**Factor scores and Scatterplot of the Countries**

Since we have named the component, it is desirable to assign scores to each country to indicate where that country stands on the component. Here scores are indicating the level of social development of the countries. The values of the scores are to be interpreted paying attention to the signs of component loadings. From Figure 5 we say that countries with high

The Basics of Linear Principal Components Analysis 203

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

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).

**Pays Life\_exp Mortality Urban Iliteracy Water Telephone Vehicles Fertility Hosp\_beds Physicians**  Albanie 72.00 25.00 40.00 16.50 76.00 31.00 27.00 2.5 3.2 1.4 Algérie 71.00 35.00 59.00 35.00 90.00 53.00 25.00 3.5 2.1 0.8 Angola 47.00 124.00 33.00 41.00 32.00 6.00 18.00 6.7 1.3 0 Argentine 73.00 19.00 89.00 3.00 65.00 203.00 137.00 2.6 3.3 2.7 Arménie 74.00 15.00 69.00 2.00 99.00 157.00 0 1.3 7.6 3 Australie 79.00 5.00 85.00 3.00 99.00 512.00 488.00 1.8 8.5 2.5 Autriche 78.00 5.00 65.00 2.00 100.00 491.00 481.00 1.3 9.2 2.8 Azerbeidjan 71.00 17.00 57.00 3.00 97.00 89.00 36.00 2 9.7 3.8 Bangladesh 59.00 73.00 23.00 60.00 84.00 3.00 1.00 3.1 0.3 0.2 Bélarus 68.00 11.00 71.00 0.5 100.00 241.00 2.00 1.3 12.2 4.3 Belgique 78.00 6.00 97.00 2.00 100.00 500.00 435.00 1.6 7.2 3.4 Bénin 53.00 87.00 41.00 61.50 50.00 7.00 7.00 5.7 0.2 0.1 Bolivie 62.00 60.00 61.00 15.50 55.00 69.00 32.00 4.1 1.7 1.3 Botswana 46.00 62.00 49.00 24.50 70.00 65.00 15.00 4.2 1.6 0.2 Brésil 67.00 33.00 80.00 16.00 72.00 121.00 88.00 2.3 3.1 1.3 Bulgarie 71.00 14.00 69.00 1.50 99.00 329.00 220.00 1.1 10.6 3.5 Burkina Faso 44.00 104.00 17.00 77.50 42.00 4.00 4.00 6.7 1.4 0 Burundi 42.00 118.00 8.00 54.00 52.00 3.00 2.00 6.2 0.7 0.1 Cambodge 54.00 102.00 15.00 61.50 13.00 2.00 5.00 4.5 2.1 0.1 Cameroun 54.00 77.00 47.00 26.50 41.00 5.00 7.00 5 2.6 0.1 Canada 79.00 5.00 77.00 35.00 99.00 634.00 455.00 1.6 4.2 2.1 Centrafrique 44.00 98.00 40.00 55.50 19.00 3.00 0 4.8 0.9 0.1 Tchad 48.00 99.00 23.00 60.00 24.00 1.00 3.00 6.4 0.7 0 Chili 75.00 10.00 85.00 4.50 85.00 205.00 71.00 2.2 2.7 1.1 Chine 70.00 31.00 31.00 17.00 90.00 70.00 3.00 1.9 2.9 2 Hong Kong 79.00 3.00 100.00 7.50 100.00 558.00 56.00 1.1 1.3 Colombie 70.00 23.00 73.00 9.00 78.00 173.00 21.00 2.7 1.5 1.1 Congo Démocratique 51.00 90.00 30.00 41.00 27.00 - 9.00 6.3 1.4 0.1 Congo 48.00 90.00 61.00 21.50 47.00 8.00 14.00 6 3.4 0.3 Costa Rica 77.00 13.00 47.00 5.00 92.00 172.00 85.00 2.6 1.9 1.4 Cote d'Ivoire 46.00 88.00 45.00 55.50 72.00 12.00 18.00 5 0.8 0.1 Croatie 73.00 8.00 57.00 2.00 63.00 348.00 17.00 1.5 5.9 2 République Tchèque 75.00 5.00 75.00 3.00 97.00 364.00 358.00 1.2 9.2 2.9

**8. Conclusion** 

**9. Appendix** 

we do not recommend them.

**9.1 Data for the case study** 

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 under-developed group.

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.

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

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

$$SI\_i = \frac{F\_i - F\_{\text{min}}}{F\_{\text{max}} - F\_{\text{min}}} \times 100\tag{32}$$

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 underdeveloped countries, whereas higher scores identify socially developed countries.
