UNDERSTANDING
FACTOR ANALYSIS *
By R.J. Rummel
Note for Rummel web site visitors: Many of the
statistical analyses on this web site use factor analysis to dimensionalize
data or to uncover underlying causes or factors. A number of these are
consolidated in the "Dimensions of
Democide, Power, Violence, and Nations" part of
the site. This article (a summary of Rummel's Applied Factor Analysis, 1970) may therefore be helpful to those
who would like to understand better this method in association with the
democide and conflict results presented here, or to apply it themselves.
INTRODUCTION
Thousands of variables
have been proposed to explain or describe the complex variety and
interconnections of social and international relations. Perhaps an equal number
of hypotheses and theories linking these variables have been suggested.1
The few basic variables
and propositions central to understanding remain to be determined. The systematic
dependencies and correlations among these variables have been charted only
roughly, if at all, and many, if not most, can be measured only on
presence-absence or rank order scales. And to take the data on any one variable
at face value is to beg questions of validity, reliability, and comparability.
Confronted with
entangled behavior, unknown interdependencies, masses of qualitative and
quantitative variables, and bad data, many social scientists are turning toward
factor analysis to uncover major social and international patterns.2 Factor analysis can simultaneously manage
over a hundred variables, compensate for random error and invalidity, and
disentangle complex interrelationships into their major and distinct
regularities.
Factor analysis is not
without cost, however. It is mathematically complicated and entails diverse and
numerous considerations in application. Its technical vocabulary includes
strange terms such as eigenvalues,
rotate, simple structure, orthogonal, loadings, and communality. Its results usually absorb a dozen or so pages in a
given report, leaving little room for a methodological introduction or
explanation of terms. Add to this the fact that students do not ordinarily
learn factor analysis in their formal training, and the sum is the major cost
of factor analysis: most laymen, social scientists, and policy-makers find the
nature and significance of the results incomprehensible.
The problem of
communicating factor analysis is especially crucial for peace research.
Scholars in this field are drawn from many disciplines and professions, and few
of them are acquainted with the method. As our empirical knowledge of conflict
processes, behavior, conditions, and patterns become increasingly expressed in
factor analytic terms, those who need this knowledge most in order to make
informed policy decisions may be those who are most deterred by the packaging.
Indeed, they are unlikely to know that this knowledge exists.3
A conceptual map,
therefore, is needed to guide the consumers of findings in conflict and
international relations through the terminological obstacles and quantitative
obstructions presented by factor studies. The aim of this paper is to help draw
such a map. Specifically, the aim is to enhance the understanding and
utilization of the results of factor analysis. Instead of describing how to
apply factor analysis or discussing the mathematical model involved, I shall
try to clarify the technical paraphernalia which may conceal important
substantive data, propositions, or scientific laws.
By way of orientation,
the first section of this paper will present a brief conceptual review of
factor analysis. In the second section the scientific context of the
method will be discussed. The major uses of factor analysis will be listed and
its relation to induction and deduction, description and inference, causation
and explanation, and classification and theory will be considered. To aid
understanding, the third section will outline the geometrical and
algebraic factor models, and the fourth section will define the factor matrices
and their elements--the vehicles for presenting factor results. Since
comprehending factor rotation is important for interpreting the findings, the fifth and final section is devoted to
clarifying its significance.
A bibliography of factor
analysis texts and applications to conflict and international relations is
given in an appendix.
1. CONCEPTUAL OVERVIEW
Factor analysis is a
means by which the regularity and order in phenomena can be discerned. As
phenomena co-occur in space or in time, they are patterned; as these
co-occurring phenomena are independent of each other, there are a number of
distinct patterns. Patterned phenomena are the essence of workaday concepts
such as "table," "chair," and "house," and--at a
less trivial level--patterns structure our scientific theories and hypotheses.
We associate a pattern of attitudes, for example, with businessmen and another
pattern with farmers. "Economic development" assumes a pattern of
characteristics, as does the concept of "communist political system."
The notion of conflict itself embodies a pattern of elements, i.e., two or more
parties and a perception of mutually exclusive or contradictory values or
goals. And to mention phenomena that everyone talks about, weather also has its
patterns.
What factor analysis does
is this: it takes thousands and potentially millions of measurements and
qualitative observations and resolves them into distinct patterns of
occurrence. It makes explicit and more precise the building of fact-linkages
going on continuously in the human mind.
Let us look at a
concrete example. Table 1 presents information on fourteen nations
for ten characteristics. The nations are selected to reflect major regional,
political, economic, and cultural groupings; the characteristics reflect
different facets of each nation, including domestic instability and foreign
conflict. The table thus contains 14 X 10, or 140 pieces of information for
1955. Factor analysis addresses itself to this question: "What are the
patterns of relationship among these data?"
These patterns can be
viewed from two perspectives. One can look at the pattern of variation of nations across their characteristics, and then
group the nations by their profile similarity. One might group together nations
which are all high on GNP per capita, low on trade, high on power, etc. When
applied to discern patterns of profile similarity of individuals, groups, or
nations, the analysis is called Q-factor analysis.4
The regularity in the
data of Table 1 can be looked at from a second
perspective, however. The focus now is the patterns of variation of characteristics. In Table 1, for example, nations high on GNP per
capita also appear low on trade and power. There is a regularity, therefore, in
the nation values on these three characteristics, and this regularity is
described as a pattern of variation. Many of our social concepts define such
patterns. For example, the concept of 11 economic development" involves
(among other things) GNP per capita, literacy, urbanization, education, and
communication; it is a pattern because these characteristics are highly
intercorrelated with each other. Factor analysis applied to delineate patterns
of variation in characteristics is called R-factor analysis.5
What actual patterns of
characteristics are revealed for the data in Table 1 by factor analysis? Figure 1 displays the four major kinds of
regularity in the interrelationships between the characteristics: power, US voting agreement, foreign
conflict, and international law. They
involve, respectively, 27.6, 21.0, 16.2, and 15.3 percent of the variation6 in the 140 pieces of information in Table
1; added together, these
patterns indicate that 80.1 percent of this information has an underlying
regularity.
Each pattern in Figure
1 is laid out in three
isobars. The central isobar includes characteristics with at least 75 percent
of their variation involved in the pattern. These are most central to interpreting
the pattern. The two remaining isobars define characteristics related to the
pattern in the range of 50-74 percent and 25-49 percent of their variation,
respectively. These groups of isobars
show
· what patterns exist in the data and how
they overlap,
· what characteristics are involved in what
pattern and to what degree, and
· what characteristics are involved in more
than one pattern.
To display another
perspective, Figure 2 plots these four patterns as profiles for
the nations in Table 1. On the horizontal axis, nations are
ordered from low to high power pattern values. Magnitudes on the vertical axis
are in standard scores, which is to say that the average score is zero and 95.5
percent of the fourteen nations will (if normally distributed) fall between
scores of +2.00 and -2.00; 68.3 percent of them will fall between scores of
+1.00 and -1.00. Each pattern has a different shape, which illustrates what is
meant by saying that factor analysis divides the regularity in the data into
its distinct patterns. If each of the ten characteristics in Table
1 were plotted as was done
for the patterns in Figure 2, and those characteristics with
similarly-shaped plots were grouped together, there would be four major groups,
and the modal plot within each group would correspond to each of the patterns
shown. Figure 1 and Figure 2 are alternative representations of the results of factoring Table 1.
2. FACTOR ANALYSIS AND
SCIENTIFIC METHOD
Factor analysis can be
applied in order to explore a content area, structure a domain, map unknown
concepts, classify or reduce data, illuminate causal nexuses, screen or
transform data, define relationships, test hypotheses, formulate theories,
control variables, or make inferences. Our consideration of these various
overlapping usages will be related to several aspects of scientific method:
induction and deduction; description and inference; causation, explanation, and
classification; and theory.
2.1 Uses Of Factor Analysis
This section will
outline factor analysis applications relevant to various scientific and policy
concerns. Many of the uses described below overlap. My aim is not to avoid
redundancy but explicitly to relate factor analysis to the diverse interests of
readers.
Interdependency and pattern delineation. If a scientist has a table of data--say,
UN votes, personality characteristics, or answers to a questionnaire--and if he
suspects that these data are interrelated in a complex fashion, then factor
analysis may be used to untangle the linear relationships into their separate
patterns. Each pattern will appear as a factor delineating a distinct cluster
of interrelated data.
Parsimony or data reduction. Factor analysis can be useful for reducing
a mass of information to an economical description. For example, data on fifty
characteristics for 300 nations are unwieldy to handle, descriptively or
analytically. The management, analysis, and understanding of such data are
facilitated by reducing them to their common factor patterns. These factors
concentrate and index the dispersed information in the original data and can
therefore replace the fifty characteristics without much loss of information.
Nations can be more easily discussed and compared on economic development,
size, and politics dimensions, for example, than on the hundreds of
characteristics each dimension involves.
Structure. Factor analysis may be employed to
discover the basic structure of a domain. As a case in point, a scientist may
want to uncover the primary independent lines or dimensions--such as size,
leadership, and age--of variation in group characteristics and behavior. Data
collected on a large sample of groups and factor analyzed can help disclose
this structure.
Classification or description. Factor analysis is a tool for developing
an empirical typology.7 It
can be used to group interdependent variables into descriptive categories, such
as ideology, revolution, liberal voting, and authoritarianism. It can be used
to classify nation profiles into types with similar characteristics or
behavior. Or it can be used on data matrices of a transaction type or a
social-choice type to show how individuals, social groups, or nations cluster
on their transactions with or choices of each other.
Scaling. A scientist often wishes to develop a scale on which individuals, groups,
or nations can be rated and compared. The scale may refer to such phenomena as
political participation, voting behavior, or conflict. A problem in developing
a scale is to weight the characteristics being combined. Factor analysis offers
a solution by dividing the characteristics into independent sources of
variation (factors). Each factor then represents a scale based on the empirical
relationships among the characteristics. As additional findings, the factor
analysis will give the weights to employ for each characteristic when combining
them into the scales. The factor score results (see Section 4.5 below) are actually such scales,
developed by summing characteristics times these weights.
Hypothesis testing. Hypotheses abound regarding dimensions of
attitude, personality, group, social behavior, voting, and conflict. Since the
meaning usually associated with "dimension" is that of a cluster or
group of highly intercorrelated characteristics or behavior, factor analysis
may be used to test for their empirical existence. Which characteristics or
behavior should, by theory, be related to which dimensions can be postulated in
advance and statistical tests of significance can be applied to the factor
analysis results.
Besides those relating
to dimensions, there are other kinds of hypotheses that may be tested. To
illustrate: if the concern is with a relationship between economic development
and instability, holding other things
constant, a factor analysis can be done of economic and instability
variables along with other variables that may affect (hide, mediate, depress)
their relationship. The resulting factors can be so defined (rotated) that the
first several factors involve the mediating measures (to the maximum allowed by
the empirical relationships). A remaining independent factor can be calculated
to best define the postulated relationships between the economic and
instability measures. The magnitude of involvement of both variables in this
pattern enables the scientist to see whether an economic
development-instability pattern actually exists when other things are held
constant.
Data transformation. Factor analysis can be used to transform
data to meet the assumptions of other techniques. For instance, application of
the multiple regression technique assumes (if tests of significance are to be
applied to the regression coefficients) that predictors--the so-called
independent variables--are statistically unrelated (Ezekiel and Fox, 1959, pp.
283-84). If the predictor variables are correlated in violation of the
assumption, factor analysis can be employed to reduce them to a smaller set of uncorrelated factor scores. The scores
may be used in the regression analysis in place of the original variables, with
the knowledge that the meaningful variation in the original data has not been
lost.8 Likewise, a large number of dependent
variables also can be reduced through factor analysis.
Exploration. In a new domain of scientific interest
like peace research, the complex interrelations of phenomena have undergone
little systematic investigation. The unknown domain may be explored through
factor analysis. It can reduce complex interrelationships to a relatively
simple linear expression and it can uncover unsuspected, perhaps startling,
relationships. Usually the social scientist is unable to manipulate variables
in a laboratory but must deal with the manifold complexity of behaviors in
their social setting. Factor analysis thus fulfills some functions of the
laboratory and enables the scientist to untangle interrelationships, to
separate different sources of variation, and to partial out or control for
undesirable influences on the variables of concern.9
Mapping. Besides facilitating exploration, factor analysis also enables a
scientist to map the social terrain. By mapping I mean the systematic attempt
to chart major empirical concepts and sources of variation. These concepts may
then be used to describe a domain or to serve as inputs to further research.
Some social domains, such as international relations, family life, and public
administration, have yet to be charted. In some other areas, however, such as
personality, abilities, attitudes, and cognitive meaning, considerable mapping
has been done.
Theory. As will be discussed in Section 2.5 below, the analytic framework of social
theories or models can be built from the geometric or algebraic structure of
factor analysis.
2.2 Induction And Deduction
The use of
"and" rather than "versus" in the headings of this and the
following subsection emphasizes that these different ways of interpreting or
using factor analysis are not mutually exclusive. They are different sides of
the same coin. The side evident in a particular set of results depends upon the
interpretation.
Factor analysis is most
familiar to researchers as an exploratory
tool for unearthing the basic empirical concepts in a field of
investigation. Representing patterns of relationship between phenomena, these
basic concepts may corroborate the reality of prevailing concepts or may be so
new and strange as to defy immediate labeling. Factor analysis is often used to
discover such concepts reflecting unsuspected influences at work in a domain.
The delineation of these interrelated phenomena enables generalizations to be
made and hypotheses posed about the underlying influences bringing about the
relationships. For example, if a political scientist were to factor the
attributes and votes of legislators and were to find a pattern involving urban
constituencies and liberal votes, he could use this finding to develop a theory
linking urbanism and liberalism. The ability to relate data in a meaningful fashion is a prime aspect of induction
and, for this, factor analysis is useful and efficient.
Factor analysis may also
be employed deductively, in two ways. One way is to elaborate the geometric or
algebraic structure of factor analysis as part of a theory. Within the theory
the factor analysis model can then be
used to arrive at deductions about phenomena. This approach is described more
fully in Section 2.5 below.
The second deductive
approach is to hypothesize the
existence of particular dimensions and then to factor analyze the data to see
whether these dimensions emerge.10 Although factor analysis is not often
used this way, the restraint is not due to methodology but to research
tradition. If, as an example, scholars believe that ideology, power, and trade
are the primary patterns of international behavior, then this proposition can
be tested. Data can be collected on those variables that index international
relations in its greatest diversity, and those specific variables
distinguishing (by theory) the ideology, power, and trade patterns should be
defined. To test whether these patterns actually exist is the factor analysis
task.
2.3 Description And Inference
A data matrix alone may
be of primary interest. Research is then centered on describing the regularities in these data. Statistical problems
like the type of underlying frequency distribution, sample size, and randomness
of selection are not part (and need not be part) of the research design. As
cases in point, all roll call votes in a UN General Assembly session can be
analyzed to describe the voting patterns of nations for that session, as did
Alker (1964), or the voting blocs into which nations were grouped, as did
Russett (1966).
Description may be only
an intermediate goal, however. The ultimate goal may be to connect a number of
descriptive studies to make generalizations
about what patterns exist for such phenomena as, say, legislative voting,
foreign conflict, political systems, personality, or role behavior.11 Although generalization from a number of
descriptive studies is a form of inference, it need not be statistical
inference in the sense that some statistical test of significance is applied.
In fact, factor analysis is seldom employed for statistical inference, although
many social scientists consider it a statistical method. The statistical
requirement of a representative sample is usually met by the research design,
but the additional statistical assumptions such as a normal frequency distribution
are seldom satisfied. Indeed, the canonical factor model (Section 3 below) which has been formulated to allow
statistical inference is seldom used, and tests of significance for factor
loadings are virtually unknown in the applied literature.
Description, then, and
generalization from a number of descriptive studies have been the tradition in
applied factor analysis. Although tests of significance can be determined for
the factors and loadings of a particular sample, factor analysis itself does
not require such tests.12 Factor analysis is a mathematical tool as
is the calculus, and not a statistical technique like the chi-square, the
analysis of variance, or sequential analysis.
2.4 Causation, Explanation, and Classification
The idea of
"cause" has had a strange fascination for scientists. Books and
scholarly papers have been devoted to just the meaning and usage of the term.13 No wonder, then, that the relationship
between causation and factor analysis has been controversial in the factor
analysis literature. The issue centers on whether a factor pattern represents a
causal nexus.14
Modem science conceives
of causation as a temporal regularity of phenomena or, more precisely, a
functional (mathematical) relationship between phenomena. The term
"cause" is then simply an expression of uniform relationships, that
is, of a generally observed concurrence or concomitance of phenomena. Even
though this interpretation drops out interesting connotations like "to
bring about," or "to influence," it removes a fuzziness from the
concept and gives it a denotation consonant with scientific method and
philosophy.
Does factor analysis
define factors, then, that can be called causes of the patterns they represent?
The answer must be yes.15 Each of the variables analyzed is
mathematically related to the factor patterns. The regularities in the
phenomena are described by these patterns, and it is these regularities that
indicate a causal nexus. just as the pattern of alignment of steel filings near
a magnet can be described by the concept of magnetism, for example, so the
concept of magnetism can be turned around and be said to cause the alignment.
Likewise, an economic development pattern delineated by factor analysis can be
called a cause. In this sense, a gregarious personality factor causes certain
attitudes, a turmoil factor causes riots, and an urbanism factor causes liberal
voting.
The term explanation adds nothing to the term cause. Although laden in the social
sciences with a surplus meaning associated with verstehen, a feeling of understanding or getting the sense of
something,16 the explanation of phenomena is nothing
more than being able to predict or
mathematically relate phenomena. To explain an event is to be able to
predict it (see Hempel, 1965, Chapter 12 and, for contrast, Hanson, 1959). To
explain that the Roman Empire fell because of disunity and moral decay is to
say that, given the presence of these two elements in an empire with the
characteristics of the Roman Empire, the empire will break up or be conquered.
Prediction itself is
based on the identification of causal relations, i.e., regularity. Therefore,
if a factor can be called a cause, it can be called an explanation.
If one wants to avoid
controversy over causation, on the other band, factor patterns may be treated
as purely descriptive or classificatory. A factor name like "turmoil"
will then be a noun describing phenomena sharing one characteristic: appearance
in time or space with a certain uniformity. "House,"
"horse," "social group," "legislature," and
"nation" are such nouns, and factors may be conceived likewise.
"Economic development" or "size," as factors actually
delineated through factor analysis (Rummel, 1972), can be descriptive
categories subsuming a pattern of telephones per capita, GNP per capita, and
vehicles per capita as distinct from a pattern of population, area, and
national income.
2.5 Theory
The aim of science is
theory. Facts or data are meaningless in themselves. They must be linked
through propositions which confer meaning. Were we unable to perceive such
relationships, our capacity to manipulate, process, or understand facts would be
overwhelmed. Relationships recurring with high probability become scientific
laws that may be incorporated into a theory covering the domain in which they
are applicable.
A scientific theory
consists of two components:17 analytic and empirical. The analytic
component is the linking of symbolic statements through chains of reasoning
that obey logical or mathematical rules but that have little or no
operational-empirical content. The symbols involved may refer to line, atom,
dimension, force, power (mechanical or social), group, or ideology. Statements
involving these symbols may be associated through verbal reasoning, symbolic
logic, or mathematics. Whatever the symbols or mode of reasoning, this analytic
component of theories can be the creation of the scientist's imagination, the
distillation of a scholar's experience with the subject matter, or a tediously
built structure slowly erected on a foundation of numerous experiments,
investigation, and findings.
The empirical component
of theories is operational. It fastens the abstract analytic part of a theory
to the facts. While the analytic part need have no empirical interpretation,
the empirical component must verifiably link to data for a theory to apply to
"reality."
A confusion between the
empirical and analytic parts of a theory may have militated against a more
theoretical use of factor analysis. The geometric or algebraic nature of the
factor model can structure the analytic framework of theory. The factors
themselves can be postulated. From them, operational deductions with empirical
content can be derived and tested.18
The factor model
represents a mathematical formalism departing from the calculus functions of
classical physics. The analytic part of the factor model is akin to that of
quantum theory.19 Vectors and their position, linear
operators, and the dimensions (factors) of a system are the focus of concern.
Since factor analysis
incorporates analytic possibilities as a theory and empirical techniques for
connecting the theory to social phenomena, its potentiality promises much
theoretical development for the social sciences. Looking ahead for a century, I
suggest that factor analysis and the complementary multiple regression model
are initiating a scientific revolution in the social sciences as profound and
far-reaching as that initiated by the development of the calculus in physics.
3. THE FACTOR MODEL
In application, there
are not one but several factor models which differ in significant respects. A
model most often applied in psychology is called common factor analysis. Indeed, psychologists usually reserve the
term "factor analysis" for just this model. Common factor analysis is
concerned with defining the patterns of common variation among a set of
variables. Variation unique to a variable is ignored. In contrast, another
factor model called component factor
analysis is concerned with patterning all the variation in a set of
variables, whether common or unique.
Other factor models are
image analysis, canonical analysis, and alpha analysis. Image analysis has the
same purpose as common factor analysis, but more elegant mathematical
properties. Canonical analysis defines common factors for a sample of cases
that are the best estimates of those for the population; it enables tests of
significance. Alpha analysis defines common factors for a sample of variables
that are the best estimates of those in a universe of content.
It would be beyond the
purpose of this paper to discuss these models in any detail. (For such a
discussion, see Rummel, 1970, Chapter 5.) In the following sections only their
general mathematical properties will be outlined. These properties clearly
distinguish the factor analysis models from others used in the social sciences,
such as analysis of variance and multiple regression, and justify our
consideration of these properties in reference to a generalized factor model.
3.1 Geometric Model
Now, in this space each
characteristic can be considered a point located according to its value for
each nation. Such a plot is shown in Figure 3 for the GNP per capita and trade values
of the US, UK, and USSR. To make the plot explicit, projections for each point
are drawn as dotted lines to each axis.20
If for each point in Figure
3 we draw a line from the
origin to the point and top the line off with an arrowhead as shown in Figure
4, then we have a vector representation of the data. The
ten characteristics of Table 1 similarly plotted as vectors in an
imaginary space of the fourteen nations (dimensions) would describe a vector space. In this space, consider
two vectors representing any two of these characteristics for the fourteen
nations.
The angle between these vectors measures the relationship between the
two characteristics for the fourteen nations. The closer to 90o the angle is, the less the relationship is. If
two vectors are at a right angle, the characteristics they represent are
uncorrelated: they have no relationship to each other. In other words, some
nations will be high on one characteristic, say GNP per capita, and low on the
other, say trade; some nations will be low on GNP per capita and high on trade;
some nations will be high on both, and some will be low on both. No regularity
exists in their covariation.
The closer the angle
between the vectors is to zero, the stronger the relationship between the
characteristics. An angle of zero means that nations high or low on one
characteristic are proportionately high or low on the other. Obtuse angles mean
a negative relationship. At the extreme, an angle of 180o between two vectors means that the two
characteristics are inversely related: a nation high on one characteristic is
proportionately low on the other.21
Let the ten
characteristics of Table 1 be projected in the fourteen-dimensional
space defined by the fourteen nations as suggested in Figure 5(a). The configuration of vectors will then reflect the data
interrelationships. Characteristics that are highly interrelated will cluster
together; characteristics that are unrelated will be at right angles to each
other. By inspecting the configuration we can discern the distinct clusters of
vectors (if such clusters exist), and these
clusters index the patterns of relationship in the data: each cluster is a
pattern.
Were we dealing with
characteristics of two or three nations, patterns could be found by simply
plotting the characteristics as vectors. What factor analysis does geometrically is this: it enables the
clusters of vectors to be defined when the number of cases (dimensions) exceeds
our graphical limit of three. Each factor
delineated by factor analysis defines a distinct cluster of vectors.22
Consider Figure
5(a) again. Factor
analysis would mathematically lay out such a plot and then project an axis through
each cluster as shown in Figure 5(b). This is analogous to giving each
vector point in a cluster a mass of one and letting the factor axes fall
through their center of gravity.23 The projection
of each vector point on the factor axes defines the clusters. These
projections are called loadings and
the factor axes are often called factors or
dimensions.
Figure 5(c) pictures the power and foreign
conflict patterns of Table 1. For simplicity, the configuration of
points is shown, rather than vectors, and the two factor axes are indicated (as
actually derived from a factor analysis). The loadings of each characteristic
(i.e., each point in space) on each axis are also displayed. This figure may
clarify how factor loadings as a set of numbers can define
·
a pattern of relationships
and
· the association of each characteristic
with each pattern.
We will consider this
geometrical perspective again when the factor matrices are described.
3.2 Algebraic Model
For the more
symbolically oriented reader it may be helpful to present the algebraic model
involved. (Others may wish to skip this Section.)
A traditional approach
to expressing relationships is to establish the mathematical function f(X, W,
Z) connecting one variable, Y, with the set of variables X, W, and Z. Such a
function might be Y = 2X + 3Z - 2W, or Y = 4XW/Z. The variables on both the
right and the left side of the equation are known, data are available, and it
is only a question of determining the best function for describing the
relationships.24
Let us say, however,
that we have a number of variables, Y1, Y2, Y3, and so on, but that we know neither the
variables to enter in on the right side of the equation nor the functions
involved. This might be the situation with UN voting, for example. We may know
the votes of nations on one roll-call (Y1), a second roll-call (Y2), etc., but not know what nation characteristics are
related to what roll-calls in what way. Moreover, we may not be able to measure
well the characteristics, like nationalism, ideology, and democracy, that we
feel might be most related to UN voting. In other words, we have data that we
wish to explain mathematically but the variables that would give us this
explanation are unknown or unmeasureable. We are then in the same dilemma the
nuclear physicist was in decades ago in describing quantum phenomena; and like
him, we resort to an untraditional mathematical approach.25
Let us assume that our Y
variables are related to a number of functions operating linearly. That is,
Equation 1:
Y1 =
11F1 +
12F2 + . . . +
1mFm,
Y2 =
21F1 +
22F2 + . . . +
2mFm,
Y3 =
31F1 +
32F2 + . . . +
3mFm,
Yn =
n1F1 +
n2F2 + . . . +
nmFm,
where:
Y = a variable with known data
F = a function, f ( ) of some unknown variables.
It is crucial in
understanding factor analysis to remember that F stands for a function of variables and not a variable. For
example, the functions might be F 1 = XW +
2Z, and F2 = 3X2Z/W1/2. The unknown variables entering into each
function, F, of Equation 1 are related in unknown ways, although the
equations relating the functions themselves
are linear.25 To take our UN voting example again, two
functions, F, related to voting behavior may be ideology and nationalism. But
each of these functions itself may be the result of a complex interaction between socioeconomic and
political variables.
Within this algebraic
perspective, what does factor analysis do? By application to the known data on
the Y variables, factor analysis defines
the unknown F functions. The loadings emerging from a factor analysis are
the a constants. The factors are the F functions. The size of each loading for
each factor measures how much that specific function is related to Y. For any
of the Y variables of Equation 1 we may write
Equation 2:
Y =
1F1 +
2F2 +
3F3 + . . . +
mFm,
with the F's representing factors and the
's representing loadings.
We may find that some of
the F functions are common to several variables. These are called group factors and their delineation is often
the goal of factor analysis. For UN voting with each Y variable being a UN
roll-call, for example, Alker and Russett (1965) found
"supernationalism" and "cold war" as group factors, among
others, related to voting.
Besides determining the
loadings,
,
factor analysis will also generate data (scores) for each case (individual,
group, or nation) on each of the F functions uncovered. These derived values
for each case are called factor scores. They,
along with the data on Y and Equation 1, give a mathematical relationship among
data as useful and important as the classical equations like Y = 2X + 3Z
Let us look at the data of
Table
1 in the context of this
Section. The table lists data on ten variables representing characteristics of
fourteen nations. A factor analysis of these data brought out four functions,
F, as linearly related to two or more variables. These results enable us to
give content to Equation 1. Leaving out those functions, F, that are
multiplied by small or near-zero loadings,
,
the findings are:
When the results are
arranged in this fashion the patterns of relationship are well brought out; a
pattern is now defined as a number of variables similarly related to the same F
function.
4. INTERPRETING FACTOR
TABLES
Factor results are
usually displayed in one or more tables. These tables consolidate more
information than the length of a research report may allow to be discussed or
highlighted. When a factor analysis is reported for, say, fifty variables for
ninety nations, none but the results most salient to the purpose of the
analysis can be evaluated. Often this only consists of describing the distinct
patterns that have been found. The reader, however, may have other interests.
He may wish to know how a particular variable (say, GNP per capita or riots)
relates to these patterns; how two particular variables (say, trade and mail)
interrelate; or how two nations (say, France and Britain) compare on their
pattern profiles. This Section, therefore, will describe the format and aspects
of typical tables containing factor results, so that the reader may interpret
those findings of most concern to him.
4.1 Correlation Matrix
The most often employed
techniques of factor analysis--centroid and principal axis--are applied to a
matrix of correlation coefficients among all the variables. The matrix is
analogous to a between-city mileage table, except that for cities we substitute
variables, and for mileage we have a coefficient of correlation. Such a matrix
for the data in Table 1 is shown in Table 2.
The full correlation
matrix involved in the factor analysis is usually shown if the number of
variables analyzed is not overly large. Often, however, the matrix is presented
without comment. The factor analysis and not the correlation matrix is the aim,
and it is on the factors that the discussion will focus. Nevertheless, the
correlation matrix contains much useful knowledge and the reader can peruse it
for relationships between pairs of variables (see Understanding Correlation for the meaning and nature of correlation
coefficients). Specifically, the correlation matrix has the following features.
·
The
coefficients of correlation express the degree of linear relationship between the
row and column variables of the matrix. The closer to zero the coefficient, the
less the relationship; the closer to one, the greater the relationship. A
negative sign indicates that the variables are inversely related.27
· To interpret the coefficient, square it
and multiply by 100. This will give the percent
variation in common for the data on the two variables. Thus, in Table
2, the correlation of .36
between GNP per capita and foreign conflict means that .362 X 100 = 13 percent of the variation of the
fourteen nations in Table 1 on these two characteristics is in common.
In other words, if one knows the nation values on one of the two variables one
can produce (predict, account for, generate, or explain) 13 percent of the
values on the other variable. Consider the correlation of .62 between GNP per
capita and stability as another example. This correlation implies that 38.4
percent (.622 X 100) of the stability of these fourteen
nations can be predicted from their GNP per capita. Assuming that the sample of
nations is random, if a fifteenth nation were randomly added to the sample and
only its GNP per capita were known, then its foreign conflict could be
predicted within 13 percent and its stability within 38.4 percent of the true
value.
· The correlation coefficient between two
variables is the cosine of the angle between the variables as vectors plotted
on the cases (coordinate axes). Thus, the correlation of .93 between GNP per
capita and trade in Table 2 can be interpreted as a cosine of .93 (an
angle of 21.3o)
for the two vectors plotted on the fourteen-nation coordinate axis. (This
assumes that the data are standardized.) Section 3.1, above, discusses the geometry of this
interpretation.
· In Table 2 the principal diagonal of the correlation
matrix is indicated in italics. The principal diagonal usually contains the
correlation of a variable within itself, which is always 1.0. Often, however,
when the correlation matrix is to be factored (using the common factor analysis
model), the principal diagonal will contain communality
estimates instead. These measure the variation of a variable in common with
all the others together.
One estimate commonly
employed for the communality measure is the squared
multiple correlation coefficient (SMC) of one variable with all the others.
The SMC multiplied by 100 measures the percent of variation that can be
produced (predicted, accounted for, generated, or explained) for one variable
from all the others. To refer to our example again: Table 2 has SMC values in the principal diagonal.
For foreign conflict this is .61. This means that 61 percent of the foreign
conflict data in Table 2 can be predicted from (is dependent upon)
data on the remaining nine characteristics. By knowing a nation's data on the
nine characteristics we could determine the incidence of foreign conflict
behavior for that nation within 61 percent of the true value, on the average.
With an understanding of
the key interpretations just given, the reader should be able to consult a
correlation matrix and test a number of hypotheses and theories. Many of our
social hypotheses involve relations between two variables, and it is in the
correlation matrix that such empirical relations are described.
4.2 Unrotated Factor Matrix
Two different factor
matrices are often displayed in a report on a factor analysis. The first is the
unrotated factor matrix; it is usually given without comment. The second is the
rotated factor matrix; it is generally the object of interpretation. The
rotated matrix will be considered in Section 4.3.
Figure 6 displays the format of an unrotated
factor matrix. The columns define the factors; the rows refer to variables. In
the intersection of row and column is given the loading for the row variable on
the column factor. The h2 column on the right of
the table, and the rows beneath the table for total and common variance and
eigenvalues, give additional information to be described below. The features of
the matrix which are useful for interpretation are as follows:
· The number of factors (columns) is the
number of substantively meaningful independent (uncorrelated) patterns of
relationship among the variables.28 Again considering the ten national
characteristics, Figure 6 presents their unrotated matrix. As can
be seen from the number of factors, there are four independent patterns of
relationship in the data. These may be thought of as evidencing four different
kinds of influence (causes) on the data, as presenting four categories by which
these data may be classified, or as illuminating four empirically different
concepts for describing national characteristics.
· The loadings,
,
measure which variables are involved in which factor pattern and to what degree
(see Equation
1 and Equation
2).29 They can be interpreted like correlation
coefficients (see Section 4.1). The square of the loading multiplied by
100 equals the percent variation that a variable has in common with an
unrotated pattern.
One
can look at this percent figure as the percent of data on a variable that can
be produced or predicted by knowing the values of a case (such as a nation) on
the pattern or on the other variables involved in the same pattern. Another
perspective is that the percent figure is the reliability of prediction of a
variable from the pattern or from the other variables in the same pattern. By
comparing the factor loadings for all factors and variables, those particular
variables involved in an independent pattern can be defined, and those variables
most highly related to a pattern can also be seen.
For
example, consider the unrotated factor loadings for the ten characteristics as
shown in the first section of Table 3. Let a pattern be limited to those
variables with 25 percent or more of their variation involved in a pattern
(loading of .50, squared and multiplied by 100). Then the first pattern of
interrelationships30 in Table
3 involves high GNP per
capita (.96), trade (.94), power (.58), stability (.69), US agreement (.56),
and defense budgets (.79).
·
The
first unrotated factor pattern delineates the largest pattern of relationships
in the data; the second delineates the next largest pattern that is independent
of (uncorrelated with) the first; the third pattern delineates the third
largest pattern that is independent of the first and second; and so on. Thus
the amount of variation in the data described by each pattern decreases
successively with each factor; the first pattern defines the greatest amount of
variation, the last pattern the least. Note that unrotated factor patterns are
uncorrelated with each other.31
· The column headed "h2"
displays the communality of each
variable. This is the proportion of a variable's total variation that is
involved in the patterns. The coefficient (communality) shown in this column,
multiplied by 100, gives the percent of variation of a variable in common with
each pattern.
This
communality may also be looked at as a measure of uniqueness. By subtracting the percent of variation in common with
the patterns from 100, the uniqueness of a variable is determined. This
indicates to what degree a variable is unrelated to the others--to what degree
the data on a variable cannot be derived from (predicted from) the data on the
other variables. In Table 3, for example, foreign conflict has a
communality of .55. This says that 55 percent of the foreign conflict behavior
as measured for the fourteen nations can be predicted from a knowledge of
nation values on the four patterns; and that 45 percent of it is unrelated to
the other nine characteristics.
The
h2 value for a variable is calculated by summing
the squares of the variable's loadings. Thus for power in Table 3 we have (.58)2 + (-.42)2 + (-.42)2 + (.43)2 = .87, the h2 value.
· The ratio of the sum of the values in the
h2 column to the number of variables, multiplied
by 100, equals the percent of total variation in the data that is patterned.
Thus it measures the order, uniformity, or regularity in the data. As can be
seen in Table 3, for the ten national characteristics the
four patterns involve 80.1 percent of the variation in the data. That is, we
could reproduce 80.1 percent of the relative variation among the fourteen
nations on these ten characteristics by knowing the nation scores on the four
patterns.
· At the foot of the factor columns in Figure
6, the percent of total variance figures show
the percent of total variation among the variables that is related to a factor
pattern. This figure thus measures the relative variation among the fourteen
nations in the original data matrix that can be reproduced by a pattern: it
measures a pattern's comprehensiveness and strength.
The
sum of these figures for each pattern equals the sum of the column of h2 multiplied by 100. Looking along the row of
percent of total variance figures and up the column of h2, one can see how the order
in the data is divided by pattern and by variable.
The
percent of total variance figure for a factor is determined by summing the
column of squared loadings for a factor, dividing by the number of variables,
and multiplying by 100.
· The percent
of common variance figures indicate how whatever regularity exists in the
data is divided among the factor patterns. The percent of total variance figures, discussed above, measure bow much of the
data variation is involved in a pattern; the percent of common variance figures measure how much of the variation accounted for by all the patterns is involved in each pattern. These latter figures
are calculated in the same way as the percent of total variance, except that
the divisor is now the sum of the column of h2 values, which measures the common variation
among the data.
·
The
eigenvalues equal the sum of the
column of squared loadings for each factor. They measure the amount of
variation accounted for by a pattern. Dividing the eigenvalues either by the
number of variables or by the sum of h2 values and multiplying by 100 determines the
percent of either total or common variance, respectively, Often only the
eigenvalues are displayed at the foot of factor tables.32
Not
all factor studies present the h2 values or
the percent of common or total variance. From the points just made, however,
the reader should be able to calculate them himself. In conjunction,
information on the factor loadings and on communalities should enable the
reader to relate the findings in an unrotated matrix to his particular
concerns.
4.3 Rotated Factor Matrix
The rotated factor
matrix should not differ in format from the unrotated factor matrix, except
that the h2 may not be given and eigenvalues are
inappropriate.33
The unrotated factors
successively define the most general patterns of relationship in the data. Not
so with the rotated factors. They delineate the distinct clusters of relationships, if such exist. This is mentioned here to
alert the reader to this difference. The distinction is clarified with
illustrations in Section 5.
The following features
characterize the rotated matrix:
· If the rotated matrix is orthogonal, that is mentioned in the
title of the matrix (e.g., "orthogonally rotated factors"), or else
the word varimax or quartimax appears in the title (these
are techniques for orthogonal rotation). An orthogonally rotated matrix appears
in the second section of Table 3, for the ten national characteristics of Table
1. The unrotated factor
matrix from Figure 6 is also given for comparison (first
section of Table 3). For an orthogonally rotated matrix the
following aspects should be noted:
· Several features of the unrotated matrix
are preserved by the orthogonally rotated matrix. These are the features
described in Section 4.2 under the first point on the number of factors indicating
the number of patterns, the second point on interpreting loadings, the sixth point on the percent of total variance,
and the seventh point on the percent of common variance.
· The h2 values given for the unrotated factors do not
change with orthogonal rotation, given the number of factors in each case is
the same. Hence they may be given with either the unrotated or the rotated
factor matrix.
· In the unrotated matrix, factor patterns
are ordered by the amount of data variation they account for, with the first
defining the greatest degree of relationship in the data. In the orthogonally
rotated matrix, no significance is attached to factor order.
· Factors are uncorrelated (refer back to Note
31). For example, in Table
3, the first orthogonally
rotated pattern--which might be labeled a power pattern--is uncorrelated with
the second pattern, that of UN agreement with the US.
· If the rotated matrix is oblique rather than orthogonal, the
title or description of the matrix will indicate this. The title may also
contain strange terms like covarimin,
quartimin, or biquartimin. These
refer to various criteria for the rotation and need not trouble us here.
Oblique rotation means that the best definition of the uncorrelated and
correlated cluster patterns of interrelated variables is sought. Orthogonal
rotation defines only uncorrelated patterns;
oblique rotation has greater flexibility in searching out patterns regardless of their correlation. This difference is
elaborated with geometric illustrations in Section 5.
Oblique
rotation takes place in one of two coordinate systems: either a system of primary axes or a system of reference axes. The reference axes give
a slightly better definition of the clusters of interrelated variables than do
the primary ones. For each set of axes there are two possible matrices: factor structure and factor pattern matrices. It is irrelevant to
the consumer of factor results whether oblique primary or reference factors are
given. There is an important difference, however, between the pattern matrix
and the structure matrix.
· The primary
factor pattern matrix and the reference
factor structure matrix delineate the oblique patterns or clusters of
interrelationship among the variables. Their loadings define the separate
patterns and degree of involvement in the patterns for each variable. Unlike
the unrotated or the orthogonally rotated factors, however, their loadings
cannot be strictly interpreted as the correlation of a variable with a pattern,
and the squared loadings do not precisely give the percent of variation of a
variable involved in a pattern. Nevertheless, as in the orthogonal factor
matrix, their loadings are zero when a variable is not involved in a pattern,
and close to 1.0 when a variable is almost perfectly related to a factor
pattern.34 The less correlated the oblique patterns
are with each other, the more their loadings are like correlations of variables
with patterns. With this understanding in mind, the reader might roughly interpret the primary pattern
matrix or reference structure matrix loadings as correlations. By squaring them
and multiplying by 100 to get an idea of the approximate percent of variation involved, the reader will have a
conceptual anchor for understanding the configuration of loadings.
The
third section of Table 3 displays the (primary) oblique pattern
factor matrix for the ten national characteristics. These may be compared with
the orthogonally rotated factors shown in the second section. Note how much
more distinct the patterns are when defined by oblique rotation (the pattern
matrix) than by orthogonal rotation. There are fewer moderate loadings and more
high and low loadings, thus giving a better definition of the pattern of
relationships.
· The primary
factor structure matrix and the reference
factor pattern matrix give the correlation of each variable with each
pattern. The loadings are strictly interpretable as correlations. They can be
squared and multiplied by 100 to measure the percent of variation of a variable
accounted for by a pattern. The last section of Table 3 shows the (primary) oblique structure
factors matrix for the ten national characteristics. The basic difference
between the primary structure and pattern matrices (or reference pattern
and structure matrices) relevant for interpretation is that the primary pattern
loadings best show what variables are highly involved in what clusters. The
primary pattern loadings distinctly display the patterns. The primary structure
loadings, however, do not display them well; instead, they measure the
correlation of variables with the patterns. Note in Table 3 how much better the patterns among the
ten national characteristics are differentiated by the pattern matrix loadings
than by the structure matrix.
By
this time, the many distinctions mentioned may have created more confusion than
understanding. Table 4 shows the important differences for the
several matrices considered. The difference between primary and reference
matrices is one of geometric perspective. Reference matrices give a slightly
better definition of the oblique patterns and are preferred by psychologists.
Because of a simpler geometrical representation, however, I often use the primary
matrices.
·
The
oblique factors will have a correlation among them as shown in a factor
correlation matrix. This matrix is discussed in
Section
4.4, below.
· Figures for percent of common variance and
percent of total variance are not given for the oblique factors. In order to
get some measure of the strength of the separate oblique factor patterns, the
sum of a column of squared factor loadings may be computed. This has been done
in Table
3 for the oblique factors
for the ten national characteristics.
4.4 Factor Correlation Matrix
This is a correlation
matrix between oblique factor patterns found through oblique rotation. Some studies
may call this a matrix of factor cosines. The cosines, however, can be read as
correlations between patterns, and vice versa. The characteristics of a
correlation matrix described in Section 4.1 apply equally well here.
What does a nonzero
correlation between two factors mean? It means that the data patterns
themselves have a relationship, to the degree measured by the factor
correlations. The idea that patterns can be related is not strange, since we
continually deal with such notions in social theorizing. Weather patterns are
related to transportation patterns, for example, and a modernization pattern is
related to cultural patterns. Factor analysis makes these links explicit through
oblique rotation and the factor correlation matrix.
Table 5 presents the factor correlations for the
oblique factors shown in Table 1. From Table 5 it can be seen that voting agreement with
the US and foreign conflict patterns are in fact orthogonal (uncorrelated) to
each other. The foreign conflict pattern does have some positive relationship
(.31) to the power pattern, however.
Sometimes the factor
correlation matrix can itself be factor analyzed, as was the variable
correlation matrix. This will uncover the pattern of relationships among the
factors; the interpretation of these patterns does not differ from those found
for the variable correlations. The reduction of factor interrelationships to
their patterns is called higher order
factor analysis.
4.5 Factor Score Matrix
The factor matrix
presents the loadings
,
by which the existence of a pattern for the variables can be ascertained. The
factor score matrix gives a score for each case (such as a nation) on these
patterns.
The factor scores are
derived in the following way: Each variable is weighted proportionally to its
involvement in a pattern; the more involved a variable, the higher the weight.
Variables not at all related to a given pattern--like the case of defense
budget as percent of GNP, a variable unrelated to the orthogonally rotated
first pattern in Table 3--would be weighted near zero. To
determine the score for a case on a pattern, then, the case's data on each
variable is multiplied by the pattern weight for that variable. The sum of
these weight-times-data products for all the variables yields the factor score.
Cases will have high or low factor scores as their values are high or low on
the variables entering a pattern.35 For an economic development pattern
involving GNP per capita, telephones per capita, and vehicles per capita, for
example, the factor scores derived from the weighted summation of data of
nations on these variables would place the United States as the highest, Japan
as moderate, and Yemen near the bottom.
How are factor scores to
be interpreted? Simply as data on any variable are interpreted. GNP as a
variable, for example' is a composite of such variables as hog production,
steel production, and vehicle production. Similarly, population is a composite
of population subgroups. In the same fashion, factor scores on, say, an
economic development pattern are a composite. The composite variables represented
by factor scores can be used in other analyses or as a means of comparing cases
on the patterns. But the factor scores have one feature that may not be shared
by many other variables. They embody phenomena with a functional unity: the
phenomena are highly interrelated in time or space.
Table 6 displays the factor scores for the
fourteen nations on the (orthogonally rotated) four patterns of Table
3. These scores are
standardized, which means they have been scaled so that they have a mean of
zero and about two-thirds of the values lie between +1.00 and -1.00. Those
scores greater than +1.00 or less than -1.00, therefore, are unusually high or
low. Figure 2 plots these factor scores for the four
patterns separately, and Figure 7 plots scores on the power and foreign
conflict patterns against each other.
4.6 Interpreting Factors
The loadings and factor
scores describing the patterning of the data are found by the analysis. Once
the patterns are determined, the scientist will study them and attach an
appropriate label. These labels facilitate the communication and discussion of
the results; they also serve as instrumental tags for further manipulation,
mnemonic recall, and research. The scientist may label the patterns in any one
of three ways: symbolically, descriptively,
or causally.
Symbolic labels are
simply any symbols without substantive meaning of their own. Their purpose is
merely to denote the patterns. Three factor patterns, for example, may be
labeled DI, D2, and D3, or A, B, and C. A label such as D, can be made
equivalent to a given pattern without fear of adding surplus meaning.
Alternatively, to name a pattern "economic development" or
"totalitarianism" might have different connotations for different
people.
Although symbolic tags
are precise and help avoid confusion, they also create problems in
communicating research findings and comparing studies. At the present stage of
research in the social sciences, symbolic tags have yet to acquire agreed-on
meanings reflecting a well-tested set of patterns, as has happened with
vitamins (e.g., vitamin C).
By contrast, descriptive labels like "agreement
with the US," once defined, can be easily remembered and referred to
without redefinition. They are clues to factor content perhaps similar to those
found by other studies. A descriptive interpretation of a pattern comprises
selecting a concept that will reflect the nature of the phenomena involved. If,
for example, a factor analysis of nations uncovers a pattern of intercorrelated
data on total area, total population, total GNP, and magnitude of resources,
the pattern might be named "size." The descriptive label is meant to
categorize the findings.
In causal naming of
patterns, the scientist reasons from the discovered patterns to the underlying
influences causing them. The causal tag is a capsule explanation of why a
pattern involves particular variables. For example, a factor pattern comprising
coups and purges may be symbolically labeled "C," descriptively named
"revolution," or causally termed "modernization." In the
last case, the scientist may believe that the occurrence and intercorrelation
among these revolutionary actions results from the social disruption of a rapid
shift from a traditional society to a modem industrial nation. As another example,
a factor analysis of Congressional roll call votes. may uncover a highly
intercorrelated pattern of foreign policy issues. A descriptive label could be
"foreign policy" pattern. Causally, however, it might be called an
"isolationist" pattern by reasoning that a common isolationist
attitude underlies the uniformity in foreign policy voting.
The approach to the
interpretation of factor patterns is a matter of personal taste, communication,
and long-run research strategy. The scientist may wish to use concepts that are
congenial to the interests of the reader to facilitate communication, encourage
thought about the findings, and make their use easier. There is always the
danger, however, of the fallacy of misplaced concreteness. The interpretations
of the findings within the research and lay community may be as much a result
of the tag itself as of what the tag denotes.
5. FACTOR ROTATION
Almost all published
factor analyses employ rotation. A section on rotation, therefore, may aid in
comprehending the nature of the results. The character of the unrotated
solution will be discussed first, and after that the nature and rationale for
rotation.
5.1 Character of Unrotated Factors
For the most popular
factor analysis techniques (centroid and principal axes), the factor patterns
define decreasing amounts of variation in the data. Each pattern may involve
all or almost all the variables, and the variables may therefore have moderate
or high loadings for several factor patterns. To uncover the first pattern, a factor
is fitted to the data to account for the greatest regularity; each successive
factor is fitted to best define the remaining regularity. The result of this is that the first unrotated factor may be located
between independent clusters of interrelated variables. These clusters
cannot be distinguished in terms of their loadings on the first factor,
although they will have loadings different in sign on the second and subsequent
factors.
This situation may be
illustrated in a two-factor, eight-variable case by Figure 8. Part (a) of this figure shows the eight
hypothetical variables plotted according to their data for, say, 50 cases.
These fifty cases are the coordinate axes of the space.36 As shown in Figure 8(b), the first factor, F1, falls
between the two clusters of interdependent variables labeled I and II. In this
position F1 maximally reflects the variation of (i.e., has
maximum loadings for) all eight variables. Another way of saying this is that
the first factor lies along the center of gravity of all the points
representing the variables. Observe that the separate loadings (dotted lines)
of these variables on the first factor does not enable the clusters to be
distinguished. Table 7 gives the factor loadings for the eight
variables on unrotated F1.
Figure 8(c) shows the variable loadings on the
second factor, which is placed at right angles (orthogonal) to the first; Table
7 also gives these
loadings on unrotated F2.
The first unrotated
factor delimits the most comprehensive classification, the widest net of
linkages, or the greatest order in the data. For comparative political data, a
first factor could be a "political institutions" pattern, and a
second might define the democratic and totalitarian poles. For international
relations, the first factor could be participation in international relations,
and a second factor might reflect a polarization between cooperation and
conflict. For variables measuring heat, the first factor could be temperature
and a second might delineate the extremes of hot and cold. For physiological
measurements on adults, the first factor could be size and a second might
mirror a polarization between height and girth.
5.2 Character of Rotated Factors
A scientist may rotate
factors to see if a hypothesized cluster of relationships exists. This can be
done by postulating the loadings of a hypothetical factor matrix and then
rotating the factors to a best fit with this matrix. The truth of the
hypothesis is tested by the difference between the fitted and hypothesized
factor loadings.
Alternatively, a
scientist may rotate the factors to control for certain influences on the
results. He may rotate the first factor to a variable or group of variables and
then rotate the subsequent factors to be at right angles (uncorrelated) with
the first. This removes the effects of variables highly loaded on the first
factor and enables us to assess the patterns independent of them.
Most often, however, a
scientist rotates his factors to a simple structure solution. When a factor
matrix is entitled "rotated factors," this almost always means a
simple structure rotation. That is, each factor has been rotated until it
defines a distinct cluster of interrelated variables. Through this rotation the
factor interpretation shifts from unrotated factors delineating the most
comprehensive data patterns to factors delineating the distinct groups of
interrelated data.
Consider again the unrotated
factors shown in Figure 8(c). A simple structure rotation would be
equivalent to that shown in Figure 9. The new factor positions F*1 and F*2 now clearly distinguish the two clusters. This
rotated factor matrix is shown in Table 7 alongside the unrotated factors.
A simple structure
rotation has several characteristics of interest here:
· Each variable is identified with one or a
small proportion of the factors. If the factors are viewed as explanations,
causes, or underlying influences, this is equivalent to minimizing the number
of agents or conditions needed to account for the variation of distinct groups
of variables.
· The number of variables loading highly on
a factor is minimized. This changes the unrotated factor patterns from being
general to the largest number of variables to patterns involving separate
groups of variables. The rotation attempts to define a small number of distinct
clusters of interrelated phenomena. The simple structure type of matrix is
illustrated in Table 8. The moderate and large factor loadings
are indicated by x and small loadings are left blank.
· A major ontological assumption underlying
the use of simple structure is that, whenever possible, our model of reality
should be simplified. If phenomena can be described equally well using simpler
factors, then the principle of parsimony is that we should do so. Simple
structure maximizes parsimony by shifting from general factors involving all
the variables to group factors involving different sets of variables.
· A goal of research is to generalize factor
results. The unrotated factor solution, however, depends on all the variables.
Add or subtract a variable from the study and the results are altered. The
unrotated solution should be adjusted, then, so that the factors will be invariant
of the variables selected. An invariate factor solution will delineate the same
clusters of relationships regardless of the extraneous variables included in
the analysis.
One of the chief
justifications for simple structure rotation is that it determines invariant
factors. This enables a comparison of the factor results of different studies.
Very seldom do different scientists study exactly the same variables. But when
variables overlap between studies and each study employs simple structure
rotation, tests can be made to see if the same patterns are consistently
emerging.
5.3 Orthogonal Simple Structure Rotation
One important type of
simple structure rotation is an orthogonal one. A second type is oblique simple
structure (discussed in Section 5.4 below). Factors rotated to orthogonal
simple structure are usually reported simply as "orthogonal factors."
Occasionally, the varimax or quartimax criteria for achieving the
rotation are used to designate the factors.
Orthogonality is a
restriction placed on the simple-structure search for the clusters of
interdependent variables. The total set of factors is rotated as a rigid frame,
with each factor immovably fixed to the origin at a right angle (orthogonal) to
every other factor. This system of factors is rotated around the origin until
the system is maximally aligned with the separate clusters of variables. If all
the clusters are uncorrelated with each other, each orthogonal factor will be
aligned with a distinct cluster. The more correlated the separate clusters are,
however, the less clearly can orthogonal rotation discriminate them. Simple
structure can then only be approximated, not achieved.
Whether or not
uncorrelated clusters of relationship exist in the data, orthogonal rotation
will still define uncorrelated patterns of relationships. These patterns may
not completely overlap with the distinct clusters, but the delineation of these
uncorrelated factors is useful. Results involving uncorrelated patterns are
easier to communicate, and the loadings can be interpreted as correlations.
Moreover, orthogonal factors are more amenable to subsequent mathematical
manipulation and analysis.
5.4 Oblique Simple Structure Rotation
Whereas in orthogonal
simple structure rotation the final factors are necessarily uncorrelated, in
oblique rotation the factors are allowed to become correlated. In orthogonal
rotation the whole factor structure is moved around the origin as a rigid frame
(like the spokes of a wheel around the hub) to fit the configuration of
clusters of interrelated variables. In oblique rotation to simple structure,
however, the factors are rotated individually to fit each distinct cluster. The
relationship between the resulting factors then reflects the relationship
between the clusters. Figure 10(a) shows a two-factor orthogonal simple
structure rotation; for comparison, Figure 10(b) displays a simple structure oblique
rotation to the same clusters.37
Orthogonal rotation is a
subset of oblique rotations. If the clusters of relationships are in fact
uncorrelated, then oblique rotation will result in orthogonal factors.
Therefore, the difference between orthogonal and oblique rotation is not in
discriminating uncorrelated or correlated factors but in determining whether
this distinction is empirical or imposed on the data by the model.
Controversy exists as to
whether orthogonal or oblique rotation is the better scientific approach.
Proponents of oblique rotation usually advocate it on two grounds:
· it generates additional information; there
is a more precise definition of the boundaries of a cluster, and the central
variables in a cluster can be identified by their high loadings;
· the correlations between the clusters are
obtained, and these enable the researcher to gauge the degree to which his data
approximate orthogonal factors.
Besides yielding more
information, oblique rotation is justified on epistemological grounds. One
justification is that the real world should not be treated as though phenomena
coagulate in unrelated clusters. As phenomena can be interrelated in clusters,
so the clusters themselves can be related. Oblique rotation allows this reality
to be reflected in the loadings of the factors and their correlations. A second
justification is that correlations between the factors now allow the scientific
search for uniformity to be carried to the second order (see Section
4.4). The factor
correlations themselves may be factor analyzed to determine the more general,
the more abstract, the more comprehensive relationships and the more pervasive
influences underlying phenomena.
NOTES
* Scanned from
"Understanding Factor Analysis," The
Journal of Conflict Resolution (December 1967): 444-480. Typographical
errors have been corrected, clarifications added, and style updated. This was
an invited paper prepared in connection with research supported by the National
Science Foundation, GS-1230. For many helpful comments made on a draft of that
published, I wish to thank Henry Kariel, Michael Haas, Robert Hefner, Woody
Pitts, and J. David Singer. This article is a summary of Rummel (1970).
2. For a bibliography of applications of factor
analysis in the social sciences (excluding psychology), see Rummel (1970). A bibliography
of applications to conflict and international relations is given in the appendix below.
3. How many readers know that over a decade ago
Raymond Cattell (1949) gave us the first comprehensive findings on the extent
to which foreign and domestic conflict behaviors have been correlated with many
socioeconomic and political characteristics of nations?
4. For a Q-factor analysis of UN voting, see
Russett (1966). A Q-factor analysis of nations on many of their characteristics
has been reported by Banks and Gregg (1965).
5. Most factor analysis done on nations has been
R-factor analysis. As one example out of many, see Tanter (1966). R- and
Q-factor analyses do not exhaust the kinds of patterns that may be considered.
Other possible patterns of variation are those in characteristics over time
units for a specified nation (this identifies similar time periods); in nations
over time units for a characteristic (this identifies nations similarly
changing on a characteristic); and in time units over nations for a
characteristic ( this identifies similar time periods for nations changing on a
characteristic). For a discussion of these varieties of analysis, see Rummel
(1970, Chapter. 8).
7. For example, see Borgatta and Cottrell's
classificatory work on groups (1955) and Schuessler and Driver's on tribes
(1956). Selvin and Hagstrom (1963) show, through an example, how to use factor
analysis to develop a classification of groups. Using factor analysis, Russett
classifies nations into their regional groups (1967) and their UN voting blocs
(1966).
9. On this and related points, see the
particularly excellent Chapters 19 and 20 in Cattell (1952). Cattell (1966) has
recently elaborated the position that factor analysis is, among other things,
an experimental method.
10. See the discussion on
the relationship between hypotheses and factor analysis in Cattell (1952, pp.
13-14). For an application of factor analysis to test a hypothesis about the
supposed dimensions of urban areas, see van Arsdol, Camilleri, and Schmid
(1958).
11. An extended discussion
of description and explanation with regard to factor analysis in psychology is
given by Henrysson (1957). Thurstone (1947, Chapter 6) discusses factors as
explanatory concepts in terms of a demonstration problem involving the
dimensions of cylinders. His illustration of this problem is helpful for
understanding factor analysis in practice.
12. The distinction being
drawn here is between descriptive and inferential statistics, not between
description and statistics.
13. Some of the more
excellent treatments are those by Frank (1955, Chapter 1), Kaufmann (1958,
Chapter 6), the essays by Russell, Feigl, and Nagel in Part V of Feigl and
Brodbeck (1953), and Nagel (1961).
14. "It would seem
that in general the variables highly loaded in a factor are likely to be the
causes of those which are less loaded, or at least that the most highly loaded
measures--the factor itself--is causal to the variables that loaded on it"
(Cattell, 1952, p. 362). Cattell and Sullivan (1962) conducted a demonstration
experiment by factoring data on cups of coffee to determine whether patterns
corresponding to known causal influences could be delineated. They found a
strong correlation between the known patterns of influences and those defined
by the factor analysis. With like results a similar experiment was conducted on
the dynamics of balls (Cattell and Dickman, 1962). These artificial experiments
are helpful in understanding applied factor analysis.
16. See the clear and
explicit analysis by Abel (1953) of the operation of verstehen in the social sciences.
17. One of the best
discussions of theory is given by Nagel (1961, Chapter 6). That theory
construction consists of two parts is argued by Einstein. See the essays on
Einstein's philosophy by Frank, Lenzen, and Nortbrup in Schilpp (1949).
18. An exciting theoretical
use of factor analysis has been published by Cattell. (1962). He describes a
role behavior model potentially rooted in empirical data, tying together
personality, structure, and syntal group dimensions. A theoretical embodiment of
factor analysis to relate the attributes and behavior of social units is
described in Rummel (1965).
19. The relationship
between classical physics and quantum theory, or between Cartesian analysis and
Hilbertian analysis as related to factor analysis, is discussed by Ahmavaara
(Ahmavaara and Markkanen, 1958, pp. 48-63). This analysis is the most
refreshing and provocative that I have read on the subject. See also Note 25, below.
20. Prior to plotting, the
data would have to be made comparable through some standardization procedure.
21. The cosine of this
angle between vectors is, with minor qualifications, equal to the product
moment correlation coefficient between the characteristics represented by the
vectors. Thus, a correlation of 1.00 between two variables on twenty cases
means that the angle is zero between the two vectors (variables) plotted in the
space of twenty dimensions (cases).
22. I am referring to the
results of the factor analysis research design, which include the application
of a factoring technique plus simple structure rotation. (See Section 5.2) For those familiar with linear algebra, it may be helpful to
know that a factor analysis defines a set
of basis dimensions for the column vectors of a data matrix. Each basis
dimension of a rotated set uniquely generates an independent subset of the
original vectors. The basis dimensions of an unrotated set are ordered by their
contributions to generating all the vectors.
23. The configuration of
vectors in Figure
5 is four-dimensional. Therefore, although the placement of the two
independent axes is the best (orthogonal) definition of the two clusters in
four-dimensional space, the two-dimensional figure can only display this fit
imperfectly.
24. This is where
curve-fitting techniques like multiple linear and curvilinear regression
analysis are helpful.
25. The factor analysis
model has much in common with quantum theory. This is one reason I have argued,
as I do in Section
2.5 above, that factor analysis is a theoretical
structure as well as a data analysis technique. See Margenau (1950, Chapter 17)
for a clear and simple description of quantum theory. Burt (1941) and Ahmavaara
in Ahmavaara and Markkanen (1958) have also drawn the comparison of factor
analysis with quantum theory.
26. Confusion on this score
has caused much unfounded criticism of factor analysis as delineating only
linear relationships.
27. The idea of a
correlation coefficient gives another perspective on factor analysis. The
patterns discovered by a factor analysis consist of those variables highly
intercorrelated. Thus, if variable A is highly correlated with both B and C,
and if B and C are highly correlated with each other, then A, B, and C form a
correlation cluster. If A, B, and C are not correlated with other variables,
then they form an independent pattern that factor analysis will delineate.
28. There is some question
about the criteria for determining the exact number of factor patterns for a
set of data. Variation in the number of patterns defined by different criteria
is usually small and, at any rate, normally concerns the minor patterns. The
larger patterns, involving many variables with high loadings, will ordinarily
be found and reported regardless of the criteria employed.
29. Note how the organization
of a factor matrix is like the layout of Equation 1. Rather than explicitly organizing the factor results in
equations, factor analysts use the matrix format, where the first column refers
to the F1 function, the second column to the F2 function, etc., and the elements (loadings) of
the matrix are the constants,
, that have been found by
the analysis.
30. These patterns differ
from those given as examples for these variables in Section 1. This is because we are now discussing unrotated patterns. The
reason for the differences will be discussed below.
31. To say the factors are
uncorrelated means that the factor scores
(to be discussed in Section 4.5) on the factor patterns are uncorrelated, and not necessarily the
factor loadings. Factor loadings are, however, independent (orthogonal)
32. The eigenvalues are
extracted only if the principal axes method of factor analysis is used. An
eigenvalue is the root of the characteristic equation [R -
I] = 0, where R is the
correlation matrix,
is an eigenvalue, I is an identity matrix, and
the brackets mean that the determinant is being computed. Let X be an
orthogonal matrix with columns determined such that XR =
X. Then the various roots,
, are the eigenvalue
solutions to the equation and X is the matrix of eigenvectors. The factor
matrix is equal to the eigenvectors times the reciprocal square root of their
associated eigenvalues.
33. Although equal to the
sum of squared factor loadings, the eigenvalue is technically a solution of the
characteristic equation (see Note 32) for the unrotated factors.
The rotated factors are derived from these by transformation (rotation).
34. The pattern matrix
loadings are best understood as regression coefficients of the variables on the
patterns.
35. These factor scores
then give values for each case on the functions, F, of Equations 1, 2, and 3 in Section
3.2. With the constant,
, defined by the factor
matrix, and the factor scores defining the value of the function, F, the factor
equations are completely specified.
37. Various mathematical
criteria are employed to achieve oblique simple structure, and have such exotic
names as quartimin, covarimin,
biquartimin, binormin, promax, and maxplane.
These may sometimes appear in the title of an oblique factor matrix.
REFERENCES
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Ahmavaara, Yrjö, and
Tourco Markkanen. The Unified Factor
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Alker, Hayward, Jr.
"Dimensions of Conflict in the General Assembly." American Political Science Review, 58
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________ and Bruce
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Banks, Arthur S., and
Phillip M. Gregg. "Grouping Political Systems: Q-Factor Analysis of 'A
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BERRY, Brian J. L.
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Buckatzsch, E. J.
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Burt, C. The Factors of the Mind. New York:
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________, and K. Dickman.
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________, and William
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APPENDIX
BIBLIOGRAPHY OF FACTOR ANALYSIS
IN CONFLICT AND INTERNATIONAL STUDIES
Adelman, Irma, and
Cynthia T. Morris. "Factor Analysis of the Interrelationship Between
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Banks, Arthur S., and
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