Some Effects of Ignoring Correlated Measurement Errors in Straight Line Regression and Prediction

Instructive expressions are developed for expected values of the least squares regression coefficient, sample residual mean squared error, and squared sample correlation coefficient in terms of a general measurement error model allowing for correlated measurement errors. These expressions are useful in assessing the importance of measurement error in given applications, and can be inverted to give simple estimators of measurement error model parameters based on the usual least squares estimates. They show that, in certain circumstances, the correlation of variables contaminated with measurement errors can be greater than that of the uncontaminated variables. These simple expressions should be useful in promoting the use of measurement error models in applications. An example demonstrates that even in some cases where the least squares regression coefficient is affected little by measurement error, valid estimation of the standard error of prediction may require the use of a measurement error model.