Ch5-4 - Outline 5.1 Interpreting Parameters in Logistic...

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Chapter 5. Logistic Regression Deyuan Li School of Management, Fudan University Feb. 28, 2011 1 / 113 Outline 5.1 Interpreting Parameters in Logistic Regression 5.2 Inference for Logistic Regression 5.3 Logit Models with Categorical Predictors 5.4 Multiple Logistic Regression 5.5 Fitting Logistic Regression Models 2 / 113 5.1 Interpreting Parameters in Logistic Regression 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 + β x ) 1+exp( α + β x ) . Equivalently, the log odds, called the logit , has the linear relationship { Insert . ..... } This equates the logit link function to the linear predictor. 3 / 113 5.1.1 Interpreting β : odds, probabilities, and linear approximations The sign of β determines whether π ( x ) is increasing or decreasing as x increases. The | β | determines the rate of climb or descent. β 0: the curve flattens to a horizontal straight line. β : Y is independent of X . For quantitative x with β> 0, the curve for π ( x )hastheshapeo f the cdf of the logistic distribution (recall Section 4.2.5). Since the logistic density is symmetric, π ( x ) approaches 1 at the same rate that it approaches 0. 4 / 113
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β and odds Exponentiating both sides of the logit equation shows that the odds are an exponential function of x : odds = π ( x ) / [1 π ( x )] = exp( α + β x )= e α e β x . Interpretation for the magnitude of β : the odds increase multiplicatively by e β for every 1-unit increase in x ; e β is an odds ratio, i.e., odds at X = x +1 odds at X = x = π ( x +1) / [1 π ( x +1)] π ( x ) / [1 π ( x )] = e α e β ( x +1) e α e β x = e β . β and linear approximation Since π ( x ) changes with x and ∂π ( x ) /∂ x = βπ ( x )[1 π ( x )], the rate of change in π ( x ) per unit change in x varies. This expression also shows that the logistic density is symmetric, i.e., π ( x ) approaches a value v at the same rate that it approaches (1 v ). 5 / 113 A straight line drawn tangent to the logistic regression curve at a particular x value describes the rate of change at that point (see Figure 5.1). FIGURE 5.1 Linear approximation to logistic regression curve. 6 / 113 Example π ( x )s l o p e = ( x )[1 π ( x )] 1 / 2 β/ 4 0 . 9o r0 . 10 . 09 β 1o 0 The steepest slope occurs when π ( x )=1 / 2 odds = 1 logit = 0 α + β x =0 x = α/β. This x value ( α/β ) is sometimes called the median efective level and denoted by EL 50 . In toxicology studies it is called LD 50 (LD = lethal dose), i.e., the dose with a 50% chance of a lethal result. 7 / 113 From this linear approximation, near x where π ( x / 2, a change in x of 1 corresponds to change in π ( x ) of roughly (1 )( 4) = 1 / 4; that is, 1 approximates the distance between x values where π ( x )=0 . 25 or 0 . 75 (in reality, 0 . 27 and 0 . 73) and where π ( x . 50. The linear approximation works better for smaller changes in x .
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Ch5-4 - Outline 5.1 Interpreting Parameters in Logistic...

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