16LinMixedEffects(1)

# 16LinMixedEffects(1) - LINEAR MIXED-EFFECT MODELS Seedling...

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LINEAR MIXED-EFFECT MODELS Studies / data / models seen previously in 511 assumed a single source of “error” variation y = X β + . β are fixed constants (in the frequentist approach to inference) is the only random effect What if there are multiple sources of “error” variation? c 2011 Dept. Statistics (Iowa State University) Stat 511 section 16 1 / 21 Examples: Subsampling Seedling weight in 2 genotype study from Aitken model section. Seedling weight measured on each seedling. Two (potential) sources of variation: among flats and among seedlings within a flat. Y ijk = μ + γ i + T ij + ijk T ij N ( 0 , σ 2 F ) ijk N ( 0 , σ 2 e ) where i indexes genotype, j indexes flat within genotype, and k indexes seedling within flat σ 2 F quantifies variation among flats, if have perfect knowledge of the seedlings in them 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 Influence of two factors: temperature and a catalyst on fermentation of dry distillers grain (byproduct of EtOH production from corn) Response is CO 2 production, measured in a tube Catalyst (+/-) randomly assigned to tubes Temperature randomly assigned to growth chambers 6 tubes per growth chamber, 3 +catalyst, 3 -catalyst all 6 at same temperature Use 6 growth chambers, 2 at each temperature 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 Two sizes of experimental unit: tube and temperature Y ijkl = μ + α i + γ ik + β j + αβ ij + ijkl γ ik N ( 0 , σ 2 G ) Var among growth chambers ijkl N ( 0 , 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 σ 2 G quantifies variation among g.c., if have perfect knowledge of the tubes in them σ 2 quantifies variation among tubes c 2011 Dept. Statistics (Iowa State University) Stat 511 section 16 4 / 21

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Examples: Gauge R&R study Want to quantify repeatability and reproducibility of a measurement process Repeatability: variation in meas. taken by a single person or instrument on the same item. Reproducibility: variation in meas. made by different people (or different labs), if perfect repeatability One possible design: 10 parts, each measured twice by 10 operators. 200 obs.
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