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.

A Potentiometric Evaluation of Stability Constants of Two-Step Overlapping Equilibria via a Bilogarithmic Hyperbolic Cosine Method

“…Thus, a representation of the left term of Equation (12) against the term into brackets of the right hand should give a straight line (Y = a 0 + a 1 X), obtained by linear regression [19]- [22], whose slope is the unity and the intercept with the X-axis is equal to ( )…”

Fitting Nonlinear Calibration Curves: No Models Perfect

“…A linear trend (descending or ascending) may indicate that an additional term in the model is needed. The "fan-shaped" residual pattern shows that experimental error increases with mean response (heteroscedasticity) so the constant variance assumption is inappropriate [21]. This last should be approach by weighted least squares method or by transforming the response.…”

“…The residuals are the vertical distances indicated in the y-direction between the points and the regression line (which gives a minimum sum of their squares) [21]. No rigorous mathematical treatment is required.…”

“…Among the various statistical ways of numerically measuring some of the observed discrepancies, to date the most widely used method is still the visual examination of the residual plots because it gives more information in a direct way [21]. The simplest model or the model with the minimum number of parameters that adequately fit the data in question is usually the best choice [25]: "Non sunt multiplicanda entia praetor necessitaten" (Occam's razor).…”

“…So, residuals provide a convenient means of checking whether the calibration data is actually linear [21] [22] [23] [24].…”