Leverage points

The several variants deriving from the items 1 to 4 are represented in the flow sheet given in Fig. 6.6. Common calibration by Gaussian least squares estimation (OLS) can only be applied if the measured values are independent and normal-distributed, free from outliers and leverage points and are characterized by homoscedastic errors. Additionally, the error of the values in the analytical quantity x (measurand) must be negligible compared with the errors of the measured values y. 

If the basic conditions for the use of least squares fitting are not fulfilled (Fig. 6.6), especially if strongly deviating calibration points appear ( outliers or, more exactly, leverage points), the OLS method fails, i.e., the estimated calibration are biased and, therefore, are not representative for the relation between x and y. Whereas normality of the measured values can be frequently obtained by a suitable transformation, especially in the case of outlying calibration points, robust calibration has to be applied (Rousseeuw and Leroy [1987] Danzer [1989] Danzer and Currie [1998]). 

Robust calibration corresponds in most cases to the problem of outlying calibration points (leverage points). In consideration of that, attention must be directed to the linearity of the relationship in general and the randomness of the residuals. 

SD, larger than this cutoff value are leverage points (good or bad leverage points, depending on the orthogonal distance). 

Outliers may heavily influence the result of PCA. Diagnostic plots help to find outliers (leverage points and orthogonal outliers) falling outside the hyper-ellipsoid which defines the PCA model. Essential is the use of robust methods that are tolerant against deviations from multivariate normal distributions. 

Chatterjee S, Hadi AS (1986) Influential observations, high leverage points, and outliers in linear regression. Stat Sci 1(3) 379—416... 

Figure 6.5 illustrates this outlier map on the stars data. We see that star 9 is a vertical outlier because it only has an outlying residual. Observation 14 is a good leverage point it has an outlying surface temperature, but it still follows the linear... 

Note that the most commonly used diagnostics to flag leverage points have traditionally been the diagonal elements / of the hat matrix H = X(X7X) X f These are equivalent to the Mahalanobis distances M/Xx,) because of the monotone relation... 

The breakdown value of all regression M-, L-, and R-estimators is 0% because of their vulnerability to bad leverage points. If leverage points cannot occur, as in fixed-design studies, a positive breakdown value can be attained [33],... 

Vertical outliers and bad leverage points highly influence the least-squares estimates in multivariate regression, and they can make the results completely unreliable. Therefore, robust alternatives have been developed. 

To detect leverage points and vertical outliers, the outlier map can be extended to multivariate regression. Then the final robust distances of the residuals, ResD, (Equation 6.14) are plotted vs. the robust distances RD(x ) of the xl (Equation 6.6). This yields the classification as given in Table 6.2. 

Figure 6.6 shows the outlier map of the Shell foam data. Observations 215 and 110 lie far from both the horizontal cutoff line at [Pg.185]

Small OD Regular observation Good PCA-leverage point... 

Next, we can construct outlier maps as in Sections 6.5.5 and 6.4.2.3. ROBPCA yields the PC A outlier map displayed in Figure 6.12a. We see that there are no PCA leverage points, but there are some orthogonal outliers, the largest being 23, 7, and 20. The result of the regression step is shown in Figure 6.12b. It exposes the robust distances of the residuals (or the standardized residuals if q = 1) vs. the score... 

---