MAXIMIN CLUSTERS FOR NEAR-REPLICATE REGRESSION LACK OF FIT TESTS. BY FORREST R. MILLER, JAMES W. NEILL AND BRIAN W. SHERFEY Kansas State University

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1 The Annals f Statstcs 1998, Vl. 6, N. 4, MAXIMIN CLUSTERS FOR NEAR-REPLICATE REGRESSION LACK OF FIT TESTS BY FORREST R. MILLER, JAMES W. NEILL AND BRIAN W. SHERFEY Kansas State Unversty T assess the adequacy f a nnreplcated lnear regressn mdel, Chrstensen ntrduced the cncepts f rthgnal between- and wthncluster lack f ft wth crrespndng ptmal tests. Hwever, the prpertes f these tests depend n the chce f near-replcate clusters. In ths paper, a graph theretc framewrk s presented t represent canddate clusterngs. A clusterng s then selected accrdng t a prpsed maxmn pwer crtern frm amng the clusterngs cnsstent wth a specfed graph n the predctr settngs. Examples are gven t llustrate the methdlgy. 1. Intrductn. Chrstensen Ž 1989, derved unfrmly mst pwerful nvarant tests fr detectng rthgnal between- and wthn-cluster lack f ft n lnear regressn mdels. These tests can be useful fr assessng mdel adequacy fr the cmmn crcumstance n whch replcate measurements are nt avalable. Green Ž 1971., Breman and Mesel Ž 1976., Atwd and Ryan Ž 1977., Lyns and Prctr Ž 1977., Shllngtn Ž 1979., Danel and Wd Ž 1980., Utts Ž 198., Nell and Jhnsn Ž and Jglekar, Schuenemeyer and LaRcca Ž have als prpsed lack f ft tests fr the case f nnreplcatn. Hwever, Chrstensen s apprach s f partcular nterest snce the lack f ft space was characterzed as a sum f rthgnal subspaces wth crrespndng ptmal tests. All f the precedng tests requre that the data be gruped nt clusters and, dependng n the test statstc, ne f tw appraches can be used as a bass fr cluster selectn. The near-replcate apprach grups bservatns accrdng t measures f nearness n the predctr space. The ratnale behnd ths apprach parallels the reasnng whch mtvates the classcal lack f ft test wth replcatn as ntrduced by Fsher Ž 19.. Alternatvely, a secnd apprach determnes clusters s that wthn each grup the respnse s well apprxmated by the hyptheszed mdel r a plynmal mdel n the specfed predctrs. An estmate f errr varance s determned by plng the resdual sums f squares btaned frm the lcal least squares mdel fttngs wthn each cluster. Jglekar, Schuenemeyer and LaRcca Ž summarzed the varus grupng methdlges, whch can be classfed as ne f the tw prevus appraches. They nted that data-drected clusterngs, whch are generally Receved December 1995; revsed September AMS 1991 subject classfcatns. Prmary 6J05; secndary 6F03. Key wrds and phrases. Regressn, lack f ft, nnreplcatn, between clusters, wthn clusters, maxmn pwer, graph thery. 1411

2 141 F. R. MILLER, J. W. NEILL AND B. W. SHERFEY nherent t the methds based n the lcal fttngs apprach, lead t ntractable dstrbutn thery prblems. In addtn, current dstance measures used fr determnng near-replcate clusters d nt always admt an explct, much less ptmal, relatn t the pwer f the crrespndng lack f ft tests. Nnparametrc regressn technques have als been develped t test the adequacy f parametrc lnear mdels Hart Ž The chce f smthng parameter n ths cntext parallels the prblem f cluster selectn as dscussed abve. A lack f ft test fr generalzed lnear mdels whch avds the need t cluster bservatns was suggested by Su and We Ž The test s based n the supremum f a partal sum prcess determned by the resduals. Hwever, n the absence f relatvely lng sequences f resduals wth lke sgn, ths test may be expected t have dmnshed pwer. In addtn, a ratnal way f rderng the cases befre cmputng partal sums s needed. The tests derved by Chrstensen nvlve mdels based n the near-replcate apprach t clusterng. These tests are n fact unfrmly mst pwerful nvarant fr detectng rthgnal between- and wthn-cluster lack f ft, gven a specfc grupng f the data nt near replcates. Thus, prpertes f the tests depend heavly n the chce f such clusters. In ths paper we address the questn f hw t select near-replcate clusters whch prvde maxmn pwer prpertes asscated wth the ptmal tests. In Sectn, the mdels and tests fr rthgnal between- and wthncluster lack f ft are revewed. A graph theretc framewrk s presented n Sectn 3 t represent a cllectn f canddate grupngs. A maxmn pwer crtern s prpsed n Sectn 4 n rder t then select an ptmal clusterng frm amng the canddate grupngs. The methdlgy s llustrated n Sectn 5.. Orthgnal between- and wthn-cluster lack f ft tests. The nrmal thery lnear regressn mdel s gven by Ž 1. Y X, where Y s an n-dmensnal randm respnse vectr, X s a knwn, nnrandm n p matrx f predctr varables, s an unknwn parameter n R P Ž and s an n-dmensnal randm errr vectr dstrbuted as N 0, I n. wth unknwn 0. Lack f ft s sad t exst when the lnear structure X n mdel Ž 1. des nt adequately descrbe the mean f Y, that s, EY X. In such case we may suppse that a true mdel whch accunts fr the nadequacy f mdel Ž 1. s f the frm Ž. Y X Q, Ž where C X CQ and N 0, I.. The ntatn CŽ A. n dentes the clumn space f a matrx A. In practce, a true mdel, and thus Q, are f curse nt knwn. Als, n the usual paradgm fr testng lack f ft, there are n addtnal predctr varables avalable fr nclusn n mdel Ž 1.. Hence, a lack f ft vectr Q n

3 LACK OF FIT TESTS 1413 mdel Ž. s knwn nly t be cntaned n the lack f ft space CŽ X.. A decmpstn f CŽ X. nt rthgnal subspaces was btaned by Chrstensen Ž 1989, The characterzatn prvdes the bass fr the cnstructn f ptmal tests fr assessng the exstence f varus types f lack f ft. In partcular, the lack f ft space can be wrtten as CŽ X. CŽ X. CŽ Z. CŽ X. CŽ Z. S, where S dentes the rthgnal cmplement f the sum f the frst tw subspaces wth respect t CŽ X.. The n c matrx Z cntans ndcatr varables fr the near-replcate clusters, and thus cntans nly zeres and nes. The clumn dmensn represents the number f grups fr a specfed near-replcate clusterng f the bservatns. Thus, the nnzer values n the th clumn f Z crrespnd t the bservatns n the th cluster f near replcates, 1,,..., c. A clusterng determned by such a Z wll als be called a grupng r parttn f the bservatns. In addtn, the frst tw subspaces n the precedng decmpstn f CŽ X. are called the rthgnal between- and wthn-cluster lack f ft subspaces, respectvely. Ths termnlgy crrespnds t ne-way analyss f varance n whch clusters f replcates are dentfed wth dfferent treatment grups. In the specal case that replcatn exsts n the rw structure f X, that s, CŽ X. CZ, the tests fr between and wthn cluster lack f ft cmpare mdel Ž 1. wth mdel Ž., wth Q replaced by lack f ft vectrs n CZ and CZ, respectvely. Thus, the test fr between-cluster lack f ft reduces t the classcal lack f ft test n the case f replcatn. Als nte that the test fr wthn-cluster lack f ft wuld, fr example, allw fr detectn f a trend n tme wthn each grup f replcates whenever the replcates are bserved n a tme sequence. The nterpretatn fr the case f near replcatn generalzes the precedng cncepts. Fr cmputatnal purpses, prjectns nt the lack f ft subspaces can be determned by the fllwng lemma. The prf f the lemma s straghtfrward and thus mtted. The ntatn PA dentes the rthgnal prjectn peratr nt a subspace A f R n. LEMMA 1. Let U and V be subspaces f R n and suppse W s the subspace n f R gven by W P ŽV.. Then P P P. U U V U W By lettng U CZ and V C X n the precedng lemma, where X Z P versus 0 CŽ Z. P P P Z, CŽ X. CŽ Z. CŽ Z. CŽ X. 0 X. Thus, the lkelhd rat test statstc fr testng H B : EŽ Y. CŽ X. B H a : EŽ Y. CŽ X. CŽ X. CŽ Z. and EŽ Y. CŽ X.

4 1414 F. R. MILLER, J. W. NEILL AND B. W. SHERFEY s gven by B P P Z CŽZ. CŽ X. Y r 0 1 B B Ž P P P Z CŽ X. Ž CŽ Z. CŽ X... Y r 0 F B Ž Z. B B where r1 c dmensn C X0 and r n p r1 wth c dmensn CZ and p dmensn CŽ X.. Als, fr x R n, x dentes the squared Eucldean length x T x f x. Snce F F B B where B r 1, r B P B CŽ X. CŽ Z. EŽ Y., a unfrmly mst pwerful nvarant sze test f H B versus Ha B rejects H B prvded F F B B r, r B. 1 The ntatn Fd, d represents the upper pnt f a central F dstrbutn 1 wth d1 and d degrees f freedm. By symmetry, the lkelhd rat test statstc FW and ts dstrbutn F W WŽ. fr testng r 1,r W versus H W : EŽ Y. CŽ X. W H a : EŽ Y. CŽ X. CŽ X. CŽ Z. and EŽ Y. CŽ X. can be btaned by replacng CZ wth CZ n F and F B BŽ. B r, r B, respec- 1 tvely. The fllwng sectns specfcally address the prblem f chsng a grupng matrx Z t effectvely test H B aganst Ha B. Althugh symmetry cnsderatns can be used n part t derve ptmal clusters fr testng H W aganst Ha W, a separate paper wll be wrtten t dscuss ths prblem mre fully. 3. Graph theretc representatn f canddate grupngs. Fr the maxmn pwer crtern, as defned n the fllwng sectn, t be cmputatnally feasble ne must generally restrct the number f ptental grupngs under cnsderatn. Fr example, wth n 16, the number f all pssble grupngs s 10,480,14,147, as calculated by the recurrence relatns satsfed by Strlng numbers f the secnd knd Cnstantne Ž The fllwng graph theretc framewrk may be vewed n part as a devce t elmnate absurd grupngs drectly and thus reduce the number f parttns under cnsderatn. In addtn, the framewrk based n graph thery prvdes a useful representatn f the cllectn f canddate grupngs t whch the maxmn crtern s appled. Suppse the rws f the n p matrx f predctr varables X are dented by r R P, 1,..., n. A graph G may be defned wth n vertces v,..., v 1 n dentfed wth r,..., r, respectvely. Furthermre, an edge s sad t exst 1 n between tw predctr settngs r and rj f and nly f the crrespndng

5 LACK OF FIT TESTS 1415 bservatns are canddates fr near-replcate status. Several appraches may be cnsdered fr determnng the edge set f G. Fr example, a nearness parameter 0 can be specfed alng wth a dstance functn d n R P R P. If dr, Ž r. then the th and jth bservatns are canddates fr near j replcate status, and thus an edge exsts between vertces v and v j. Mre generally, dfferent nearness parameters k, k 1,..., p, can be used fr predctrs wth dfferent scales. In such a case an edge exsts between v and Ž k k. k v prvded d r, r, k 1,..., p, where r dentes the kth cmpj k j k nent f r, and dk s a specfed dstance functn apprprate fr the scalng f the kth predctr varable. A dstance functn d n R P R P can als be generated as d wkdk wth the dk as descrbed abve and fr selected weghts w k, k 1,..., p. Alternatvely, a famly f verlappng subsets 4 P Ž P1 S,..., S may be specfed n R R n case X has a clumn f nes r 1 m ts equvalent.. An edge exsts between vertces v and vj f r and rj le n Sk fr sme k 1,..., m. Fr example, the subsets may be verlappng grds n R P whch are chsen n a manner that excludes extreme parngs f the predctr settngs frm beng clustered tgether. Each f the precedng appraches fr determnng the edge set s llustrated belw wth a smple example n R. In addtn, the verlappng grd apprach s used n the mre cmplex settng f Example n Sectn 5. In ether case, a graph s determned wth a crrespndng cllectn f grupngs. Specfcally, the cllectn f grupngs asscated wth a graph are naturally defned as n Defntn 1 belw. In ths defntn, let G be a graph wth vertex set V and recall that a subgraph f G nduced by V V s the graph btaned by deletng all vertces nt n V frm the graph n V, tgether wth all edges that d nt jn tw vertces f V. In addtn, a subgraph s cmplete f there s an edge between every tw f ts vertces. Nte that sngletns are cnsdered t be cmplete subgraphs. DEFINITION 1. The cllectn f grupngs cnsstent wth a graph G cnssts f all parttns P A,..., A 4 f V v,..., v 4 1 k 1 n such that each A determnes a cmplete subgraph f G. T llustrate the cncepts gven n the precedng paragraph and defntn, cnsder a smple example wth T X and suppse the graph n the fve vertces Ž rws f X. has edge set determned by Eucldean dstance n R and a nearness parameter 1, as descrbed abve. The resultng graph s gven by

6 1416 F. R. MILLER, J. W. NEILL AND B. W. SHERFEY TABLE 1 Cnsstent grupngs ( ) Dmensn C Z Vertex Nte that the same graph s btaned by specfyng verlappng subsets n 1 R gven by the ntervals S Ž 1,., S Ž 1.5, 3. and S Ž.5, Fr example, an edge exsts between vertces v and v3 snce 1.8 and.6 le n S. Hwever, there s n edge between v and v snce 1.3 and.6 d nt le n a 1 3 cmmn S fr sme 1,, 3. Fr n 5 there are 5 pssble grupngs, 15 f whch are cnsstent wth ths graph. The cnsstent grupngs are lsted n Table 1 and classfed accrdng t the dmensn f CZ. The ntatn used n the table represents a partcular grupng matrx Z by a crrespndng n-tuple z Ž z. 1,..., zn T where z s the cluster number fr the th rw f X. It shuld be emphaszed that specfcatn f a graph nly ndcates whch pars f bservatns are ptental near replcates. The specfcatn f a nearness parameter, fr example, des nt determne a unque grupng but rather a cllectn f grupngs cnsstent wth the asscated graph. The maxmn pwer crtern s then appled t ths cllectn. One s stll generally chsng a maxmn clusterng frm a large number f canddate grupngs represented by the cllectn f grupngs cnsstent wth the specfed graph. Of curse the cmplete graph may be selected, n whch case the cllectn f cnsstent grupngs cnssts f all pssble grupngs. Hwever, fr mst practcal prblems, the number f all pssble grupngs s enrmus, as ndcated abve. 4. Maxmn pwer crtern fr clusterng near replcates Defntn f the crtern. The tests dscussed n Sectn are ptmal fr rthgnal between- and wthn- cluster lack f ft, whch depend n the chce f the grupng matrx Z. Fr the purpse f clusterng n bservatnal experments, the matrx X may be regarded as fxed whle the matrx Z remans t be chsen. T dentfy clusterngs that allw the crrespndng ptmal lack f ft test t dscrmnate effectvely between B mdel 1 and a type f lack f ft represented by mdel H a, a maxmn strategy s prpsed. In partcular, snce the pwer f the F test fr lack f ft s an ncreasng functn f the nncentralty parameter, a maxmn crtern wll be appled t B t prvde a cluster selectn asscated wth the ptmal test. It wll be shwn belw that the degrees f freedm parameters f the F

7 LACK OF FIT TESTS 1417 dstrbutn fr the ptmal test, crrespndng t a maxmn clusterng, are nherently n cncrdance wth the bjectve f maxmal pwer. Jnes and Mtchell Ž 1978., Atknsn and Fedrv Ž and Atknsn Ž 197. dscuss desgn crtera whch als use a maxmn pwer apprach fr detectng lack f ft. Hwever, these crtera are cncerned wth the selectn f desgn pnts whch determne the rws f X. In addtn, the alternatve mdels upn whch these crtera are based are f the frm f mdel Ž. where Q s a matrx f knwn functns f the settngs f the predctr varables n X. Fr ths frm f mdel Ž., Sheltn, Khur and Crnell Ž als suggested a maxmn pwer crtern t select check pnts fr use n assessng lack f ft. Mdel dscrmnatn desgns fr plynmal regressn were dscussed by Dette Ž 1994, Specfcally, ptmal desgns whch maxmze the mnmum pwer f a gven set f alternatves were dscussed, n addtn t generalzed c-ptmal desgns fr plynmal regressn. In rder t present the maxmn crtern fr cluster selectn, a set f canddate grupngs s requred. Assume a graph G has been specfed alng wth the cllectn f grupngs cnsstent wth the graph as defned n Sectn 3. Ths cllectn must be restrcted accrdng t the dmensn cnsderatns gven n the fllwng defntn. DEFINITION. Let G dente the set f grupng matrces Z whch gve parttns cnsstent wth G and satsfy CŽ X. CŽ Z. 0 4, excludng the trval parttns whch cluster all r nne f the bservatns tgether. Fr example, returnng t the X matrx and asscated graph ntrduced n Sectn 3, G cnssts f thse parttns cnsstent wth the graph and havng dmensn CZ equal t three r fur. Ths bservatn fllws frm Z the fact that dmensn C X C Z c rank X0 where c dmensn Z CZ, and ntng that rank X0 fr all the matrces Z ndcated n Table 1. Thus, excludng the trval parttn crrespndng t c 5, nly clusterngs wth c 3, 4 prvde rthgnal between-cluster lack f ft subspaces wth pstve dmensn. Nte that the trval parttn whch clusters all f the bservatns tgether, that s, the case c 1, s autmatcally excluded fr ths example snce X has a clumn f nes and thus CŽ X. CŽ Z Suppse the alternatve mdel s assumed t be f the frm gven by Ha B fr sme Z G, and ne wshes t test the adequacy f the H B mdel. A maxmn crtern wll next be dscussed n the present cntext t determne an apprprate Z G, and hence a clusterng. T explan the basc cncept underlyng the maxmn crtern fr cluster selectn, let dente a quadratc frm defned n the lack f ft space CŽ X.. As seen n the fllwng dscussn, must be pstve defnte n each f the lack f ft subspaces CŽ X. CŽ Z. fr Z G. A specfc class f such frms whch allw fr nearness cnsderatns s gven n Sectn 4.. Fr fxed Z G, determne the vectr n the crrespndng rthgnal between-cluster lack f ft space

8 1418 F. R. MILLER, J. W. NEILL AND B. W. SHERFEY whch mnmzes the nncentralty parameter B, subject t the restrctn that the vectr les n a set f the frm 4 fr sme 0. We want t determne a Z G whch maxmzes such mnmum nncentralty parameter values. Specfcally, a Z s sught whch maxmzes G 4 nf v : v C X C Z, v Z fr sme fxed 0. Next, lettng 5 v Ž 3. lz nf : v CŽ X. CŽ Z., v 0 ½ Ž v. and ntng that sv Ž sv. v Ž v. fr nnzer v CŽ X. CŽ Z. and any nnzer scalar s, t fllws that v 1 lz nf : v CŽ X. CŽ Z., Ž v. Z. Ž v. ½ 5 Cnsequently, a clusterng matrx Z G whch maxmzes Z des nt depend n. Accrdngly, a maxmn clusterng matrx s frmally defned as fllws. DEFINITION 3. A clusterng matrx Z G whch maxmzes lz as gven by Ž 3. s defned t be a maxmn clusterng matrx frm amng the canddate grupngs n G. Nte that a maxmn clusterng s nvarant under reparametrzatns and des nt depend n the data vectr Y. Als, mplct n the abve dscussn s the fact that the quadratc frm allws cmparsns f vectrs n the lack f ft subspaces CŽ X. CŽ Z. fr dfferent Z G. In partcular, suppse CŽ X. CŽ Z. and CŽ X. CŽ Z. where Z, Z and Ž. 0 0 G. Furthermre, f Z s a maxmn clusterng matrx and 0 Z 0 and, then Z. Thus, fr lack f ft vectrs pssessng the precedng prpertes, the crrespndng nncentralty parameter values based n a maxmn clusterng are maxmal as cmpared t crrespndng values based n nnmaxmn clusterngs. In fact, by the defntn f Z t fllws that v fr any v CŽ X. CŽ Z. wth Ž v. and 0 Z. A calculatnal frm fr Ž equvalently l. s derved n the Appendx. Z 4.. Cntur quadratc frms based n pwer and nearness. A specfc class f quadratc frms fr use n Z wll next be defned. Frst nte that permutng the clumns f a specfed grupng matrx Z results n dfferent Z matrces, but each such Z crrespnds t the same clusterng. Such Z matrces wll be cnsdered as beng equvalent. Thus, when a set f grupng matrces s specfed, a set f equvalence classes s nherently specfed. In partcular, the ntatn Ý 4 Z, where s a cllectn f grupng matrces, wll ndcate that the sum s ver all dstnct grupngs determned by. That s, each equvalence class f cntrbutes just ne term t the sum. Z

9 LACK OF FIT TESTS 1419 Nw nte that the pssble alternatve mdels are f the frm Y X P CŽ X. CŽ Z. u, where Z G and u C X. T mtvate a chce fr, let u C X be fxed. If P u fr all Z where 0 s a small preas- CŽ X. CŽ Z. G sgned number, then ne may suppse that u wll nt cntrbute t any detectable between-cluster lack f ft. Next let E G cnsst f thse grupngs determned by a sngle edge n the edge set E f the specfed graph G. That s, each edge f G determnes a clusterng n G, whch clusters nly the tw cnnected vertces, wth all ther vertces beng sngletn clusters. Fr example, nte that E fr the example ntrduced n Sectn 3 cnssts f thse parttns wth dmensn CZ equal t fur. Nw bserve that P u fr all Z f and nly f P CŽ X. CŽ Z. G CŽ X. CŽ Z. u fr all Z E. Nte that the f part f the precedng clam fllws snce fr any Z there exsts a Z such that CZ CZ, and thus G 1 E 1 P u P CŽ X. CŽ Z. CŽ X. CŽ Z. u. The pnt f ntrducng E and the 1 precedng equvalence s that the cardnalty f E s generally much smaller than that f G, frm whch cmputatnal advantages are btaned. Thus, based n the precedng, cnsder defned by Ž 4. Ž u. Ý w P Z CŽ X. CŽ Z. u Z E fr u CŽ X. wth w 0 and Ý w 1. Snce Ž u. Z Z Z s a cnvex cmb- E natn, P u fr all Z mples that Ž u. CŽ X. CŽ Z. E. Hence, Ž u. mples that P CŽ X. CŽ Z. u fr at least ne Z E, and u may cntrbute t detectable between-cluster lack f ft fr sme Z E. T ncrprate nearness as measured by X X Z 0 nt the weght w Z, let Z Z w X X X X. Ý Z 0 0 Z* E The ntatn A dentes the squared matrx nrm defned by Ý n 1Ýj1 p a j where a s the Ž j. j th element f an n p matrx A. The mtvatn fr the abve measure f nearness s derved frm the fact that the jth rw f the matrx X0 Z PCŽ Z. X s btaned by averagng ver the rws f X crrespndng t the cluster, as determned by Z, whch cntans the jth bservatn, j 1,..., n. Thus, as we llustrate belw, the effect f usng the precedng weghts wz n s t penalze thse grupng matrces Z whch cluster bservatns crrespndng t rws f X whch are far apart. Cnsder the case wth T 1, 1,...,1 X x, x,..., x, 1 n and nte that reduces t Ž u. Ý w P Z CŽ X. CŽ Z. u, x, x je j j j

10 140 where Z and j F. R. MILLER, J. W. NEILL AND B. W. SHERFEY s the grupng matrx that clusters the th and jth vertces nly Ý w Ž x x. Ž x x.. Zj j k m x k, x me km In the fllwng dscussn, suppse there are nly tw values x and x j n the secnd clumn f X whch are relatvely far part. If there are mre than tw such values, then an analgus argument hlds n rder t btan the fllwng cnclusns. Nw nte that f Z G grups x and x j where Ž x. x j s relatvely large, then wz 1. As a result, we clam that j l 1 fr such Z. T see ths, let u C X CŽ Z. Z and wthut lss f generalty suppse u 1. Then, snce CŽ X. CŽ Z. CŽ X. CŽ Z. j, P u u 1 and thus w Ž u. 1. Cnsequently, Ž u. CŽ X. CŽ Z j. Z j 1 fr u CŽ X. CŽ Z. wth u 1, and hence lz 1. Ths result ndcates that Z matrces whch cluster bservatns crrespndng t rws f X whch are far apart are nt favred accrdng t the maxmn clusterng crtern. The precedng cnclusn fllws, snce lz 1 fr every Z G when has the general frm gven by Ž 4.. Of curse may be defned wth alternatve chces fr the weghts w Z. Fr example, a unfrm weght may be taken wth w Ž. Z cardnalty E 1. Fr the maxmn apprach t clusterng t be peratve, the quadratc frm must be pstve defnte n each subspace CŽ X. CŽ Z. fr Z G. The fllwng prpstn prvdes a cndtn t ensure that any f the frm gven by Ž 4. s n fact pstve defnte n CŽ X.. The prf f the prpstn s gven n the Appendx. PROPOSITION 1. s pstve defnte n CŽ X. prvded Ý Z E 5 C X C Z C X, where cnssts f thse Z such that w 0. E E E Z 4.3. Refnements and atms. As wll be shwn, the maxmn apprach t clusterng necessarly results n a cluster selectn frm amng thse parttns whch grup as many bservatns tgether as pssble. T facltate the fllwng dscussn, the cncepts f refnement and atms are ntrduced n the next defntn. These cncepts wll be llustrated later n ths subsectn n the cntext f the example ntrduced n Sectn 3. DEFINITION 4. Fr Z 0, Z1 G, Z1 s sad t be a refnement f Z0 prvded CZ CZ 0 1. In addtn, Z0 s an atm f G prvded Z0 s nt a refnement f any ther member f G. Let 0 dente the set f atms n G.

11 LACK OF FIT TESTS 141 Nte that f Z1 s a refnement f Z0 n G then v : v C Ž X. C Ž Z. 0, v 0 ½ Ž v. 5 ½ 5 v : v CŽ X. CŽ Z 1., v 0. Ž v. Hence, lz l Z, s that maxmzatn f lz can be made wth respect t the 0 1 atms Z. Snce atms represent thse parttns cnsstent wth the graph whch grup as many bservatns tgether as pssble, the clam made at the begnnng f ths subsectn has been verfed. It shuld als be nted that ths dscussn cncernng refnements hlds fr any quadratc frm whch s pstve defnte n each subspace CŽ X. CŽ Z. fr Z G. Based n the precedng, n rder t determne a maxmn clusterng fr a gven X, the set f atms n G must be determned. The fllwng therem characterzes the atms asscated wth a graph. The prf f Therem 1 s gven n the Appendx. In Therem 1 and n subsequent dscussns, a clque f a graph G s defned as a maxmal cmplete subgraph f G. In addtn, the defntn f a mnmal cver fr a set s recalled fr use n determnng the atms f a graph. DEFINITION 5. A cver f a set S s a cllectn f sets A 4 whse unn s S. A cver A 4 f S s mnmal prvded A A fr each. Nte that ths s equvalent t the case that fr every there s an element f S whch s n A but nt A fr. THEOREM 1. Let G be a graph wth vertex set V dentfed wth the rws f X as ndcated at the begnnng f Sectn 3. Let C dente the set f clques f G. Als assume that dmensn CŽ X. p k where k s the cardnalty f C. Ž. If Z determnes an atm f and c dmensn CŽ Z. G G, then c k. Ž. If G has the prperty that C s a mnmal cver fr V, then the atms f cnsst f exactly thse Z wth dmensn CŽ Z. k. G G T llustrate the use f Therem 1 fr determnng atms, recall the example ntrduced n Sectn 3 and frst nte by nspectn that the set f clques fr the graph f ths example s gven by C 1, 4,, 3, 44, 3, 4, 544. In addtn, nte that C s nt a mnmal cver fr the vertces f the graph snce the set, 3, 44 des nt cntan an element f V whch s nt cntaned n 1, 4 r 3, 4, 54. Thus, Therem 1Ž. s nt applcable. Hwever, Therem 1Ž. ensures that the atms f have dmensn CZ G 3, where 3 s the cardnalty f C. Snce as shwn prevusly there are n Z G wth dmensn CZ 3, the set f atms 0 cnssts f thse Z G wth dmensn CZ 3. As argued abve, a maxmn clusterng wuld be chsen

12 14 F. R. MILLER, J. W. NEILL AND B. W. SHERFEY frm amng thse parttns n 0. Example n Sectn 5 wll be used t llustrate the case n whch the set f clques s a mnmal cver fr the vertces f the graph, and thus allws applcatn f Therem 1Ž. t determne the set f atms. Next suppse that a specfed graph G breaks up nt k p Žk and p as defned n Therem 1. dsjnt cmplete subgraphs G 1,..., Gk wth n edges between the vertces f G and Gj fr, j 1,..., k and j. Let Z0 be a clusterng matrx whch grups nly the vertces f G tgether fr 1,..., k. Snce dmensn CŽ X Z. 0 p fr every clusterng matrx Z, dmensn C X CŽ Z. dmensn CZ dmensn CŽX Z k p. Then, snce k p, Z0 s n G and every clusterng n G s a refnement f Z 0. Thus, by Defntn 4, Z0 s the nly atm n G. Hence, when there s a clear grupng, as n ths case, the maxmn pwer crtern chses t. Hwever, the utlty f the crtern s mre fully realzed when there s nt ne clear grupng. In such cases, we want t allw suffcently many ptental nearreplcate pars s that the graph s cnnected, as n Examples 1 and f Sectn 5. Recall that a graph G s cnnected f fr every tw dstnct vertces v, vj V there s a cntnuus path cnsstng f a fnte sequence f dstnct edges jnng v t v. Fr the case f dscnnected graphs, t s pssble, j dependng n, that the maxmn clusterng crtern prvdes n dscrmnatn between the canddate grupngs. Ths pnt s further dscussed n the Appendx. Hwever, t shuld be emphaszed that cnnected graphs prvde the apprprate clusterng pssbltes n case there s nt ne clear grupng, and thus cnnected graphs are f mst nterest Pwer cnsderatns and mplementatn. The pwer functn f the ptmal sze test fr testng rthgnal between-cluster lack f ft s gven by B B Ž B B 1 B B r, r B. 1 q r, r,, P F F fr a specfed Z G. Nte that q s an ncreasng functn f r B fr fxed values f r1 B and B and a decreasng functn f r1 B fr fxed values f r B and Ghsh Ž B. Nw f Z G and Z0 s an atm n G wth Z a refnement f Z 0, then lz lz as shwn n Sectn 4.3. Cnsequently, maxmzatn f lz can be restrcted t the atms f G. Furthermre, snce CZ B s a subspace f CZ, t fllws that r1 Z dmensn C X CŽ Z. dmensn CŽ X. CŽ Z. r B Ž Z. and r B Ž Z. n p r B Ž Z. 1 1 n p r B Ž Z. r B Ž Z. 1. Thus, the degrees f freedm parameters fr the dstrbutn f F B, based n clusterngs crrespndng t the atms, are nherently n cncrdance wth the bjectve f maxmal pwer. If the degrees f freedm parameters r B Ž Z. and r B Ž Z. 1 are cnstant n the atms Z, then the atms can be cmpared fr pwer n the bass f the lz values alne fr Z. An atm whch maxmzes lz wth respect t Z thus crrespnds t a maxmn clusterng as defned n Sectn 4.1.

13 LACK OF FIT TESTS 143 Fr mplementatn purpses, the cnstancy f r B Ž Z. and r B Ž Z. 1 s determned by drectly evaluatng dmensn CŽ X. CŽ Z. fr each atm Z. Furthermre, snce dmensn CŽ X. CŽ Z. dmensn CZ dmensn CŽ X Z., there are tw ways ths cnstancy may nt btan. In partcular, f dmensn CZ s nt cnstant n the atms Z then cnstancy f the degrees f freedm parameters may nt hld. On the ther hand, f dmensn CZ s cnstant but dmensn CŽX Z. s nt cnstant n the atms then cnstancy f the degrees f freedm may nt fllw. Hwever, accrdng t Therem 1, f the number f clques fr the specfed graph G s greater than dmensn CŽ X. and the clques frm a mnmal cver fr the vertces f G, then dmensn CZ s equal t the cardnalty f the set f clques fr every atm Z. In addtn, we clam that fr mst predctr matrces X, f dmensn CZ s cnstant n the atms then dmensn C X CŽ Z. s cnstant n the atms as well. Ths clam, whch s justfed by Therem belw, leads t the cncept f a generc predctr matrx as defned next. The prf f Therem s gven n the Appendx. DEFINITION 6. Let X be an n p matrx f predctr varables wth dmensn CŽ X. p and let c dmensn CZ fr Z a grupng matrx. If p, f p c, Z dmensn CŽ X. ½ c, f p c, fr all grupng matrces Z then X s generc. THEOREM. Cnsder the n p matrx X Ž x,..., x. 1 p, where x1 s a clumn f nes r ts equvalent, t be determned by an element f R nž p1.. Wth ths dentfcatn, the set f generc X matrces s pen and dense n R nž p1.. If X des nt prvde fr an ntercept n the mdel, then the precedng hlds wth R np n place f R nž p1.. Therem justfes and makes mre precse the clam made abve. In partcular, except fr nngenerc n pxmatrces whch cnsttute a set f np Ž nž p1. Lebesgue measure zer n R R n case f an ntercept., f dmensn CZ s cnstant n the atms then dmensn CŽ X. CŽ Z. s als cnstant n the atms. Hwever, t shuld be emphaszed that fr mplementatn f the maxmn clusterng crtern, we d nt need t check whether X s generc. Rather, as ndcated prevusly, we smply check whether dmensn CŽ X. CŽ Z. s cnstant n the atms. The sgnfcance f Therem s based n the assurance that the cnstancy f dmensn CŽ X. CŽ Z. n the atms wll n fact rdnarly be realzed whenever dmensn CZ s cnstant n the atms. The latter wll hld, fr example, when the clques frm a mnmal cver fr the vertces f the specfed graph. In ths case, dmensn CŽ X. CŽ Z. k p fr all Z whenever X s generc wth p k and k dentes the number f clques f G. Fr cmpleteness we remark that n case the specfed graph prduces atms such that dmensn CŽ X. CŽ Z,.

14 144 F. R. MILLER, J. W. NEILL AND B. W. SHERFEY Z, s nt cnstant then ne can prceed as fllws. Frst grup the atms nt classes accrdng t dmensn CŽ X. CŽ Z.. Next chse an atm frm each class wth maxmum lz value, and then select frm amng such atms the clusterng whch prvdes a dmnant pwer curve Cnsstency f the test based n maxmn clusterng. In addtn t atm determnatn, the clques f a graph can als be related t the cnsstency f the test based n F B. Suppse that as the ttal number f bservatns n, the number f clusters c dmensn CZ n n such that cnn, 0 1. In addtn, assume the true mdel s f the frm gven by mdel Ž. wth dmensn CŽ X, Q. fxed and dmensn CŽ X. n n n p fr all n. Under the precedng asympttc scheme and assumng ndependent zer mean randm errrs wth cmmn secnd, thrd and furth mments, Chrstensen Ž prved cnsstency f the test based n FW when the alternatve nly hlds n the lmt. By symmetry, analgus cndtns fr cnsstency f the test based n FB can be gven. In partcular, usng the same asympttc scheme and methd f prf, the fllwng prpstn hlds. PROPOSITION. If Q c 0 and P Q Ž n c. n n CŽ X n.cž Z n. n n 0 then F 1 as n. B p Nte that the cndtns f Prpstn ensure that the true mdel cntans lack f ft Qn, whch remans substantal as n and s asymp- ttcally between clusters. Next suppse that Gn and Vn dente a graph and asscated vertex set crrespndng t the generc predctr matrx X. Als n let C represent the set f clques fr G, and suppse C frms a mnmal n n n cver fr V. Assume the predctr space s unbunded n R P and let n cn cardnalty Cn as n wth cnn, 0 1. Fnally, sup- pse the lack f ft vectr Q CŽ X. CŽ Z. fr all n and Q c n n n n n 0asn. Then, by Prpstn, the lack f ft test based n FB s cnsstent. Fr example, suppse that Q nb n B fr sme pstve cnstants b and B fr all n. Then Q n cn b 0asns that the lack f ft remans substantal fr cnsstency purpses. 5. Examples. EXAMPLE 1. Sme fundamental cncepts f the maxmn clusterng crtern have been llustrated n Sectns 3 and 4 based n the smple matrx X and asscated graph gven n Sectn 3. In partcular, the set f atms G was fund t be the cllectn f grupng matrces cnsstent wth the graph wth dmensn CZ 3, as lsted n Table 1. Accrdngly, a maxmn clusterng s selected frm ths cllectn and the atms can be cmpared fr pwer by usng the maxmn nncentralty parameter alne. Utlzng the calculatnal frm f lz gven n the Appendx, max Z lz crrespndng t the maxmn grupng matrx Z represented by

15 LACK OF FIT TESTS 145 z Ž 1, 1,,, 3. T. As n Table 1, lke crdnates ndcate that the crrespndng bservatns are t be gruped tgether. EXAMPLE. The data fr ths example are taken frm Draper and Smth Ž 1981., page 15, whch reprts percentage f prperly sealed bars f sap Ž y., sealer plate clearance Ž x. and sealer plate temperature Ž x. wth 1 x T 1 Ž130, 174, 134, 191, 165, 194, 143, 186, 139, 188, 175, 156, 190, 178, 13, 148., x T Ž190, 176, 05, 10, 30, 19, 0, 35, 40, 30, 00, 18, 0, 10, 08, 5.. Let X 116, x 1, x. Fgure 1 gves a plt f the predctr settngs n R wth the th rw f X crrespndng t n the fgure, 1,...,16. T llustrate the verlappng grd apprach fr determnng a graph, let I 1 130, 171 and I 153, 194 be tw verlappng ntervals whse unn ncludes the range f values n x. Smlarly, let J 176, and J 199, 40 cver the values n x. The crrespndng verlappng grd elements n R are then gven by Sj I Jj fr, j 1,. That s, usng the cell desgnatns gven n Fgure 1, S11 A1 A A4 A 5, S1 A A3 A5 A 6, S1 A4 A5 A7 A8 and S A5 A6 A8 A 9. The S fr, j 1, determne a graph G wth vertces V crrespndng t the j FIG. 1. Overlappng grd elements S A fr, j 1,, where I 1,, 4, 54 j I k 11, I1 j 4 4 4, 3, 5, 6, I1 4, 5, 7, 8 and I 5, 6, 8, 9, f the predctr space n R and maxmn grupng fr Example.

16 146 F. R. MILLER, J. W. NEILL AND B. W. SHERFEY 16 rws f X and edges as dscussed n Sectn 3. Fr example, an edge exsts between vertces v1 and v3 snce v1 and v3 le n S 11. Hwever, there s n edge between v and v snce v and v d nt le n a cmmn S fr sme j, j 1,. In addtn, the clques f G are the sets S V fr, j 1,. A j methd s gven n the Appendx t shw ths s the case. The methd s als applcable t Example 1, althugh the clques n the smple settng f that example can be determned by nspectn. Thus, the set f clques fr the graph G s gven by C 1,3,15 4,,4,6,11,14 4, 3,5,7,9,1,15,16 4, 4, 5, 8, 10, 11, 1, 13, Furthermre, C s a mnmal cver fr V snce v1 s n the frst set but n ther, v s n the secnd set but n ther, v9 s n the thrd set but n ther and v s n the furth set but n ther. Thus, by Therem 1Ž. 8, the set f atms G cnssts f thse grupngs cnsstent wth G and havng dmensn CZ 4, where 4 s the cardnalty f C. Specfcally, the atms fr ths example are the grupngs that have bservatn 1 cnfned t cluster 1, bservatns and 6 cnfned t cluster, bservatns 7, 9 and 16 cnfned t cluster 3 and bservatns 8, 10 and 13 cnfned t cluster 4. Nte that the cluster desgnatns are arbtrary. Als nte that bservatn 1 s cnfned t cluster 1 snce ths bservatn les n a prtn f S Ž 11 n partcular, the A cell. 1 whch des nt verlap wth any f the remanng grd elements S 1, S1 r S. Thus, there can be n edge between v1 and any vertex whch les n the A 3, A7 r A9 cells by cnstructn f the graph accrdng t the verlappng grd apprach. Analgus reasnng can be used t justfy the clams made fr the ther three clusters. The remanng bservatns le n ne f the fur clusters descrbed abve. As determned by cmputer enumeratn, there are 18 atms fr ths example whle the cardnalty f the set f all pssble grupngs s 10,480,14,147 as ndcated n Sectn 3. A Frtran prgram has been wrtten and mplemented fr general use t calculate max Z l Z, whch fr ths example s crrespndng the maxmn grupng matrx Z represented by z Ž 1,, 1, 4, 3,, 3, 4, 3, 4,, 3, 4, 4, 1, 3. T Ž ntatn as n Table 1. and shwn n Fgure 1. APPENDIX A calculatnal frm fr lz s derved n Part A f the Appendx. Prfs f Prpstn 1 and Therems 1 and are gven n Parts B, C and D, respectvely. Dscnnected graphs are dscussed n Part E and a methd fr determnng the clques f a graph derved by the verlappng grd apprach s develped and appled t Example n Part F. Part A. T calculate l r equvalently fr Z, let e,..., e 4 Z Z G 1 d be an rthnrmal bass fr CŽ X. CŽ Z. where d dentes the dmensn f Ž 1 d. d C X C Z. Thus, lettng v,..., v be the d-tuple n R representng

17 v C X C Z wth respect t ths bass, and where LACK OF FIT TESTS 147 d v Ý Ž v. 1 d d j Ž v. Ý Ý vvb j, 1 j1 bj Ž e, ej. fr, j 1,..., d. Next let and, 1,..., d, represent the egenvalues and crrespndng rthnrmal egenvectrs, respectvely, f the matrx B Ž b. j. Hence, let- tng x y dente the usual Eucldean nner prduct n R n, and where Thus, Z d Ý 1 v d Ž v. Ý Ž., 1 v fr 1,..., d. may be cmputed by cnsderng the cnstraned extrema prblem: d Ý 1 mnmze fž x. x subject t gž x. d d fr x x 1,..., x d R, x 0, where g x Ý1x j and s a pstve cnstant. Crtcal pnts f the functn FŽ x. fž x. Ž gž x.. are determned by f g and gven by x, x 0 fr j, 1,..., d, ( j where dentes the Lagrange multpler. Thus, the mnmum value f f subject t g s max where max s the largest egenvalue f the matrx B. Hence, Z max and lz 1 max. Part B. The prf f Prpstn 1 fllws Lemma. The prf f Lemma s straghtfrward and thus mtted. LEMMA. Let U and V be subspaces n R n and suppse W1 and W are n the subspaces f R gven by W P ŽV. 1 U and W PU V, respectvely. Then: Ž. U V U W 1, Ž U U V. W.

18 148 F. R. MILLER, J. W. NEILL AND B. W. SHERFEY Ž. PROOF OF PROPOSITION 1. Frst nte that by Lemma, ž / CŽ X. CŽ X. P CŽ Z. CŽ X. CŽ Z. fr Z. Thus, by cndtn Ž 5., E Ý ž / Ž. CŽ X. Z E 6 C X P C Z. Next nte that 6 s equvalent t CŽ X. 4 Ž 7. P CŽ Z. 0. Ths equvalence wll be establshed belw. Nw nte that Ž 7. mples that Z E s pstve defnte n CŽ X.. T see ths, let v CŽ X. and suppse that P v 0 CŽ X. CŽ Z. fr all Z. Thus, v C X CZ fr all Z. By Lemma Ž. E, v CŽ X. CŽ X. CŽ Z. P CŽ Z. fr all Z. Hence, E E CŽ X. CŽ X. Ž Ž.. Z E v P C Z s that v 0byŽ 7.. Thus, fr v CŽ X., Ž v. 0 mples that v 0. The equvalence f Ž 6. and Ž 7. fllws by frst bservng that Ž 7. hlds f and nly f n 8 R Ý ž P CŽ X. Ž C Z./. Z E Thus, 8 mples 6 mmedately. In addtn, snce fr all Z, we have that E Thus, assumng that 6 hlds, P CŽ X. CŽ Z. CŽ X. Ý ž CŽ X. / Z E CŽ X. P CŽ Z.. Ý ž / n R CŽ X. CŽ X. P CŽ X. CŽ Z.. Z E Hence, 6 mples 8 and the equvalence f 6 and 7 fllws. Part C. The prf f Therem 1 fllws drectly frm Lemmas 3 and 4 belw. The prf f Lemma 3 s straghtfrward and thus mtted. LEMMA 3. Let G be a graph wth vertex set V. Let C C,..., C 4 1 k be the cllectn f clques f G. Then we have the fllwng.

19 LACK OF FIT TESTS 149 Ž. C s a cver fr V. Ž. If P A,..., A 4 1 m s a parttn f V whch s cnsstent wth G, then fr each 1,..., m, A C fr sme j 1,..., k 4. j LEMMA 4. Let C C,..., C 4 1 k be a cver f a set T. Suppse L s the cllectn f all parttns P f T satsfyng: f P A,..., A 4 1 m then fr each 1,..., m there exsts j 1,..., k4 such that A C. Ž. Let P L wth cardnalty f P equal t m k 1. Then there exsts P* L such that P s a refnement f P* and the cardnalty f P* s equal t m 1. Ž. Suppse the cver C s mnmal. If P A,..., A 4 1 m L, then the cardnalty f P s greater than r equal t k. PROOF. Ž. Snce m k, there exsts 1 a, b m wth a b and 1 j k such that A C and A C. Thus, P* A a j b j A P, 1 m, a, b4 A A 4 a b L wth cardnalty m 1. Ž. Fr each 1,..., m, chse j 1,..., k4 such that A C j. Let : 1 m4 j1 j k4 where Ž. j. Fr each j 1,..., k, chse t C C. Defne : j1 j k4 1 m4 j j a j a by tj A Ž j.. Snce tj Ca fr 1 a k and a j, t fllws that A Ž j. Ca fr a j. Thus Ž Ž j.. j. Hence, s nt and m k. Part D. The prf f Therem fllws drectly frm Therems 3 and 4 n belw. In the fllwng, let X x 1,..., x p where each x s a vectr n R wth p n, and dentfy X wth a pnt n R np. As n Sectn, let P A: R n R n dente the rthgnal prjectn nt a subspace A R n. The prf f Lemma 5 s straghtfrward and thus mtted. N LEMMA 5. Suppse P z 1,..., zn s a plynmal functn n R whch s N 4 N nt dentcally zer. Then the set z R : P z 0 s pen and dense n R. LEMMA 6. There exsts an pen dense subset O R np such that X O mples dmensn C X p. np PROOF. Cnsder X as a varable pnt n R and let PŽ X. be the determnant f the upper p rws f X. Snce there are X fr whch P s nt np zer, the set f X such that dmensn C X p s pen and dense n R by Lemma 5. LEMMA 7. Suppse B s a subspace f R n wth dmensn B p n. Then 4 np the set f X fr whch B C X 0 cntans an pen dense subset n R. PROOF. Let v,..., v 4 1 k be a bass fr B and extend ths cllectn t a lnearly ndependent set dented by v,..., v, v,..., v 4 1 k k1 np. Next let V be the n Ž n p. matrx wth clumns gven by v, j 1,..., n p, and j j

20 1430 F. R. MILLER, J. W. NEILL AND B. W. SHERFEY defne PŽ X. as the determnant f the n n matrx Ž V, X.. Snce there are X fr whch P s nt zer, the set f X such that B CŽ X. 04 cntans an pen dense subset n R np by Lemma 5. In the fllwng prfs, the fact that n a cmplete metrc space the ntersectn f fntely many pen dense subsets s pen and dense wll be used. LEMMA 8. Suppse A s a subspace f R n wth dmensn A c p. Then there exsts an pen dense subset OA R np such that X OA mples dmensn P CŽ X. dmensn CŽ X. p. A PROOF. Frst nte that dmensn PCX dmensn CŽ X. A f and nly 4 f A C X 0, wth dmensn C X p c and dmensn A n c. Ths fllws by cnsderng P : CŽ X. A s that dmensn CŽ X. A dmen- Ž. Ž sn kernel P C X dmensn PCX dmensn A CŽ X.. A A dmensn PCX. A Next nte by Lemma 7 there exsts an pen dense subset np V R such that X V mples A CŽ X. 0 4 A A. Hence, dmensn PCX dmensn CŽ X. A. Nw take OA VA O where, by Lemma 6, np O R s pen and dense such that X O mples dmensn CŽ X. p. LEMMA 9. Suppse A s a subspace f R n wth dmensn A c p. Then there exsts an pen dense subset OA R np such that X OA mples dmensn P CŽ X. c and dmensn CŽ X. p. A ˆ nc PROOF. Let X x 1,..., x c R. By Lemma 8, there exsts an pen ˆ nc dense subset O R such that Xˆ Oˆ mples dmensn PCX Ž ˆ. A A A Ž ˆ. ˆ nž pc. dmensn C X c. Nw take OA OA R O where, by Lemma 6, np O R s pen and dense such that X O mples dmensn CŽ X. p. np Nte als that O s pen and dense n R, and snce PCX Ž ˆ. PCX A A A A, dmensn PCX c. A THEOREM 3. Suppse A 1,..., As are subspaces f R n. Then there exsts an np pen dense subset O R such that X O mples dmensn CŽ X. p and dmensn P CŽ X. p f p dmensn A, whle dmensn P CŽ X. A A dmensn A f p dmensn A, 1,..., s. PROOF. O s 1 Let O A, 1,..., s, be gven by Lemma 8 r Lemma 9, and take O. A The fllwng generalzatn f Therem 3 may be used t cver, fr example, the case n whch the predctr matrx prvdes fr an ntercept n the mdel. THEOREM 4. Suppse A 1,..., As are subspaces f R n and als let Y be a subspace f R n where Y A fr each 1,..., s. Suppse dmensn Y k

21 LACK OF FIT TESTS 1431 and cnsder X Ž x,..., x. Ž y,..., y, z,..., z. where y,..., y 4 1 p 1 k 1 pk 1 k s a fxed bass fr Y, k p n. Then there exsts an pen dense subset O nž pk. R such that Z Ž z,..., z. O mples dmensn CŽ X. 1 pk p and dmensn P CŽ X. p f p dmensn A, whle dmensn P CŽ X. A A dmensn A f p dmensn A, 1,..., s. PROOF. Frst nte by Lemma 7 there exsts an pen dense subset O 1 nž pk. R such that Z O mples Y CZ 0 4. Then CŽ X. Y CZ 1 4 Ž and A C X 0 f and nly f A Y. CZ 0, 4 1,..., s. These bservatns allw the analyss fr Therem 3 t be used t cnclude the prf f Therem 4. Part E. Cnsder a dscnnected graph G Ž V, E. whch breaks up nt q cnnected cmpnents wth V, E V, E V, E V, E. 1 1 q q In ths case there s a subspace R whch s cmmn t CZ fr every Z cnsstent wth G. In fact, R CZ CZ Z Z and dmensn G R q. If q p dmensn CŽ X. then Z C X C Z C X R 0 4. Furthermre, R CZ R where ZR s the grupng matrx crrespndng t the parttn V, V,..., V 4 1 q. Nte that ZR need nt be an element f G. Thus, C X R s cmmn t every alternatve mdel f the B type H under cnsderatn and CŽ X. ŽCŽ X. CŽ Z.. CŽ X. Ž a R. Ž C X R CŽ X. CŽ Z.. fr all Z G. Cnsequently, lz 1 fr all Z whenever has the frm gven by Ž 4. G and G s dscnnected as specfed abve, and hence prvdes n dscrmnatn between the canddate grupngs. Thus, n cmparng the atms cnsstent wth the graph, lz may be mdfed as v lz nf : v R CŽ X. CŽ Z., v 0. ½ Ž v. Nte that s nt affected by R. 5 Part F. Let G be a graph determned by a famly f verlappng subsets 4 p S,..., S n R as dscussed n Sectn 3. Let V v,..., v 4 1 m 1 n dente the set f vertces f G and let V S V, 1,..., m. Cndtns wll be gven under whch V,..., V 4 1 m s the cllectn f clques f G. We assume that Ž 1. V V and Ž. V Vj wth VVj and VjV fr j. Next recall that fr v, vj V there s an edge jnng v t vj f and nly f v, vj Vk fr sme k 1,..., m. Thus, the sets V, 1,..., m, themselves frm cmplete subgraphs. The questn that remans s whether every cmplete subgraph s cntaned n sme V. That s, V,..., V 4 1 m s nt the cllectn f clques f G f and nly f there exsts a subset O V whch nduces a cmplete subgraph f G and O s nt cntaned n any f the sets V. Suppse such subsets O exst. Frm all such subsets chse O0 t be f mnmal cardnalty. The cardnalty f O must be at least three snce any 0

22 143 F. R. MILLER, J. W. NEILL AND B. W. SHERFEY set f cardnalty tw whch has ts pnts jned by an edge must le n ne f the sets V by cnstructn f the graph accrdng t the verlappng grd apprach. Next chse dstnct vertces v 01, v0 and v03 frm O 0, and nte by the mnmal cardnalty f O0 that there exst sets V 01, V0 and V03 frm the cllectn V,..., V 4such that 1 m O0v014 V 01, v01 V 01, O0v04 V 0, v0 V 0, O0v034 V 03, v03 V 03. The precedng dscussn establshes the fllwng lemma. LEMMA 10. If the set f clques f G s nt the cllectn C V,..., V 4 1 m then there exsts three dstnct vertces v 01, v0 and v03 and three dstnct sets V 01, V0 and V03 frm C such that v 0, v034 V 01, v01 V 01, v 01, v034 V 0, v0 V 0, v 01, v04 V 03, v03 V 03. The fllwng crllary fllws drectly frm Lemma 10 and prvdes cndtns under whch V,..., V 4 s the cllectn f clques f G. 1 m COROLLARY 1. If there des nt exst such a cnfguratn Žv 01, v 0, v 03, V, V, V as descrbed n Lemma 10, then the cllectn f clques f G s gven by V,..., V 4. 1 m The precedng methdlgy fr determnng the clques f a graph derved by the verlappng grd apprach s nw appled t Example. In ths example there are fur verlappng grd elements n R gven by S 11, S 1, S1 and S. The crrespndng sets V, 1,..., 4, may be dentfed as V1 S11 V 1, 3, 154, V S1 V, 4, 6, 11, 144, V3 S1 V 3, 5, 7, 9, 1, 15, 164, V4 S V 4, 5, 8, 10, 11, 1, 13, 144. Nte fr example that V1 V4 and V V3 s that by chsng any three dstnct sets V, V, V frm the cllectn V,..., V there exsts at least ne par wth empty ntersectn. Thus, there cannt exst a cnfguratn f the type descrbed n Lemma 10. Hence, by Crllary 1, the cllectn f clques f G n Example s gven by V,..., V

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