APPROXIMATE THEORY-AIDED ROBUST EFFICIENT FACTORIAL FRACTIONS UNDER BASELINE PARAMETRIZATION

Transcription

1 APPROXIMAE HEORY-AIDED ROBUS EFFICIEN FACORIAL FRACIONS UNDER BASELINE PARAMERIZAION Rahul Muerjee an S. Hua Inian Institute of Management Calcutta Department of Statistics an OR Joa, Diamon Harbour Roa Faculty of Science, Kuwait Uniersity Kolata , Inia P.O. Box-5969, Safat-3060, Kuwait Abstract: With reference to a baseline parametrization, we explore highly efficient fractional factorial esigns for inference on the main effects an, perhaps, some interactions. Our tools inclue approximate theory together with certain carefully eise iscretization proceures. he robustness of these esigns to possible moel misspecification is inestigate using a minimaxity approach. Examples are gien to emonstrate that our technique wors well een when the run size is quite small. Key wors: A-criterion, bias, binary esign, irectional eriatie, iscretization, minimaxity, moel misspecification, multiplicatie algorithm, nonorthogonality, requirement set. Mathematics Subject Classification: 6K5.. Introuction Design of optimal or efficient factorial fractions has receie significant attention in recent years; see Muerjee an Wu (006), Wu an Hamaa (009) an Xu, Phoa an Wong (009) for sureys an further references. A ast majority of the existing wor on fractional factorial esigns centers aroun the well-nown orthogonal parametrization (Gupta an Muerjee, 989) where orthogonal arrays play a ey role in the construction of optimal fractions. In recent years, howeer, a baseline parametrization has starte gaining popularity in factorial experiments. It has foun use in microarray experiments (Yang an Spee, 00) an can also arise naturally in many other situations wheneer each factor has a control or baseline leel. Following Kerr (006), an example is gien by a toxicological stuy with binary factors, each representing the presence or absence of a toxin, where the state of absence is a natural baseline leel of each factor. Also, in agricultural or inustrial experiments, the currently use leel of each factor can well constitute the baseline leel. Uner baseline parametrization, optimal paire comparison esigns for full factorials hae been stuie by seeral authors in the context of microarray experiments; see Banerjee an Muerjee (008), Zhang an Muerjee (03), an the references therein. As for esigning efficient or optimal factorial fractions uner this parametrization, not much wor has been reporte so far beyon the results in Muerjee an ang (0) an Li, Miller an ang (04) on the two-leel case. Inee, the lac of orthogonality in baseline parametrization maes the combinatorics for general fractional factorials extremely inole for example, unlie what happens in orthogonal parametriza-

2 tion, beyon two-leel factorials, orthogonal arrays cease to remain optimal for main effects een in the absence of interactions; see Muerjee an ang (0). Gien this ifficulty, one naturally woners if, pening the eelopment of perfect optimality results, it is possible to hae esigns with assure high efficiency, say oer 0.90 or 0.95, which woul suffice in most applications. From this perspectie, we eelop a technique for fining, uner the baseline parametrization, esigns which are highly efficient uner a gien moel an, at the same time, enjoy robustness to moel misspecification. his is one ia the approximate theory which in a sense exploits the unerlying nonorthogonality to our aantage an proies a benchmar for assuring the efficiency of exact esigns. It is, howeer, seen that iscretization of the resulting optimal esign measure ia a commonly use rouning off technique can hae isastrous consequences. As a remey, we propose certain proceures that yiel exact esigns with high efficiency. We a that our approach to robustness attempts to aress an issue raise by Yin an Zhou (04) who wore with the orthogonal parametrization an stresse on the nee for similar stuies uner nonorthogonal parametrizations, lie baseline. Section introuces the baseline parametrization an formulates the esign objectie with reference to a moel which is ept quite flexible. It inclues the baseline effect, all main effects an perhaps some interactions. hen in Section 3, we show how approximate theory, in conjunction with certain iscretization proceures, can yiel highly efficient fractions of general factorials with an arbitrary number of leels for each factor an arbitrary run size. he robustness of these esigns to moel misspecification is explore in Section 4. Examples are gien in Section 5 to emonstrate that our approach wors well een when the run size is quite small.. Baseline parametrization an esign objectie. Baseline parametrization Consier an m... m n factorial with n factors F,..., Fn. For i n F i, the leels of are coe as 0,,, m i, where 0 is the control or baseline leel. hus there are n m...m treatment combinations j... ( 0 j m, i n ). Let ( j... j ) enote the treatment effect of the treatment j n i i n combination j... j n. hen the baseline parametrization for a full factorial moel is gien by j... j ) = u 0, j }... u {0, j } ( u... u ), (.) ( n { n n n for eery j... j n, where the sum on each u i is oer ui { 0, ji}, i.e., ui i s fixe at 0 if ji 0, while u i ran ges oer only 0 an j i if j i 0. In (.), (0... 0) is the baseline effect, while ( u... u n ) is a main or interaction effect parameter for eery u..., epening on which u i s are nonzero. For instance, with a 3 factorial, (.) yiels u n

3 ( 00) (00), j 0) (00) ( 0) ( j, ), 0 j ) (00) (0 ) ( j, ), ( j ( j j j ( j ) (00) ( j 0) (0 j ) ( j ) j, j,), (.) ( where (00) is the baseline effect, main effect F is represente by (0), (0), main effect is represente by (0), (0), an interaction F is represente by (), (), (), (). It is F F easy to see that the parametrization (.) is nonorthogonal if the equations (.) arising from the treatment combinations are written in matrix notation, then the columns of the coefficient matrix in the right-han sie are not mutually orthogonal een when they correspon to ifferent factorial effects. We begin by consiering a reuce ersion of (.) which inclues the baseline effect an a collection of possibly significant factorial effects, such as all main effects an perhaps some interactions as well. he factorial effects ept in the moel constitute the requirement set, enote by R. Write 0 for the baseline effect (0...0), an for the ector of parameters representing the factorial effects in R. Interest lies in inference on, while 0 is a nuisance parameter. Let q be the imension of. hus in the 3 example, if R consists of main effects F an F, then parameters representing F F o not appear in the moel. So, (.) reuces to j j ) ( 00) ( j 0) (0 ) for j j,, but remains ( j, unchange when j 0 or j 0. In this case, = ( (0), (0), (0), (0)) consists of the main effects parameters, with the superscript enoting transposition, an q = 4. In orer to streamline the presentation, we now label the treatment combinations,..., in the lexicographic orer, e.g., in a 3 factorial, the lexicographic orering of the = 9 treatment combinations is 00, 0, 0, 0,,, 0,,, an these are labele,,, 9, respectiely. More generally, in an m... m n factorial, any treatment combination j... j n is labele where j..., (.3) n jn an m... m ), i n. With as aboe, we simply write for j... j ). Clearly, then for i /( i ( n the moel introuce aboe, = 0 z, where z is a nown q ector, ; e.g., in the example, the treatment combinations 0 an are labele 4 an 8, respectiely, an if R consists of 3 the two main effects then, with as shown in the preious paragraph, z 4 (, 0, 0, 0) an z 8 ) (0,,, 0). Let (,...,. hen our moel can be expresse as 0 Z, (.4) where is the ector of ones, an Z is the q matrix with rows z,..., z. 3

4 In what follows, for any ector b ( b,..., b l ), with B b... bl > 0, we efine D b) iag( b,..., b ), (b) ( l iag( b b B bb. (.5),..., l ) Also, for any positie integer l, we write 0 for the l ector of zeros, for the l ector of ones, l l an for the ientity matrix of orer l. I l. Design objectie Consier now an N-run esign, consisting, say, of treatment combinations labele ( ),..., ( N) which are not necessarily istinct. Let Y enote the N obserational ector arising from. Assume that the obserational errors are uncorrelate with a common ariance to (.4), we hae the moel. hen, corresponing E( Y ) 0 N Z, co( Y ) I N, (.6) where Z is the N q matrix with rows z ( ),..., z ( ). Let LN = I N N N. hen uner (.6), N the information matrix for the parametric ector of interest is gien by N H Z L Z. (.7) N We consier only esigns which eep estimable, i.e., hae a nonsingular H ; one can chec that the conition N q is necessary for this purpose. By (.6), for such a esign, we hae where ˆ H Z L Y, co ( ˆ) H, (.8) ˆ is the best linear unbiase estimator of. N We aim at fining a esign so as to minimize the aerage ariance of the elements of ˆ among all N-run esigns (the issue of moel robustness is taen up later). By (.8), this calls for minimizing tr( H ). So, we go by the A-criterion which is justifie here because interest lies in the parameters in as they stan, an not in their linear functions; cf. Kerr (0). As hinte in the introuction, this exact esign problem is combinatorially ery complex, ue to the nonorthogonality of the baseline parametrization. Use of approximate theory, howeer, allows us to mae consierable progress. 3. Approximate theory an iscretization proceures 3. Approximate theory o motiate the ieas, we consier an alternatie expression for. For, let the treatment H combination labele appear r ( 0) times in, where r... r N. he esign is calle binary if each is 0 or. Let r r,..., r ). hen from (.5), (.7) an the efinition of Z, r ( 4

5 H ( ) Z L Z Z ( r ) Z NM ( p ), (3.) N ( ) ( ) ( ) where p N r an M ( p ) = Z ( p ) Z. he iscreteness of r inuces the same on p, ( ) but in orer to employ the approximate theory we, for now, treat the elements of ( ) p as nonnegatie continuous ariables which a up to. Any such p ( p,..., ) p is calle a esign measure assigning masses p,..., p on the treatment combinations,,. By (3.), then the esign problem reuces to fining an optimal esign measure which minimizes ( p) oer all possible p, where ( p) = tr{ M ( p)}, if M ( p ) is nonsingular, with =, otherwise, (3.) M ( p) Z ( p) Z. (3.3) Due to elimination of the nuisance parameter 0, the expression (3.3) for M ( p) is not linear in the elements of p. Lemmas an below show that the still the basic ieas of approximate theory continue to hol in our setup. he first of these establishes the conexity of ( p) an the secon one characterizes p so as to minimize ( p). heir proofs appear in the appenix. Lemma : For any esign measures p an ~ p an any 0, ( ) ( p) ( ~ p) (( ) p ~ p). Lemma. A esign measure p minimizes ( p) if an only if M ( p) is nonsingular an ( z Z p) { M ( p)} { M ( p)} ( z Z p) tr{ M ( p)},. Lemma leas to a multiplicatie algorithm for numerical etermination of the optimal esign measure. It loos lie that in Zhang an Muerjee (03) who consiere paire comparison esigns for full factorials, but is more elaborate than theirs ue to the nonlinearity of our [0] M ( p). he algorithm starts with the uniform measure p (/,...,/ ), an fins p = ( p,..., p ), h,,... recursiely as [h] p = [ h] p ( z Z [ h] p ) { M ( p [ h]) )} tr{ M ( p [ h] { M ( p )} [ h] )} ( z [h] Z p [ h] [ h] ) [ h],, (3.4) till a esign measure [h] p, satisfying ( z Z p [ h] [ h] [ h] [ h] ) { M ( p )} { M ( p )} ( z Z p [ ] ) tr { M ( p h )} t,, (3.5) 5

6 is obtaine, where t (> 0) is a preassigne small quantity. If we enote the terminal esign measure meeting (3.5) by p ( pˆ,..., ˆ ) ˆ p, then arguments similar to those in Siley (980, p. 35) an the proof of our Lemma show that ( pˆ ) t, where is the minimum of ( p) oer all possible p. We tae t. hen pˆ represents the optimal esign measure with accuracy up to nine places of 0 0 ecimals, Een at this leel of accuracy, the algorithm in (3.4) an (3.5) wors quite fast. In all the examples in Section 5 an many others, it terminates almost instantaneously. In iew of the foregoing iscussion, for an N-run exact esign, it follows from (3.) an (3.) that tr( H ) s / N, (3.6) where s tr{ M ( pˆ)} t, with t. hus a lower boun to the efficiency of uner moel (.6) is obtaine as 0 0 eff = s /{ Ntr( )} (3.7) Of course, t is so small that its inclusion or otherwise in s has effectiely no impact on (3.7), as seen in our examples an many others. We remar that the boun (3.7), relatie to the optimal esign measure, is typically unattainable in an exact setup. So, it acts as a conseratie benchmar an the actual efficiency of among N-run exact esigns is often higher than this boun. 3. Discretization proceures Approximate theory exploits to our aantage the two features of the baseline parametrization that hiner the eelopment of exact optimality results, namely, nonorthogonality an lac of symmetry between the baseline an other leels of the factors. Because of these features, much unlie what often happens uner orthogonal parametrization, our optimal esign measure pˆ turns out to be far from uniform. For instance, in a 5 3 factorial, with the requirement set consisting of all main effects an interactions FF 6, F F6, it assigns masses , an to treatment combinations , 0000 an 00, respectiely. his lac of uniformity proies useful guiance to fining goo exact esigns e.g., it is intuitiely clear that an efficient exact esign shoul inclue those treatment combinations where the optimal esign measure assigns greater masses. Notwithstaning the aboe, serious ifficulties remain in translating the optimal esign measure pˆ H ( p ˆ,..., ˆ ) p to efficient exact esigns. Consier, for instance, the common rouning off technique where, with a gien run size N, one (i) multiplies pˆ,..., ˆ p by a positie constant c so chosen that the quantities cˆ p,..., cˆ, when roune off to nearest integers, say r,...,r, a up to N, (ii) obtains an p 6

7 exact esign where the treatment combination labele appears r times,, an (iii) hopes to be highly efficient with the belief that r,...,r are approximately proportional to pˆ,..., pˆ. his often fails to wor, especially for smaller run sizes which are of practical interest from the iewpoint of experimental economy. First, there may not exist any constant c such that the resulting a up to N. Secon, een when such a c exists, the esign obtaine as aboe may turn out to be of poor efficiency, if not outright singular. hus in the off fails to prouce a nonsingular exact esign for any N 3. r,...,r 5 3example mentione aboe, it is seen that rouning For large N, on the other han, the consequences of rouning off are negligible compare to N, an hence rouning off wors well. his enables us to eise proceures for iscretization of manner that the resulting exact esigns retain high efficiency een for smaller N. Proceure A: in such a I. Start with an N -run exact esign ( N ), obtaine from pˆ by rouning off, such that (N ) has ery high efficiency, say with eff 0.98, where eff is gien by (3.7). Here can be much larger N pˆ than the target run size N an ( N ) can be non-binary. II. For i 0,,..., N N, obtain an ( N i ) -run esign ( N i ) from an ( N i) -run esign ( N i) as follows: (a) Consier all possible eletions of one run from (N i). Among the resulting ( N i ) -run esigns, let * be the one with largest eff. If eff 0.95 for *, then tae ( N i ) as *. (b) Else, consier all possible eletions of two runs from ( N i) couple with all possible aitions of one run from amongst the treatment combinations. ae ( N i ) as the esign with the largest eff among all ( N i ) -run esigns so generate. he aboe proceure attempts to guar against eff going below 0.95 at any stage. Een if this is not always achiee, the final esign (N), of run size N, typically has high efficiency, with eff 0.90, een when N is rather small. Seeral examples in Section 5 illustrate this point. Our computations suggest that Zhang an Muerjee s (03) approach, inoling only eletion or only aition of runs, oes not often wor here in ensuring high efficiency for smaller N. his why, our proceure A is ifferent from theirs, allowing both eletion an aition. Inee, one may woner about arious moifications of A, such as eleting three runs an aing two in Step II (b), but we fin that these only increase the computational buren significantly without entailing much gain in efficiency. 7

8 Since the initial esign ( N ) in proceure A is often non-binary, the final output (N) can also be so, especially when N is not too small. Howeer, as seen in the next section, non-binary esigns are unliely to be robust to moel misspecification. So, we also consier two ariants of this proceure which always lea to binary esigns while aiming at high efficiency. Proceure B: I. Start with the full factorial esign where each of the treatment combinations appears once. Set =, an call this esign ( N ). II. Same as in A with the only change that if possibility (b) arises then the ae run is so chosen that only binary esigns are entertaine. Note that starting with a binary ( N i), one necessarily gets a N binary ( N i ) if (a) arises. Proceure B: I. Start with the full factorial esign. Set N =, an call this esign N ). II. For i 0,,..., N N, consier all possible eletions of one run from ( ( N i). ae ( N i ) as the esign with the largest While the optimal esign measure pˆ eff among all ( N i ) -run esigns so generate. has no role in Step I of B an B, it oes influence the final outcome through the use of eff in Step II. Proceure B is ery fast since it inoles only eletion. Hence there is no harm in trying it first an if one is not happy with the eff alue of the resulting (N), one can employ B which is not much time consuming either. In fact, B can help in maing B een faster. One can employ B to quicly fin an N -run binary esign ( N), with eff 0.98, an initiate B from this ( N ) rather than the full factorial. ypically, such N is large, but still smaller than, an this may expeite B in some situations. Illustratie examples follow in Section Robustness to moel misspecification With reference to the moel (.4), write X = [ Z]. Let C(X) enote the column space of X an C (X ) the orthocomplement thereof in the -imensional Eucliean space. he imensions of C(X) an C ( X ) are q an q, respectiely, since X has full column ran, a fact which is not har to see from a matrix representation of the baseline parametrization; cf. Banerjee an Muerjee (008). Moel (.4) amounts to assuming that C(X ). In the absence of this moel assumption, is any ector in the -imensional Eucliean space an hence can be represente as 0 Z P, (4.) 8

9 C where P is a ( q ) matrix whose columns form an orthonormal basis of (X ) an is a column ector of orer q. he choice of P is non-unique but, as seen below, this oes not affect our finings. Corresponing to (4.), the moel (.6) for an N-run exact esign gets moifie to E( Y ) 0 N Z P, co( Y ) I N, (4.) where P is relate to P exactly in the same way as Z is relate to Z. Uner (4.), ˆ in (.8) no longer remains unbiase for. It has now bias H Z LN P an its mean square error matrix equals MSE ( ˆ) = H H Z L P P L Z H. (4.3) A moel robust efficient esign aims at eeping N N tr{mse ( ˆ)} small, a problem which is complicate by the fact that is unnown. o oercome this ifficulty, in the spirit of Muerjee an ang (0), we aopt a Bayesian inspire approach without a ery explicit prior specification. Motiate by the hope that een if the assume moel (.4) is inaequate, it is not too far from the true moel (4.), we consier the class of priors such that max ( )} { E, (4.4) where max stans for largest eigenalue an E enotes expectation with respect to. With a iew to protecting against the worst scenario oer all possible priors satisfying (4.4), we procee to fin a esign which eeps max E [tr{mse ( ˆ)}] (4.5) small. his minimaxity approach is inspire by Wilmut an Zhou (0), Lin an Zhou (03) an Yin an Zhou (04), who wore with the orthogonal parametrization. Howeer, although (4.4) has essentially the same motiation as the conition that they impose, it is not ientical to theirs an seems to be more suite to the present context. We return to this point later but, at this stage, remar that (4.4) inariant of the choice of P. o see this, note that if P ~ is another matrix with orthonormal ~ ~ columns an the same column space as P, then P P for some ~, an P = P ~ K for some orthogonal matrix K. herefore, K an (4.4) is equialent to its counterpart with change to ~ ~. Lemma 3, proe in the appenix, helps in fining explicitly. Lemma 3. (a) Let V H Z ( r ) ( r ) ZH an W { Z ( ) Z}. hen Z LN P P LN Z H H = V W. 9

10 (b) he matrix V H is nonnegatie efinite (nn), an V H if is binary. q By (4.4), I E ( ) is nn for any. Hence by (4.3), for any such, E [tr{mse ( ˆ)}] tr( H ) tr{ H Z L P P L Z H }, (4.6) an this upper boun is attaine if E ( ) Iqwhich also meets (4.4). hus in (4.5) equals right-han sie of (4.6), an inoing Lemma 3(a), it follows that N = tr( H ) {tr( V ) tr( W )}, (4.7) which is inariant of the choice of P. By (3.6), (4.7) an Lemma 3(b), ( )tr( ) tr( W ) ( )( s/ N) tr( W ), (4.8) H an the first inequality in (4.8) becomes an equality for binary esigns. hus, as our computations also confirm, binary esigns play a major role in eeping N small. From (4.7) an (4.8), a lower boun to the efficiency of, uner possible moel uncertainty as consiere here, is gien by / eff ( ) = ( )( s / N) tr( W ), (4.9) tr( H ) {tr( V ) tr( W )} where, an V an W are as shown in Lemma 3. Note that W oes not epen on the esign. Just as the boun in (3.7), the one in (4.9) is a conseratie benchmar; the actual efficiency of among N-run exact esigns, in the present setup, is often higher than this boun. he examples in the next section illustrate how proceures A, B an B introuce earlier can yiel esigns with impressie alues of eff ( ) een for relatiely small run sizes. We now compare our approach in some etail with that in Wilmut an Zhou (0), Lin an Zhou (03) an Yin an Zhou (04), who explore binary esigns uner orthogonal parametrization. Instea of assigning a prior on, they stuie minimaxity uner a conition which amounts to (4.0) in our setup. By (4.3) an Lemma 3(a), the counterpart of, uner (4.0), is foun to be ~ = ˆ max tr{mse ( )} = tr( H ) max ( V W ). : In contrast to what happens uner an orthogonal parametrization, een with all factors at two leels, the matrix W has a complex form in our setup an hence, unlie in (4.7), no further splitting up of ~ is possible ue to nonlinearity of max in its arguments. Moreoer, ue to lac of ifferentiability of max, een the approximate theory oes not help in fining a useful lower boun on ~. hus, with 0

11 conition (4.0), exhaustie or near exhaustie search seems to be the only iable esign strategy an, computationally, this may be quite formiable unless an N are rather small. On the other han, our conition (4.4) seres essentially the same purpose as (4.0) but, aie by approximate theory, eeps the computations manageable. Incientally, a connection between ~ an emerges if we rewrite (4.0) as (, showing that ~ = max E [tr{mse ( ˆ )}], where the maximum is oer all max ) egenerate priors meeting (4.4). On the ot her han, our in (4.5) or (4.7) is the corresponing maximum oer the wier class all priors, egenerate or not, satisfying (4.4). 5. Examples For ease in presentation, we show the exam ples in the form of tables. ables -7 exhibit how in a wie ariety of situations, approximate theory, in conjunction with proceures A, B an B, yiels efficient esigns which are also moel robust, haing eff eff an eff ( ) alues 0.90 or higher. Since eff an ( ) are only conseratie lower bouns, the true efficiencies of these esigns, among exact e- signs with the same run size, shoul be een better. hroughout, we tae an 5. he choice is along the lines of Yin an Zhou (04), while the choice 5 shes light on the robustness of the esigns when the eparture from the assume moel is possibly een larger. Satisfyingly, our computations show that in binary esigns eff ( ) falls off rather slowly with increase in. For non-binary esigns, howeer, the fall is quite fast as anticipate from the finings in the last section. Inee, all the esigns reporte in ables -7 are binary. In each table, we inicate the requirement set R an the alue of q, the smallest run size neee to eep the factorial effect parameters ictate by R estimable uner the assume moel. Designs are reporte for eight consecutie alues of N, close to q. We continue to write (N) to enote a esign with run size N. In orer to sae space, the treatment combinations in a esign are liste by their labels following (.3). Moreoer, if one esign is similar to another, then we only mention how they iffer. hus in able, the esign for N = 6 is escribe as (5) + (5, 9) 4, to inicate that it can be obtaine from (5) by aing the treatment combinations labele 5 an 9 an eleting the treatment combination labele 4. In each table, after fining the optimal esign m easure ia the algorithm in (3.4) an (3.5), all three proceures A, B an B are applie for eery N consiere, an we report the best esign also inicating the corresponing proceure. Since none of A, B an B is ery computation intensie, we suggest that this be one in other applications as well. In the tables, wheneer proceure A or B is mentione, we inicate the initial esigns that they start from.

12 able. Robust efficient esigns for a factorial [ R F,..., F } { F F : i,,3 an j 4,5,6}, q = 6] { 6 i j 6 N (N) Proceure eff eff () eff (5) 6 9,, 4, 5, 7, 0,, 3, 33, 36, 38, 39, 57, B , 6, 63 7 (6) + B (7) + 49 B (8) + 6 B (9) + 4 B (0) + 7 B () + 4 B () + 5 B able. Robust efficient esigns for a 5 3factorial [ R { F,..., F6, F F6, F F6}, q = ; A initiate from (304) with eff = 0.995; B initiate from the full factorial] N (N) Proceure eff eff () eff (5) 3 4 0, 3, 0, 4, 7, 9, 3, 5, 53, 55, 76, 9, 96 (3) + (40, 67) 55 A A (4) + 4 A (5) + (5, 9) 4 A (6) + (8, 7) 0 A (7) + (, 88) 76 A , 3, 5, 7, 8, 9, 3, 6, 8, 33, 46, 50, 5, 57, 70, B , 90, 9, 95 0, 4, 6, 7, 8, 9,, 34, 36, 37, 4, 44, 49, 60, 70, 7, B , 79, 88, 90 able 3. Robust efficient esigns for a 3 4 factorial [ R { F,..., F5}, q = 0; A initiate from (80) with eff = ; B initiate from the full fac- torial] N (N) Proceure eff eff () eff (5) 4 8, 0, 3, 9, 8, 33, 39, 57, 66, 77, 86, 07, B , 3 5 (4) + 37 B (5) + (9, 5) 33 B (6) + B , 6, 4, 5, 33, 40, 43, 50, 53, 73, 8, 87, 89, 04, A , 7, 34, 43 9 (8) + 4 A (9) + 80 A , 9, 3, 9, 5, 8, 38, 39, 47, 57, 66, 7, 77, 8, B , 99, 07, 09,, 8, 37

13 8 able 4. Robust efficient esigns for a factorial [ R { F,..., F8, F F, F F3, F F F3}, q = ; A initiate from (88) with eff = ; B initiate from the full factorial] N (N) Proceure ef f eff ( ) eff ( 5) 4,, 60, 9, 00, 5, 37, 5, 67, 86, A , 09, 30, 5 5 (4) + (47, 60) 5 A (5) + (63, 96) 9 A (6) + (58, 89) 60 A , 0, 44, 49, 74, 83, 03, 8, 30, 37, 48, 59, B , 90, 04, 3, 3, 49 9 (8) + (6, 3) 5 B (9) + (0, 5) 3 B (0) + (34, 9) 30 B able 5. Robust efficient esigns for a 3 factorial [ R { F,..., F5}, q = ; A initiate from (306) with eff = ; B initiate from the full factorial] N (N) Proceure ef f eff ( ) eff ( 5) 5 0, 3, 7, 3, 39, 6, 65, 83, 94, 5, 54, 65 A , 08, 3 6 (5) + (4, 97) 0 A (6) + (4, 39) 4 A (7) + (36, 6) 54 A , 0, 3, 4, 35, 46, 57, 66, 99, 0,, 9, B , 63, 65, 05, 3, 30, 4 0 (8) + (7, 46, 66, 00) (39, 3) A (9) + (74, 06, 67, 07) (66, 05) B () + (58, 38) 39 B able 6. Robust efficient esigns for a factorial [ R { F,..., F6, F5 F6 }, q = 6; B initiate from (74) with eff = , as gien by B] N (N) Proceure ef f eff ( ) eff ( 5) 0, 4, 8, 9, 50, 63, 73, 93, 05, 3, 33, 34, B , 6, 6, 63, 64, 66, 67, 80 (0) + 34 B () + (85, 89) 9 B () + (0, 57) 85 B (3) + (7, 9) 67 B (4) + (9, 75) 63 B (5) + (54, 38) 6 B (6) + (0, 76) 64 B

14 4 3 able 7. Robust efficient esigns for a 3 factorial [ R { F,..., F7, F F, F F3, F F3, F F F3}, q = 5; B initiate from (98) with eff = , as gien by B] N (N) Proceure ef f eff ( ) eff ( 5) 0 6, 38, 5, 77, 90, 9, 4, 37, 59, 84, 9, B , 37, 56, 7, 34, 34, 353, 389, 4 (0) + (343, 356) 353 B () + (3, 5) 84 B () + (, 75) 3 B (3) + (47, 53) 56 B (4) + (5, ) 4 B (5) + (49, 06) 5 B (6) + (83, 99) 90 B For smaller N lie the ones in the tables, proceure A often yiels binary esigns an completes well with B een uner moel misspecification. Proceure B tens to perform worse though there are exceptions as in able. For N larger than those in the tables, B an B yiel esigns with een higher eff an eff ( ). hus, in the setup of able 5 with N = 8, proceure B, initiate from the full factorial, yiels a esign with eff =0.9608, eff () = an eff (5) = , while in the setup of able 6 with N = 33, proceure B leas to a esign with eff = 0.973, eff () = an eff (5) = For such larger N, howeer, proceure A often yiels non-binary esigns which are ery efficient uner the assume moel but perform poorly uner moel misspecification. he smallest N, say N0, in ables -7 is a little larger than q. For q N N0, in these examples none of A, B or B yiels esigns haing eff an eff ( ) alues 0.90 or higher. Howeer, since the bouns are conseratie, this oes not necessarily mean that these proceures, especially A an B, lea to inefficient esigns for such alues of N. Simply, the bouns o not assure high efficiencies of these esigns an complete enumeration seems to be the only way of assessing their efficiencies. his is again infeasible because is quite large in these examples. o gie a flaor of what may actually happen when N is ery close to q an the efficiency bouns are not impressie, we iscuss below a few situations where is small an hence complete enumeration is possible. 4 (a) factorial, R F,..., F, F F, F }, q = 7; N = 7, 8, 9, 0; { 4 3F4 (b) 3 3factorial; R F,..., F, F F, F }, q = 0; N = 0, ; { 4 4 F4 (c) 3 4 factorial; R F, F, F, F }, q = 3; N = 3, 4. { 3 F3 In the situations consiere in (a)-(c) aboe, proceures A, B an B yiel binary esigns but none of these has eff or eff ( ) alues 0.90 or higher. Howeer, a complete enumeration of binary esigns 4

15 shows that A an B prouce esigns with smallest possible tr( H ) as well as smallest possible for = an 5, in each of these situations except the one in (a) with N = 9. In this last situation, both lea to a esign with true efficiencies , an , for = 0, an 5, respectie ly, among all binary esigns with the same run size; note that 0 correspons to the efficiency uner assume moel. hus A an B are capable of proucing highly efficient, if not optimal, exact esigns een when this is not reflecte in the alues of the corresponing efficiency bouns. As mentione earlier, our approach turns to aantage the two features of the baseline parametrization that mae the exact esign problem har, namely, nonorthogonality an lac of symmetry among factor leels. Because of these ery features, the optimal esign measure becomes highly nonuniform an this helps us. Similar approach shoul be useful also in other situations which exhibit sufficient asymmetry so as to entail such nonuniformity of the optimal esign measure. We conclue with the hope that the present wor will generate interest in problems pertaining to situations of this in. Appenix Proof of Lemma. By (3.), the lemma is obious if either M ( p) or M ( ~ p) is singular. Suppose they are both nonsingular. hen by (.5) an (3.3), on simplification, M (( ) p ~ p) ( ~ ( ) M ( p) M p) + ( ) Z ( ~ p p)( ~ p p) Z. (A.) i.e., M (( ) p ~ p ) {( ) M ( p) M ( ~ p)} is nn. Since M ( p) an M ( ~ p) are both nonsingular, the result now follows from (3.). Proof of Lemma. We procee along the lines of Siley (980, pp. 8-0), with more elaborate arguments to cope with the nonlinearity of M ( p) in the elements of p. Let ~ p ( ~ p,..., ~ ) an p be any esign measures such that M ( p) is nonsingular. he lemma will follow arguing as in Siley (980) if we can show that ( ~ p) ( p) lim { (( ) p ~ p ) ( p)}/. (A.) 0 an ~ ) lim { (( ) p p ( p)}/ = ~ [tr { ( )} { ( )} { ( )} p M p e M p M p e ], (A.3) 0 where e z Z p,. he truth of (A.) is not har to see from Lemma. It remains to proe (A.3). o that effect, obsere that by (A.), M (( ) p ~ p) ( ) M ( p) Q( ), where Q ( ) = ~ M ( p) + ( ) Z ( ~ p p)( ~ p p) Z, is nn, for 0. Write g Z ( ~ p p) an note that by (.5) an (3. 3), p 5

16 M ( ~ p ) ~ ~ )( ~ ) ~ ) p ( z Z p z Z p p ( e g)( e g. herefore,, where columns ~ / Q ( ) U ( ) U ( ) U ( ) consists of the p ( e g) / ( ) g. hus for 0, the inerse o f M (( ) p ~ p) is gien by where ( ) [{ M ( p) } { M ( p)} U ( ){ I U ( ) ( M ( p)) U ( )}, U ( ) { M ( p)}, an /( ). Hence by (3.), after some simplification, the left-han sie of (A.3) equals 0 0 tr{ M ( p)} tr[{ M ( p)} U U { M ( p )} where U. Since ~ 0 lim 0 U ( ) p e g an, as a consequence, ], 0 U 0 U ~ p truth of(a.3) is now eien t. e ], e, the Proof of Lemma 3. (a) Note the followings facts: (i) As in (3.), Z LN P Z ( r ) P. (ii) If X [ N Z ], then LN X [ 0N LN Z ], so that as in (3.), Z ( r ) X Z LN X [ 0q H ], using (.7); the refore, H Z ( r ) X [ 0 I ]. (iii) By the efinition of P, the matrix PP is the orthogo- nal projector on Z LN P LN Z H = H Z ( r ){ I X ( X X ) X } ( r ) ZH = V [0 I ]( X X ) [0 I ] = V, wh ere W is the square submatrix of ( X ) as gien by its last q rows an columns. Part (a) is now (b) By (.5), after some algebra, C (X ), i.e., q X ( X X ) H P = H Z ( r ) PP ( r ) ZH * immeiate, noting that W * q PP = I X X. By (i)-(iii), W, by (.5) an the efinition of X. q q q q * W ( r ) ( r ) ( r ) ( I N r ){ ( ) ( ) ( )}( D r D r D r I N r ), (A.4) r which is nn, because D( r ) D( r ) D( r ) iag( r r,..., r r ) an each ri is a nonnegatie integer. herefore, recalling (3.), V H = H Z { ( r ) ( r ZH ) ( r )} is nn. If is binary,. then (A.4) anishes, an so V H Acnowlegement. he wor of RM was supporte by the J.C. Bose National Fellowship of the Go- ernment of Inia an a grant from the Inian Institute of Management Calcutta. he wor of SH was supporte by a grant from Kuwait Uniersity. References Banerjee,. an Muerjee, R. (008). Optimal factorial esigns for cdna microarray experiments. Ann. Appl. Statist.,

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