Fitting Straight Lines with Replicated Observations by Linear Regression. III. Weighting Data

Abstract

The purpose of this article is to stress the importance of weighting in fitting straight lines with replicated observations. Nevertheless, single response data are also taken into account. Although the concept of weighting is treated on chemometric texts, a detailed procedure is not given. For this reason the present review covers the information concerning this topic. Ignoring non-constant variance (heterocedasticity) often leads to improper estimation and inference in a statistical model which quantifies a given relationship. There are two main solutions to remedy this problem: transform the data or perform a weighted least-squares regression analysis. Weighting with replication in homocedastic and heterodedastic condition, including transformation depending weights, and normalization of the weights are considered. Weighting of observations, however, presents a more difficult problem that has commonly been recognized. The review covers briefly topics as random errors and noise, modelling the variance as a function of the independent variable and variation of precision with concentration. By transforming variable it is possible to introduce non-linear terms to the mathematical framework of linear regression, in order to improve fit as to satisfy the necessary assumptions such as homocedasticity. However, transformation data, the analysis of variance and summary data analysis will be the subject of a future report. A number of applications concerning the uses of weighting in analytical chemistry and weighted linear regression are given in tabular form. The analytical, pharmaceutical, biochemical and clinical literature has been thoroughly revised.

Keywords:

• least squares method
• replicated observations
• weighting

Notes

*Examples in physical sciences indicate that the constant variance assumption may often be inappropriate.

*s _i ² is the estimate of σ _i ².