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 xtrue,
we provide simple expressions for errors in predictors (EIP) estimators
\hbox{$\Hat{{\beta }}_{0, {\rm EIP}} $}β̂0, EIP for β0 and \hbox{$\Hat{{\beta }}_{1, {\rm EIP}} $}β̂1, EIP for β1 and for an
approximation to covariance
(\hbox{$\Hat{{\beta }}_{0, {\rm EIP}} $}β̂0, EIP,
\hbox{$\Hat{{\beta }}_{1, {\rm EIP}} $}β̂1, EIP). It is assumed that there are measured data
x = xtrue + ex,
and
y = ytrue + ey
with errors ex in x and
ey in y and the
variances of the errors ex and
ey are allowed to depend on
xtrue and ytrue, respectively.
This paper also investigates the accuracy of the estimated cov(\hbox{$\Hat{{\beta }}_{0, {\rm EIP}} $}β̂0, EIP,
\hbox{$\Hat{{\beta }}_{1, {\rm EIP}} $}β̂1, 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.