The correlation between regression coefficients: combined significance testing for calibration and quantitation of bias

Abstract

In the analytical sciences regression methods have two main uses – in calibrations in instrumental analysis, and in testing for bias in method comparison studies. In first order (straight line) regression the true values of the intercept α and the slope β are independent of each other but their estimated values and are not independent. This can be appreciated visually by considering the straight lines joining all the individual pairs of points; those with a large slope will have a small intercept and vice versa, so the correlation between and is negative and possibly substantial. This correlation has important consequences when the estimated coefficients are used for significance testing in the interpretation of the regression line.