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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 π ( x ) , reflecting its dependence on values x = ( x 1 ,..., x p ) of predictors. The variance of Y is Var ( Y ) = π ( x )( 1 π ( 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 π ( x ) = α + β 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 π ( x ) and x . In practice, nonlinear relationships between π ( x ) and x are often monotonic, with π ( x ) increasing continuously or π ( x ) decreasing continuously as x increase. The Sshaped curves are typical. The most importance curve with this shape has the model formula π ( x ) = exp ( α + β x ) 1 + exp ( α + β x ) This is the logistic regression model. As x → ∞ , π ( x ) → 0 when β < 0 and π ( x ) → ∞ when β > 5 / 18 Logistic regression model The link function for which logistic regression is a GLM. For the odds are π ( x ) 1 π ( x ) = exp ( α + β x ) The log odds has the linear relationship log ( π ( x ) 1 π ( x ) ) = α + β 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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 Fall '11
 Staff
 Binomial, Normal Distribution, Probability theory, Cumulative distribution function, Discrete probability distribution

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