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Unformatted text preview: LINEAR MIXEDEFFECT MODELS I Studies / data / models seen previously in 511 assumed a single source of “error” variation I y = X β + . β are fixed constants (in the frequentist approach to inference) is the only random effect I What if there are multiple sources of “error” variation? c 2011 Dept. Statistics (Iowa State University) Stat 511 section 16 1 / 21 Examples: Subsampling I Seedling weight in 2 genotype study from Aitken model section. I Seedling weight measured on each seedling. I Two (potential) sources of variation: among flats and among seedlings within a flat. Y ijk = μ + γ i + T ij + ijk T ij ∼ N ( , σ 2 F ) ijk ∼ N ( , σ 2 e ) where i indexes genotype, j indexes flat within genotype, and k indexes seedling within flat I σ 2 F quantifies variation among flats, if have perfect knowledge of the seedlings in them I σ 2 e quantifies variation among seedlings c 2011 Dept. Statistics (Iowa State University) Stat 511 section 16 2 / 21 Examples: Split plot experimental design I Influence of two factors: temperature and a catalyst on fermentation of dry distillers grain (byproduct of EtOH production from corn) I Response is CO 2 production, measured in a tube I Catalyst (+/) randomly assigned to tubes I Temperature randomly assigned to growth chambers 6 tubes per growth chamber, 3 +catalyst, 3 catalyst all 6 at same temperature I Use 6 growth chambers, 2 at each temperature I The o.u. is a tube. What is the experimental unit? c 2011 Dept. Statistics (Iowa State University) Stat 511 section 16 3 / 21 Examples: Split plot experimental design  2 I Two sizes of experimental unit: tube and temperature Y ijkl = μ + α i + γ ik + β j + αβ ij + ijkl γ ik ∼ N ( , σ 2 G ) Var among growth chambers ijkl ∼ N ( , σ 2 ) Var among tubes where i indexes temperature, j indexes g.c. within temp., k indexes catalyst, and l indexes tube within temp., g.c., and catalyst I σ 2 G quantifies variation among g.c., if have perfect knowledge of the tubes in them I σ 2 quantifies variation among tubes c 2011 Dept. Statistics (Iowa State University) Stat 511 section 16 4 / 21 Examples: Gauge R&R study I Want to quantify repeatability and reproducibility of a measurement process I Repeatability: variation in meas. taken by a single person or instrument on the same item. I Reproducibility: variation in meas. made by different people (or different labs), if perfect repeatability I One possible design: 10 parts, each measured twice by 10 operators. 200 obs....
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 Spring '08
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 Regression Analysis, Random effects model, Block matrix, STATE UNIVERSITY

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