**8. Conclusion**

202 Principal Component Analysis

positive scores on the first component demonstrate higher level of social development relatively to countries with negative scores. In Figure 6 we can see that countries such as Burkina Faso, Niger, Sierra Leone, Tchad, Burundi, Centrafrique and Angola belong to the

SPSS does not provide directly the scatterplot for subjects. Since factor scores have been created and saved as variables, we can use the Graph menu to request a scatterplot. This is an easy task on SPSS. The character variable Country is used as an identifier variable. Notice that in SPSS factor scores are standardized with a mean zero and a standard deviation of 1.

A social development index is most useful to identify the groups of countries in connection with their level of development. The construction of this index assigns a social

> min max min 100 *<sup>i</sup>*

where Fmin and Fmax are the minimum and maximum values of the factor scores *F*. Using the rescaled-scores, countries are sorted in ascending. Lower scores identify socially under-

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development-ranking score to each country. We rescale factor scores as follows:

developed countries, whereas higher scores identify socially developed countries.

*i F F SI F F*

under-developed group.

Fig. 6. **Scatterplot of the Countries**

Principal components analysis (PCA) is widely used in statistical multivariate data analysis. It is extremely useful when we expect variables to be correlated to each other and want to reduce them to a lesser number of factors. However, we encounter situations where variables are non linearly related to each other. In such cases, PCA would fail to reduce the dimension of the variables. On the other hand, PCA suffers from the fact each principal component is a linear combination of all the original variables and the loadings are typically nonzero. This makes it often difficult to interpret the derived components. Rotation techniques are commonly used to help practitioners to interpret principal components, but we do not recommend them.

Recently, other new methods of data analysis have been developed to generalize linear PCA. These include Sparse Principal Components Analysis (Tibshirani, 1996; Zou, Hastie & Tibshirani, 2006), Independent Component Analysis (Vasilescu & Terzopoulos, 2007), Kernel Principal Components Analysis (Schölkopf, Smola & Müller, 1997, 1998), and Multilinear Principal Components Analysis (Haiping, Plataniotis & Venetsanopoulos, 2008).
