Multidimensional Nonlinear Descriptive Analysis

329 Pages·2006·1.666 MB·\329

by  Shizuhiko Nishisato (Author)

MultidiMensional
nonlinear
descriptive analysis
Shizuhiko Nishisato

MultidiMensional
nonlinear
descriptive analysis
Shizuhiko Nishisato
Chapman & Hall/CRC
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To
Lorraine, Ira, Samantha
and
In memory of George William Ford

v
Preface
This book is intended for those involved in data analysis in diverse
areas of research. Unlike in a well-controlled and well-designed
statisticalexperiment,manyofusfacedatatowhichthenotionof“data
being a random sample from the normal population” does not apply.
“Normal distribution” means that data must be continuous, but most
datawedealwithinthesocialsciencesarecategoricalornon-numerical.
Furthermore,weknowthattherelationbetweentwonormallydistributed
variables is by design linear. In practice, however, we encounter many
nonlinear relations such as “the strength of the body is generally a
concave function of age.” Such a phenomenon exists and cannot be
ignoredforthesakeofusingthenormaldistributionassumption.
Quantification of categorical or non-numerical data is indeed a
ubiquitousproblem. Yet most courses for data analysis are devoted to
thetraditionalstatisticsbasedonnormaltheorywhichaffordsanelegant
andsophisticatedinferentialframeworksuchasaconfidenceintervalfor
aparameterandhypothesistesting.Thecurrentbookisintendedtoserve
theneedsforthosewhomustfacetherealityoftypicaldataanalysis:data
arediscreteornon-numerical,notnecessarilysampledrandomlyfroma
populationandinvolvenotonlylinearbutalsononlinearrelations.
Thebookstartswithsome discussionofwhyand howthisbookcan
respond to typical needs of data analysis. The discussion is hopefully
illuminating.Thenthereaders are exposedtoconceptualpreliminaries,
again innon-technicalterms, and totechnical preliminaries needed for
data analysis in the remaining chapters. Part One of the book
covers these topics, which hopefully is useful to understand data
analysis discussed in subsequent chapters. Part Two and Part Three
contain applications of what we call “multidimensional nonlinear
descriptive analysis,” abbreviated as MUNDA, to diverse data types,
witheach chapterbeingdevotedtoonetypeofcategorical data, abrief
historicalcommentandbasicskillspeculiartothedatatypes.Thebook
willendwiththechapterentitled“FurtherPerspectives,”whichpresents
severalproblemsthatneedtobesolvedinthenearfutureforthecurrent
methodology.Thus,thisbookisanoverviewofMUNDAofdiscretedata
withsuggestionsforfuturedevelopments. The mainpartofthebookis
similar in topics to Nishisato(1994), but the topics are discussed with
much more insightthan in the 1994 book, as well as with new results
sincethen.
This book is for students in the social and biological sciences and
researchers in such fields as marketing research, education, health and
vi
medical sciences, psychology, sociology, biology,ecology, agriculture,
economics, political science, criminology, archaeology, geology and
geography.Thelevelofthebookisintermediate,butthewritingstyleis
straightforwardsothatonemayreadthebookwithrelativeease, andat
thesametimeitisintendedtobebeneficialforwell-seasonedresearchers
engagedindataanalysis.
Work related to MUNDA started in the early years of the
twentieth century in ecology and biology under the name of gradient
andordinationmethods.Thecontributionstothisareabyecologistsand
biologistsfrommany countriesareoverwhelming,andtheyhave ledto
anenormousnumberofapplicationstoproblemsintheirdisciplinesup
to the present moment. Outside ecology and biology, we see familiar
formulations of such methods as the method of reciprocal averages,
simultaneouslinearregressions and appropriatescoringbystatisticians
and psychologists, starting in the 1930s and 1940s, particularly in the
United Kingdom and the United States. After the 1950s, the number
of publicationson MUNDA and related studies increased, resultingin
its wide applications, particularlyin Japan and France, led by eminent
researchers ChikioHayashi inJapan andJean Paul Benze´cri inFrance.
Theresearchspreadnotonlytomanyothercountriesbutalsotodiverse
disciplines.Itisthustimelynowtopauseandreviewthegeneralareaof
MUNDA.
Finally,a personal note. I have devoted my entire research career to
MUNDA, which I coined as “dual scaling.” R. Darrell Bock was my
mentor at the University of North Carolina who introduced me to his
optimal scaling (Bock, 1960). Also instrumental to my successful
graduateworkatthePsychometricLaboratoryinChapelHillwereLyle
V. Jones, Director, Masanao Toda, mentor in Japan, the late Mr. and
Mrs.ArthurRingwalt,hostfamily,theFulbrightCommissionandmany
dear fellow students and other faculty members. Michael W. Browne
introducedmetotheFrenchcorrespondenceanalysisinearly1970sand
drewmyattentiontotheproblemthatthejointplotofrowweightsand
columnweightsinthesamespacewasnotmathematicallysound.Atmy
retirementsixyearsago,JohnC.Gowerencouragedmewiththewords
“Lifeexistsafterretirement.”
During my career, I have had the pleasure of personally knowing
researchers in many countries: Austria, Australia, Belgium, Brazil,
Britain,Bulgaria,Canada,China,France,Germany,Greece,HongKong,
India,Italy,Japan,TheNetherlands,Pakistan,Russia,Singapore,South
Korea, Spain, Sweden, Switzerland, Taiwan, USA and The West
Indies.Togetherwithmanycolleagues,amongothers,RossTraub,Gila
vii
Hanna, Richard Wolfe, Merlin Wahlstrom, Leslie McLean, Donald
Burrill,VincentD’Oyley,RoderickMcDonald,PhilipNagy,GlenEvans,
Dennis Roberts, Ruth Childs, Tony Lam, Joel Weiss, Alexander Even,
DomerEllis,SabirAlvi,TahanyGaddala,JoanPreston,thelateHoward
Russell, the late Raghu Bhargava, the late Shmuel Avital, the late Sar
Khan and the late Dorothy Horn, I also taught students from such
countries as Argentine, Australia, Bahrain, Brazil, Canada, Central
Africa, China(Mainland,Hong Kong),Egypt, Ethiopia,Greece, India,
Iran, Israel, Japan, Malawi, Malaysia, New Zealand, Nigeria, Pakistan,
Peru, Philippines, Singapore, South Africa, South Korea, Sri Lanka,
Taiwan,Thailand,USA,VenezuelaandTheWestIndies.Otherthanmy
careeratMcGillUniversityinMontrealandtheUniversityofToronto,I
was a visitingprofessorattheUniversityofKarlsruhe, Germany (Host
Professor Wolfgang Gaul), Institute of Statistical Mathematics, Japan
(Host Professor Yasumasa Baba), Kwansei Gakuin University, Japan
(Host Professors Takehiro Fijihara, Masao Nakanishi, Akihiro Yagi
(twice), and Shoji Yamamoto) and Doshisha University, Japan (Host
Professors ShigeoTatsukiand HirotsuguYamauchi). Iowe allofthese
dearfriends,colleaguesandstudentsagreatdealfortheirmanyyearsof
associationandfriendship.
Forthecurrentbook,Ihavethreeregrets.Thefirstregretisaboutthe
partialcoverage ofthegeneraltopicsofMUNDA.Duringtheliterature
search,Iwasoverwhelmedbytheenormouscontributionsbyecologists
andbiologistssincetheearlypartofthetwentiethcenturyallthewayto
date andIwasaware ofonlysome oftheircontributionsuntilrecently.
Because of this late discovery, I could not cover their contributionsin
a deserving way in the current book. In addition,jointcorrespondence
analysis(Greenacre,1988;TateneniandBrowne,2000)isnotdiscussed.
The departure of the procedure from correspondence analysis is
comparabletothatoffactoranalysisfromprincipalcomponentanalysis,
and as such it is very importantfor the development of MUNDA. The
secondregretisaboutthenotation.Sincethecurrentbookisasummary
ofwhatIhavebeenworkingonunderthenamedualscaling,Iusedmy
own notationof some antiquity.This was done in spiteof the fact that
MichaelGreenacreandJo¨rgeBlasiushavebeentryinghardtounifythe
notationforMUNDA-relatedpublications.Ifounditdifficulttochange
my notation in many of my old papers to the new one. My third and
greatestregretisthefactthatinspiteofmyfinalstatementinmy1980
book that “Inferential aspects of dual scaling deserve immediate
attentionbyresearchers,”Ididnotincludeinference-relatedtopicsinthis
book,notablyworkonconfidenceregions,associationmodels(loglinear