# 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) = π ( X ): reflects its dependence on value of the predictor, X . – A linear probability model: P ( Y = 1) = π ( X ) = α + βX ∗ Major structural defect: probabilities fall between 0 and 1. But linear functions take values over entire real line. ∗ Interpretation of β : is the change in π for a one-unit increase in X . – We want to construct our model so that: ∗ Predicted value of P ( Y = 1) = π ( X ) is bounded between 0 and 1. ∗ The regression coefficient β 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 π as a linear function of X . logit ( π ( X )) = log π ( X ) 1 − π ( X ) = α + βX = ⇒ π ( X ) = exp ( α + βX ) 1 + exp ( α + βX ) . If α = 1 and β = 2, ∗ π ( X ) = α + βX . ∗ π ( X ) = exp ( α + βX ) 1+ exp ( α + βX ) . ∗ π ( X ) = exp ( α − βX ) 1+ exp ( α − βX ) . ••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • ••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••...
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lect04 - Categorical Data Analysis Lei Sun 1 CHL 5210...

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