MAXIMIN POWER DESIGNS IN TESTING LACK OF FIT Douglas P. Wiens 1

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1 MAXIMIN POWER DEIGN IN TETING LACK OF FIT Douglas P. Wens Absrac We nd desgns wc maxmze e mnmum power, over a broad class of alernaves, of e es for Lack of F, n dscree desgn spaces. Ts complemens prevous resuls n connuous spaces, bu s less conroversal. AM 2 ubjec Class caons: Prmary 62K5; econdary 62F35, 62J5. Key words and prases Bas, F-es, ne desgn space, maxmn, mnmax, power. Inroducon In Wens (99), encefor referred o as [W], we suded e unform desgn, as appled o desgn spaces a are subses of R q nervals, ypercubes, ec. We call suc desgn spaces dense, o dsngus em from e ne, dscree desgn spaces consdered n s arcle. Te unform desgn on s e absoluely connuous measure, w consan densy R dx. Of course suc a desgn mus be approxmaed n order o mplemen n an acual expermen. A conrbuon of [W] was a, n a sense made precse ere, e unform desgn possesses an opmaly propery n e class of all desgns on maxmzes e mnmum power of e sandard F-es for Lack of F (lof) of a ed lnear regresson model, w e mnmum aken over a broad class of alernaves. Te eory n [W] as been adaped o jusfy e use of dscree unform desgns n numerous applcaons n e scences. For s applcaon o drug combnaon sudes see e seres of papers Tan, Fang, Tan and Hougon (2), Fang, Ross, ausvlle and Tan (28), Tan, Fang and Tan (29), Fang, Tan, L and Tan (29) and Fang, Cen, Pe, Gran and Tan (25). Te deas n [W] ave ganed racon n e eory of ar cal neural neworks see Zang, Lang, Jang, Yu and Fang (998) and reduced suppor vecor macnes see Lee and Huang (27). Te eory as been exended o nonparamerc regresson models Xe and Fang (2) and, also allowng for eeroscedascy, by Bedermann and Dee (2) and Bsco and Mller (26). Te dense naure of n s conex as been conroversal. Indeed Bsco (2) argues a allows for classes of alernave regresson models as used bo n [W] and n Bedermann and Dee (2) a are oo broad for e opmaly propery o be asympocally meanngful (wen e connuous unform Deparmen of Maemacal and ascal cences; Unversy of Albera, Edmonon, Albera; Canada T6G 2G. e-mal: doug.wens@ualbera.ca

2 2 Douglas P. Wens desgn s vewed as e lm of dscree unform desgns); e proposes a resrced nerpreaon. Ta e rcness of classes of alernaves as n [W] makes dscree desgns nadmssble was noed n Wens (992, p. 355), were we sae Our aude s a an approxmaon o a desgn wc s robus agans more realsc alernaves s preferable o an exac soluon n a negbourood wc s unrealscally sparse. Ts remans our vew. Noneeless, n s arcle we sugges an alernae approac a we feel s less conroversal. We ake a ne desgn space fx ; :::; x N g ere N can be arbrarly large, allowng for a leas a close approxmaon of e space of neres n an ancpaed applcaon. We oban exac desgns for small values of N ese are non-unform and nd a for only moderaely large values of N e desgns are, o e lms of numercal accuracy, unform on. In e nex secon we oulne e maemacal framework, and provde a reducon of e maxmn problem o a smpler mnmax problem. ome soluons are gven n 3. Proofs are n e Appendx. Te compung code s avalable from e auor s personal web se. 2 Prelmnares As far as possble we use noaon as n [W], o wc we refer e reader for background maeral relang e sandard F-es for lack of o properes of e desgn. We denoe by e unform probably measure on, vz. (x ) N; ; :::; N. For a desgn on we wre (x ). An mplemenable desgn w n observaons requres a n be an neger; we sall loosen s resrcon and allow o be any probably dsrbuon on. 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 + ; () (e full models, n lof ermnology) from e ed ( reduced ) regresson model E [Y (x)] z (x) : (2) Te class F + s as a (), () on p. 28 of [W], wren ere n a manner a re ecs e dscreeness of e desgn space: Z Z f 2 (x) d (x) f 2 (x ) 2 ; (3a) N z (x) f (x) d (x) N z (x ) f (x ) p : (3b)

3 Maxmn Power Desgns n Tesng Lack of F 3 Condon (3a) enforces a separaon beween e ed and alernae models, so a e es as posve power, and (3b) ensures e den ably of e regresson parameers under (), va def arg mn (E [Y (x )] z (x ) ) 2 : Ts de nes unquely, n e presence of (3b) and e requremen, made ere, a e 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 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 a B s non-sngular. Ten as a (2.2) of [W] e non-cenraly parameer (ncp) of e F-sasc for esng e lof of e ed model (2), w alernaves of e form (), and usng a desgn, s proporonal o B (f; ) f D f b f;b b f; : Te power of e es s an ncreasng funcon of e non-cenraly parameer, as long as e F-sasc s socascally ncreasng n s parameer. Ts monooncy s well known o old n ne samples under a Gaussan error dsrbuon, and s a leas asympocally vald oerwse, under mld condons. In s alernae form B (f; ) P N f (x ) z (x ) B 2 b f;, we see a B (f; ) s e L 2 () dsance from f o e neares funcon of e form (2). Tus mn f B (f; ) s a naural measure of e dscrepancy beween e full and reduced models beng compared, w larger values leadng o a more e ecve es. I s reasonable o nk a e followng conjecure, exendng a resul of [W] o dscree desgn spaces, s vald. Conjecure For any desgn on we ave a mn F + B (f; ) 2 mn F + B (f; ), so a e unform desgn maxmzes e mnmum power of e F-es of lof. Despe s plausbly s conjecure urns ou o false a rs counerexample s furnsed n Example below. To express e conjecure more explcly, we rs caracerze e classes F + paramercally. Va e qr-decomposon we can consruc a marx Q : N p w muually orogonal columns formng a bass for col (Z), e column space of Z. Consruc as well : N (N p) wose columns form an orogonal bass for e orogonal complemen col (Z)? col (Q )? ; us Q NN [Q. ]

4 4 Douglas P. Wens s orogonal. Condon (3b) requres f o le n col ( ): f p N d for some d N p, and en (3a) requres kdk for f 2 F +. W we nd a P def p pnd ND 2 2 Q Q. p ND 2 H def P (P P ) P D 2 Q (Q D Q ) Q D 2 ; Te conjecure us asks a we solve max mn kdk B (f; ) 2 d P 2 (I N H) P 2 d: def B (f; ) 2 max c mn [P 2 (I N H) P 2 ] ; and sow a e soluon s. Te problem s grealy smpl ed by e followng lemma. P.P 2 ; Lemma Te followng are equvalen: (a) A desgn s maxmn w respec o e power of e es of lof, n a arg max c mn [P 2 (I N H) P 2 ] : (b) A desgn places posve mass a eac pon of and s a mnmax desgn, n a arg mn c max Q 2 D : I s broug ou n e proof of Lemma a f e desgn s no suppored on all of en ere wll be deparures n F + for wc e ncp s zero. Agans ese e e power of e es s no greaer an e sze. Ts s a re econ of e rcness of F + ; an neresng open problem s o nd a smaller bu sll realsc class n wc s d culy s avoded. Te followng example sows a e soluon s no, n general, e unform desgn, and ses e sage for e soluon presened n e nex secon. Example. uppose N 2, p, (; ) w and ; >. Ten ere s only one egenvalue, gven by Q 2D : ( ) Ts s mnmzed by ; ( + ) ;2, w Q 2D ( + ) Ts mproves on, for wc Q 2D for all ;, w src nequaly unless p 2, n wc case.

5 Maxmn Power Desgns n Tesng Lack of F 5 3 Mnmax desgns We are o mnmze L () def c max Q 2 D over e se of desgns a place posve mass on eac pon of. Ts se s no closed, and s poses ecncal d cules wc wll become evden. Tus we sall rs mnmze nsead over e closed, convex se " of desgns a place mass of a leas " > on eac pon of. In mos cases urns ou a e mnmax desgn les n e neror of s se, so a e resrcon o " s moo and e soluon olds for all of. Wen e maxmum egenvalue of Q 2D s smple, e mnmax desgn as a paramerc form. Teorem Denoe by q ; :::; q N e rows of. For a posve consan and a vecor x N p de ne a desgn by jq ; (; x) max xj p ; " : (4) If and x sasfy ; (; x) ; (5) x egenvecor belongng o c max Q 2D (;x) ; (6) and f s maxmum egenvalue s smple, en mnmzes c max Q 2 D, ence maxmzes mn f B (f; ), n ". Remark: If, wen " we ave a q x 6 for all, en as poned ou above we may ake " and e soluon n s ; jq x ( )j p. In s case (5) becomes p P N jq xj, and (6) s L ( (x)) x Mx P N (q x)2 ; p P N jq xj. Example connued. In s example we may ake ". W q, q 2, we ave x, (jj + jj) 2, and ; jj (jj + jj) ;2, exendng e soluon obaned earler, wen we ook ; >. Example 2. Here we consder cubc regresson (p 4) w N 7 and desgn space f :424; :522; :269; :3358; :445; :4594; :5628g. Ten e regressors are z (x) (; x; x 2 ; x 3 ). We mplemen Teorem by usng parcle swarm opmzaon (pso) n malab s brngs one que close o e nal values followed by a consraned nonlnear mnmzer. Te mnmax desgn s found o be f:86; :356; :9; :77; :78; :863; :36g :

6 6 Douglas P. Wens N N N N Fgure : Maxmn lof desgns for esng e of a quadrac model (p 3); f + ( )(N )j ; :::; Ng; N 4; 6; 8; N N N N Fgure 2: Maxmn lof desgns for esng e of a cubc model (p 4); f + ( )(N )j ; :::; Ng; N 5; 7; 9;. Te egenvalues of Q 2D are f5:9799; 5:697; 5:54g, so a Teorem apples (w " ). Te maxmzng egenvecor (6) s x (:4223; :334; :8437), and L ( ) 5:9799. I urns ou o be que rare for Teorem o old n mos cases e maxmum egenvalue s no smple. Ts s llusraed n e followng example. Example 3. Take N 3, p, Q I 3, so a " > and [.I 2 ]. Ten Q 2D dag ( 2 ; 3 ), and L () mn ( 2 ; 3 ). Te problem of maxmzng e mnmum of ( 2 ; 3 ) subjec o " leads o ;2 ;3 ( ") 2, so a e maxmum egenvalue s no smple and Teorem does no apply. Wen Teorem does no apply we mnmze c max Q 2 D drecly over ( ; :::; N ), subjec o e consran a ese form a probably dsrbuon, usng pso followed by a consraned nonlnear mnmzer. Wle s mg no be feasble for very large desgn spaces, we ave found a for all bu very small values of N e desgns are, o e lms of numercal accuracy, unform on. Wle asympoc unformy s o be expeced n lg of e resul n [W], we were surprsed a e speed w wc s occurred. ee Fgures and 2 for llusraons.

7 Maxmn Power Desgns n Tesng Lack of F 7 4 Appendx: Proofs Proof of Lemma : Noe a P 2 (I N H) P 2 N nd D Q (Q D Q ) Q D o ; (A.) and a Q D Q Q Q 2 D Q Q D Q Q D Q 2D Q Q 2D ; so a jd j jq D Qj jq D Q j N Q 2D Q 2D Q (Q D Q ) Q D N p jq D Q j jp 2 (I N H) P 2 j ; (A.2) usng (A.). By assumpon e egenvalues of Q D Q are posve, so a f (and only f) all are srcly posve en c mn [P 2 (I N H) P 2 ] >. Tus a maxmn desgn places posve mass on every pon n oerwse s beaen by suc a desgn. We en ave a (Q D Q) Q D Q! P 2 (I N H)P 2 ; so a P 2 (I N H) P 2 N Q 2D. Tus c mn [P 2 (I N H) P 2 ] Nc mn N Q 2D Proof of Teorem : For a xed desgn pu ( 2 " and 2 [; ]. For a Lagrange mulpler pu F (; ) c max Q 2D + D : N c max Q 2 D : ) + for any oer Noe a F (; ) s convex n : we ave a D ( ) D + D, so a c max Q 2D c max Q 2 ( ) D + D c max Q 2 ( ) D + c max Q 2 D ( ) c max Q 2D + c max Q 2D :

8 8 Douglas P. Wens In order a be e mnmzng desgn su ces a F (; ) be mnmzed uncondonally a and sasfy e sde condon. (Ta all " s obaned wou e use of a mulpler.) A necessary and su cen condon for a mnmum a s a (dd) F (; ) j, for all 2 ". W ( ) and x x ( ) beng e maxmum egenvalue (assumed smple) and correspondng egenvecor of un norm of Q 2D we ave, usng Teorem of Magnus (985), a wence d d F (; ) j d d ( ) j x ( ) d Q d 2D x ( ) ; j x ( ) Q d 2 d D x ( ) + d d D j x ( ) Q 2D D D D x ( ) + D D " # q 2 ; x ( ) : ; Ts olds for all 2 " f (and only f) s of e form (4), w and x deermned by (5) and (6). ; Acknowledgemens Ts work was carred ou w e suppor of e Naural cences and Engneerng Researc Councl of Canada. We are graeful for e ncsve commens of e anonymous referees. References Bedermann,., Dee, H. (2), Opmal desgns for esng e funconal form of a regresson va nonparamerc esmaon ecnques, ascs and Probably Leers, 52, Bsco, W. (2), An mprovemen n e lack-of- opmaly of e (absoluely) connuous unform desgn n respec of exac desgns, moda, Bsco, W., Mller, F. (26), Opmal desgns wc are e cen for lack of ess, Te Annals of ascs, 34, Fang, H. B., Cen, X., Pe, X. Y., Gran,., Tan, M. (25), Expermenal desgn and sascal analyss for ree-drug combnaon sudes, ascal Meods n Medcal Researc,

9 Maxmn Power Desgns n Tesng Lack of F 9 Fang, H. B., Ross, D. D., ausvlle, E., Tan, M. (28), Expermenal desgn and neracon analyss of combnaon sudes of drugs w log-lnear dose responses, ascs 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 Boparmaceucal ascs, 9, Fang, K. T., L, R., udjano, A. (25), Desgn and Modelng for Compuer Expermens, CRC Press. Lee, Y. J., Huang,. Y. (27), Reduced suppor vecor macnes: A sascal eory, IEEE Transacons on Neural Neworks, 8, -3. Magnus, J. R.: On D erenang Egenvalues and Egenvecors. Economerc Teory, 79-9 (985). Tan, M., Fang, H. B., Tan, G. L., Hougon, P. J. (23), Expermenal desgn and sample sze deermnaon for esng synergsm n drug combnaon sudes based on unform measures, ascs n Medcne, 22, Tan, M. T., Fang, H. B., Tan, G. L. (29), Dose and sample sze deermnaon for mul-drug combnaon sudes, ascs n Boparmaceucal Researc,, Wens, D. P. (99), Desgns for approxmaely lnear regresson: wo opmaly properes of unform desgns, ascs and Probably Leers, 2, Wens, D. P. (992), Mnmax desgns for approxmaely lnear regresson, Journal of ascal Plannng and Inference, 3, Xe, M. Y., Fang, K. T. (2), Admssbly and mnmaxy of e unform desgn measure n nonparamerc regresson model, Journal of ascal Plannng and Inference, 83, -. Zang, L., Lang, Y. Z., Jang, J. H., Yu, R. Q., Fang, K. T. (998), Unform desgn appled o nonlnear mulvarae calbraon by ANN, Analyca Cmca Aca, 37,