Calibration of ¹⁰⁹Cd KXRF systems for in vivo bone lead measurements: The guiding role of the assumptions for least-squares regression in practical problem solving

The use of least-squares regression to probe the level of lead contamination of plaster of Paris standards in the calibration of ¹⁰⁹Cd KXRF systems for bone lead measurement, as well as the use of iteratively reweighted least-squares (IRLS) in the case of violation of the assumptions for ordinary least-squares (OLS), is discussed here. One common violation is non-uniform residual variance, which makes the use of OLS inappropriate due to strong influence of points with large variance on the calibration line and variance of the slope and intercept. Comparison between OLS and IRLS in that case showed that IRLS estimates of the intercept are significantly smaller and more precise than OLS estimates, while a less marked increase in the calibration slope is observed when IRLS is used. Moreover, OLS underestimates bone lead concentrations at low levels of lead exposure and overestimates those concentrations at higher levels. These discrepancies are smaller in magnitude than the measurement uncertainty of conventional systems, except for high concentrations. For the newly developed cloverleaf systems, the suggested differences at bone lead concentrations below 17 ppm are comparable to the minimum detection limit, but are larger than the measurement uncertainty for bone lead concentrations above 60 ppm.