Statistical methods for research workers.docx

Statistical methods for research workers.docx

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Statisticalmethodsforresearchworkers

Statisticalmethodsforresearchworkers

STATISTICALMETHODSFORRESEARCHWORKERS

ByRonaldA.Fisher（1925）

PostedApril2000

VI

THECORRELATIONCOEFFICIENT

30. Noquantityismorecharacteristicofmodernstatisticalworkthanthecorrelationcoefficient,andnomethodhasbeenappliedsuccessfullytosuchvariousdataasthemethodofcorrelation.Observationaldatainparticular,incaseswherewecanobservetheoccurrenceofvariouspossiblecontributorycausesofaphenomenon,butcannotcontrolthem,hasbeengivenbyitsmeansanaltogethernewimportance.Inexperimentalworkproperitspositionismuchlesscentral;itwillbefoundusefulintheexploratorystagesofanenquiry,aswhentwofactorswhichhadbeenthoughtindependentappeartobeassociatedintheiroccurrence;butitisseldom,withcontrolledexperimentalconditions,thatitisdesiredtoexpressourconclusionintheformofacorrelationcoefficient.

Oneoftheearliestandmoststrikingsuccessesofthemethodofcorrelationwasinthebiometricalstudyofinheritance.Atatimewhennothingwasknownofthemechanismofinheritance,orofthestructureofthegerminalmaterial,itwaspossiblebythismethodtodemonstratetheexistenceofinheritance,andto[p.139]"measureitsintensity";andthisinanorganisminwhichexperimentalbreedingcouldnotbepractised,namely,Man.Bycomparisonoftheresultsobtainedfromthephysicalmeasurementsinmanwiththoseobtainedfromotherorganisms,itwasestablishedthatman'snatureisnotlessgovernedbyhereditythanthatoftherestoftheanimateworld.Thescopeoftheanalogywasfurtherwidenedbydemonstratingthatcorrelationcoefficientsofthesamemagnitudewereobtainedforthementalandmoralqualitiesinmanasforthephysicalmeasurements.

Theseresultsarestilloffundamentalimportance,fornotonlyisinheritanceinmanstillincapableofexperimentalstudy,andexistingmethodsofmentaltestingarestillunabletoanalysethementaldisposition,butevenwithorganismssuitableforexperimentandmeasurement,itisonlyinthemostfavourablecasesthattheseveralfactorscausingfluctuatingvariabilitycanberesolved,andtheireffectsstudied,byMendelianmethods.Suchfluctuatingvariability,withanapproximatelynormaldistribution,ischaracteristicofthemajorityoftheusefulqualitiesofdomesticplantsandanimals;andalthoughthereisstrongreasontothinkthatinheritanceinsuchcasesisultimatelyMendelian,thebiometricalmethodofstudyisatpresentalonecapableofholdingouthopesofimmediateprogress.

WegiveinTable31anexampleofacorrelationtable.Itconsistsofarecordincompactformofthestatureof1376fathersanddaughters.（PearsonandLee'sdata.）Themeasurementsaregroupedin[p.140-141][table][p.142]inches,andthosewhosemeasurementwasrecordedasanintegralnumberofincheshavebeensplit;thusafatherrecordedasof67incheswouldappearas1/2under66.5and1/2under67.5.Similarlywiththedaughters;inconsequence,whenbothmeasurementsarewholenumbersthecaseappearsinfourquarters.Thisgivesthetableaconfusingappearance,sincethemajorityofentriesarefractional,althoughtheyrepresentfrequencies.Itispreferable,ifbiasinmeasurementcanbeavoided,togrouptheobservationsinsuchawaythateachpossibleobservationlieswhollywithinonegroup.

Themostobviousfeature of thetableisthatcasesdonotoccurinwhichthefatherisverytallandthedaughterveryshort,and viceversa ;theupperright-handandlowerleft-handcornersofthetableareblank,sothatwemayconcludethatsuchoccurrencesaretooraretooccurinasampleofabout1400cases.Theobservationsrecordedlieinaroughlyellipticalfigurelyingdiagonallyacrossthetable.Ifwemarkouttheregioninwhichthefrequenciesexceed10itappearsthatthisregion,apartfromnaturalirregularities,issimilar,andsimilarlysituated.Thefrequencyofoccurrenceincreasesfromallsidestothecentralregionofthetable,whereafewfrequenciesover30maybeseen.Thelinesofequalfrequencyareroughlysimilarandsimilarlysituatedellipses.Intheouterzoneobservationsoccuronlyoccasionally,andthereforeirregularly;beyondthiswecouldonlyexplorebytakingamuchlargersample.

Thetablehasbeendividedintofourquadrantsby[p.143]markingoutcentralvaluesofthetwovariates;thesevalues,67.5inchesforthefathersand63.5inchesforthedaughters,arenearthemeans.Whenthetableissodivideditisobviousthatthelowerright-handandupperleft-handquadrantsaredistinctlymorepopulousthantheothertwo;notonlyaremoresquaresoccupied,butthefrequenciesarehigher.Itisapparentthattallmenhavetalldaughtersmorefrequentlythantheshortmen,and viceversa. Themethodofcorrelationaimsatmeasuringthedegreetowhichthisassociationexists.

Themarginaltotalsshowthefrequencydistributionsofthefathersandthedaughtersrespectively.Thesearebothapproximatelynormaldistributions,asisfrequentlythecasewithbiometricaldatacollectedwithoutselection.Thismarksafrequentdifferencebetweenbiometricalandexperimentaldata.Anexperimenterwouldperhapshavebredfromtwocontrastedgroupsoffathersof,forexample,63and72inchesinheight;allhisfatherswouldthenbelongtothesetwoclasses,andthecorrelationcoefficient,ifused,wouldbealmostmeaningless.Suchanexperimentwouldservetoascertaintheregressionofdaughter'sheightinfather'sheight,andsotodeterminetheeffectonthedaughtersofselectionappliedtothefathers,butitwouldnotgiveusthecorrelationcoefficientwhichisadescriptiveobservationalfeatureofthepopulationasitis,andmaybewhollyvitiatedbyselection.

Justasnormalvariationwithonevariatemaybespecifiedbyafrequencyformulainwhichthe[p.144]logarithmofthefrequencyisaquadraticfunctionofthevariate,sowithtwovariatesthefrequencymaybeexpressibleintermsofaquadraticfunctionofthevaluesofthetwovariates.Wethenhaveanormalcorrelationsurface,forwhichthefrequencymayconvenientlybewrittenintheform

Inthisexpression x and y arethedeviationsofthetwovariatesfromtheirmeans, σ1 and σ2 arethetwostandarddeviations,and ρ isthecorrelation between x and y. Thecorrelationintheaboveexpressionmaybepositiveornegative,butcannotexceedunityinmagnitude;itisapurenumberwithoutphysicaldimensions.If ρ=0,theexpressionforthefrequencydegeneratesintotheproductofthetwofactors

showingthatthelimitofthenormalcorrelationsurface,whenthecorrelationvanishes,ismerelythatoftwonormallydistributedvariatesvaryingincompleteindependence.Attheotherextreme,when p is+1or-1,thevariationofthetwovariatesisinstrictproportion,sothatthevalueofeithermaybecalculatedaccuratelyfromthatoftheother.Inotherwords,weceasestrictlytohavetwovariates,butmerelytwomeasuresofthesamevariablequantity.

Ifwepickoutthecasesinwhichonevariatehasanassignedvalue,wehavewhatistermedanarray;[p.145]thecolumnsandrowsofthetablemay,exceptasregardsvariationwithinthegrouplimits,beregardedasarrays.Withnormalcorrelationthevariationwithinanarraymaybeobtainedfromthegeneralformula,bygiving x aconstantvalue,（say） a,anddividingbythetotalfrequencywithwhichthisvalueoccurs;thenwehave

showing（i.）thatthevariationof y withinthearrayisnormal;（ii.）thatthemeanvalueof y forthatarrayis ρaσ2/σ1,sothattheregressionofy on x islinear,withregressioncoefficient

and（iii.）thatthevarianceof y withinthearrayis σ22（1-ρ2）,andisthesamewithineacharray.Wemayexpressthisbysayingthatofthetotalvarianceof y thefraction（1-ρ2）isindependentof x,whiletheremainingfraction, ρ2,isdeterminedby,orcalculablefrom,thevalueof x.

Theserelationsarereciprocal,theregressionof x on y islinear,withregressioncoefficient ρσ1/σ2;thecorrelation ρ isthusthegeometricmeanofthetworegressions.Thetworegressionlinesrepresentingthemeanvalueof x forgiven y,andthemeanvalueof y forgiven x,cannotcoincideunless ρ=[plusorminus]1.Thevariationof x withinanarrayinwhich y isfixed,isnormalwithvarianceequalto σ12（1-ρ2）,sothatwemaysaythatofthevarianceof x thefraction（1-ρ2）[p.146]isindependentof y,andtheremainingfraction, ρ2,isdeterminedby,orcalculablefrom,thevalueof y.

Sucharetheformalmathematicalconsequencesofnormalcorrelation.Muchbiometricdatacertainlyshowsageneralagreementwiththefeaturestobeexpectedonthisassumption;thoughIamnotawarethatthequestionhasbeensubjectedtoanysufficientlycriticalenquiry.Approximateagreementisperhapsallthatisneededtojustifytheuseofthecorrelationasaquantitydescriptiveofthepopulation;itsefficacyinthisrespectisundoubted,anditisnotimprobablethatinsomecasesitaffordsacompletedescriptionofthesimultaneousvariationofthevariates.

31.TheStatisticalEstimationoftheCorrelation

Justasthemeanandthestandarddeviationofanormalpopulationinonevariatemaybemostsatisfactorilyestimatedfromthefirsttwomomentsoftheobserveddistribution,sotheonlysatisfactoryestimateofthecorrelation,whenthevariatesarenormallycorrelated,isfoundfromthe"productmoment."If x and y representthedeviationsofthetwovariatesfromtheirmeans,wecalculatethethreestatistics s1, s2, r bythethreeequations

ns12 =S（x2）, ns22 =S（y2）, nrs1s2 =S（xy）;

then s1 and s2 areestimatesofthestandarddeviations σ1,and σ2,and