Bias Correction in Estimation of the Population Correlation Coefficient

Transcription

1 Kasetsart J. (Nat. Sc.) 47 : (3) Bas Correcto Estmato of the opulato Correlato Coeffcet Juthaphor Ssomboothog ABSTRACT A estmator of the populato correlato coeffcet of two varables for a bvarate ormal dstrbuto was proposed ad evaluated usg comparsos wth the earso correlato coeffcet ad a estmator of Olk ad ratt, coducted usg a smulato study. It was foud that for a small sample sze of, the absolute bas of the proposed estmator was less tha those of the earso correlato coeffcet ad a estmator of Olk ad ratt. I addto, the mea square errors of those estmators seemed to have o dfferece each stuato for ths study. Keywords: earso correlato coeffcet, bvarate ormal dstrbuto, absolute bas, mea square error, estmator. Itroducto The earso correlato coeffcet s oe of the most frequetly used tools of researchers for correlato coeffcet vestgato as metoed by Rodgers ad Ncewader (988) ad Huso et al. (7). Ufortuately, Neter et al. (996) ad Zmmerma et al. (3) cosdered that t was a based estmator for the populato correlato coeffcet (ρ). Furthermore, the bas decreased whe the sample sze creased ad t was zero whe the populato correlato coeffcet was zero or oe. I addto, ths coformed to the research of Ssomboothog (a). A covetoal estmate of the earso correlato coeffcet s lkely to uderestmate the populato correlato coeffcet (Gorsuch ad Lehma, ; Adolph ad Hard, 7; Zmmerma et al., 3) because the dstrbuto of ths estmator s asymmetrcal (Fsher, 9). Therefore, research actvty to fd a correcto for bas the estmato of the correlato coeffcet has already bee udertake. Fsher (95) publshed a approxmately ubased estmator of the populato correlato coeffcet samples from a deftely large populato. Later, Olk ad ratt (958) developed the earso correlato coeffcet to decrease the amout of bas of ths estmator for two varables havg a bvarate ormal dstrbuto wth equal varaces. The Zmmerma et al. (3) demostrated that the bas of the earso correlato coeffcet was almost elmated by Fsher s estmator ad a related estmator proposed by Olk ad ratt (958). I the preset study, a estmator of the populato correlato coeffcet of two varables for a bvarate ormal dstrbuto was proposed ad the jackkfe method (Queoulle, 949; Tukey, 958) was appled for bas reducto Departmet of Statstcs, Faculty of Scece, Kasetsart Uversty, Bagkok 9, Thalad. E-mal: fscjps@ku.ac.th Receved date : 4//3 Accepted date : 8/5/3

2 454 Kasetsart J. (Nat. Sc.) 47(3) (Efro ad Tbshra, 993; Smth ad otus, 6; Ssomboothog, a, b). Furthermore, comparsos of the absolute bases ad mea square errors of three estmators the proposed estmator, the earso correlato coeffcet, ad a estmator of Olk ad ratt were performed by a smulato study. Materals ad Methods Ths study proposed a estmator of the populato correlato coeffcet ad appled the jackkfe method for bas reducto of the earso correlato coeffcet. I order to emprcally evaluate the valdty ad relablty of the proposed estmator, a smulato study was coducted for 5 stuatos. The, comparsos of the absolute bas ad mea square error of the proposed estmator ad two sample correlato coeffcets the earso correlato coeffcet ad the Olk ad ratt estmator were emprcally performed. Sample correlato coeffcet Let (x, y ),..., (x, y ) be a radom sample from a bvarate ormal dstrbuto wth meas μ, μ varaces σ, σ ad the populato correlato coeffcet ρ. It s well kow that the maxmum lkelhood estmator of ρ, deoted by ρ, s gve by Equato ρ x ad y ( x x)( y y) ( x x) ( y y) x y () (Neter et al., 996; Aderso, 3). Ths estmator s ofte called the earso correlato coeffcet. It s a based estmator of ρ (uless ρ or ), whch s usually small whe the sample sze s large (Neter et al., 996; Zmmerma et al., 3; Ssomboothog, a). Later, Olk ad ratt (958) recommeded the correcto estmator the form of O + ( 3) as a more early ubased estmator of ρ. Zmmerma et al. (3) usg smulato demostrated that the bas of ρ ˆO was less tha that of ρ for a small sample sze. roposed estmator Ths secto proposes a estmator of ρ ad apples the jackkfe method for bas reducto of ρ as follows: ) Suppose we have a radom sample from a bvarate ormal dstrbuto wth mea vector ad a varace covarace matrx Σ σ σ σ ρ σ σ ad t s σ σ gveby S ((x, y ), (x, y ),..., (x, y )). I addto, a estmator of ρ s show Equato. δ( S) x ad y ( x x)( y y) ( x x) ( y y) x y. ()

3 Kasetsart J. (Nat. Sc.) 47(3) 455 ) The th jackkfe sample, S (-), cossts of the dataset wth the th observato removed. ( S ( x, y ),( x, y ),..., ( x, y ),( x, y ), ( ) ( ) ( ) ( + ) ( + ) for,,,. 3) Let δ(s (-) ) be the th jackkfe replcato of δ(s) ad t s gve by Equato 3 ( ) ˆ δ S ρ x ad the form J by J ( ) ( ) j ( ) y ( ) j ( x x )( y y ) j ( ) j ( ) ( x x ) j ( ) j x j y j for,,,. ( yj y ( ) ) j j (3) 4) Calculato of the pseudo values J δ( S) ( ) δ( S( ) ) ( ). ( ) 5) The proposed estmator of ρ s gve J J ( ) ( ) ˆ ρ ( ) ( ) for ρ ad ρ (-) are gve by the format of Equatos ad 3, respectvely. RESULTS I order to emprcally evaluate the valdty ad relablty of the proposed estmator, a smulato study was coducted. I the study, two populatos of sze,, cotag ordered pars of x ad y, were each geerated accordg to a bvarate ormal dstrbuto wth μ, μ 4 wth a equal varace (σ 8,σ 8) for the frst case ad a uequal varace (σ,σ ) for the secod case. I addto, the correlato coeffcets (ρ) of the two varables, x ad y, were set at -, -.8,,,,.,,, thus creatg 38 populatos for ths smulato study. A small sample sze of ad large sample szes of 3, 5 ad 6 were take from each populato by usg smple radom samplg wth replacemet wth, repettos, thus creatg 5 stuatos for the smulato study. The, the absolute bas ad mea square error (MSE) comparsos of ρ, ρ ˆJ ad ρ ˆO were performed emprcally. The smulato results preseted Fgure ad Fgure reveal the absolute bases of ρ, ρ ˆJ ad ρ ˆO. Furthermore, they cofrm the bas reducto of ρ from the proposed estmator. For the uequal varace of two populatos, the absolute bas of ρ was less tha those of ρ ˆJ ad ρ ˆO for a small sample sze,, at all levels of the populato correlato coeffcet. I the case of a equal varace for the two populatos, the absolute bas of ρ was less tha those of ρ ˆJ ad ρ ˆO whe the populato correlato coeffcets fell betwee - ad.4. I addto, the absolute bas of ρ seemed to have o dfferece from ˆJ that of ρ ˆO whe the sample szes were ot less tha 3 at all levels of the populato correlato coeffcet for uequal ad equal varaces of the two populatos.

4 456 Kasetsart J. (Nat. Sc.) 47(3) Eve for the large sample szes of 3, 5 ad 6, the absolute bases of ρ ad ρ ˆJ ˆO were less tha that of ρ whe the populato correlato coeffcet dd ot approxmate zero ad there was a uequal varace for the two populatos, as there were equal varaces for the two populatos, the absolute bases of ρ ad ρ ˆJ ˆO were lkely to be lower tha that of ρ whe the populato correlato coeffcet was postve. I addto, the absolute bases of ρ, ρ ˆJ ad ρ ˆO Absolute bas Absolute bas Absolute bas Absolute bas Sample sze of.3 Sample sze of opulato correlato (ρ) opulato correlato (ρ) Sample sze of 3. Sample sze of earso.4 Olk&ratt. Juthaphor opulato correlato (ρ) opulato correlato (ρ) opulato correlato (ρ) opulato correlato (ρ) Sample sze of 6.5 Sample sze of earso. Olk&ratt Juthaphor. opulato correlato (ρ) Absolute bas Absolute bas Absolute bas Absolute bas..5 Sample sze of 5.3 Sample sze of earso Olk&ratt Juthaphor earso Olk&ratt Juthaphor opulato correlato (ρ) earso Olk&ratt Juthaphor earso Olk&ratt Juthaphor earso Olk&ratt Juthaphor earso Olk&ratt Juthaphor Fgure Absolute bases of ρ ˆJ, ρ ad ρ ˆO whe σ ad σ. Fgure Absolute bases of ρ ˆJ, ρ ad ρ ˆO whe σ 8 ad σ 8.

5 Kasetsart J. (Nat. Sc.) 47(3) 457 seemed to decrease wheever the sample sze creased. Fgure 3 ad Fgure 4 dcate that the MSE of ρ seemed to have o dfferece from ˆJ those of ρ ad ρ ˆO each stuato for ths study. Furthermore, the MSEs of ρ, ρ ˆJ ad O seemed to decrease wheever the sample sze creased, regardless of the populato correlato coeffcet. Ths smulato study foud that the proposed estmator, ρ, almost completely ˆJ elmated the bas, ad the performace of ths estmator seemed to be better tha those of ρ ad ρ for the small sample sze of. ˆO Mea square error Sample sze of earso Olk&ratt Juthaphor Mea square error Sample sze of earso Olk&ratt Juthaphor opulato correlato (ρ) opulato correlato (ρ) Mea square error Sample sze of 3 earso Olk&ratt Juthaphor Mea square error Sample sze of 3 earso Olk&ratt Juthaphor opulato correlato (ρ) opulato correlato (ρ) Mea square error Sample sze of 5 earso Olk&ratt Juthaphor Mea square error Sample sze of 5 earso Olk&ratt Juthaphor opulato correlato (ρ) opulato correlato (ρ) Mea square error Sample sze of 6 earso Olk&ratt Juthaphor Mea square error Sample sze of 6 earso Olk&ratt Juthaphor opulato correlato (ρ) opulato correlato (ρ) Fgure 3 Mea square errors of ρ ˆJ, ρ ad ρ ˆO whe σ ad σ. Fgure 4 Mea square errors of ρ ˆJ, ρ ad ρ ˆO whe σ 8 ad σ 8.

6 458 Kasetsart J. (Nat. Sc.) 47(3) DISCUSSION The smulato results showed that ρ seemed to be a based estmator as metoed by Neter et al. (996) ad Zmmerma et al. (3). The proposed estmator, ρ ˆJ, was modfed from ρ ad the jackkfe method was appled for bas reducto. The results of ths smulato study showed that the bas of the proposed estmator was reduced to zero all stuatos. I addto, these results also showed that the varaces of two populatos do ot affect the bas reducto for all estmators as studed by Olk ad ratt (958). These fdgs ca be appled to research psychology, the behavoral sceces, ecology ad other felds. I addto, t s possble to use computer programmg to calculate ρ ˆJ wthout dffculty. Cocluso Ths paper proposed a estmator of the populato correlato coeffcet for a bvarate ormal dstrbuto. The proposed estmator provded a approxmately ubased estmator of the populato correlato coeffcet. The results of a smulato study dcated that the performace of ρ ˆJ seemed to be better tha those ρ ad O for a small sample sze,, regardless of the populato correlato coeffcets. I addto, the MSE of ρ ˆJ seemed to have o dfferece from those of ρ ad ρ each stuato for ths ˆO study. ACKNOWLEDGEMENTS The author would lke to thak the Departmet of Statstcs, Faculty of Scece, Kasetsart Uversty for the provso of ecessary facltes durg the research. LITERATURE CITED Adolph, S.C. ad J.S. Hard. 7. Estmatg pheotypc correlatos: Correctg for bas due to tradvdual varablty. Fuct. Ecol. (): Aderso, T.W. 3. A Itroducto to Multvarate Statstcal Aalyss. 3rd ed. Wley. Hoboke, NJ, USA. 7 pp. Efro, B. ad R.J. Tbshra A Itroducto to the Bootstrap. Chapma & Hall/CRC, part of the Taylor ad Fracs group. Lodo, UK. 45 pp. Fsher, R.A. 95. Frequecy dstrbuto of the values of the correlato coeffcet samples from a deftely large populato. Bometrka (4): O the probable error of a coeffcet of correlato deduced from a small sample. Metro : 3 3. Gorsuch, R.L. ad C.S. Lehma.. Correlato coeffcets: Mea bas ad cofdece terval dstortos. Joural of Methods ad Measuremet the Socal Sceces (): Huso, L.W., Bostatstcs Group ad F.H. La- Roche. 7. erformace of some correlato coeffcets whe appled to zero-clustered data. J. Mod. Appl. Stat. Methods 6: Neter, J., M.H. Kuter, C.J. Nachtshem ad W. Wasserma Appled Lear Statstcal Models. 4th ed.. Irw. Chcago, IL, USA.,43 pp. Olk, I. ad J.W. ratt Ubased estmato of certa correlato coeffcets. A. Math. Statst. 9:. Queoulle, M.H Approxmate test of correlato tme-seres. J. R. Statst. Soc. B : Rodgers, J.L. ad W.A. Ncewader Thrtee ways to look at the correlato coeffcet. Am. Stat. 4():

7 Kasetsart J. (Nat. Sc.) 47(3) 459 Ssomboothog, J. a. Estmato of the correlato coeffcet for a bvarate ormal dstrbuto wth mssg data. Kasetsart J. (Nat. Sc.) 45(4): Ssomboothog, J. b. Jackkfe maxmum lkelhood estmates for a bvarate ormal dstrbuto wth mssg data. Joural of Tha Statstcal Assocato 9(): Smth, C.D. ad J.S. otus. 6. Jackkfe estmator of speces rchess wth S LUS. J. Stat. Softw. 5:. Tukey, J.W Bas ad cofdece otqute large samples. A. Math. Statst. 9: Zmmerma, D.W., B.D. Zumbo ad R.H. Wllams. 3. Bas estmato ad hypothess testg of correlato. scológca 4: