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Unformatted text preview: 1 Onefactor models Factors A factor is a “categorical” variable. • the values of the factor are called “levels” – the factor “machine” might have three levels, machine 1, machine 2, machine 3 – the variable “temperature” might have four levels, 300 o , 325 o , 350 o , 375 o (treated as categories) – the individual batches from a production process are levels of the factor “batch” • typically, the number of levels is small, even 2 Single factor experiments: data Notation: • There are I levels of a single factor • There are n i observations at the i th level • The responses are Y ij , i = 1 , . . . , I and j = 1 , . . . , n i Single factor experiments: model • The statistical model is Y ij = μ i + ij = μ + α i + ij • α i = μ i μ • typically, one assumes that the ij are independent N (0 , σ 2 ) • the assumptions about the α i depend on whether these param eters are viewed as fixed or random 1.1 Fixed and random effects Fixed effects From previous page Y ij = μ + α i + ij α 1 , . . . , α I are called the “effects” of the factor • Fixed effects – Example: the levels are the only three suppliers of silicon wafers used by the company – Example: the levels are the only four operators employed by the company – the “levels” are the factor are of intrinsic interest – we do not want to generalize to a larger population – α 1 , . . . , α I are viewed as fixed parameters Random effects From previous page Y ij = μ + α i + ij • Random effects – Example: the levels are a sample of silicon wafers from a supplier – Example: the levels are a sample of operators from a large pool of workers – the “levels” of the factor are a sample from a larger popu lation – we want to generalize our conclusions to the larger popu lation 2 – the particular levels in the sample are not of much interest – α 1 , . . . , α I are assumed to be independent N (0 , σ 2 α ) – the parameter of interest is σ 2 α More about the α i α i = μ i μ (deviation of i th mean from overall mean) • Fixed effects: α 1 + ··· + α I = 0 • Random effects: α i independent N (0 , σ 2 α ) Variance components of a random effects model • σ 2 α and σ 2 are called variance components • they are the components of the variance of Y ij : Var ( Y ij ) = Var ( μ + α i + ij ) = σ 2 α + σ 2 . Example: Example: The levels are batches of a product. Y ij is measured on the j item from the i th batch • σ α is the standard deviation of the batch means • σ is the withinbatch standard deviation of the product • p σ 2 α + σ 2 is the overall standard deviation of the product Quality control In quality control: 3 •...
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 Spring '08
 D.RUPPERT,P.JACKS
 Normal Distribution, Standard Deviation, Variance, Yij, Simple wafer example

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