New Nonparametric Rank-Based Tests for Paired Data

Transcription

1 Ope Joural of Statstcs, 04, 4, Publshed Ole August 04 ScRes. New Noparametrc Rak-Based Tests for Pared Data Guoge Sha Departmet of Evrometal ad Occupatoal Health, Epdemology ad Bostatstcs Program, Uversty of Nevada Las Vegas, Las Vegas, NV, USA Emal: Receved May 04; revsed 6 Jue 04; accepted 9 July 04 Copyrght 04 by authors ad Scetfc Research Publshg Ic. Ths work s lcesed uder the Creatve Commos Attrbuto Iteratoal Lcese (CC BY). Abstract We propose a ew oparametrc test based o the rak dfferece betwee the pared sample for testg the equalty of the margal dstrbutos from a bvarate dstrbuto. We also cosder a modfcato of the ovel oparametrc test based o the test proposed by Baumgarter, Weβ, ad Schdler (998). A extesve umercal power comparso for varous parametrc ad oparametrc tests was coducted uder a wde rage of bvarate dstrbutos for small sample szes. The two ew oparametrc tests have comparable power to the pared t test for the data smulated from bvarate ormal dstrbutos, ad are geerally more powerful tha the pared t test ad other commoly used oparametrc tests several mportat bvarate dstrbutos. Keywords BWS Test, Noparametrc Test, Pared Data, Power Study, Rak Dfferece, Wlcoxo Sged Rak Test. Itroducto Pared data are very commo statstcal ad medcal research. A typcal example s a clcal tral where subjects are measured pror to a treatmet, say for elevated systolc blood pressure, ad the measured aga after the treatmet wth a drug to lower the blood pressure. Aother example s the use of matched cases ad cotrols. Oe sample from the the case group ad aother matched sample from the cotrol group may be used to form a pared sample by usg addtoal varables that are measured addto to the varable of terest. Pared data are ofte used to reduce varablty ad to make more precse comparsos wth fewer subjects, ad ths has resulted attractg may statstcas to develop more effcet tests ad fereces for pared data. X, Y, X, Y,, X, Y be radom samples from a bvarate dstrbuto wth cotuous ed- Let ( ) ( ) ( ) How to cte ths paper: Sha, G. (04) New Noparametrc Rak-Based Tests for Pared Data. Ope Joural of Statstcs, 4,

2 pots. The margal dstrbutos of X ad Y follow FX ( x ) ad FY ( ) y, respectvely. The ull hypothess of terest s H : 0 FX = FY. Ths problem ofte occurs appled research for testg the equalty of the margal dstrbutos. For example, a oe-arm Ocolgy study, the tumor sze of each patet s measured before ad after treatmet. If the cacer treatmet s effectve o patets, the tumor szes the majorty of patets are expected to be smaller after the treatmet tha the basele measuremet. Therefore, a approprate alteratve hypothess s gve as H : F F <. Oe mportat ty- Y = X + for all x, where θ > 0. The X dstrbuto has a postve shft compared to that of the Y dstrbuto. The two sample pared t test s a commoly used parametrc approach for comparg the meas of two dstrbutos. It computes the dfferece betwee the two measuremets of each subject a X Y wth at least oe pot z such that FX ( z) FY ( z) pcal case the above problem s the locato problem, that s, F ( x) F ( x θ ) D = X Y, =,,,, ad the tests whether the average of these dffereces s sgfcatly dfferet from zero by usg the test statstc µ D PT =, (.) s D where µ D ad s D are the sample mea ad the stadard devato, respectvely. The two sample t test makes certa assumptos, such as the ormalty of the sample dfferece whch eeds to be checked by ormalty tests [] [] before applyg the pared t test. If oe or more of these assumptos ca t reasoably be met, the the pared t test may be ot approprately appled. A alteratve to the two sample pared t test s the Wlcoxo sged rak (WSR) test [3], whch s a commoly used oparametrc test for pared data whe at least oe of the assumptos s ot satsfed. The Wlcoxo rak sum test (also kow as the Ma-Whtey test) [3] [4] s a oparametrc statstcal test for assessg whether the two depedet samples are from the same dstrbuto. It may be ot be sutable for testg pared data wthout some modfcato. Later, Lam ad Logecker [5] proposed a modfcato of the Wlcoxo rak sum (MWRS) test by troducg a cosstet varace estmator for assessg the equalty of the margal dstrbutos of a bvarate dstrbuto. The MWRS test was compared to other tests based o Mote Carlo smulato wth small sample szes, ad was show to be as powerful as the two sample pared t test for the bvarate ormal data, ad more powerful tha both the two sample pared t test ad the WSR test for the Farle-Gumbel-Morgester dstrbuto wth expoetal margals. We propose a ew rak dfferece (RD) test for pared data based o the rak dfferece betwee the pared sample to capture the sample dfferece. We also troduce the modfed Baumgarter, Weβ, ad Schdler (MBWS) test proposed by Sha et al. [6] for pared data. A dscusso o choosg betwee the parametrc ad oparametrc tests may be foud Fay ad Proscha [7]. The remader of ths artcle s orgazed as follows. I Secto, we brefly revew the two exstg oparametrc tests for pared data ad troduce the two ew oparametrc tests. I Secto 3, we compare the performace of the competg tests, studyg the smulated power of the tests uder a wde rage of bvarate dstrbutos. A real example s gve to llustrate the applcato of the parametrc ad oparametrc tests Secto 4. Secto 5 s gve to dscusso.. Noparametrc Tests A oparametrc couterpart to the two sample pared t test s the WSR test for pared samples. The WSR test begs by trasformg each dfferece D to ts absolute value D, the the absolute dffereces are raked from the lowest to the hghest R = Rak ( D ). For cotuous edpots, there s o te betwee measuremet, ad all D s are used the rakg precess. The WSR test statstc s the expressed as = ( ) WSR = sg D R. (.) The value of the WSR test statstc s a o-egatve teger betwee 0 ad ( + ). The upper boud would be reached whe all sged values are ether postve or egatve. The stadardzed WSR test statstc WSR asymptotcally follows a stadard ormal dstrbuto. The asymptotc dstrbuto ca be ( )( ) 496

3 used to calculate the p-value ad to fd the threshold values. But, for small sample szes, the exact dstrbuto of the WSR test provdes accurate ad relable results. The exact samplg dstrbuto of the WSR test ca be obtaed by eumeratg all possble combatos of the postve ad egatve sgs. For example, f we have subjects the study, the the absolute dffereces, D, =,,,, produce the order of raks,,,. All possble combatos of plus ad mus sgs that could be dstrbuted amog these raks are. The, the exact p-value of a gve data s the proporto the combatos whose WSR test statstc s as extreme as that of the gve data. Aother oparametrc test cosdered s the MWRS test proposed by Lam ad Logecker [5] for assessg the equalty of the margal dstrbutos of a bvarate dstrbuto. Let S ad T deote the rak for X ad Y the combed sample, ad U be the rak for X the X sample ad V be the rak for Y the Y sample. The the MWRS test s defed as where σ ( ˆ ρ ) W =, S ( ) S + = MWRS = +, σw ˆ ρ S UV = ( ) ( + ) (.3) 3 = s the Spearma s coeffcet of rak correlato. The asymptotc dstrbuto of MWRS s a stadard ormal dstrbuto due to the cosstecy of the varace estmator [5]. The MWRS test was show to have comparable power to pared t test ad the WSR test. Two Proposed Noparametrc Tests Two steps are mplemeted the Wlcoxo sged rak test: calculato of the absolute dfferece followed by the rakg of these dffereces. The ew proposed RD test calculates the test statstc by revsg the order the the two steps the WSR test: rakg the observatos followed by the dfferece of the raks. Specfcally, the assocated test statstc of the RD test s RD = ( S T). (.4) = ( )( ) The value of the RD test statstc s a teger betwee +, whch - cludes the sample space of the WSR test. A larger sample space could potetally have a less dscrete type I error rate studes wth small to medum sample szes. The sg of S T s the same as that of sg( D) R the WSR test. The ew proposed RD test captures ot oly the dfferece wth each subject, but also the rak of the observatos wth each subject. Recetly, Baumgarter, Weß, ad Schdler (BWS) [8] proposed a ovel oparametrc test for two depedet sample problem, whch s based o the squared value of the dfferece betwee the two emprcal dstrbuto fuctos weghted by the respectve varace. Ths weghtg places more emphasze o the tals of the dstrbuto fuctos. Ths ew test s ot sutable for a oe sded problem due the ature of the costructo of the test statstc. For ths reaso, Neuhauser [9] proposed a modfed BWS test usg the sg of the dfferece of the rak ad the mea of the rak to eable the oe sded problem. It was the further modfed by Sha et al. [6] wth the exact mea ad varace estmates of raks [0] for a oe sded two depedet sample problem. We cosder ths MBWS test [6] for pared data, ad the test statstc s of the form where B X ad ( )( ) MBWS = ( BX BY ), (.5) + + S S + + = = ( + )

4 ad B Y + + Tj j Tj j + + =. j= j j ( + ) Although the asymptotc dstrbuto of the test statstc for the MBWS test may ot be easly derved, a exact permutato test or a smulato based test ca readly be performed order to calculate the p-value for a gve data set. It should be ote that all the oparametrc procedures aforemetoed ca be used for data wth or wthout tes; the case of tes the raks are defed to be the mdraks. 3. Numercal Study To evaluate the performace of the parametrc ad oparametrc test, sample sze = 0, sgfcace level of α = 0.05 ad 0,000 smulated teratos were used the Mote Carlo exact smulato. Fve dfferet tests were competed for each plot: ) the RD test; ) the MBWS test; 3) the MWRS test; 4) the WSR test; ad 5) the two sample pared t test. The two sample pared t test s the oly parametrc test ths artcle, ad all the other four tests are oparametrc approaches. Four dfferece bvarate dstrbutos were examed: ) the bvarate ormal dstrbuto; ) the bvarate dstrbuto wth gamma margal dstrbutos; 3) the bvarate geeralzed expoetal dstrbuto; ad 4) the bvarate dstrbuto wth a gamma ad a expoetal margal dstrbutos. The frst cosdered bvarate dstrbuto s a bvarate dstrbuto wth mea ( θ, θ ) ad varace cova- σ ρσσ race matrx, where ρ s the correlato coeffcet, 0 ρ. Fgure shows the ρσσ σ power plots for the bvarate ormal dstrbuto of dfferet meas wth a fxed covarace matrx. Equal varaces are assumed σ = σ = σ, ad four dfferet ρ values are cosdered the fgure: 0, 0., 0.4, ad 0.7. The 95% threshold value was smulated from the bvarate ormal dstrbuto wth θ = θ θ = 0, σ =, ad a gve ρ for each plot Fgure. As see, the smulated power of each test s a creasg fucto of θσ. The two sample pared t test s the most powerful test as expected due to the fact that ths s the uform most powerful ubased test for ths problem whe the data s from the bvarate ormal dstrbuto of dfferet meas for a gve covarace matrx. The ew proposed RD test ad the MBWS test are compatble wth regard to the power, ad both are geerally more powerful tha the WSR test. The MBWS test has greater power tha the MWRS test for a small to medum ρ, ad the RD test s geerally more powerful tha the MWRS test. Gve a large ρ, the MWRS could be more powerful tha the proposed MBWS test, but less powerful tha the RD test. Fgure shows the power plots of the correlato coeffcet ρ gve equal varaces σ = σ = σ = ad the rato of mea dfferece ad varace θσ= 0.3. Smlar results are observed as the results from Fgure. It should be oted that the pared t test s oly approprate whe the dfferece D follows a ormal dstrbuto. The other four tests cosdered ths artcle are oparametrc approaches that are applcable to ay cotuous dstrbutos wth fewer assumptos. We also compare the bvarate ormal dstrbuto wth equal meas but dfferet varaces gve the same covarace ρσσ = 0.6. The power plots as a fucto of σ /σ are show Fgure 3. The threshold value s smulated from a bvarate ormal dstrbuto wth equal varaces. The pared t test, the WSR test, ad the MWRS test appear to have less power tha the two ew proposed tests. The MBWS test s clearly more powerful tha the other proposed RD test. The two ew proposed tests are able to detect the varace chage the dstrbuto, whle others do ot. I addto to the bvarate ormal dstrbuto, we also cosder other bvarate dstrbutos. Oe example s the bvarate dstrbuto wth gamma margal dstrbutos G ( κη, ), where κ ad η are the shape ad scale parameters, respectvely. The data may be geerated from the fucto rmvdc the R package copula. The two margal gamma dstrbutos wth the same scale parameter but dfferet shape parameters are cosdered,.e., G ( κ, ) ad G ( κ,). Fgure 4 shows the power plot as a fucto of the rato of the shape parameters κ κ. The two proposed tests have the hghest power, followed by the MWRS test, the WSR test, ad the pared t test. The two ew proposed tests domate other tests ad the power gas are substatal. 498

5 Fgure. Power study for a bvarate ormal dstrbuto wth dfferece mea gve four dfferet covarace matrces. Fgure. Power study for a bvarate ormal dstrbuto wth the same equal varaces ad the rato of mea dfferece ad varace but dfferet ρ. 499

6 Fgure 3. Power study for a bvarate ormal dstrbuto wth the same mea but dfferet varaces σ, σ gve the covarace 0.6. Fgure 4. Power study for a bvarate dstrbuto wth gamma G κ ad G( κ,). margal dstrbutos, ( ), Aother bvarate dstrbuto examed here s the bvarate geeralzed expoetal dstrbuto [] wth the jot cumulatve dstrbuto fucto τ ( ) 3 τ ( ) ( ) τ x y ( ) ( ) m xy < < =, P X xy, y e e e, where τ, τ, ad τ 3 are the three parameters the dstrbuto. The margal dstrbutos for X ad Y are geeralzed expoetal dstrbutos wth parameters ( τ + τ, 3 ) ad ( τ,) + τ 3, respectvely. The thrd parameter the geeralzed expoetal dstrbuto s gve as τ 3 = the smulato study. The ull dstrbuto s smulated wth equal τ ad τ,.e., τ = τ =. The power plot s draw as a fucto of τ τ, see Fgure 5. The sged rak test s very lower power as compared to other procedures; the two ew proposed tests are ot as powerful as the pared t test ad the MWRS test. 500

7 Fgure 5. Power study for a bvarate geeralzed expoetal τ, τ, τ =. dstrbuto wth parameters ( ) 3 For further comparso, we examed the bvarate dstrbuto wth dfferet types of margal dstrbutos, for example, oe margal dstrbuto follows a gamma dstrbuto G ( κ,) ad the other s a expoetal dstrbuto Exp( ). Equal mea s assumed uder the ull hypothess wth κ = the gamma dstrbuto. The power plots as a fucto of κ are dsplayed Fgure 6. The pared t test ad the WSR test are less powerful tha the other three tests. The proposed MBWS test s geerally more powerful tha the MWRS test uder large κ alteratves. 4. Example We cosder a example ad apply the fve dfferet tests dscussed ths artcle: ) the pared t test; ) the WSR test; 3) the MWRS test; 4) the RD test; ad 5) the MBWS test. Suppose a pharmaceutcal compay wats to assess the effcacy of a drug lowerg systolc blood pressure. The systolc blood pressure readg mmhg for 0 subjects were measured before ad after the admstrato of the drug, ad the assocated data ca be foud Atosamy et al. []. The systolc blood pressure s expected to be lower after the drug treatmet, therefore a oe sded alteratve s approprate for ths study. The p-value of the WSR test was calculated based o the exact permutato approach, the p-value of the pared t test was computed usg the asymptotc approach, ad the p-values of all the other three oparametrc tests were calculated based o the 00,000 Mote Carlo exact smulato. The p-values are reported Table. All fve tests coclude that the drug s effectve lowerg the systolc blood pressure at the sgfcace level of Cocluso I ths artcle, we troduce two ew oparametrc tests for testg whether pared samples come from the same populato. The two ew proposed oparametrc tests are comparable to the pared t test for testg the mea dfferece for the bvarate ormal dstrbuto gve a covarace matrx, ad much more powerful tha the pared t test ad aother two oparametrc tests for the dfferece varaces for the bvarate ormal dstrbuto. Extesve umercal power comparso was coducted for varous other mportat bvarate dstrbutos. The proposed RD test ad the MBWS test have greater power tha other tests several mportat scearos, ad the power gas are substatal. These two proposed tests are recommeded for use practce due the power gas as compared to other compettors. Oe lmtato of the MBWS test s the dffculty to fd the asymptotc dstrbuto. However, permutato-based or smulato-based tests ca always be used for the p-value calculato. We cosder exact testg procedures as future work [3]-[0]. The exteso of the RD test 50

8 Fgure 6. Power study for a bvarate dstrbuto wth G( κ,) ad Exp() as margal dstrbutos. Table. p-values for the example from the systolc blood pressure study. Tests Pared t test WSR MWRS RD MBWS ad the MBWS test to the k-sample depedet ad depedet problems []-[4] s curretly uderway. Ackowledgemets The author s research s partally supported by a Faculty Opportuty Awards from UNLV. Refereces [] Shapro, S.S. ad Wlk, M.B. (965) A Aalyss of Varace Test for Normalty (Complete Samples). Bometrka, 5, [] Sha, G.G., Vexler, A., Wldg, G. ad Hutso, A. (0) Smple ad Exact Emprcal Lkelhood Rato Tests for Normalty Based o Momet Relatos. Commucatos Statstcs: Smulato ad Computato, 40, [3] Wlcoxo, F. (945) Idvdual Comparsos by Rakg Methods. Bometrcs Bullet,, [4] Ma, H.B. ad Whtey, D.R. (947) O a Test of Whether Oe of Two Radom Varables Is Stochastcally Larger tha the Other. Aals of Mathematcal Statstcs, 8, [5] Lam, F.C. ad Logecker, M.T. (983) A Modfed Wlcoxo Rak Sum Test for Pared Data. Bometrka, 70, [6] Sha, G.G., Ma, C.X., Hutso, A.D. ad Wldg, G.E. (03) Some Tests for Detectg Treds Based o the Modfed Baumgarter Weß Schdler Statstcs. Computatoal Statstcs & Data Aalyss, 57, [7] Fay, M.P. ad Proscha, M.A. (00) Wlcoxo-Ma-Whtey or t-test? O Assumptos for Hypothess Tests ad Multple Iterpretatos of Decso Rules. Statstcs Surveys, 4, -39. [8] Baumgarter, W., Weß, P. ad Schdler, H. (998) A Noparametrc Test for the Geeral Two-Sample Problem. Bometrcs, 54, [9] Neuhäuser, M. (00) Oe-Sded Two-Sample ad Tred Tests Based o a Modfed Baumgarter-Weß Schdler 50

9 Statstc. Joural of Noparametrc Statstcs, 3, [0] Murakam, H. (006) A k-sample Rak Test Based o Modfed Baumgarter Statstc ad Its Power Comparso. Joural of the Japaese Socety of Computatoal Statstcs, 9, [] Kudu, D. ad Gupta, R.D. (009) Bvarate Geeralzed Expoetal Dstrbuto. Joural of Multvarate Aalyss, 00, [] Atosamy, B., Chrstopher, S. ad Samuelso, P. (00) Bostatstcs: Prcples ad Practce. McGraw-Hll Educato, New York. [3] Wldg, G.E., Sha, G. ad Hutso, A.D. (0) Exact Two-Stage Desgs for Phase II Actvty Trals wth Rak- Based Edpots. Cotemporary Clcal Trals, 33, [4] Sha, G., Ma, C., Hutso, A.D. ad Wldg, G.E. (0) A Effcet ad Exact Approach for Detectg Treds wth Bary Edpots. Statstcs Medce, 3, [5] Sha, G. ad Ma, C. (0) Ucodtoal Tests for Comparg Two Ordered Multomals. Statstcal Methods Medcal Research, Publshed Ole. [6] Sha, G. (03) More Effcet Ucodtoal Tests for Exchageable Bary Data wth Equal Cluster Szes. Statstcs & Probablty Letters, 83, [7] Sha, G. (03) A Note o Exact Codtoal ad Ucodtoal Tests for Hardy-Weberg Equlbrum. Huma Heredty, 76, [8] Sha, G. ad Ma, C. (04) Exact Methods for Testg the Equalty of Proportos for Bary Clustered Data from Otolarygologc Studes. Statstcs Bopharmaceutcal Research, 6, 5-. [9] Sha, G., Ma, C., Hutso, A.D. ad Wldg, G.E. (03) Radomzed Two-Stage Phase II Clcal Tral Desgs Based o Barard s Exact Test. Joural of Bopharmaceutcal Statstcs, 3, [0] Sha, G. (04) Exact Approaches for Testg No-Iferorty or Superorty of Two Icdece Rates. Statstcs & Probablty Letters, 85, [] Jockheere, A.R. (954) A Dstrbuto-Free k-sample Test agast Ordered Alteratves. Bometrka, 4, [] Terpstra, T.J. (95) The Asymptotc Normalty ad Cosstecy of Kedall s Test agast Tred, Whe Tes Are Preset Oe Rakg. Idgatoes Mathematcae, 4, [3] Sha, G., Hutso, A.D. ad Wldg, G.E. (0) Two-Stage k-sample Desgs for the Ordered Alteratve Problem. Pharmaceutcal Statstcs, [4] Page, E.B. (963) Ordered Hypotheses for Multple Treatmets: A Sgfcace Test for Lear Raks. Joural of the Amerca Statstcal Assocato, 58,

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