OPTIMAL DESIGNS FOR COMPLETE INTERACTION. By Rainer Schwabe. Freie Universitat Berlin

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1 1 OPTIMAL DESIGNS FOR COMPLETE INTERACTION STRUCTURES By Rainer Schwabe Freie Universitat Berlin In linear models with complete interaction structures product type designs are shown to be q -optimal for the maximal interactions, 0 q 1, and D- optimal simultaneously for the whole parameter vector and for testing against any complete submodel. 1. Introduction. We consider a general multi-factor linear model E(X(t 1 ; : : : ; t K )) = H (t 1 ; : : : ; t K ) = P N H2H k2h f k (t k ) T H ; t k 2 T k, k = 1; : : : ; K, with interaction structure H P(f1; : : : ; Kg) := fh; H f1; : : : ; Kgg. Throughout this paper we assume that the interaction structure H is complete, i. e. each interaction H 2 H comes with all its lower order interactions: Definition 1.1. H is complete if P(H) H for every H 2 H. Models of these type include product models, H = P(f1; : : : ; Kg), considered by Hoel (1965), additive models, H = f;; f1g; : : : ; fkgg, (see e. g. Schwabe and Wierich, 1995) and complete M-factor interaction models, H = fh; H f1; : : : ; Kg; jhj Mg, (see e. g. Schwabe, 1995). Particular examples will be given in section 4. The aim of the present paper is to establish that, in the setting of approximate designs (see e. g. Kiefer, 1974), product type designs are simultaneously optimal for higher order interactions if their components are optimal in the corresponding marginal models k (t k ) = 0 + f k (t k ) T k, t k 2 T k. Definition 1.2. H 2 H is maximal in H if H 0 62 H for every H 0 H, H 0 6= H. We will show that the product type design = N K k is q-optimal for maximal interactions H, i. e. minimizes the q-\norm" of the eigenvalues of the covariance matrix of the least squares estimator ^ H of H, if its components k are q -optimal for k in the marginal models, 0 q 1. Note that the q -criteria include the common A-optimality, q = 1, and the D- and E-optimality as limiting cases, q! 0 and q! 1, respectively. Moreover, for the D-criterion, the product type The research was supported by grant Ku719/2-1 of the Deutsche Forschungsgemeinschaft. AMS 1991 subject classication. 62K05. Key words and phrases. Product type designs, multi-factor linear models, q -optimality, D- optimality, complete interactions, maximal interaction. Abbreviated title. Designs for interactions.

2 2 rainer schwabe design will be shown to be simultaneously optimal for the whole parameter vector = ( H ) H2H as well as for testing every complete model H against every complete submodel H, H H. 2. Notations and auxiliary results. We recall that a subset H = L H of parameters is identiable under a design in a linear model (t) = a(t) T if the rows of the selection matrix L H are contained in the space spanned by the regression function a evaluated at the support points t of or, equivalently, spanned by the columns of the information matrix I() = R aa T d. In the present setting the regression functions considered are a H (t 1 ; :::; t K ) = ( N k2h f k (t k )) H2H. If H is identiable the corresponding covariance matrix is given by C( H ; ) = L H I()? L T. For larger subsets H H = ( H) 0 H2H 0 we have L H 0 = diag(l H ) H2H 0 and C( H 0; ) = L H 0I()? L T if each H H, H 2 H 0, is identiable under. In case H 0 = H 0 we write C() = C(; ), for short. In the corresponding marginal models the parameters are (k) = ( 0 ; k T ) T and the information matrices I k ( k ) = R a k a T d k k where a k = (1; f T k )T. The marginal covariance matrices for (k) and k = L k (k) are denoted by C k ( k ) = I k ( k )?1 and C k ( k ; k ) = L k C( k )L T k respectively. Note that k is identiable under k i I k ( k ) is regular. As usual max and tr denote the largest eigenvalue and the trace of a matrix. For every pair of designs H on k2h T k and H on k62h T k the resulting product design on K T k is denoted by H ( H ; H ). Lemma 2.1. Let (t) = f 0 (t) T 0 + f 1 (t) T 1 + f 2 (t) T 2 and let 0 (t) = f 0 (t) T 0 + f 1 (t) T 1 be a submodel of. (i) If 1 is identiable under in, then C 0 ( 1 ; ) C( 1 ; ). (ii) If 1 and 2 are identiable under in, then det C( 1 ; 2 ; ) = det C 0 ( 1 ; ) det C( 2 ; ). Proof. (i) is a common renement statement (see e. g. Kunert, 1983). For (ii) we partition the information matrix! I() = I 0() I 12 I T 12 I 2 in according to the submodel, and the covariance matrix! C C( 1 ; 2 ; ) = 1 C 12 C12 T C 2 according to the components 1 and 2, C j = C( j ; ). Further let L be the selection matrix for 1 = L( T 0 ; T 1 ) T in 0. By the formula for generalized inverses of partitioned matrices C 1 = L(I 0 ()? + I 0 ()? I 12 C 2 I T 12I 0 ()? )L T C 12 = LI 0 ()? I 12 C 2

3 optimal designs for complete interaction structures 3 and hence C 1? C 12 C?1 2 C T 12 = LI 0 ()? L T = C 0 ( 1 ; ): The result now follows from det C( 1 ; 2 ; ) = det(c 1? C 12 C?1 2 C T 12) det C 2. 2 Remark 2.1. If f 0 1 (or f 0 0), then det C() = det C 0 () det C( 2 ; ) from Lemma Optimality of product type designs. The rst result deals with the optimality of product type designs for the highest interaction in total interactions models Theorem 3.1. Let H = P and let H = f1; : : : ; Kg. If k is q-optimal for k in the kth marginal model, k = 1; : : : ; K, then = N K k is q-optimal for H. Proof. Note that H = L where L = N K L k and k = L k (k). Hence the covariance matrix C( H ; ) = N K C k ( k ; k) factorizes. The q -optimality can now be obtained by means of the appropriate equivalence theorems. For q < 1 see Rafaj lowicz and Myszka (1988). For the E-citerion (q = 1) a more implicit equivalence theorem (Pukelsheim, 1993, p. 182) has to be used: k is E-optimal for k i there exists a non-negative denite, symmetric matrix M k, tr M k = 1, such that a k (t k ) T C k ( k )LT k C k ( k ; k )?1 M k C k ( k ; k )?1 L k C k ( k )a k(t k ) 1= max (C k (; k )) for all t k 2 T k. Let M = N K M k. Then M is non-negative denite, symmetric, tr M = 1, and a(t) T C( )L T C( H ; )?1 MC( H ; )?1 LC( )a(t) 1= max (C( H ; )): Thus the equivalence theorem proves the E-optimality of for H. 2 Note that H = f1; : : : ; Kg is the only maximal element in H = P. For general, complete interaction structures H a maximal interaction H will be a proper subset of f1; : : : ; Kg. Then, conditions are only required on those factors k belonging to H. Theorem 3.2. Let H be complete, and let H be maximal in H. If k is q- optimal for k in the kth marginal model, k 2 H, then = H ( N k2h k ; H) is q -optimal for H, for every design H on k62h T k. Proof. Consider the model P(H) of total interactions in the factors involved in H. Then P(H) can be regarded as a submodel of H. Moreover, as H is maximal, C P(H) ( H ; N k2h k ) = C( H; );

4 4 rainer schwabe and by Theorem 3.1 the design N k2h k is q-optimal for H in P(H). Hence, the result follows by a renement argument (cf Lemma 2.1 (i)). 2 The main result on q?optimality is the following extension to more than one maximal interaction: Theorem 3.3. Let H be complete, H 0 fh; H maximal in Hg. If k is q -optimal for k in the kth marginal model, k 2 H(H 0 ) := S 0 H2H H; then = H(H0 )( N k2h(h 0 ) ; k H(H )) is 0 q -optimal for H 0, for every design H(H0 ) on k62h(h0 )T k. Proof. The key observation is that the covariance matrix is block diagonal: C( H 0; ) = diag(c( H ; )) H2H 0 because each H 2 H 0 is maximal. By Fan's (1954) result on the eigenvalues of block matrices the q -optimality of can be deduced for H 0 from the q -optimality of k for every H, H 2 H 0, established in Theorem For additive models, H = f;; f1g; : : : ; fkgg all direct eects fkg are maximal, and we revover a special case of Theorem 3.2 by Schwabe (1994). As indicated there no general results can be expected for q -optimality, q > 0, of product type designs if larger parts of the parameter vector are of interest including interactions which are not maximal. However, for D-optimality (q = 0) a stronger result can be obtained. Theorem 3.4. Let H be complete, and let H 0 H, HnH 0 complete. If k is D- optimal in the kth marginal model, k 2 H(H 0 ); then = H(H0 )( N k2h(h 0 ) ; k H(H )) 0 is D-optimal for H 0, for every design H(H0 ) on k62h(h 0 )T k. Proof. First note that k is D-optimal i k is D-optimal for k (see e. g. Schwabe and Wierich, 1995, Lemma 6). Hence is D-optimal for any set of maximal interactions by Theorem 3.3. The proof will be completed by a nite induction: Therefore assume that the theorem is valid for H 00. Let H be maximal in H n H 00. Then it suces to show that the theorem is also valid for H 00 [ fhg, because every set H 0 can be generated successively by adding relatively maximal interactions. In fact, by Lemma 2.1 (ii), det C( H 00 [fhg; ) = det C( H 00; ) det C HnH 00( H ; ) det C( H 00; ) det C HnH 00( H ; ); where the inequality follows by Theorem 3.2 and the assumption. 2 In particular, for H 0 = H, the design N K k is D-optimal if each of its components k is D-optimal in the corresponding marginal model. Finally, note that

5 optimal designs for complete interaction structures 5 the full product type design N K k is D-optimal for H 0 simultaneously in H and H 0 H when H and H n H 0 are complete. 4. Examples Multilinear regression on [?1; 1] K. The full 2 K factorial design is simultaneously q -optimal for every set of maximal interactions in every complete interactions model H (t 1 ; : : : ; t K ) = P H2H H Qk2H t k : Moreover, the 2 K factorial is D-optimal for the whole parameter vector as well as for any set H 0 of higher interactions, such that the remaining interaction structure H n H 0 is complete. In particular, in the model of complete M-factor interactions the 2 K factorial is the unique D-optimal design (the unique D-optimal design for all interactions of m up to M factors, i. e. H 0 = fh; H f1; : : : ; Kg; m jhj Mg; the unique q -optimal design for all M-factor interactions) if 2M K. In case 2M < K also 2 d fractions may be optimal, 2M d < K. For example, in the additive model (M = 1) there exist optimal 2 d fractional factorials for d log 2 (K + 1), K 3, constructed from Hadamard matrices K-way layout. Every complete K-way layout H (i(1); : : : ; i(k)) = P H2H (H) H(i(1);:::;i(K)) ; i(k) = 1; : : : ; I k, k = 1; : : : ; K, where H(i(1); : : : ; i(k)) = (i(k 1 ); : : : ; i(k jhj )) is the projection of (i(1); : : : ; i(k)) onto the components k 1 ; : : : ; k jhj of H, can be reparametrized to the form considered above, if suitable identiability conditions P i(k1 );:::;i(k jhj) = 0, H 2 H, are imposed. (Here the summation is to be considered over every set of jhj? 1 components). Hence the equireplicated design is D-optimal for testing every model with complete interaction structure H against each submodel with complete interactions structure H n H Qualitative and quantitative factors. Following Kurotschka (1988) those models are of particular interest in applications which contain both kinds of factors: qualitative ones which may vary only over a nite number of levels (see e. g. 4.2) and quantitative ones which vary over a whole continuum of levels (see e. g. 4.1). To be more specic, in a two-way layout with additional straight line regression on [?1; 1] the model may be either additive or may have total interactions (i; j; t) = (1) i + (2) j + t (i; j; t) = ij + ij t or, as indermediate models, the additive model is supplemented by some or all of the two-factor interactions ij, (1) i t and (2) j t. For all those 9 dierent models the

6 6 rainer schwabe equireplicated design on f1; : : : ; Ig f1; : : : Jg f?1; 1g is D-optimal for the whole parameter vector (under suitable identiability conditions) as well as for estimating (testing) the higher interaction terms. REFERENCES Fan, K. (1954). Inequalities for eigenvalues of Hermitian matrices. Nat. Bur. Standards, Appl. Math. Ser {139. Hoel, P. G. (1965). Minimax designs in two dimensional regression. Ann. Math. Statist {1106. Kiefer, J. (1974). General equivalence theory for optimum designs (approximate theory). Ann. Statist {879. Kunert, J. (1983). Optimal design and renement of the linear model with applications to repeated measurement designs. Ann. Statist {257. Kurotschka, V. G. (1988). Characterizations and examples of optimal experiments with qualitative and quantitative factors. In Model-Oriented Data Analysis, Proceedings Eisenach (V. Fedorov and H. Lauter, eds.) 53{71. Springer, Berlin. Pukelsheim, F. (1993). Optimal Design of Experiments. Wiley, New York. Rafaj lowicz, E. and Myszka, W. (1988). Optimum experimental design for a regression on a hypercube - generalization of Hoel's result. Ann. Inst. Statist. Math {827. Schwabe, R. (1994). Optimal designs for additive linear models. Statistics (to appear). Schwabe, R. (1995). Experimental design for linear models with higher order interaction terms. In Symposia Gaussiana. Proceedings of the 2nd Gauss Symposium, Conference B: Statistical Sciences, Munchen (H. Schneewei and V. Mammitzsch, eds.) 281{288. DeGruyter, Berlin. Schwabe, R. and Wierich, W. (1995). D-optimal designs of experiments with noninteracting factors. J. Statist. Plann. Inference {384. Freie Universitat Berlin 1. Mathematisches Institut Arnimallee 2{ Berlin Germany