Test for Intraclass Correlation Coefficient under Unequal Family Sizes

Transcription

1 Journal of Modern Appled Statstcal Methods Volume Issue Artcle Test for Intraclass Correlaton Coeffcent under Unequal Famly Szes Madhusudan Bhandary Columbus State Unversty, Columbus, GA, bhandary_madhusudan@colstate.edu Koj Fujwara North Daota State Unversty, Fargo, ND, oj.fujwara@ndsu.edu Follow ths and addtonal wors at: Part of the Appled Statstcs Commons, Socal and Behavoral Scences Commons, and the Statstcal Theory Commons Recommended Ctaton Bhandary, Madhusudan and Fujwara, Koj (03) "Test for Intraclass Correlaton Coeffcent under Unequal Famly Szes," Journal of Modern Appled Statstcal Methods: Vol. : Iss., Artcle 9. DOI: 0.37/jmasm/ Avalable at: Ths Regular Artcle s brought to you for free and open access by the Open Access Journals at DgtalCommons@WayneState. It has been accepted for ncluson n Journal of Modern Appled Statstcal Methods by an authorzed edtor of DgtalCommons@WayneState.

2 Journal of Modern Appled Statstcal Methods November 03, Vol., No., Copyrght 03 JMASM, Inc. ISSN Test for Intraclass Correlaton Coeffcent under Unequal Famly Szes Madhusudan Bhandary Columbus State Unversty Columbus, GA Koj Fujwara North Daota State Unversty Fargo, ND Three tests are proposed based on F-dstrbuton, Lelhood Rato Test (LRT) and large sample Z-test for ntraclass correlaton coeffcent under unequal famly szes based on a sngle multnormal sample. It has been found that the test based on F-dstrbuton consstently and relably produces results superor to those of Lelhood Rato Test (LRT) and large sample Z-test n terms of sze for varous combnatons of ntraclass correlaton coeffcent values. The power of ths test based on F-dstrbuton s compettve wth the power of the LRT and the power of Z-test s slghtly better than the powers of F-test and LRT when = 5, but the power of Z-test s worse n comparson wth the F-test and LRT for = 30,.e. for large sample stuaton, where = sample sze. Ths test based on F-dstrbuton can be used for both small sample and large sample stuatons. An example wth real data s presented. Keywords: Lelhood rato test, Z-test, F-test, ntraclass correlaton coeffcent. Introducton Suppose t s requred to estmate the correlaton coeffcent between blood pressures of chldren on the bass of measurements taen on p chldren n each of n famles. The p measurements on a famly provde p(p-) pars of observatons, (x,y) --- x beng the blood pressure of one chld and y that of another. From the n famles a total of np(p-) pars are generated from whch a correlaton coeffcent s computed n the ordnary way. The correlaton coeffcent thus computed s called an ntraclass correlaton coeffcent. It s mportant to have statstcal nference concernng ntraclass correlaton, because t provdes nformaton regardng blood pressure, cholesterol, etc., n a famly wthn some race n the world. Madhusudan Bhandary s a Professor n the Department of Mathematcs. Emal hm at: bhandary_madhusudan@colstate.edu. Koj Fujwara s a graduate student n the Department of Statstcs. Emal hm at: oj.fujwara@ndsu.edu. 7

3 TEST FOR INTRACLASS CORRELATION COEFFICIENT The ntraclass correlaton coeffcent ρ has a wde varety of uses n measurng the degree of ntrafamly resemblance wth respect to characterstcs such as blood pressure, cholesterol, weght, heght, stature, lung capacty, etc. Several authors have studed statstcal nference concernng ρ based on a sngle multnormal sample ( Scheffe, 959; Rao, 973; Rosner et.al, 977, 979,; Donner and Bull, 983; Srvastava, 984; Konsh, 985; Gohale and SenGupta, 986; SenGupta, 988; Velu and Rao, 990). Donner and Bull (983) dscussed the lelhood rato test for testng the equalty of two ntraclass correlaton coeffcents based on two ndependent multnormal samples under equal famly szes. Konsh and Gupta (987) proposed a modfed lelhood rato test and derved ts asymptotc null dstrbuton. They also dscussed another test procedure based on a modfcaton of Fsher s Z-transformaton followng Konsh (985). Huang and Snha (993) consdered an optmum nvarant test for the equalty of ntraclass correlaton coeffcents under equal famly szes for more than two ntraclass correlaton coeffcents based on ndependent samples from several multnormal dstrbutons. For unequal famly szes, Young and Bhandary (998) proposed Lelhood rato test, large sample Z-test and large sample Z * - test for the equalty of two ntraclass correlaton coeffcents based on two ndependent multnormal samples. For several populatons and unequal famly szes, Bhandary and Alam (000) proposed Lelhood rato test and large sample ANOVA test for the equalty of several ntraclass correlaton coeffcents based on several ndependent multnormal samples. Donner and Zou (00) proposed asymptotc test for the equalty of dependent ntraclass correlaton coeffcents under unequal famly szes. However, none of the above authors derved any test for a sngle sample and unequal famly szes. It s an mportant practcal problem to consder a sngle sample test for ntraclass correlaton coeffcent under unequal famly szes. Ths artcle consders three tests for ntraclass correlaton coeffcent based on a sngle multnormal sample under unequal famly szes. Condtonal analyss s conducted assumng famly szes fxed though unequal. It could be of nterest to examne the blood pressure or cholesterol or lung capacty, etc., among famles n U.S.A. or among some other races, and therefore t s necessary to develop a sngle sample test for ntraclass correlaton coeffcent under unequal famly szes. Three tests are proposed: F-test, LRT and large sample Z-test. These three tests are compared n Secton 3 usng smulaton technque. It has been found on the bass of smulaton study that the test based on F-dstrbuton consstently and 7

4 BHANDARY & FUJIWARA relably produced results superor to those of Lelhood Rato Test (LRT) and large sample Z-test n terms of sze for varous combnatons of ntraclass correlaton coeffcent values. The power of ths test based on F-dstrbuton s compettve wth the power of the LRT, and the power of Z-test s slghtly better than the powers of F-test and LRT when =5; but the power of Z-test s worse n comparson wth the F-test and LRT for =30,.e. for large sample stuaton, where = sample sze. Ths test based on F-dstrbuton can be used for both small sample and large sample stuatons. An example wth real data s presented n Secton 4. Tests of H 0ρ= : ρ 0 Versus H ρ¹ρ : 0 Lelhood Rato Test: Let X = ( x, x,..., x ) be a px vector of observatons from th famly; p =,,...,. The structure of mean vector and the covarance matrx for the famlal data s gven by the followng ( Rao 973) : µ µ and σ = Σ = p... xp ρ ρ ρ... ρ ρ ρ... () where s a px vector of s, µ ( < µ < ) s the common mean and σ ( σ > 0) s the common varance of members of the famly and ρ, whch s called the ntraclass correlaton coeffcent, s the coeffcent of correlaton among the members of the famly and max ρ. p It s assumed that x Np ( µ, );,..., Σ =, where N p represents p - varate normal dstrbuton and µ, Σ ' s are defned n (). 73

5 TEST FOR INTRACLASS CORRELATION COEFFICIENT Let u = ( u, u,..., u ) = Qx () p where Q s an orthogonal matrx. Under the orthogonal transformaton (), t can be seen that * * u N ( µ, Σ ); =,..., p where µ = ( µ,0,...,0) and * px * Σ = σ η ρ ρ and η = p { + ( p ) ρ} The transformaton used on the data from x to u above s ndependent of ρ. One can use Helmert s orthogonal transformaton. Srvastava (984) gves the estmator of ρ and σ under unequal famly szes whch are good substtutes for the maxmum lelhood estmators and are gven by the followng: ˆ γ ˆ ρ = ˆ σ = ˆ ˆ + = = ˆ σ ( ) ( u µ ) γ ( a ) ˆ γ = = r= ˆ µ = = = p r ( p ) u u (3) and a = p Now, consder a random sample of famles from a populaton. 74

6 BHANDARY & FUJIWARA Let x = ( x, x,..., x ) be a px vector of observatons from th famly; =,,, K p x N p µ Σ µ = µ Σ = σ and (, ), where, ρ... ρ ρ... ρ ρ ρ... (4) and max ρ. p Usng orthogonal transformaton, the data vector can be transformed from x to u as follows: u = u u u N Σ = (,,..., ) ( *, * );,..., p p µ where, * * µ = ( µ,0,...,0), Σ = σ η ρ ρ { p ρ} η = p + ( ) (5) The transformaton used on the data above from x to u s ndependent of ρ. Under the above setup, lelhood rato test statstc for testng H0 : ρ = ρ0 Vs. H: ρ ρ0 s gven by the followng: { ρ } log Λ= log p + ( p ) + ( p ) log( ρ ) 0 0 = = 75

7 TEST FOR INTRACLASS CORRELATION COEFFICIENT { p + ( ˆ [ ]} ) ( ) 0 r / ( 0) ˆ p u µ + p ρ + u ρ σ = = r= log p { ( ) ˆ p ρ} - + = = ( p ) log( ˆ ρ) p ˆ + + σ = = r= { p ( ˆ [ ˆ] } ˆ u µ ) ( p ) ρ ur / ( ρ) (6) where, Λ = lelhood rato test statstc, ˆρ = estmate of ntraclass correlaton coeffcent under H, ˆ σ = estmate of σ and ˆµ s the estmate of mean. The estmators ˆ ρσ, ˆ and ˆµ can be obtaned from Srvastava s estmator gven by (3). It s well-nown from asymptotc theory that log Λ has an asymptotc ch-square dstrbuton wth degree of freedom. Large Sample Z-test: A large sample Z-test s proposed as follows : Z ˆ ρ ρ 0 = (7) Var where, ˆρ = the estmator of ρ from the sample usng Srvastava (984) and Var under H 0 ( usng Srvastava and Katapa (986) ) s as follows : Var = ( ρ0) ( p ) + c ( ρ0)( p ) ( a ) (8) = where, = number of famles n the sample 76

8 BHANDARY & FUJIWARA p = p = c a a p a = ( ρ0) + ( ρ0) + ( ) = = a = a = and a = p It s obvous ( usng Srvastava and Katapa (986) ) to see that under H 0, the test statstc Z gven by (7) has an asymptotc N(0,) dstrbuton. Because the alternatve hypothess s H: ρ ρ0, the above Z-test s a two sded test. F-test: Usng (5), u N µση (, ) where η = p { + ( p ) ρ} Hence = Also from (5), ( u ˆ µ ) ση χ p ur = r= χ σ ( ρ) ( p ) = p ur ( u ˆ µ ) = r= and and are ndependent. = ση σ ( ρ) χ n denotes Ch-square dstrbuton wth n degrees of freedom. Hence, an F-test s proposed as: 77

9 TEST FOR INTRACLASS CORRELATION COEFFICIENT F = ( u ˆ µ ) = p 0 p ur = r= p ( ρ0) = { + ( p ) ρ } ( ( )) ( ) (9) Under H0, F F, ( p ) = (0) The performance of the three tests gven by (6), (7) and (9) s dscussed n terms of sze and power n the next secton usng smulated data. Smulaton Results: Multvarate normal random vectors were generated usng R program n order to evaluate the power of the F statstc as compared to the LRT statstc and Z- statstc. Fve and thrty vectors of famly data were created for the populaton. The famly sze dstrbuton was truncated to mantan the famly sze at a mnmum of sblngs and a maxmum of 5 sblngs. The prevous research n smulatng famly szes ( Rosner et al. (977), Srvastava and Keen (988) ) determned the parameter settng for FORTRAN IMSL negatve bnomal subroutne wth a mean =.86 and a success probablty = Here, a mean =.86 and a theta = 4.55 were set. All parameters were set the same except the value of ρ whch too on all combnatons possble over the range of values from 0. to 0.9 at ncrements of 0.. The R program produced estmates of ρ along wth F statstc, the LRT statstc and the Z- statstc 3,000 tmes for each partcular populaton parameter ρ. The frequency of rejecton of each test statstc at α =0.05 was noted and the proporton of rejectons are calculated for varous combnaton of ρ. The szes for the LRT statstc, F statstc and Z statstc for varous combnaton of ρ were also calculated. On the bass of ths study, t was found that the test based on F-dstrbuton consstently and relably produced results superor to those of Lelhood Rato Test (LRT) and large sample Z-test n terms of sze for varous combnatons of ntraclass correlaton coeffcent values. 78

10 BHANDARY & FUJIWARA The power of ths test based on F-dstrbuton s compettve wth the power of the LRT and the power of Z-test s slghtly better than the powers of F-test and LRT when =5 but the power of Z-test s worse n comparson wth the F-test and LRT for =30.e. for large sample stuaton. Hence, overall recommendaton should be to use F-test gven by (9). The LRT has a large bas under H 0 and ths bas s probably due to the fact that log Λmay tae some negatve values (whch t should not tae) and those negatve values are deleted for the calculaton of sze and power. But, for the F- test or Z-test, ths problem does not arse. alpha 0.30 power rho = rho0 PowerF PowerLRT PowerZ rho PowerF PowerLRT PowerZ Fgure. Alpha Levels ( = 5, alpha = 0.05) Fgure 3. Power (alpha = 0.05, = 5, rho0 = 0.) alpha 0.0 power rho = rho0 PowerF PowerLRT PowerZ rho PowerF PowerLRT PowerZ Fgure. Alpha Levels ( = 30, alpha = 0.0) Fgure 4. Power (alpha = 0.05, = 30, rho0 = 0.) 79

11 TEST FOR INTRACLASS CORRELATION COEFFICIENT The test based on F-dstrbuton can be used for both small sample and large sample stuatons. Hence, the F test s strongly recommended for use n practce. Example wth Real Lfe Data The three tests are compared usng real lfe data collected from Srvastava and Katapa (986). Table gves the values of pattern ntensty on soles of feet n fourteen famles, where values for daughters and sons are put together. In the Table below, for example, for famly #, the chldren (both sons and daughters) have feet szes 5,3,4,4 respectvely. Table. Pattern ntensty values on soles of feet for 4 famles. Sample Famly Sblngs A, 4 A 0 4, 5, 4 A 9 5, 6 A, A 4,,,, A 5 6, 6 A 8, 4, 7, 4, 4, 7, 8 A 3,, A 6 4, 3, 3 A 4,, A 7,, 3, 6, 3, 5, 4 A, 3 A 5, 3, 4, 4 A 3 4, 3, 3, 3 The data on the chldren from Table was used to analyze the case of testng ntraclass correlaton coeffcent. The above data were collected for the purpose of havng nference on famlal correlaton. Srvastava and Katapa (986) used the above data to estmate the ntraclass correlaton coeffcent for unequal famly szes. 80

12 BHANDARY & FUJIWARA Frst, the data s transformed by multplyng each observaton vector by Helmert s orthogonal matrx Q where, Q =... p p p p ( p )... p( p ) p( p ) p( p ) p( p ) pxp Ths results n transformed vectors u for =,,...,. Here, =4. Srvastava s formula gven by (3) s used to compute ntraclass correlaton coeffcent and varance. The computed values of ntraclass correlaton coeffcent and varance are ˆ ρ = 0.88 and ˆ σ = Because ˆ ρ = 0.88 s estmated from the above sample, t s necessary to now whether the ntraclass correlaton coeffcent ρ n the populaton from whch the sample came s close to 0.8, and therefore necessary to test H : ρ = 0.8 Vs. H : ρ Formulae (6) and (7) and (9) are used to obtan the values of the test statstcs for testng H 0 : ρ = 0.8 Vs. H : 0.8 ρ. The computed values of the LRT statstc, Z statstc and F statstc obtaned from formula (6), (7) and (9) respectvely are as follows: LRT statstc = 33.3, Z statstc = and F statstc = The crtcal values at α = 0.05 and 0.0 for the tests are as follows: LRT = 3.845; Z =.96; F =.8506; F = ; LRT =.7055; Z =.645; F =.3973; F = Hence, the null hypothess s accepted by F- test and Z- test at 5% and 0% levels, whereas t s rejected by LRT at 5% and 0% levels. 8

13 TEST FOR INTRACLASS CORRELATION COEFFICIENT If the F-test s used t provdes nformaton that the populaton ntraclass correlaton coeffcent s 0.8 whch should be true ntutvely. But, LRT gves nformaton that the populaton ntraclass correlaton coeffcent s not 0.8. Therefore, t s mportant to choose the rght test to produce the correct nformaton. It was found before that LRT s based n terms of sze; therefore the recommendaton s to use F-test gven by (9) n real practce. References Bhandary, M., & Alam, M. K. (000). Test for the equalty of ntraclass correlaton coeffcents under unequal famly szes for several populatons. Communcatons n Statstcs-Theory and Methods, 9(4), Donner, A., & Bull, S. (983). Inferences concernng a common ntraclass correlaton coeffcent. Bometrcs, 39, Donner, A., & Zou, G. (00). Testng the equalty of dependent ntraclass correlaton coeffcents. The Statstcan, 5(3), Gohale, D. V., & SenGupta, A. (986). Optmal tests for the correlaton coeffcent n a symmetrc multvarate normal populaton. J. Statst. Plann Inference, 4, Huang, W., & Snha, B. K. (993). On optmum nvarant tests of equalty of ntraclass correlaton coeffcents. Annals of the Insttute of Statstcal Mathematcs, 45(3), Konsh, S. (985). Normalzng and varance stablzng transformatons for ntraclass correlatons. Annals of the Insttute of Statstcal Mathematcs, 37, Konsh, S., & Gupta, A. K. (987). Testng the equalty of several ntraclass correlaton coeffcents. J. Statst. Plann. Inference,, Rao, C.R. (973). Lnear Statstcal Inference and Its Applcatons. Wley, New Yor. Rosner, B., Donner, A., & Henneens, C.H. (977). Estmaton of ntraclass correlaton from famlal data. Appled Statstcs, 6, Rosner, B., Donner, A., & Henneens, C.H. (979). Sgnfcance testng of nterclass correlatons from famlal data. Bometrcs, 35, Scheffe, H. (959). The Analyss of Varance. Wley, New Yor. SenGupta, A. (988). On loss of power under addtonal nformaton an example. Scand. J. Statst., 5,

14 BHANDARY & FUJIWARA Srvastava, M. S. (984). Estmaton of nterclass correlatons n famlal data. Bometra, 7, Srvastava, M.S., & Katapa, R. S. (986). Comparson of estmators of nterclass and ntraclass correlatons from famlal data. Canadan Journal of Statstcs, 4, 9-4. Srvastava, M. S., & Keen, K. J. (988). Estmaton of the nterclass correlaton coeffcent. Bometra, 75, Velu, R., & Rao, M. B. (990). Estmaton of parent-offsprng correlaton. Bometra, 77(3), Young, D., & Bhandary, M. (998). Test for the equalty of ntraclass correlaton coeffcents under unequal famly szes. Bometrcs, 54(4),