Least-squares fitting with errors in the response and predictor

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 ytrue = β0 + β1 x true, we provide simple expressions for errors in predictors (EIP) estimators \hbox{$\Hat{{\beta}}-{0, {\rm EIP}} $} for β0 and \hbox{$\Hat{{\beta}}-{1, {\rm EIP}} $} for β1 and for an approximation to covariance (\hbox{$\Hat{{\ beta}}-{0, {\rm EIP}} $}, \hbox{$\Hat{{\beta}}-{1, {\rm EIP}} $}). It is assumed that there are measured data x = xtrue + ex, and y = ytrue + ey with errors e x in x and ey in y and the variances of the errors e x and ey are allowed to depend on xtrue and ytrue, respectively. This paper also investigates the accuracy of the estimated cov(\hbox{$\Hat{{\beta}}-{0, {\rm EIP}} $}, \hbox{$\Hat{{\beta}}-{1, {\rm EIP}} $}) 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. © EDP Sciences 2012.