STA 4107/5107
Chapter 3
Factor Analysis
1
Key Terms
Please review and learn these terms.
2
What is Factor Analysis?
Factor analysis is an interdependence technique (see chapter 1) that primarily uses metric
variables, however, nonmetric variables can be included as “dummy variables. The basic idea
is to describe the set of
P
variables
X
1
, X
2
,
· · ·
, X
P
in our data as linear combinations of a
smaller number of
factors
, and in the process get a better understanding of the relationships
between our variables.
Factor analysis is based on a statistical model, unlike principal
components (which we will cover later). Mathematically, factor analysis can be represented
thusly:
x
1
=
l
11
F
1
+
l
12
F
2
+
· · ·
+
l
1
m
F
m
+
²
1
x
2
=
l
21
F
1
+
l
22
F
2
+
· · ·
+
l
2
m
F
m
+
²
2
.
.
.
x
p
=
l
p
1
F
1
+
l
p
2
F
2
+
· · ·
+
l
pm
F
m
+
²
p
Here, the
X
i
are the original variables, the
l
ij
are called the
loadings
and the
F
j
are the
common factors
, and the
²
i
are the
unique factors
. In factor analysis, the loadings or weights
are chosen to maximize the correlation between the variable and the factor.
Our motive for studying factor analysis in this course is that we will extend the concept
to
structural equation modeling
, which is also called
confirmatory factor analysis
near the
end of the term.
3
Some History and Examples
Factor analysis was developed at the turn of the 20
th
century by psychologist Charles Spear
man, who hypothesized that a person’s score on a wide variety of tests of mental ability–
mathematical skill, vocabulary, other verbal skills, artistic skills, logical reasoning ability,
etc.–could all be explained by one underlying “factor” of general intelligence that he called
g
. From a data set of test scores of boys in preparatory school, he noticed that any two rows
in the table of correlations were approximately proportional across the different variables. So
for example,
Classics
and
English
in the table below have the ratios
.
83
.
67
≈
.
70
.
64
≈
.
66
.
54
≈
.
63
.
51
.
Upon observing this, he hypothesized that it was due to a common factor,
g
.
1
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Classics
French
English
Mathematics
Pitch
Music
Classics
1.0
.83
.78
.70
.66
.63
French
.83
1.0
.67
.67
.65
.57
English
.78
.67
1.0
.64
.54
.51
Mathematics
.70
.67
.64
1.0
.45
.51
Pitch
.66
.65
.54
.45
1.0
.40
Music
.63
.57
.51
.51
.40
1.0
It was an interesting idea, but it turned out to be wrong. Today the College Board testing
service operates a system based on the idea that there are at least three important factors of
mental ability–verbal, mathematical, and logical abilities–and most psychologists agree that
many other factors could be identified as well.
3.1
Examples
1. Consider various measures of the activity of the autonomic nervous system–heart rate,
blood pressure, etc. Psychologists have wanted to know whether, except for random
fluctuation, all those measures move up and down together–the “activation” hypothesis.
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 Spring '08
 Staff

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