Characterization of c-, L- and φ k -optimal designs for a class of non-linear multiple-regression models

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1 J. R. Statist. Soc. B (2019) 81, Part 1, pp Characterization of c-, L- and φ k -optimal designs for a class of non-linear multiple-regression models Dennis Schmidt Otto-von-Guericke-Universität Magdeburg, Germany [Received July Final revision August 2018] Summary. Optimal designs for multiple-regression models are determined. We consider a general class of non-linear models including proportional hazards models with different censoring schemes, the Poisson and the negative binomial model. For these models we provide a complete characterization of c-optimal designs for all vectors c in the case of a single covariate. For multiple regression with an arbitrary number of covariates, c-optimal designs for certain vectors c are derived analytically. Using some general results on the structure of optimal designs for multiple regression, we determine L- and φ k -optimal designs for models with an arbitrary number of covariates. Keywords: c-optimality; L-optimality; Multiple regression; φ k -optimality; Poisson model; Proportional hazards model with censored data 1. Introduction The proportional hazards model is one of the most commonly used models in survival analysis, where a typical feature is censoring. We consider a class of non-linear multiple-regression models, that includes the proportional hazards model with type I and random censoring as well as the Poisson and negative binomial model, and determine c-, L- and φ k -optimal designs. This class was first considered by Konstantinou et al. (2014), who determined D- and c-optimal designs, the latter for the effect parameter, in the case of a single covariate. Schmidt and Schwabe (2015) computed D-optimal designs for a discrete design region and Schmidt and Schwabe (2017) determined D-optimal designs for multiple regression, obtaining the results of Russell et al. (2009) for the Poisson model as a special case. For the Poisson and negative binomial model with a single covariate, Rodríguez-Torreblanca and Rodríguez-Díaz (2007) determined D-optimal designs and c-optimal designs for certain vectors c. Further work on D-optimal designs for the Poisson model was done by Wang et al. (2006) and for proportional hazards models with censoring by Müller (2013), who also gave an overview of existing literature for planning lifetime experiments, and by López-Fidalgo et al. (2009). In some recent references by Yang and Stufken (2009), Yang (2010), Dette and Melas (2011), Yang and Stufken (2012) and Dette and Schorning (2013) complete-class results were obtained for models with two parameters in the first and for models with an arbitrary number of parameters in the last four papers. This means that a subclass of designs was identified, so that for every design ξ there is a design in this subclass whose information matrix dominates that of ξ in the Loewner ordering. In all these references models with only a single covariate were consid- Address for correspondence: Dennis Schmidt, Institut für Mathematische Stochastik, Otto-von-Guericke- Universität Magdeburg, PF 4120, Magdeburg, Germany. dennis.schmidt@ovgu.de 2018 Royal Statistical Society /19/81101

2 102 D. Schmidt ered, but not with multiple covariates. We note that complete-class results based on Chebyshev systems cannot easily be extended to multiple regression (see Hu et al. (2015)). For the case of multiple covariates Yang et al. (2011) gave complete-class results for logistic and probit models, which have a symmetric efficiency function, and they determined D-, A- and E-optimal designs. However, many models in survival analysis such as proportional hazards models with censored data have monotonic efficiency functions. In this paper we focus on models with such efficiency functions. There is only a little literature on optimal design for non-linear regression models with multiple covariates, especially for other criteria than D-optimality. Although D-optimal designs can often be constructed from the D-optimal designs in the marginal models with a single covariate (Schmidt and Schwabe, 2017), it is not clear how optimal designs for other optimality criteria can be analytically determined. Some general results concerning multiple regression are given in Section 2. In Sections 3 5 we analytically determine c-, L- and φ k -optimal designs for models with one covariate and for multiple regression with an arbitrary number of covariates. In Section 6 we investigate the effect of censoring on the optimal designs for the various optimality criteria. The optimal designs depend on the unknown parameters since the models are non-linear. Such designs, which are optimal for a prespecified parameter value, are called locally optimal in accordance with Chernoff (1953). All proofs and some helpful lemmas are deferred to Appendix A. 2. Model specifications and optimal design The Fisher information matrix I.x, β/ depends on the covariate vector x =.x 1, :::, x p 1 / T. Under mild regularity conditions, the maximum likelihood estimator for the parameter vector β =.β 0, :::, β p 1 / T R p is asymptotically efficient and the inverse of the information matrix is equal to the asymptotic covariance matrix. Therefore, we want to minimize some aspect of the inverse of the information matrix or else to maximize the information matrix in some equivalent sense, to find the optimal values for the covariates and thus to be able to estimate the parameters as precisely as possible. We determine approximate designs x1 x ξ = 2 ::: x m,.2:1/ ω 1 ω 2 ::: ω m where x 1, :::, x m are distinct values of the covariates, which can be chosen from a given design region X, and ω 1, :::, ω m are the corresponding weights, which satisfy 0 ω i 1fori = 1, :::, m and Σ m i=1 ω i = 1 (see Silvey (1980), page 15). The information matrix M.ξ, β/ of a design ξ is defined by (see Silvey (1980), page 53) M.ξ, β/ = I.x, β/ ξ.dx/ = m ω i I.x i, β/:.2:2/ We consider models with Fisher information matrices of the form X i=1 I.x, β/ = Q{f.x/ T βf.x/f.x/ T,.2:3/ where Q 0 is the efficiency function (see Fedorov (1972), page 39), f is a vector of known regression functions, x R p 1 is the vector of covariates and β R p is the vector of p unknown parameters. Throughout this paper, multiple regression with additive linear effects of the covariates in the linear predictor is considered, i.e. f.x/ =.1, x T / T. Such information matrices occur for proportional hazards models with censored data (see Konstantinou et al. (2014) and Schmidt and Schwabe (2017)) and for generalized linear models.

3 Characterization of Optimal Designs 103 Table 1. Efficiency functions for various models Proportional hazards model Poisson Negative binomial model model Type I Random U(0, c) 1 exp{ c exp.θ/ Q.θ/ 1 exp{ c exp.θ/ 1 c exp.θ/ exp.θ/ exp.θ/ exp.θ/ + λ We consider two types of censoring: type I and random censoring. For type I censoring all individuals enter the experiment simultaneously and the time to the event of interest is observed until a fixed censoring time c, at which the experiment is terminated. For random censoring the censoring times follow some probability distribution; here we consider a uniform distribution U.0, c/. The efficiency functions for the proportional hazards model with a constant baseline hazard are given in Table 1 for both censoring schemes. These efficiency functions satisfy the following conditions (see Konstantinou et al. (2014)), which are also satisfied for other censoring distributions (see Schmidt and Schwabe (2017)) and by the efficiency functions for the Poisson and negative binomial model given in Table 1. Assumption 1. Q.θ/ is positive for all θ R and twice continuously differentiable. Assumption 2. Q.θ/ is positive for all θ R. Assumption 3. The second derivative g.θ/ of the function g.θ/ = 1=Q.θ/ is injective. Assumption 4. The function Q.θ/=Q.θ/ is an increasing function. Optimality criteria are real-valued functions of the information matrix, which are maximized or minimized with respect to the design. If interest is in estimating a particular linear combination c T β with minimum variance, then the appropriate optimality criterion is c-optimality. A linear combination c T β is identifiable for a design ξ, if there is a vector y R p such that c = M.ξ, β/ y (see Silvey (1980), page 25). A design ξ Å is a c-optimal design, if c T β is identifiable and if c T M.ξ Å, β/ c c T M.ξ, β/ c holds for all designs ξ for which c T β is identifiable. The matrix M.ξ, β/ is a generalized inverse of M.ξ, β/. Let B = AA T be a positive definite p p matrix. A design ξ Å with regular information matrix M.ξ Å, β/ is an L-optimal design, if tr{m.ξ Å, β/ 1 B tr{m.ξ, β/ 1 B holds for all designs ξ with regular information matrix. L-optimality aims at estimating several linear combinations A T β with minimum average variance. A special case of L-optimality is A-optimality, which is obtained for B = I p, the identity matrix. An A-optimal design minimizes the average of the variances of the parameter estimates (see Atkinson et al. (2007), pages ). A general class of optimality criteria is the φ k -criteria of Kiefer (1974) with k [, 1]. For a symmetric positive definite p p matrix M the matrix mean is defined as {.1=p/tr.M k / 1=k for k,0, φ k.m/ = det.m/ 1=p for k = 0,.2:4/ λ min.m/ for k =, where λ min.m/ denotes the smallest eigenvalue of M (see Pukelsheim (1993), page 141). A design ξ Å k with regular information matrix M.ξÅ k, β/ is a φ k-optimal design, if φ k {M.ξ Å k, β/ φ k {M.ξ, β/ holds for all designs ξ with regular information matrix. The cases k = 0 and k = 1 correspond to D- and A-optimality. The popular D-optimality criterion can be used, when the

4 104 D. Schmidt full parameter vector is to be estimated as precisely as possible. A D-optimal design minimizes the volume of the confidence ellipsoid for the parameters. Each optimality criterion can be transformed into a maximization problem with an isotonic criterion function Φ, i.e. Φ.M 1 / Φ.M 2 / holds for M 1 M 2 (see Pronzato and Pázman (2013), page 118). If even Φ.M 1 />Φ.M 2 / for M 1 M 2 with M 1 M 2, then Φ is called strictly isotonic. For example the φ k -criteria, k., 1], are strictly isotonic on the set of symmetric positive definite matrices (see Pukelsheim (1993), page 151). One of the most important tools for proving the optimality of a design is the equivalence theorems (see Kiefer (1974) and Pukelsheim (1993), page 180). Theorem 1. A design ξ Å or ξ Å k with regular information matrix respectively M.ξÅ, β/ or M.ξ Å k, β/ is a c-optimal, L-optimal or φ k-optimal design for k., 1] respectively if and only if (a) Q { f.x/ T β f.x/ T M.ξ Å, β/ 1 cc T M.ξ Å, β/ 1 f.x/ c T M.ξ Å, β/ 1 c,.2:5/ (b) Q { f.x/ T β f.x/ T M.ξ Å, β/ 1 BM.ξ Å, β/ 1 f.x/ tr { M.ξ Å, β/ 1 B,.2:6/ (c) Q { f.x/ T β f.x/ T M.ξ Å k, β/k 1 f.x/ tr { M.ξ Å k, β/k.2:7/ holds for all x X. At the support points of ξ Å and ξ Å k there is equality. Schmidt and Schwabe (2017) considered multiple-regression models with p 3, f.x/ =.1, x T / T and polytopes as design region X, which are defined as the convex hull of finitely many points. They showed that, for optimality criteria for which the condition in the equivalence theorem is of the form d.x/ := Q { f.x/ T β f.x/ T Af.x/ b.2:8/ with positive definite matrix A and constant b>0, the support points of an optimal design are at the edges of X, since the maximum of the sensitivity function d.x/ is attained only there. To show the optimality of a design, it is sufficient to check condition (2.8) only on the edges. This result can be extended to polyhedra (see Schmidt and Schwabe (2017)), which are defined as the intersection of finitely many half-spaces and may be unbounded. Theorem 2. Let the design region X be a multi-dimensional polyhedron and let the condition in the equivalence theorem be of the form (2.8) with positive definite matrix A. Let the hyperplanes H η = {x R p 1 : f.x/ T β = η be bounded on X for all η R. Then a design ξ Å is optimal if and only if condition (2.8) is satisfied on the edges of X. If such a design ξ Å exists, then its support points are at the edges of X. The requirement of the boundedness of X H η ensures, in the case of an unbounded polyhedron, that the sensitivity function is maximal at the edges. Otherwise, there is no optimal design, since the sensitivity function is unbounded above. If the matrix A is only positive semidefinite, optimal designs with support points in the interior of the design region can exist. This is so with c-optimality and is shown in Section 3. In what follows, the relation for matrices is meant in the Loewner ordering sense, i.e. M 1 M 2 means that M 1 M 2 is positive semidefinite. Theorem 3. Let X be a multi-dimensional polytope. For each design ξ there is a design ξ, whose support points are only on the edges of X, with M.ξ, β/ M.ξ, β/. Remark 1. Theorem 3 shows the existence of an optimal design with all support points on the edges. For strictly isotonic criteria it follows from the proof of theorem 3 that the support points of an optimal design must be located on the edges of X.

5 Characterization of Optimal Designs 105 The following result provides a bound for the number of support points of an optimal design. Lemma 1. Let X be a polyhedron and let assumptions 1 and 3 be satisfied. If the condition in the equivalence theorem is of the form (2.8) with positive semidefinite matrix A, then an optimal design has at most two support points per edge. Lemma 2. The information matrix given by equations (2.2) and (2.3) can be decomposed as M.ξ, β/ = X T D ω X with diagonal matrix D ω = diag.ω 1, :::, ω m / and X =. f.x 1 /, :::, f.x m // T, where f.x/:=[ Q{f.x/ T β] f.x/. X can be written as X=Q 1=2 X with X=.f.x 1 /, :::, f.x m // T and Q 1=2 = diag[ Q{f.x 1 / T β, :::, Q{f.x m / T β]. On the basis of this decomposition of the information matrix, we obtain the optimal weights for different optimality criteria from Pukelsheim and Torsney (1991). It is assumed that the vectors of the regression functions, which are the rows of X, are linearly independent. Theorem 4. The c-optimal weights for a design with m p support points, L-optimal weights for a design with minimal support (i.e. with m = p support points) and φ k -optimal weights, k., 1], for a design with minimal support are given respectively by (a) ω Å i = v i =Σ m j=1 v j for i = 1, :::, m with v :=. X X T / 1 Xc,.2:9/ (b) ω Å i = s ii =Σ p j=1 sjj for i = 1, :::, p with S :=. X T / 1 B X 1,.2:10/ (c) ω Å i = s ii =Σ p j=1 sjj for i = 1, :::, p with S :=. X T / 1 M.ξ, β/ k+1 X 1 :.2:11/ Here v i are the entries of the vector v and s ii are the diagonal entries of the matrix S. 3. c-optimal designs First, we consider models with a single covariate, so we have p = 2, f.x/ =.1, x/ T and β =.β 0, β 1 / T. Rodríguez-Torreblanca and Rodríguez-Díaz (2007) determined c-optimal designs for the Poisson model and the negative binomial model for the vectors c =.1, 0/ T and c =.0, 1/ T for negative β 1 and design region in the positive range. For the class of models that are considered here, which includes both above-mentioned models, Konstantinou et al. (2014) computed c- optimal designs for the vector c =.0, 1/ T. Now, we determine c-optimal designs for an arbitrary vector c =.c 1, c 2 / T and an arbitrary design region X = [u, v] by using Elfving s theorem, which is given in Appendix A. Let the function ψ p,a. / : R R with a =.a 1, :::, a p 1 / T for general p-dimensional regression function f be defined by ψ p,a.x/ := x 2 Q{ f.a/ T β x Q { f.a/ T β x [ Q{f.a/ T β x Q{f.a/ T β ] + 1 :.3:1/ Lemma 3. Let assumptions 1, 2 and 4 hold. The function ψ p,a is strictly increasing, continuous and one to one. Hence the inverse function ψp,a 1 exists with the same properties. Theorem 5. Let X = [u, v] and let assumptions 1, 2 and 4 be satisfied. Let g.x/ :=.c 1 x c 2 / 2. For β 1 >0 let x Å =max{u, v ψ2,v 1.0/=β 1, a =v and I =[x Å, v]. For β 1 <0 let x Å =min{v, u ψ2,u 1.0/=β 1, a = u and I = [u, x Å ]. (a) If c 2 =c 1 I, the one-point design with support point c 2 =c 1 is the unique c-optimal design.

6 106 D. Schmidt (b) If c 2 =c 1 I, the unique c-optimal design is given by { ξ Å x Å = {Q.β0 +β 1 a/g.a/ {Q.β0 +β 1 x Å /g.x Å /+ {Q.β 0 +β 1 a/g.a/ a {Q.β0 +β 1 x Å /g.x Å / {Q.β0 +β 1 x Å /g.x Å /+ {Q.β 0 +β 1 a/g.a/ :.3:2/ Remark 2. For c 1 = 0 in theorem 5 design (3.2) is the c-optimal design. For the Poisson model with Q.θ/ = exp.θ/ the equation ψ 2,a.x/ = 0 has an analytic solution: ψ2,a 1.0/ = 2[1 + W{exp. 1/] 2:557. Here, W denotes the principal branch of the Lambert W -function, which is the inverse function of g.w/ = w exp.w/ for w 1 (see Corless et al. (1996)). Now, we consider multiple-regression models with p 1 covariates and rectangular design region. We have p 3, f.x/ =.1, x T / T and x =.x 1, :::, x p 1 / T X. Theorem 6. Let X = [u 1, v 1 ] ::: [u p 1, v p 1 ] and let assumptions 1, 2 and 4 be satisfied. The vector c is given by c =.0, c T / T with c =.c 1, :::, c p 1 / T.Fori = 1, :::, p 1 let β i 0. Let either β i c i 0fori = 1, :::, p 1orβ i c i 0fori = 1, :::, p 1. Define a i = v i,ifβ i > 0 and a i = u i,ifβ i < 0. Let x Å = f.a/ T β ψp,a 1.0/ with a =.a 1, :::, a p 1 / T. (a) If c i ψp,a 1.0/ ct β.v i u i / holds for i = 1, :::, p 1, then the following two-point design is a c-optimal design: ξ Å a ψ 1 p,a.0/ = c T β c a Q{f.a/ T β Q.x Å :.3:3/ / Q.x Å / + Q{f.a/ T β Q.x Å / + Q{f.a/ T β (b) For i = 1, :::, m 1 let x Å i X {x R p 1 : f.x/ T β = x Å be different points such that the convex combination of the x Å i equals the first support point of the c-optimal design (3.3): Σ m 1 i=1 λ ix Å i = a ψ 1 p,a.0/=.ct β/ c. Then the following m-point design is also a c-optimal design: { ξ Å x Å 1 ::: x Å m 1 a = Q{f.a/ λ T β : 1 Q.x Å /+ Q{f.a/ T β ::: λ m 1 Q{f.a/ T β Q.x Å /+ Q{f.a/ T β Q.x Å / Q.x Å /+ Q{f.a/ T β The inequalities c i ψp,a 1.0/ ct β.v i u i /, i = 1, :::, p 1, ensure that the first support point of the c-optimal two-point design (3.3) is inside the design region. The second support point a is the point of the design region where the efficiency function is maximized. The information matrix of this design is singular. To estimate all the parameters, designs with regular information matrices are required. Therefore, the c-optimal m-point designs in theorem 6, part (b), are of interest. They exist because the points on the hyperplane f.x/ T β = x Å form a linear surface of the Elfving locus { f.x/ x X, and this surface is on the boundary of the Elfving set. From the proof of theorem 6 it follows that besides discrete c-optimal designs there also exist c-optimal designs with continuous parts. Theorem 7. Let the assumptions of theorem 6 hold. Let ω Å 1 and ωå 2 be the weights of the c-optimal two-point design (3.3). Let ξ 1 be an arbitrary (not necessarily discrete) design on X {x R p 1 : f.x/ T β = x Å with E.ξ 1 / = a ψp,a 1.0/=.cT β/ c and let ξ 2 be the one-point design with support point a. Then the design ξ Å = ω Å 1 ξ 1 + ω Å 2 ξ 2 is a c-optimal design. In what follows e i denotes the ith standard unit vector in R p 1. Fig. 1 shows the structure of the c-optimal designs for the model with two covariates (p = 3). All support points except a of

7 Characterization of Optimal Designs 107 Fig. 1. Structure of the c-optimal designs the c-optimal designs of theorem 6 are on the hyperplane f.x/ T β = x Å, which in this case is a straight line. With c =.0, e i / T we obtain from theorem 6, part (a), for i = 1, :::, p 1 the c-optimal designs for β i with support points a {ψp,a 1.0/=β ie i and a. These designs are unique, because the first support point lies on an edge of the design region and cannot therefore be represented as a convex combination of further support points on the hyperplane f.x/ T β = x Å. Corollary 1. Let the assumptions of theorem 6 be satisfied and let ψp,a 1.0/ β i.v i u i / for all i for which c i 0. Let ξ Å i be the unique c-optimal designs for β i with support points a {ψp,a 1.0/=β ie i and a, and let λ i = β i c i =.c T β/ for i = 1, :::, p 1. Then the design ξ Å = Σ p 1 i=1 λ iξ Å i, whose support points are only on the edges of the design region, is a c-optimal design for the vector c =.0, c 1, :::, c p 1 / T. Corollary 1 follows from theorem 6, part (b), since the convex combination of the support points a {ψp,a 1.0/=β ie i of the designs ξ Å i equals the first support point of the c-optimal twopoint design (3.3). The c-optimal designs may have a support point in the interior of the design region, since there is no strictly isotonic version of c-optimality and the matrix A in the equivalence theorem for c-optimality is only positive semidefinite. By theorem 3 and corollary 1, however, a c-optimal design with all support points on the edges of the design region exists. In toxicity studies the design variables are concentrations or doses of toxicants, which are non-negative. The parameters β 1, :::, β p 1 are typically negative, since the toxicants have a negative effect on the number of organisms of interest, which is often assumed to be Poisson distributed (see Wang et al. (2006)). Such a situation is considered in the following example Example 1 The Poisson model with two covariates is considered. Let X = [0, 10] [0, 10] and β =.0, 1, 1/ T. We are looking for the c-optimal design for c =.0, 1, 1/ T.Wehaveψ2, / = 2[1 + W{exp. 1/] 2:557. By theorem 6, part (a), the two-point design ξ Å.1:278, 1:278/.0, 0/ = 0:782 0:218 is a c-optimal design. Let ξ Å i for i = 1, 2 be the unique c-optimal designs for β i with support points 2:557e i and 0 2 =.0, 0/ and corresponding weights 0:782 and 0:218. By corollary 1 the following design with support points only on the edges of X, which has a regular information matrix, is also a c-optimal design for c =.0, 1, 1/ T :

8 108 D. Schmidt ξ Å = 1 2 ξå :557, 0/.0, 2:557/.0, 0/ 2 ξå 2 = : 0:391 0:391 0: L-optimal designs We consider a multiple-regression model with p 1 covariates, including the case p = 2 with one covariate, so p 2. We have f.x/ =.1, x T / T and x =.x 1, :::, x p 1 / T X. In what follows, B is a symmetric positive definite matrix. Theorem 8. Let X = [u 1, v 1 ] ::: [u p 1, v p 1 ] and let assumptions 1 4 be satisfied. For i = 1, :::, p 1 let a i = v i if β i > 0 and a i = u i if β i < 0. Let the matrix S =.s i,j / i,j=1,:::,p be defined as S =. X T / 1 B X 1 as in expression (2.10) for a design with support points a.x 1 =β 1 /e 1, :::, a.x p 1 =β p 1 /e p 1 and a =.a 1, :::, a p 1 / T. For i = 1, :::, p 1 let x Å i be the unique solutions of the following system of equations in the interval.0, /: x i 2 Q{ f.a/ T [ β x i Q{f.a/ Q T β x i f.a/ T 1 β x i Q{f.a/ T β s 1, i+1.s1,1 s i+1,i+1 / ] = 0:.4:1/ If x Å i β i.v i u i / holds for i = 1, :::, p 1, then the design with the support points a.x Å 1 =β 1/e 1, :::, a.x Å p 1 =β p 1/e p 1 and a and corresponding weights in expression (2.10) is the unique L-optimal design. We note that in equations (4.1) the entries s i,j of the matrix S depend on the variables x 1, :::, x p 1. From the proof of theorem 8 it follows that the unique solutions x Å i of the system of equations (4.1) lie in the interval.0, 4Q{f.a/ T β=q {f.a/ T β/. This system of equations must be solved numerically. It can be written as a fixed point equation x = k.x/. Starting with some vector x 0, we can use the fixed point iteration x n+1 = k.x n / to determine a solution. In numerical investigations this method always converged to the fixed point. The L-optimal designs in theorem 8 have the vertex a of the design region as a support point, at which the efficiency function is maximized. For the censored data models that are considered here, a is the point of the design region where censoring is most unlikely. The other support points lie on those edges of the design region that are incident to a with distance x Å i = β i. In contrast with the D-optimality criterion, the L-optimal weights will generally not be equal. For the model with a single covariate, i.e. for p = 2, it can be easily shown that the L-optimal design has both boundary points u and v as its support points, when the design region is so small that the solution of equation (4.1) is outside the design region Example 2 We consider the situation of example 1. We are looking for the A-optimal design, so B = I 3. The entries of the matrix S of theorem 8 are given by s 1,1 = 1 + x1 2 + x2 2, s 1,i+1 = exp.x i =2/xi 2 and s i+1,i+1 = exp.x i =2/xi 2 for i = 1, 2. The solutions of the system of equations (4.1) are given by x Å 1 = xå 2 = 2:245. By theorem 8 the following design is an A-optimal design: ξ Å.2:245, 0/.0, 2:245/.0, 0/ = : 0:349 0:349 0:302.

9 5. φ k -optimal designs Characterization of Optimal Designs 109 In this section we determine φ k -optimal designs, k.,1/, for a model with p 1 covariates, p 2, f.x/ =.1, x T / T and x =.x 1, :::, x p 1 / T X. Theorem 9. Let X = [u 1, v 1 ] ::: [u p 1, v p 1 ] and let assumptions 1 4 be satisfied. For i = 1, :::, p 1 let a i = v i if β i > 0 and a i = u i if β i < 0. Let the matrix S be defined as S =. X T / 1 M.ξ, β/ k+1 X 1 as in expression (2.11) for a design with support points a.x 1 =β 1 /e 1, :::, a.x p 1 =β p 1 /e p 1 and a =.a 1, :::, a p 1 / T. If a solution exists, let x Å i.0, / and ωå i for i = 1, :::, p 1 be the unique solutions of the common system of equations (2.11) and (5.1): [ x i 2 Q{f.a/T β x i Q{f.a/ T ] β x i s 1, i+1 Q {f.a/ T 1 β x i Q{f.a/ T = 0:.5:1/ β.s1,1 s i+1,i+1 / If x Å i β i.v i u i / holds for i = 1, :::, p 1, then the design ξ Å k with support points a.x Å 1 =β 1/e 1, :::, a.x Å p 1 =β p 1/e p 1 and a and corresponding weights ω Å 1, :::, ωå p is the unique φ k -optimal design. Theorem 9 is similar to theorem 8 for L-optimality but differs in the definition of the matrix S. Since S depends on the weights, the system of equations to be solved additionally includes the equations for the optimal weights. This common system of equations must be solved numerically. The existence of a solution is not proved; however, in numerical investigations a solution always existed. As in the case of L-optimality, the system of equations can be written as a fixed point equation and solved with iterative methods. Remark 3. For D-optimality we have S=diag.ω 1, :::, ω p / and so s 1,i+1 =0fori=1, :::, p 1. Thus, the system of equations (5.1) simplifies to p 1 identical equations of the form φ a.x i / = 0 with solutions x i = φ 1 a.0/, where the function φ a is defined by φ a.x/ := x 2Q{f.a/ T β x=q {f.a/ T β x. This result coincides with that of Schmidt and Schwabe (2017) for D-optimality. In particular, the D-optimal design is unique Example 3 We consider the situation of example 1. We are looking for the φ k -optimal design for k = 2. The common system of equations (2.11) and (5.1) has the solutions x Å 1 =xå 2 =2:369, ωå 1 =ωå 2 =0:352 and ω Å 3 = 0:296. The φ 2-optimal design is given in Table 2. For comparison, the D-, A- and c-optimal designs (see examples 1 and 2), the last for c =.0, 1, 1/ T,aregiveninTable2for the same model. As c-optimal design, the three-point design with all support points on the boundary of the design region was chosen. To compare the various designs, the efficiency is calculated, which sets the value of the criterion function of a design ξ in relation to the value of the criterion function for the optimal design ξ Å and can thus take values between 0 and 1. For example, the φ k -efficiency is given by eff φk.ξ, β/ = φ k {M.ξ, β/=φ k {M.ξ Å, β/. All designs have relatively high efficiencies with respect to the other optimality criteria, which can be explained by the similar structure of the designs. 6. Application to censoring In this section we investigate the behaviour of the optimal designs in the case of censoring for various optimality criteria, numbers of covariates, amounts of censoring and parameter values. We consider the proportional hazards model with type I and uniform random censoring, for

10 110 D. Schmidt Table 2. criteria Comparison of optimal designs for the Poisson model for various optimality Criterion Optimal design Efficiencies D-optimality (φ 0 ) c-optimality A-optimality (φ 1 ) φ 2 -optimality {.2, 0/.0, 2/.0, 0/ :557, 0/.0, 2:557/.0, 0/ 0:391 0:391 0:218 {.2:245, 0/.0, 2:245/.0, 0/ 0:349 0:349 0:302 {.2:369, 0/.0, 2:369/.0, 0/ 0:352 0:352 0:296 D c A φ which the efficiency functions are given in Table 1. In the case of no censoring the Fisher information matrix is given by I.x/ = f.x/f.x/ T (see Konstantinou et al. (2014)), which is equal to the Fisher information for the linear model. A common situation in survival analysis is that the effect parameters are negative and the design variables can take only non-negative values. For example, in clinical trials the doses of drugs administered to patients are non-negative and the effect parameters are negative, because the drug is expected to decrease the hazard rate. Let X = [0, 3] p 1 and β =.3, 2:5 1 T p 1 /T, where 1 p 1 =.1, :::,1/ T R p 1. First, we investigate how censoring affects the optimal design for the case of one, two and three covariates, so p {2, 3, 4. Forp = 2 let ξ 1 be the design with equally weighted support points 0 and 3. For p>2 let ξ p 1 be the p-fold product design ξ p 1 = ξ 1 ::: ξ 1 with all vertices of the design region as support points and equal weights. For example, for p=3, the design ξ 2 has the support points.0, 0/,.3, 0/,.0, 3/ and.3, 3/ with weights 4 1. These designs are D-optimal designs in the linear model (see Schwabe (1996)). They are also c-optimal designs for β 1 in the linear model, but not A-orφ 2 -optimal designs. For A- and φ 2 -optimality the support points of an optimal design stay the same, but the weights differ from equal allocation. The amount of censoring q, which is the overall probability of censoring, is given by q = 1 Σω j P.Y j <C/ (see Kalish and Harrington (1988)), where Y j is the survival time and C is the censoring distribution. The censoring time c is chosen such that q is equal to 60% for the two-covariate case if the balanced design ξ 2 had been used. We obtain c =32 for type I censoring and c =69 for random censoring. The amounts of censoring for p=2 and p=4 are given by 35% for ξ 1 and 76% for ξ 3 for both models. The optimal designs in the proportional hazards model with type I and random censoring obtained from theorems 6, 8 and 9 as well as the efficiencies of the designs ξ p 1 are given in Tables 3 6 for the various optimality criteria. The results for type I and random censoring are quite similar. The optimal designs have the vertex 0 p 1 as support point, which is the point of the design region where censoring is most unlikely. The distance of the other support points to 0 p 1 can be seen as the result of the trade-off between minimizing the amount of censoring and maximizing the distance between the support points. For both censoring schemes the efficiencies of the balanced designs ξ p 1 decrease with increasing number of covariates for all optimality criteria, and they are quite low for two or more covariates.

11 Characterization of Optimal Designs 111 Table 3. D-optimal designs for proportional hazards models and D-efficiencies of ξ p 1 p D-optimal design D-efficiency D-optimal design D-efficiency for type I censoring of ξ p 1 for random censoring of ξ p { 2: { 2 2.2:337, 0/.0, 2:337/.0, 0/ { :337e1 2:337e 2 2:337e { 2: { 2 2.2:385, 0/.0, 2:385/.0, 0/ { :385e1 2:385e 2 2:385e Table 4. c-optimal designs for proportional hazards models and c-efficiencies of ξ p 1 p c-optimal design c-efficiency c-optimal design c-efficiency for type I censoring of ξ p 1 for random censoring of ξ p : :553 0:447.2:565, 0/.0, 0/ 0:553 0:447.2:565, 0, 0/.0, 0, 0/ 0:553 0: : :572 0:428.2:628, 0/.0, 0/ 0:572 0:428.2:628, 0, 0/.0, 0, 0/ 0:572 0: Table 5. A-optimal designs for proportional hazards models and A-efficiencies of ξ p 1 p A-optimal design A-efficiency A-optimal design A-efficiency for type I censoring of ξ p 1 for random censoring of ξ p 1 2 2: :302 0: :426, 0/.0, 2:426/.0, 0/ 0:224 0:224 0: :422e 1 2:422e 2 2:422e :178 0:178 0:178 0: : :315 0:685.2:483, 0/.0, 2:483/.0, 0/ 0:231 0:231 0:538 2:478e1 2:478e 2 2:478e :183 0:183 0:183 0: To examine the effect of the amount of censoring q on the optimal designs, we consider the two-covariate case and change the censoring time c to obtain overall censoring probabilities of 40% and 80% for ξ 2. Such amounts of censoring are quite common in survival analysis (see Klein and Moeschberger (2003), chapter 1). Table 7 summarizes the results. We observe that with increasing amounts of censoring the optimal designs shift the support points on the edges towards the vertex 0 p 1, where censoring is most unlikely. The overall probability of censoring under the optimal design, which is given in Table 8 for the various optimality criteria, is less than for the balanced design ξ 2. The efficiencies of the balanced designs usually decrease with increasing amounts of censoring and are quite low for q = 0:8. Interestingly, for φ 2 -optimality the efficiencies are higher for q = 0:6 than for q = 0:4. We now consider a different set of parameter values with a larger covariate effect, so let β =.3, 3, 3/ T. The results for the two-covariate case with c = 32 for type I censoring and

12 112 D. Schmidt Table 6. φ 2 -optimal designs for proportional hazards models and φ 2 -efficiencies of ξ p 1 p φ 2 -optimal design for φ 2 -efficiency φ 2 -optimal design for φ 2 -efficiency type I censoring of ξ p 1 random censoring of ξ p 1 2 2: :260 0: :486, 0/.0, 2:486/.0, 0/ 0:193 0:193 0: :478e 1 2:478e 2 2:478e :154 0:154 0:154 0: : :274 0:726.2:545, 0/.0, 2:545/.0, 0/ 0:202 0:202 0:597 2:537e1 2:537e 2 2:537e :160 0:160 0:160 0: Table 7. Optimal designs for proportional hazards models for various amounts of censoring and efficiencies of ξ 2 Criterion q Optimal design Efficiency Optimal design Efficiency for type I censoring of ξ 2 for random censoring of ξ 2 D 0.4 D 0.8 c 0.4 c 0.8 A 0.4 A 0.8 φ φ :773, 0/.0, 2:773/.0, 0/ { :876, 0/.0, 0:876/.0, 0/ { :979, 0/.0, 0/ 0:541 0:459.1:124, 0/.0, 0/ 0:747 0:253.2:844, 0/.0, 2:844/.0, 0/ 0:204 0:204 0:591.1:034, 0/.0, 1:034/.0, 0/ 0:376 0:376 0:249.2:899, 0/.0, 2:899/.0, 0/ 0:168 0:168 0:663.1:073, 0/.0, 1:073/.0, 0/ 0:376 0:376 0: :878, 0/.0, 2:878/.0, 0/ { :936, 0/.0, 0:936/.0, 0/ { , 0/.0, 0/ 0:545 0:455.1:196, 0/.0, 0/ 0:726 0:274.2:962, 0/.0, 2:962/.0, 0/ 0:209 0:209 0:583.1:098, 0/.0, 1:098/.0, 0/ 0:363 0:363 0:274.3, 0/.0, 3/.0, 0/ 0:173 0:173 0:655.1:141, 0/.0, 1:141/.0, 0/ 0:363 0:363 0: c = 69 for random censoring are given in Table 9. The larger covariate effect causes an increase in the overall probability of censoring to 71% for both models. Hence the results are basically similar to the previous case, where a higher amount of censoring was investigated. The support points of the optimal design move closer to the support point at the vertex and the efficiencies of the balanced designs ξ 2 decrease. Our results show that using the balanced designs, which are also the D- and c-optimal designs in the linear model, can lead to very poor performance, especially if the amount of censoring is high. Hence, censoring should be taken into account when planning the experiment and the optimal designs that are derived in this paper should be used. 7. Discussion Until now, for non-linear multiple-regression models only few results concerning optimal de-

13 Characterization of Optimal Designs 113 Table 8. Censoring probabilities under the optimal designs for various optimality criteria for p D 3 Results for q = 0.4 Results for q = 0.6 Results for q = 0.8 D c A φ 2 D c A φ 2 D c A φ 2 Type I Random Table 9. Optimal designs for proportional hazards models for β D.3, 3, 3/ T and efficiencies of ξ 2 Criterion Optimal design for Efficiency Optimal design for Efficiency type I censoring of ξ 2 random censoring of ξ 2 D c.2:137, 0/.0, 0/ {.1:948, 0/.0, 1:948/.0, 0/ { :553 0:447 A.2:031, 0/.0, 2:031/.0, 0/ 0:240 0:240 0:520 φ 2.2:081, 0/.0, 2:081/.0, 0/ 0:214 0:214 0: {.1:987, 0/.0, 1:987/.0, 0/ { :190, 0/.0, 0/ 0:572 0:428.2:079, 0/.0, 2:079/.0, 0/ 0:248 0:248 0:505.2:130, 0/.0, 2:130/.0, 0/ 0:223 0:223 0: sign are known for D-optimality and even less is known for c-, L- and φ k -optimality. In this paper optimal designs for a large class of commonly used regression models were determined. The structure of these designs is similar in terms of location of support points for many criteria, at least for D-, L- and φ k -optimality. The optimal designs have a minimal support, which is particularly advantageous with many covariates, since such designs can be more easily run in practice than designs with more support points like full factorial designs. Theorems 8 and 9 require some conditions to be satisfied, which ensure that the support points are inside the design region. If these conditions are not satisfied, then designs with more support points may be optimal. However, the design, which has the vertices for those components, where the corresponding inequality is not satisfied, as support points, is often optimal or has a very high efficiency. For c-optimality it is not clear which design is optimal, if the given conditions are not satisfied, and this is a topic for further research. Whereas for D-optimality only the unique solution of a non-linear equation must be found, a non-linear system of equations must be solved for L- and φ k -optimality. Although the system of equations, particularly for φ k -optimality, may be a difficulty in computing an optimal design and finally must be solved numerically, these results offer advantages over a direct numerical approach for the computation of optimal designs. If a solution of the system of equations is found, there is certainty that the constructed design is indeed optimal. Numerical optimization methods may yield only a local maximum or minimum. Moreover, these optimization methods are generally more complex to establish. A natural extension of the present results is the determination of c-optimal designs for the vectors c that were not considered here. Furthermore, the multiple-regression design problem for the case k = of φ k -optimality corresponding to E-optimality remains open. The locally

14 114 D. Schmidt optimal designs that were determined in this paper can be used to compute standardized maximin optimal designs, which maximize the minimal efficiency with respect to the parameters (see Dette (1997) and Konstantinou et al. (2014)). Such designs are robust to parameter misspecifications and are a method to overcome the problem of the parameter dependence of locally optimal designs. Acknowledgements The authors thanks the two referees and Associate Editor for their constructive and helpful comments. Appendix A Theorem 10 (Elfving, 1952). Let f.x/ := [ Q{f.x/ T β] f.x/. A design ξ Å with support points x 1, :::, x m and corresponding weights ω 1, :::, ω m is a c-optimal design for c T β if and only if there are " 1, :::, " m { 1, 1 and γ > 0 such that γc is a boundary point of the Elfving set R := conv[{ f.x/ x X { f.x/ x X] and γc = Σ m i=1 ω i" i f.x i /. Lemma 4. Let Q satisfy assumptions 1, 2 and 4 and let p be an arbitrary polynomial. Then Q.θ/ decreases exponentially fast to 0 as θ tends to : lim θ p.θ/q.θ/ = 0. Proof. Let n 0 be the degree of p.θ/. First, we note that h.θ/ := θ +.n + 1/Q.θ/=Q.θ/ is a strictly increasing and bijective function by assumption 4. It has exactly one root, which is located at θ < 0, since Q.θ/ and Q.θ/ are positive. We now consider the function r.θ/ := θ n+1 Q.θ/. Its derivative is given by r.θ/ =.n + 1/θ n Q.θ/ + θ n+1 Q.θ/ = θ n Q.θ/h.θ/. This derivative has a root at 0 and a further root at θ < 0. The function r cannot have a saddle point at θ, since its derivative changes sign at θ. Ifnis odd, r has a maximum at θ, because r.0/ = 0 and r.θ/>0 for all θ 0. If n is even, r has a minimum at θ, because r.0/ = 0 and r.θ/<0 for all θ < 0. Hence r.θ/ is monotonic for θ < θ and bounded by zero for θ.it follows that r.θ/ converges for θ and we conclude that lim θ p.θ/q.θ/ = 0. A.1. Proof (theorem 3) Let x be a support point which is not on an edge and let η =f. x/ T β. Let x 1, :::, x m be the intersection points of the hyperplane H η = {x R p 1 : f.x/ T β = η with the edges of the design region. Then x = Σ m i=1 λ ix i is a convex combination of the x i with λ i.0, 1/ for at least two i {1, :::, m, and we have f. x/ = Σ m i=1 λ if.x i /. By the inequality in lemma 8.4 in Pukelsheim (1993), pages , it follows that I. x, β/ = Q.η/f. x/f. x/ T Q.η/ m λ i f.x i /f.x i / T = m λ i I.x i, β/ and in particular I. x, β/ Σ m i=1 λ ii.x i, β/. Replacing those support points of a design ξ, which do not lie on an edge, by such convex combinations of points on the edges as above, we obtain a design ξ with M.ξ, β/ M.ξ, β/. The support points of ξ are the support points of ξ on the edges with their original weights and the points on the edges from the convex combinations with weights λ i ξ. x/. If some of these support points occur multiple times, then their weights are added. i=1 i=1 A.2. Proof (lemma 1) Let p.x/ := f.x/ T Af.x/ and h.x/ := p.x/ b=q{f.x/ T β. Then condition (2.8) is equivalent to h.x/ 0. We consider the function h on an arbitrary edge K of X, which can be written as K = {x R p 1 : x = s + yr, y I R, where s is a point on K, r is the direction vector of K and I is an interval. Now, p.y/ := p.s + yr/ is a quadratic polynomial whose second derivative is constant. Since the second derivative of 1=Q is injective by assumption 3, it follows that the second derivative of h.y/ := h.s + yr/ is also injective and has at most one root. By Rolle s theorem h can have no more than two roots, so h has at most two extrema. An optimal design can therefore have no more than two support points per edge.

15 A.3. Proof (theorem 4) Let ( ) 1 0 T = : β 0 β 1 Characterization of Optimal Designs 115 With the canonical transformation f.x/ f.z/=tf.x/ the minimization of c T M.ξ, β/ c is equivalent to the minimization of cz TM z.ξ/ c z with information matrix M z.ξ/=σ m i=1 ω iq.z i /f.z i /f.z i / T, c z =Tc, transformed parameter vector β z =.0, 1/ T and induced design region Z =β 0 +β 1 X=:[ũ, ṽ] (see Ford et al. (1992)). We first show that the curve f.z/ = Q.z/ f.z/ = Q.z/.1, z/ T, z [ũ, ṽ], is strictly convex. For this purpose, we define x.z/ := Q.z/ and y.z/ := Q.z/z. Then we have dy dx = z + 2Q.z/ Q.z/ and d 2 { y dx = d 2 dz 2 Q.z/ Q.z/ : Q.z/ Q.z/ The function Q.z/=Q.z/ is increasing by assumption 4, so its derivative is non-negative. Since Q.z/ and Q.z/ are positive, it follows that d 2 y=dx 2 > 0, which yields the strict convexity. Hence there is a unique point f.z Å / such that the boundary of the Elfving set consists of the two line segments joining f.ṽ/ to f.z Å / and joining f.ṽ/ to f.z Å / and of the curves f.z/ and f.z/, z [z Å, ṽ]. Fig. 2 shows the geometry of the Elfving set. The line segment joining f.ṽ/ and f.z Å / must be the tangent at the point f.z Å /, provided that the design region is sufficiently large. Then z Å is given by the solution of Q.z Å /z Å { Q.ṽ/ṽ Q.z Å = z Å + 2Q.zÅ / / { Q.ṽ/ Q.z Å / ṽ zå 2Q.zÅ { / Q.z Å / Q.z Å + 1 = 0: / Q.ṽ/ With z =ṽ z Å this equation is given by ψ 2,ṽ.z/ = 0 with solution z = ψ 1 2,ṽ.0/. Hence zå =ṽ ψ 1 2,ṽ.0/. If the line that is defined by the vector c z intersects the boundary of the Elfving set at the line segment joining f.ṽ/ and f.z Å / or at the line segment joining f.ṽ/ and f.z Å /, then the design with support points z Å and ṽ is the c-optimal design by Elfving s theorem. This is so if and only if c z,2 =c z,1 [z Å, ṽ]. The c-optimal weights follow from expression (2.9). For c z,2 =c z,1 [z Å, ṽ] a one-point design is the c-optimal design. Its support point is c z,2 =c z,1 since c z f.c z,2 =c z,1 /. If ṽ ψ 1 2,ṽ.0/<ũ, then the boundary of the Elfving set consists of the two line segments joining f.ṽ/ to f.ũ/ and joining f.ṽ/ to f.ũ/ and of the curves f.z/ and f.z/, z [ũ, ṽ]. The c-optimal designs follow with the same reasoning as above. The back-transformation yields the c-optimal designs of theorem 5. A.4. Proof (theorem 6) A.4.1. Part (a) We give the proof for the case β i c i 0, i = 1, :::, p 1. Since c-optimality is independent of the sign of the Fig. 2. Elfving set

16 116 D. Schmidt vector c, the results are also valid if β i c i 0fori = 1, :::, p 1. Let T =.e 1,p, β, D/ T, where e j,p denotes the jth standard unit vector in R p, D =.0 p 2, 0 p 2, diag.β 2, :::, β p 1 // T and 0 p 2 =.0, :::,0/ T R p 2. With the canonical transformation f.x/ f.z/ = Tf.x/ the minimization of c T M.ξ, β/ c is equivalent to the minimization of cz TM z.ξ/ c z with M z.ξ/ = Σ m j=1 ω jq.z j,1 /f.z j /f.z j / T, c z = Tc =.0, Σ p 1 i=1 β ic i, β 2 c 2, :::, β p 1 c p 1 / T and transformed parameter vector β z = e 2,p (see Ford et al. (1992)). In the transformed model the components z i, i = 2, :::, p 1, of a covariate z can take values from the interval [ũ i, ṽ i ], where [ũ i, ṽ i ]:=[β i u i, β i v i ]forβ i > 0 and [ũ i, ṽ i ]:=[β i v i, β i u i ]forβ i < 0. For the component z 1 we have z 1 [Σ p 1 i=1 ũi, Σ p 1 i=1 ṽi], where [ũ 1, ṽ 1 ]:=[β 0 + β 1 u 1, β 0 + β 1 v 1 ]forβ 1 > 0 and [ũ 1, ṽ 1 ]:=[β 0 + β 1 v 1, β 0 + β 1 u 1 ]for β 1 < 0. These components are not independent of each other, since z 1 = f.x/ T β = β 0 + β 1 x 1 + Σ p 1 i=2 z i.in what follows let ṽ :=.Σ p 1 i=1 ṽi, ṽ 2, :::, ṽ p 1 /. The surface f.z/ = Q.z 1 / f.z/ = Q.z 1 /.1, z 1, :::, z p 1 / T coincides for the first two components with the two-dimensional curve for the case of a single covariate. Hence, we conclude by analogy with the proof of theorem 5 that the surface connecting the point f.ṽ/ with the points f.z Å, z 2, :::, z p 1 / is on the boundary of the Elfving set, where z Å = Σ p 1 i=1 ṽi ψ 1 2, ṽ 1 + ::: +ṽ p 1.0/ with the function ψ for the model with a single covariate and β =.0, 1/ T. For the components z i, i = 2, :::, p 1, we must have z Å Σ p 1 i=2 z i [ũ 1, ṽ 1 ] because of the dependence on the first component, which has the value z Å. Let z Å i :=ṽ i.β i c i /=.Σ p 1 j=1 β jc j / ψ 1 2, ṽ 1 + ::: +ṽ p 1.0/ for i = 1, :::, p 1 and ω Å ( p 1 ) 1 Q /{ ( = p 1 ) ṽ i Q.z Å / + Q ṽ i i=1 i=1 and ω Å 2 = Q.z Å / ( /{ p 1 ) Q.z Å / + Q ṽ i : i=1 For the point z Å :=.z Å, z Å 2, :::, zå p 1 / to lie in the design region, we must have zå i >ũ i for i=2, :::, p 1 and z Å Σ p 1 i=2 zå i = z Å 1 > ũ 1. Then f. z Å / as well as every point on the line segment to f.ṽ/ is on the boundary of the Elfving set. It follows that ω Å 1 { f. z Å / + ω Å 2 f.ṽ/ = { ( Q.z Å p 1 ) /Q ṽ i i=1 ( p 1 ) Q.z Å / + Q ṽ i i=1 ψ 1 2, ṽ 1 + ::: +ṽ p 1.0/ c p 1 z = γc z, β j c j j=1 where γ is the prefactor. By Elfving s theorem the design with the two support points z Å and ṽ and corresponding weights ω Å 1 and ωå 2 is a c-optimal design for the vector c z. A.4.2. Part (b) There may be an arbitrary number of further c-optimal designs with m 3 support points. Let.z Å, z1 T /, :::,.z Å, zm 1 T / X {z Rp 1 : f.z/ T β z = z 1 = z Å be m 1 different points such that Σ m 1 i=1 λ i.z Å, zi T/ = zå holds with λ i [0, 1] and Σ m 1 i=1 λ i = 1. It follows that Σ m 1 i=1 λ iω Å 1 { f.zå, zi T/ + ωå f.ṽ/ 2 = γc z and hence the design ξ Å {.z Å, z T = 1 / :::.z Å, zm 1 T / ṽ λ 1 ω Å 1 ::: λ m 1 ω Å 1 ω Å 2 is also a c-optimal design. The back-transformation yields the result of theorem 6. A.5. Proof (theorem 8) First, we consider the canonical design problem with parameter vector β =.β 0,1,:::,1/ T and the extended

17 design region.,0] p 1,sox i.,0]fori = 1, :::, p 1. Let p i.x/ := f.xe i / T M.ξ, β/ 1 BM.ξ, β/ 1 f.xe i / Characterization of Optimal Designs 117 for i=1, :::, p 1. Since the hyperplanes H η ={x R p 1 :f.x/ T β =η are bounded on.,0] p 1, a design ξ is an L-optimal design by theorems 1 and 2 if and only if Q{f.x i e i / T β p i.x i / tr{m.ξ, β/ 1 B 0 holds for all x i 0 and for i = 1, :::, p 1. This condition is equivalent to h i.x i / := p i.x i / tr{m.ξ, β/ 1 B 0:.A.1/ Q{f.x i e i / T β From the positive definiteness of the matrix M.ξ, β/ 1 BM.ξ, β/ 1 it follows that the quadratic polynomials p i.x i / are positive for all x i R. They tend to for x i, and so does h i.x i /, because Q is a strictly increasing function. By lemma 4 it follows that lim x i h i.x i / = lim x i p i.x i /Q.β 0 + x i / tr{m.ξ, β/ 1 B = : Q.β 0 + x i / From the proof of lemma 1 it follows that h i.x i / has at most two extrema, so an L-optimal design can have at most two support points per edge. Since for an L-optimal design equality holds in expression (A.1) at the support points, the limit considerations imply that one of the two possible support points on an edge must be located at the vertex 0 p 1 =.0, :::,0/ T. At the other support point there must be a maximum of h i.x i /. Between these two support points there is a minimum of h i.x i /. Further extrema of h i.x i / do not exist. Hence, the support points of an L-optimal design ξ Å must be given by x Å 1 e 1, :::, x Å p 1 e p 1 and 0 p 1 with L-optimal weights in expression (2.10), where x Å i < 0fori = 1, :::, p 1. According to lemma 2, the information matrix of such a design ξ Å can be written in the form M.ξ Å, β/ = X T D ω X = X T Q 1=2 D ω Q 1=2 X with diagonal matrices Q 1=2 =diag{ Q.β 0 /, Q.β 0 + x Å 1 /, :::, Q.β 0 + x Å p 1 / and D ω = diag.ω Å 1, :::, ωå p /. The matrix X and its inverse are given by ( ) 1 0 T X = p 1 1 p 1 diag.x Å / and ( ) 1 0 X 1 T = p 1 y Å diag.y Å / with vectors x Å =.x Å 1, :::, xå p 1 /T, 1 p 1 =.1, :::,1/ T and y Å =.1=x Å 1, :::,1=xÅ p 1 /T. With S =. X T / 1 B X 1 defined as in expression (2.10) for the design ξ Å, we obtain p i.x i / = f.x i e i / T. X T D ω X/ 1 B. X T D ω X/ 1 f.x i e i / =.1, x i e T i / X 1 D 1 ω. X T / 1 B X 1 D 1 ω. X T / 1.1, x i e T i /T =.1, x i e T i /X 1 Q 1=2 D 1 ω SD 1 ω Q 1=2.X T / 1.1, x i e T i /T : We have.1, x i e T i /X 1 =.1 x i =x Å i,.x i=x Å i /et i /,so p i.x i / =.1 x i=x Å i /2 s 1,1.ω Å 1 /2 Q.β 0 / ( p = j=1 +.x i=x Å i /2 s i+1,i+1.ω Å i+1 /2 Q.β 0 + x Å i / x i =x Å i /.x i=x Å i /s 1,i+1 ω Å 1 ωå i+1 Q.β0 / Q.β 0 + x Å i / sjj ) 2 {.1 xi =x Å i /2 Q.β 0 / +.x i=x Å i /2 Q.β 0 + x Å i / x i=x Å i /.x i=x Å i /s 1,i+1=.s1,1 s i+1,i+1 / Q.β0 / Q.β 0 + x Å i / : We also have tr{m.ξ Å, β/ 1 B = tr{ X 1 D 1 ω. X T / 1 B = tr.d 1 ω S/ = p i=1 s ii ω Å i ( ) 2 p = sjj : j=1