Unweighted regression methods

After outliers have been purged from the data and a model has been evaluated visually and/or by, e.g. residual plots, the model fit should also be tested by appropriate statistical methods [2, 6, 9, 10, 14], The fit of unweighted regression models (homoscedastic data) can be tested by the ANOVA lack-of-fit test [6, 9]. A detailed discussion of alternative statistical tests for both unweighted and weighted calibration models can be found in Ref. [16]. The widespread practice to evaluate a calibration model via its coefficients of correlation or determination is not acceptable from a statistical point of view [9]. 

It should also be noted that, in contrast to the situation described in the previous paragraph, results can be transformed to produce data that can be treated by unweighted methods. Data of the form y = bx with y-direction errors strongly dependent on X are sometimes subjected to a log-log transformation the errors in log y then vary less seriously with log x, so the transformed data can reasonably be studied by unweighted regression equations. 

Figure 4 shows an example of weighted and unweighted (u j = I) methods of cumulants regressions of PCS data for a Stober [68,69] silica suspension. This example employs a quadratic cumulant (QC) expansion given by... 

Most published method evaluations fail to apply regression analysis in a rigorous fashion. This section considers both the use of OLR instead of Deming regression and the use of unweighted analysis in the setting of proportional random errors. 

According to current practice in method comparison studies, it is usual to apply unweighted forms of regression analysis (i.e., OLR and the Deming procedure), even though the SDs vary with the measured concentration, as occurs... 

However, when the number of replicates is small, as is usually the case, the estimated variance can be quite erroneous and unstable. Nonlinear regression estimates using this approach are more variable than their unweighted least-squares counterparts, unless the number of replicates at each level is 10 or more. For this reason, this method cannot be supported and the danger of unstable variance estimates can be avoided if a parametric residual variance model can be found. 

We must now discuss how the terms y and are obtained in practice when a regression line is used for calibration as described in the preceding sections. A fundamental assumption of the unweighted least-squares method is that each point on the plot (including the point representing the blank or backgroimd) has a normally distributed... 

Because this chapter aims at introducing the basics of unweighted LS regression and at clarifying several difficulties rather frequent in its routine appKcation, it was decided to retain standard nomenclature. Therefore, it has to be considered in the following discussions that the k-mode is the method of choice and that the calibration function will be developed in one way and used in another way (this has been termed frequently the inverse use of standardization ) and this will have consequences in the calculation of several statistics, as will be discussed throughout the sections. 

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