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University of Michigan School of Public Health The University of Michigan Department of Biostatistics Working Paper Series

Year  Paper 

Binary isotonic regression procedures, with application to cancer biomarkers

Debashis Ghosh∗ Moulinath Banerjee† Pinaki Biswas‡

∗University of Michigan, [email protected] †University of Michigan, [email protected] ‡Univeristy of Michigan, [email protected] This working paper is hosted by The Berkeley Electronic Press (bepress) and may not be commer- cially reproduced without the permission of the copyright holder. http://biostats.bepress.com/umichbiostat/paper38 Copyright c 2004 by the authors. Binary isotonic regression procedures, with application to cancer biomarkers

Debashis Ghosh, Moulinath Banerjee, and Pinaki Biswas

Abstract

There is a lot of interest in the development and characterization of new biomark- ers for screening large populations for disease. In much of the literature on di- agnostic testing, increased levels of a biomarker correlate with increased disease risk. However, parametric forms are typically used to associate these quantities. In this article, we specify a monotonic relationship between biomarker levels with disease risk. This leads to consideration of a nonparametric regression model for a single biomarker. Estimation results using isotonic regression-type estimators and asymptotic results are given. We also discuss confidence set estimation in this setting and propose three procedures for computing confidence intervals. Meth- ods for estimating the receiver operating characteristic (ROC) curve are also de- scribed. The finite-sample properties of the proposed methods are assessed using simulation studies and applied to data from a pancreatic cancer biomarker study.

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½ˆ Ñ uˆ ÔAwAq ìuˆ Ô:qZ¼ŽD uˆ rAqZÑA¢ Ÿuˆ ÓoÑArD uˆ wA½sµ¢ ìuˆ ÓoroutD uˆ raÔAÓo¢

q ˆ u uˆ ÔAwAq ìuˆ ÔAuowtD uˆ raÔAwo¢ Ÿuˆ ÓAwouD uˆ rAwora¢ ìuˆ ÓowoutD uˆ wouaÑA¢

Ñtˆ u uˆ rouou ìuˆ ÑAwo½tD>soˆ uouaÔA¢ Ÿuˆ ÓAro½D uˆ wtsµÔo¢ ìuˆ ÔAuaԎD uˆ waÑoÔA¢

Óˆ u soˆ uouou ¤soˆ uououtD>soˆ uououo¢ Ÿuˆ r:qarDŠsoˆ>svÑo¼o¢ ìuˆ ÔA½oÓtD>soˆ uououo¢ ¼A½

Hosted by The Berkeley Electronic Press 1.0 0.8 0.6 F(x) 0.4 0.2 0.0

0 2 4 6 8 10 12

x

Á=KmoI Q,5so𕗨_`^ =KQ,=Kf78AJ V =>9,MNQ,=K6 ItMN=K@AL”@Ac 4 Ÿ6 JK8AfdY‰JK=>L 5v¢-Q,57JK8:MN=Šjo5gMN@"MN; 5U@oQ<5EMN=>f€8AJJ>=K_`=>Mb Q<5UVƒJ>=KL 5€¢ ¼:q

http://biostats.bepress.com/umichbiostat/paper38 (a) 0.8 0.4 True Positive Rate 0.0

0.0 0.2 0.4 0.6 0.8 1.0

False Positive Rate

(b) 0.8 0.4 True Positive Rate 0.0

0.0 0.2 0.4 0.6 0.8 1.0

False Positive Rate

Á=KmoI Q,5X¼tðHžg "¡ f7I Q¤jo5H579,M<=K_ƒ8:MN5UVÌI9<=KLm“^G@t@AJŠ„ 8:VeR<8Af75ULaM”jt=>@oJ‡8eMN@oQ,9|8AJKmo@AQ<=>M<; _ Ÿ9<@AJK=KVÂJ>=KL 5€¢

n

8:L VœJK@omA=K9,M<=KfQ<5UmoQ<5U9<9<=>@oL“ŸV @:M@o_†8:Q

n

¡ sµ¼oÑtˆ ¼oÑ

Hosted by The Berkeley Electronic Press (a) 0.8 0.4 True Positive Rate 0.0

0.0 0.2 0.4 0.6 0.8 1.0

False Positive Rate

(b) 0.8 0.4 True Positive Rate 0.0

0.0 0.2 0.4 0.6 0.8 1.0

False Positive Rate

Á=KmoI Q,5{½ðbÁ 5€8€jo5]„'@oL 5]„'@oItM¦f7Q,@o9<9 „˜j:8AJ>=KV8:M<57V–žg "¡ f7I Q¤jo5‰5U9,MN=>_†8eMN57VHI 9<=>L ml^G@t@AJŠ„ 8:VeR<8Af75ULaMjŽ=DŽ

@AJ‡8:M<@oQ<98AJKmA@oQ<=ŠMN; _ŝ 9<@oJ>=KVƒJK=KL5v¢*8AL VlJ>@omo=K9¤MN=>fQ,57moQ,579,9<=K@AL£ŸV @AM,MN5UVœJK=KL5v¢•c @oQ^8:L f7Q,5€8:M<=Kf}f78AL fU57Q

n n

6=K@o_ƒ8AQ

http://biostats.bepress.com/umichbiostat/paper38