﻿Simultaneous Confidence Bands in Linear Regression-大连理工大学数学科学学院（新）
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# Simultaneous Confidence Bands in Linear Regression

2018-03-20

Title: Simultaneous Confidence Bands in Linear Regression

Reporter: Prof. HAN Yang  (The University of Manchester)

Time: March 25, 2018 (Sunday)  PM 3:00-3:40

Location: 213 room, Comprehensive Teaching Building 1

Abstract: Construction of simultaneous confidence bands for a percentile line has been considered by several authors. However only conservative symmetric bands, which use critical constants over the whole covariate range (-∞,∞), are available in the literature. Our new methods allow the construction of exact symmetric bands for a percentile line over a finite interval of the covariate x. The exact symmetric bands can be substantially narrower than the corresponding conservative symmetric bands. Several exact symmetric confidence bands are compared under the average band-width criterion. Furthermore, new asymmetric confidence bands for a percentile line are proposed. They are uniformly (and can be very substantially) narrower than the corresponding exact symmetric bands. Therefore, asymmetric bands should always be used under the average band-width criterion. The proposed methods are illustrated with a real example of a drug stability study.

Many modern medicines are targeted therapies, targeting specific pathways. A biomarker that is informative of how sick a patient in the targeted pathway is could be sufficiently predictive of the effect on the patients to allow such medicine to be personalized. Baseline HbA1c for diabetic patients is an example of such potential biomarkers. If a candidate biomarker is continuously valued, it is typically dichotomized to classify patients into target (marker-positive) and non-target (marker-negative) subgroups. The question, for each potential cut-point, is therefore whether the drug has sufficient efficacy in the overall population, or only in the marker-positive patients, or neither. This question can be fully answered by providing simultaneous confidence intervals on the effect of the drug on the marker-positive patients, on the marker-negative patients, and on their mixture. To confidently decide whether a continuously-valued biomarker is useful for targeting patients, such simultaneous confidence intervals need to be further adjusted for the multiplicity of searching through all possible cut-point values. This presentation gives, for continuously-valued outcome measures such as reduction in HbA1c for Type II Diabetes, a neat method providing exact (fully) multiplicity-adjusted simultaneously confidence intervals. An app is also available, which conveniently implements the method we propose.