18anovaLME - THE ANOVA APPROACH TO THE ANALYSIS OF LINEAR...

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THE ANOVA APPROACH TO THE ANALYSIS OF LINEAR MIXED EFFECTS MODELS I A model for expt. data with subsampling y ijk = μ + τ i + u ij + e ijk , ( i = 1 , ..., t ; j = 1 , ..., n ; k = 1 , ..., m ) β = ( μ, τ i , ..., τ t ) 0 , u = ( u 11 , u 12 , ..., u tn ) 0 , ± = ( e 111 , e 112 , ..., e tnm ) 0 , β IR t + 1 , an unknown parameter vector , ± u ± ² N ³± 0 0 ² , ± σ 2 u I 0 0 σ 2 e I ²´ σ 2 u , σ 2 e IR + , unknown variance components c ± 2011 Dept. Statistics (Iowa State University) Stat 511 section 18 1 / 37
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I This is the commonly-used model for a CRD with t treatments, n experimental units per treatment, and m observations per experimental unit. I We can write the model as y = X β + Z u + ± , where X = [ 1 |{z} tnm × 1 , I |{z} t × t 1 |{z} nm × 1 ] and Z = [ I |{z} tn × tn 1 |{z} m × 1 ] I Consider the sequence of three column spaces: X 1 = 1 tnm × 1 , X 2 = [ 1 tnm × 1 , I t × t 1 nm × 1 ] , X 3 = [ 1 tnm × 1 , I t × t 1 nm × 1 , I tn × tn 1 m × 1 ] I These correspond to the models: X 1 : Y ijk = μ + ± ijk X 2 : Y ijk = μ + τ i + ± ijk X 3 : Y ijk = μ + τ i + u ij + ± ijk c ± 2011 Dept. Statistics (Iowa State University) Stat 511 section 18 2 / 37
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ANOVA table for subsampling Source SS df df treatments y 0 ( P 2 - P 1 ) y rank ( X 2 ) - rank ( X 1 ) t - 1 eu ( treatments ) y 0 ( P 3 - P 2 ) y rank ( X 3 ) - rank ( X 2 ) t ( n - 1 ) ou ( eu , treatments ) y 0 ( I - P 3 ) y tnm - rank ( X 3 ) tn ( m - 1 ) C . total tnm - 1 I In terms of squared differences: Source df Sum of Squares trt t - 1 t i = 1 n j = 1 m k = 1 ( y i .. - y ... ) 2 eu ( trt ) t ( n - 1 ) t i = 1 n j = 1 m k = 1 ( y ij . - y i .. ) 2 ou ( eu , trt ) tn ( m - 1 ) t i = 1 n j = 1 m k = 1 ( y ijk - y ij . ) 2 C . total tnm - 1 t i = 1 n j = 1 m k = 1 ( y ijk - y ... ) 2 c ± 2011 Dept. Statistics (Iowa State University) Stat 511 section 18 3 / 37
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I Which simplifies to: Source df Sum of Squares trt t - 1 nm t i = 1 ( y i .. - y ... ) 2 eu ( trt ) tn - t m t i = 1 n j = 1 ( y ij . - y i .. ) 2 ou ( eu , trt ) tnm - tn t i = 1 n j = 1 m k = 1 ( y ijk - y ij ... ) 2 C . total tnm - 1 t i = 1 n j = 1 m k = 1 ( y ijk - y ... ) 2 I Each line has a MS = SS / df I MS’s are random variables. Their expectations are informative. c ± 2011 Dept. Statistics (Iowa State University) Stat 511 section 18 4 / 37
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Expected Mean Squares, EMS E ( MS trt ) = nm t - 1 t i = 1 E ( y i .. - y ... ) 2 = nm t - 1 t i = 1 E ( μ + τ i + u i . + e i .. - μ - τ. - u .. - e ... ) 2 = nm t - 1 t i = 1 E ( τ i - τ. + u i - u .. + e i .. - e ... ) 2 = nm t - 1 t i = 1 [( τ i - τ. ) 2 + E ( u i - u .. ) 2 + E ( e i .. - e ... ) 2 ] = nm t - 1 [ t i = 1 ( τ i - τ ) 2 + E ( t i = 1 ( u i - u .. ) 2 ) + E ( t i = 1 e i . - e ) 2 )] I u 1 ,..., u t . i
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This note was uploaded on 02/11/2012 for the course STAT 511 taught by Professor Staff during the Spring '08 term at Iowa State.

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18anovaLME - THE ANOVA APPROACH TO THE ANALYSIS OF LINEAR...

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