Calibration of ¹⁰⁹Cd KXRF systems for in vivo bone lead measurements: Weighted least-squares regression with different weighting functions

The use of iteratively reweighted least squares (IRLS) has recently been described as an alternative to ordinary least squares with heteroscedastic data, in the calibration of ¹⁰⁹Cd KXRF systems for in vivo bone lead measurements. This work addresses the use of weighted least squares (WLS) with two different weighting functions and no iteration, with that same data set. The functions are defined as the inverse of the variance of observed ratios of lead to coherent peak amplitudes and the inverse of the square of the error reported by the Marquardt fitting program for these ratios. The results show that if no iteration is implemented when using WLS, then the two weighting functions are highly inefficient in homogenizing the residual variance. Moreover, both methods estimate much more imprecise calibration intercepts and slopes than did the IRLS method. Work is in progress to investigate the implementation of IRLS with these weighting functions, with the focus on the selection of the best function for residuals to be used in each iteration stage.