note17 - STAT5044: Regression and Anova Inyoung Kim 1 / 17...

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STAT5044: Regression and Anova Inyoung Kim 1 / 17
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Outline 1 Generalized Linear Model 2 / 17
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GLM Consider the simple linear regression model E ( Y ) = β 0 + β 1 x where Y is normally distributed Denoting E ( Y ) = μ , we can write μ = β 0 + β 1 x For the logistic regression model ln ( π 1 - π ) = β 0 + β 1 x where π is the mean of a bernoulli random variable Y . By letting μ = π , we can also write ln ( μ 1 - μ ) = β 0 + β 1 x 3 / 17
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GLM More generally, we can think of a model of the form g ( μ ) = β 0 + β 1 x We call the generalized linear model g function is called the link function because it connects the mean μ and the linear predictor x. g is usually one-to-one, and differentiable. As we have just seen in, the identity link g ( t ) = t and logit link g ( t ) = ln ( t / 1 - t ) are examples of link functions. 4 / 17
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GLM For Poisson random variable Y (count data), the natural link function is log: g ( t ) = log ( t ) In that case, the model is of the form log ( μ ) = β 0 + β 1 x and it called the Poisson loglinear model (or Poisson regression model) Although the usual linear regression model may be used to handle count data, it poses problems because the linear predictor can be negative while the count response is strictly positive. The problem does not happen with the Poisson regression model. 5 / 17
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Generalized linear models (GLMs) extend ordinary regression models to encompass nonnormal response distributions and modeling function of the mean. Three components specify a GLM: A random component identifies the response variable Y and its probability distribution A systematic component specifies explanatory variables used in a linear predictor function A link function specifies the function of E ( Y ) that the model equates to the systematic component. 6 / 17
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This note was uploaded on 01/02/2012 for the course STAT 5044` taught by Professor Staff during the Fall '11 term at Virginia Tech.

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note17 - STAT5044: Regression and Anova Inyoung Kim 1 / 17...

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