true concentration x = 0). An unbiased sample estimate of (i.e., the variance of deviations from the population regression line) is given by
When variance is not constant, as is typically the case in the calibration setting, then the previous OLS solution for constant or “homoscedastic” errors no longer applies. There are several approaches to this problem, but in general the most widely accepted approach is to model the variance as a function of true concentration x and to then use the estimated variances as weights in estimating the calibration parameters, which are now denoted as and .
The weighted least squares regression of measured concentration or instrument response (y) on true concentration (x) is denoted by
2This section is adapted from Gibbons and Bhaumik, 2001.