Calibration and Regression with Nonconstant Error Variance

Ordinary least squares regression analysis is generally inappropriate for calibration and regression problems when the usual assumption ofconstant variance across all observations doesn't hold. Estimators of regression parameters are of relatively poor quality and the resulting inference can be misleading. The use of standard data transformations is a common alternative but may not provide enough flexibility for some cases. The use of weighted regression with weights estimated from replicates is generally unreliable for reasonable sample sizes. However, when the error variance changes systematically with the mean response or other variables, generalized least squares (GLS) and variance function estimation (VFE) methods can be used. The GLS-VFE approach allows the experimenter to specify a model for the systematic change in variance, estimate unknown parameters, and to use this information to provide more efficient estimates of the regression parameters. In this tutorial, GLS-VFE methods are introduced and described in the context of calibration and regression. An example of calibration for a chemical assay is used to motivate discussion and illustrate the implementation of these methods using standard software packages.