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Unformatted text preview: PUBH 7430 Lecture 9 J. Wolfson Division of Biostatistics University of Minnesota School of Public Health October 4, 2011 The Generalized Linear Model The Generalized Linear Model generalizes the standard linear model by: 1 Letting a function of the mean μ be a linear combination of the covariates: g ( μ ) = X β 2 Letting the variance be a function of the mean: Var ( Y  X ) = V ( μ ) GLMs: Coefficient interpretation Coefficient interpretation must take into account the link function g : “A oneunit increase in X is associated with a βunit increase/decrease in g ( μ )” Can work backwards to get familiar cases: • Poisson regression: “A oneunit increase in X is associated with an exp( β ) percent increase/decrease in the mean of Y ” • Logistic regression: “A oneunit increase in X is associated with an exp( β ) 1+exp( β ) percent increase/decrease in the odds of Y ” Aside: Log transformation • When data are positive and display evidence of nonnormality, log transformation is often recommended before proceeding with inference. • If using a linear model, we are assuming: E [log( Y i )  x i ] = x i β Var [log( Y i )  x i ] = σ 2 • An alternative is to fit a GLM with a log link and our choice of variance function: log( E [ Y i  x i ]) = x i β Var [ Y i  x i ] = V ( μ i ) Aside: Log transformation...
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 Fall '04
 Prof.Eberly
 Regression Analysis, GLM, variance function

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