Leverage plots

Leverage is discussed in Chapter 7. It is a measure of influence in regression models, but the idea of calculating Squared Mahalanobis distances can also be used for PARAFAC or Tucker loadings. For the peat example (5 x 7 x 21) the Squared Mahalanobis distances hA, hB and hc as they are defined in Chapter 7, may be calculated for the peat types, for the treatments and for the particle sizes. 

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Studentized Concentration Residual vs. Sample Leverage Plot (Sample Diagnostic) This diagnostic is u.sed to more closeh examine individual samples. The sample ie erage for the /th sample is computed by... 

Sample leverage plot (leverage vs. sample number)... 

Studentized concentration residual vs. sample leverage plot... 

The Studentised residuals vs leverage plot is a two-dimensional graph... 

Figure 4.24 Studentised residuals leverage plot to detect outlier samples on the calibration set (a) general rules displayed on a hypothetical case and (b) diagnostics for the data from the worked example (Figure 4.9, mean centred, four factors in the PLS model).

Figure 12.20 The rate of oxidation of pyrite (r = rf(FeS2l/t/< in mo /m s) near 25"C and 1 bar pressure. Whole model and leverage plots for multiple linear regression analysis of published and measured rate data for the aqueous oxidation of pyrite (a) Oxidation of pyrite by dissolved oxygen (b) Oxidation of pyrite by ferric iron under an N2 atmosphere and (c) Oxidation of pyrite by ferric iron in the presence of dissolved oxygen. Reprinted from Geochim. et Cosmochim. Acta, 58, M. A. Williamson and J. D. Rimstidt, The kinetics and electrochemical rate-determining step of aqueous pyrite oxidation, 5443-54, 1994, with permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington 0X5 1GB, U.K.

The design matrix (including actual and model predicted responses) generated for the Box-Behnken study is shown in Table 3.1. Here, three center point experiments were incorporated to compute an estimate of the error term that does not depend on the fitted model. Figure 3.1a shows the whole model leverage plot of actual-versus-predicted responses (based on aU effects) with the quality of fit expressed by the coefficient of determination (r ). This coefficient is variation in the response around the mean that can be attributed to terms in the model rather than to random error. 

FIGURE 3.1. (a) Whole model leverage plot of actual-versus-predicted responses and (b) model generated contour plots showing injection time-versus-capiUary length. (Reprinted with permission from Reference 51.)... 

The order of the topics treated is basically as plots would be used in an ongoing analysis scree plots, line plots, scatter plots, special plots as an aid in understanding the model, residual plots, leverage plots. 

Figure 10 shows the leverage plot of this experimental design. The leverage can be computed at every point of the experimental domain (it depends on the experimental matrix and on the postulated model), and its value. 

FIGURE 10 Leverage plot of the Face-Centred Design reported in Table 12. 

Figure 11 shows the plots of the semi-amplitude of the 95% confidence interval for each of the two responses. These plots, derived from the leverage plot, are anyway much more easily interpretable as they give a direct idea of the uncertainty of the predicted value. So, we can immediately see that for the first response the uncertainty is between about 2 and about 3, while for the second respmise it is in the range of 0.1 to 0.2. It is very important to note that the cmifidence interval changes according to the position in the experimental domain, and that the shape of this surface depends on the distribution of the experiments in the experimental domain. Again, what is important is which experiments are performed, not how many. 

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