See next slide 47 esl chapter 4 linear methods for

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Unformatted text preview: o oo oooooo o ooooo o o oooooo oo ooo o ooooooo o o oooooo o o oo ooo oo o ooo ooo o o oooo ooo ooooo o ooo o oo oo o o ooo oo o o ooo oo ooooo ooo o ooooooo o o oooo o oo o ooo o oooooo o oo oo ooooo oo oo o oo o o oooooo o oo oooo o o oooo o o o oo ooooo o ooooooo oo o age 20 0 50 100 0.0 0.4 famhist 20 40 60 A scatterplot matrix of the South African heart disease data. Each plot shows a pair of risk factors, and the cases (160) and controls (302) are color coded (red is a case). The variable famhist (family history of heart disease) is binary (yes or no). 44 ESL Chapter 4 — Linear Methods for Classification Trevor Hastie and Rob Tibshirani Results from a logistic regression fit to the South African heart disease data. Coefficient Standard Error Z Score (Intercept) −4.130 0.964 −4.285 sbp 0.006 0.006 1.023 tobacco 0.080 0.026 3.034 ldl 0.185 0.057 3.219 famhist 0.939 0.225 4.178 obesity -0.035 0.029 −1.187 alcohol 0.001 0.004 0.136 age 0.043 0.010 4.184 45 ESL Chapter 4 — Linear Methods for Classification Trevor Hastie and Rob Tibshirani Model building ˆ • Deviance: dev(y, p) = −2 (β ) ˆ • H0 : First q components of β are non-zero • H1 : β is unrestricted • under H0 , dev(y, p0 ) − dev(y, p1 ) ∼ χ2−q asymptotically (as ˆ ˆ p N → ∞) • “Chi-square statistic”—quadratic approximation to deviance: n n wi (zi − i=1 ˆ xT β ) i = i=1 (yi − pi )2 ˆ pi (1 − pi ) ˆ ˆ ˆ • β ∼ N (β, (XT WX)−1 ) asymptotically, if the model is correct. 46 ESL Chapter 4 — Linear Methods for Classification Trevor Hastie and Rob Tibshirani Case-control sampling and logistic regression • In South African data, there are 160 cases, 302 controls — π = 0.35 ˜ are cases. Yet the prevalence of MI in this region is π = 0.05. • With case-control samples, we can estimate the regression parameters βj accurately; the constant term β0 is incorrect. • We can correct the estimated intercept by a simple transformation ˆ∗ ˆ β0 = β0 + log π ˜ π − log 1−π 1−π ˜ • Often cases are rare and we t...
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