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# 52 esl chapter 4 linear methods for classication name

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Unformatted text preview: ake them all; up to ﬁve times that number of controls is sufﬁcient. See next slide 47 ESL Chapter 4 — Linear Methods for Classiﬁcation Trevor Hastie and Rob Tibshirani Prospective vs retrospective sampling Exposure=0 Exposure=1 control (0) p00 p01 case (1) p10 p11 • β in logistic regression estimates prospective odds ratio ORp = p11 /p01 p10 /p00 But this equals the retrospective odds ratio: p11 /p10 ORr = p01 /p00 So logistic regression can be applied to either sampling scheme. 48 ESL Chapter 4 — Linear Methods for Classiﬁcation Trevor Hastie and Rob Tibshirani 0.07 0.06 0.05 0.04 Coefficient Variance 0.08 0.09 Simulation Theoretical 2 4 6 8 10 12 14 Sampling more controls than cases reduces the variance of the parameter estimates. But after a ratio of about 5 to 1 the variance reduction ﬂattens out. Control/Case Ratio 49 ESL Chapter 4 — Linear Methods for Classiﬁcation Trevor Hastie and Rob Tibshirani Risk estimates and classiﬁcation ˆ • We can estimate the risk for a new observation x0 via η (x0 ) = xT β ˆ 0 ˆ ˆ ˆ and Pr(Y = 1|X = x0 ) = eη(x0 ) /(1 + eη(x0 ) ). • To obtain a 95% conﬁdence interval for Pr(Y = 1|X = x0 ), we ﬁrst ˆ obtain one for η (x0 ) (using the estimated covariance of β ). We then apply the sigmoid transformation to the lower and upper values. ˆ • To classify a new observation, we threshold Pr(Y = 1|X = x0 ) at 0.5. Other thresholds change the sensitivity and speciﬁcity, and are used to construct ROC curves. 50 ESL Chapter 4 — Linear Methods for Classiﬁcation Trevor Hastie and Rob Tibshirani “Receiver Operating Characteristics” or ROC curve 0.6 0.4 0.0 0.2 True positive rate 0.8 1.0 ROC Curve 0.0 0.2 0.4 0.6 0.8 1.0 False positive rate ˆ Sensitivity = P (G = 1|Y = 1) = True positive rate. ˆ Speciﬁcity = P (G = 0|Y = 0) = 1 - False positive rate. 51 ESL Chapter 4 — Linear Methods for Classiﬁcation Trevor Hastie and Rob Tibshirani Predicted Class − or Null + or Non-null Total True − or Null True Neg. (TN) False Pos. (FP) N Class + or N...
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