**Eigenvalues and number of meaningful components**

Table 9 displays the eigenvalues, percent of variance and cumulative percent of variance from the observed data. Earlier it was stated that the number of components computed is equal to the number of variables being analyzed, necessitating that we decide how many components are truly meaningful and worthy of being retained for interpretation.

Here only component 1 demonstrates an eigenvalue greater than 1.00. So the Kaiser eigenvalue-one criterion would lead us to retain and interpret only this component. The first component provides a reasonable summary of the data, accounting for about 72% of the total variance of the 10 variables. Subsequent components each contribute less than 8%.


Table 9. **Eigenvalues**

The Basics of Linear Principal Components Analysis 201

1 2

Component

Life\_exp .911 -.268 Mortality -.926 .287 Urbanisation .809 -.093 Iliteracy -.848 .200 Water .850 -.139 Telephone .862 .355 Vehicles .780 .483 Fertility -.911 .183 Hosp\_beds .713 .396 Physicians .850 .087

Table 10. **Component Matrix**

Fig. 5. **Scatterplot of variables** 

**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 scree plot is displayed in Figure 4. From the second component on, we observe that the line is almost flat with a relatively large break following component 1. So the scree test would lead us to retain only the first component. The components appearing after the break (2-10) would be regarded as trivial (less than 10%).

#### Fig. 4. **Scree Plot**

In conclusion, the dimensionality of the data could be reduced to 1. Nevertheless, we shall add the second component for representation purpose. Plot in a plane is easier to interpret than a three or 10-dimensional plot. Note that by default SPSS uses the Kaiser criterion to extract components. It belongs to the user to specify the number of components to be extracted if the Kaiser-criterion under-estimate the appropriate number. Here we specified 2 as the number of components to be extracted.

### **Component loadings**

Table 10 displays the loading matrix. The entries in this matrix are correlations between the variables and the components. As can be seen, all the variables load heavily on the first component. It is now necessary to turn to the content of the variables being analyzed in order to decide how this component should be named. What common construct do variables seem to be measuring?

In Figure 5 we observe two opposite groups of variables. The right-side variables are positively correlated one with another, and deal with social status of the countries. The leftside variables are also positively correlated one with another, and talk about another aspect of social life. It is therefore appropriate to name the first component the "social development" component.

200 Principal Component Analysis

The scree plot is displayed in Figure 4. From the second component on, we observe that the line is almost flat with a relatively large break following component 1. So the scree test would lead us to retain only the first component. The components appearing after the break

In conclusion, the dimensionality of the data could be reduced to 1. Nevertheless, we shall add the second component for representation purpose. Plot in a plane is easier to interpret than a three or 10-dimensional plot. Note that by default SPSS uses the Kaiser criterion to extract components. It belongs to the user to specify the number of components to be extracted if the Kaiser-criterion under-estimate the appropriate number. Here we specified 2

Table 10 displays the loading matrix. The entries in this matrix are correlations between the variables and the components. As can be seen, all the variables load heavily on the first component. It is now necessary to turn to the content of the variables being analyzed in order to decide how this component should be named. What common construct do

In Figure 5 we observe two opposite groups of variables. The right-side variables are positively correlated one with another, and deal with social status of the countries. The leftside variables are also positively correlated one with another, and talk about another aspect of social life. It is therefore appropriate to name the first component the "social

(2-10) would be regarded as trivial (less than 10%).

Fig. 4. **Scree Plot**

**Component loadings** 

variables seem to be measuring?

development" component.

as the number of components to be extracted.


Table 10. **Component Matrix**
