anova1_article - 1 One-factor models Factors A factor is a...

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Unformatted text preview: 1 One-factor 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 = i + ij = + i + ij i = i- typically, one assumes that the ij are independent N (0 , 2 ) the assumptions about the 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 = + i + ij 1 , . . . , 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 1 , . . . , I are viewed as fixed parameters Random effects From previous page Y ij = + 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 1 , . . . , I are assumed to be independent N (0 , 2 ) the parameter of interest is 2 More about the i i = i- (deviation of i th mean from overall mean) Fixed effects: 1 + + I = 0 Random effects: i independent N (0 , 2 ) Variance components of a random effects model 2 and 2 are called variance components they are the components of the variance of Y ij : Var ( Y ij ) = Var ( + i + ij ) = 2 + 2 . Example: Example: The levels are batches of a product. Y ij is measured on the j item from the i th batch is the standard deviation of the batch means is the within-batch standard deviation of the product p 2 + 2 is the overall standard deviation of the product Quality control In quality control: 3...
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anova1_article - 1 One-factor models Factors A factor is a...

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