Rank-Score Tests in Factorial Designs with Repeated Measures

Transkript

1 Joural of Multivariate Aalysis 70, (1999) Article ID jmva , available olie at httpwww.iealibrary.com o Rak-Score Tests i Factorial Desigs with Repeate Measures Egar Bruer* a Ulrich Muzel* Uiversita t Go ttige, Go ttige, Germay a Maa L. Puri - Iiaa Uiversity Receive December 19, 1996 Noparametric factorial esigs for multivariate observatios are cosiere uer the framework of geeral rak-score statistics. Ulike most of the literature, we o ot assume the cotiuity of the uerlyig istributio fuctios. The moels stuie iclue geeral repeate measures esigs, compou symmetry esigs, a esigs for logituial ata. I particular, esigs for orere categorical ata are iclue. The vectors of the multivariate observatios may have ifferet legths. Moreover, our geeral framework iclues missig values a sigular covariace matrices which occur quite frequetly i practical ata aalysis problems. The asymptotic properties of the propose statistics are stuie uer geeral oparametric hypotheses as well as uer a sequece of oparametric cotiguous alteratives. L -cosistet estimators for the ukow covariace matrices are give a two types of quaratic forms are cosiere for testig the oparametric hypotheses. The results are applie to a two-way mixe moel assumig compou symmetry a to a factorial esig for logituial ata. The maiea of the proofs is base o some momet iequalities for empirical istributio fuctios i mixe moels. The etails are provie i the Appeix Acaemic Press AMS 1991 subject classificatios 6G10, 6G0, 6H10, 6H1, 6H15. Key wors a phrases rak tests; score fuctios; ties; orere categorical ata; oparametric hypotheses; mixe moel; logituial ata. * Supporte i part by DFG, Br , Partly supporte by the Office of Naval Research Cotract N J X Copyright 1999 by Acaemic Press All rights of reprouctio ay form reserve. 86

2 RANK TESTS FOR REPEATED MEASURES INTRODUCTION I may biological experimets a ecological, psychological, or meical stuies, the subjects are observe repeately uer ifferet or uer the same coitios. Such esigs ca be escribe by several more or less complicate moels which come uer the framework of mixe moels. They iclue growth curves, logituial ata, or repeate measures esigs where a special structure for the epeecies of the multivariate observatios, e.g., the compou symmetry, may or may ot be assume. The theory for parametric as well as for semi-parametric mixe moels is well evelope a is excelletly escribe i some recet textbooks (see, e.g., Daviia a Giltia, 1995; Diggle, Liag, a Zeger, 1994; Kshirsagar a Smith, 1995; or Lisey, 1993). For oparametric moels, however, where oly the istributio fuctios of the observatios are use to efie treatmet effects or to express hypotheses, the theory is less evelope, particularly whe several factors are preset i the trial. The first ieas to use he so-calle margial moel to efie treatmet effects i a oparametric mixe moel ate back to Hollaer, Pleger, a Li (1974) a Goviarajulu (1975) a were extee later o a stuie i more etail by Bruer a Neuma (198), Thompso (1990, 1991), a Bruer a Deker (1994). I this margial moel, a treatmet effect is efie through the margial istributios F j,,..., of X k =(X k1,..., X k )$ where X k is the vector of observatios for subject k. The observatios X kj a X k$j$ comig from ifferet subjects k a k$ are assume to be iepeet while the observatios X kj a X kj$ from the same subject may be epeet. The, a treatmet effect ca be efie i the same way as for the WilcoxoMaWhitey statistic or for the KruskalWallis statistic with iepeet observatios. Both statistics are base o cosistet estimates of the quatities p j =HF j, where H= &1 F j eotes the mea of the margial istributios. The quatities p j are calle relative treatmet effects, because they measure the effect of the treatmet j with respect to the mea H of all istributios i the experimet, a are estimate i a atural way by replacig the istributios by their empirical couterparts F j a H respectively. It is well kow that the estimator p^ j= H F j ca be compute from the raks R kj over all (epeet a iepeet) observatios X kj,,...,,,...,. Thus, the cocept of the margial moel leas to the problem of cosierig the istributio of rak statistics which use a rakig over epeet a iepeet observatios. For historical review of rak methos for mixe moels, we refer to Bruer a Puri (1996) or Akritas a Bruer (1997). A geeral formulatio of hypotheses i oparametric moels was suggeste by Akritas a Arol (1994) who itrouce the iea to formulate

3 88 BRUNNER, MUNZEL, AND PURI the hypotheses for the margial moel i terms of the istributio fuctios a erive the relevat asymptotic istributio theory uer the ull hypothesis as well as uer a sequece of oparametric cotiguous alteratives. The above results, however, were obtaie uer the assumptio of the cotiuity of the istributio fuctios which meas that ties were ot allowe. This is rather a urealistic assumptio for applicatios. Base o the ieas of Muzel (1994, 1998), Bruer, Puri a Su (1995) use the ormalize versio of the istributio fuctio F(x)= 1 [F + (x)+f & (x)] a of the empirical istributio fuctio to, erive the asymptotic results for two-sample rak statistics i mixe moels icluig the case of arbitrary ties. Here, F + (x) eotes the right cotiuous versio a F & (x) eotes the left cotiuous versio of the istributio fuctio of a raom variable. Akritas a Bruer (1997) geeralize these results to factorial esigs usig the cocept of Akritas a Arol (1994) to formulate oparametric hypotheses. The avatage of this cocept to formulate the hypotheses i terms of the istributio fuctios is that it iclues moels with cotiuous istributio fuctios as well as moels with iscrete observatios (e.g. orere categorical ata). Sice the proceures use rakigs over all observatios, they are robust to outliers as well as ivariat uer strictly mootoe trasformatios of the ata. I orer to make these proceures broaly applicable for the aalysis of real ata sets, several improvemets have to be worke out. First, the case of missig values has to be stuie a also the situatios where sigular covariace matrices appear i a atural way have to be aresse. For example, i moels with orere categorical ata, sigular covariace matrices appear quite ofte. If a treatmet is very effective, the (at a certai time poit) all subjects may be rate i the same category of the graig scale for this treatmet while the graig scores for the subjects i the cotrol group may be ifferet. Cosier the ice ata set of the shouler tip pai trial (see, e.g., Lumley, 1996) a assume that the observatios of patiet o. 113 at time poit 4 a of patiet o. 119 at time poit 5 woul be missig. The the estimate variace for the male patiets uer the active treatmet at time poit 4 a the estimate variace for the female patiets uer the active treatmet at time poit 5 becomes 0 a hece, the estimate covariace matrix becomes sigular. It caot be regare as reasoable to elete the two time poits from the aalysis of the trial i orer to be able to perform the computatios. Hece, also sigular covariace matrices have to be iclue i the cocept. I aitio, the special moel of compou symmetry where the covariace matrix has a simple structure, has to be aresse. Moreover, some alterative versio of the Wal-type statistic (WTS) has to be stuie sice it is well kow that the performace of the WTS is rather poor for small samples sizes (see

4 RANK TESTS FOR REPEATED MEASURES 89 Akritas a Bruer, 1997). It is the aim of the preset paper to aress these problems from both the applie as well as the theoretical viewpoit uer the framework of liear rak statistics with geeral scores. Orgaizatio of the Paper. The paper is orgaize as follows I Sectio, the otatios are itrouce a the relative treatmet effects a the estimators are efie. These estimators are liear cotrasts of rak statistics. The asymptotic properties of the estimators uer the hypothesis as well as uer a sequece of oparametric alteratives is cosiere i Sectio 3. Cosistet estimators for the ukow covariace matrices i the multivariate moel as well as i the compou symmetry moel are also erive i Sectio 3. Fially, the asymptotic properties of the WTS a of a propose simple moificatio, which is calle ANOVA-type statistic (ATS), are cosiere i this sectio. Both statistics are applie i Sectio 4 to ifferet esigs. Some techical Lemmas which are eee to prove the theorems i the boy of the paper are provie i the Appeix.. THE GENERAL MIXED MODEL AND NONPARAMETRIC HYPOTHESES I the geeral mixe moel, r treatmet groups (the so-calle whole-plot factor) are cosiere where every treatmet group i cotais,..., iepeet (raomly chose) subjects. These = r +1 i subjects are observe repeately uer,..., ifferet (fixe) situatios (``treatmets,'' the so-calle sub-plot factor) with s=1,..., replicatios for subject k uer the treatmet combiatio (i, j). Thus, there are M ik = m ijk repeate measures for each subject where the subjects are repeately observe uer the same treatmet as well as uer ifferet treatmets. This geeral mixe moel ca be writte by iepeet raom vectors X ik =(X$ i1k,..., X$ ik )$, i=1,..., r,,...,, where X ijk =(X ijk1,..., X ijkmijk )$,,...,, (.1) a where X ijks tf ij (x)= 1 [F + ij (x)+f & ij (x)], i=1,..., r,,...,, k= 1,...,, s=1,..., (the sig t meas ``is istribute as''). Here, F + ij (x)= P(X ijks x) is the right cotiuous versio a F & ij (x)=p(x ijks <x) is the left cotiuous versio of the istributio fuctio. The ormalize versio F ij (x) (see Lag, 1993, p. 89) iclues cotiuous as well as iscotiuous istributio fuctios a is eee for halig ties. To erive the geeral results, we o ot assume ay particular structure for the epeecies of the compoets of the vectors X ik except that the X ik are iepeet,

5 90 BRUNNER, MUNZEL, AND PURI i=1,..., r,,..., a that the bivariate margial istributio fuctios of (X ijks, X ij$ks$ ) o ot epe o k, s a s$, i.e., (X ijks, X ij$ks$ )tf ijj$ (x, y). This assumptios reasoable sice the observatios with k{k$ are iepeet replicatios a the observatios with s{s$ for the same j a k are epeet replicatios of the experimet. We ote that this assumptio o the bivariate margial istributio fuctios is eee to erive the asymptotic results uer cotiguous alteratives (see Theorem 3.4). The rather geeral otatiotrouce above, covers a lot of esigs which are commoly use i practice. For example, if r=1 group of,..., subjects is observe uer,..., treatmets, the F ij #F j, j+1,...,. This esigs calle ``oe-group repeate measures esig.'' If there are r groups of subjects the this is the so-calle split-plot esig. I the two-fol este classificatio with i=1,..., r treatmets,,..., iepeet subjects are observe uer treatmet i where every subject receives oly oe treatmet but is observe repeately s=1,..., m ik times i orer to get a more accurate measuremet for the variable of iterest. I total, there are N= r i=1 m ik observatios of = r i=1 iepeet subjects. Higher-way layouts are easily escribe i this setup by splittig the iices i or j ito sub-iices i$, i",... or j$, j",..., respectively. Thus, higherway layouts with repeate measures or logituial ata are covere by the geeral moel efie i (.1). Note that i most cases with logituial ata, =1 or =0 (if the observatios missig). The case of 1 typically occurs whe some material, tissue or a set of iiviuals is split ito several homogeeous parts a the compou symmetry moel ca be use as a appropriate moel for this esig. Sice o parameters are ivolve i this setup, we use the istributio fuctios F ij (x) to escribe a effect (e.g. time effect or treatmet effect). To this e, we cosier the so-calle relative treatmet effects p ij =H(x) F ij (x) where H(x)=N &1 r i=1 m ijkf ij (x) is the average of all N= r i=1 m ijk istributio fuctios i the experimet. The relative effects p ij may be weighte iepeetly of i a j by a score fuctio J(u) u #(0,1)R with boue seco erivative, i.e., &J"& =sup 0u1 J"(u) < a we efie the relative score effect p ij (J)=J[H(x)] F ij (x). We eote by p(j)=(p 11 (J),..., p r (J))$ the vector of these relative effects which are estimate by replacig H(x) a F ij (x) by their empirical couterparts. To iclue also the case of missig values, let * ijk = {0, 1, if =0, if 1, (.),...,, i=1,..., r,,...,

6 RANK TESTS FOR REPEATED MEASURES 91 a let * ijk =0 if =0. Moreover, a sum is uerstoo to be 0, if the upper summatio limit equals 0. Further, let c(x&x ijks )=0, if the observatio X ijks is missig. The the empirical fuctios are efie by F ij(x)= 1 * ijk F ijk(x) * ij } * ijk = 1 * ij } m ijk H (x)= 1 N r i=1 s=1 s=1 c(x&x ijk, ), * ij } = c(x&x ijks ). * ijk, (.3) Here, c(u)= 1 [c+ (u)+c & (u)] is the ormalize versio of the coutig fuctios c + (u) a c & (u) where c + (u)=0 or 1 accorig as u< or0 a c & (u)=0 or 1 accorig as u or >0. Note that F ij(x) is the uweighte mea of F ijk(x)=m &1 ijk c(x&x s=1 ijks),,..., where F ijk (x)=0 if =0. The relative treatmet effects p ij (J) are estimate by p^ ij(j)= J(H ) F ij= 1 * ijk J[H (X ijks )] * ij } m ijk s=1 * ijk = 1, ijks, (.4) * ij } m ijk s=1 where, ijks =J[H (X ijks )]=J[1N(R ijks & 1 )] is the rak-score of X ijks a R ijks is the mi-rak of X ijks amog all epeet a iepeet observatios. The quatities J[H(X ijks )] shall be calle asymptotic rak-score trasform of X ijks sice E[J(H )&J(H)] 0 uer suitable coitios (see Lemma A. i the Appeix) a H (X ijks )=1N(R ijks & 1 ). The ormalize versio H (x) of the combie empirical istributio fuctio leas automatically to a symmetric efiitio of H (x), sice H (x)#[1n, 1&(1N)] a thus, ulike i the literature so far, we shall cosier J[H (x)] istea of J[(NN+1) H (x)]. I the sequel, we will rop the expressio ``ormalize versio'' for brevity a whe usig the above quote fuctios, the ``ormalize versio'' is uerstoo if ot state otherwise. I the oparametric setup itrouce above, hypotheses are formulate by the istributio fuctios F ij.letf=(f 11,..., F r )$ eote the vector of the istributio fuctios a let C eote a cotrast matrix, i.e., C1=0 where 1=(1,..., 1)$ a 0=(0,..., 0)$. The a oparametric hypothesis i its most geeral form is writte as H F 0 CF=0. For mixe moels, such hypotheses have beetrouce by Akritas a Arol (1994) a have

7 9 BRUNNER, MUNZEL, AND PURI bee further evelope a iscusse by Akritas a Bruer (1997) a by Bruer a Puri (1996). For etails, we refer to these papers. Some examples for oparametric hypotheses are give Sectio ASYMPTOTIC RESULTS I this sectio, we erive the asymptotic istributio of - Cp^ (J)= - C( p^ 11(J),..., p^ r(j))$ uer the hypothesis H F 0 CF=0 as well as uer a sequece of cotiguous alteratives. Our results exte those of Akritas a Bruer (1997) to mixe moels with missig values. We also cosier the cases with sigular covariace matrices as well as compou symmetry moels. All these problems are stuie uer oe geeral framework usig the liear rak statistics with geeral scores. Moreover, for testig the oparametric hypothesis H F 0 CF=0, we ivestigate the properties of the ANOVA-type statistic ATS a compare it with the Wal-type statistic WTS Cosistecy a Basic Asymptotic Equivalece Note that * ij }, is the umber of subjects withi treatmet i with o missig observatios i the jth level of the sub-plot factor. Let 0 = mi 1ir, 1j * ij }. (3.1) The asymptotic results are erive uer the followig assumptios Assumptios. (A1) 0 m 0 <, i.e., the umber of replicatios of a fixe treatmet combiatio (i, j) withi oe subject is uiformly boue for all subjects a treatmets, (A) * ij } N 0 <, i=1,..., r,,...,, i.e., the ratio of the total umber of subjects a the umber of subjects with o missig values for treatmet combiatio (i, j) is uiformly boue, (A3) &J"& <, i.e., the score fuctio J( } ) has a boue seco erivative. First, we show that p^ ij(j) give (.4), is cosistet for p ij (J)= J(H) F ij i the sese of the followig propositio. Propositio 3.1. Let X ik be as efie i (.1) a let J(u) eote a score fuctio with boue first erivative, i.e., &J$& <. Further let * ijk a 0 be as efie i (.) a (3.1), respectively. If 0, the uer the assumptio (A1), E( p^ ij(j)&p ij (J)) 0, i=1,..., r,,...,.

8 RANK TESTS FOR REPEATED MEASURES 93 Proof. Let.(X ijks )=J[H(X ijks )]&J(H) F ij a ote that E[.(X ijks ).(X i$j$k$s$ )]=0 if (i, k){(i$, k$). The by the c r -iequality, Jese's iequality, Lemma A.() (i the Appeix) a by iepeece, E( p^ ij(j)&p ij (J)) =E _ J(H ) F ij& ij& J(H) F =E \ [J(H )&J(H)] F ij+ J(H) (F ij&f ij ) + m * ijk ijk E(J[H (X * ij } m ijks )]&J[H(X ijks )]) ijk s=1 + m i * ijk * ijk $ ijk$ E[.(X * iks ).(X ijk$s$ )] ij } $ k$=1 s=1 s=1 m 0 &J$& + &J& =O \ 1 0 r Next, the basic asymptotic equivalece is state. Theorem 3.. Let X ik be as i Propositio 3.1, let F=(F 11,..., F r )$ a F =(F 11,..., F r)$ where F ij is give (.3). Further, let = r =1 i eote the total umber of subjects a let 0 be as efie i (3.1). If 0, the, uer the assumptios (A1)(A3), - J(H ) (F &F)=.. - K J(H) (F &F) =- \Y }}(J)& J(H) F +, where Y }}(J)=(Y $ 1} } (J),..., Y $ r }} (J))$, Y i }}(J)=(Y i1} }(J),..., Y i }}(J))$, Y ij }}(J)= 1 m i * ijk ijk Y ijks (J), * ij } m ijk s=1 Y ijks (J)=J[H(X ijks )]. (3.) (Here a i the sequel, the sig =.. meas ``asymptotically equivalet.'')

9 94 BRUNNER, MUNZEL, AND PURI It suffices to cosier the (i, j) th compoet, which is ecom- Proof. pose as - J(H ) (F ij&f ij ) =- J(H) (F ij&f ij )+- [J(H )&J(H)] (F ij&f ij ). To prove the theorem, it suffices to show that E(- [J(H )&J(H)] (F ij&f ij )) 0as 0. Usig Taylor's expasio, we obtai J(H )&J(H)=J$(H)(H &H)+ 1 J"(% N)(H &H), where % N is betwee H a H. Thus, [J(H )&J(H)] (F ij&f ij )=B 1 +B where a B 1 = J$(H)(H &H) (F ij&f ij ) B = 1 J"(% N)(H &H) (F ij&f ij ). It follows from the Appeix, Lemma A.3 that E(- B k ) =O( &1 0 ),, which completes the proof. K 3.. Asymptotic Distributios Asymptotic Distributio uer H F 0. To erive the asymptotic istributio of - Cp^ (J) uer the hypothesis H F 0 CF=0 as well as uer the sequece of alteratives cotiguous to H F 0, we ee some regularity assumptio for the covariace matrix of - Y }}(J) give (3.). Let Y ik } (J)=(Y i1k } (J),..., Y ik } (J))$, where Y ijk } (J)=m &1 ijk Y s=1 ijks(j) asi (3.) a Y ijk } (J)=0 if =0. Further let V ij =Cov(Y ik } (J)) a i1k 4 ik = } iag {*,..., * ik * i1} * i }= a V i, =Cov(- Y i }}(J)). (3.3) The, Y i }}(J)= 4 iky ik } (J) a, by iepeece, r V =Cov(- Y }}(J))= i=1 r V i, i = i=1 i 4 ik V ik 4 ik. (3.4) Assumptio. (A4) V V{0 as, where V is a matrix of costats. (This assumptioclues the case of sigular covariace matrices as well.)

10 RANK TESTS FOR REPEATED MEASURES 95 Theorem 3.3. Let X ik be as i Propositio 3.1 a let V be as give (3.4). If 0, the uer the assumptios (A1)(A4) a uer H F 0 CF=0, the statistic - Cp^ (J)=- C J(H ) F has asymptotically ( 0 ) as multivariate ormal istributio with mea 0 a covariace matrix CV C$. Proof. First, we ote that uer H F 0, - C J(H ) (F &F)=- C J(H ) F =- Cp^ (J)=.. - CY }}(J), by Theorem 3. where Y }}(J)=(Y $ 1} } (J),..., Y $ r }} (J))$. Note that the vectors Y i }}(J) are iepeet a that Y i }}(J)= &1 i 4 iky ik } (J) is the mea of iepeet raom vectors Z ik (J)= (* i1k Y i1k } (J)* i1,..., * ik Y ik } (J)* i } )$. From the multivariate Cetral Limit Theorem, it follows that - Y i }}(J) has asymptotically a multivariate ormal istributio with mea 0 a covariace matrix V i sice Z ik (J) is uiformly boue by the assumptios of the theorem, i.e., &Z ik (J)&N 0 &J& -. IfV i =0 the the asymptotic istributios a multivariate oe-poit istributio which ca be regare as a egeerate ormal istributio. K 3... Cotiguous Alteratives. To ivestigate the set of alteratives for which the test statistics base o - Cp^ (J) are cosistet, efie a sequece of alteratives for the istributios of the bivariate margials, viz. which implies that (X ijks, X ij$ks$ )tf, ijj$ (x, y) =F ijj$ (x, y)+ 1 - (K ijj$(x, y)&f ijj$ (x, y)) (3.5) X ijks tf, ij (x)=f ij (x)+ 1 - (K ij(x)&f ij (x)), i=1,..., r,,...,,,...,, s=1,...,. (3.6) Let F =(F,11,..., F, rc )$ = F+ &1 (K& F) eote the vector of the margial istributio fuctios of this sequece where F=(F 11,..., F rc )$ such that CF=0 a K=(K 11,..., K rc )$ is some vector of (alterativeoe-imesioal istributio fuctios CK{0. Further, let &(J)= - C HF =J(H)&(CK). Theorem 3.4. Let X ik be as i Propositio 3.1 where X ijks tf, ij (x) as efie i (3.6),,...,, s=1,...,, i=1,..., r,,...,. Further-

11 96 BRUNNER, MUNZEL, AND PURI more, let F ij(x) a H (x) eote the empirical istributio fuctios as efie i (.3) a let C be a suitable cotrast matrix such that CF=0. Let H(x)=N &1 r i=1 ijk F ij (x) eote the weighte average istributio fuctio of F 11,..., F r a let H (x)= 1 N r i=1 F, ij (x) =H(x)+ 1 r - i=1 N [K ij(x)&f ij (x)] eote the weighte average istributio fuctio of F,11,..., F, r. If 0, the uer the sequece of alteratives (3.5) a uer the assumptios (A1)(A4), (1) the statistics &^ (J)=- Cp^ (J)=- C J(H ) F a - CY }}(J) =- C J(H) F are asymptotically equivalet, () &^ (J) has asymptotically a multivariate ormal istributio with mea &(J)=J(H) (CK) a covariace matrix CV C$. The proof is give the Appeix Estimatio of Covariace Matrices I most practical examples, the covariace matrices V i, i = &1 i 4 ikv ik 4 ik efie i (3.4) are ukow a must be estimate from the ata. To erive a cosistet estimator for V i, i, i=1,..., r, we istiguish two moels. The multivariate moel oes ot assume ay special patter for the bivariate margial istributio fuctios while the compou symmetry moel assumes the equality of certai margial istributio fuctios uer the hypothesis which is state i etails i the last part of this subsectio. I what follows, we provie the estimators for V i, i i both moels a we prove the cosistecy of these estimators Multivariate Moel. I the multivariate moel, let v i ( j) eote the iagoal elemets of V i, i a let v i ( j, j$) eote the off-iagoal elemets. Note that by (.), v i ( j)= * ij } * ijk v il ( j), where v ik ( j)=var(y ijk } (J)), (3.7)

12 RANK TESTS FOR REPEATED MEASURES 97 v i ( j, j$)= * ijk * ij$k v ik ( j, j$), * ij } * ij$} where v ik ( j, j$)=cov(y ijk } (J), Y ij$k } (J)). (3.8) Let, ijks =J[1N(R ijks & 1 )] eote the rak scores efie i (.4) a let, ijk=m &1 ijk, s=1 ijks a, ij }}=* &1 ij } * ijk, ijk } eote the uweighte meas of the, ijk } where, ijk } =0 if =0. The we efie the estimator V i, with iagoal elemets v^ i( j) a off-iagoal elemets v^ i( j, j$) give by v^ i( j)= * ij } (* ij } &1) v^ i( j, j$)= (* ij } &1)(* ij$} &1)+4 i, jj$ &1 _ * ijk (, ijk } &, ij }}), (3.9) * ijk * ij$k (, ijk } &, ij }})(, ij$k } &, ij$} }), (3.10) where 4 i, jj$ = * ijk* ij$k. We ote that V i, may ot be positive semiefiite (p.s..) if the umber of missig values is large. I this case, a p.s.. estimator for V i, i ca be obtaie by the geeral metho give by Staish, Gilligs, a Koch (1978). Theorem 3.5. Let v^ i( j) a v^ i( j, j $) be as give (3.9) a (3.10), respectively. If 0, the uer the assumptios (A1)(A4), E[v^ i( j)& v i ( j)] 0 a E[v^ i( j, j$)&v i ( j, j$)] 0. The proof is give the Appeix Compou Symmetry Moel. To state the results for the compou symmetry moel, we ee some further otatios a we assume that 1, for simplicity. We ote however, that balace icomplete esigs ca also be treate withi this framework; but for brevity, we o ot cosier this case here. I the compou symmetry moel, it is assume that uer H F 0 F i1 =}}}=F i, (1) the bivariate margial istributio fuctios of (X ijks, X ij$ks$ )o ot epe o j, j$, k, s or s$, i.e., (X ijks, X ij$ks$ )tf ijj$ (x, y)=f *(x, i y), j{ j$=1,...,, () the bivariate margial istributio fuctios of (X ijks, X ijks$ ), s{s$=1,..., o ot epe o j, i.e. (X ijks, X ijks$ )tf ij **(x, y)= F i *(x, y),,...,.

13 98 BRUNNER, MUNZEL, AND PURI Thus, uer H F 0, the variaces a the covariaces are give by _ i #_ ij =Var(Y ijks(j)),,...,, c i *=Cov(Y ijks (J), Y ij$ks$ (J)), j{ j$=1,...,,,...,, s=1,...,, s$=1,..., m ij$k, c i **=Cov(Y ijks (J), Y ijks$ (J)),,...,,,...,, s{s$=1,...,. We ote that _ i <, sice Y ijks(j) = J[H(X ijks )] &J&, which follows immeiately from the Assumptio (A3) i Subsectio 3.1. Thus, the fatess of the tails of F ij (x) oes ot have ay impact o the existece of these variaces. Let Y ik (J)=(Y i1k1 (J),..., Y i1kmi 1k (J),..., Y ik1 (J),..., Y ikmik (J))$. Thet follows that where Cov(Y ik (J))=7 ik = V ik =Cov(Y ik } (J))= \ [(_ i &c i**) I mijk +(c i **&c i *) J mijk ]+c i *J Mik, 1 1$ mijk+ 7 ik\ 1 = _ (_ i &c i **) +(c m i **&c i *) &+c i*j ijk V i, i = 1 { ij = mijk+ V ik =iag[{ i1,..., { i ]+c i *J =D i, i +c i *J, { ijk a { ijk = 1 (_ i +(&1) c i **)&c i *. (3.11) Compou symmetry is oly assume for hypotheses regarig the subplot factor, i.e. for hypotheses which ca be writte as H F 0 [I r C ) F=0 where C is a suitable cotrast matrix for the sub-plot factor. Thus we ee oly to estimate D i, i sice (I r C ) V (I r C$ )= r i=1 C D i, i C$. Theorem 3.6. Let D i, i =iag[{ i1,..., { i ] where { ij is give (3.11). Let D i eote the matrix correspoig to D i, i where { ij is replace by {^ ij= &1 i {^ ijk a {^ ijk is efie below. Furthermore, let, ijks eote the rak-scores as efie i (.4) a let, i } k } = &1, ijk }. The the compou symmetry moel, uer the assumptios (A1)(A3) a uer H F F 0 i1= }}}=F i, &D i&d i, i & 0 as mi i.

14 RANK TESTS FOR REPEATED MEASURES 99 If is ot equivalet to a costat, the {^ ijk= 1 M ik &_ 1 & 1 t=1 1 m itk& S + 1 ik1 &1 S, (3.1) ik where S ik1 = s=1 (, ijks&, ijk } ), S ik = (, ijk } &, i } k } ), a M ik =. If #m, the D i={^ ii where {^ i= 1 (&1) (, ijk } &, } k } ). (3.13) The proof is give the Appeix. Remark. Note that the results of the Theorems 3.5 a 3.6 may ot be sufficiet to erive the asymptotic istributios of the statistics (to be efie i the ext subsectio) whe V i, i or D i, i are replace by V i, or D i, respectively. Further assumptios o the covariace matrices will be iscusse for the ifferet statistics separately Statistics To test the oparametric hypothesis H F 0 CF=0, we cosier two statistics, amely the rak versios of the WTS a of the ATS. Other statistics which are commoly use i multivariate aalysis are ot iscusse here sice they require the equality of the covariace matrices. I a oparametric setup, however, this assumptios oly justifie i a few special cases. Note that i geeral ay assume homosceasticity of the paret istributio fuctios is ot trasferre to the asymptotic rakscore trasform Y ijks (J)=J[H(X ijks )] because H( } ) is a o-liear trasformatio Wal-Type Statistics (WTS). Let V i, eote the L -cosistet estimator of the covariace matrices V i, i, i=1,..., r as give Theorem 3.5. The r V = i=1 V i, (3.14) is a L -cosistet estimator of the covariace matrix V give (3.4) i the sese of Theorem 3.5. Let [ } ] + eote a symmetric reflexive g-iverse of a matrix. The we efie the rak versio of the WTS for testig H F 0 CF=0 by Q W (C)=p^ $(J) C$[CV C$] + Cp^ (J). (3.15) The asymptotic istributio of Q W (C) is give the ext theorem.

15 300 BRUNNER, MUNZEL, AND PURI Theorem 3.7. Let V a V be as efie i (3.14) a (3.4), respectively. Assume that V V{0 as 0 such that rak(cv )=rak(cv). The, uer the assumptios (A1)(A4) a (1) uer the sequece of alteratives (3.5) cotiguous to H F 0 CF=0, Q W (C) wl Zt/ f (*), where / f(*) is the chi-square istributio with egrees of freeom f =rak(cv) a ocetrality parameter *=&$(J)[CVC$] + &(J) where &(J)=C J(H) K, () uer H F CF=0, 0 QW(C) wl Z 0 t/ f (0), with f =rak(cv). Proof. First ote that - Cp^ (J) has asymptotically a multivariate ormal istributio with mea &(J)=J(H) (CK) a covariace W matrix CVC$ by Theorem 3.4 a the assumptio (A.4). Let Q (C)= p^ $(J) C$[CVC] + Cp^ (J). Thet follows from Theorem 9..3 i Rao a W Mitra (1971) that Q (C) has asymptotically a ocetral /(*)-istribu- tio with f =rak(cv) egrees of freeom a ocetrality parameter f *=&$(J)[CVC$] + &(J). Next, ote that &CV C$&CV C$& 0 by Theorem 3.5 a that CV C$&CVC$ 0 by assumptio. Thus, it follows that (CV C$) + & (CVC$) + 0 because rak(cv )=rak(cv) by assumptio. Hece, Q W(C)&Q W (C) 0iL a the result state i (1) follows. Statemet () follows immeiately from (1) because &(J)=0 uer H F CF=0. K ANOVA-Type Statistic (ATS). Let M=C$[CC$] & C where [}] & eotes some geeralize iverse of A matrix. The, we efie the rak versio of the ATS by Q A (C)=p^ $(J) Mp^ (J). (3.16) Note that M is a projectio matrix a that MF=0 CF=0 because C$[CC$] & is a geeralize iverse of C. Thus, it is also reasoable to use Q A(C) as a test statistic for testig the hypothesis H F 0 CF=0. The asymptotic istributio of Q A (C) is give the ext theorem. Theorem 3.8. Let M=C$[CC$] & C a let V a V be as i (3.4) a (3.14) respectively. Let &(J)= J(H) K be as efie i Theorem 3.4. Uer the assumptios (A1)(A4), if 0, the (1) uer the sequece of alteratives (3.5) cotiguous to H F 0 CF=0 the statistic Q A (C) give (3.16) has asymptotically the same istributio as r i=1 *, ijz, ij where the *, ij are the characteristic roots of MV

16 RANK TESTS FOR REPEATED MEASURES 301 a the Z, ij are iepeet ocetral / 1 (+, ij )-istribute raom variables where r i=1 *, ij+ =&$(J)M&(J),, ij () uer H F CF=0, the statistic 0 QA (C) give (3.16) has asymptotically the same istributio as r i=1 *, ijz 0, ij where the *, ij are the characteristic roots of MV a the Z 0, ij are iepeet cetral / 1-istribute raom variables. Proof. The proof follows from Theorem 3.4 a well kow theorems o the istributio of quaratic forms (see, e.g., Mathai a Provost, 199, Chap. 4). The istributio of r i=1 *, ijz 0, ij ca be approximate by a scale / -istributio. Approximatio Proceure by a Cetral F( f, )-Distributio. (1) Assume that tr(mv )t 0 >0. The, uer H F 0, the first two momets of the asymptotic istributio of Q A (C)tr(MV ) a of the F( f, )-istributio coicie for f =[tr(mv )] tr(mv MV ). () The ukow traces tr(mv ) a tr(mv MV ) ca be estimate cosistetly by replacig V with V give Theorems 3.5 a 3.6, respectively. This fially leas to the statistic where F (C)= 1 tr(mv ) QA (C)t * * F( f, ), f = [tr(mv )] tr(mv MV ). (3.17) (Here a i the sequel, the sig t* * meas ``approximately istribute as.'') Derivatio. From Theorem 3.8, Q A (C) has asymptotically the same istributio as r i=1 *, ijz 0, ij. We wat to approximate the istributio of Q A(C) byg } Z where Z is a raom variable with a cetral /-istribu- tio a g is a costat such that the first two momets coicie. Uer f H F 0, we obtai E(Q A (C))= r Var(Q A (C))= } r i=1 i=1 =Var(g } Z)=g f. *, ij =tr(mv )=E(g } Z)=gf, *, ij = } tr(mv MV )

17 30 BRUNNER, MUNZEL, AND PURI Fially, ote that tr(mv )t 0 >0 implies that tr(mv MV )t 1 >0. Thus, Q A (C)(gf )t * * F( f, ) a the result follows from Theorem 3.5 a Theorem 3.6, respectively. Remark. The approximatio proceure goes back to Box (1954) a turs out to be fairly goo for iepeet observatios (see Bruer, Dette, a Muk, 1997). For repeate measures, f may be biase for small samples Compariso of the Two Statistics a Cosistecy of the Test. The mai avatage of the WTS Q W (C) is that its asymptotic istributio uer H F is a kow istributio fuctio, amely a 0 / -istributio. The geeral rawback of Q W (C) is that it coverges extremely slowly to its asymptotic istributio resultig i rather liberal ecisios for small or moerate sample sizes. Moreover, the restrictive assumptio that V V such that rak(cv )=rak(cv) caot be checke. I the case of a sigular covariace matrix, the set of alteratives which ca be etecte by Q W (C) epes o the structure of the covariace matrix a o the hypothesis uer cosieratio. Thus, there exists a set of fixe alteratives which may ot be etecte by Q W (C). This set is give by where z is a arbitrary vector such that p=[i&vc$(cvc$) + C] z, (3.18) Cp=[I&CVC$(CVC$) + ] Cz{0. (3.19) To see this, ote that *=p$c[cvc$] + Cp=0 C$[CVC$] + Cp=0 sice [CVC$] + is positive semiefiite a symmetric. Furthermore observe that V is a g iverse if C$[CVC$] + C. Thus, the solutio space of the homogeeous system of liear equatios C$[CVC$] + Cp=0 is give by (3.18) a the restrictio (3.19) follows if the test is require to be cosistet for alteratives of the form Cp{0. From this, it is easy to see that a test base o Q W (C) is cosistet for all alteratives Cp{0 if, e.g., either V is o-sigular a C is of full row rak or if rak(cv)=rak(c). Note, however, that these simple coitios are oly sufficiet. The ATS F (C) has the mai isavatage that its asymptotic istributio uer H F 0 cotais ukow quatities, amely the characteristic roots of MV which are ukow geeral a must be estimate. We suggest to use the Box-approximatio which is kow to be fairly accurate. However, ote that eve the asymptotic case, the / f f-istributios a approximatio of the true istributio of the statistic uer H F. Oe 0 avatage is that it is either ecessary to assume the covergece of the covariace matrix V to a costat matrix V or that the rak of CV s

18 RANK TESTS FOR REPEATED MEASURES 303 preserve i the limit CV. The oly aitioal assumptio to (A1)(A4) which is eee for the ATS is that tr(mv ){0 which meas thatregarig the hypothesis of iterestthere is at least some variatio amog the observatios of the experimet. This is close to a trivial assumptio. The set of alteratives etecte by a test base o F (C) is give by &(J)=Cp K (J)=C J(H) K{0 sice E(Q A (C))= r i=1 *, ij (1++, ij )=tr(mv )+&$(J) M&(J) uer the sequece of alteratives (3.5) cotiguous to H F 0 CF=0. This set oes ot epe o the rak of the covariace matrixulike as for the WTS. The mai avatage of F (C) is that the approximatio by the / f f-istributio works also fairly well for rather small sample sizes (for etails, see, e.g., Bruer a Lager, 1999). We recomme the use of the ATS for small a moerate sample sizes a for the case where the estimate covariace matrix is sigular or close to beig sigular which may happe frequetly with orere categorical ata. I the ext sectio, the geeral results give here will be applie to ifferet two- a three-way layouts. 4. APPLICATIONS Numerous examples for the applicatio of the WTS i the oparametric multivariate moel (Wilcoxo-scores, o missig values, osigular covariace matrices) have bee give by Akritas a Bruer (1997). Sice they i ot cosier the compou symmetry moel, either the ATS or missig values, we ow give two examples which are cocere with these situatios. The applicatio of the WTS is cosiere for a two-way layout with raom iteractio where compou symmetry is assume. The ATS is applie i a three-way layout with two fixe factors with missig values i a multivariate moel Two-way Layout with Raom Iteractio, Compou Symmetry First, we apply the results of the previous sectio to a cross-classifie esig with oe raom factor (,..., levels) a with oe fixe factor (,..., levels). For every subject k a treatmet j, the umber of replicatios m kj is assume to be uiformly boue, i.e. m ij m 0 <. Let X k =(X k11,..., X kmk )$=(X$ k1,..., X$ k )$,,..., be iepeet raom vectors where X kj =(X kj1,..., X kjmkj )$ eotes the vector of observatios for subject k a where X kjs tf j (x),,...,,,...,,

19 304 BRUNNER, MUNZEL, AND PURI s=1,..., m kj. It is reasoable to assume compou symmetry if subjects are split ito homogeeous parts a if there is o treatmet effect. This meas that the parts of each subject are ``iterchageable'' uer the hypothesis. However uer a treatmet effect, the compou symmetry structure may ot be preserve. The property of iterchageable parts of the subjects is reflecte by the iterchageability of the raom variables X kjs a X kj$s$ for j{ j$=1,..., a \s, s$ uer the hypothesis of o treatmet effect. The raom variables X kjs a X kjs$, s, s$=1,..., m kj are always iterchageable sice they escribe replicatios of the same experimet uer the same treatmet for the same subject. Treatmet effects are escribe by the relative treatmet effects p j (JJ)=J(H) F j,,..., where H=N &1 N jf j a N= N j, N j = m kj. Let p(j)=j(h) F where F=(F 1,..., F )$. We erive the results from Theorem 3.3 a Theorem 3.6 for r=1, = a =m kj. Note that the subjects are umbere by k which is the first iex i this otatio. Let p^ (J)=(p^ 1(J),..., p^ (J))$=J(H ) F where p^ j(j)= J(H ) F j= 1 m kj 1, kjs =, } j } m kj s=1 a where, kjs =J[H (X kjs )]=J[1N(R kjs & 1 )] a R kjs is the rak of X kjs amog all the N raom variables. Accorig to the otatio Theorem 3.6, let D=iag[{ 1,..., { ], { j = &1 { kj, a let D = iag[{^ 1,..., {^ ], {^ j= &1 {^ kj where {^ kj= 1 M k &_ 1 m kj & 1 t=1 1 m kt& S + 1 k1 &1 S, k S k1 = m kj (, s=1 kjs&, kj }), S k = (, kj } &, k }}) a M k = m kj. The the WTS for testig H F P 0 F=0 is erive from (3.15) for C=P &1 &1 a V =D. Thus, CV C$=P D P. Note that W =D [I &J D &1 tr(d )] is a g-iverse of P&D P a ote that P W P =W. Let, kj } =m &1 kj m kj, s=1 kjs eote the meas of the rak-scores for subject k,,..., a let, }}}= 1 (1{^ j), } j } {^ j eote the weighte mea of the estimate relative treatmet effects p^ j(j)=, } j }. The, from (3.15) a Theorem 3.7, Q W(P )= } p^ $(J) P [P D P] & P p^(j)= 1 (, } j } &, }}}) {^ j

20 RANK TESTS FOR REPEATED MEASURES 305 has asymptotically a cetral / &1 -istributio uer the hypotheses H F 0 if D is of full rak a if D D such that rak(d )=rak(d)=. Note that i this case, the quaratic form Q W (P ) has a cetral / f -istributio with f =rak(p )=&1 for ay choice of the g-iverse [P D P ] &. I the case of equal cell frequecies, m kj #m, {^ j #{^ = 1 (&1) t=1 (, kt } &, k }}),, }}}= 1 N m s=1, kjs a the statistic simplifies to (&1) Q W (P )= (, (, kj } &, k }}) } j } &, }}}), which has bee give by Bruer a Neuma (198) for the case of cotiuous istributio fuctios a for Wilcoxo-scores, kjs =1N(R kjs & 1 ). 4.. Three-way Layout, Multivariate Moel I a seco example, we apply the geeral theory evelope i Sectio 3 to a three-way layout with two fixe factors. Let X ik =(X i1k,..., X ibk )$, i=1,..., a,,..., be = a =1 i iepeet raom vectors where X ijk tf ij (x), i=1,..., a,,..., b,,...,. This moel is appropriate for a trial where i=1,..., a groups (whole-plot factor A) of (iepeet) subjects are observe repeately uer,..., ifferet situatios (subplot factor B). The subjects are este uer the levels of factor A a are crosse with factor B. I esigs with logituial ata, typically the time is the sub-plot factor B. For simplicity, we apply the geeral results erive i Sectio 3 oly to the case of #m=1 replicatio for each subject k a combiatio (i, j) of the factor levels of A a B. We amit however that some observatios are missig (at raom) where the otatiotrouce i (.) is use. We shall give statistics for testig the oparametric hypotheses H F (A) 0 of o group effect, H F(B) of o sub-plot factor effect, a H F 0 0 (AB) ofo iteractio betwee the groups a the sub-plot factor. Let P = I &(1) J where is the imesio of the ietity matrix I a of the matrix J =1 1$ of 1's. Let F=(F 11,..., F ab )$ eote the vector of the margial istributio fuctios F ij of the vectors X ik. The the oparametric hypotheses are expresse as H F (A) C 0 AF=0, H F (B) C 0 bf=0 a H F 0(AB) C AB F=0, respectively, where C A =P a (1b) 1$ b, C B = (1a) 1$ a P b a C AB =P a P b. Let R ijk eote the (mi)-rak of X ijk amog all the N= a i=1 b * ijk raom variables a let, ijk =J[1N(R ijk & 1 )] eote the rak-score

21 306 BRUNNER, MUNZEL, AND PURI of X ijk. Furthermore let, ik =(, i1k,...,, ibk )$ eote the vector of the rakscores for subject k i group i a let, i } =(, i1},...,, ib } )$, where, ij } = 1 * ijk, ijk * ij } eote the mea vector of the scores i group i, i=1,..., a. Fially, let, } =(, $ 1},...,, $ a } )$ eote the vector of all a } b meas of the rak-scores. A estimator of the covariace matrix is erive from Theorem 3.5, a V = i=1 V i,, (4.1) where the elemets of V i, are give (3.9) a (3.10), respectively. Below, the statistics for testig H F(A), H F(B) a H F (AB) are give. They are erive from Theorem 3.8 a from the approximatio proceure. To erive the statistic for H F(A) C 0 AF=0, let, i }}=b &1 b, ij },, }}}=a &1 a, i=1 i }} a _^ =1$ i bv i, 1 b = b v^ j{ j$ i( j, j$)+ b v^ i( j). Further let a 7 a= i=1 _^, a T i A=C$ A [C A C$ A ] & C A =P a 1 i b J b. Note that T A is a projectio matrix with rak(t A )=a&1 a with ietical iagoal elemets (a&1)(ab). Sice 7 a is a iagoal matrix, a improve approximatio as give Bruer, Dette, a Muk (1997) ca be use. This approximatios rather accurate for 7, i=1,..., a. Uer H F 0 (A), the istributio of F (T A )= } a } b (a&1) tr(7 a),$ a } b a } T A, } = (, (a&1) a i }}&, }}}) i=1 _^ i is approximately the cetral F( f A, f 0)-istributio with estimate egrees of freeom i=1 f A= [(a&1) a i=1 _^ i ] a tr(p a 7 ap a 7 a) a f 0= [a _^ =1 i i] a i=1 (_^ i ) ( &1), where tr( } ) eotes the trace of a square matrix. It is easy to see that f A=1 if a=. To test H F 0(B) C B F=0, let, }}=((1a) 1$ a I b ), } =(, }1,...,, } b } )$ eote the vector of the average scores, } j } =a &1 a, i=1 ij } for the b

22 RANK TESTS FOR REPEATED MEASURES 307 time poits. Let V B= a i=1 &1 i V i, a let T B =C$ B (C B C$ B ) + C B = (1a) J a P b. The, uer H F 0 (B), the statistic F (T B )= a tr(p b V B),$ }}P b, }}= a tr(p b V B) b (, } j } &, }}}) has asymptotically a cetral F( f B, )-istributio with f B=[tr(P b V B)] tr(p b V BP b V B). To test H F(AB), let T 0 AB=P a P b. Uer H F 0 (AB), the statistic F (T AB )= tr(t AB V ) a i=1 b (, ij } &, i }}&, } j } +, }}}) has approximately a cetral F( f AB, )-istributio with f AB= [tr(t AB V )] tr(t AB V T AB V ). APPENDIX Lemma A.1. Let X ijks tf ij, i=1,..., r,,...,,,...,, s= 1,..., a let X ijks a X i$j$k$s$ be iepeet for (i, k){(i$, k$). The (1) E[c(x&X ijks )&F ij (x)]=0, i=1,..., r,,...,,,...,, s=1,...,, () E[c(X ijks &X i$j$k$s$ )&F i$j$ (X ijks )]=0, if (i, k){(i$, k$), (3) F ij F ij = 1. Proof. (1) By efiitio, E[c(x&X ijks )&F ij (X)]=P(X ijks <x)+ 1 P(X ijks=x)&f ij (x) = 1 [F + ij (x)+f & ij (x)]&f ij (x)=0. () This follows by otig that E([c(X ijks &X i$j$k$s$ )&F i$j$ (X ijks )] X ijks =x)=0 a applyig Fubii's theorem for (i, k){(i$, k$). (3) The result follows usig by parts. K To prove the asymptotic results, we first give some momet iequalities for empirical processes i the mixe moel which are eee i the boy of the paper. Lemma A.. Let X ik be as efie i (.1) a let J(u) eote a score fuctio with boue first erivative, i.e., &J"& <. Further let * ijk a 0

23 308 BRUNNER, MUNZEL, AND PURI be as efie i (.) a (3.1), respectively. If 0, the uer the assumptio (A1), (1) E[J[H (x)]&j[h(x)]] m 0 r 0 &J$&, () E[J[H (X ijks )]&J[H(X ijks )]] m 0 r 0 &J$&, (3) E[F ij(x)&f ij (x)] 1 0, (4) E[F ij(x ijks )&F ij (X ijks )] 1 0, (5) E[H (x)&h(x)] 4 0 \m r 0+, (6) E[H (X ijks )&H(X ijks )] 4 0 \m r 0+. Proof. To prove (1), ote that by the mea value theorem, [J[H (x)]&j[h(x)]] &J$& [H (x)&h(x)]. Next, ote that by iepeece, E[H (x)&h(x)] = 1 r N r $ m i $j $k$ i=1 i$=1 j$=1 k$=1 s=1 s$=1 _E([c(x&X ijks )&F ij (x)][c(x&x i$j$k$s$ )&F i$j$ (X)]) 1 r N m ij $k 1 i=1 j$=1 s=1 s$=1 m 0 N m 0 r 0. Statemet () follows i the same way by otig that E([(X ijks &X stuv )&F st (X ijks )][c(x ijks &X s$t$u$v$ )&F s$t$ (X ijks )])=0 if (s, t){(s$, t$) sice either (i, k){(s, t) or(i, k){(s$, t$) i this case.

24 RANK TESTS FOR REPEATED MEASURES 309 Statemet (3) follows i the same way as statemet (1), sice E[F ij(x)&f ij (x)] = k$=1 * ijk * ijk$ * m ij } ijk$ s=1 $ s$=1 _E([c(x&X ijks )&F ij (x)][c(x&x ijk$s$ )&F ij (x)]) * ijk * ij } m ijk 1 * ij } 1 0. s=1 s$=1 E[c(x&X ijks )&F ij (x)] Statemet (4) follows by the same argumet as use to prove (). To prove (5) a (6), the iices i a k are collapse to oe iex l, say, where l=1,..., = r i=1. Thus, the umber of replicatios for each subject l uer treatmet j is relabele from to m lj,,...,. Similarly, the iices j a s are collapse to oe iex t, say, where t=1,..., M l = m lj. Thus, the raom variables X ijks are relabele to X lt, l=1,...,, t=1,..., M l. Note that N= l=1 M l a that X lt a X l$t$ are iepeet if l{l$. Further let.(x lt )=c(x&x lt )&F lt (x). The, E[H (x)&h(x)] 4 = 1 N 4 l=1 l$=1 l"=1 l$$$= 1 M l t=1 M l$ t$=1 M l$$ t"=1 _E[.(X lt ).(X l$t$ ).(X l"t" ).(X l$$$t$$$ )]. M l$$$ t$$$=1 If oe of the iices l, l$, l", l$$$ is ifferet from the three other iices, thet follows that the expectatio o the right ha sie is 0 sice the raom variables are iepeet if the first iices are ifferet. It suffices to cout the umber of cases where ot oe of the iices l, l$, l", l$$$ is ifferet from the three others. This happes if the iices are either all equal or if they are pairwise equal, e.g., l=l${l"=l$$$. Thus, E[H (x)&h(x)] 4 << 1 N 4 l=1 l$=1 M l t=1 M l t$=1 M l$ t"=1 M l$ t$$$=1 _E[.(X lt ).(X lt$ ).(X l$t" ).(X l$t$$$ )] << 1 N 4\ M l+ l=1 1 max N 4\ 1l M l M l=1 l+ = 1 N max M. l 1l

25 310 BRUNNER, MUNZEL, AND PURI returig to the origial otatio where the raom variables X ijks, i=1,..., r,,...,,,...,, s=1,..., are cosiere, it follows that M l m 0, l=1,..., a that Nr 0.Thus, E[H (x)&h(x)] 4 << 1 N max 1l M l m 0 r 0 which proves (5). The proof for (6) follows by coitioig o X ijks a proceeig as i (5). K To prove Theorem 3., we ee the followig lemma. Lemma A.3. Let B 1 =J$(H)(H &H) (F ij&f ij ) a B = 1 J"(% N) (H &H) (F ij&f ij ) a cosier the otatios of Theorem 3.. The, uer the assumptios (A1)(A3), E(- B k ) =O( &1 0 ),,. Proof. First cosier B 1 =(1N* ij } ) s=1 r a=1 a t=1 u=1 m aut (* v=1 ijk )[. 1 (X ijks, X autv )&. (X autv )], where. 1 (X ijks, X autv )=J$[H(X ijks )][c(x ijks &X autv )&F au (X ijks )],. (X autv )= J$[H(x)][c(x&X autv)&f au (x)] F ij (x). The, E(- B 1 ) = N * ij } k$=1 s=1 $ s$=1 r a=1 r a$=1 a t=1 a $ t$=1 u=1 u$=1 m aut _ v=1 m a $u $t $ v$=1 * ijk * ijk$ $ E([. 1 (X ijks, X autv )&. (X autv )] _[. 1 (X ijk$s$, X a$u$t$v$ )&. (X a$u$t$v$ )]) << &J$& N * ij } s=1 s$=1 << &J$& m 0N 0 r 0 =O \ 1 0+ r a=1 a t=1 u=1 m aut m au $t v=1u$=1 v$=1 * ijk m ijk by argumets similar to those use i the proof of lemma A., (6) where the Viograov symbol ``<<'' is use istea of the O( } )-otatio. Next cosier B =B 1 +B where B 1 = 1 J"(% N)(H &H) F ij

26 RANK TESTS FOR REPEATED MEASURES 311 a B =& 1 J"(% N)(H &H) F ij. The two terms are estimate separately. * ijk E(- B 1 ) &J"& E[H (X 4* ij } m ijks )&H(X ijks )] 4 ijk s=1 N 0m 0 &J"& 4r 0 =O \ 1 Usig Jese's iequality a Lemma A. (6). I the same way it follows that 0+ which completes the proof. e(- B ) &J"& 4 E[H (x)&h(x)]4 F ij N 0m 0 &J"& 4r 0 =O \ 1 0+ K To prove the results for cotiguous alteratives, we ee the followig lemma. Lemma A.4. Uer the assumptios (A1)(A4), (1) (J[H(x)]&J[H (x)]) (1) &J$&, () E(J[H (x)]&j[h(x)]) (4m 0 r 0 ) &J$&, (3) E(J[H(X ijks )]&J[H (X ijks )]) (1) &J$&. Proof. To prove (1), we ote that H(x)&H (x) 1 - r i=1 N F ij(x)&k ij (x) 1 - a that J[H(x)]&J[H (x)]= H(x) J(s)=J$(% 1 )[H(x)&H (x)], H (x) where % 1 ; is betwee H (x) a H(x) a the result follows.

27 31 BRUNNER, MUNZEL, AND PURI Statemet () follows from J[H (x)]&j[h(x)]=j$(% )[H (x)&h(x)] where % is betwee h(x) a H (x). Thus, =J$(% )[H (x)&h (x)+h (x)&h(x)], (J[H (x)]&j[h(x)]) &J$& ([H (x)&h (x)] +[H (x)&h(x)] ). Sice E(H (x))=h (x), Lemma A.(1) is still vali if H is replace by H. Hece, it follows from Lemma A. (1) that E(H (x)&h (x)) m 0 (r 0 ). Together with statemet (1), it follows that E(J[H (x)]&j[h(x)]) 4m 0 r 0 &J$&. Statemet (3) follows aalogously usig (1). Proof of Theorem 3.4. ecompose where K Proceeig as i the proof of Theorem 3., we &^ (J)=- C J(H ) F =- C J(H) F +a 1+a &a 3, a 1 =C [J(H )&J(H)] K, a =- C [J(H )&J(H )] (F &F ), a 3 =- C [J(H)&J(H )] (F &F ). To prove the statemet i (1), it suffices to cosier the (i, j) th compoets of a j,,, 3 a it will be show that (i) " ij" [J(H )&J(H)] K =O \ 1 0+, (ii) " - [J(H )&J(H )] (F ij&f, ij ) " =O \ 1 0+, (iii) " - [J(H)&J(H )] (F ij&f, ij ) " =O \ 1 0+.

28 RANK TESTS FOR REPEATED MEASURES 313 Statemet (i) follows easily by Jese's iequality, E \ ij+ [J(H )&J(H)] K E[J(H )&J(H)] 4m 0 &J$& r 0 a usig lemma A.4 (). To prove (ii), we ote that uer the sequece of alteratives (3.6), E(F ij(x))=f, ij (x) a E(H (x))=h (x) (1.1) a thus, Lemma A.3 still hols if F ij is replace by F, ij a H is replace by H. Hece, the result follows i the same way as i the proof of theorem 3. usig Lemma A.3. To prove (iii), let.(u)=j[h(u)]&j[h (u)] a cosier E \-.(x) [F ij(x)&f, ij (x)] + = * ij } k$=1 s=1 $ s$=1 * ijk * ijk$ $ _E \_.(X ijks)&.(x) F, ij(x) &_.(X ijk$s$)&.(x) F, ij(x) &+ * ij } * ijk &J$& 1 &J$& 0 which follows by iepeet, (1.1), a Lemma A.4. This completes the proof of statemet (1). To prove statemet (), first ote that E(- CJ(H) F )=J(H) (CK)=&(J) a V*=Cov(- J(H) F )is the covariace matrix of - J(H) F uer the sequece of alteratives efie i (3.5). It is easily see that V* a V efie i (3.4) are asymptotically equivalet a the result follows from Theorem 3.3. K Proof of Theorem 3.5. First, we cosier the iagoal elemets v i ( j) a we efie a ``estimator'' v~ i ( j) with the uobservable raom varibales Y ijk } (J), v~ i ( j)= 1 * ij } &1 i * ij } * ijk (Y ijk } (J)&Y ij }}(J)). Recall that * ijk=* ijk by efiitio a ote that E[v~ i ( j)]=v i ( j). To show the cosistecy of v~ i ( j), cosier the ifferece

29 314 BRUNNER, MUNZEL, AND PURI v~ i ( j)&v i ( j) = 1 * ij } * ij } * ijk [(Y ijk } (J)&p ij (J)) &v ik (J)] + 1 * ijk [(Y ijk } (J)&Y ij }}(J)) &(Y ijk } (J)&p ij (J)) ] * ij } * ij } + 1 * ij } v~ i ( j). Thus, by iepeece, Jese's iequality, a by the assumptios (A) a (A3), E[v~ i ( j)&v i ( j)] 3 * ij } + 3 * ij } 0 =O &J&4 \N 0 +. N 0 * ijke[(y ijk } (J)&p ij (J)) &v ik (J)] N 0 * ijk(4 &J& ) E[Y ij }}(J)&p ij (J)] +O \ 1 0+ ext, the uobservable raom variables Y ijks (J)=J[H(X ijks )] are replace by the observable rak-scores, ijks =J[H (X ijks )]=J[1N(R ijks & 1 )] a we cosier the ifferece v^ i( j)&v~ i ( j). It follows that [v^ i( j)&v~ i ( j)] = \ 1 & i * ijk * ij } (* ij } &1) [, ijk } &Y ijk } (J)] i * ij } &1 [, &Y (J)] ij }} ij }} + 8 &J& i (* ij } &1) _ * ij } + [J(H )&J[H]] F ij& * ijk [J(H )&J[H]] F ijk 3N 0 (N 0+1) &J& [J(H )&J[H]] F ij

30 RANK TESTS FOR REPEATED MEASURES 315 by the c r -iequality, Jese's iequality a the assumptio (A) a (A3). Fially, by Lemma A.(), it follows that E[v^ i( j)&v~ i ( j)] 3N 0(N 0 +1) &J& &J$& } m 0 r 0 =O \ 1 0+ which completes the proof for the iagoal elemets of V i, i. For the off-iagoal elemets set i v~ i ( j, j$)= (* ij } &1)(* ij$} &1)+4 i, jj$ &1 _* ijk * ij$k (Y ijk } (J)&Y ij }}(J))(Y ij$k } (J)&Y ij$} }(J)) a ote that E[v~ i ( j, j$)]=v i ( j, j$). The, the cosistecy for the offiagoal elemets follows i a similar way as for the iagoal elemets. K Proof of Theorem 3.6. As i the proof of Theorem 3.5, we efie a ``estimator'' {~ ij for { ij efie i (3.11) where the uobservable raom variables Y ijks (J)) are use. Let Y i } k } (J)= &1 Y ijk } (J) a let S = ik1 (Y s=1 ijks(j)&y ijk } (J)) a S ik = (Y ijk } (J)&Y i } k } (J)). Thet follows by Lacaster's Theorem that E(S )=(M ik1 ik&) i a E(S ik)=(&1) ; i +(1& &1 ) i m &1 ijk where i =_ i &c i ** a ; i =c i **&c i *. Now let c ijk = 1 M ik &_ 1 & 1 t=1 1 m itk& a let {~ ij = &1 i {~ ijk where {~ ijk =c ijk S + 1 ik1 &1 S ik are iepeet, uiformly boue raom variables,,...,, sice c ijk 1, S ik1m 0 &J& a S ik &J&. Moreover, E({~ ijk )={ ijk a hece, by iepeece, E({~ ij &{ ij ) = 1 i E({~ ijk &{ ijk ) =O \ 1 +. Fially, it is show that E({^ ij&{~ ij ) 0as. By Jese's iequality a by the c r -iequality,