MINIMAX OPTIMAL DESIGNS IN NONLINEAR REGRESSION MODELS

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1 Statistica Sinica 8(998, MINIMAX OPTIMAL DESIGNS IN NONLINEAR REGRESSION MODELS Holger Dette and Michael Sahm Ruhr-Universität Bochum Astract: We consider the maximum variance optimality criterion of Elfving (959 in the context of (nonlinear response models. Some practical guidelines for the construction of minimax optimal designs are given. In some cases this criterion yields one point designs as a consequence of different scales of the elements in the Fisher information matrix. As an alternative a standardized maximum variance criterion is introduced and applied to the calculation of efficient designs. The results are illustrated for inary response models and it is demonstrated that in these models standardized minimax optimality should e prefered to ordinary minimax optimality. Key words and phrases: Binary response models, maximum variance criterion, nonlinear regression models, optimal design, standardized variances.. Introduction Consider a response variale y having a distriution from an exponential family p(y x, θ with Fisher information I(θ, x forθ given an oservation at x. Here θ =(θ,...,θ m T denotes an m-dimensional vector of unknown parameters and x is the explanatory variale which varies in a design space X IR k. A design is a proaility measure on X with finite support and the matrix I(θ, ξ = I(θ, xdξ(x IR m m X is called the Fisher information matrix of ξ. A (locally optimal design maximizes an optimality criterion depending on I(θ, ξ (see Silvey (980, Atkinson and Donev (99 or Pukelsheim (993. In this paper we consider the prolem of constructing optimal designs for inary response models with respect to the partial maximum variance criterion introduced y Elfving (959 Φ(ξ =max i M et i I (θ, ξe i, (. where ξ is a design for which the parameters {θ i } i M are estimale, e i IR m denotes the ith unit vector (i =,...,m, M {,...,m} and I (θ, ξ denotes

2 50 HOLGER DETTE AND MICHAEL SAHM a generalized inverse of I(θ, ξ. In many cases the elements of the Fisher information matrix are of different scale. Therefore we also consider the minimum efficiency criterion ψ(ξ =max i M e T i I (θ, ξe i e T =/ min i I (θ, ηe eff θ i (ξ, (. i i M min η which compares the variances relative to their optimal value otainale y the choice of a design. More precisely, the quantities eff θi (ξ in (. are the efficiencies of the design ξ, relative to the optimal design for estimating the parameter θ i. Designs minimizing (. are called minimax optimal for the parameters {θ i i M}and have een discussed y many authors (see e.g. Murty (97, Torsney and López-Fidalgo (995, Krafft and Schaefer (995 or Dette and Studden (994. Following Dette (997 we call the designs minimizing (. standardized minimax optimal for the parameters {θ i i M}. If M = {,...,m} we riefly talk aout minimax or standardized minimax optimal designs. It is worthwhile to mention that a similar criterion was considered y Müller (995, who determined designs maximizing the minimum of A-efficiencies, where the minimum is taken with respect to the parameter θ. Note that in many cases of practical interest the optimality criteria (. and (. produce locally optimal designs which require prior knowledge of the unknown parameter θ. Some arguments in favour of local optimality criteria can e found in Ford, Torsney and Wu (99. For alternative optimality criteria see Wu (985, Chaloner and Larntz (989, Sitter (99. In Section we provide some theoretical ackground regarding minimax and standardized minimax optimality which is useful for the construction of optimal designs. We also present a procedure for determining minimax and standardized minimax optimal designs. Some guidelines for the construction of minimax optimal designs in inary response models are given in Section 3.. It turns out that for some parameter cominations the minimax optimal designs ecome inefficient in the sense that the numer of different dose levels is less than the numer of parameters in the model. This motivates the consideration of standardized minimax optimal designs for inary response models which are discussed in Section 3.. Finally, in Section 4 the results are illustrated for two inary response models, the doule exponential and the logistic regression model.. Constructing Minimax and Standardized Minimax Optimal Designs The equivalence theory of the minimax optimality criteria can e otained from Pukelsheim (993, page 75. More precisely, assume that {I(θ, x x X} is compact, consider a suset M {,...,m} and a design ξ for which e T j θ is

3 MINIMAX OPTIMAL DESIGNS IN NONLINEAR REGRESSION MODELS 5 estimale for all j M. Define ρ k =min η e T k I (θ, ηe k (where the minimum is taken over all designs for which e T k θ is estimale and { } N (ξ:= j M e T j I (θ, ξe j =max i M et i I (θ, ξe i { } Ñ (ξ:= j M ρ j e T j I (θ, ξe j =max ρ ie T i i M I (θ, ξe i ; then the design ξ is minimax optimal for the parameters {θ i i M} if and only if there exists a generalized inverse G of I(θ, ξ and nonnegative numers α j,j N(ξ, with sum equal to such that j N (ξ α j e T j GI(θ, xge j e T j I (θ, ξe j, (. for all x X, where equality holds on the support of ξ. Similarly the design ξ is standardized minimax optimal for the parameters {θ i i M}if and only if there exists a generalized inverse G of I(θ, ξ and nonnegative numers α j,j Ñ (ξ, with sum equal to such that j Ñ (ξ α j e T j GI(θ, xge j e T j I (θ, ξe j (. with equality on the support of ξ. The following two Lemmata are ovious from these considerations. Lemma.. A minimax optimal design ξ for the parameters {θ i i M}is also minimax optimal for the parameters {θ i i N(ξ }. A standardized minimax optimal design ξ for the parameters {θ i i M}is also standardized minimax optimal for the parameters {θ i i Ñ ( ξ}. Lemma.. If for all A Mwith #A k<#m the (standardized minimax optimal design for the parameters {θ i i A}is not (standardized minimax optimal for {θ i i M}, then in the generalized inverse of the Fisher information matrix of the (standardized minimax optimal design for {θ i i M}the largest diagonal entry has at least multiplicity k +. Lemma. suggests an iterative procedure for determining minimax optimal designs for the parameters {θ i i M}. For n =,...,#M we proceed as follows.. For all susets M n Mwith #M n = n we calculate the minimax optimal designs ξ n for the parameters {θ i i M n } under the constraint that the corresponding entries in the diagonal of the inverse fisher information matrix

4 5 HOLGER DETTE AND MICHAEL SAHM are all equal and check if ξ n is already minimax optimal for the parameters {θ i i M}. By (. this is equivalent to showing max e T j j M I (θ, ξn e j =max n j M et j I (θ, ξn e j. (.3. If (.3 is satisfied, then ξn is minimax optimal for the parameters {θ i i M} and the procedure stops. Otherwise we put n = n + and repeat step (. Although this procedure appears to e cumersome (ecause in the worst case one has to go #M steps, the results otained in the literature (see Murty (97, Krafft and Schaefer (995, Dette and Studden (994 suggest that in many regression models the algorithm stops after the first or second step. For the standardized minimax optimality criterion the procedure is similar, replacing (.3 y max ρ j e T j j M I (θ, ξn e j =max ρ je T j n j M I (θ, ξn e j. Moreover, the following Lemma shows that for standardized minimax optimality one can always start with n =. Lemma.3. If ξ is a standardized minimax optimal design for the parameters {θ i i M}and #M, then #N (ξ. Proof. Let ξ denote the standardized minimax optimal design and assume #Ñ (ξ =. Without loss of generality we put Ñ (ξ ={}. By Lemma. ξ is also optimal for estimating the parameter θ ;inotherwords =ρ e T I (θ, ξ e =max ρ je T j I (θ, ξ e j min ρ je T j I (θ, ξ e j. j M j M This proves Ñ (ξ =M, contradicting the assumption #M. 3. Minimax and Standardized Minimax Optimal Designs for Binary Response Models In this section we demonstrate the calculation of minimax and standardized minimax optimal designs in inary response models. Consider a inary response variale y with proaility of success p(x, θ, where θ is a two dimensional parameter. We discuss two different parametrizations p (x, θ=f ((x µ θ =(µ,, x IR (3. p (x, θ=f (a + x θ =(a,, x IR (3. that are commonly used in practice. Here F is a known distriution function with symmetric density function f, which is assumed to e differentiale for x 0. Define f (x h(x = F (x( F (x ; (3.3

5 MINIMAX OPTIMAL DESIGNS IN NONLINEAR REGRESSION MODELS 53 then the Fisher information for the parameter θ of a design ξ is given y ( I (θ, ξ = h((x µ (x µ (x µ (x µ dξ(x (3.4 if the parametrization (3. is used, and y ( x I (θ, ξ = h(a + x x x dξ(x if we use parametrization (3.. Locally optimal designs in these models for various differentiale optimality criteria have een discussed y several authors (see e.g. Wu (988, Adelasit and Placket (983, Sitter and Wu (993, Gaudard, Karson, Lindner and Tse (993. Note that the numer of support points of (locally optimal designs depends sensitively on the form of the function h in (3.3 (see e.g. Sitter and Wu (993. We will now give some general guidelines for the determination of minimax and standardized minimax optimal designs in these models; illustrative examples are presented in the following section. Throughout this paper we will assume that max h(t =h(0 (3.5 t IR and k =max t IR t h(t =c h(c,c >0 (3.6 which is satisfied for most of the commonly used inary response models. Moreover, the point c where the function t h(t attains its maximum is usually unique (up to its sign and can e otained from Tale 4 in Ford, Torsney and Wu ( Minimax optimal designs We concentrate on the parametrization (3.. Standard arguments show that there is a minimax optimal design symmetric with respect to the LD 50. Moreover, y the following lemma it suffices to consider symmetric designs with at most 4 support points. Lemma 3.. In a inary response model with parametrization (3. there exists a minimax optimal design with at most 4 support points which is symmetric to the LD50 µ. The proof is deferred to the appendix. Note, however, that this result does not depend on the inary response model ut holds for any regression model with two paramters having a corresponding symmetry property.

6 54 HOLGER DETTE AND MICHAEL SAHM Following the procedure in Section we first determine the optimal designs ξ µ and ξ for estimating the LD 50 and the slope. These are easily otained from Elfving s theorem (Elfving (95 and assumptions (3.5, (3.6 oserving ρ =inf ξ et I (θ, ξe = h(0 (3.7 ρ =inf ξ et I (θ, ξe = k. (3.8 This gives for ξ µ a one point design at µ and for ξ a two point design ( µ c ξ = µ + c, (3.9 where c>0 is defined y (3.6. Now ξ is minimax optimal if and only if e T I (θ, ξ e e T I (θ, ξ e which is equivalent to c. In the other case the inverse Fisher information matrix must have equal diagonal elements. We start with a two point design where the equality of the diagonal elements readily yields ( ξ0 µ µ+ =. (3.0 / / The minimax optimality of ξ0 can e checked via the inequality (. where the α j (N (ξ0 ={, } have to e chosen (due to equality in (. at the support points of the minimax optimal design and the differentiaility of h as α = α = h ( h(. If inequality (. is not fulfilled we increase the numer of support points successively. Thus in the next step we consider ξ w = ( µ z µ µ+ z w w w, (3. where z>0and h(z(z 4 w = 4 [h(0 h(z] + z h(z due to equality of the diagonal elements in I (θ, ξw. The optimal z minimizes h(0 z h(z z (3.

7 MINIMAX OPTIMAL DESIGNS IN NONLINEAR REGRESSION MODELS 55 and the weights α,α in the equivalence theorem are given y α = α = [ z 4 z h(z ]. (3.3 4 h (z Note that the expression in (3. is independent of the parameters µ and and the minimizing z is usually also unique. If (. holds for these weights, then ξ w is minimax optimal, otherwise the procedure has to e continued. Thus in the last step we have to consider designs of the form ( µ ξ z = µ z µ + z µ + z w w The equality of the diagonal entries of the Fisher information matrix yields w h(z ( 4 z w = h(z ( 4 z +h(z (z 4. The optimal values for z and z maximize the function w h(z h(z (z z h(z ( 4 z +h(z (z 4 under the constraint that z > >z > 0 (i.e. 0 <w<. Furthermore, the differentiaility of h implies that either h(z =h(z andh (z =h (z =0or. z (h(z h (z = z (h(z h (z and h(z h (z h(z (h(z h(z = z z. z The quantities α and α in the equivalence theorem are given y 4 (h(z h(z α = α = h(z ( 4 z +h(z (z 4. Recent results in the literature suggest that for most inary response models the symmetric optimal designs will have either two or three support points and are given y (3.9, (3.0 or (3., respectively (see Sitter and Wu (993 for some results regarding A-, D- and Fieller-optimality and also for the construction of a model with a symmetric D-optimal design supported at four points. For the parametrization (3. the approach is very similar and illustrated in the second part of Section Standardized minimax optimal designs We demonstrate in Section 4 that minimax optimal desgins for inary response models may ecome inefficient in the sense that the numer of different

8 56 HOLGER DETTE AND MICHAEL SAHM dose levels of the optimal design is less than the numer of parameters in the model. This disadvantage is usually caused y the different size of the elements in the Fisher information matrix. The standardization used in the criterion (. makes elements of different scale etter comparale and does not yield degenerate optimal designs (see Lemma.3. It turns out that in inary response models the standardized minimax optimal designs can e easily otained from the minimax optimal designs for special parameter cominations. More precisely, for the parametrization (3. a minimax optimal design ξ can e otained from ξ (x =η((x µ where η minimizes c c 0 c max{ c,c 0 }, (3.4 c j = y j h(ydη(y (j =0,,. Similarly, the standardized minimax optimal design ξ is given y ξ(x = η((x µ where η minimizes c c 0 c max{h(0c,c 0 k } = h(0k { h(0 c 0 c c max c, k k h(0 c 0 }. (3.5 This follows easily from definitions (., (3.7 and (3.8. Note that the optimization prolem in (3.5 does not depend on the parameters µ and. Moreover, the maximum of (3.5 is otained y maximizing (3.4 for = k/ h(0 which proves the following result. Lemma 3.. Let ξµ, denote the minimax optimal design for a inary response model with parametrization (3. such that (3.5 und (3.6 are satisfied. Then the standardized minimax optimal design ξ for this model is given y ξ(x =ξ 0,s ( s (x µ, where s = k / (h(0 /4 and k is defined in (3.6. For the parametrization (3. a similar result can e estalished. The proof is omitted for the sake of revity. Lemma 3.3. Let ξa, denote the minimax optimal design for a inary response model with parametrization (3. such that S = { ( } h(ω ω IR ω (3.6 is convex and lim ω ω h(ω =0. (3.7

9 MINIMAX OPTIMAL DESIGNS IN NONLINEAR REGRESSION MODELS 57 The standardized minimax optimal design for this model is given y ( x ξ(x =ξa,s, s where s = mc, c is defined in (3.6 and m = m (a =h(c inf ξ et I (θ, ξe = a c if a >c h(c h(a if a c. (3.8 The conditions (3.6 and (3.7 are satisfied for the commonly used regression models (see Sitter and Wu (993 or Ford, Torsney and Wu (99. It is also worthwile to mention that for oth parametrizations (3. and (3. the standardized minimax optimal designs are scale invariant in the sense that a scaling of the explanatory variale x transfers to the standardized minimax optimal design. The experimenter only has to scale the support points of the standardized minimax optimal design according to the transformation of the x- space. Moreover, y Lemma 3., the same argument applies for translations in inary response models with parametrization (3.. In this case the standardized minimax optimal designs are invariant with respect to linear transformations of the explanatory variale. This is a strong advantage of parametrization (3. over the alternative parametrization (3. in inary response models. 4. Examples In this section we illustrate the result with two examples; a doule exponential model with parametrization (3. and a logistic regression with parametrization (3.. We also demonstrate that the maximum variance criterion can produce inefficient designs for the inary response model in the sense that the numer of different dose levels of the minimax optimal design is less than the numer of parameters in the model. 4.. Doule exponential model Let F (t = +sign(t sign(t e t and consider the parametrization (3.. We calculate h(t =(e t and the values for k and c in (3.6 are easily otained as c.844 k (4. We now apply the general procedure of Section 3.. If c, the optimal design ξ for estimating the slope is also minimax optimal and given y (3.9. In a

10 58 HOLGER DETTE AND MICHAEL SAHM second step we consider the case c and check whether the design in (3.0 is minimax optimal using α = α = h ( /h( in the inequality (.. A straightforward calculation shows that (. is satisfied if and only if v 0 c where v 0 is the unique positive solution of v+e v =,v Finally, if <v 0 the minimax optimal design is given y (3. where z and w are defined y v 0 and w = (v 0 4 h(v 0 h(v 0 (v , (4. respectively (note that h(0 =. This can e shown y the equivalence theorem (. oserving the representation of α and α in (3.3. The minimax optimal designs are listed in Tale 4.. The standardized minimax optimal design are easily otained y Lemma 3.. More precisely we otain, for s = k /,(h(0 /4 = c h(c < v 0, which shows that the standardized minimax optimal design is always supported at three points, i.e. ξ = µ v 0 µ µ+ v 0 w w w where v and w is otained from (4. as, (4.3 w = (v 0 k h(v 0 k + h(v 0 (v0 k This design has equal efficiencies eff (ξ =eff µ (ξ =( w(v0 h(v 0(c h(c for the individual parameters and µ. Tale 4.. Minimax optimal designs ξ for the doule exponential model with parametrization (3. (first row. The second row gives the value α in the equivalence theorem (., while the third and fourth rows contain the efficiencies for estimating the individual parameters µ and. The tale shows the influence of the size of the slope parameter on the minimax optimal design. The weight w is given y (4., the constant c y (4. and v 0 minimizes (3. [c.844,v ]. ( µ v 0 ξ <v 0 v 0 c >c µµ+ v0 w w w α ( µ µ+ ( µ c µ + c ( + h( eff µ h( h(c h( eff c h(c

11 MINIMAX OPTIMAL DESIGNS IN NONLINEAR REGRESSION MODELS 59 A rief comparison of the minimax and standardized minimax optimal designs in the doule exponential model might e appropriate at this point. The standardized minimax optimal designs are ased on more points than there are parameters in the model. As a consequence tests of goodness-of-fit of the doule exponential model can e performed. Note also that the minimax optimal design for a small slope quickly degenerates to a one point design (see Tale 4.. For example, if = 0., the minimax optimal design advises the experimenter to take 99.9% of the oservations at the LD 50 µ. However, this design does not allow any statistical inference regarding the slope parameter. These oservations can e explained y oserving the different scales of the elements in the Fisher information matrix (3.4. For small values of the slope the dominating element in I (θ, ξ is the element at position (,. Consequently the minimax optimal design is close to the optimal design for estimating the LD 50, which is usually the one point design concentrated at µ. Note that these phenomena contradict the intuition. If the proaility of success is slowly increasing with the dose level, a good design should cover a large range of dose levels. The standardization of the maximum variance criterion in (. produces efficiencies, which are of the same scale and therefore etter comparale than the pure diagonal elements of I (θ, ξ. As a consequence we do not oserve degenerate standardized minimax optimal designs at all. For example, if =0., the standardized minimax optimal design advises the experimenter to take 6.7% of the oservations at µ±5.93 and the remaining 46.6% of the oservations at the LD50 µ (see equation ( Logistic regression As a second example consider the logistic regression model with parametrization (3., i.e. F (t =[+e t ] e t and h(t = ( + e t, t IR and the value c>0maximizing t h(tisgivenyc We first determine the optimal designs for ξ a and ξ for estimating the individual coefficients a and using Elfving s theorem (Elfving (95. This gives inf ξ et I (θ, ξe = e T I (θ, ξ a e = m h(a inf ξ et I (θ, ξe = e T I (θ, ξ e = c h(c, (where m is defined y (3.8 and the optimal designs are given y ( 0 if a c ( a ξ a = ( a c a + c if a >c c a + c a ξ = c a + c.

12 60 HOLGER DETTE AND MICHAEL SAHM From the procedure in Section it follows that ξ is minimax optimal if and only if e T I (θ, ξ e e T I (θ, ξ e which is equivalent to a + c. Similarly we otain ξ a is minimax optimal a c. In all other cases ( a <c we must have equality of the diagonal elements of the inverse Fisher information matrix of the minimax optimal design. A reasonale guess for a design at two points is ( a ξ = v a + v w w v>0,w (0, (4.4 which implies for the weight if a =0 w = + v +(a 4av if a 0. (4.5 The point v>0can e found y maximizing the function h(xg(x forx>0, where g(x =x (a + x 4 (a, (4.6 and the optimality of the design in (3.0 is estalished using the equivalence theorem (.. The resulting cases are listed in Tale 4. which also contains the corresponding quantities α j, used for checking the minimax optimality y (.. Note that the tale does not include the case a =0, <c. In this case the minimax optimal design has equal masses at the points and which also follows from (. using α =. Tale 4.. Locally minimax optimal designs for the logistic regression model with parametrization (3.. The tale shows the dependence of the locally optimal design on the size of the quantity a. Here g(x isdefined y (4.6, the constant c =argmax{x h(x} y (3.6, v = arg max{h(xg(x} and w y (4.5. minimax opt. design a c a <c c a ( c a c a α 0 Φ(ξ c h(c ( v a v a w w 4 (a v g(v 4a h(vg(v ( c a a c a c a a+c a a c h(c

13 MINIMAX OPTIMAL DESIGNS IN NONLINEAR REGRESSION MODELS 6 Finally, the standardized minimax optimal designs for the logistic regression with parametrization (3. are otained from Lemma 3.3 and given y ( a ξ = ṽ a + ṽ w w (4.7 ṽ =argmax{h(x[x (a + s x 4 (a s ]; x>0} (4.8 if a =0 w = (4.9 + ṽ +(a s 4aṽ if a 0, where s = s(a is defined in Lemma 3.3. Note that the standardized minimax optimal design is scale invariant, ecause ṽ and w defined in (4.8 and (4.9 do not depend on the slope parameter. In Tales 4.3 and 4.4 the minimax and standardized minimax optimal designs for the logistic regression model with parametrization (3. are calculated for some specific values of a and as well as their efficiencies with respect to the optimal designs for the individual parameters a and. We oserve again that for many parameter cominations the minimax optimal designs are inefficient for estimating the slope, especially for small values of the slope parameter. The standardized minimax optimal designs are equally efficient for estimating the individual parameters. This efficiency does not depend on the parameter and is increasing with the intercept a. Tale 4.3. Locally minimax optimal designs ξ for logistic regression with parametrization (3. for various values of a and. The designs are otained from (4.4-(4.6 and put masses w and w at the points a/ v/ and a/+v/. The last two columns give the efficiencies of the minimax optimal design for estimating the individual parameters. The tale illustrates how sensitively the locally minimax optimal design depends on the slope parameter of the inary response model. In particular it shows that for small values of the slope, the minimax optimal design ecomes inefficient for estimating the slope parameter. a v w Φ(ξ eff a (ξ eff (ξ c c c c c c c

14 6 HOLGER DETTE AND MICHAEL SAHM Tale 4.4. Locally standardized minimax optimal designs for the logistic regression model with parametrization (3. for various values of a and aritrary. The last column gives the efficiencies of the minimax optimal design for estimating the individual parameters. The designs are otained from (4.7-(4.9 and put masses w and w at the points a/ ṽ/ and a/ + ṽ/. The locally standardized minimax optimal designs and its efficiencies are invariant with respect to the slope parameter. Thus the tale illustrates the impact of the intercept on the locally standardized minimax optimal designs. Note that these designs never ecome inefficient for estimating the individual parameters. a ṽ w Ψ( ξ eff a ( ξ =eff ( ξ c Conclusions In this paper we considered the prolem of constructing minimax optimal designs for inary response models which minimize the asymptotic maximum variance of the estimates for the individual parameters. Some general guidelines for the determination of these designs are given and illustrated in the doule exponential and logistic regression model. It turns out that minimax optimal designs may ecome inefficient in the sense, that the numer of their dose levels is less than the numer of parameters in the model. For this reason a standardized version (standardized minimax optimality of Elfving s partial minimax criterion is also considered, which is ased on the variances of the estimators for the individual coefficients relative to the est variances otainale y the choice of an experimental design, i.e. efficiencies of the estimates are considered. The standardized minimax optimal designs for inary response models can e otained from the minimax optimal designs and produce equal efficiencies for estimating the individual parameters in the model. In some cases the standardized minimax optimal design has more support points than the minimax optimal design and a goodness-of-fit test of the inary response model can e performed. Because the standardized minimax optimal designs are scale invariant and ecause they cannot degenerate to one point designs, we would recommend their application instead of minimax optimal designs. Acknowledgement The authors are grateful to an unknown Associate Editor and referee for their valuale comments and suggestions which led to a sustantial improvement of the original manuscript.

15 MINIMAX OPTIMAL DESIGNS IN NONLINEAR REGRESSION MODELS 63 Appendix Proof of Lemma 3.. Let z,...,z k e aritrary nonnegative numers. We show that there is a symmetric minimax optimal design with at most four support points in the class of all symmetric designs with support in the set {µ ± z i / i =,...,k}. Therefore, let ξ e a symmetric measure on {µ ± z i / i =,...,k} with weights w i /atthepointsµ ± z i /. Due to the symmetry of ξ the Fisher information matrix is diagonal, i.e. k k I(θ, ξ =diag ( w i h(z i, w i zi h(z i. i= Optimizing with respect to the criterion (. is therefore equivalent to maximizing k k } min { w i h(z i, w i zi h(z i i= over all possile weights, that is maximizing on the k-simplex { k } w,...,w k w i 0, w i =. i= Since oth functions are linear in the weights the Lemma follows from a general result from linear programming. More precisely, the maximum value of the minimum of two linear functions on the k-simplex is always attained at a convex comination of two extreme points. The extreme points of the k-simplex represent the symmetric two point designs with weight / atthepointsµ ± z i0 /, i 0 k. That means there is always a minimax optimal design in the considered class with at most 4 support points. References Adelasit, K. M. and Plackett, R. L. (983. Experimental design for inary data. J. Amer. Statist. Assoc. 78, Atkinson, A. C. and Donev, A. N. (99. Optimum Experimental Designs. Oxford: Oxford University Press. Chaloner, K. and Larntz, K. (989. Optimal Bayesian design applied to logistic regression experiments. J. Statist. Plann. Inference, Dette, H. (997. Designing experiments with respect to standardized optimality criteria. J. Roy. Statist. Soc. Ser. B 59, Dette, H. and Studden, W. J. (994. Optimal designs with respect to Elfving s partial minimax criterion in polynomial regression. Ann. Inst. Statist. Math. 46, Elfving, G. (95. Optimum allocation in linear regression theory. Ann. Math. Statist. 3, Elfving, G. (959. Design of linear experiments. In Proaility and Statistics. The Harald Cramér Volume (Edited y U. Grenander, Stockholm: Almagrist and Wiksell. i= i=

16 64 HOLGER DETTE AND MICHAEL SAHM Ford, I., Torsney, B. and Wu, C. F. J. (99. The use of a canonical form in the construction of locally optimal designs for non-linear prolems. J. Roy. Statist. Soc. Ser. B 54, Gaudard, M. A., Karson, M. J., Linder, E. and Tse, S. K. (993. Efficient designs for estimation in the power logistic quantal response model. Statist. Sinica 3, Krafft, O. and Schaefer, M. (995. Exact Elfving-minimax designs for quadratic regression. Statist. Sinica 5, Müller, C. (995. Maximin efficient designs for estimating nonlinear aspects in linear models. J. Statist. Plann. Inference 44, 7-3. Murty, V. N. (97. Minimax designs. J. Amer. Statist. Assoc. 66, Pukelsheim, F. (993. Optimal Experimental Design. New York: Wiley. Silvey, S. D. (980. Optimal Design: An Introduction to the Theory for Parameter Estimation. Chapman and Hall, London. Sitter, R. R. (99. Roust design for inary data. Biometrics 48, Sitter, R. R. and Wu, C. F. J. (993. Optimal designs for inary response experiments: Fieller-, D- and A-criteria. Scand. J. Statist. 0, Torsney, B. and López-Fidalgo, J. (995. MV-optimization in simple linar regression. In MODA4-Advances in Model-Oriented Data Analysis (Edited y C. P. Kitsos, W.G. Müller, Wu, C. F. J. (985. Efficient sequential designs with inary data. J. Amer. Statist. Assoc. 80, Wu, C. F. J. (988. Optimal design for percentile estimation of a quantal response curve. In Optimal Design and Analysis of Experiments (EditedyJ.Dodge,V.V.Fedorov,H.P. Wynn, North Holland, Amsterdam. Ruhr-Universität Bochum, Fakultät für Mathematik, Universitätsstr. 50, D Bochum, Germany. holger.dette@rz.ruhr-uni-ochum.de michael.sahm@ruhr-uni-ochum.de (Received May 997; accepted Octoer 997