The effects of errors in measuring the independent variable in least-squares regression analysis

It is well known that linear regression analysis gives unbiased estimates of the slope and intercept of a straight line if the dependent variable y is subject to random errors of measurement while the independent variable x is not. It is much less well known that, if x is also subject to random errors of measurement, linear regression analysis yields an underestimate of the slope and a correspondingly biased estimate of the intercept. These errors cannot be removed by weighted regression. Similar errors arise in non-linear regression when the independent variable is afflicted by random errors of measurement.