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When regression analyses relate an outcome (y) to an explanatory factor (x), it has long been recognised that even purely random errors (or “noise”) in the measured values of the explanatory factor x systematically weaken the slope of the regression line of y on x, diluting the explanatory power of x (regression dilution). 

It has also long been recognised that, when multiple regression is used to correct for a confounding factor (x), purely random errors in the measured values of the confounding factor x systematically weaken the magnitude of that correction, so some confounding by that particular confounding factor remains (residual confounding), along with any confounding by unmeasured factors.

The formulae used to help quantify these two potentially substantial biases are similar, as both biases depend in the same simple way on r2, where r is the correlation coefficient between the exact values and the noisy values of x.

Sir Richard Peto, FRS, is Emeritus Professor of Medical Statistics and Epidemiology at the University of Oxford. He was made a Fellow of the Royal Society of London in 1989 for introducing meta-analyses of randomised trials, was knighted by Queen Elizabeth in 1999 for services to epidemiology, and received in 2010 and 2011 the Cancer Research UK and the BMJ Lifetime Achievement Award.

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