Calibration regression outliers

Fig. 13. Unweighted calibration including suspicious regression outliers.

A second approach is to define if the calibration standard falling outside the acceptance criteria is or is not an outlier. All the calibration standards identified statistically as outliers must be rejected and not used in the standard curve regression. 

In a previous section we mentioned that outliers and highly deviating values in a series of measurements are known to have a severe elfect on most tests. In regression models also, the parameters are most sensitive to the response values near the borders of the calibration range. In order to moderate the influence of possible outliers one should try robust techniques. These so-called nonparametric regression statistics start from the common model ... 

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]. 

In multivariate regression the samples are grouped into a defined space much like the clusters in figure 11.3. An outlier or unknown interference (which was not present during calibration) will place the sample outside the region of the calibration samples. By measuring the distance away from the... 

When the laboratory value is plotted against the NIR predicted value for the calibration sample set it may well be noted that some points lie well away from the computed regression line. This will, of course, reduce the correlation between laboratory and NIR data and increase the SEC or SEP. These samples may be outliers. The statistic hi describes the leverage or effect of an individual sample upon a regression. If a particular value of hi is exceeded this may be used to determine an outlier sample. Evaluation criteria for selecting outliers, howevei are somewhat subjective so there is a requirement for expertise in multivariate methods to make outlier selection effective. 

The guidelines (FDA 2001) continue with a discussion of the LLOQ and state The analyte response at the LLOQ should be at least five times the response compared to a blank response , and Analyte peak (response) should be identifiable, discrete and reproducible with a precision of 20% and accuracy of 80-120% . With respect to what types of regressions are acceptable and how residual outliers should be handled in practical apphcation, the guidehnes state The simplest model that adequately describes the concentration-response relationship should be used. Selection of weighting and use of a complex regression equation should be justified. The following condition should be met in developing a calibration curve 20 % deviation of the LLOQ from nominal concentration and 15 % deviation of standards other than the LLOQ from nominal concentration. At least four out of six nonzero standards should meet the... 

In this section we return to a problem already discussed in Chapter 3, the occurrence of outliers in our data. These anomalous results inevitably arise in calibration experiments, just as they occur in replicate measurements, but it is rather harder to deal with them in regression statistics. One difficulty is that, although the individual yj-values in a calibration experiment are assumed to be independent of one another, the residuals (y - pj) are not independent of one another, as their sum is always zero. It is therefore not normally permissible to take the residuals as if they were a conventional set of replicate measurements, and apply (for example) a Q-test to identify any outliers. (If the number of y-values is large, a condition not generally met in analytical work, this prohibition can be relaxed.)... 

From time to time, in practice, measuring values are obtained that deviate significantly from the others, without an obvious reason. However, as a matter of principle, calibration data have to be free of outliers. Outlier tests [32] aid in deciding if the value is part of a homogeneous data material. However, at first a suitable regression model has to be determined, because the validity of the chosen regression model is assumed when applying outlier tests. 

The F statistic can also be useful in recognizing suspected outliers within a calibration sample set if the F value decreases when a sample is deleted, the sample was not an outlier. This situation is the result of the sample not affecting the overall fit of the calibration line to the data while at the same time decreasing the number of samples (N). Conversely, if deleting a single sample increases the overall F for regression, the sample is considered a suspected outlier. F is defined as the mean square for regression divided by the mean square for residual (see statistical terms in this table). Statistic Coefficient of Multiple Determination Abbreviations R or r ... 

The PLSR is a bilinear regression method that extracts a small number of factors, ta, a = 1,2,..., A that are linear combinations of the KX variables, and use these factors as regressors for y. In addition, the X variables themselves are also modeled by these regression factors. Therefore outliers with abnormal spectra x in the calibration set or in future objects can be detected. 

This two-block predictive PLS regression has been found very satisfactory for multivariate calibration and many other types of practical multivariate data analysis. This evaluation is based on a composite quality criterion that includes parsimony, interpretability, and flexibility of the data model lack of unwarranted assumptions wide range of applicability good predictive ability in the mean square error sense computational speed good outlier warnings and an intuitively appealing estimation principle. See, for example. Reference 6, Reference 7, and References 15-17. 

On the basis of the above interpretation of this first regression, a second one was done with two differences (a) The same three wavelengths were forced into the equation (to avoid waiting several more hours for a new calibration) and (b) the outlier (which was flagged in the printout) was removed from the input data. The consequence of deleting this outlier is shown in the following table ... 

As for SVC, optimisation of the SVM parameters for regression is far from simple and, as mentioned, a trade-off is implicit in the use of the loss function. The penalty parameter (C) and e must be considered in order to obtain a robust regression, insensitive to the presence of outliers. However, if non-linear kernel models are used (typically, the RBF) another term, the width of the Gaussian function, must be taken into account. Further, as e defines the radius of the -tube around the regression function it also defines the number of SVs that are finally selected to construct the function. An excessively large value of e results in fewer SVs (more experimental data within the a-tube) and, therefore, the model may over-fit the calibration data. In this respect, Brereton and Loyd stressed how easy it is to obtain over-fitted models that yield errors that are too large when unknowns are to be predicted. More details and practical examples can be found in both references. 

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