anova1_article

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

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 = 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 within-batch 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 •...
View Full Document

Page1 / 13

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online