Scree test

The number of retained PCs should be assessed by the scree test. 

The last approach is widely used for practical data exploration and yields a biplot in which PC eigenvalues are plotted against PC numbers (Figure 15.12). Usually the PCs retained are those on the slope of the graph before the decrease in eigenvalues levels off to the right of the plot. In the example presented, three PCs can be retained according to the scree test. 

Cattell RB, The scree test for the number of factors, Multivariate Behavioral Research, 1966, 1, 245-276. 

This method is adapted from the scree test. ... 

The scree test is based on the phenomenon of the residual variance leveling off when the proper number of PCs is obtained. Visually, the residuals, or more often the eigenvalues, are plotted against the number of components in a scree plot. The component number is then derived from the leveling off in this dependence. Figure 5.3 demonstrates the scree plot for the hair data. The slope can be seen to change between the second and third components. 

Determination of the Number of Significant Factors To decide on the number of common factors, some criteria have been introduced already in connection with PCA, such as the eigenvalue-one test or the scree test. Those criteria can be overtaken in FA, if the determination of loadings is performed by a PCA. 

If 6 PCs are retained for further evaluation, a residual variance in the experimental values remains beyond consideration. Other methods for estimating a reasonable size of w are the eigenvalue-one criterion [13, 14], the Scree-test [13, 14], and cross validation (cf. Section 22.3.3). 

Scree-test Based on the idea that residual variance... 

Due to the high standard deviations reported above, a cluster analysis was carried out based on 5 variables representing the most important statements towards contracting contr 1-5). Euclidian Distance serves as proximity measure. The optimal number of clusters is first defined using Ward method. A four cluster solution is chosen based on scree test, dendrogram and plausibility considerations. In order to refine this solution in a second step, a K-means cluster analysis is conducted. 

This empirical test is based on the so-called Scree-plot which represents the residual variance as a function of the number of eigenvectors that have been extracted [42]. The residual variance V of the r -th eigenvector is defined by ... 

The scree plots described above are also used in regression models of the type y = Xb + ey where y is decomposed into a model Xb and a residual ey. Plots of the percentage variation explained as a function of the number of PCA orPLS components are used to study the fit of the regression model. With results from cross-validation or a test-set, a similar plot can be used to select the rank of the regression model. 

Similar to the PCA, only the significantly different sensory descriptors were used in the CVA. As CVA is a classification technique, an a priori grouping is needed. We chose the most basic model, and used a MANOVA model with only the sample effect. Bartlett s test for the determination of significant canonical dimensions revealed that only the first CV was significantly different (P<0.05). However, a knee in the scree plot was observed after the first two CVs, thus, the first two dimensions were kept for interpretation (data not shown). 

Using the significantly different elements, a CVA biplot was created and is shown in Fig. 8.4. The Bartlett s test revealed that the first four CVs were significantly different from each other, but in the scree plot a knee was observed after the second CV, thus, only the first two CVs are used for further interpretation (data not shown). 

Several criteria and rules of thumb have been formulated [26,28,46] to answer the question How many PCs In EMDA, criteria based on statistical inference, that is, on formal tests of hypothesis, should be avoided as we do not want to assume, in the model estimation phase, our PCs to follow a specific distribution. In this context, more intuitive criteria, albeit not formal, but simple and working in practice, are preferable, especially graphics-based criteria, such as sequential exploration of scores plots and/or inspection of residuals plots plots of eigenvalues (scree plots [47]) or cumulative variance versus number of components. Different consideration holds when PCA is used to generate data models that are further used, for example, for regression, classification tasks or process monitoring [48,49] (Section 3.1.5), where PCA model validatiOTi, for example, by cross-validation, in terms of performance on the assessment of future samples has to be taken into account. 

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