7 Transformations and Weighted Least Squares

7 Transformations and Weighted Least Squares - Statistics...

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Unformatted text preview: Statistics 191: Introduction to Applied Statistics Jonathan Taylor Department of Statistics Stanford University Statistics 191: Introduction to Applied Statistics Weighted Least Squares, Transformations Jonathan Taylor Department of Statistics Stanford University February 17, 2010 1 / 1 Statistics 191: Introduction to Applied Statistics Jonathan Taylor Department of Statistics Stanford University Outline Today’s class Transformations to achieve linearity. Transformations to stabilize variance. Correcting for unequal variance: weighted least squares. 2 / 1 Statistics 191: Introduction to Applied Statistics Jonathan Taylor Department of Statistics Stanford University Transformations Transformations to achieve linearity We have been working with linear regression models so far in the course. Many models are nonlinear, but can be transformed to a linear model. 3 / 1 Statistics 191: Introduction to Applied Statistics Jonathan Taylor Department of Statistics Stanford University Bacteria death R code 4 / 1 Statistics 191: Introduction to Applied Statistics Jonathan Taylor Department of Statistics Stanford University Transformations Exponential growth model Suppose the expected number of cells grows like E ( n t ) = n e β 1 t , t = 1 , 2 , 3 , .. . If we take logs of both sides log E ( n t ) = log n + β 1 t . (Reasonable ?) model: log n t = n + β 1 t + ε t , ε t ∼ N (0 , σ 2 ) independent 5 / 1 Statistics 191: Introduction to Applied Statistics Jonathan Taylor Department of Statistics Stanford University Transformations Logarithmic transformation This is slightly different than original model: E (log n t ) ≤ log E ( n t ) but may be approximately true. If ε t ∼ N (0 , σ 2 ) then n t = n · t · e β 1 t . t = e ε t is called a log-normal random (0 , σ 2 ) random variable. 6 / 1 Statistics 191: Introduction to Applied Statistics Jonathan Taylor Department of Statistics Stanford University Bacteria death, fitted values R code 7 / 1 Statistics 191: Introduction to Applied Statistics Jonathan Taylor Department of Statistics Stanford University...
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7 Transformations and Weighted Least Squares - Statistics...

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