The Correlation Coefficients

Transcription

1 Jounal of Moen Apple Statstcal Methos Volume 6 Issue Atcle The Coelaton Coeffcents Ruy A Geon Unvesty of Montana Follow ths an atonal woks at: Pat of the Apple Statstcs Commons, Socal an Behavoal Scences Commons, an the Statstcal Theoy Commons Recommene Ctaton Geon, Ruy A (007) "The Coelaton Coeffcents," Jounal of Moen Apple Statstcal Methos: Vol 6 : Iss, Atcle 6 DOI: 037/jmasm/ Avalable at: Ths Regula Atcle s bought to you fo fee an open access by the Open Access Jounals at DgtalCommons@WayneState It has been accepte fo ncluson n Jounal of Moen Apple Statstcal Methos by an authoze eto of DgtalCommons@WayneState

2 Jounal of Moen Apple Statstcal Methos Copyght 007 JMASM, Inc Novembe, 007, Vol 6, No, /07/$9500 The Coelaton Coeffcents Ruy A Geon Unvesty of Montana A genealze metho of efnng an ntepetng coelaton coeffcents s gven Seven coelaton coeffcents ae efne thee fo contnuous ata an fou on the anks of the ata A quck calculaton of the ank base coelaton coeffcents usng a 0- gaph-matx s shown Examples an compasons ae gven Key wos: Peason, Speaman, Kenall, Gn, Geatest Devaton, mean, absolute value, nonpaametcs, coelaton, te values Intoucton Defntons Ths atcle ntouces a system of estmaton that has numeous avantages ove cuent pactce Among these avantages s the global te value poceue fo nonpaametc o ank base coelaton coeffcents makng estmaton functonal ove all ata an avance statstcal methos, such as multple egesson; the cuently use local te value poceue s vey estctve Ths system has pouce a way of vewng coelaton that has allowe othe coelaton coeffcents to be efne In patcula, the new contnuous absolute value an mean coelaton coeffcents shoul be use fo L methos o the MAD scale estmate It s geneal an poves a obust estmaton poceue n coelaton analyss an n avance statstcal poceues f obust coelaton s use (wwwmathumteu/geon) Ruy Geon eceve the PhD n Statstcs n 970 une John Gulan at the Unvesty of Wsconsn Hs acaemc caee began n the Depatment of Mathematcal Scences at the Unvesty of Montana n 970; he ete fom the Depatment n June of 005 He has woke extensvely wth Mastes an Doctoal stuents as well as on a multtue of vaous apple statstcal pojects Hs pme goal n etement s to ssemnate hs ognal coelaton estmaton system that encompasses basc statstcal methos To make the efntons of the coelaton coeffcents moe natual, Peason's s efomulate Ths e-expesson of also makes possble a natual efnton of paametc an nonpaametc coelaton coeffcents base on absolute values an means Let CC an NP stan fo coelaton coeffcent an fo nonpaametc Some NPCCs ae efne base on countng technques A 0- gaph-matx s use to establsh elatonshps Fnally, some ata s analyze to examne the elatve obustness of the NPCCs Let ( x, y ),,, n be a bvaate ata set The usual mean notaton wll be use * * an x x x, y y y,,, n ae the centee ata The sample covaance s popotonal to x * y * To pepae fo late efntons, ths covaance s ewtten as * * * * * * y ( ( x y ) ( x y ) ) x / 4 In the uncentee notaton, ths can be wtten as ( ( x x y y) x x y y ( ) /4 Ths fom of the covaance functon appeae as an ntepetaton of Peason's n Roges an Ncewate (988), when the escale vaance ntepetatons wee ae Heustc motvaton ) 57

3 THE CORRELATION COEFFICIENTS 58 fo ths fom as a measue of the elatonshp between the x-y ata s now gven an t hols fo all CCs that ae to be efne When thee s postve coelaton the * * tems ( x y ) ( x x y y) wll ten to be lage, because the two evatons wll ten to be n the same ecton The stance fom a negatve elatonshp s lage, so the coelaton woul be postve The tems * * ( x y ),,, n wll have some cancelng effect, so they wll ten to be small The net effect s that the covaance wll be lage The stance fom a postve elatonshp s small so that the coelaton woul be postve When x an y ae nepenent vaables, a smla amount of cancelng occus n both tems an the covaance wll fluctuate aoun zeo When thee s negatve coelaton the stance fom postve coelaton wll be lage as the * * ( x y ),,, n tems wll ten to be lage, but cancellaton wll be occung n the * * ( x y ),,, n tems, so the stance fom negatve coelaton s small Thoughout ths atcle the tem stance oes not mean just Euclean stance, but s meant to escbe the numecal measues of evatons fom pefect postve o negatve coelaton These concepts ae next elaboate n Euclean n-space Fo ths paagaph x an y ae the n-mensonal vectos of the centee ata, nomalze so that each has Euclean length one, x y Conse the vecto x y n n-space; the fathe ths vecto s fom the ogn (fo ths vecto the ogn epesents pefect negatve coelaton) the moe postve s the coelaton Fo pefect postve coelaton, cos(x, y) an x y ; that s, stance fom the ogn s maxmum Conse the vecto x y The close ths vecto s to the ogn, the moe postve the coelaton Fo x y, the ogn epesents pefect postve coelaton an hence, x y small means stance fom pefect postve coelaton s small Thoughout ths atcle the tem stance oes not mean just Euclean stance, but s meant to escbe the numecal measues of evatons fom pefect postve o negatve coelaton To estate, fo x y the suface of the centee n-mensonal ball of aus epesents pefect negatve coelaton, so x y lage means stance fom pefect postve coelaton s lage Fo pefect negatve coelaton, cos(x, y ), an x y 0, so the stance fom the ball of aus s a maxmum Anothe way to expess ths, n tems of paametes, s that thee s postve coelaton when V(XY) > V(X Y) an negatve coelaton when the nequalty goes n the othe ecton The connecton between stance away fom negatve coelaton an V(XY) an also fo stance away fom postve coelaton an V(X Y) s now llustate fo a bvaate nomal stbuton Let Z an Z be stanaze nomal anom vaables wth CC ρ Note that E(ZZ) ρ [V(ZZ) V(Z Z)] / 4 The tem V(ZZ) equals stance fom pefect negatve coelaton an s a lnea functon of ρ, namely ρ Fo ρ ths stance s zeo but fo ρ, ths stance s 4 Smlaly, V(Z Z) s stance fom pefect postve coelaton an t s ρ Fo ρ, ths stance s 4, but fo ρ, ths stance s 0 Note that these stances ae monotonc functons of ρ an the oveall coelaton V(ZZ) V(Z Z) combnes to equal 4 ρ Howeve, fo some of the othe coelaton coeffcents ths combnng of the stance measues oes not smplfy Also note that n the case of Fshe s nomal tansfomaton, V( Z Z) ρ ln ln V( ZZ) ρ V( Z Z) tanh ρ ln V( Z Z ) It s possble that a smla nomalzng concept woul wok fo othe coelaton coeffcents

4 59 RUDY A GIDEON Atonally, a coelaton coeffcent coul be base on the ato, V(XY) / V(X Y), whch woul be less than one fo negatve coelaton, one fo nepenent anom vaables, an geate than one fo postve coelaton Peason s an othe coelaton coeffcents base on absolute values an means can now be efne Let SSx stan fo a centee sum of squaes an SAx stan fo the sum of absolute values about the mean; e, SA x x x Contnuous coelaton coeffcents Defnton : Peason s xy (, ) * * * * x y x y () ( ) ( ) 4 SSx SSy SSx SS y {(stanaze stance fom pefect negatve coelaton) (stanaze stance fom pefect postve coelaton)} ve by a constant, that puts the value between an Defnton : An absolute value CC, av av( x, y) x y x y SA SA SA SA * * * * x y x y * x SA whee y,e x y SA * y () Defnton 3: The Mean Absolute Devaton coelaton coeffcent Fo the fnal contnuous coelaton, a coelaton analog of the MAD, mean absolute evaton estmate of vaaton, s gven an enote by ma Fo a anom sample, efne MAD x me x me( x ) an smlaly fo the ata fom Y A mean-type coelaton coeffcent s efne as ma xme( x) yme( y) me MAD x MAD y (3) xme ( x) yme ( y) me MAD x MAD y It s not tue that ma Let * x x me(x) *, an smlaly fo y MAD Now, x * * me x me y The poof that ma beaks own s because the mean of the sum of two sets of nonnegatve numbes s not always less than the sum of the means It woul be tue f the followng equaton hel fo ma ( ) mex y me x y mex mey * * * * * * Howeve, the secon nequalty oes not always hol The compute package S has been use to examne ma, an values slghtly geate than one wee occasonally obtane Smulaton stues of ma show t to behave vey much lke othe coelaton coeffcents even wth the anomaly of occasonally beng geate than one The spea of the stbuton s vey close to othe coelatons, an only when the populaton coelaton s vey nea one can ma become slghtly geate than one In the case when X, Y have a bvaate nomal stbuton wth paametes, μ x,μ y,σ x,σ y,ρ, the populaton value s known to ρ ρ be ρma Substtute y fo x n fomula (3) an essentally MAD s ecovee Note that the same heustc motvaton fo Peason s hols fo ths absolute value CC Rank base coelaton coeffcents The fst NPCC base on absolute values s now efne In the same way that Speaman's CC s motvate fom Peason s by usng ect substtuton of anks, so s ths new

5 THE CORRELATION COEFFICIENTS 50 coelaton coeffcent obtane fom Defnton by substtuton of anks An nteestng hstocal note s that the NPCC n Defnton 4 was foun fst an av etemne fom t Fst ewte ( x x) ( y y) as x y ( x y) Replacng the ata by the anks an oeng the bvaate ata by the x ata, gves the ata n ank fom ths s just befoe the (, p ),,, n Thus p equals the ank of the y fo the x wth ank The means of the n anke ata ae, so x y becomes n The anks p ae hee assume stnct; te values wll be hanle late In Defnton, wth anks substtute, the tems SAx an SAy ae equal an can be factoe fom expesson () The value s n SAx SA y p n n Fo n o, n can be shown to be n n an fo n even t becomes ; fo ethe n even o o n, t s, the geatest ntege n n Thus the enomnato n () becomes n SA x n Defnton 4: Speaman s mofe footule coelaton coeffcent, Gn (94), Beto (993) mf ( x, y) n / ( n p p ) (4) The attempt by Speaman (906) to make an absolute value ank CC was also ocumente n Kenall an Gbbons (990) Speaman te to make a computatonally smple an obust CC an base t on one summaton The ea n ths atcle s that all o at least most coelatons shoul be a ffeence of two functons that measue stance fom postve an negatve coelaton, whch contasts wth Kenall s metho n Chapte n Kenall an Gbbons (990) Thee, Kenall avance the ea that some type of nne pouct shoul be use to efne all CCs The above two absolute value CCs cannot be efne usng Kenall s nne pouct concept Ths ffeence of two functons gves the necessay symmety to a CC The enomnato ases fom the absolute value of the numeato whch occus when p (coelaton ), o when p n (coelaton ) Note agan that the same heustc motvaton apples The foumulaton of Speaman's coelaton coeffcent base on Defnton s: Defnton 5: Speaman's coelaton coeffcent, Speaman (906) s (, x y) nn ( ) 3 ( ( n p ) ( p ) ) 6 ( p ) n( n ) (5) The lnea estcton that allows s to smplfy as shown oes not hol fo mf Two moe CCs ae to be efne Kenall s, fo whch a lnea estcton oes allow a smplfcaton of the efnng fomula an one base on maxmum o geatest evatons fo whch no smplfcaton occus Agan the natual efntons ae base on the ffeence of

6 5 RUDY A GIDEON two functons that measue stance fom pefect postve an negatve coelaton an makes the stbuton of the CCs symmetc about zeo fo the case when x an y ae nepenent, e the null case It wll also be shown that mf can be compute fom the quanttes efne fo the numeato of the Geatest Devaton CC Both Kenall s CC (k), usually calle Tau, an the one base on geatest evatons (g) use a countng technque that can be efne wth an ncato functon Let I () f the agument s tue 0 f false Recall that the ata ae assume oee by the x ata an fo the th lagest element of x, the ank of the coesponng y ata s p Fo Kenall s coelaton coeffcent, let n I ( p j > p ) nc, j count the numbe of concoances an n j I( p j < p ) n, count the numbe of scoances at poston (ecall that no te values ae yet allowe) The lage the numbe of concoances the smalle the numbe of scoances Let nc an n be the sum ove,,,, n- of the concoances an scoances, espectvely The concoance functon, nc, s a countng functon that measues stance of the anke ata fom a pefect negatve monotone elatonshp, wheeas n s a smla scete measue of the anke ata fom a pefect postve monotone elatonshp Defnton 6: Kenall s k coelaton coeffcent, see eg Kenall an Gbbons (990) n n n k ( x, y) nc, n, / (6) ( n c n n ) / (4 n /( n( n))) c (4 n /( n( n ))), n because nc n n The quantty means n choose Ths summaton of nc an n wll be shown n the next secton to be n choose usng a 0- gaphmatx fomulaton of the calculaton of k Fo the Geatest Devaton CC let j I( p > ), a functon that s lage j when thee s negatve coelaton an small f not; that s, the measue s lage f stance fom postve coelaton s geat Let j I( n p > ) Ths s a measue that s lage f stance fom negatve coelaton s geat Defnton 7: The Geatest Devaton coelaton coeffcent, g; Geon an Hollste (987) an n Geon, Pentce, an Pyke (989) n g ( x, y) (max max ) / (7) n n n whee s the geatest ntege n n/; t's value s the maxmum value of the ffeence n the numeato Ths completes the efntons of the coelaton coeffcents une conseaton The next secton gves some nsghtful examples; the wok s conseably ease usng a computatonal a that allows computatons of the fou nonpaametc coelaton coeffcents j

7 THE CORRELATION COEFFICIENTS 5 fom an augmente plot of the ata wth a 0- matx, calle a gaph-matx Methoology Computatons usng the gaph-matx The ata n ank fom ae (, p ),,, n Let e (,,,n) an p ( n p, p,, p ) be the ata n vecto fom The gaph of the anke ata wll have e plotte on the hozontal axs an p plotte on the vetcal axs The YMCA basketball ata that wee use n llustatng the Geatest Devaton CC (Geon & Hollste, 987) s use hee agan These ata occue as anks an they wll now be use to calculate all fou of the NPCCs that have been efne The e contans the anks of the won-lost ecos of the 6 teams that wee n the ffth gae league n Mssoula, Montana n 980 Rank one s the team wth the best eco Thoughout the season, afte each game, each coach was aske to ate the spotsmanshp of the opposng team an at the en of the season the cumulatve atngs wee pesente n ank fom wth ank one beng the team wth the hghest ate spotsmanshp These anks wee p (4,,6,,,3,7,9,0,3,8,,5,6,4,5) Note that n geneal the teams wth the best won-lost ecos ha the lowe spotsmanshp atngs The coelaton coeffcents put a measue on the elatonshp between wnnng an spotsmanshp The gaph-matx appeas n the mle of Fgue suoune by auxlay nfomaton The two leftmost an the two ghtmost columns as well as the two bottom ows ae ntemeate calculatons explane below Boeng the ata plot ae the axes labels The *s ncate the plotte ponts (, p ),,, n an unlke a scatteplot, the Catesan pouct, e x e, on the gaph s flle n wth 0s above each of the plotte ponts an s below The combnaton of these *s, 0s, an s ae use to calculate all fou NPCCs whch appea on the thee boes Although the efntons of the coelaton coeffcents may seem unwely, the countng technque s easy an quck to use It s eally moe convenent to use the metho f the agonals to the ata plot ae awn n, whch s ease one by han The lne of slope one s enote sl ; ths s the lne though (, ),,,, n The lne of slope mnus one, sl -, goes though ponts (, n ),,,, n Immeately below the gaph ae two ows that gve the values necessay to calculate the Speaman an Absolute Value CCs The uppe ow counts fom the * to the lne sl - wth a mnus sgn f the * s below sl - The lowe ow counts fom the * to the lne sl agan wth a mnus sgn f the * s below the lne It s ealy appaent that ths countng technque ectly coespons to the summans n the fomulas of Defntons 4 an 5 The sum of the absolute values of these two ows ae gven just to the ght of them (56, 06), followe by the sum of squaes of them (348, 0) To the ght of the gaph-matx ae two columns that gve the nvual concoances an scoances n Kenall s Tau as gven n Defnton 6 Statng at a * n poston (, p ), a 0 appeas n column j> (to the ght of the *) f an only f the ank of that column p j s n scoance ( p > p j ) an a appeas n a column to the ght of the * f an only f the ank of that column s n concoance ( p < p j ) To obtan the scoances, count the 0s to the ght of the * n each column, an to obtan the concoances count the s to the ght of each * n each column These esults appea n the two columns to the ght of the gaph The sums of the two columns, the total numbes of con- an scoances, ae gven below the columns as (38, 8) Note that the oeng wthn the two columns oes not match the stana algothm use to calculate Kenall s Tau, k To the left of the gaph ae two columns heae by an They label the values fo whch the maxmums nee to be taken n Defnton 7 of the Geatest Devaton coelaton coeffcent Fo each element n the column count all the 0's on an to the left of

8 53 RUDY A GIDEON YMCA basketball ata: coelaton computatons left: Geatest Devaton bottom: Speaman an Absolute Value ght: Kenall vetcal axs: spotsmanshp ankngs hozontal axs: won an lost stanngs * * * * * * * * * * * * * * 0 5 * 0 0 * g mf Fgue n c n the sl - lne To obtan each element n the column count all the 's on an to the left of the sl lne Fo example, 7 7 I( p j > 7) j, p, p3, p5, p 5, because exactly p 6 ae geate than 7 Usng the gaph, thee ae exactly 5 s on o to the left of sl n ow 7, coesponng pecsely to the fve p ' s mentone above, because n that pat of the plane, the secon coonate excees the fst Smlaly, 7 7 j I ( < 7 7 0) because only p 4 p j an p 7 ae less than 0 Now fo, the tem n p j > n the ncato functon means p j < n ; that s, count all the zeoes at n on the vetcal axs on an to the left of the sl - lne So fo 7, count all the zeoes at 7-70 on the vetcal axs on an to the left of sl - ; the 0s appea only n columns 4 an 7 coesponng to p 4 an p 7 beng less than 0 Just below the an columns ae the maxmums fo g an the below them ae the sums of these two columns It wll be shown that these sums can be use to compute mf

9 THE CORRELATION COEFFICIENTS 54 Note that twce 53 s 06 an twce 8 s 56, the numbes neee fo mf Fom the statstcs gven n Fgue, the ffeences n the numeatos of the fou coelaton coeffcents can be obtane an the enomnatos ae n n 8, n ( n ) / 3 360,, 0 n mf 03906, s k , g Note that the two numbes n the numeato fo s an k a to the enomnato ( s: , an k: ), the wellknown lnea estcton, but ths oes not occu fo g an mf as g: > 8, an mf: > 8 Specal fom fo calculaton of g If only g s ese, thee s a convenent algothm to compute the an values Wte own fo,,, n the thee ows vectos, p, n p ) Compute ( by placng a make just to the ght of the th poston an count left n the p ow an note all the anks geate than Compute by keepng the same make, but countng left n the n p ow notng all the anks geate than Ths s one n Table Note that n Fgue an Table appea n the same oe wheeas, the values ae evese Thee theoems ae gven below whch show some atonal usefulness of ths gaphmatx appoach The fst shows the elatonshp between the statstcs use n g an mf Theoem : p n p an, all sums fom to n Poof: Fst the n elatonshp s establshe Clealy ( p ) 0; that s, the sum of the evatons about the sl s zeo Thus, ( p ) ( p ) Now ( p ) p < p > p > just counts all the s on o above the sl lne But, j I( p > ) counts all the s n ow j I that ae on o above the sl lne so that ( p ) ( p ) o p > p < ( p ) p p > These equaltes ae emonstate n Fgue The bottom two ows cay sgns to allow these equaltes to be easly seen The poof of the elatonshp follows n a smla manne Theoem : The numbe of s on o to the ght of the sl - lne n ow - equals the numbe of 0s on o to the left of sl - n ow,,3,,n The numbe of 0s on o to the ght of the sl lne n ow equals the numbe of s on o to the left of the sl lne n ow -,, 3,,n (In ths theoem ow efes to the vetcal axs, whch ae anks; eg ow coespons to the bottom ow of the 0-gaph-matx) Fgue poves a guelne fo the poof The symmety splaye n ths theoem shows that the Geatest Devaton CC coul have been equvalently efne n a ght-hane fashon; e nstea of countng 0s an s fom the left to the agonal lnes, countng coul have been one fom the ght wth a sutable ajustment n

10 55 RUDY A GIDEON Theoem 3: Fo Kenall s CC, n n n c Poof: If the ata postons (*s) fell on the agonal of the gaph-matx t s clea that thee woul be a total of n n 0s an s wth complete ant-symmety The pemutaton of the columns to epct the actual ata oes not change ths total an hence, the total numbe of 0s an s to the left of the *s must equal the total numbe to the ght Thus, n n n nc n Futhe, the numbe of s to the ght (38 n Fgue ) equals the numbe of 0s to the left an the numbe of 0s to the ght (8 n Fgue ) equals numbe of s to the left Results Whch coelaton coeffcents ae outle esstant? In ths secton two examples ae gven to llustate that the fou NPCCs can have qute ffeent values on the same ata The maxmum ffeences between k an s appea on page 34 of Kenall an Gbbons (990) The examples below suggest that g an mf ae the most obust, k next, but that Speaman's s s not vey obust Let e an p be the ank vectos The calculaton of the coelaton coeffcents s left to the eae The values of the NPCCs fo n 0 an p (5,4,3,,,0,9,8,7,6 ) ae 6 7 mf 0500, s 055, k 0, g The values of the CCs now wth p (0,,,3,4,5,6,7,8,9) ae 4 6 mf 0800, s 008, k 0444, g It s known that fo the bvaate nomal stbuton, the NPCCs estmate a functon that s less than the coelaton paamete, ρ When the CCs ffe geatly, t suggests that thee ae stange obsevatons n the ata Hee, g an mf gve the lagest ncaton of a postve Table Calculaton of the Geatest Devaton CC max p n-p

11 THE CORRELATION COEFFICIENTS 56 Table 3 Speaman ata Peson aton soun D 3 I H 3 B 4 45 J 5 45 E 6 A 7 6 K 8 9 F 9 8 C 0 0 G 7 () (p) elatonshp fo the stange ata of these two examples Hence, they may be the most esstant to outles o to any unusual ata (Wok n pogess shows them moe eslent) Pobabltes an asymptotcs fo the ank coelaton coeffcents Some aspects of the ank CCs wll be compae by usng an example fom Speaman (906) concenng the elatonshp between the ablty of people to a numbes quckly an accuately an the ablty to stngush between two soun tones Speaman use ths example to llustate hs footule CC The ata wee fo eleven stuents of psychology; Speaman anke the ablty n ptch scmnaton an a secon peson anke nepenently fo aton ablty The ata ae oee by the aton vaable an note the two te values wth the usual conventon use, whch coul be calle a local conventon as oppose to a moe useful global efnton gven below Speamans s footule CC s f 6 ( p ) p 6(85) > 057 n 0 Because ths footule only nvolve stance fom pefect postve coelaton, t s not a val coelaton coeffcent It s nteestng fom a hstocal pespectve He compae ths numbe to pobable eo (eve n hs atcle) of 03 an conclue because 057/03 438, the faculty of ang numbes an that of scmnatng ptch s just about lage enough to be beyon all easonable suspcon of mee chance concence (p 96) Speaman not use a table of ctcal values but nstea state a heustc value fo the above ato to be sgnfcant The fou nonpaametc CCs an the coesponng pobablty values ae now compute fo ths ata Refeng to what s now known as the Speaman CC (the ank equvalent of Peason s CC; e, s) Speaman sa, the effect of squang s to gve moe weght to the exteme ffeences as compae wth the mean ones Ths s pobably a conseable avantage n most physcal measuements But n othe fels of eseach, an pehaps above all n Psychology, these exteme cases ae just the ones of most suspcous valty, so that the squang s hee moe lkely to o ham than goo (p 99) Thus, Speaman wante a obust CC fo hs ata Ths example llustates the efnton of a ank CC when te values ae pesent In

12 57 RUDY A GIDEON avance wok on the use of CCs n estmaton, the cuent local methos of te value calculatons ae not aequate an hence a global metho fst ntouce n Hollste an Geon (987) s pesente In ths metho, the calculatons ae one twce: fst when Peson B s assgne ank 4 fo soun an Peson J s assgne ank 5 fo soun, favong postve coelaton; n the secon calculaton tes ae boken n the evese ecton to favo negatve coelaton Note that g s the only CC wthout a change Each CC can be efne unquely by aveagng the values of the two exteme coelaton coeffcents In Table 4, g emans at but mf becomes ( )/ 0767 A geneal global efnton fo an altenatve te value poceue s now gven Defnton: The global values of ank CC when tes ae pesent Let ( x, y) be a set of ata, an ( I, P ) be the coesponng anks whch ae assgne among the te values n the way that most favo postve coelaton, an let ( I, P ) the coesponng anks assgne among the te values n the way to most favos negatve coelaton I becomes e an P an P ae pemutatons of e Then a ank coelaton coeffcent,, s efne unquely fom the two extemes, P an P Its value s ( x, y) ( ( e, P ) ( e, P )) / (8) The quanttes ( e, P ) an ( e, P ) ae abbevate to an -, espectvely As an example, let ( x, y) ((,,,4,5), (,,,,3)) Then P (,,4,3,5) an P - (3,4,,,5) Thus, fo g, / ( / ) g 0 Retun to the level of sgnfcance fo the Speaman example The numeatos an values of the fou NPCCs as compute by the 0- gaph-matx metho ae gven n Table 4 The enomnatos ae 5, 55, 60, ( ) Tal pobabltes ae obtane fom Neave (978) fo k an s, fom Geon an Hollste (987) fo g, an fom Beto (993) fo mf The table values ae compae to the asymptotc values compute fom the asymptotc stbutons whch ae gven n Kenall an Gbbons (990) fo s an k an n Geon, Pentce, an Pyke (989) fo g The asymptotc null stbutons ( ρ 0) of the fou CCs ae gven fst These ae n s N(0,) ; n s N (0,4 / 9) ; s n g s N (0,) ; n s N(0, / 3) Fo completeness the exact vaances of each CC s gven; V ( s ) /( n ) ; V ( k ) (n 5) /(9n( n )) ; V ( g ) s unknown; V ( mf ) ( n ) /(3n ( n )) fo n even an ( n 3) /(3( n )( n )) fo n o The one te s neglecte an the ata fo the most coelaton case, P, s use Fst, fom tables, 000 P ( 07636) 0005 ; s 00 P ( 05636) 005 ; k 00 P ( 06000) 005 ; g P ( / ) 0003 an mf P ( 7 / ) 0004 mf Thus, all of the CCs ae sgnfcant wth s an mf beng the most sgnfcant These esults ae now compae to the asymptotc appoxmatons usng the notaton of Z as N(0,) mf k

13 THE CORRELATION COEFFICIENTS 58 Table 4 Speaman s 906 Data an Coelatons The pas of numbes n the numeatos show stances fom an coelaton Coelatons ae n secon ow of the name coelaton most most - aveage 5-5- g mf k s Table 5: Some asymptotc compasons P ( 07636) PZ ( 0(07636) s 447) (05636) P ( k 05636) PZ ( /3 6734) P ( 06000) PZ ( (06000) g 9900) (07333) P ( mf 07333) PZ ( /3 840) 0003

14 59 RUDY A GIDEON All of these appoxmate esults ae easonably goo All fou coelatons suppot Speaman s concluson that hs footule CC gave Speaman ew hs concluson by compang hs footule value of 057 to the pobable eo, whch he gave as 03 Thus, 057/ Ths example s conclue by compang the value of mf, the mofe footule CC, 07333, to ( 3) V( mf ) 3(0)( ) Now, 07333/ an by Speaman s ule of satsfactoy emonstaton that ths ato be at least 4, ha Speaman foun the coect fomulaton, mf, he woul have awn the opposte concluson (p 96) Agan fo ths example t shoul be ponte out that s an k have a lnea estcton but mf an g o not Hence, the tems n the numeato, when ae gve the enomnato fo s an k but not fo mf an g Fo s: an fo k: wheeas fo mf: > 60 an fo g: 5 7 > 5 Concluson By vewng coelaton boaly as the ffeence between measues of stance fom pefect negatve an pefect postve coelaton, many new fomulatons of coelaton may be efne Two new contnuous coelaton coeffcents ae base on absolute values an means The mean one s an extenson of the MAD scale measuement an the absolute value one pouces Gn s CC when ata anks ae substtute A 0- gaph-matx was ntouce as an extenson to the plot of the bvaate ank ata an use to compute all fou nonpaametc coelaton coeffcents an exhbt some elatonshps Seveal examples suggest whch of the coelatons ae most obust: the Geatest Devaton an Gn A ata set fom Speaman was use to emonstate the applcaton of the asymptotc stbutons, to compae the coelatons on the same ata, an to llustate a global te value poceue Ths poceue oes not seem ctcal hee, but fo late evelopments on the use of coelaton coeffcents n estmaton t s essental Seveal tmes the nomal stbuton was selecte to set up notaton but ths s not necessay, as any stbuton fom the class of bvaate t stbutons woul suffce The fou nonpaametc coelaton coeffcents woul be stbuton-fee on ths class of bvaate stbutons wth ellptcal shape contous, nclung the Cauchy stbuton Refeences Beto, B (993), On the Dstbuton of Gn's Rank Coelaton Assocaton Coeffcent, Communcatons n Statstcs: Smulaton an Computaton,, No, Geon, R A & Hollste, R A (987), A Rank Coelaton Coeffcent Resstant to Outles, Jounal of the Amecan Statstcal Assocaton 8, no398, Geon, R A, Pentce, M J, & Pyke, R (989) The Lmtng Dstbuton of the Rank Coelaton Coeffcent g, appeas n Contbutons to pobablty an statstcs (Essays n Hono of Ingam Olkn) ete by Glese, L J, Pelman, M D, Pess, S J, & Sampson, A R NY: Spnge- Velang, p 7-6 Geon, R A wwwmathumteu/geon Gn, C (94), L'Ammontae c la Composzone ella Rcchezza ella Nazon, Bocca, Tono Kenall, M G & Gbbons, J D (990), Rank coelaton methos, 5th e Oxfo Unvesty Pess, o also Kenall, M G (96), Rank coelaton methos, 3 e GB: Hafne Publ Co Neave, H R (978), Statstcal tables, Lonon: Geoge Allen & Unwn Publshes, Lt/ Roges, J L & Ncewate, W A (988), Thteen Ways to Look at the Coelaton Coeffcent, The Amecan Statstcan, 4, no, Rousseeuw, P J & Coux, C (993), Altenatves to the Mean Absolute Devaton, Jounal of the Amecan Statstcal Assocaton, 88, Scasn, M (984), On Measues of Conoance, Stochastca, 8, No 3, 0-8 Schweze, B & Wolfe, E F (98), On Nonpaametc Measues of Depenence fo Ranom Vaables, The Annals of Statstcs, 9, Speaman, C (906), 'Footule' fo Measung Coelatons, Btsh Jounal of Psychology,, 89-08