Regression plots, outliers detection

One problem which may sometimes be most easily detected using plots of the data is that of detecting "outliers , or "bad" data points. These may have resulted from improper application of experimental techniques, incorrect measurements, or other factors not accounted for in the experimental design. Such data may be excluded from the regression analysis. However, care should be taken to not exclude legitimate data points arising from random variation in a functional property or from variation due to the consistent Influence of variable factors which should have been Included in the analysis (factors the Influence Of which could not have been excluded). 

It should be stressed that LTS regression does not throw away a certain percentage of the data. Instead, it finds a majority lit, which can then be used to detect the actual outliers. The purpose is not to delete and forget the points outside the tolerance band, but to study the residual plot in order to find out more about the data. For instance, we notice the star 7 intermediate between the main sequence and the giants, which might indicate that this star is evolving to its final stage. 

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. 

An outlier is a sample which looks so different from the other that either is not well described by the model or influences the model too much. In regression, there are many ways for a sample to be classified as an outlier. It may be outlying according to the X-variables only, or to the Y-variables only, or to both. It may also not be an outlier for either separate set of variables, but become an outlier when you consider the (X,Y) relationship. Outliers can be detected in the Unscrambler software using X-Y relation outliers, Y-residual vs. predicted Y, the influence plot, the score plots, Y residuals, leverages and normal probability plot. 

For an analysis of the AD of regression models, the author has always used the Williams plot, which is now widely applied by other authors and commercial software. The Williams plot is the plot of standardized cross-validated residuals (R) versus leverage (Hat diagonal) values (h from the HAT matrix). It allows an immediate and simple graphical detection of both the response outliers i.e., compounds with cross-validated standardized residuals greater than 2-3 standard deviation units) and structurally anomalous chemicals in a model (h>h, the critical value being h = 3p /n, where p is the number of model variables plus one, and n is the number of the objects used to calculate the model).40,62,66... 

Fig. 2. A simulation study illustrating the use of model population analysis to detect whether a dataset contains outliers. Plot A and Plot B shows the data simulated without and with outliers, respectively. 1000 linear regression models computed using 1000 sub-datasets randomly selected and the slope and intercept are presented in Plot C and D. Specifically, the distribution of slope for these two simulated datasets are displayed in Plot E and Plot F.

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