note18 - STAT5044: Regression and Anova Inyoung Kim 1 / 18...

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Unformatted text preview: STAT5044: Regression and Anova Inyoung Kim 1 / 18 Outline 1 Logistic regression for Binary data 2 Poisson regression for Count data 2 / 18 GLM Let Y denote a binary response variable. Each observation has one of two outcomes, denoted by 0 or 1, binomial for a single trial. The mean E ( Y ) = P ( Y = 1 ) We denote P ( Y = 1 ) by ( x ) , reflecting its dependence on values x = ( x 1 ,..., x p ) of predictors. The variance of Y is Var ( Y ) = ( x )( 1- ( x ) the binomial variance for one trial. In introducing GLMs for binary data, for simplicity we use a single explanatory variable. 3 / 18 Linear Probability model For a binary response, the regression model ( x ) = + x is called a linear probability model With independent observations it is a GLM with binomial random component and identity link function. 4 / 18 Logistic regression model Usually, binary data result from a nonlinear relationship between ( x ) and x . In practice, nonlinear relationships between ( x ) and x are often monotonic, with ( x ) increasing continuously or ( x ) decreasing continuously as x increase. The S-shaped curves are typical. The most importance curve with this shape has the model formula ( x ) = exp ( + x ) 1 + exp ( + x ) This is the logistic regression model. As x , ( x ) 0 when < 0 and ( x ) when > 5 / 18 Logistic regression model The link function for which logistic regression is a GLM. For the odds are ( x ) 1- ( x ) = exp ( + x ) The log odds has the linear relationship log ( ( x ) 1- ( x ) ) = + x Thus, the appropriate link is the log odds transformation, the logit Logistic regression models are GLMs with binomial random component and logit link function....
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note18 - STAT5044: Regression and Anova Inyoung Kim 1 / 18...

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