The FACTOR Procedure

This example analyzes socioeconomic data provided by Harman (1976). The five variables represent total population (Population), median school years (School), total employment (Employment), miscellaneous professional services (Services), and median house value (HouseValue). Each observation represents one of twelve census tracts in the Los Angeles Standard Metropolitan Statistical Area.

You conduct a principal component analysis by using the following statements:

data SocioEconomics;
   input Population School Employment Services HouseValue;
   datalines;
5700     12.8      2500      270       25000
1000     10.9      600       10        10000
3400     8.8       1000      10        9000
3800     13.6      1700      140       25000
4000     12.8      1600      140       25000
8200     8.3       2600      60        12000
1200     11.4      400       10        16000
9100     11.5      3300      60        14000
9900     12.5      3400      180       18000
9600     13.7      3600      390       25000
9600     9.6       3300      80        12000
9400     11.4      4000      100       13000
;

proc factor data=SocioEconomics simple corr;
run;

You begin with the specification of the raw data set with 12 observations. Then you use the DATA= option in the PROC FACTOR statement to specify the data set in the analysis. You also set the SIMPLE and CORR options for additional output results, which are shown in Output 35.1.2 and Output 35.1.3, respectively.

By default, PROC FACTOR assumes that all initial communalities are 1, which is the case for the current principal component analysis. If you intend to find common factors instead, use the PRIORS= option or the PRIORS statement to set initial communalities to values less than 1, which results in extracting the principal factors rather than the principal components. See Example 35.2 for the specification of a principal factor analysis.

For the current principal component analysis, the first output table is displayed in the Output 35.1.1.

In Output 35.1.1, the input data type is shown to be raw data. PROC FACTOR also accepts other data type such as correlations and covariances. See Example 35.4 for the use of correlations as input data. For the current raw data set, PROC FACTOR reads in 12 records and all these 12 records are used. When there are missing values in the data set, these two numbers might not match due to the dropping of the records with missing values. The last row of the table shows that 12 is used in the significance tests conducted in the analysis.

The SIMPLE option specified in the PROC FACTOR statement generates the means and standard deviations of all observed variables in the analysis, as shown in Output 35.1.2.

The ranges of means and standard deviations for the analysis are quite large. Variables are measured on quite different scales. However, this is not an issue because PROC FACTOR basically analyzes the standardized scales (that is, the correlations) of the variables.

The CORR option specified in the PROC FACTOR statement generates the output of the observed correlations in Output 35.1.3.

The correlation matrix shown in Output 35.1.3 is analyzed by PROC FACTOR.

The first step of principal component analysis is to look at the eigenvalues of the correlation matrix. The larger eigenvalues are extracted first. Because there are five observed variables, five eigenvalues can be extracted, as shown in Output 35.1.4.

In Output 35.1.4, the two largest eigenvalues are 2.8733 and 1.7967, which together account for 93.4% of the standardized variance. Thus, the first two principal components provide an adequate summary of the data for most purposes. Three components, which explain 97.7% of the variation, should be sufficient for almost any application. PROC FACTOR retains the first two components on the basis of the eigenvalues-greater-than-one rule since the third eigenvalue is only 0.2148.

To express the observed variables as functions of the components (or factors, in general), you consult the factor loading matrix as shown in Output 35.1.5.

The factor pattern is often referred to as the factor loading matrix in factor analysis. The elements in the loading matrix are called factor loadings. There are at least two ways you can interpret these factor loadings. First, you can use this table to express the observed variables as functions of the extracted factors (or components, as in the current analysis). Each row of the factor loadings tells you the linear combination of the factor or component scores that would yield the expected value of the associated variable. Second, you can interpret each loading as a correlation between an observed variable and a factor or component, provided that the factor solution is an orthogonal one (that is, factors are uncorrelated), such as the current initial factor solution. Hence, the factor loadings indicate how strongly the variables and the factors or components are related.

In Output 35.1.5, the first component (labeled “Factor1”) has large positive loadings for all five variables. Its correlation with Services (0.9324) is especially high. The second component is basically a contrast of Population (0.8064) and Employment (0.7261) against School (–0.5448) and HouseValue (–0.5582), with a very small loading on Services (–0.1043).

The total variance explained by the two components are shown in Output 35.1.6.

The first and second component account for 2.8733 and 1.7967, respectively, of the total variance of 5. In the initial factor solution, the total variance explained by the factors or components are the same as the eigenvalues extracted. (Compare the total variance with the eigenvalues shown in Output 35.1.4.) Due to the dropping of the less important components, the sum of these two numbers is 4.6700, which is only a little bit less than total variance 5 of the original correlation matrix.

You can also look at the variance explained by the two components for each observed variables in Output 35.1.7.

In Output 35.1.7, the final communality estimates show that all the variables are well accounted for by the two components, with final communality estimates ranging from 0.8802 for Services to 0.9878 for Population. The sum of the communalities is 4.6700, which is the same as the sum of the variance explained by the two components, as shown in Output 35.1.6.