Table Of ContentSAUNDERS BOOKS IN PSYCHOLOGY
ROBERT D. SINGER, Consulting Editor
GORSUCH FACTOR ANALYSIS
HARDYCK AND
PETRINOVICH INTRODUCTION TO STATISTICS FOR THE
BEHAVIORAL SCIENCES
JOHNSON AGGRESSION IN MAN AND ANIMALS
KAUFMANN INTRODUCTION TO THE STUDY OF HUMAN
BEHAVIOR
L'ABATE AND
CURTIS TEACHING THE EXCEPTIONAL CHILD
SATTLER ASSESSMENT OF CHILDREN'S INTELLIGENCE
SINGER AND
SINGER PSYCHOLOGICAL DEVELOPMENT IN CHILDREN
SKINNER NEUROSCIENCE: A Laboratory Manual
STOTLAND AND
CANON SOCIAL PSYCHOLOGY: a Cognitive Approach
I
WALLACE PSYCHOLOGY: A Social Science
RICHARD L. GORSUCH
TEXAS CHRISTIAN UNIVERSITY
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84168707
W. B. SAUNDERS COMPANY
PHILADELPHIA
LONDON
TORONTO
1974
W. B. SaundersCompany: WestWashingtonSquare
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© 1974 by,W,'B, Saunders-Company. Copyrightunderthe InternationalCopyright Union.
All rights reserved; This bookIs protected by copyright. No partofit may bereproduced,
storedinaretrievalsystem.'ortransmittedinanyfonnorby anymeans.electronic,mechanical.
photocopying, recording, orotherwise, withoutwritten permissionfromthepublisher, Made
in the United States ofAmerica. PressofW. B. SaundersCompany. LibraryofCongress
catalogcard number72-95830.
Lastdigitis the PrintNumber: 9 8 7 6 5 4 3 2
FactorAnalysis began the way many books do. When asked to teach a
courseinfactoranalysisinthe late1960's,Ifelttherewasnotrulyappropriate
text. The high-quality texts which were availableseemed to me nottostress
points I deemed critical when factor-analyzing data or appeared to be too
y difficultfor graduate psychologystudents with only an averagemathematical
background. My response Wasto develop a set ofdittoed notes for use in
my classes. The notes were part of my effort to make my course effective
in teaching graduate students when and how to use factor analysis in their
substantiveresearch. Tobe frank, I also felt thatusingmy own notes would
keep the course from unduly imposing upon my substantive research,since
itisofteneasierto presentone's own ideas than it isto integratethoseideas
with someone else's work. Once the notes had begun, they tookon alifeof
their own. Revising the notes each time the course was taught eventually
led to this book.
The first purpose of the present book is to enable an investigator to
properly utilize factor analysis as a research tool. Such utilization requires
an understanding of when the differences in possible mathematical pro
cedures may have a major impact upon the substantive conclusions and
when the differences might not be relevant for a given research study. In
addition, one should also know when factor analysis is not the best pro
cedure to be used.
Stressing the aspects of factor analysis which are important for re
searchdoes notmean, however, thatthe mathematicalfoundations offactor
analysis are ignored, sinceabasicunderstandingofthemathematicalmodels
is necessary to understand the application ofthose models. Hence, deriva
tions are given for many aspects offactor analysis; however, no calculus
is used. Ifone has a working knowledge ofbasic algebra and patience, he
willbe able to follow the mathematics.
Any p rson completing the usual first year ofgraduate level statistics
will probab have the background necessary to pursue this text. I have
assumed tha the reader is familiar with correlation coefficients, the
rudiments of ultiple correlation, and the basic concepts of significance
testing. Some iliarity with social science research is also helpful sothat
the importanceofthe research-orienteddiscussions maybe readilyapparent. vii
viii Preface Preface lx
Some sections are of greater interest to the more mathematically Among those who have helped with the manuscript itselfare Anthony
oriented or advanced reader. These sections, which can be skipped on the Conger, Jolin Horn, and Jum Nunnally. The thoughtful, detailed reviews
first reading, are identified by an asterisk preceding the section title. which they providedofan earlier draft are deeply appreciated. Their com
In addition to increasing the student's sophisticationinfactor analysis, ments have certainly resulted in a higher-qualityproduct; ifI havefailed to
it is hoped that this book will increase the reader's generalresearch sophis achieve my goals in writing this book, it is probably because I have not
tication as well. Studyingfactor-analytic procedures requires one to askand properly interpreted their reactions.
to explore basic questions abouthow an areais to be conceptualized.These I also deeply appreciate the contributions which my associates and
basic questions exist in every scientific endeavor, whether or not factor friends have made to my thinking intheareaoffactoranalysis. Inparticular,
analysis is used as a formal tool. Studying factor analysis may therefore numerous students have contributed their reactions to earlier drafts of the
help the reader to better understand the basic questions and provide pos materials included here. They are, as is usually the case, the unsungheroes
sible rationales for improving scientific models. An increase in research in the development of this book.
sophistication has been one of the prime benefits which I have personally In turning ideas into a manuscript, numerous errands need to be run,
derived from my contact with factor analysis. pages typed, and proofreading conducted. Without the'able assistance of
There are several ways in which this bookcan be used. First,it can be Mrs. Betty Howard, such tasks would have kept the book from being pub
used to help decide iffactor analysis is relevant to one's research program. lished. Her professional expertise as well as her cooperative, responsible
If this is your interest, I suggest that you read Chapter 1carefully and then attitudes are assets which mostauthors need in secretarial support, but only
scan Chapters 17 and 18. These three chapters can be read more or less a few are privileged to have.
independently of Chapters 2 through 16 and should provide a basis for During the period that this book was being written, I held a Kennedy
deciding iffactor analysis would be useful in your research program. The Professorship in the John F. Kennedy Centerfor Research on Education
rest ofthe textcanthenbe readiffactoranalysis islikelyto be used. and Human Development at George Peabody College for Teachers. I
Second, the present book can serve as the.text for a graduate level deeply appreciate the gracious support provided by theJoseph P. Kennedy
course on factor analysis orfor a section ofa course in multivariate statis Jr. Foundation.
tical techniques. In some situations, it may be appropriate to use the book Several ofthe book's examples are based on re-analyses ofpreviously
in its entirety. In other courses, Chapters 14 and 15 may be skipped along published data. Therefore, I would like to thank the University ofChicago
with the starredsections. Thelatteroption will allowmoretimeforpursuing Press for their permission to use data from the Holzinger and Swineford
outside readings, practice problems, and so forth. Chapters 11, 13,.16, 17 (1939) monograph, as well as the British Journal ofPsychiatry for their
and 18 are essential to a complete understanding of factor analysis and permission to use datafrom an article by Rees (1950). .
should not be treated lightly simply because they occur later. They are
criticalfor understanding the role offactor analysis in research. RICHARD L. GORSUCH
The third way in which this book can be used is as a reference. It is
hoped that the detailed Table of Contents, the Subject Index, the starred
sections, the references.to and discussions of computer programs, the ap
pendix on computer program libraries, and Chapters suchas 14 and 15 will 'Ii
be particularly useful to investigators actively engaged in factor-analytic
research. To increase the book's usefulness as a reference, citations are
made to thefactor-analytic literature so thatfurtherpursuitofanyparticular
topic is possible.
In writing about factor analysis, I find that I have been extensively
influenced by my former professors. I am especially indebted to Raymond
B. Cattell. Not only did he involve me in factor analytic-research, but the
types of questions which he raised concernirig both the factor-analytic
procedures themselves and the usefulness of factor analysis in theory
building have had a lasting impact upon me. The ideas of my former pro
fessors are so interwoven with mine that I find it difficult to know if! have
unduly appropriated their constructs while consideringthem to be my own.
Indeed, I am sure that I have not referenced professors such as Wesley
Becker, Lloyd Humphreys, S. B. Sells, and Ledyard Tucker sufficiently
throughout the text.
1
INTRODUCTION . 1
1.1 Science and FactorAnalysis . 1
1.2 Elementary Procedures for Factoring . 4
1.3 Examples . 8
2
BASIC FACTOR MODELS . ............ 12
2.1 Multivariate Linear Models and FactorAnalysis.... 12
2.2 The Full Component Model.. 18
2.3 The Common FactorModel........................... 23
2.4 Correlated and Uncorrelated FactorModels.................. 30
2.5 Factor Analysis and the Multivariate Linear ModeL... 31
3
MATRIX ALGEBRA AND FACTOR ANALySiS.............................. 33
3.1 Matrix Definitions and Notation................................. 33
3.2 Matrix Operations... 37
3.3 Definitional Equations in Matrix Algebra Form............ 44
3.4 The Full Component Model Expressed in Matrix
Algebra : 46
3.5 The Common Factor Model in Matrix Algebra............ 47
3.6 Uncorrelated Factor Models and Matrix Algebra......... 50 xi
xii Contents Contents xiii
4 9
COlEOMETRic REPRESENTATION OF FACTOR MODELS............... 52
ROTATION AND INTERPRETATION OF FACTORS : 161
4.1 Representing Variables and Factors Geometrically...... 52 9.1 Principles for Guiding the Rotation ofFactors 163
4.2 The Uncorrelated (Orthogonal) Component Mode!......... 61 9.2 Rotating Visually 168
4.3 The Correlated (Oblique) Component Mode!..... 62 *9.3 Algebraic Principles ofRotation 177
4.4 Common Factor Models..... 65 9.4 Interpreting Factors 182
5 10
DIAGONAL AND MULTIPLE-GROUP ANALYSIS.. 66 ANALYTIC ROTATION......... 189
5.1 Diagonal Analysis.. 67 10.1 Orthogonal Rotation................................................ 190
5.2 Multiple-Group FactorAnalysis... 73 10.2 Oblique Rotation by Reference Vectors : 195
.5.3 Applications ofDiagonal and Multiple-Group Factor 10.3 Direct Oblique Rotation 203
Analysis............................................................... 80 10.4 Comparing Alternative Solutions 206
6 11
PRINCIPAL FACTOR SOLUTIONS............................................. 85 HIGHER-ORDER FACTORS ... 213
6.1 Characteristics ofPrincipal Factor Methods :... 86 11.1 Interpretation ofHigher-Order Factors .. 214
6.2 Principal Components................................................ 90 11.2 Extracting Higher-Order Factors . 216
6.3 Communality Estimation and Principal Axes...... 92 11.3 Relationship ofOriginal Variables to Higher-Order
6.4 Minimum Residual Analysis (Minres) 102 Factors . 219
6.5 Image Analysis 103 11.4 Usefulness ofHigher-Order Factors . ... 227
*6.6 Alpha FactorAnalysis 107
6.7 Applications for Principal Factors 107 12
6.8 Accuracy ofComputer Processing : 108
FACTOR SCORES .. 228
7
12.1 Procedures for Computing Factor Scores 229
MAXIMUM LIKELIHOOD AND OTHER SOLUTIONS 113 12.2. Approximation Procedures for Factor Scores 236
12.3 ClusterAnalysis ofIndividuals: Typological Scoring 240
7.1 Maximum Likelihood Solutions..: 113 12.4 Evaluating the Factor Scores 241
7.2 A Comparison ofFactor Extraction Procedures 120 12.5 Selecting Among Procedures 245
*7.3 Non-Linear Factor Analysis 125
*7.4 Non-Metric Factor Analysis 127 13
*7.5 Extension Analysis 128
RELATING FACTORS ACROSS STUDIES 246
8
13.1 Information Useful in Relating Factors 247
DETERMINING THE NUMBER OF FACTORS 130 13.2 Same Individuals and Variables but Different
Procedures 249
8.1 Adequacy ofthe Fit ofthe Model to the Data 131 13.3 Same Individuals but Different Variables : 250
8.2 Statistical Approaches to the Number of Factors 135 13.4 Same Variables but Different Individuals (R"v.
8.3 Mathematical Approaches to the Number ofFactors 143 Svrand WvrAvailable) 251
8.4 Extracting the Non-Trivial Factors 151 13.5 Same Variables but Different Individuals (RvV>
8.5 The Search for the Proper Number ofFactors 157 Svrand WvlUnavailable) 253
*Advanced topic. May be consideredoptionalreading. *Advanced topic. May beconsideredoptional reading.
xlv Contents
Contents xv
13.6 DifferentVariables and Different Individuals 257 Appendix A
13.7 Matching Factors 257
DATA FOR EXAMPLES 337
A.l Box Plasrnode: Means and Standard Derivations 337
14
A.2 Twelve Physique Variables:
Means and Standard Deviations 338
DATA TRANSFORMATIONS AND INDICES OF ASSOCiATION 259
A.3 Twenty-four Ability Variables:
14.1 Non-Continuous Data . . 259 Means, Standard Deviations and Reliabilities 338
14.2 Effects ofTransformations.. ... 264
14.3 Indices ofAssociation.. . 270
AppendixB
DEVELOPING A LIBRARY OF FACTOR-ANALYTIC
15 COMPUTER PROGRAMS . 339
B.l Desirable Characteristics ofa Library , 339
TWO- AND THREE-MODE FACTOR ANALySiS 276
B.2 Programs in Textbooks 342
15.1 Two-Mode Factor Analysis . ... 276 B.3 Other Sources '" 343
*15.2 Three-Mode Factor Analysis . ... 283 B.4 Adapting Programs for Local Use 345
REFERENCES . 347
16
AUTHOR INDEX . .r.. 361
THE REPLICATION AND INVARIANCE OF FACTORS 292
SUBJECT INDEX .
16.1 The Replication of Factors Across Random Samples 365
ofIndividuals 292
16.2 The Invariance ofFactors 297
17
FACTOR ANALYSIS AS A RESEARCH TECHNIQUE 312
17.1 Operationalization ofConstructs '" 312
17.2 FactorAnalysis ofIndependent and Dependent
. Variables ; 319
17.3 Using FactorAnalysis to Suggest New Leads for
Future Research 325
17.4 Other Uses of Factor Analysis , 326
18
EPiLOGUE . . 327
18.1 Criticisms of Present Factor-Analytic Practices 327
18.2 Recommended Steps in a Factor Analysis 330
18.3 The Future ofFactorAnalysis 335
*Advanced topic. May beconsidered optional reading.
INTRODUCTION
A basic review of the role offactor analysis in scientific endeavors
is necessary backgroundfor the presentation offactor-analyticprocedures.
The overview is begun in Section 1.1, Science and Factor Analysis, where
factoranalysisisplacedwithinageneralperspectiveofscience.InSection1.2
it is noted that factor analysis-like any statistical procedure- is simply a
logical extension of what mankind has always done in understanding his
world. Some common-sense procedures often used to achieve the goals of
factor analysis are noted. This section is concluded by pointing to the limi
tations of these elementary procedures and by indicating that some ofthe
limitations are overcome through the more powerful factor-analytic tech
niques. The chapter.concludes with a discussion of examples which are
used throughout the book (Section 1.3).
1.1 SCIENCE AND FACTOR ANALYSIS
A major objective of scientific activities is to observe events so that
the empirical relationships among those events can be efficiently sum
marized by theoretical formulations. The events that can be investigated
are almost infinite, so any general statement about scientific activities is
difficult to make.'However, it could be stated that scientists analyze the
relationships among a set of variables where these relationships are evalu
ated across a set ofindividualsunderspecifiedconditions.The variables are
the characteristics being measured and could be anything that can be ob
jectively identified or scored. For example, a psychologist mayusetests as
variables, whereas apolitical scientistorsociologist may use thepercentage
ofballots castfor different candidates as variables. The individuals are the
subjects, cases, voting precincts orotherindividual units which provide the
data by which the relationships among the variables are evaluated. The
conditions specify that which pertains to all the datacollectedand setsthis
study apart from other similar studies. Conditions can include locations in
time and space, variations in the methods ofscoring variables, treatments
after which the variables are measured, or, even, the degrees oflatitude at
which the observation ismade. An observationis a particularvariable score 1
2 Introduction [Ch.1 Sec. 1.1] Science and FactorAnalysis 3
ofa specified individual under the designated conditions. For example, the EXAMPLE
score ofJimmyonamathematicalabilitytesttakenduringhis fifth gradeyear
Table 1.1.1 presents the resultsofa typicalfactoranalysis.CattellandGorsuch
in Paul RevereSchoolmay be oneobservation,whereas Mike'sscoreon the
(1965) correlated demographic variahles across countries, and factored them by
same test at the same school might be another. (Note that observation will the iterated principal axis procedure described in Chapter6. Thefactors werethen
not be used to refer to an individual; such usage is often confusing.) rotated to visual simple structure, a procedure described in Chapter9. Thefactor
Although all sciencefalls withinthe majorobjectiveofbuildingtheories loadings relating each variable to eachfactor are given in theTable.
to explain the interrelationships ofvariables, the differences in substantive
TABLE 1.1.1 FACTOR PATTERN LOADINGS OF DEMOGRAPHIC
problems lead to variations in how the investigation proceeds. One
VARIABLES ON TWO FACTORS
variation is the extent to which events are controlled. In traditional
experimentalpsychologyandphysicsthe investigatorhasrandomlyassigned Factors
subjects to the various manipulated conditions, while astronomers have, Variables Size Development
until recently, been concerned with passive observations of naturally Population .84 -.09
occurring events. Within any broad area, some scientific attempts are No. of GovernmentMinistries .63 .00
purely descriptive of phenomena while others seek to generate and test Area .86 .57
UNICEFContributions per capita .58 .91
hypotheses. Research also varies in the .extent to which it utilizes only a MilesofRailroad per capita .41 .94
few variables or utilizes many variables; this appears to be partly afunction Incomepercapita -.17 .94
of the complexity of the area and the number of variables to which a dis Telephones per capita .00 .89
Physiciansper capita -.01 .02
cipline has access.
Regardless ofhow the investigator determines the relationships among Note:TheTable isbasedondatacollectedinthe 1950'sfrom52individualcountries.
variables under specified conditions, all scientists are united in the common Theelementsinthe tablearethe weightsgiven tothefactorstandardscorestoreproduce
or estimate the variablestandardscores. (Adapted from Cattell and Gorsuch, 1965.)
goal: they seek to summarize dataso thatthe empirical relationships canbe
grasped by the human mind. In doing so, constructs are built which are
From the set of variables having large loadings, it can be easilyseen that the
conceptually clearer than the a priori ideas, and then these constructs are factor labeled "development" is distinctfrom thefactorrepresentingthesizeofthe
integrated through the development of theories. These theories generally country. Ifit is known that the country has a high score on an operational repre
exclude minor variations in the data so that the major variations can be sentative of the development factor, then it can be concluded that the national
summarized and dealt with by finite man. income per capita, telephones, and the other variables with high loadings on this
factor will also be prominent. A different setofgeneralizations can be made ifthe
The purpose in using factor analysis is scientific in the sense outlined
size ofthe countryis known,since thatisan empiricallydifferentcontentarea.
above. Usually the aim is to summarize the interrelationships among the Some variables overlap both ofthe distinct factors, for example, the miles of
variables in a concise but accurate manner as an aid in conceptualization. railroad per capita. Others may be unrelated to either factor, as is the physician
This is often achieved by including the maximum amountofinformation rate; no generalizationscanbe madeto itfrom thefactors identifiedinthisanalysis.
from the original variables in as few derivedvariables,orfactors, as possible
to keep the solution understandable. Some Uses ofFactorAnalysis. A statisticalprocedure which gives both
In parsimoniously describing the data, factor analysts explicitly recog qualitative and quantitative distinctions can be quite useful. Some of the
nize that any relationship is limited to a particular area of applicability. purposes for which factor analysis can be used are as follows:
Areas qualitatively different, that is, areas where relatively littlegeneraliza 1. Through factor-analytic techniques, the number of variables for
tion can be made from one area to another, are referred to as separate further research can be minimized while also maximizing the
factors. Eachfactor represents an areaofgeneralization that is qualitatively amount ofinformation in the analysis. The original set ofvariables
distinct from that represented by any other factor. Within an area where is reduced to a much smaller set which accounts for most of the
datacan be summarized, i.e., within anareawheregeneralizationcanoccur, reliable variance of the initial variable pool. The smaller set of
factor analysts first represent that area by a factor and then seek to make variables can be used as operational representatives of the con
the degree ofgeneralization between each variable and the factor explicit. structs underlying the complete set ofvariables.
A measure of the degree of generalizability found between each variable 2. Factor analysis can be used to search data for possible qualitative
and each factor is calculated and referred to as a factor loading. Factor and quantitative distinctions, and is particularly useful when the
loadings reflect quantitative relationships. The farther the factor loading sheer amount of available data exceeds comprehensibility. Out of
is from zero, the more one can generalize from that factor to the variable. this exploratory work can arise new constructs and hypotheses for
Comparing loadings of the same variable on several factors provides in future theory andresearch. Thecontributionofexploratoryresearch
formation concerning how easy it is to generalize to that variable from to science is, of course, completely dependent upon adequately
each factor. pursuing the results in future research studies so as to confirm or
reject the hypotheses developed.
4 Introduction [Ch.1 Sec. 1.2] Elementary Procedures for Factoring 5
3. Ifa domain ofdata can be hypothesized to have certain qualitative sions, the next improvement is classification byjoint presence orabsence.
and quantitative distinctions, then this hypothesis can be tested by One notes the number oftimesA occurs in relation to the occurrence ofB
factor analysis. If the hypotheses are tenable, the various factors for the cases at hand which are under the specified conditions. Ifthe per
will represent the theoretically derived qualitative distinctions. If centage is high enough, the variables are considered related, i.e., they are
one variable is hypothesized to be more related to one factor than manifestations of the same factor. Ifa number ofsuch variables are found
another, this quantitative distinction can also be checked. together, then they are classified underthe same rubric andaninplicitfactor
The purposes outlined above obviously overlap with purposes served results.As Royce (1958)points out, such thinkingin the areaofpsychology
by other statistical procedures. In many cases the other statistical proce can easily be traced back at least to the Greek philosophers.
dures can be used to answer the question more efficiently than one could
answerthem with afactor analysis. Forexample, ifthehypothesisconcerns EXAMPLE
whether or not one measure ofextroversion is more related to a particular In his studies of the Old Testament, Wellhausen (1885) noticed that passages
job performance than another, the investigator can correlate the two vari fromthe Pentateuch had varying characteristics. Goingfrom onepassagetoanother
led to ashift instyle, names, and concerns. Furthermore, he noted thatthese char
ables under consideration with an operational representative of the job
acteristics covaried together; if one appeared, then another was also likely to be
performance. A simple significance test of the difference between the cor found in that passage. He identified several "factors"from his "variables."
relation coefficients would be sufficient. This is actually a small factor The variables are given inTable 1.2.1inthe form ofquestions askedofagiven
analysis in that the same statistics would result by defining the operational passageofscripture.Theanswersare enteredinthe tableunderthefactors that they
representative ofjobperformance as the first diagonalfactor (cf. Chapter5) are generally assumed to measure (e.g., Gottwald, 1959). Thefactors-E, J, P and
D-arenamed for theprincipalcharacteristicsbywhichtheyareidentified.Because
and determining the loadings of the two extroversion variables on that
of the differences in writing style, thefactors are attributed todifferent traditions or
factor. While the result would be exactly the same, proceeding factor sources which were combinedinto the presentPentateuch.In writing anOldTesta
analytically would be to utilize an unnecessarily complex approach. ment text, itwould beappropriatetodevoteseparatesections toeach ofthefactors,
Many traditional statistics can be viewed as special cases of the andafinalsectiontohowthetraditions(factors) were integratedintothePentateuch.
There is no universal agreement on the exact number of factors, or on their
factor-analytic model where, because of their limited nature, detailed
characteristics. The data can support different theories, as is true inany scientific
statisticaltheories have beendevelopedwhichgreatlyenhancethestatistic's
enterprise. Old Testament scholars cannot, unfortunately, test their rival theories
interpretability. In particular, probability testing is widely available in the by gathering new .data although they can reanalyze the existing data more
traditional statistical areas, while it is more limited in factor analysis. thoroughly.
Therefore wheneverthe decision is betweenotherstatisticalproceduresand
factor analysis, the choice will usually go to the former ifthey both answer A slightly more sophisticated implicit factoring procedure is to group
the same question. This should not, however, obscure the fact that the variables together from an examination of appropriate statistics. These
basic rationale and problems are often the same. statistics could be correlation coefficients, covariances, measures of dis
tance, or any other measure of the degree ofassociation that would sum-
1.2 .ELEMENTARY PROCEDURES FOR FACTORING
TABLE 1.2.1 PENTATEUCH FACTORS
While factor analysis is a term usually applied to the mathematical
Factors
models discussed in Chapter 2, the logical process is often utilized on an
intuitive basis. For example, common observation has identified four Variables E J P D
dimensions ofposition. Theseinclude the three dimensions ofspaceandthe
IsGod's name Elohim? Yes No No No
dimension oftime. The use ofsixorseven variables to mark the locationof 15the passageanti-monarchic? Yes No No No
a plane in flight would be redundant since four variables cangive all the IsGod's nameYahweh(GermanJahweh)? No Yes No No
unique information necessary to locate it. Quantitative comparisons canbe Isitanthropomorphic? No Yes No No
Isitconcernedwithpriestlyaffairs? No No Yes No
made within one of these dimensions but seldom across the dimensions DoesIthavealaboredstyle? No No Yes No
since the dimensions are qualitativelydistinct. Fiftyfeet ofaltitudedoesnot IsitfromDeuteronomy? No No No Yes
IsItexplicitlymonotheistic? No No No Yes
offset a change in latitude in describing the position ofan airplane.
A formalfactoranalysisis notneededto discoverthefourdimensionsof Note:The first factor, E (forElohim), ishypothesized to consist ofthe stories and
position because the distinctions are obvious. Many scientists can, fortu writingsthat came fromthe northern tribes of Israel.J (forJahweh) appears to be a
nately, operate upon such easily observed distinctions. But many others southern source givingsomewhatdifferenttraditions.Thesehistoriesarethoughttohave
been combined with the Priestly and Deuteroncimic traditions by the editors of the
find the distinctions in their areas to be difficult to identify' and turn to Pentateuch.The editors attempted to weaveallofthesetraditionsintoonemanuscript,
statistical aids. but the originalsources maybe Identifiedinthe various passages bynotingthe char
After the variables are subjectively classified along obvious dimen- acteristicsdescribedabove.
