# Robust mislabel logistic regression without modeling mislabel probabilities

Robust mislabel logistic regression without modeling mislabel probabilities IntroductionLogistic regression is one of the most widely used statistical methods for linear discriminant analysis. Let Y0 be a binary response with {0,1} values, and X be the p‐dimensional random vector of explanatory variables. Logistic regression assumes P(Y0=1|X=x) to satisfy the conditional label probability model π(x;β)=exp(β⊤x)1+exp(β⊤x), where β=(β1,…,βp)⊤, and let β0 denote the true value of β in model . The MLE is known to be the most efficient estimator for β0 when data are truly generated from model . However, in some situations we can only observe a contaminated label Y instead of the true status Y0. That is, Y is flipped from Y0 according to the mislabel probabilities η0(x)=P(Y=1|Y0=0,X=x) andη1(x)=P(Y=0|Y0=1,X=x). The success probability of Y no longer follows model , but instead has the form P(Y=1|X=x)=η0(x){1−π(x;β)}+{1−η1(x)}π(x;β). Fitting label contaminated data {(Yi,Xi)}i=1n to the uncontaminated model will produce a biased estimate of β0. To overcome the problem of mislabeling, some robustified logistic regression methods are developed based on with different modelings for ηj(x)'s. Copas () considered equal and constant mislabel probabilities, η0(x)=η1(x)=η, which we call the constant‐mislabel logistic regression. For any given η, the estimating equation of β is 1n∑i=1nwη,i(β){Yi−πη(Xi;β)}Xi=0 with πη(x;β)=η{1−π(x;β)}+(1−η)π(x;β) and the weight function wη,i(β)=1−2η{1−η+ηexp(−β⊤Xi)}{1−η+ηexp(β⊤Xi)}. Another example http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Biometrics Wiley

# Robust mislabel logistic regression without modeling mislabel probabilities

, Volume 74 (1) – Jan 1, 2018
10 pages

/lp/wiley/robust-mislabel-logistic-regression-without-modeling-mislabel-Btm2V5Q0RR
Publisher
Wiley Subscription Services, Inc., A Wiley Company
© 2018, The International Biometric Society
ISSN
0006-341X
eISSN
1541-0420
D.O.I.
10.1111/biom.12726
Publisher site
See Article on Publisher Site

### Abstract

IntroductionLogistic regression is one of the most widely used statistical methods for linear discriminant analysis. Let Y0 be a binary response with {0,1} values, and X be the p‐dimensional random vector of explanatory variables. Logistic regression assumes P(Y0=1|X=x) to satisfy the conditional label probability model π(x;β)=exp(β⊤x)1+exp(β⊤x), where β=(β1,…,βp)⊤, and let β0 denote the true value of β in model . The MLE is known to be the most efficient estimator for β0 when data are truly generated from model . However, in some situations we can only observe a contaminated label Y instead of the true status Y0. That is, Y is flipped from Y0 according to the mislabel probabilities η0(x)=P(Y=1|Y0=0,X=x) andη1(x)=P(Y=0|Y0=1,X=x). The success probability of Y no longer follows model , but instead has the form P(Y=1|X=x)=η0(x){1−π(x;β)}+{1−η1(x)}π(x;β). Fitting label contaminated data {(Yi,Xi)}i=1n to the uncontaminated model will produce a biased estimate of β0. To overcome the problem of mislabeling, some robustified logistic regression methods are developed based on with different modelings for ηj(x)'s. Copas () considered equal and constant mislabel probabilities, η0(x)=η1(x)=η, which we call the constant‐mislabel logistic regression. For any given η, the estimating equation of β is 1n∑i=1nwη,i(β){Yi−πη(Xi;β)}Xi=0 with πη(x;β)=η{1−π(x;β)}+(1−η)π(x;β) and the weight function wη,i(β)=1−2η{1−η+ηexp(−β⊤Xi)}{1−η+ηexp(β⊤Xi)}. Another example

### Journal

BiometricsWiley

Published: Jan 1, 2018

Keywords: ; ; ; ;

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