Tests for Correlation on Bivariate Non-Normal Data

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Jounl of Moden Applied Sttisticl Methods Volume 0 Issue Aticle 9 --0 Tests fo Coeltion on Bivite Non-Noml Dt L. Bevesdof Noth Colin Stte Univesity, lounneb@gmil.

edu/jmsm Pt of the Applied Sttistics Commons, Socil nd Behviol Sciences Commons, nd the Sttisticl Theoy Commons Recommended Cittion Bevesdof, L.

1 Jounl of Moden Applied Sttisticl Methods Volume 0 Issue Aticle Tests fo Coeltion on Bivite Non-Noml Dt L. Bevesdof Noth Colin Stte Univesity, lounneb@gmil.com Ping S Univesity of Noth Floid, ps@unf.edu Follow this nd dditionl woks t: Pt of the Applied Sttistics Commons, Socil nd Behviol Sciences Commons, nd the Sttisticl Theoy Commons Recommended Cittion Bevesdof, L. nd S, Ping (0) "Tests fo Coeltion on Bivite Non-Noml Dt," Jounl of Moden Applied Sttisticl Methods: Vol. 0 : Iss., Aticle 9. DOI: 0.37/jmsm/30680 Avilble t: This Emeging Schol is bought to you fo fee nd open ccess by the Open Access Jounls t DigitlCommons@WyneStte. It hs been ccepted fo inclusion in Jounl of Moden Applied Sttisticl Methods by n uthoized edito of DigitlCommons@WyneStte.

2 Jounl of Moden Applied Sttisticl Methods Copyight 0 JMASM, Inc. Novembe 0, Vol. 0, No., //$95.00 Tests fo Coeltion on Bivite Non-Noml Dt L. Bevesdof Ping S Noth Colin Stte Univesity, Rleigh, NC Univesity of Noth Floid, Jcksonville, FL Two sttistics e consideed to test the popultion coeltion fo non-nomlly distibuted bivite dt. A simultion study shows tht both sttistics contol type I eo tes well fo left-tiled tests nd hve esonble powe pefomnce. Key wods: Sddlepoint ppoximtion, Fishe s tnsfomtion, tests fo coeltion, bivite nonnoml distibution. Intoduction Bivite dt e dt in which two vibles e mesued on n individul. If the vibles e quntittive, eseche my be inteested in descibing the eltionship between them. One mesue used to descibe the stength of line eltion between two quntittive vibles is the line coeltion coefficient, denoted by ρ. The tue eltionship between two vibles of inteest is lwys unknown. Diffeent estimtos hve been poposed fo ρ nd two of them e used fequently: () the Spemn Rnk Ode Coeltion, which is used fo odinl dt, nd () the Peson Poduct Moment Coeltion, which is pplied to intevl nd tio dt. The mximum likelihood estimto of ρ is the Peson poduct-moment Ping S is Pofesso of the Deptment of Mthemtics nd Sttistics. She eceived he Ph.D. in Sttistics fom the Univesity of South Colin in 990. She hs published moe thn 0 ppes. He ecent scholly ctivities hve involved esech in multiple compisons, qulity contol nd sttisticl infeence fo nonnoml dt. Emil: ps@unf.edu. Ms. Bevesdof eceived he Mste s degee in Sttistics fom the Univesity of Noth Floid in 008. She is cuently Ph. D. cndidte t Noth Colin Stte Univesity. Emil: lounneb@gmil.com. coeltion coefficient. When the dt is not bivite noml nd the smple size exceeds 0, the nonpmetic Spemn nk coeltion is useful. Little wok hs been done fo cses when the distibution of the dt is unknown nd the smple size is eltively smll. The most popul ρ estimto is the Peson Poduct Moment Coeltion Coefficient,, which is bised point estimto fo ρ, howeve, the bis is smll when n (smple size) is lge. Given two vibles Y nd Y, the sttistic is: ( Yi Y)( Yi Y) i n / ( Yi Y) ( Yi Y) i n whee ( Y i, Yi ) is the i th obsevtion of the bivite dt (Y, Y ),,(Y n,y n ), Y is the smple men of Y nd Y is the smple men of Y. Reseches hve done intensive wok on the distibution of when the popultion is bivite noml (Fishe, 95; Stut & Od, 994). It hs been found tht, when n, the distibution of cn be egded s n exteme cse of U-shped distibution, fo n 3 the density is still U-shped, but if n 4 the distibution is unifom when ρ 0 nd J-shped othewise. Fo n > 4 the density function is unimodl nd hs incesed skew s ρ inceses, this follows fom the fct tht the, 699

3 TESTS FOR CORRELATION ON BIVARIATE NON-NORMAL DATA mode moves with ρ nd. Fo ny ρ, the distibution of slowly tends to nomlity s n (Stut & Od, 994). When the popultion is bivite noml nd hs equl vince, test sttistic t * n cn be deived to test H 0 : ρ 0. Unde H 0, () follows the Student s t-distibution with (n ) degees of feedom, denoted t (n-). Disdvntges of this test include the need fo eltively lge smple o bivite noml dt nd the bility to test only fo ρ 0. When the popultion is not bivite noml nd the smple size exceeds 0, nonpmetic sttistic, the Spemn Rnk Coeltion Coefficient (Spemn), is typiclly used to mesue the ssocition between two vibles when no tnsfomtion fo the dt cn be found to ppoximte bivite noml distibution. Spemn, denoted by s, is then defined s the odiny Peson poduct-moment coeltion coefficient bsed on dt nking: s ( Ri R )( Ri R ) ( Ri R ) ( Ri R ) [ ] / whee (R i, R i ) e the nks of ( Y i, Yi ) espectively; nd R is the men of the nks of R i, i,, n, nd R is the men of the nks of R i, i,, n. Spemn cn lso be used to test the ssocition between the two vibles with the null hypothesis, H 0, stting: thee is no ssocition between Y nd Y. When smple size n, exceeds 0, the test sttistic: t * s s, * t s n, () cn be used. t * s is ppoximtely t-distibution with n degees of feedom unde H 0. This is nonpmetic test nd thus my esult in lowe powe pefomnce. Agin this test cn only be used fo testing whethe n ssocition exists. The pupose of this study is to test H 0 : ρ ρ 0, whee ρ 0 cn be vlues othe thn zeo, fo bivite non-noml dt. Fishe s Z- tnsfomtion nd sddlepoint tnsfomtion e investigted nd tested. Methodology Two sttistics fo testing the coeltion coefficient of bivite non-noml popultions e investigted: () Fishe s z-tnsfomtion, denoted F, nd () the sddlepoint ppoximtion, denoted L. These methods e used on bivite non-noml dt sets with smll smple sizes. The gol is to detemine if eithe of the two methods is ppopite fo hypothesis testing bout the popultion coeltion coefficient, specificlly fo bivite non-noml dt sets with smll smple size. Fishe s Z-Tnsfomtion The smpling distibution of is complicted when ρ 0 even when the popultion is bivite noml. Fishe (9) deived n ppoximtion pocedue bsed on + tnsfomtion of, z ' log nd it tends to nomlity much fste thn. Afte stnddizing, the sttistic fo Fishe s clssicl tnsfomtion is given by: F + + ρ ρ log log n 3. ρ ( n ) (3) Sddlepoint Appoximtion Sddlepoint ppoximtions wee intoduced by Dniels (954). Howeve, computtions of these ppoximtions only ecently becme fesible with the vilbility of inexpensive computing powe. In pctice, sttisticl infeence often involves test sttistics with noml distibutions, which e vlid s smple sizes incese. Fo smll smple size poblems, these distibutions tend to povide inccute esults. Sddlepoint methods offe ppoximtions tht e ccute to highe ode thn fist-ode ppoximtions nd thei 700

4 BEVERSDORF & SA ccucy holds fo extemely smll smple sizes (Huzubz, 999). Sddlepoint ppoximtions lso povide good estimtes to vey smll til pobbilities o to the density in the tils of the distibutions. Jensen (995) tnsfoms the Peson coeltion coefficient using Lplce tnsfomtions to deive function of tht cn be nomlized nd he clims tht L is nomlly distibuted to high ccucy. Assuming bivite noml dt set with coeltion ρ, the sddlepoint ppoximtion, denoted L, povided by Jensen (995) is: L u v + log, (4) v v whee ρ v sgn ( ρ ) ( n 4)log, ρ 3 ρ ρ u n 4, ρ nd sgn(.) is the sign of ( ρ ). Poposed Test A new test is equied to investigte the hypothesis H: 0 ρ ρ0 vesus thee possible ltentive hypotheses, H:ρ ρ0, H:ρ > ρ 0 nd H:ρ < ρ 0, when dt set is bivite nonnoml nd smple size is eltively smll to modete. Although both the Fishe nd sddlepoint tnsfomtions e deived fo bivite noml dt, little wok hs been done to investigte if they cn lso be used fo nonnoml bivite dt; thus, the two ppoximtions, F in (3) nd L in (4), e used s the test sttistics fo the hypothesis H: 0 ρ ρ0. Note tht ρ 0 should be used in both equtions wheneve ρ is pesent. The decision ule to eject the null hypothesis fo the twotiled, uppe-tiled nd lowe-tiled tests is F > z α/ o L > z α/, F, L > z α, nd F, L < -z α,, espectively. Simultion Study: Geneting Bivite Non- Noml Dt Fleishmn (978) deived method fo geneting univite non-noml ndom vibles. Fleishmn s method is bsed on the vible Y defined s Y bz + cz dz (5) whee Z is stndd noml ndom vible, nd, b, c nd d e constnts chosen in such wy tht Y hs the desied coefficients of skewness nd kutosis, γ nd γ, espectively. Fleishmn showed tht c nd the constnts b, c nd d e detemined by simultneously solving the following thee equtions: b + 6bd + c + 5d 0 cb 4bd 05d 0 ( ) γ bd + c ( + b + 8bd ) + d ( + 48bd + 4c + 5d ) 4 γ 0 (6) Using these equtions, non-noml ndom vible Y cn be obtined by geneting stndd noml vible Z nd using the eqution (5). Vle nd Muelli (983) poposed geneting multivite non-noml ndom vibles with specified coeltion stuctue bsed on Fleishmn s method. Fo bivite nonnoml ndom dt, (Y, Y ) with desied coefficients of skewness nd kutosis, (γ nd γ ) fo Y nd (γ nd γ ) fo Y, solutions to the system of equtions (6) given in Fleishmn s method must be found. Let Z, Z be two stndd noml coelted vibles. Y nd Y cn be clculted with the following equtions: Y + bz + cz + d Z, Y b Z c Z d Z (7) 70

5 TESTS FOR CORRELATION ON BIVARIATE NON-NORMAL DATA Vle nd Muelli (983) found tht the coeltion coefficient between Y nd Y is: ρ Y, Y ρ ( bb + 3bd + 3b d + 9d d ) Z, Z 3 ρz, Zcc ρz, Z6dd + + (8) Fo desied coeltion, ρ, the intemedite y, y coeltion, ρ, cn be detemined by solving Z,Z the bove cubic eqution. The bivite nonnoml ndom vite (Y, Y ) cn then be obtined by fist geneting set of bivite stndd noml ndom vite with coeltion ρ, nd then using eqution (7). Z,Z Simultion Desciption Diffeent vlues of skewness nd kutosis wee chosen fo the simultion study in ode to eflect diffeent popultion distibutions. Fou vlues of skewness, 3,,, 3 nd thee vlues of kutosis, 3, 7, 5 wee used, esulting in 78 possible pis of popultions. A eltively smll smple size of 0 nd modete smple size of 0 wee used in the study nd the test sttistics L nd F wee investigted fo type I eo tes with left-tiled, ight-tiled nd two-tiled tests with the nominl levels of 0.0 nd 0.05 fo ech smple. Compisons in the simultion study use L nd F ginst thee citicl vlues, z α, t (n-, α), nd (z α +t (n-, α) )/, to dw conclusions. Fou ρ 0 vlues 0, 0.5, 0.7 nd 0.9 wee evluted s the hypothesized vlues fo H: 0 ρ ρ0. Whenρ 0 0, the t * in () nd t * s in () e lso included in the study fo compison puposes. The simultion study hs two pts: the type I eo te compisons nd the powe study. The steps of the simultion e: Dt Genetion: Steps () (5) ) Input the five popultion pmetes: skewness nd kutosis fo ech of the two popultions nd the desied popultion coeltion; ) Solve the system of equtions (6) to clculte coefficients, b, c nd d fo the two popultions; 3) Solve ρ by eqution (8); z z 4) Genete n bivite stndd noml vibles (Z, Z ) with coeltion ρ ; z z 5) Apply the tnsfomtion in (7) to obtin the non-noml smple dt Y nd Y ; Evlution: Steps (6) (8) 6) Evlute L nd F nd compe to citicl vlues z α, t (n-, α), nd (z α +t (n-, α) )/; if ρ0 0, * t nd t * s e evluted nd comped to t (n- ) citicl vlue; 7) Repet steps (4) (6) 99,999 times; 8) Clculte type I eo te fo ech method by finding the popotion of ejection in the 00,000 smples. In the powe study, n ext pmete ρ (which is diffeent fomρ 0 ) is input in step () nd used to genete the dt s the tue popultion coeltion, howeve, ll test sttistics in step (6) e evluted unde ρ 0. All othe steps in the powe study e identicl to the type I eo te study. All the simultions wee un with Fotn 77 fo Windows on Toshib Stellite-A05 Lptop Compute. Results Type I Eo Rte Compison Tbles -4 povide compisons of type I eo tes with smple size n 0. The set of popultion pmetes fo skewness nd kutosis e in the fist column with the fist popultion s pmetes in the fist ow nd the second in the second ow. Compisons wee mde between the tests fo sddlepoint nd Fishe s tnsfomtion, given in the tble s the two djcent numbes within given coeltion column, L nd F, espectively. Thee citicl 70

6 BEVERSDORF & SA zα + t points t n, α n, α, nd z α wee used fo the two poposed methods. The esults e the fist, second nd thid numbes in the espective column. Peson nd Spemn e evluted with citicl vlue tn, α fo ρ 0 only, nd the type I eo tes e epoted in the fist column with Peson fist nd Spemn undeneth. Due to simil esults in the study, only pis of popultions nd the smll smple size n 0 e epoted in the tbles. Also, lthough ll the tests e done with levels of significnce 0.05 nd 0.0, both levels e epoted hee only fo the left-tiled tests. (Fo complete simultion esults, plese contct the fist utho.) Left-Tiled Type I Eo Rtes Left-tiled type I eo tes e given in Tbles nd. Tble uses significnce level of 0.05 nd Tble uses significnce level of 0.0. It cn be obseved tht only slight diffeences in type I eo tes e pesent between the esults fo the sddlepoint nd Fishe s tnsfomtions. This sme esult ws obseved thoughout the simultion study. Results using the t citicl vlue chieve vey good type I eo tes fo ll of the distibutions. The z citicl vlue esults in few slightly inflted type I eo tes nd only by the sddlepoint ppoximtion. The wost cse found in the study, poduced by the sddlepoint ppoximtion, is fo the pi popultions with the sme (skewness, kutosis) (3, 5) unde ρ 0.9 using z α s the citicl point. The type I eo te fo this cse is Howeve, fte zα + t n, α the citicl point ws chnged to, the type I eo te decesed to nd it futhe decesed to when tn, α is used. The Fishe s tnsfomtion, by contst, contols the type I eo tes popely fo nely ll cses consideed. Fo the impotnt cse when ρ 0, esults show tht both the L nd F sttistics contol type I eo tes using ny of the thee citicl vlues t the 0.05 significnce level. When the significnce level is loweed to 0.0, some of the type I eo tes using the z citicl vlue e slightly inflted but within cceptble nge. Supisingly, Peson contols the type I eo tes bette thn the Spemn method. It pefomed vey well fo the 0.05 significnce level; howeve, those involving popultion with lge kutosis e slightly inflted when the significnce level is loweed to 0.0. Spemn hs some slightly inflted type I eo tes t both significnce levels. Ovell, it is fi to sy tht essentilly ll cses studied poduced contolled type I eo tes fo the left-tiled test. Right-Tiled & Two-Tiled Type I Eo Rtes Right-tiled type I eo tes e shown in Tbles 3 with significnce level of (Although the 0.0 level of significnce is lso studied, the tble is omitted due to the simil esults.) With the ight-tiled test, most type I eo tes e inflted, the only vlues tht stnd out e fo tests whee the t citicl vlues wee used nd both the skewness nd kutosis wee eltively smll. A get esult is found fo the t citicl vlues when ρ 0, type I eo tes e contolled fo both the L nd F. As opposed to the left-tiled test, the Spemn t-test woks bette thn the Peson; howeve, esults e still not s good s the coesponding esults by L nd F. Ovell, both Sddlepoint nd Fishe s sttistics e bette cndidtes fo testing H:ρ > 0. The t citicl vlue poduces moe stble esults then the z citicl vlue, lthough the two sttistics cn lso be used fo othe ρ 0 vlues if the popultions hve smll kutosis with t citicl points, in genel the two sttistics e not ecommended fo ight-tiled test. Two-tiled type I eo tes e shown in Tble 4. As expected, the esults of the twotiled tests e moe contolled thn tht of the ight-tiled test. Howeve, becuse the methods essentilly filed fo the ight-tiled tests, they e not ecommended to be used to pefom two-tiled test. Powe Results Tble 5 summizes the esults of the powe study fo left-tiled tests with H: o ρ 0.7 vesus vious ρ vlues such tht 703

7 TESTS FOR CORRELATION ON BIVARIATE NON-NORMAL DATA ρ < 0.7. Five diffeent ρ vlues nd two levels of significnce wee investigted, but only thee ρ nd α 0.05 esults e epoted hee. Powe esults fo both methods show esonble te of convegence to pobbility. As expected, the z citicl vlues hve highe powe thn the othe two tests. (Fo complete simultion esults, plese contct the fist utho.) Conclusion This study poposed nd exmined two sttistics, the sddlepoint tnsfomtion, L, nd Fishe s tnsfomtion, F, fo testing coeltion which my o my not be zeo fo ny bivite non-noml popultion. The simultion study indictes tht the two sttistics pefom similly. They both hve vey good obust pefomnce fo ll the distibutions studied when testing left-tiled test; they mintin the type I eo tes close to the nominl level nd show esonbly good powe. The two sttistics e not ecommended fo testing ight-tiled test o two-tiled test unless the pctitione knows fo cetin tht the popultions hve both smll skewness nd kutosis. In these cses, the two test sttistics with t citicl point cn popely contol the type I eo tes. The two sttistics cn lso be used fo testing H o : ρ 0 vesus ny of the thee possible ltentive hypotheses. They contol type I eo tes bette thn the existing Peson nd Spemn t-tests. Becuse the two sttistics e deived bsed on bivite noml popultion, smple size of t lest 0 is ecommended. Refeences Dniels, H. E. (954). Sddlepoint ppoximtions in sttistics. Annls of Mthemticl Sttistics, 5, Fishe, R. A. (95). Fequency distibution of the vlues of the coeltion coefficient in smples fom n indefinitely lge popultion. Biometik, 0(4), Fishe, R. A. (9). On the pobble eo of coefficient of coeltion deduced fom smll smple. Meton, (4),. Fleishmn, A. (978). A method fo simulting non-noml distibution. Psychometik, 43, Huzubz, S. (999). Pcticl sddlepoint ppoximtion. The Ameicn Sttisticin, 53(3), 5-3. Jensen, J. (995). Sddlepoint ppoximtion. New Yok: Oxfod Univesity Pess, Inc. Stut, A., & Od, J. K. (994). Kendll s dvnced theoy of Sttistics, Vol., 6 th Ed. New Yok: Hlsted Pess. Vle, C., & Muelli, V. (983). Simulting multivite non-noml distibutions. Psychometik, 48,

8 BEVERSDORF & SA Skewness Kutosis Tble : Type I Eo Rtes fo Left-Tiled Test, 0.05 Level of Significnce RHO 0 RHO 0.5 RHO 0.7 RHO 0.9 Peson Spemn L F L F L F L F

9 TESTS FOR CORRELATION ON BIVARIATE NON-NORMAL DATA Skewness Kutosis Tble : Type I Eo Rtes fo Left-Tiled Test, 0.0 Level of Significnce RHO 0 RHO 0.5 RHO 0.7 RHO 0.9 Peson Spemn L F L F L F L F E-05 5E E-05 5E

10 BEVERSDORF & SA Skewness Kutosis Tble 3: Type I Eo Rtes fo Right-Tiled Test, 0.05 Level of Significnce Peson Spemn RHO 0 RHO 0.5 RHO 0.7 RHO 0.9 L F L F L F L F

11 TESTS FOR CORRELATION ON BIVARIATE NON-NORMAL DATA Skewness Kutosis Tble 4: Type I Eo Rtes fo Two-Tiled Test, 0.05 Level of Significnce Peson Spemn RHO 0 RHO 0.5 RHO 0.7 RHO 0.9 L F L F L F L F

12 BEVERSDORF & SA Tble 5: Powe Results fo Left-Til Test when ρ 0.7, 0.05 Level of Significnce RHO 0.7 RHO 0.5 RHO 0.3 RHO 0. Skewness Kutosis L F L F L F L F

( ) D x ( s) if r s (3) ( ) (6) ( r) = d dr D x

( ) D x ( s) if r s (3) ( ) (6) ( r) = d dr D x SIO 22B, Rudnick dpted fom Dvis III. Single vile sttistics The next few lectues e intended s eview of fundmentl sttistics. The gol is to hve us ll speking the sme lnguge s we move to moe dvnced topics.