On generalized Simes critical constants

Transcription

1 Biometrial Joural , DOI: 0.00/bimj O geeralized Simes ritial ostats Jiagtao Gou ad Ajit C. Tamhae, Departmet of Statistis, Northwester Uiversity, 006 Sherida Road, Evasto, IL 6008, USA Departmet of Idustrial Egieerig ad Maagemet Siees, Northwester Uiversity, 45 Sherida Road, Evasto, IL 6008, USA Reeived 4 November 03; revised 6 Marh 04; aepted 8 April 04 We osider the problem treated by Simes of testig the overall ull hypothesis formed by the itersetio of a set of elemetary ull hypotheses based o ordered p-values of the assoiated test statistis. The Simes test uses ritial ostats that do ot eed tabulatio. Cai ad Sarkar gave a method to ompute geeralized Simes ritial ostats whih improve upo the power of the Simes test whe more tha a few hypotheses are false. The Simes ostats a be viewed as the first order requirig solutio of a liear equatio ad the Cai-Sarkar ostats as the seod order requirig solutio of a quadrati equatio ostats. We exted the method to third order requirig solutio of a ubi equatio ostats, ad also offer a extesio to a arbitrary kth order. We show by simulatio that the third order ostats are more powerful tha the seod order ostats for testig the overall ull hypothesis i most ases. However, there are some drawbaks assoiated with these higher order ostats espeially for k > 3, whih limits their pratial usefuless. Keywords: Multiple hypotheses; Power; Simes test; Type I error. Additioal supportig iformatio may be foud i the olie versio of this artile at the publisher s web-site Itrodutio Cosider ull hypotheses, H,...,H, ad deote their assoiated p-values by p,...,p.let p p deote the ordered p-values ad H,...,H, the orrespodig ull hypotheses. I this paper we osider the problem of testig the overall ull hypothesis H 0 = i= H i. We assume that the p i are idepedet uiform 0, ] radom variables uder H 0. The depedee ase will be studied i a separate paper. The Simes 986 test is based o the idetity P i= { p i iα } = α, where the probability is omputed uder H 0 as are all the type I error probabilities i this paper. Thus it rejets H 0 at level α 0, if at least oe p i iα/ i. It is more powerful tha the Boferroi test, whih rejets H 0 if at least oe p i α/. Correspodig author: atamhae@orthwester.edu, Phoe: , Fax: C 04 WILEY-VCH Verlag GmbH & Co. KGaA, Weiheim

2 036 J. Gou ad A. C. Tamhae: O geeralized Simes ritial ostats Cai ad Sarkar 008 defied geeralized Simes ritial ostats as ay set of i i that satisfy { P pi i α } = α i= subjet to the mootoiity oditio:. 3 I this otatio, the Simes ritial ostats are i = i/ ote that we use a differet otatio for ritial ostats from that used by Cai ad Sarkar. The test based o the geeralized ostats rejets H 0 if p i i α for at least oe i i. 4 The mootoiity oditio 3 is eessary for this test to be valid as will be see i the sequel. I the method give by Cai ad Sarkar 008 to ompute these ostats, the Simes ostats a be viewed as the first order requirig solutio of a liear equatio ad the Cai-Sarkar ostats as the seod order requirig solutio of a quadrati equatio ostats. By reursive appliatio of the Cai-Sarkar method we derive third order ostats ad study their properties i detail. We also preset a geeral result o the kth order ostats. Fially, we ompare differet hoies of ostats i terms of power via simulatio ad show that the third order ostats improve the power of the test ompared to the first ad seod order ostats i a majority of the ases studied. Berhard, Klei, ad Hommel 004 have give a ie review of the literature o global ad multiple test proedures based o p-values. The followig global tests disussed there use speial ases of geeralized Simes ostats. I the ase of idepedet p-values, Bauer 989 proposed the soalled, k,α-test whih uses = = k = 0ad k = = = where > 0 is determied from the equatio i=k α i α i = α, i where k is prespeified. Röhmel ad Streitberg 987 showed that if the p-values are arbitrarily depedet the the α-level is otrolled if i i /i. i= The ostats that satisfy this oditio are i Boferroi: = = = /, ii Rüger 978: = = k = 0ad k = = = k/ where k is prespeified ad iii Hommel 983: i = i/ j= j. Geerally, a global test does ot otrol the familywise error rate FWER if used as a multiple test proedure MTP. For example, the Simes test does ot otrol the FWER if used to rejet ay H i for whih p i iα/ i. A MTP a be derived by ostrutig a losed proedure Marus, Peritz, ad Gabriel, 976 whih uses a α-level global test for all itersetio hypotheses. Wei 996 showed uder what oditios this losed proedure has a stepwise shortut. Toward this ed, deote i by i to idiate its depedee o. Wei 996 showed that, if the losed proedure uses 4 to test all oempty subset itersetios of H i s of size m with ostats i = im, the im = m i m is a eessary ad suffiiet oditio for the losed proedure to have a step-dow shortut ad im = i+,m+ i m to have a step-up shortut. The Holm 979 proedure is the step-dow shortut to a losed proedure that uses the Boferroi test for all C 04 WILEY-VCH Verlag GmbH & Co. KGaA, Weiheim

3 Biometrial Joural itersetio hypotheses. The Hommel 988 proedure is the losed proedure that uses the Simes test for all itersetio hypotheses. But the Simes ostats do ot satisfy either of Wei s oditios; hee the Hommel proedure does ot have a simple stepwise shortut. Hohberg s 988 step-up testig proedure a be show to be based o a oservative hoie of the ostats, im = /m i +, whih satisfy Wei s oditio. The outlie of the paper is as follows. I Setio we review the derivatio of seod order ostats. I Setio 3 we exted the method to third order ostats ad study their properties. Setio 4 gives a geeral result about the kth order ostats. Setio 5 gives tables of the seod ad third order ostats for seleted values of ad,, Setio 6 ompares the powers of the geeralized Simes test for differet hoies of ostats. Colusios are give i Setio 7. Proofs of all the results are give i the Appedix. Seod order geeralized Simes ostats We assume throughout that the geeralized Simes ostats satisfy the type I error rate oditio. Defie the probabilities: Pp > α,...,p > α i = 0, A i = Pp i i α, p i+ > i+ α,...,p > α i =,...,, 5 Pp α i =. Note that i=0 A i = ad hee A i = A 0 = P {p i i α} = α. 6 i= i= I the sequel we use a reursio whih ivolves, for fixed, expressig A i i terms of A i, A i, et. These lower dimesioal probabilities are give by Pp > m+ α,...,p m > α i = 0, A m i = Pp i m+i α, p i+ > m+i+ α,..., p m α i =,..., m, 7 Pp m α i = m. Note that whe omputig A m i for m <, p is ompared with m+ α, ot with α; p is ompared with m+ α, ot with α, et. The latter would be the ase if we hage the otatio so that the idex of i is haged from i to i +. Fier ad Roters 994 showed that uder the mootoiity oditio 3, the followig reurree relatio holds: A i = i α A i i i =,...,. 8 Sie this reurree relatio lies at the ore of the omputatio of geeralized Simes ostats, their validity i terms of otrollig the type I error requires that the mootoiity oditio 3 must be satisfied. By substitutig this reurree relatio i 6 we get i α A i i = α. 9 If we set i= i i = β i, 0 C 04 WILEY-VCH Verlag GmbH & Co. KGaA, Weiheim

4 038 J. Gou ad A. C. Tamhae: O geeralized Simes ritial ostats ad ote that i= A i = i=0 A i = the we obtai from 9 that αβ A i = αβ = α = β =. i=0 Substitutig β = bak i 0 yields the Simes ostats i = i/. Observe that they do ot require predetermiig ay i s. Cai ad Sarkar 008 applied the reurree relatio 8 a seod time by puttig A i = i i αa i i =,..., i 9 to obtai the equatio α + α i i i i A i = α. i= If we set i i i i = β ad ote that i= A i =, we obtai from that β = α. Substitutig β bak i we obtai the quadrati equatio: i ii i i α = 0. The roots of this equatio deped o α ulike the Simes ostats. Furthermore, they deped o, whih eeds to be speified. Cai ad Sarkar 008 limited the rage of to 0 / i whih ase the admissible root is give by i = i + i 4 + α ii. 3 We a show that the rage of a be exteded to / + α] > /. However, the seod order ostats obtaied by this extesio do ot result i ay sigifiat power gai. Therefore, we omit the details of this extesio. Note that if we put = / i 3 the we get the Simes ostats i = i/ ad if we put = 0theweget i = ii ]/ α]. 3 Third order geeralized Simes ostats I this setio we show how third order ostats a be obtaied by applyig the reurree relatio 8 a seod time. Substitute A i = i i αa 3 i 3 i = 3,..., C 04 WILEY-VCH Verlag GmbH & Co. KGaA, Weiheim

5 Biometrial Joural i to obtai α = α + α A 0 + α 3 i i Further substitutig A 0 = i=3 A i i=3 = α i=3 i=3 i ii i i i i A 3 i 3 i the seod term of 4 ad olletig the terms we get α = α + α + α 3 i i i ii i i + ] A 3 i 3. 4 ] A 3 i 3. Now set i i i ii i i i + ] = β 3 5 i the above equatio ad use the fat that i=3 A 3 i 3 = to obtai α = α + α + α 3 β 3. Solvig this equatio we get β 3 = α α ]. Substitutig this value of β 3 bak i 5 we obtai the followig ubi equatio for i : where f i i = 3 i + q i i + r i i + s i = 0, 6 q i = i, r i = ii ii i s i = α, 7 α ]. C 04 WILEY-VCH Verlag GmbH & Co. KGaA, Weiheim

6 040 J. Gou ad A. C. Tamhae: O geeralized Simes ritial ostats Note that the solutios of the ubi Eq. 6 deped o α as well as,, whih must be speified. The followig speial ases are worth otig: If we use the same, for the third order ostats as for the seod order ostats with = + α, 8 from 3, the the third order ostats 3,..., obtaied by solvig the ubi Eq. 6 are the same as the seod order ostats. As a speial ase of the above, if we put = / ad = /, the the solutios to the ubi Eq. 6 are i = i/ 3 i, whih are the Simes ostats. This result exteds the orrespodig result for seod order ostats where, if we put = /, the we get the Simes ostats. 3 If we put = = 0 the the solutio is ii i i = 3 3 i. 9 α We ow state the mai theorem about the third order ostats. Theorem. If the followig oditios hold: α, 0, α, the the ubi Eq. 6 has a uique positive root i ad,..., satisfy the mootoiity oditio 3. Furthermore, all i α /3 < /α so that the p i s are ompared with i α<. Remark. The reasos for the three oditios are as follows. First, α / holds beause geerally α /. Seod, the rage of abeextededto 0 α α, where the upper limit is > /. Similarly, the lower boud o a be exteded to 3 > + 3, whih is < ; see A. i the proof of Result i the Appedix. However, if we exted these rages, the ubi Eq. 6 does ot have a uique positive root, whih poses diffiulties i hoosig the partiular positive root ad showig the mootoiity of the hose set of roots i 3 i. This poit will beome learer i the proof of Theorem give i the Appedix. Cai ad Sarkar 008 showed that if α<i ]/i ] the the seod order ostats i are dereasig futios of for eah i =,...,. The followig theorem gives a extesio of this result for the third order ostats: C 04 WILEY-VCH Verlag GmbH & Co. KGaA, Weiheim

7 Biometrial Joural Theorem. If α ]/6 ] ad 0, /] is fixed the the i are dereasig futios of for eah i = 3,...,for ], + + α. 0 Furthermore, if,,..., are third order ostats ad,,..., are seod order ostats suh that = ad is stritly less tha the upper limit of the iterval 0 the < ad i > i for i = 3,...,. Note that the upper boud o α i this theorem equals for = 3 ad is a dereasig futio of, equalig for = 5, for = 0 ad approahig /6as. Thus this theorem holds for all pratial values of α ad. As a result of this theorem, give ay set of seod order ostats, we a fid third order ostats suh that they are larger for i = 3,...,, whih would make them more powerful i may ases. We will illustrate this umerially i Setios 5 ad 6. 4 kth order geeralized Simes ostats We a apply the suessive reursio proess employed i the previous setio k times to obtai the kth order ostats for ay k <. They require oe to speify the first k ritial ostats from whih the remaiig oes a be determied by solvig a kth degree polyomial equatio. As suh, this geeralizatio is ot of muh pratial use but we give it here for theoretial iterest. Theorem 3. I geeral, oe a determie the ostats where /, whih satisfy the type I error requiremet by speifyig,..., k subjet to ertai ostraits ad the by solvig for k+i 0 i k from the kth degree polyomial equatio: β k i + α γ k = 0, α k where β k i ad γ k are defied reursively by the followig set of equatios: Let The δi, = i,β i i = δi +, i = 0,..., ad γ = 0. β k+ i = β k i + β k 0]δi +, k i = 0,..., k ad γ k+ = γ k + α k β k 0. 5 Tables of geeralized Simes ostats I this setio we give the seod ad third order geeralized Simes ostats for α = 0.05 ad = 3, 4, 5 i Tables, 3, ad 4, respetively. The Simes ostats i = i/, whih are idepedet of α, are also iluded for ompariso purposes ad are tabulated uder Colum I. The other six olums list two hoies of seod order ostats uder olums labeled II, III ad four hoies of third order ostats uder olums labeled IV VII. These hoies are show as labeled poits I VII i the admissible regio of, for third order ostats show i Fig.. The upper boudary of this regio gives the admissible values of, for seod order ostats. C 04 WILEY-VCH Verlag GmbH & Co. KGaA, Weiheim

8 04 J. Gou ad A. C. Tamhae: O geeralized Simes ritial ostats Table Costats, for seve hoies. No. Type I Simes II Seod order 0 III IV Third order 0 0 V 0 VI VII 3 α + α 3 + α + α Table Geeralized Simes ostats = 3,α = i Simes Seod order Third order I II III IV V VI VII Table 3 Geeralized Simes ostats = 4,α = i Simes Seod order Third order I II III IV V VI VII Table 4 Geeralized Simes ostats = 5,α = i Simes Seod order Third order I II III IV V VI VII C 04 WILEY-VCH Verlag GmbH & Co. KGaA, Weiheim

9 Biometrial Joural Figure Feasible regio of,. These hoies were determied as follows. First, I, II, ad IV are the orer poits of the admissible regio show i Fig.. Next III, V, ad VI are the midpoits of the three boudaries of the regio. Fially, VII is the etroid of the triagle formed by the verties I, II, ad IV. The, -values for olums I VII are listed i Table. Note that the upper boud o all the seod order ostats a be show to be α / = 4.47, whih is ahieved for uder olum II whe = 0. I Theorem, the upper boud o all the third order ostats a be show to be α /3 = whih is ahieved for uder olum IV whe = = 0. 6 Simulatio omparisos Sie all ostats satisfy, they all otrol the type I error. So we fous o omparig their powers. Note that here power is simply the probability of rejetig H 0 = i= H i whe at least oe H i is false. Let m be the umber of false ull hypotheses. We studied the followig ofiguratios: = 0 ull hypotheses, α = 0.05, ad m = 0. For eah ofiguratio we made a total of 0 9 simulatio rus. I eah ru we geerated m values of N0, ad m values of Nδ i, radom variates where the meas δ i were hose i two differet ways:. Costat Meas Cofiguratio: δ i = δ i m where δ = 0.5,.0,.5.. Liear Meas Cofiguratio: δ i = iγ i m where the slope γ = δ/m + ad δ = 0.5,.0,.5. The slope for the liear meas ofiguratio is hose so that the average of the δ i sisδ, the same as for the ostat meas ofiguratio ase. Next we trasformed the ormal variates to p-values ad the used these same set of p-values to test H 0 with differet hoies of ostats. The Simes ostats hoie I were used as the basis for ompariso. The simulated powers for the ostat ofiguratio ase are give i Table 5 ad those for the liear meas ofiguratio ase are give i Table 6. The differees i powers of the six hoies, II through VII, of geeralized Simes ostats with respet to the Simes ostats are plotted i Fig. ad i Fig. 3 for ostat ad liear meas ofiguratios, respetively, for the remaiig six hoies of ostats as bar harts with the bars labeled as II VII. The Simes power is oted at the top of eah bar hart. The followig olusios a be draw from these bar harts. C 04 WILEY-VCH Verlag GmbH & Co. KGaA, Weiheim

10 044 J. Gou ad A. C. Tamhae: O geeralized Simes ritial ostats Table 5 Power ompariso ostat meas ofiguratio, = 0,α = m δ Simes Seod order Third order I II III IV V VI VII m = umber of false hypotheses Table 6 Power ompariso liear meas ofiguratio, = 0,α = m δ Simes Seod order Third order I II III IV V VI VII m = umber of false hypotheses. C 04 WILEY-VCH Verlag GmbH & Co. KGaA, Weiheim

11 Biometrial Joural Figure Power gais of seod ad third order geeralized Simes ritial ostats over the first order Simes ritial ostats ostat δ i ofiguratio.. The origial Simes ostats ompare favorably i power with higher order ostats oly whe m = hypotheses are false. This result agrees with that observed by Cai ad Sarkar for seod order ostats.. For eah fixed δ, the power gais of both the seod order ad third order ostats irease as the umber of false ull hypotheses ireases. 3. Third order ostats geerally yield higher powers tha seod order ostats. 4. Maximum power gais by seod order ad third order ostats are attaied at δ =.0. This is atural sie as δ dereases, all powers approah α ad as δ ireases, all powers approah. So the maximum power gais are ahieved at a medium value of δ. 5. Geerally, hoies IV ad V have the highest power gais but they are less powerful tha the Simes ostats whe m = adδ =.5, so they are reommeded i other ases. O the other had, hoies VI ad VII have uiformly high power gais although ot always the highest i all ases, ad are thus robust to ukow umber of false hypotheses, with hoie VII beatig hoie VI i all ases. Thus hoie VII, whih is approximately the etroid of the admissible regio of,, is reommeded. C 04 WILEY-VCH Verlag GmbH & Co. KGaA, Weiheim

12 046 J. Gou ad A. C. Tamhae: O geeralized Simes ritial ostats Figure 3 Power gais of seod ad third order geeralized Simes ritial ostats over the first order Simes ritial ostats liear tred δ i ofiguratio. 7 Coludig remarks We have show how higher order geeralized Simes ostats a be derived ad omputed. The origial Simes ostats ompare favorably i power with higher order geeralizatios oly whe a few hypotheses are false. Whe more hypotheses are false both the seod order ad third order ostats are sigifiatly more powerful with the third order ostats beig more so. Although ot reported here due to spae ostraits, we also made power omparisos betwee Bauer s 989, k, α-test ad the geeralized Simes test ad foud that the latter provides a more powerful test. The details are available from the authors. Assoiated with these higher powers there are also some drawbaks: First, oe eeds to speify for seod order ostats ad, for third order ostats more geerally,,..., k for the kth order ostats for k <. Seod, the kth order ostats require solvig a kth degree polyomial equatio ad hoosig a suitable positive root satisfyig the mootoiity oditio whih is ot easy. All the omparisos i this paper are restrited to the idepedee ase. We have made some prelimiary simulatio studies uder depedee whih suggest that the higher order ostats otrol the type I error uder egative depedee but ot uder positive depedee. This is opposite of the behavior of the Simes ostats Samuel-Cah 996, Sarkar ad Chag 997, Sarkar 998. We C 04 WILEY-VCH Verlag GmbH & Co. KGaA, Weiheim

13 Biometrial Joural have developed a method to robustify the seod ad third order ostats so that they approximately otrol the type I error, while still ahievig substatial power gais over the Simes ostats. We will report these developmets i a separate paper. Fially, oe ould obtai MTPs by applyig the losure method usig these geeralized Simes ostats. These MTPs will ot have a step-up shortut sie the ostats do ot satisfy Wei s 996 oditio; however they will be more powerful tha the Hohberg, Hommel or Rom MTPs. Akowledgmets We thak two referees for their ommets whih helped to improve the paper. Coflitofiterest The authors have delared o oflit of iterest. Appedix Proof of Theorem. The proof is i a umber of parts stated as Results. For ompatess of otatio we will deote the ubi f i i defied i 6 by f x, droppig the subsript i from f i x, q i, r i, ad s i util eeded i the fial part of the proof. The proof ivolves studyig the ritial poits where the derivative f x of f x is zero ad roots of f x where f x = 0. Basially, we show that uder the three oditios stated i the theorem, f x has oly oe positive root ad f 0 0. The possible shapes of f x are show i Fig. A. We wat to rule out the ases a ad b. Result. Whe, the ubi f x has two real ritial poits orrespodig to the four ubi urves show i Fig. A. Proof of Result. The derivative of f x is f x = 3x + q + r = 3x i x ii. The disrimiat of this quadrati is = 4i i + 3i ] = i 3i i 3 ]. The f x = 0 will have two real roots if ad oly if >0 3i >i 3 i 3 > 3i sie i 3 > + 3i. Sie the above must be true for all i = 3,...,, we must have i 3 3 > max + 3 i 3i = + 3. A. C 04 WILEY-VCH Verlag GmbH & Co. KGaA, Weiheim

14 048 J. Gou ad A. C. Tamhae: O geeralized Simes ritial ostats Figure A Differet ases of the ubi futio of f x. Figure A Cubi urves f x with oe positive root ad f 0 0. But 3 > + 3, thus satisfyig the iequality A.. This ompletes the proof of Result. Result. The ubi f x has oe positive ad oe opositive ritial poits if orrespodig to the last three ubi urves i Fig. A. Proof of Result. Let x ad x deote the two roots of the quadrati f x = 0. The we have x + x = 3 i > 0adx x = 6 ii. Sie x + x > 0, at least oe of the roots must be positive. If the x x 0 ad so oe root must be positive ad the other must be opositive. Note that if < the x x > 0 ad so both roots must be positive, a ase that we have exluded. Result 3. If ad s 0 the f x has exatly oe positive root orrespodig to the last two ubi urves i Fig. A. Proof of Result 3. If, the ubi f x has oe opositive ritial poit x ad oe positive ritial poit x suh that x 0 < x. Beause the oeffiiet of x 3 i f x is positive, we kow that x is a loal maximum ad x is a loal miimum. Beause f 0 = s 0 ad the loal miimum x > 0, we kow that f x <0. Sie lim x f x =, by usig the itermediate value theorem, we olude that a positive root x + x, exists. If the ubi equatio f x = 0 has oe real root ad two omplex ojugate roots, it is lear that f x has exatly oe positive root. If the roots of f x = 0 are all real, the roots x, x,adx+ satisfy that x x x < x < x+ as show i Fig. A. Note that x x ad x x+ beause f x <0. C 04 WILEY-VCH Verlag GmbH & Co. KGaA, Weiheim

15 Biometrial Joural Fially, we must have x x 0. That x 0 follows from x 0. If x > 0, the for ay x x, x, f x >0. Sie 0 x, x we have f 0 >0, whih is a otraditio. Hee we olude that x 0. Beause x x 0, we olude that x+ is the oly positive root of f x = 0. The ext result shows uder what oditios is s 0. Result 4. We have that ii i s = α α ] + α 0 if ad oly if + ad α α α α or + α A.. A.3 Proof of Result 4. Note that s 0 α + α 0 A.4 + α 0. A.5 For this quadrati i to have real roots, its disrimiat must be 0. So α 0 α + α 0. The two roots of this quadrati iequality i are α ± Itheabove,wehaveusedthefatthat α ] 4 α 8 α = α 0. = α ± α. A.6 Furthermore, the quadrati is ovex ad symmetri about = /α /. Hee it follows that the iequality A.6 will be satisfied if is either the smaller root or the larger root. C 04 WILEY-VCH Verlag GmbH & Co. KGaA, Weiheim

16 050 J. Gou ad A. C. Tamhae: O geeralized Simes ritial ostats Returig to A.5, we see that must lie iside the iterval ± α. Sie, the limits A. o follow. Result 5. Uder the three oditios stated i Theorem, we have i for i = 3,...,. Proof of Result 5. Defie gx = f x s = x 3 i x ii x. The g = 3 i ii ] ii = ii. Note that whe i 3, ii > 0. Usig the oditio that, we dedue g = 3 ii i < 0, ] ii where the last step follows from the fat that i 3. Sie s < 0 from Result 3, it follows that f = g + s < 0. However, f i = 0. Give that f x has oly oe positive root, amely, x = i, it follows that i > ad this is true for all i = 3,...,. We are ow ready to omplete the proof of Theorem. First we prove a lemma. Lemma. Let h x ad h x be two otiuous futios o the iterval,, with otiuous first derivatives. Suppose that h h <0 ad h x h x for all x,. Further suppose that h x ad h x have uique roots x ad x, respetively. The x x. Proof of Lemma. First we show that h x 0. Write h x h = x h xdx x h xdx = h x h = h h. Hee h x 0. Sie there is a uique x that satisfies h x = 0, it follows that x x. C 04 WILEY-VCH Verlag GmbH & Co. KGaA, Weiheim

17 Biometrial Joural Returig to the proof of the theorem, we kow from Results 4 ad 5 that there is a uique i, suh that f i i = 0. Note that whe i j, so q i q j 0, r i r j 0ads i s j 0, f i = 3 + p i + q i + r i 3 + p j + q j + r j = f j. By usig Result 5, f i <0 for all i = 3,...,, so f i f j <0ifi j. Meawhile, whe x,, f i x = 3x + q i x + r i 3x + q j x + r j = f j x. By usig Lemma, we olude that i j if i j. Therefore the mootoiity oditio 3 holds. Fially, we show that all i α /3.If = = 0, the it is easy hek that = α /3 by substitutig i = i 9. It is also easy to hek that f α /3 = α α /3 + α α /3, ad f α /3 0 beause aordig to Coditio 3. Sie f 0 0, we olude that α /3 for ay hoie of ad by the itermediate value theorem subjet to the three oditios stated i Theorem. Beause the i s are mootoe it follows that all i <α /3. This ompletes the proof of Theorem. Proof of Theorem. For third order ostats, the ubi equatio for i is give by 6 where p i, q i, r i are give by 7. The seod order ase is a speial ase of the third order ase whe is give by 8. First, we eed to show that if α /6 the we have i i /α for i = 3,...,. This is equivalet to showig that i = i /i /α for i = 3,...,. The ubi equatio for i is f i x = x 3 + q i x + r i x + s i = 0, A.7 where q i = q i /i, r i = r i /i,ad s i = s i /i 3. Usig the formulae 7, it is easy to hek that q i, r i,ad s i are ireasig futios of i for i 3. Uder the oditios i Theorem, by followig a similar argumet, we olude that f i x = 0 has a uique positive root i. If i j, the q j q i 0, r j r i 0, ad s j s i 0. We have f j 0 = s j s i = f i 0 0, ad f j x = 3x + q j x + r j 3x + q i x + r i = f i x for ay x 0. By usig the Lemma, we olude that the positive root i j. Beause i j, i order to show that i /α for i 3, we oly eed to show this result for i = 3, that is, that 3 = 3 /α. Note that 3 α f 3 0. α Now, f 3 = α α α 6 α C 04 WILEY-VCH Verlag GmbH & Co. KGaA, Weiheim

18 05 J. Gou ad A. C. Tamhae: O geeralized Simes ritial ostats = = 6 α α + α α α 6 α α , α 3 6α α ] ] ] beause / / whe 3adα /6. We olude that i i /α for ay i = 3,...,. Next, let deote two values of satisfyig 8. Further deote the ubi futio based o, as f i x ad the ubi futio based o, as fi x. The for ay 0 x i /α, wehave fi x fi x = x 3 i x ii x ii i α x 3 + i x + ii x ii i + α ] + α α ] α + α = i ii α x i ii α x = ] i ii α x 0. Note that the uique positive root i of the ubi equatio f i x = 0isi0,i /α ], ad fi x fi x o the rage 0,i /{α }], so we olude that the root i is less tha the root i. Next, let,,..., deote the third order ostats ad,,..., deote the seod order ostats suh that =. Note that give by 8 is the upper limit of the iterval 0, so <. As oted just before Theorem, if we hoose = the the third order ostats are the same as the seod order ostats, that is, i = i for i =,...,. But, sie <, ad sie the i are dereasig futios of, it follows that i > i for i = 3,...,. C 04 WILEY-VCH Verlag GmbH & Co. KGaA, Weiheim

19 Biometrial Joural Proof of Theorem 3. First ote that A i = αδi, A i, wehave α = A 0 = = γ + α A i = α i= δi, A i i= β i A i = γ + α β ia i. i= Now assume the idutio hypothesis that α = γ k + α k k i=0 β k ia k i. The we have k α = γ k + α k β k ia k i i=0 k = γ k + α k β k 0A k 0 + α k β k ia k i k k = γ k + α k β k 0 A k i + α k β k ia k i i= i= i= i=0 k = γ k + α k β k 0 + α k β k i β k 0A k i i= k = γ k+ + α k+ β k i β k 0δi, ka k i i= k = γ k+ + α k+ β k i + β k 0δi +, ka k i i=0 k = γ k+ + α k+ i=0 β k+ ia k i. Thus assumig the idutio hypothesis for k, we have show it to be true for k +. Seod, by settig β k i = β k as a ostat ad otig that k i=0 A k i = yields β k = α γ k α k. Substitutig for β k we obtai the polyomial equatio for k+i β k i + α γ k α k = 0. It is lear that this polyomial equatio has degree k by usig idutio o k to hek the degree of β k i. C 04 WILEY-VCH Verlag GmbH & Co. KGaA, Weiheim

20 054 J. Gou ad A. C. Tamhae: O geeralized Simes ritial ostats Referees Bauer, P Sequetial tests of hypotheses i oseutive trials. Biometrial Joural 3, Berhard, G., Klei, M. ad Hommel, M Global ad multiple test proedures usig ordered p-values a review. Statistial Papers 45, 4. Cai, G. ad Sarkar, S. K Modified Simes ritial values uder idepedee. Statistis & Probability Letters 78, Dalal, S. R. ad Mallows, C. L. 99. Buyig with exat ofidee. Aals of Applied Probability, Fier, H. ad Roters, M O the limit behavior of the joit distributio of order statistis. Aals of Istitute of Statistial Mathematis 46, Hohberg, Y A sharper Boferroi proedure for multiple sigifiae testig. Biometrika 75, Holm, S A simple sequetially rejetive multiple test proedure. Sadiavia Joural of Statistis 6, Hommel, G Tests of the overall hypothesis for arbitrary depedee strutures. Biometrial Joural 5, Hommel, G A stagewise rejetive multiple test proedure based o a modified Boferroi test. Biometrika 75, Liu, W Multiple tests of a o-hierarhial fiite family of hypotheses. Joural of Royal Statistial Soiety, Ser. B 58, Marus, R., Peritz, E. ad Gabriel, K. R O losed testig proedures with speial referee to ordered aalysis of variae. Biometrika 63, Rom, D. M A sequetially rejetive test proedure based o a modified Boferroi iequality. Biometrika 77, Rüger, B Das maximale Sigifikaziveau des Tests: Lehe H 0 ab, we k uter gegebee Tests zur Ablehug führe. Metrika 5, Samuel-Cah, E Is the Simes improved Boferroi proedure oservative? Biometrika 83, Sarkar, S Some probability iequalities for ordered MTP radom variables: A proof of the Simes ojeture. Aals of Statistis 6, Sarkar, S. ad Chag, C. K Simes method for multiple hypothesis testig with positively depedet test statistis. Joural of the Ameria Statistial Assoiatio 9, Simes, R. J A improved Boferroi proedure for multiple tests of sigifiae. Biometrika 73, C 04 WILEY-VCH Verlag GmbH & Co. KGaA, Weiheim