14-AitkenModel(1)

14-AitkenModel(1) - THEAITKENMODEL y = X β ∼,σ 2 V...

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Unformatted text preview: THEAITKENMODEL y = X β + , ∼ ( ,σ 2 V ) IdenticaltotheGauss-MarkovLinearModelexceptthat Var ()= σ 2 V insteadof σ 2 I . V isassumedtobea known nonsingularVariancematrix. TheNormalTheoryAitkenModeladdsanassumptionof normality: ∼ N ( ,σ 2 V ) Observationsarenowcorrelated,orhaveunequalvariances. butthecorrelations,orunequalvariances,followaknownpattern c 2011Dept.ofStatistics(IowaStateUniversity) Stat511,section14 1/19 Examples-1 Analysisofaverages Thedatatobeanalyzedareaveragesofunequalnumbersof observations. Y i isanaverageof n i observations.Var Y i = σ 2 / n i first4rowsandcolumnsofVarare: ⎡ ⎢⎢ ⎣ σ 2 / n 1 σ 2 / n 2 σ 2 / n 3 σ 2 / n 4 ⎤ ⎥⎥ ⎦ = σ 2 ⎡ ⎢⎢ ⎣ 1 / n 1 1 / n 2 1 / n 3 1 / n 4 ⎤ ⎥⎥ ⎦ c 2011Dept.ofStatistics(IowaStateUniversity) Stat511,section14 2/19 Examples-2 Analysisofdataonapedigree(geneticrelationshipsamong parents,children,grandchildren,...) Geneticcorrelationsbetweenparentsandchildren,among children,...,allknown. Y = X β + FirstfourrowsofVar: 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 istheknowngeneticcorrelationamongindividuals i , j c 2011Dept.ofStatistics(IowaStateUniversity) Stat511,section14 3/19 Examples-3 Regressionondatacollectedovertime: Y i = β + β 1 X i + i X i =year(1990,1991,...) Assumeerrorsfollowanautoregressiveprocess(morelater), first4rowsandcolumnsofVarare: ⎡ ⎢⎢ ⎣ 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 ⎤ ⎥⎥ ⎦ Aitkenmodelif ρ known. c 2011Dept.ofStatistics(IowaStateUniversity) Stat511,section14 4/19 Examples-4 Splitplotexperimentwithtwoeu’s:growthchamberandpot two“error”variances: σ 2 c and σ 2 p iftwopotsperchamber,firstfourrowsofVarare: ⎡ ⎢⎢ ⎣ σ 2 c + σ 2 p σ 2 c σ 2 c σ 2 c + σ 2 p σ 2 c + σ 2 p σ 2 c σ 2 c σ 2 c + σ 2 p ⎤ ⎥⎥ ⎦ Aitkenmodelif σ 2 p /σ 2 c known, = k σ 2 c ⎡ ⎢⎢ ⎣ 1 + k 1 1 1 + k 1 + k 1 1 1 + k ⎤ ⎥⎥ ⎦ Manyotherwaysofrepresenting V .TobeAitken,allrequire knownrelationshipbetween σ 2 c and σ 2 p . c 2011Dept.ofStatistics(IowaStateUniversity) Stat511,section14 5/19...
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14-AitkenModel(1) - THEAITKENMODEL y = X β ∼,σ 2 V...

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