factor A. Then the raw sum of squares of the factor A is NA TA = [y. 3) i=1 where NA is the number of levels of the factor A. 1. , Appendix M) of NA matrices [1/n(Ai )]Jn(Ai ) , i = 1, 2, . . , 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.

The assumptions imply that Var(Y ) = i=1 σi Ui Ui = p 2 i=1 σi Vi where Vi = Ui Ui . Let Pi (i = 1, 2, . . , p) be the orthogonal projection operator on the column space of (X, U1 , U2 , . . , Ui ). Note that Pp = IN . , P0 = X(X X)−1 X . Finally, let Qj be the orthogonal projection on the orthogonal complement of the column space of (X, U1 , U2 , . . , Uj −1 ) (for j = 1, the column space of X). Note that Qj = Pj −Pj −1 (j = 1, 2, . . , p) and Qp = Pp −Pp−1 = IN −Pp−1 . Now, consider the following orthogonal decomposition of Y : p Y = P0 Y + Qj Y , j =1 which implies that p Y Y = Y P0 Y + Y Qj Y .

The user can specify the type of output required, which may range from estimates of variance components plus standard errors to a complete list of all parameters and standard errors of differences between all pairs including linear functions and ratios of linear functions of variance components such as heritability. Current releases of SAS® , SPSS® , BMDP® , and S-PLUS® compute the ML and REML estimates with great speed and accuracy simply by specifying the model in question (see, Appendix O). 9 BEST QUADRATIC UNBIASED ESTIMATION The variance component analogue of the best linear unbiased estimator (BLUE) of a function of fixed effects is a best quadratic unbiased estimator (BQUE), that is, a quadratic function of the observations that is unbiased for the variance component and has minimum variance among all such estimators.

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Analysis of Variance for Random Models, Volume 2: Unbalanced Data: Theory, Methods, Applications, and Data Analysis by Hardeo Sahai


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