lect04 - Categorical Data Analysis - Lei Sun 1 CHL 5210 -...

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Unformatted text preview: Categorical Data Analysis - Lei Sun 1 CHL 5210 - Statistical Analysis of Qualitative Data Topic: Logistic Regression Outline Single predictor (2 levels, > 2 levels, quantitative). Parameter interpretation. Inference of the parameter. Multiple predictors (next lecture). Categorical Data Analysis - Lei Sun 2 Introduction to logistic regression (logit model). So far: two-way contingency tables. Most studies: several explanatory variables (categorical and continuous). Goal: describe their effects on response variables, including relevant in- teractions, smoothed estimates of response probabilities. Most important case: logistic regression (logit model), a linear model for the logit transformation of a binomial parameter. Also: log-linear model (Poisson regression model), a linear model for the log of a Poisson mean. We will study the following. Single predictor (2 levels, > 2 levels, quantitative). Parameter interpretation. Inference of the parameter. Multiple predictors. Interaction. Model building (model fit and selection). Categorical Data Analysis - Lei Sun 3 Logistic regression with one covariate with two levels. Background. Y : binary response, outcomes denoted by 0 and 1. e.g. birth weight, normal (0) or low weight (1). X : binary predictor, two levels denoted by 0 and 1. e.g. smoking status, no (0) or yes (1). We are interested in defining E [ Y ] = P ( Y = 1) = ( X ): reflects its dependence on value of the predictor, X . A linear probability model: P ( Y = 1) = ( X ) = + X Major structural defect: probabilities fall between 0 and 1. But linear functions take values over entire real line. Interpretation of : is the change in for a one-unit increase in X . We want to construct our model so that: Predicted value of P ( Y = 1) = ( X ) is bounded between 0 and 1. The regression coefficient is measured on a meaningful scale. We can construct statistical inferences which approximately model the Y i , i = 1 , ..., n , as Bernoulli random variables. Categorical Data Analysis - Lei Sun 4 Consider logit of as a linear function of X . logit ( ( X )) = log ( X ) 1 ( X ) = + X = ( X ) = exp ( + X ) 1 + exp ( + X ) . If = 1 and = 2, ( X ) = + X . ( X ) = exp ( + X ) 1+ exp ( + X ) . ( X ) = exp ( X ) 1+ exp ( X ) ....
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lect04 - Categorical Data Analysis - Lei Sun 1 CHL 5210 -...

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