Least-squares fitting with errors in the response and predictor

¹ Statistical Sciences, Los Alamos National Laboratory, USA
² Safeguards Science and Technology, Los Alamos National Laboratory, USA
³ Department of Physics, Alma College, USA

^⋆ Correspondence: tburr@lanl.gov

Received: 10 August 2011
Accepted: 22 March 2012

Abstract

Least squares regression is commonly used in metrology for calibration and estimation. In regression relating a response y to a predictor x, the predictor x is often measured with error that is ignored in analysis. Practitioners wondering how to proceed when x has non-negligible error face a daunting literature, with a wide range of notation, assumptions, and approaches. For the model y_true = β₀ + β₁   x_true, we provide simple expressions for errors in predictors (EIP) estimators for β₀ and for β₁ and for an approximation to covariance (, ). It is assumed that there are measured data x = x_true + e_x, and y = y_true + e_y with errors e_x in x and e_y in y and the variances of the errors e_x and e_y are allowed to depend on x_true and y_true, respectively. This paper also investigates the accuracy of the estimated cov(, ) and provides a numerical Bayesian alternative using Markov Chain Monte Carlo, which is recommended particularly for small sample sizes where the approximate expression is shown to have lower accuracy than desired.