高尔顿发明回归Word格式.docx

高尔顿发明回归Word格式.docx

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2001byJeffreyM.Stanton,allrightsreserved.

Thistextmaybefreelysharedamongindividuals,butitmaynotberepublishedinanymediumwithoutexpresswrittenconsentfromtheauthorandadvancenotificationoftheeditor.

KeyWords:

Correlation;

FrancisGalton;

Historyofstatistics;

KarlPearson.

Abstract

AnexaminationofpublicationsofSirFrancisGaltonandKarlPearsonrevealedthatGalton'

sworkoninheritedcharacteristicsofsweetpeasledtotheinitialconceptualizationoflinearregression.SubsequenteffortsbyGaltonandPearsonbroughtaboutthemoregeneraltechniquesofmultipleregressionandtheproduct-momentcorrelationcoefficient.Moderntextbookstypicallypresentandexplaincorrelationpriortointroducingpredictionproblemsandtheapplicationoflinearregression.ThispaperpresentsabriefhistoryofhowGaltonoriginallyderivedandappliedlinearregressiontoproblemsofheredity.Thishistoryillustratesadditionalapproachesinstructorscanusetointroducesimplelinearregressiontostudents.

1.Introduction

ThecompletenameofthecorrelationcoefficientdeceivesmanystudentsintoabeliefthatKarlPearsondevelopedthisstatisticalmeasurehimself.AlthoughPearsondiddeveloparigoroustreatmentofthemathematicsofthePearsonProductMomentCorrelation（PPMC）,itwastheimaginationofSirFrancisGaltonthatoriginallyconceivedmodernnotionsofcorrelationandregression.Galton,acousinofCharlesDarwinandanaccomplished19thcenturyscientistinhisownright,hasoftenbeencriticizedinthiscenturyforhispromotionof"

eugenics"

（plannedbreedingofhumans;

see,forexample,Paul（1995）.Historianshavealsosuggestedthathiscousin'

slastingfameunfairlyovershadowedthesubstantialscientificcontributionsGaltonmadetobiology,psychologyandappliedstatistics（see,forexample,FitzPatrick1960）.Galton'

sfascinationwithgeneticsandheredityprovidedtheinitialinspirationthatledtoregressionandthePPMC.

ThethoughtsthatpromptedthedevelopmentofthePPMCbeganwithathenvexingproblemofheredity--understandinghowstronglythecharacteristicsofonegenerationoflivingthingsmanifestedinthefollowinggeneration.Galtoninitiallyapproachedthisproblembyexaminingcharacteristicsofthesweetpeaplant.Hechosethesweetpeabecausethatspeciescouldself-fertilize;

daughterplantsexpressgeneticvariationsfrommotherplantswithoutcontributionfromasecondparent.Thischaracteristiceliminated,oratleastpostponed,havingtodealwiththeproblemofstatisticallyassessinggeneticcontributionsfrommultiplesources.Galton'

sfirstinsightsaboutregressionsprangfromatwo-dimensionaldiagramplottingthesizesofdaughterpeasagainstthesizesofmotherpeas.Asdescribedbelow,Galtonusedthisrepresentationofhisdatatoillustratebasicfoundationsofwhatstatisticiansstillcallregression.Thegeneralizationoftheseeffortsintotheproduct-momentcorrelationandthemorecomplexmultipleregressioncamemuchlater.Currenttextbooksofbehavioralsciencestatisticstypicallyreversethisorder:

thePPMCispresentedfirstandlinearregressioniscoveredlater.Manyinstructorsmayalsofeelmorecomfortablestartingwithcorrelationandbuildinguptoregression.

Thepresentpaperprovideshistoricalbackgroundandillustrativeexamplesthatstatisticsinstructorsmayfindusefulinintroducingtheseconceptstocollegelevelclassesinappliedstatistics.Bybrieflytracingthehistoricaldevelopmentofregressionandcorrelation,thispapershowshowintroductorystatisticsinstructorscanuseengagingandhistoricallyaccurateexamplestointroduceregressionandcorrelationtostudents.Anumberofarticlesconcerningtheteachingofregressionandcorrelationindicatethatstudentsoftenhavedifficultyunderstandingtheseconceptsandtheconnectionbetweenthem（see,forexample,Williams1975;

Duke1978;

Karylowski1985;

GoldsteinandStrube1995;

）.Thepresentarticleprovidesnewideasforinstructionbasedonthehistoricaloriginsofthesestatisticaltechniques.

2.Galton'

sEarlyConsiderationsofRegression

BesideshisroleasacolleagueofGalton'

sandaresearcherinGalton'

slaboratory,KarlPearsonalsobecameGalton'

sbiographerafterthelatter'

sdeathin1911（Pearson1922）.Inhisfour-volumebiographyofGalton,Pearsondescribedthegenesisofthediscoveryoftheregressionslope（Pearson1930）.1875年,Galton把7包甜豌豆（sweetpea）种籽分发给7位朋友，每包里的种子是重量一样的，但是包间重量差别很大（alsoseeGalton1894）,。

朋友们种下这些种籽，又把收获的豆子寄还Galton（seeAppendixA）。

Galtonplottedtheweightsofthedaughterseedsagainsttheweightsofthemotherseeds.Galtonrealizedthatthemedianweightsofdaughterseedsfromaparticularsizeofmotherseedapproximatelydescribedastraightlinewithpositiveslopelessthan1.0:

"

Thushenaturallyreachedastraightregressionline,andtheconstantvariabilityforallarraysofonecharacterforagivencharacterofasecond.Itwas,perhaps,bestfortheprogressofthecorrelationalcalculusthatthissimplespecialcaseshouldbepromulgatedfirst;

itissoeasilygraspedbythebeginner."

（Pearson1930,p.5）

Thesimple,specialcasethatPearsonreferredtois,ofcourse,boththeroughlyequivalentvariabilityofthetwomeasuresandtheiridenticalunitsofmeasurement.Figure1usesasimple,inventeddatasettoillustrateGalton'

searliestfindings.TheparentsweetpeasizeontheX-axisandtheoffspringsweetpeasizeontheY-axishaveapproximatelyequalvariability.Thus,theslopeofthelineconnectingthemeansofthedifferentcolumnsofpointsisequivalentbothtotheregressionslopeandthecorrelationcoefficient.ForGalton'

spurposes,anyslopesmallerthan1.0indicatedregressiontothemeanforthatgenerationofpeas.Thephenomenonofregressiontothemeanisillustratedbytheconfigurationofpoints:

They-coordinatesofmostofthepointsinFigure1areclosertothehorizontaloffspringmeanthantheirx-coordinatesaretotheverticalparentmean.Galton'

sfirstdocumentedstudyofthistypesuggestedaslopeof0.33（obtainedthroughcarefulinspectionofhisscatterplots）,whichindicatedtohimthatextremelylargeorsmallmotherseedstypicallygeneratedsubstantiallylessextremedaughterseeds.Thisfindingis,ofcourse,prototypicalofregressiontothemean:

Formanyvariables,naturalprocessesworkto"

dampen"

extremeoutliersandbringthemclosertotheirrespectivemeans.

Figure1.

Figure1.Connectingthemeansoftheindividualcolumnsofdataprovidesacrudeapproximationoftheregressionline.Theslopeisexactly0.50andthecorrelationisapproximatelyr=0.51.Many,thoughnotall,ofthepointsareclosertotheoffspringpeasizemeanof9ontheY-axisthantotheparentalpeasizemeanof10ontheX-axis.ThenumericdataappearinAppendixB.

Nonetheless,onlyahorizontallinewouldhaveindicatednoheritabilityinseedsizewhatsoever,soGalton'

sfindingaffirmedhisbasicassumptionsconcerningtheheritabilityof"

characters."

Figure1,greatlysimplifiedfromGalton'

soriginalgraph,illustrateshowalineconnectingthemeansofthecolumnsofdatapointsindicatesthedegreetowhichextremevaluesinthefirstgeneration（ontheX-axis）tendtoregresstowardthemeanofthesecondgeneration（ontheY-axis）.Iinventedthedatapoints,whicharelistedinAppendixB,tosimplifyhandcalculationinaclassroomsetting.IncontrasttoFigure1,Galton'

soriginaldatadidnotproduceaperfectlysmoothline,buthewasabletodraw,byhand,asinglelinethatfitallthedatareasonablywell（Galton'

sfirstregressionlinewaspresentedatalecturein1877;

see（Pearson1930）.Theslopeofthislinehedesignated"

r"

forregression.OnlyunderPearson'

slatertreatmentdidrcometostandforthecorrelationcoefficient（Pearson1896）.

Galton'

sprogresswasbotheasedandhobbledbyhischoicesfordescriptivestatistics;

heusedthemedianasameasureofcentraltendencyandthesemi-interquartilerangeasameasureofvariability.Oneadvantageofthesemeasureslayinthesimplicityofobtainingthem.Galtonwasnearlyfanaticalaboutgraphingandtabulatingeveryavailabledatapoint.Thesedescriptivevaluescouldemergefromaninspectionoftheresultingfigureortablewithaminimumofcomputation.Itisunderstoodnowthatthemedianandsemi-interquartilerangedonothavethefavorablemathematicalpropertiesofthemeanandstandarddeviation（forexample,theycannotbemanipulatedusingcovariancealgebra）.ButGaltonwasnotasophisticatedenoughmathematiciantorecognizethedeficiency.SoGalton'

sprogresstowardamoregeneralimplementationofregressionwasdelayedbyhischoiceofdescriptivestatistics.InNaturalInheritance（Galton1894）,GaltonexpendedapageortwomakingvariousargumentsabouttheexactvalueoftheslopeofaregressionlineascalculatedwithvarioustechniquestoestimatethechangeinYversusthechangeinXonthescatterplot.Atthatpointintime,hiseffortslackedthemathematicalfoundationtoderivetheslopefromthedatathemselves.Asaninterestingfootnote,inthelate1870s,Galtondidnothaveaccesstoamechanicalcalculatingmachine,whereasPearsonhadoneforpersonaluseonhisdesknolaterthan1910（Pearson1938）.

3.Galton'

sRecognitionoftheGeneralityofRegressionSlope

Evenwithhispoorchoiceofdescriptivestatistics,Galtonwasabletogeneralizehisworkoveravarietyofheredityproblems.Hetackledpersonalitytemperament,artisticability,anddis