SAUNDERS 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 \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ 84168707 W. B. SAUNDERS COMPANY PHILADELPHIA LONDON TORONTO 1974 W. B. SaundersCompany: WestWashingtonSquare Philadelphia, PA 19105 12DyottStreet London.WCIA lOB 833 OxfordStreet Toronto. Ontario M8Z 5T9, Canada 841687-'()? DOACAO ~NICHJUFMG TO ERIC AND KAY ~Cl.~'i ,.,IJ'-''-' {'\C)Cl". ""A~Cl " v ' ';;;',"1 ,r}r;" ;-;,';'S 1'> FactorAnalysiS-.-·~~';,!<' I,>_l.:,~;' ISBN 0-7216-4170,9 'fI ,.r'i',H: :,r,~i"'·~;_ :;,t ,,: © 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.