MAXIMIN POWER DESIGNS IN TESTING LACK OF FIT Douglas P. Wiens 1. July 30, 2018

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1 MAXIMIN POWER DESIGNS IN TESTING LACK OF FIT Douglas P. Wens July 3, 28 Absrac In a prevous arcle (Wens, 99) we esablshed a maxmn propery, wh respec o he power of he es for Lack of F, of he absoluely connuous unform desgn on a desgn space whch s a subse of R q wh posve Lebesgue measure. Here we dscuss some ssues and conroverses surroundng hs resul. We nd desgns whch maxmze he mnmum power, over a broad class of alernaves, n dscree desgn spaces of cardnaly N. We show ha hese desgns are suppored on he enre desgn space. They are n general no unform for xed N, bu are asympocally unform as N!. Several examples wh N xed are dscussed n hese we nd ha he approach o unformy s very quck. AMS 2 Subjec Class caons: Prmary 62K5 Secondary 62F35, 62J5. Key words and phrases F-es, ne desgn space, maxmn, mnmax, noncenraly, power. Inroducon In Wens (99), henceforh referred o as [W], we suded he unform desgn, as appled o desgn spaces S c ha are subses of R q nervals, hypercubes, ec. wh posve Lebesgue measure. We call such desgn spaces connuous, o dsngush hem from he ne, dscree desgn spaces consdered n hs arcle. The unform desgn on S c s he absoluely connuous measure, wh consan densy R S c dx. Of course such a desgn mus be approxmaed n order o mplemen n an acual expermen. A conrbuon of [W] was ha, n a sense made precse here and dealed n 2 below, he unform desgn possesses an opmaly propery n he class of all desgns on S c maxmzes he mnmum power of he sandard F-es for Lack of F (lof) of a ed lnear regresson model, wh he mnmum aken over a broad class of alernaves. The heory n [W] has been adaped o jusfy he use of dscree unform desgns n numerous applcaons n he scences. For s applcaon o drug combnaon sudes see he seres of papers Tan, Fang, Tan and Houghon (23), Fang, Ross, Sausvlle and Tan (28), Tan, Fang and Tan (29), Fang, Tan, L and Tan (29) and Fang, Chen, Pe, Gran and Tan (25). The deas n [W] have ganed racon n he heory of ar cal neural neworks see Zhang, Lang, Jang, Yu and Fang (998) and reduced suppor vecor machnes see Lee and Huang (27). Deparmen of Mahemacal and Sascal Scences Unversy of Albera, Edmonon, Albera Canada T6G 2G. e-mal: doug.wens@ualbera.ca

2 2 Douglas P. Wens The heory has been exended o nonparamerc regresson models Xe and Fang (2) and, also allowng for heeroscedascy, by Bedermann and Dee (2) and Bscho and Mller (26). The connuous naure of S c n hs conex has been conroversal. Indeed Bscho (2) argues ha allows for classes of alernave regresson models as used boh n [W] and n Bedermann and Dee (2) ha are oo broad for he opmaly propery o be asympocally meanngful (when he connuous unform desgn s vewed as he lm of dscree unform desgns) he proposes a resrced nerpreaon. Tha he rchness of classes of alernaves as n [W] makes dscree desgns nadmssble was noed n Wens (992, p. 355), where we sae Our aude s ha an approxmaon o a desgn whch s robus agans more realsc alernaves s preferable o an exac soluon n a neghbourhood whch s unrealscally sparse. Ths remans our vew. Noneheless, n hs arcle we sugges an alernae approach ha we feel s less conroversal. We ake a ne desgn space S fx ::: x N g here N can be arbrarly large, allowng for a leas a close approxmaon of he space of neres n an ancpaed applcaon. We oban exac desgns for small values of N hese are non-unform and show ha he maxmn desgns are asympocally unform, as N!. Theory and examples show ha hs lm s approached very quckly. In he nex secon we oulne he mahemacal framework, provde a reducon of he maxmn problem o a smpler mnmax problem, and prove he asympoc opmaly of he unform desgn. Some soluons wh N xed are gven n 3. Proofs are n he Appendx. The compung code s avalable from he auhor s personal web se. 2 Prelmnares As far as possble we use noaon as n [W], o whch we refer he reader for background maeral relang he sandard F-es for lack of o properes of he desgn. We denoe by he unform probably measure on S, vz. (x ) N ::: N. To faclae comparsons wh [W] we now wre c (x) for he connuous unform desgn on a connuous desgn space S c. For a desgn on S we wre (x ). An mplemenable desgn wh n observaons requres ha n be an neger we shall loosen hs resrcon and allow o be any probably dsrbuon on S. In parcular, we nclude as a possble desgn. For p-dmensonal regressors z (x) we eneran a class of deparures E [Y (x)] z (x) + f (x) f 2 F + () (he full models, n lof ermnology) from he ed ( reduced ) regresson model E [Y (x)] z (x) : (2)

3 Maxmn Power Desgns n Tesng Lack of F 3 In [W] we de ned hs class of funcons on S c by () R S c f 2 (x) d c (x) 2 and () R S c z (x) f (x) d c (x) p. We now adop an analogous de non of F + on he dscree desgn space S, vz., Z Z S f 2 (x) d (x) N S z (x) f (x) d (x) N f 2 (x ) 2 z (x ) f (x ) p : (3a) (3b) Condon (3a) enforces a separaon beween he ed and alernae models, so ha he es has posve power, and (3b) ensures he den ably of he regresson parameers under (), va def arg mn (E [Y (x )] z (x ) ) 2 : Ths de nes unquely, n he presence of (3b) and he requremen, made here, ha he marx Z Np [z (x )..z (x N )] be of full column rank. We wre f f (x ) and de ne f (f ::: f N ) and D dag ( ::: N ). De ne as well b f B Z Z S S z (x) f (x) d (x) z (x) z (x) d (x) z (x ) f (x ) Z D f z (x ) z (x ) Z D Z and assume ha B s non-sngular. Then as a (2.2) of [W] he non-cenraly parameer (ncp) of he F-sasc for esng he lof of he ed model (2), wh alernaves of he form (), and usng a desgn, s proporonal o B (f ) f D f b fb b f : The power of he es s an ncreasng funcon of he ncp, as long as he F-sasc s sochascally ncreasng n hs parameer. Ths monooncy s well known o hold n ne samples under a Gaussan error dsrbuon, and s a leas asympocally vald oherwse, under mld condons. In s alernae form B (f ) f (x ) z (x ) B b f 2 we see ha B (f ) s he L 2 () dsance from f o he neares funcon of he form (2). Thus mn f B (f ) s a naural measure of he dscrepancy beween he full

4 4 Douglas P. Wens and reduced models beng compared, wh larger values leadng o a more e ecve es. I s reasonable o hnk ha he followng conjecure should hold. If rue, exends he man resul of [W] o dscree desgn spaces. Conjecure For any desgn on S we have ha mn F + B (f ) 2 mn F + B (f ), so ha he unform desgn maxmzes he mnmum power of he F-es of lof. Despe s plausbly hs conjecure urns ou o be false a rs counerexample s furnshed n Example below. To express he conjecure more explcly, we rs characerze he classes F + paramercally. Wre he qr-decomposon of Z as Z. R where he columns of : N p form an orhogonal bass for col (Z), he column space of Z, and he columns of : N (N p) form an orhogonal bass for he orhogonal complemen col (Z)? col ( )?. Then def. s an orhogonal marx. Condon (3b) requres f o le n col ( ),.e. f p N d for some d N p, and hen (3a) requres kdk for f 2 F +. Wh we nd ha P def p pnd ND 2 2. p ND 2 H def P (P P ) P D 2 ( D ) D 2 B (f ) 2 d P 2 (I N H) P 2 d: def P.P 2 We denoe by he se of all desgns on S, and by ch mn and ch max he mnmum and maxmum egenvalues of a marx. The conjecure hen asks ha we solve max 2 mn kdk B (f ) 2 max 2 ch mn [P 2 (I N H) P 2 ] and show ha he soluon s. The problem s grealy smpl ed by he followng heorem. Theorem The followng are equvalen: (a) A desgn s maxmn wh respec o he power of he es of lof, n ha arg max 2 ch mn [P 2 (I N H) P 2 ] : (b) A desgn places posve mass a each pon of S and s a mnmax desgn whn he se + of all such desgns, n ha arg mn 2 + ch max 2 D : If 2 + hen ch mn [P 2 (I N H) P 2 ] Nch max 2 D.

5 Maxmn Power Desgns n Tesng Lack of F 5 To oban he requred maxmn/mnmax desgn we are o mnmze L () def ch max 2 D over he se + of desgns ha place posve mass on each pon of S. Noe ha L () N he followng heorem shows ha we canno expec much mprovemen on hs, for large N. Theorem 2 For any desgn 2 +, L () N p, so ha p N mn 2 + L () L () : In a sense made precse by Theorem 2 he unform desgn s asympocally opmal, as N!. The followng example shows ha hs opmaly does no n general hold for ne N. Example. Suppose N 2, p, ( ) wh and >. Then here s only one egenvalue, gven by 2D : ( ) Ths s mnmzed by ( + ) 2, wh L ( ) ( + ) Ths mproves on, for whch L () for all. There s src nequaly unless p 2, n whch case. 3 Mnmax desgns for xed N The se + s no closed, and hs poses echncal d cules whch wll become evden. Thus we shall rs mnmze nsead over he closed, convex se " of desgns ha place mass of a leas " > on each pon of S. In mos cases urns ou ha he mnmax desgn les n he neror of hs se, so ha he resrcon o " s moo and he soluon holds for all of +. When he maxmum egenvalue of 2D s smple, he mnmax desgn has a paramerc form. Theorem 3 Denoe by q ::: q N he rows of. For a posve consan and a vecor x N p de ne a desgn by jq ( x) max xj p " : (4)

6 6 Douglas P. Wens () If and x sasfy ( x) (5) h x egenvecor belongng o ch max 2D (x) (6) and f hs maxmum egenvalue s smple, hen mnmzes ch max 2 D n ". () If q x 6 for all, hen we may ake " n (4) and he soluon n + s jq xj p wh p P N jq xj and L ( ). Example connued. In hs example Theorem 3() apples. Wh q, q 2, we have x, (jj + jj) 2, and jj (jj + jj) 2, exendng he soluon obaned earler, when we ook >. Example 2. Here we consder cubc regresson (p 4) wh N 7 and (randomly generaed) desgn space S f :424 :522 :269 :3358 :445 :4594 :5628g : Then he regressors are z (x) ( x x 2 x 3 ). We mplemen Theorem 3 va a consraned nonlnear mnmzer n malab. The mnmax desgn s found o be f:86 :356 :9 :77 :78 :863 :36g : The egenvalues of 2D are f5:9799 5:697 5:54g, so ha Theorem 3() apples we hen check numercally ha () does as well. The maxmzng egenvecor (6) s x (:4223 :334 :8437), and L ( ) 5:9799. I urns ou o be que rare for Theorem 3 o hold n mos cases he maxmum egenvalue s no smple. Ths s of course expeced asympocally, snce all egenvalues are hen equal, and s llusraed for xed N n he followng example. Example 3. Take N 3, p, I 3, so ha " > and [.I 2 ]. Then 2D dag ( 2 3 ), and L () mn ( 2 3 ). The problem of maxmzng he mnmum of ( 2 3 ) subjec o " leads o 2 3 ( ") 2, so ha he maxmum egenvalue s no smple and Theorem 3 does no apply. When Theorem 3 does no apply we mnmze L () drecly over +, usng randomly generaed desgns on S as sarng values, whch are hen used n a consraned nonlnear mnmzer. Whle hs mgh no be feasble for very large desgn spaces, we have found ha he lmng behavour mpled by Theorem 2 s approached very quckly - for all bu very small values of N he desgns are, o he lms of numercal accuracy, unform on S. See Fgures and 2 for llusraons.

7 Maxmn Power Desgns n Tesng Lack of F N N N N Fgure : Maxmn lof desgns for esng he of a quadrac model (p 3) S f + ( )(N )j ::: Ng N N N N N Fgure 2: Maxmn lof desgns for esng he of a cubc model (p 4) S f + ( )(N )j ::: Ng N An open problem Our resuls have poned ou he lmaons of any desgn, suppored on a proper subse of S, n provdng robusness agans alernaves n F +. Indeed, s brough ou n he proof of Theorem ha f he desgn s no suppored on all of S hen here wll be deparures n F + for whch he ncp s zero. Such deparures may be pahologcal and nconsequenal, bu agans hem he power of he es s no greaer han he sze. Ths s a re econ of he rchness of F + an neresng open problem s o nd a smaller bu sll realsc class n whch hs d culy s avoded. 4 Appendx: Proofs Proof of Theorem : Noe ha P 2 (I N H) P 2 N nd D ( D ) D o (A.)

8 8 Douglas P. Wens and ha D 2 D D D 2D 2D so ha jd j j D j j D j N 2D N p j D j jp 2 (I N H) P 2 j 2D ( D ) D usng (A.). Noe ha P 2 (I N H) P 2, snce I N H s dempoen, hence nonnegave de ne. The egenvalues of D, assumed non-sngular, are posve. Hence f, and only f, all are srcly posve s jp 2 (I N H) P 2 j >, equvalenly ch mn [P 2 (I N H) P 2 ] >. Thus a maxmn desgn places posve mass on every pon n S oherwse s beaen by such a desgn. We hen have ha ( D ) D! P 2 (I N H)P 2 so ha P 2 (I N H) P 2 N 2D. Thus ch mn [P 2 (I N H) P 2 ] Nch mn h N 2D N ch max 2 D : Proof of Theorem 2: De ne K 2, an dempoen marx whose dagonal elemens fk j ::: Ng le n [ ] and sum o rk ( ) N p. Snce he average egenvalue of a posve de ne marx canno exceed he maxmum, we have ha L () N p r 2D N p r D K k N p : We now vew fk (N p)g as a probably dsrbuon and apply Jensen s Inequaly o he convex funcon f () o oban k N p P N k N p P N N p N p:

9 Maxmn Power Desgns n Tesng Lack of F 9 Proof of Theorem 3: () Pu ( undeermned mulpler pu ) + for any and 2 [ ]. For an F ( ) ch max h 2D + D : Snce D for desgns, su ces o show ha F ( ) s mnmzed uncondonally a for xed and any 2 ", and ha he sde condons ensurng ha 2 " are sas ed. (Tha all " s obaned whou he use of a mulpler.) Noe ha F ( ) s convex n : we have ha D ( ) D + D by he convexy of marx nverson, so ha h ch max 2D ch max h 2 ( ) D + D ch max h 2 ( ) D + ch max h 2 D h h ( ) ch max 2D + ch max 2D : A necessary and su cen condon for a mnmum a s hen ha (dd) F ( ) j hs mus hold for all 2 ". Wh ( ) and x x ( ) beng he maxmum egenvalue (assumed smple) and correspondng egenvecor of un norm of 2D we have, usng Theorem of Magnus (985), ha whence d d F ( ) j d d ( ) j x ( ) d h d 2D x ( ) j x ( ) d 2 d h D x ( ) + d d D j x ( ) 2D D D D x ( ) + D D " # q 2 x ( ) : If (5) and (6) hold hen 2 " and for any 2 " he nal lne above s d d F ( ) j X " " # q " 2 2 x ( ) p " 2 " snce jq x ( )j p " when ", and he summands vansh when jq x ( )j p > ". () If we have ha q x 6 for all, hen he mnmzng desgn les n he neror of + and so s he soluon n " for any " mn :::N jq xj, and hence for all

10 Douglas P. Wens of +. In hs case (4) and (5) become jq xj p, wh p P N jq xj, and hen (6) yelds L ( ) x 2D x (q x) 2 p jq xj : Acknowledgemens Ths work was carred ou wh he suppor of he Naural Scences and Engneerng Research Councl of Canada. We hank he wo anonymous referees for her ncsve commens. References Bedermann, S., Dee, H. (2), Opmal desgns for esng he funconal form of a regresson va nonparamerc esmaon echnques, Sascs and Probably Leers, 52, Bscho, W. (2), An mprovemen n he lack-of- opmaly of he (absoluely) connuous unform desgn n respec of exac desgns, moda, Bscho, W., Mller, F. (26), Opmal desgns whch are e cen for lack of ess, The Annals of Sascs, 34, Fang, H. B., Chen, X., Pe, X. Y., Gran, S., Tan, M. (25), Expermenal desgn and sascal analyss for hree-drug combnaon sudes, Sascal Mehods n Medcal Research, Fang, H. B., Ross, D. D., Sausvlle, E., Tan, M. (28), Expermenal desgn and neracon analyss of combnaon sudes of drugs wh log-lnear dose responses, Sascs n Medcne, 27, Fang, H. B., Tan, G. L., L, W., Tan, M. (29), Desgn and sample sze for evaluang combnaons of drugs of lnear and loglnear dose-response curves, Journal of Bopharmaceucal Sascs, 9, Lee, Y. J., Huang, S. Y. (27), Reduced suppor vecor machnes: A sascal heory, IEEE Transacons on Neural Neworks, 8, -3. Magnus, J. R.: On D erenang Egenvalues and Egenvecors. Economerc Theory, 79-9 (985).

11 Maxmn Power Desgns n Tesng Lack of F Tan, M., Fang, H. B., Tan, G. L., Houghon, P. J. (23), Expermenal desgn and sample sze deermnaon for esng synergsm n drug combnaon sudes based on unform measures, Sascs n Medcne, 22, Tan, M. T., Fang, H. B., Tan, G. L. (29), Dose and sample sze deermnaon for mul-drug combnaon sudes, Sascs n Bopharmaceucal Research,, Wens, D. P. (99), Desgns for approxmaely lnear regresson: wo opmaly properes of unform desgns, Sascs and Probably Leers, 2, Wens, D. P. (992), Mnmax desgns for approxmaely lnear regresson, Journal of Sascal Plannng and Inference, 3, Xe, M. Y., Fang, K. T. (2), Admssbly and mnmaxy of he unform desgn measure n nonparamerc regresson model, Journal of Sascal Plannng and Inference, 83, -. Zhang, L., Lang, Y. Z., Jang, J. H., Yu, R.., Fang, K. T. (998), Unform desgn appled o nonlnear mulvarae calbraon by ANN, Analyca Chmca Aca, 37,