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Unformatted text preview: ’ & ST3241 Categorical Data Analysis I Logistic Regression An Introduction and Some Examples 1 ’ & Example Applications • Business Applications – The probability that a subject pays a bill on time may use predictors such as the size of the bill, annual income, occupation, mortgage and debt obligations, percentage of bills paid on time in the past, and other aspects of an applicants credit history. – A company that relies on catalog sales may determine whether to send a catalog to a potential customer by modeling the probability of a sale as a function of indices of past buying behavior . • Genetics – Modeling the probability that an offspring inherits an allele of one type instead of another type as a function of phenotypic values on various traits for that offspring. – To model the probability that affected siblings pairs (ASP) have identitybydescent allele sharing and tested its heterogeneity among the research centers. 2 ’ & A Simple Logistic Regression Model • For a binary response variable Y and an explanatory variable X , let π ( x ) = P ( Y = 1  X = x ) = 1 P ( Y = 0  X = x ) . • The logistic regression model is π ( x ) = exp( α + βx ) 1 + exp( α + βx ) • Equivalently, the log odds, called the logit , has the linear relationship logit [ π ( x )] = log π ( x ) 1 π ( x ) = α + βx • This equates the logit link function to the linear predictor. 3 ’ & Interpretation of Parameters • The parameter β determines the rate of increase or decrease of the Sshaped curve. • The sign of β indicates whether the curve ascends or descends. • The rate of change increases as  β  increases. • When the model holds with β = 0 , then π ( x ) is identical at all x , so the curve becomes a horizontal straight line, and Y is then independent of X . 4 ’ & Interpretation of Parameters • Liner Approximation Interpretation 5 ’ & Linear Approximation Interpretations • A straight line drawn tangent to the curve at a particular x value, describes the rate of change at that point. • For logistic regression parameter β , that line has slope equal to βπ ( x )[1 π ( x )] . • For example, the line tangent to the curve at x for which π ( x ) = 0 . 5 has slope β (0 . 5)(0 . 5) = 0 . 25 β . • By contrast when π ( x ) = 0 . 9 or . 1 , it has slope . 09 β 6 ’ & Linear Approximation Interpretations • The slope approaches as the probability approaches 1 . or . • The steepest slope of the curve occurs at x for which π ( x ) = 0 . 5 ; that x value is x = α/β . • This value of x is sometimes called the median effective level and is denoted by EL 50 . • It represents the level at which each outcome has a 50% chance....
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