14-AitkenModel

# 14-AitkenModel - THE AITKEN MODEL y = X(0 2 V Identical to...

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THE AITKEN MODEL I y = X β + ± , ± ( 0 , σ 2 V ) I Identical to the Gauss-Markov Linear Model except that Var ( ± ) = σ 2 V instead of σ 2 I . I V is assumed to be a known nonsingular Variance matrix. I The Normal Theory Aitken Model adds an assumption of normality: ± N ( 0 , σ 2 V ) I Observations are now correlated, or have unequal variances. I but the correlations, or unequal variances, follow a known pattern c ± 2011 Dept. of Statistics (Iowa State University) Stat 511, section 14 1 / 19

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Examples - 1 I Analysis of averages The data to be analyzed are averages of unequal numbers of observations. Y i is an average of n i observations. Var Y i = σ 2 / n i ﬁrst 4 rows and columns of Var ± are: σ 2 / n 1 0 0 0 0 σ 2 / n 2 0 0 0 0 σ 2 / n 3 0 0 0 0 σ 2 / n 4 = σ 2 1 / n 1 0 0 0 0 1 / n 2 0 0 0 0 1 / n 3 0 0 0 0 1 / n 4 c ± 2011 Dept. of Statistics (Iowa State University) Stat 511, section 14 2 / 19
Examples - 2 I Analysis of data on a pedigree (genetic relationships among parents, children, grandchildren, . ..) I Genetic correlations between parents and children, among children, . .., all known. Y = X β + ± First four rows of Var ± : σ 2 1 ρ 12 ρ 13 ρ 14 ρ 12 1 ρ 23 ρ 24 ρ 13 ρ 23 1 ρ 34 ρ 14 ρ 24 ρ 34 1 where ρ ij is the known genetic correlation among individuals i , j c ± 2011 Dept. of Statistics (Iowa State University) Stat 511, section 14 3 / 19

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Examples - 3 I Regression on data collected over time: Y i = β 0 + β 1 X i + ± i X i = year (1990, 1991, . ..) I Assume errors follow an autoregressive process (more later), ﬁrst 4 rows and columns of Var ± are: σ 2 σ 2 ρ σ 2 ρ 2 σ 2 ρ 3 σ 2 ρ σ 2 σ 2 ρ σ 2 ρ 2 σ 2 ρ 2 σ 2 ρ σ 2 σ 2 ρ σ 2 ρ 3 σ 2 ρ 2 σ 2 ρ σ 2 = σ 2 1 ρ ρ 2 ρ 3 ρ 1 ρ ρ 2 ρ 2 ρ 1 ρ ρ 3 ρ 2 ρ 1 I Aitken model if ρ known. c ± 2011 Dept. of Statistics (Iowa State University) Stat 511, section 14 4 / 19
I Split plot experiment with two eu’s: growth chamber and pot I two “error” variances: σ 2 c and σ 2 p I if two pots per chamber, ﬁrst four rows of Var ± are: σ 2 c + σ 2 p σ 2 c 0 0 σ 2 c σ 2 c + σ 2 p 0 0 0 0 σ 2 c + σ 2 p σ 2 c 0 0 σ 2 c σ 2 c + σ 2 p I Aitken model if σ 2 p 2 c known, = k σ 2 c 1 + k 1 0 0 1 1 + k 0 0 0 0 1 + k 1 0 0 1 1 + k I Many other ways of representing V . To be Aitken, all require known relationship between σ 2 c and σ 2 p . c

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14-AitkenModel - THE AITKEN MODEL y = X(0 2 V Identical to...

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