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Ch4-4 - Outline 4.1 Generalized Linear Models Chapter 4...

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Chapter 4. Introduction to Generalized Linear Models Deyuan Li School of Management, Fudan University Feb. 28, 2011 1 / 107 Outline 4.1 Generalized Linear Models 4.2 Generalized Linear Models for Binary Data 4.3 Generalized Linear Models for Counts 4.4 Moments and Likelihood for Generalized Linear Models 4.5 Inference for Generalized Linear Models 4.6 Fitting Generalized Linear Models 4.7 Quasi-likelihood and Generalized Linear Models 2 / 107 4.1 Generalized Linear Models Generalized linear models (GLMs) extend ordinary regression models to encompass nonnormal and nonidentical response distributions and modeling functions of the mean. 4.1.1 Components of generalized linear models random component; systematic component; link function. 1) The random component consists of a response variable Y with independent observations ( y 1 , . . . , y N ) from a distribution in the natural exponential family . This family has probability density or mass function of form { Insert ...... } The value of the parameter θ i may vary for i = 1 , . . . , N , depending on values of explanatory variables. The term Q ( θ ) is called the natural parameter . 3 / 107 2) The systematic component relates a vector ( η 1 , . . . , η N ) to the explanatory variables through a linear model. Let x ij denote the value of predictor j ( j = 1 , 2 , . . . , p ) for subject i . Then η i = j β j x ij , for i = 1 , . . . , N . This linear combination of explanatory variables, η i , is called the linear predictor . Usually, x i 1 = 1 for all i , for the coefficient of an intercept (often denoted by α ) in the model. 3) The link function connects the random and the systematic components. Let μ i = E ( Y i ), i = 1 , . . . , N . The model links μ i to η i by η i = g ( μ i ) , where the link function g is a monotonic, differentiable function. 4 / 107
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Thus, g links E ( Y i ) to explanatory variables through the formula { Insert ...... } The link function g ( μ ) = μ , called the identical link , has η i = μ i , i.e., a linear model for the mean itself. This is the link function for ordinary regression with normally distributed Y . The link function that transforms the mean to the natural parameter is called the canonical link . For it, g ( μ i ) = Q ( θ i ), and Q ( θ i ) = j β j x ij . In summary, a GLM is a linear model for a transformed mean of a response variable that has distribution in the natural exponential family. 5 / 107 4.1.2 Binomial logit models for binary data The Bernoulli distribution has the probability mass function f ( y ; π ) = π y (1 π ) 1 y = (1 π )[ π/ (1 π )] y = (1 π ) exp y log π 1 π for y = 0 and 1. This is in the natural exponential family, with θ = π , a ( π ) = 1 π , b ( y ) = 1 and Q ( π ) = log[ π/ (1 π )]. The natural parameter Q ( π ) = log[ π/ (1 π )] is the log odds of response y = 1, i.e., the logit of π . The canonical link function is the logit link, η = log[ π/ (1 π )]. GLMs using the logit link are often called logit models .
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