factor analysis
Chapter 14
Factor analysis
14.1
INTRODUCTION
Factor analysis is a method for investigating whether a number of variables of interest Y1 , Y2 , : : :, Yl, are linearly related to a smaller number of unobservable factors F1, F2, : : :, Fk .
The fact that the factors are not observable disquali¯es regression and other methods previously examined. We shall see, however, that under certain conditions the hypothesized factor model has certain implications, and these implications in turn can be tested against the observations. Exactly what these conditions and implications are, and how the model can be tested, must be explained with some care.
14.2
AN EXAMPLE
Factor analysis is best explained in the context of a simple example. Students entering a certain MBA program must take three required courses in
¯nance, marketing and business policy. Let Y1, Y2 , and Y3 , respectively, represent a student's grades in these courses. The available data consist of the grades of ¯ve students (in a 10-point numerical scale above the passing mark), as shown in Table 14.1.
Table 14.1
Student grades
Student
no.
1
2
3
4
5
Finance, Y1
3
7
10
3
10
Grade in:
Marketing, Y2
6
3
9
9
6
Policy, Y3
5
3
8
7
5
°Peter Tryfos, 1997. This version printed: 14-3-2001. c 2
Chapter 14: Factor analysis
It has been suggested that these grades are functions of two underlying factors, F1 and F2, tentatively and rather loosely described as quantitative ability and verbal ability, respectively. It is assumed that each Y variable is linearly related to the two factors, as follows:
Y1 = ¯10 + ¯11 F1 + ¯ 12 F2 + e1
Y2 = ¯20 + ¯21 F1 + ¯ 22 F2 + e2
Y3 = ¯30 + ¯31 F1 + ¯ 32 F2 + e3
(14:1)
The error terms e1, e2, and e3 , serve to indicate that the hypothesized relationships are not exact.
In the special vocabulary of factor analysis, the parameters ¯ij are referred to as loadings. For example, ¯12 is called the loading of variable
Factor analysis
14.1
INTRODUCTION
Factor analysis is a method for investigating whether a number of variables of interest Y1 , Y2 , : : :, Yl, are linearly related to a smaller number of unobservable factors F1, F2, : : :, Fk .
The fact that the factors are not observable disquali¯es regression and other methods previously examined. We shall see, however, that under certain conditions the hypothesized factor model has certain implications, and these implications in turn can be tested against the observations. Exactly what these conditions and implications are, and how the model can be tested, must be explained with some care.
14.2
AN EXAMPLE
Factor analysis is best explained in the context of a simple example. Students entering a certain MBA program must take three required courses in
¯nance, marketing and business policy. Let Y1, Y2 , and Y3 , respectively, represent a student's grades in these courses. The available data consist of the grades of ¯ve students (in a 10-point numerical scale above the passing mark), as shown in Table 14.1.
Table 14.1
Student grades
Student
no.
1
2
3
4
5
Finance, Y1
3
7
10
3
10
Grade in:
Marketing, Y2
6
3
9
9
6
Policy, Y3
5
3
8
7
5
°Peter Tryfos, 1997. This version printed: 14-3-2001. c 2
Chapter 14: Factor analysis
It has been suggested that these grades are functions of two underlying factors, F1 and F2, tentatively and rather loosely described as quantitative ability and verbal ability, respectively. It is assumed that each Y variable is linearly related to the two factors, as follows:
Y1 = ¯10 + ¯11 F1 + ¯ 12 F2 + e1
Y2 = ¯20 + ¯21 F1 + ¯ 22 F2 + e2
Y3 = ¯30 + ¯31 F1 + ¯ 32 F2 + e3
(14:1)
The error terms e1, e2, and e3 , serve to indicate that the hypothesized relationships are not exact.
In the special vocabulary of factor analysis, the parameters ¯ij are referred to as loadings. For example, ¯12 is called the loading of variable