Powerful Modifications of Williams Test on Trend

Transcription

1 Powerful Modfcatos of Wllams Test o Tred Vom Fachberech Gartebau der Uverstät Haover zur Erlagug des aademsche Grades ees Dotors der Gartebauwsseschafte Dr. rer. hort. geehmgte Dssertato vo Dpl. Math. Fra Bretz gebore am Vlaardge 999

2 Referet: Prof. Dr. L.A. Hothor Korreferet: Prof. Dr. L. Barghaus Tag der Promoto: 6. Jul 999

3 To my wfe Jame for her love, costat support, ad uderstadg that sometmes multple cotrast tests may come frst.

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5 Abstract (Schlagworte: multpler Kotrasttest, multvarate t Vertelug, Ordugsrestrto) Häufg stellt sch de Frage ach statstsch sgfate mootoe Wrugsverläufe quattatver Eflußgröße. West e bestmmtes Herbzd mt astegeder Doss ee verbesserte Wrug m Verglech zu eer Kotrollgruppe auf? Trete be juge Kulturpflaze mt abfalleder Temperaturbehadlug sgfat häufger Aomale auf? Fragestelluge deser Art blde de Schwerput der vorlegede Dssertato. Im Gegesatz zur herömmlche Varazaalyse wrd her e mootoes Wrugsprofl vorausgesetzt, um vo deser Aahme ausgehed mächtgere Tests zu etwcel. We der Dssertato hervorgehobe wrd, berge jedoch de lasssche Tredtests vo Bartholomew, Wllams ud Marcus z.t. erheblche Nachtele. Daruter fällt de ugelöste Problemat der umersche Verfügbaret uter der Null- oder Alteratvhypothese, bes. m wchtge ubalacertem Fall. Ferer führt de uzurechede Ketsahme der Fallzahlauftelug de Varazschätzer be Wllams ud Marcus zu eem ubefredgede Güteverhalte. Dese ud wetere Nachtele schräe de Awedug der dre lasssche Tredtests somt star e. Das Zel der Dssertato besteht dar, mttels dem Kozept der multple Kotrasttests de Problemate zumdestes telwese zu etschärfe. Herbe wrd das Maxmum über mehrere ezele Kotrasttests (stadardserte Learombatoe der Mttelwerte) betrachtet. E ezeler Kotrast st auf Grud seer Defto für ee bestmmte Wrugsurve sehr mächtg, reagert aber empfdlch auf Abwechuge derselbge. Der Maxmumtest hgege wählt de beste Teststatst aus ud st demach weger afällg gegeüber uterschedlche Doss-Wrugs-Verläufe. Darauf basered wrd der Wllamstest de Theore der multple Kotraste egebettet. Ee ausführlche Behadlug der zugrude legede multvarate t Vertelug ermöglcht see uegeschräte Awedug. Quatle ud p-werte sd auch m Ubalacerte efach zu bereche. E veräderter Varazschätzer mmt de Fallzahlauftelug besser zur Kets ud auf Grud der Kostruto der multple Kotraste hägt de Güte des eue Tests weger star vo der Wrugsfuto ab. Darüber haus wrd auch der Marcustest auf multple Kotraste verallgemeert. Ausgehed vo eer vollstädge Auftelug des Alteratvraumes bldet e drtter Zugag das Maxmum über loal güteoptmaler Ezelotraste (sotoscher Kotrast). I eer ausführlche Gütestude werde dese Tests mt de orgale Tredtests verglche. De Herletug weterer theoretscher ud umerscher Resultate ermöglche sbes. ee geschlossee Darstellug der Güteformel zur teratve Fallzahlbestmmug ud weterführede post-hoc Aalyse. De dazu beötgte chtzetrale multvarate t-vertelug st u alog zur obe erwähte zetrale Form ohe Beschräug der Korrelatosstrutur verfügbar. E weteres Ergebs reduzert de effetve Dmeso ees belebge multple Kotrasts auf de Azahl der zu utersuchede Gruppe, was zu erheblch verefachte Auswertuge führt. Abschleßed wrd vor allem praxsoreterte Fragestelluge achgegage. De bsherge Ergebsse für ormalvertelte Date werde vollstädg auf de dchotome Fall übertrage. De sch ergebede Asymptote erforder sbes. de Utersuchug der multvarate Normalvertelug, welche u m allgemee Fall zur Verfügug steht. De Betrachtug spezeller Aspete bomale Testes (Kotutätsorretur, gepoolte/ugepoolte Versoe, exate bedgte ud ubedgte Verteluge) erweter de Awedugsmöglchete. Ferer werde Asätze zur Bestmmug ausgewählter Parameter vorgestellt (z.b. de Bestmmug eer mmale effetve Doss). Wetere Awedugsmöglchete werde urz agersse (chtparametrsche Aalyse, Kofdeztervalle, höherfatorelle Alage, etc). SAS/IML ud FORTRAN Programme sd erstellt worde ud m Ahag doumetert.

6 Abstract (Keywords: multple cotrast test, multvarate t dstrbuto, order restrcted ferece) Frequetly the questo arses whether gve dose-respose shapes of quattatve varables show ay statstcally sgfcat effect. Does the effcacy of a certa herbcde deed mproves wth creasg doses whe compared to a cotrol group? Has the temperature a sgfcat fluece o the occurrece of aomales youg ohlrab plats? These ad smlar questos are aalysed the preset thess. I cotrast to the usual aalyss of varace oe assumes a mootoous dose-respose profle. Based o ths assumpto ew tests are developed, whch show a mproved power behavour. However, as t s see more detal the thess, the classcal tred tests of Bartholomew, Wllams ad Marcus bear a seres of dsadvatages. Amog other ssues these volve the usolved problem of evaluatg the dstrbuto fuctos uder the ull ad the alteratve hypotheses, partcular the mportat case of uequal replcatos. Moreover, the test statstcs of Wllams ad Marcus do ot tae the sample sze allocato suffcetly to accout. These ad other dsadvatages restrct serously the applcato of the three classcal tred tests for practcal purposes. The am of the thess s to overcome at least partally these problems by applyg the cocept of multple cotrast tests. Here, the maxmum s tae over several sgle cotrast test statstcs (stadardsed lear combatos of the meas). Due to ts defto a sgle cotrast test s very powerful for a fxed dose-respose curve. But already for small departures from t the test may bear a poor power behavour. The above metoed maxmum test, however, chooses the best test statstc ad s therefore more robust agast varyg doserespose fuctos. Hece, Wllams orgal test s embedded the theory of multple cotrast tests. A tesve dscusso of the uderlyg multvarate t dstrbuto eables a urestrcted use of the ew test. Quatles ad p-values are easly calculated ubalaced set-ups. A modfed varace estmator taes the sample sze allocato better to accout. Due to the costructo of multple cotrast tests the power of the ew test depeds less o the dose-respose shape. Moreover, Marcus orgal test s geeralsed smlarly. A thrd ew cotrast defto s provded by decomposg the alteratve space the smallest possble sub-hypotheses ad tag subsequetly the maxmum over the locally optmal sgle cotrasts (sotoc cotrast). I a detaled power study the performaces of these multple cotrast tests are compared wth the orgal tred tests. The dervato of further theoretcal ad umercal results eables the represetato of a power formula a closed form for teratve sample sze determato ad for further leadg post-hoc aalyss. Smlarly to the above metoed cetral case the arsg o-cetral multvarate t dstrbuto s ow avalable wthout restrcto of the correlato structure. A further result reduces the effectve dmesoalty of arbtrary multple cotrasts to the total umber of treatmets uder vestgato, leadg to clearly smplfed evaluatos. Fally, further mportat practcal problems are vestgated. The results obtaed so far for ormal varates are geeralsed to the dchotomous case. The arsg asymptotcs demad partcular the vestgato of the multvarate ormal dstrbuto. Its evaluato s ow avalable the geeral urestrcted case. The cosderato of specal aspects of bomal testg (cotuty correcto, pooled/upooled versos, exact codtoal/ucodtoal dstrbutos) exteds the rage of applcatos. Furthermore, approaches for the determato of certa parameters dose fdg studes are preseted (e.g. the mmum dose wth maxmum effect or the mmum effectve dose). Further applcatos are setched brefly (oparametrc aalyses, cofdece tervals, hgher factoral layouts, etc.). SAS/IML ad FORTRAN programs have bee wrtte for most applcatos ad are eclosed wth the thess.

7 Cotets Itroducto.... Survey of tred tests for ormal meas Geeral otatos Maxmum lelhood estmators uder total order restrcto The tred tests accordg to Wllams, Marcus, Bartholomew ad multple cotrast tests Wllams t - test Marcus t mod - test Lelhood rato test Multple cotrast test Example Overvew of other tred tests Multvarate ormal ad t-dstrbuto Multvarate ormal dstrbuto Defto ad basc propertes Computato of multvarate ormal probabltes Approxmato of Solow Trasformatos of Gez Calculato of orthat probabltes Coclusos Calculato of level probabltes Multvarate t dstrbuto Defto ad basc propertes Computato of multvarate t probabltes The methodologes of Somervlle Trasformatos of Gez Computato of equcoordate quatles Numercal comparsos ad coclusos... 73

8 3. Choce of approprate cotrast coeffcets Revew of cotrast deftos A Wllams-type multple cotrast test A Marcus-type multple cotrast test A ew multple cotrast defto Example Reducto of the dmesoalty of multple cotrast tests Power comparso for ormal data Power expresso of multple cotrast tests Power study Coclusos Power comparso for bomal data Notatos Power expresso of multple cotrast tests Itroducto of ew cotrast tests Cotuty correcto Upooled verso Exact codtoal ad ucodtoal versos Example Estmato of the mmum effectve dose Summary ad complemets... 6 Refereces... 7 Appedx... 79

9 I Salo e ch ee, der mch lest, ud auch Bad Nauhem das sd scho zwe. Güter Ech, Zuverscht Itroducto I may research areas the objectve of a expermet s to test whether the effcacy of a ew treatmet or drug s mproved wth respect to a certa cotrol group. A atural way of coductg these d of tests s to cosder several treatmet levels of the ew compoud, drug, fertlser, herbcde,... ad compare them wth a referece group, whch respose s assumed to be ow due to pror owledge of ts behavour. Such a referece group ca be for example a egatve cotrol group wthout ay admstrato or of a vehcle oly. I these cases the goal of the user would be to fd out whether the ew developed treatmet shows ay (statstcally) sgfcat respose at all. By choosg the referece to be a well ow stadard applcato, the am dffers. Here the scetsts wats to vestgate whether the ew treatmet s ot oly better tha a egatve cotrol but eve better tha the stadard. Formalsg the troduced terms above, we deote by C- a egatve ad by C+ a postve cotrol group. Addtoally, D, K, D stad for treatmet or dose levels. Therefore, the frst stuato metoed cossts of a aalyss of the desg C-, D, K, D. I the secod case the desg D, K, D, C+ would have bee chose stead. The umber s usually small due to practcal reasos, frequetly ³ ;, 3, 4@. More complex desgs,.e. cludg more tha oe referece group, are possble, but wll ot be cosdered throughout ths thess. For applcatos whe usg C- ad C+ smultaeously, the reader s referred to Hothor (995) ad Bauer et al. (998). The classcal statstcal approach to aalyse such + 6-sample stuatos the radomsed oe-way layout s the aalyss of varace (ANOVA). However, both the F-test of the ANOVA ad correspodg oparametrc test procedures are oly able to detect ay dfferece amog the vestgated samples. But frequetly the user s more terested

10 specfc results rather tha such geeral assessmets. For example, oe mght be terested aalysg the dose-respose depedece of the data. I these cases the goal s to detect a global tred. Therefore, more formato s requred tha usually establshed by the classcal tests. To llustrate these deas cosder the data provded by Bao ad Yamagam (989) as a example. They studed the coverso effcecy of gested food (E.C.I.) of the wood-feedg sect Eupromus ruber at fve larva stages (thrd to seveth star) ad a adult stage. The edpot was calculated as dry weght of a larva or a adult E.C.I.= dry weght of wood cosumed for each dvdual larva ad adult. The followg table summarses the ma statstcal quattes. Here, the groups,..., 5 correspod to the seveth through thrd star ad the dex s assocated wth the adult stage. The preset desg s of the form D, K, D, eve f the adult stage ca ot be regarded as a cotrol the classcal sese. 5 Stage Mea Std. dev Sample sze The ma questo of terest, from the authors pot of vew, was to vestgate whether the E.C.I. decreased mootoously over all developmet stages. Does a larva from a lower developmet stage has a sgfcat hgher E.C.I. regardg to those of a hgher developmet stage, up to the adult form? Assume that a statstcal sgfcat dose-respose relatoshp (.e. dfferet from costat) has bee detected. A further questo of terest could be the detfcato of the hghest developmet stage amog the larvae, whch stll yelds a sgfcat dfferece to the adult stage. Ths problem s closely related to the estmato of a mmum effectve dose (MED) clcal ad pre-clcal trals ad to the whole theory of

11 tred tests geeral. From the bologst pot of vew, loog at the data, these questos mght have oly oe aswer. Nevertheless, a statstcal aalyss should be coducted to assure the evdece of a possble tred wth respect to the developmet stages. Some further selected examples from the lterature uderle the mportace ad the broad feld of applcatos of detectg sgfcat treds amog several treatmets. Cosder the data provded the table below as a ext example (Savlle ad Wood, 99, p. 4). They refer to a feld expermet, whch was coducted to determe how the gra yeld of sprg sow maltg barley was affected by dfferet seedg rates. A radomsed oe-way layout was chose wth the fve treatmets represetg the fve seedg rates 5 g/ha through 5 g/ha. Each of the sx replcates were harvested from plots of sze 4 m by.5 m. I the lght of above cosderatos we frst otce that o egatve cotrol s preset. Otherwse a cotrol group wth seedg rate g/ha would have bee cluded the tral. As the descrpto of the data does ot clarfy whether a stadard seedg was cluded, we do ot assume the exstece of C+ as well. Therefore the preset desg s of the patter D, D, K, D 5. The ma questo whch aturally arses ths cotext s, whether the gra yeld creased wth creasg seedg rate. Seedg rate Gra yeld Mea Std. dev. 5 g/ha g/ha g/ha g/ha g/ha Peterse (985) descrbed a expermet order to assess whether the addto of partcular ezymes retarded the separato of froze orage juce shortly after the addto of water to the froze cocetrate. The expermet reported cossted of a cotrol wth o treatmet at all ad four levels of a certa ezyme (,, 3 ad 4 ppm). Coductg four replcatos a completely radomsed desg the arthmetc meas 6.68, 9.5, 36.8, ad 49. were obtaed (tme to separato mutes). Does the presece of the ezyme retard separato as 3

12 compared to ts absece? Is there ay dfferetal effect of the level of added ezyme? Oe further expermet provded by Savlle ad Wood (99, p. 59) was carred out to determe the effect of the weedller oxadazo o the early developmet of peach seedlgs. I a typcal radomsed desg cosstg of a cotrol, half dose, sgle dose ad trple dose wth uequal replcatos (6, 6, 5 ad 3, respectvely) the resultg heghts of the seedlgs are show the table below. Does the herbcde oxadazo deed fluece the developmet of the seedlgs? Ad f so, whch would be the statstcally sgfcat smallest dose wth such a effect? Treatmet (g/ha) Heght of seedlgs (cm) Further examples ca be foud may other textboos ad artcles. I the course of the preset thess we wll ecouter a umber of addtoal materal whch demostrates the wde rage of applcato of tred tests. To vestgate the statstcal problems setched above a ew class of tests has bee troduced the lterature the past 4 to 5 years. A varety of tred tests were proposed, may of them wth satsfactory power results for specfc costellatos. But oe ma dsadvatage of ths whole approach s that o uformly most powerful test s at had. All of the tred tests preseted later ths thess deped, sometmes stroger, sometmes weaer, o the uderlyg dose-respose shape (see also Neuhäuser, 996). Therefore, the research for powerful tred tests (yet easy to coduct) s stll ogog. Oe approach wth ths wde rage of aalyses s the lelhood rato test uder total order restrcto (LRT) accordg to Bartholomew (959, 96). Eve f o uformly best test exsts, the LRT has a reasoable power performace ad s cojectured to provde the hghest 'average' power amog the preset tred tests. However, the LRT lacs a wde use for practcal applcatos. Several artcles dscuss ths cotradcto vew of the fact of ts superor power behavour, see for 4

13 example Tag ad L (997) or Agrest ad Coull (998). As we wll see the sequel, the LRT s regarded to behave less robust agast certa types of volatos of ts assumptos, such as varace heterogeety ad o-ormalty. Apart from ths, oe crucal drawbac les the dffculty to evaluate the ull dstrbuto. Log tme the use of the LRT was restrcted to strctly balaced desgs, a assumpto whch s frequetly volated practce. The geeralsato to ubalaced set-ups got oly possble wth moder computer slls ad ew statstcal techques. Because of such practcal problems whe mplemetg the LRT, may researchers tred to develop alteratve testg procedures. Oe mportat approach s due to Wllams (97, 97). Sce ts publcato t has frequetly bee used both medcal ad o-medcal applcatos. Itroduced orgally for ormal dstrbuted data oly, several geeralsatos to dchotomous ad oparametrc set-ups ad hgher factoral layouts permt a wde rage of applcatos. Shrley (996, p. 6) emphassed accurately her lterature revew of tred tests the dstgushg features, that geerally, Wllams t - test s favoured the lterature because of ts robustess to oormalty, lac of balace, ad o-mootocty of dose-respose. Bartholomew s test comes a close secod because of ts superor power overall. It becomes clear that Wllams test s regarded as havg good robust characterstcs agast several types of volatos of ts assumptos. I partcular, as Shrley (996) pots out aga the sequel of her paper, Bartholomew s test s less robust tha Wllams verso. O the other had, t s well recogsed that the t - test has o average a lower power tha the LRT. Commo to both tests, however, are ther complcated dstrbutos uder the ull hypothess. No geeral method s avalable to compute quatles quc ad accurately for Wllams test the geeral ubalaced case. Ths restrcts the use of the t - test to strctly balaced stuatos, although t s robust agast smaller departures of the requred balace. However, t has bee show (Bretz ad Hothor, 999) that Wllams test matas less ad less a predetermed a -level as the degree of mbalace creases. 5

14 May other tred tests were proposed the lterature. Based o the sghts setched above, the search for ew tred tests s coducted from oe pot of vew oly. The geeral goal s to combe the followg ma features: good power behavour comparable to the LRT throughout the alteratve space; easy umercal mplemetato of the test statstcs, a problem of partcular mportace, for the multvarate ature of comparg several treatmets maes a easy hadlg dffcult; robustess agast specfc volatos of the assumptos the sese Hothor (989) has show for Wllams test. The preset thess should be cosdered ths cotext of developg ew procedures for statstcal fereces uder order restrcto. The am of ths thess s to fll the gap betwee the approaches of Wllams ad Bartholomew. Startg from Wllams t - test the attempt s made to derve ew, mproved test statstcs. Ths s doe by applyg the basc cocept of Wllams to the class of multple cotrast tests (MCTs) accordg to Muerjee et al. (986, 987). The resultg test combes several advatages of the volved approaches ad ca be appled to the geeral case of uequal sample szes wthout further restrctos. Improvemets o the umercal methods avalable so far result fast evaluatos of the correspodg ull hypothess. Smulato results suggest that the power behavour of the ew approach s close to that of the LRT uder a varety of codtos. Geeralsatos to oparametrc ad dchotomous set-ups, as well as robustfcatos agast outlers ad applcatos to hgher factoral expermets are possble ad straght forward. I ths sese the sprt of the preset thess s well descrbed by McDermott (998) hs abstract: The lelhood rato test for equalty of order-costraed meas s ow to have power characterstcs that are geerally superor to those of competg procedures. Dffcultes mplemetg ths test have led to the developmet of alteratve approaches, such as tests based o sgle ad multple cotrasts. 6

15 The thess s roughly outled as follows. Chapter presets a overvew of the most mportat procedures the parametrc case of ormal data for testg o equalty of several meas uder total order restrcto. Emphass s gve o Wllams t - test, the LRT of Bartholomew, Marcus (976) modfed t - test ad sgle ad multple cotrast tests. Other methods are brefly metoed ad ther ls to exstg tests are establshed. Further o, geeral otatos ad basc cocepts mportat for the readg of Chapter 3 through 7 are troduced. The example of comparg the coverso effcecy amog several larva stages of Eupromus ruber s aalysed detal ad provdes addtoal motvato for mprovg Wllams' test. As already poted out, the ull dstrbutos of multvarate tests cosdered the preset cotext are geeral dffcult to compute. Whe developg the deas of multple cotrast tests further Chapter 3 through 7 we eed the ablty of computg both the multvarate ormal ad multvarate t-dstrbuto uder several aspects. Chapter provdes a dscusso depth of ths topc. Theoretcal results, as far as requred, are cted or prove. Numercal algorthms for the calculato of both multvarate dstrbuto fuctos are troduced, whch ca be appled to a varety of dfferet problems ad stuatos. Ths chapter provdes the theoretcal ad umercal fudametals for the remag thess. Importat developmets are acheved evaluatg the ull dstrbutos of the LRT ad MCTs the geeral ubalaced set-up. I Chapter 3 we wll focus o approprate choces of cotrast sets. The problem of the emprcal determato of cotrast coeffcets s dscussed. Based o the approaches of Wllams, Marcus ad Bartholomew, three attempts of ew deftos are made. A useful result for possble hgh dmesoalty problems coecto wth MCTs s derved. Wth these ew formulated test statstcs we provde a extesve power study for ormal dstrbuted data Chapter 4. A power fucto closed form for arbtrary multple cotrast tests s derved ad optmal sample sze determato s dscussed. We compare the three metoed MCTs wth the correspodg orgal versos for a varety of scearos, cludg dfferet total sample szes, varable sample sze allocatos wth the groups ad the fluece of the choce of a predefed a amog other aspects. 7

16 I Chapter 5 geeralsatos to the bomal case are gve. We establsh asymptotc power ad sample sze fuctos closed form for sgle ad multple cotrasts. Alteratve methods are dscussed where the dervato fals to succeed. Further o, we geeralse the cocept of dchotomous cotrast tests developed so far. Amog other topcs we troduce a cotuty correcto ad dscuss ts approprate defto. We compare pooled wth upooled asymptotc versos ad develop the deas of Neuhäuser (996) further by provdg codtoal ad ucodtoal exact MCTs. Bref power ad sze comparsos are gve for each topc. Fally, a example aalysed detal llustrates ad summarses the ma deas of the chapter. We wll focus o the mportat pot of estmatg the MED Chapter 6. Istead of testg the global ull hypothess oly, we sequetally coduct several tests accordg to the closure prcple of Marcus et al. (976). Further assumptos of mootocty ad restrcted comparsos to the cotrol lead to smple testg procedures, where at each step a codtoal testg at full sze a s allowed. A outloo o other dose estmatos, such as the maxmum effectve dose, s gve. I the fal Chapter 7 we summarse our results ad try to provde advses to the practtoer as far as possble. Afterwards, further applcatos are vestgated brefly. These clude a short dscusso about other order restrctos tha the smple orderg. Addtoally, the cases of o-parametrc aalyss ad varace heterogeety are cosdered amog other topcs. Appedx A cotas the balaced cotrast sets of the proposed tests Chapter 3 up to dmeso sx. Appedx B cludes some of the algorthms used throughout the thess. Most of them wll refer to Chapter. Because of the wdespread use of the statstcal computato pacage SAS statstcs ad ts applcatos, most of the algorthms preseted ad calculatos provded were mplemeted SAS, verso 6.. The use of other software s metoed at the respectve passages. 8

17 . Survey of tred tests for ormal meas I ths chapter the most mportat procedures from lterature for testg the equalty of several meas uder total order restrcto wll be revewed. But before dog ths we troduce some basc otatos the frst secto, whch wll be vald for the whole thess. Afterwards we revew brefly the theoretcal aspects of maxmum lelhood estmato uder total order restrcto. The results stated here are fudametal for the uderstadg of the preseted tred tests Secto.3. These are the procedures due to Wllams (97), Marcus (976) ad Bartholomew (96), whch are all based o the prcple of maxmum lelhood estmato. Fally, a dfferet approach due to Muerjee et al. (987), the multple cotrast test, s also dscussed ths secto. Commo to all these four approaches s ther mportace the course of ths thess. I the last Secto.4. we touch brefly o other procedures ad try to demostrate ther relatoshps to the precedg tred tests dscussed Secto.3... Geeral otatos Suppose the followg radomsed fxed effect oe-way layout model X = m + e, =,, K,, j =, K,, j j wth oe cotrol group ad treatmet or dose levels, labelled by,,,...,, respectvely. = B be the sample values, detcally ad depedetly ormal dstrbuted wth the Let X j j uow meas m, m, K, m ad commo varace s,.e. Xj ~ Nm, s 7. The varable deotes the sample sze of the th group. For most parts of ths thess we therefore mpose o restrctos o the sample szes, but assume that the uow varaces are equal betwee the treatmet groups. We deote further the sample mea Ê Xj by X for =,, K, ad by s = Ê Xj X Ê - j = 3 8 = the pooled varace estmator wth = - + Ê j 6 degrees of freedom. Our goal s to test the ull hypothess of o effect betwee the + dose groups 9

18 H : m = m = K = m. (.) Whe applyg the classcal ANOVA, the alteratve s geerally formulated as HA : $, j: m ž m j, ž j,, j³ ; The F-test therefore states oly dffereces of ay two treatmet groups ad allows o further coclusos about ther detfcato. Ths s clearly ot a approprate way of testg our stuato. The frst modfcato s therefore to cosder comparsos to the cotrol oly. The correspodg (two-sded) alteratve would the be stated as HA : $ : m ž m, ³ ; For such stuatos the tests accordg to Duett (955, 964) the parametrc case ad to Steel (959) the o-parametrc case are stadard. But aga ths alteratve s ot sutable to the preset terestg cotext, because we assume a relatoshp amog the meas m to hold. If there s ay dfferece betwee the treatmet levels, we assume the respose to crease (or decrease) mootocally wth respect to creasg levels. I the frst example gve the Itroducto t s atural to assume that for hgher developmet stages the coverso effcecy, f at all, decreases. We therefore tae such mootoc dose-respose depedeces to accout ad restrct the alteratve space aga by formulatg the oe-sded hypothess H A : m ˆ m ˆ ˆ m, m < m K. (.) Ths meas that the m s are (ot ecessarly strctly) ordered wth respect to ˆ. If H s rejected, the we coclude due to our pror owledge that a global tred over all cluded + groups deed exsts. Wthout loss of geeralty we lmt the aalyss o creasg doserespose fuctos. Stuatos wth decreasg resposes are reduced to above costellatos by revertg the sgs of the data. Furthermore, we cosder oly oe-sded alteratves throughout ths thess. Geeralsatos to two-sded cases are possble ad mostly straght forward. The followg mportat remar seems to be approprate at ths stage. It should be oted that the corporato of the two assumptos comparso to the cotrol, oly, mootoe restrcto of the alteratve,

19 must be see the cotext of searchg for better tests terms of power. The cluso of pror formato as doe above leads to more powerful tests comparso to those whch do ot tae the orderg of the meas to accout. Tests for tred are therefore recommeded here. However, cauto s advsable, f the practtoer s ot sure, whether ths d of uderlyg dose-respose shape really holds. Bauer (997) showed that already small departures from the assumed alteratve may lead to approprate results f tred tests, such as those preseted below, are used. They are the ot useful the sese that they do ot cotrol the probablty of correctly declarg a dose to be effectve whe fact t s ot effectve. Irrespectve whch tred test s gog to be coducted, the decso for ts use should always be doe uder ths aspect ad the cotext of the applcato should be aalysed carefully before loog at the data. Geeralsatos to stuatos, whch a possble dowtur at hgh doses ca ot be excluded a-pror, are hadled by Smpso ad Margol (99) ad Pa ad Wolfe (996) amog others, but wll ot be aalysed here... Maxmum lelhood estmators uder total order restrcto The problem s to vestgate depedet radom samples from + ormal populatos wth meas m, m, K, m ad commo varace s. Recall the ull hypothess (.) of o effect ad that we restrcted the alteratve to (.) for our applcatos. The am s ow to derve the maxmum lelhood estmator (MLE) of the populato mea vector 6 uder both hypotheses. Ths was frst doe by Bru (955) uder m = m, m, K, m rather geeral aspects. The descrpto here, however, follows more closely the represetato of Robertso et al. (988, p. 6). The dervatos preseted below are fudametal for the tests of Wllams (97), Marcus (976) ad Bartholomew (96), as all three use these estmates ther proposed statstcs. Frst ote that the correspodg lelhood fucto s gve by

20 L 7 X, X, K, X m, s = ¼exp X N &- Ê Ê3 j -m 8 ), s p s = j= 3 8 % ' ( * where N = Ê s the total sample sze ad X = X, K, group. To obta the MLE uder H tae the logarthm of L 3 X 8 the data vector of the th (.3) = j = N log L =- log p -Nlog s - ÊÊ X j -m s log L6 ad set ts partal dervatve = (otce that uder H m m = K = m = : m). Ths yelds the well ow result ÊÊ3 j 8 Ê3 8 Ê 7 = j= = = X - m = X - m+ X - m+ K + X - m = X - m =. Ths meas that the vector X = X, X, K, X 7 s the urestrcted MLE of µ. We ow drect the atteto towards the MLE uder the restrcted alteratve. Loog at the log-lelhood fucto (.3) above we otce that the MLE subject to m ˆ m ˆK ˆ m s gve by the mmsato of ÊÊ 3X j - m 8 = = j= = ÊÊ3Xj - X + X - m 8 = = j= = ÊÊ3Xj - X8 + ÊÊ3Xj - X8 X - m7 + ÊÊ X - m7 = = j= = j= = j= ÊÊ3 j 8 Ê> C = j= = ÊÊ3Xj X8 Ê3X K X X8X m7 X Ê m7 = j= = = = = ÊÊ3Xj - X8 + ÊX -m 7. = j= = = X - X + X - X X - m + K + X - X X - m + X - m = = = Ê = 7

21 Because the last equato the frst term does ot deped o µ the restrcted MLE K 6 mmses m $ = m$, m$,, m$ Ê = X - m 7 (.4) wth respect to m ˆ m ˆK ˆ m. Before we cotue searchg for a explct represetato of $m we gve the followg deftos. Defto..: Let X = x, x, K, x be a fte set wth the smple (or total) order x ˆ x ˆK ˆ x. A fucto f o X s called sotoc subject to the gve orderg f f x ˆ f x ˆK ˆ f x. Defto..: Let g be defed o a fte set X = ; x, x, K, A fucto g * o X s called a sotoc regresso of g wth weghts w = w, w, K, w 6 subject to the smple orderg x ˆ x ˆK ˆ x uder the L - orm, f g * s sotoc ad mmses gx 6- f6 x wx 6 the class of all sotoc fuctos f o X. x X Ê ³ Lemma..: Wth above otatos the restrcted MLE of + ormal meas wth respect to 7 ad m ˆ m ˆK ˆ m s gve by the sotoc regresso $m of X = X, X, K, X weghts,, K, 6. Proof: Defe above dervato the + treatmet groups by D, D, K, D ad X = ; D, D, K, Let further gd6= X ad wd6= for =,, K,. The asserto follows drectly for g* = m $ from Defto.. ad above calculatos We therefore have maaged to reduce the calculato of the MLE for ordered meas to the problem of solvg the mmsato problem (.4). Wth Lemma.. we coclude further that ths s equvalet to determe the sotoc regresso the sese of Defto.. Startg wth ths termedate result we proceed forward ad mae use of varous algorthms avalable for the computato of g *,.e. $m. 3

22 The pool-adjacet-volator algorthm (PAVA) accordg to Ayer et al. (955) s the most wdely used algorthm to compute the sotoc regresso. I the cotext of searchg for restrcted MLEs the process ca be descrbed as follows. Frst loo, whether X ˆ X ˆK ˆ X. If t s the case the set m$ = X, =K,,,, ad the procedure fshes wth m $ = X,, that X K X 7 beg the sought restrcted MLE. Otherwse there s at least oe, so X. Replace X - ad X by the sgle mea > - X -X -+ X = + -,. - The ew seres s therefore reduced to the meas X, X, K, X, X, X, K, X - -, +. From ow o repeat these steps by treatg X -, as a sgle mea wth correspodg weght w-, = - + utl the remag amalgamated meas are completely ordered. At the last stage the mea X -j, K s replaced by j + meas, -, m$,, m$, m$ -j K - wth the same value as the amalgamated mea. Ths provdes the restrcted MLE, whch cossts of the fal vector of + meas m $ = m$, m$, K, m$ 6. Wth above algorthm we are fally able to calculate farly smple the restrcted MLE wth respect to the total order. The mplemetato of the algorthm s straght forward ad the computatos coducted qucly. Though we stll have o closed formula for the $m s yet. The ext lemma solves ths dsadvatage by gog oe step further. It s based o the max-m formulas for sotoc regresso, compare e.g. Robertso et al. (988, p. 3). But the equvalece betwee the followg aalytcal expresso ad the PAVA descrbed so far ca be see drectly whe wrtg the maxmum ad mmum terms out full. Wth ths last l we are ow able to state 4

23 Lemma..: For gve weghts,, K, ad ormal meas m, m, K, m the maxmum lelhood estmates $m subject to the smple order restrcto (.) are gve by $ = max m m ˆuˆ ˆvˆ v Ê j= u v Ê j= u X j j j, (.5) where X = Ê X are the sample meas for =,, K,. j j A purely aalytcal proof of the fudametal theorem that the amalgamated meas are the soluto to the sotoc regresso problem s gve Cheg (995). I the Appedx a SAS mplemetato of these max-m formulas s provded. The rather theoretcal results obtaed so far are best llustrated by applyg Lemma.. ad the PAVA to a example. Example..: Barlow et al. (97, p. 8) report the umber g6 x of days to freezg for Lae Medota/USA, whch were collected to study local evrometal flueces. Note the mssg radomsato of the uderlyg expermet ad that the example s therefore adequate for statstcal purposes. Nevertheless we use ths data set as a approprate example to expla the prcple of restrcted MLEs. The measuremets were doe each Year x gx Step { { Weght 3 3 Mea 4/3 = / = 4.5 3/ = 5 73/3 = Step { Weght 3 4 Mea /4 = Table.. Days to freezg for Lae Medota/USA. 5

24 year from 3 November o for a total perod of years begg 855. As cosderg the data for the purpose of llustrato oly we restrct the evaluato to the frst years, gve Table.. The varable x stads for the correspodg measured year x. Because the vestgatos were coducted to detect a possble warmg tred over the years, the smple order s defed as creasg values for gx 6 wth progressg years. Havg a frst loo o the data oe mmedately otces that they are ot completely ordered. We assg each year the startg weght w =, =,, K,. Already for the frst two years g6 x = 5 > 3 = gx 6,.e. the orderg s volated. We 5 3 therefore replace both by the average value g ¼ + ¼ x, 7= = 9 ad assg t the ew weght w, = + =. As gx 9, = > = gx3, the mootocty s volated aga ad 9 we replace both meas by g ¼ + ¼ x 3,, 7= = After poolg every decreasg sub- + sequece the mddle part of Table.. s yelded. It cotas all amalgamated meas at ths frst stage ad the correspodg weghts. I the secod pass through the data we compare the remag meas ad pool them, f ecessary. We ote, for example, that gx , 9, 7=. > = gx 6 ad therefore volates the smple orderg. Averagg both yelds the ew mea gx 8, 9,, 7= 35. wth the total weght w 8, 9,, = 3+ = 4 ad we replace the precedg meas by the sgle ew calculated oe. The lower part of Table.. llustrates ths last step ad presets the completely ordered amalgamated meas. Fally, the values the orgal data set are replaced by the ew meas accordg to ther weghts to obta the restrcted MLE $m = (3.33, 3.33, 3.33, 4.5, 4.5, 5, 5, 3.5, 3.5, 3.5, 3.5, 5) for ths example. Before we leave ths secto the reader s referred to the boos of Barlow et al. (97, Chapter & ) ad Robertso et al. (988, Chapter ) for a deeper approach ad uderstadg of ths subject. They cota ot oly the mssg proofs omtted here, but they troduce the sotoc regresso from a geeralsed pot of vew. As a matter of fact, we eep focusg o the smple order, as ths was the startg pot of our cosderatos ad remas beg our ma purpose. 6

25 .3. The tred tests accordg to Wllams, Marcus, Bartholomew ad multple cotrast tests.3.. Wllams t - test The startg pot of all cosderatos ths thess s the tred test accordg to Wllams (97, 97). I hs frst paper, Wllams troduced the ew test statstc for ormally dstrbuted data the balaced set-up ad provded crtcal values (upper % ad 5% pots) for dfferet umber of treatmet groups ad varyg degrees of freedom. I the follow-up paper Wllams geeralsed hs test to stuatos of uequal replcatos ad dscussed ther optmal allocato for a fxed total umber of expermetal uts. The test statstc s gve the geeral set-up by the parwse t-type statstc t, = m$ s - X + (.6) ad s easly mplemeted umercally. Here, s ad are the usual varace estmator ad degrees of freedom gve Secto.. The MLE $m s obtaed by usg Lemma.. or applyg the PAVA from Secto.., but wth the dfferece of excludg the cotrol group from the amalgamato process. Note that hs frst paper, Wllams cluded X both the descrpto of the procedure ad the gve umercal example. But whe dervg the ull dstrbuto order to determe the crtcal values, the cotrol group was excluded. Smlarly, X s omtted throughout hs follow-up paper (97). Tamhae et al. (996) otced to ths topc: Actually, both ways of calculatg the sotoc estmates lead to detcal estmates * of the MED as well as detcal t - tests for testg t. Hece, t does ot matter whch way they are calculated. 7

26 Based o theses facts we therefore cotue by excludg X from the poolg procedure. However, oe ma dsadvatage of the t - test s the arsg ull dstrbuto, whch s dffcult to compute, especally for uequal replcatos. For the balaced case upper crtcal pots for several combatos of a, ad are tabulated (e.g. Wllams, 97). For the partal balaced case,.e. equal umber of observatos the treatmet groups ( ž = = K = = : ), Wllams (97) derved a emprcal approxmato, based o the values of the balaced set-up, t w = t - -. (.7),, b w Here, w= deotes the rato betwee the umber of replcates the cotrol group ad the remag groups. The factor b s extrapolated from accurate values ad depeds o ad. Fally, for w = the relatoshp t, 6 = t, leads to the balaced quatles. I the meatme, Wllams procedure s avalable SAS (SAS Isttute Ic., 997, p. 987) for the balaced case by the call 35%&:LOOLDPVTXDQWLOHSUREDELOLW\ (.8) I ths statemet ether quatle or probablty has to be defed, whle the other value has to be set as mssg.. Up to = 5 both quatles ad p-values are calculated fast ad accurately, but for hgher dmesos the computato s very expesve ad slow. For the geeral case of uequal sample szes, however, the evaluato of Wllams dstrbuto stll seems to be a challegg tas, as o algorthm for ts computato s avalable. Several approaches for geeralsg Wllams orgal statstc to other stuatos have bee publshed the lterature ad a small overvew s gve the followg. For the geeral oparametrc set-up Shrley (977) exteded the t - procedure va rag over all groups usg the asymptotc verso of the orgal test (fte degrees of freedom). Next Wllams (986) suggested a slght modfcato of her method order to mprove the power (usg the subsetrag method stead of the -rag). House (986) provded a o-parametrc verso for radomsed bloc desgs based o Fredma-type ras. 8

27 For the dchotomous case Wllams (988) hmself exteded hs procedure ad proposed a codtoal exact test based o the multvarate hypergeometrc dstrbuto uder the ull hypothess. Mout (999) modfed the t - statstc for bomal parameters for comparg two doses to a cotrol ( = ). He derved a asymptotc dstrbuto close to, but ot the same as, a stadard ormal dstrbuto. A robustess study was carred out by Hothor (989) o the behavour of both Wllams ad Shrley s tests uder volato of the ormalty assumpto, varace heterogeety, omootoous dose-respose shapes ad uequal group szes. Tsa ad Che (995) proposed a robustfed statstc by usg robust estmates (M- ad trmmed estmators) stead of the arthmetc meas X. The ew procedures are supposed to be robust agast outlers ad devatos from ormalty..3.. Marcus t mod - test Wllams (97) already proposed a modfed verso of (.6) by replacg X by $m, where $m s obtaed by usg Lemma.. Marcus (976) succeeded the dervg the exact ull dstrbuto of the ew statstc mod m$ - m$ t, = s +. (.9) But as... the computato of exact a quatles... requres -varate umercal tegrato... we computed oly the 5% ad % quatles for = 3 ad 4. The dervato of a algorthm for calculatg quatles or p-values for geeral has ot bee solved utl Hayter et al. (999a). They maaged to decompose the volved varate tegral to a seres of ested lower order tegrals by usg a Marov property of the arsg radom varables. Recursve tegrato techques ca the be appled. From ow o we wll call t mod Marcus test. I the lterature t s also referred to as the modfed Wllams or the sotoc rage statstc. Crtcal values of ths statstc are gve Hayter et al. (999a) for several costellatos of a ad degrees of freedom. Already Marcus (976) coducted a power smulato study ad compared both t ad t mod procedures for 9

28 several parameter cofguratos. She foud out that t has a hgher power for dose-respose shapes of the type m < m = K = m (cocave profles), whereas t mod s better for m = m = K m < m (covex profles). Overall her data suggest that o average t mod s - slghtly better tha t. Cohe ad Sacrowtz (99) showed that Marcus method s admssble ad proposed a better test for = by usg the total sum of errors stead of s Lelhood rato test The lelhood rato test (LRT) for homogeety of ormal meas uder total order restrcto was frst troduced by Bartholomew (959). Wth the varace s ow, Bartholomew s 7, where X X N test statstc s c m X = Ê $ - s = = Ê= s the overall mea estmator ad $m are the MLEs accordg to Lemma.. I the followg we wll focus us, however, o the practce more mportat case of a uow commo varace, estmated by the mea square error s wth = N - - degrees of freedom. Bartholomew the showed that the LRT s based upo the statstc E = Ê 7 m$ - X m$ - X = = X - X m$ - X + m$ - X + s = ÊÊ3 j 8 = j= Ê 7 Ê 7 Ê 7 = =. (.) The E Ê= 7 volves the rato of the betwee groups sum of squares $m - X amalgamato ad the total sum of squares. It ca therefore be terpreted as a ANOVA F test aalogue uder total order restrcto. Bartholomew (96) already succeeded dervg the ull dstrbuto of E. But quotg the ma results we wll aga follow the represetato of Robertso et al. (988, pp. 68). after Recallg Defto.. of a sotoc regresso we frst troduce the level probabltes whch repeatedly arse later o. We call those subsets, where the quattes arsg (.5) are costat, level sets.

29 Defto.3.: Let µ³h ad w =,, K, 6. Further, let Y, Y, K, Y be depedet radom varables, Y ~ Nm, s 7. Further o, we set M the umber of level sets Y*, the sotoc regresso of Y = Y, Y, K, Y 6. The we call the quattes 6 6 K, Pl, + ; w = P M= l, l=,, + level probabltes. The level probablty Pl, +; w 6 s therefore the probablty that the sotoc regresso fucto Y* taes exactly l dstct values. By defto t follows that + Ê Pl, + ; w = = l 6. The followg fudametal lemma shows that the ull dstrbuto of E ca be stated as a weghted sum of F probabltes. Lemma.3.: Let µ³h ad c ³. The N - l c P E c = Ê+ + P l, + ; w P Bl- N-l c = ÊP l, + ; w P Fl N l, -,, - l --c l = l = where F s a radom varable followg a F dstrbuto wth ad v, freedom ad B ab, s a beta varable wth parameters a ad b. degrees of Proof: See for example Robertso et al. (988, pp. 7). Wth Lemma.3. we have the geeral form of the ull dstrbuto of E, but to mae use of ths result the values of the level probabltes Pl, +; w 6 have to be obtaed. These probabltes volve the evaluato of multdmesoal tegrals. The arsg umercal dffcultes are oe ma reaso for the restrcted use of the LRT throughout the lterature. I fact, there are several possbltes calculatg these tegrals, but we postpoe ther represetato to Chapter. A SAS/IML program s preseted there, whch computes for arbtrary weghts w the requred values few secods at a accuracy of more tha 7 - up to =.

30 I the passages above we have troduced the statstc of the LRT ad have gve a basc oto of ts ull dstrbuto. As already metoed the Itroducto, the LRT s supposed to have good average power throughout the alteratve space H A. Ths has bee show by several power smulato studes (see for example Marcus, 976 ad Turbull et al., 987). However, the LRT has bee regarded for a log tme as dffcult to mplemet ad therefore a great varety of smplfyg approxmatos to the LRT exsts. Leavg the cocrete umercal evaluato for Chapter we fsh ths subsecto wth some recet developmets the lterature o the LRT. For a broad overvew up to 988 the reader s referred to Robertso et al. (988, Chapter 3). It ca be show that the alteratve parameter space of the classcal LRT s a poted polyhedral coe ad that the ull hypothess s a lear subspace cotaed the boudary of the coe. Mag use of a dea datg bac to Pcus (975), Aerboom (99) ad Coaway et al. (99) depedetly used a crcular lelhood rato test (CLRT) ad foud that the power of the CLRT was close to that of the classcal LRT. Moreover, because of ts smpler geometrc ature, ts use s supposed to be easer to hadle both balaced ad ubalaced stuatos. Recetly, Tag ad L (997) developed a approxmate lelhood rato test (ALR), whch s based o a orthat alteratve coe ad has accordg to ther results good power propertes as well. Hu (998) preseted a exact algorthm for projectg a vector oto a polyhedral coe relatvely low dmesos. Fally, Wrght (988) troduced a modfed lelhood rato test (MLRT) by usg the usual mea square error stead of the total varace. He derved the ull dstrbuto of the ew test (whch s smlar to the orgal LRT) ad showed the asymptotc equvalece betwee them. A smulato study suggested that the MLRT s more robust agast volatos of the hypothessed ordergs Multple cotrast test The cocept of multple cotrast tests (MCTs) wll be very mportat the course of ths thess. Therefore much atteto s gve to ts troducto. It was frst descrbed by Muerjee et al. (986, 987). Older artcles exst whch meto or deal wth MCTs, but oe of them troduced them thoroughly (see for example Dwass, 96, Du ad Massey, 965, Koe, 976, ad Mehta et al., 984). The ma reaso developg such a ew test was to have a test wth smlar power behavour as the LRT though stll easy to use.

31 From the geometrc startg pot, whch led Muerjee et al. (987) to the ew test, the MCT... s assocated wth a set of vectors that are strategcally located wth the alteratve rego. The am of ths approach s therefore to cover most parts of the alteratve space by choosg some selected vectors wth ths space ad coduct the MCT wth respect to ths grd. However, we leave these geometrcal cosderatos ad troduce the statstc rather aalytcally. Recallg the otato of Subsecto.. we test the ull hypothess (.) of o dfferece by defg the stadardsed statstc of a sgle cotrast test (SCT) as T SC = = s Ê cx Ê = c ~ t. (.) Formulatg the statstc T SC as a quotet of a stadard ormal varable ad a depedet ch varable wth parameter, t follows by defto that T SC s uvarate cetral t- dstrbuted wth degrees of freedom. The weghts c deote the cotrast coeffcets uder the sub-codto c =. Besdes ths lmtato, the choce of the c s s free ad umerous proposals cocerg ther (optmal?) choce have bee publshed. Nevertheless, ths problem has ot bee solved satsfactorly the lterature ad s stll a ope questo of research. We leave ths ssue for Chapter 3, where a detaled revew ad dscusso follows. Istead, we llustrate the cosequeces, whe choosg a poor set of cotrast coeffcets ad o pror formato o the uderlyg dose-respose shape s avalable. Example..: Suppose that we compare = 3 doses of a compoud to a egatve cotrol. Further o we vestgate the two cotrast vectors c = -, -, -, 36 ad c = - 3,,, 6. We aalyse the power of the resultg SCTs T SC ad T SC for two dfferet dose-respose shapes:, d, d, d6 ad,,,d6, where d deotes the shft parameter. I the frst case (cocave profle) the lowest dose has already a effect of sze d comparso to C, whereas the remag doses have o addtoal fluece. I the other case (covex profle) oly the hghest dose has a effect (of sze d ) wth respect to C, but the two low doses have o creased effect at all. 3

32 .9 (a).9 (b) Power.5 Power Shft parameter Shft parameter Fgure.. Power comparso of T SC (dotted le) ad T SC (sold le), balaced case wth sample sze allocato (,,, ), α =.5, = 3 for (a) m =, d, d, d 6 ad (b) m =,,,d6. The power fucto follows a o-cetral uvarate t dstrbuto. For the represetato of the ocetralty parameter we refer to Chapter 4. From Fgure.. t becomes clear, how much SCTs may deped terms of power o the uderlyg dose-respose shape. The effect of the cotrast coeffcets T SC s that they pool the lower treatmet groups ad compare the resultg average value wth that of the hghest dose. Ths s meagful whe the effects of the pooled treatmets are smlar ad therefore T SC behaves well for covex profles. For cocave shapes, however, the poolg of groups wth dfferet effect szes has a egatve fluece o the test statstc ad therefore the power decreases maredly, resultg a loss up to 6%. Smlar argumets hold for T SC, too. From Example.. the strog shape-depedece of SCTs becomes evdet. The crucal pot ow s that these shapes are geeral uow a-pror a stuato of frequet occurrece real data examples. Igorg ths mportat fact s commo practce but ca ot be accepted. The problem of a-pror uow shapes eve creases at least two cases: testg sub-hypotheses by usg the closure prcple (see Chapter 6 for a applcato); vestgato of stratfed desgs because of varyg strata specfc shapes. It seems hghly ureasoable to assume that the same dose-respose shape holds for all subhypotheses, respectve for all strata. 4