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Unformatted text preview: Introduction The first part of Stat 511 reexamines methods from Stat 500 from a linear models perspective. A simple study: does a proprietary food additive increase milk production in dairy cows? 6 cows, housed one per stall. Randomly choose 3 to get food with the additive; other 3 without. response: average daily milk production two weeks after start of treatment Most (all?) would use a ttest and related methods to: estimate the difference of treatment means estimate the standard error of that difference construct a 95% confidence interval for that difference test whether the difference is 0 Copyright c 2011 Dept. of Statistics (Iowa State University) Statistics 511 1 / 36 Introduction (continued) One possible model: y ij = μ i + ij y ij is the response from cow j in treatment i , i = 1, 2; j = 1 ··· 3 μ i is the population mean for treatment i ij is the deviation from the population mean for cow j in trt i Assume: ij ∼ N ( , σ 2 ) Standard methods use: ¯ y 1 ¯ y 2 to estimate μ 1 μ 2 s 2 = ∑ i , j ( y ij ¯ y i ) 2 / ( 2 n 2 ) to estimate σ 2 a tdistribution to construct the confidence interval or test a hypothesis Copyright c 2011 Dept. of Statistics (Iowa State University) Statistics 511 2 / 36 Introduction (continued) Some questions: What about ttests and related estimation methods generalize to more complicated models? i.e., how is the ttest an example of y = X β + Does it matter whether you use a cell means or effects model? i.e., does it matter which X you use? Is ¯ y 1 ¯ y 2 a good estimate of μ 1 μ 2 ? Is there a more precise estimate of μ 1 μ 2 ? When does [(¯ y 1 ¯ y 2 ) ( μ 1 μ 2 )] / p s 2 * 2 / n follow a t distribution? We’ll answer these and more using a very general framework Based on linear models theory. Provides guide to when methods work and when you have a problem. Copyright c 2011 Dept. of Statistics (Iowa State University) Statistics 511 3 / 36 Introduction (continued) Notes have lots of equations and lots of matrices Will use matrices to present results Skim (or read) the Koehler notes on matrices http://www.public.iastate.edu/ ∼ kkoehler/stat511/sect2.4page.pdf Skip material on eigenvalues and other decompositions for now Will show proofs. Want you to understand concepts and results. Will not expect you to prove results on exams. Stat PhD students take 611: full semester of linear model theory with proofs. Will try to be careful about notation. Ask if you don’t understand something. Copyright c 2011 Dept. of Statistics (Iowa State University) Statistics 511 4 / 36 The GaussMarkov Linear Model y = X β + y is an n × 1 random vector of responses. X is an n × p matrix of constants with columns corresponding to explanatory variables. X is sometimes referred to as the design matrix ....
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 Spring '08
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