Optimal approximate designs for estimating treatment contrasts resistant to nuisance effects

Abstract

Suppose that we intend to perform an experiment consisting of a set of independent trials. The mean value of the response in each trial is assumed to be equal to the sum of the effect of the treatment selected for that trial and some nuisance effects, e.g., the effect of a time trend or blocking. In this model, we examine optimal approximate designs for the estimation of a system of treatment contrasts, with respect to a wide range of optimality criteria. We show that it is necessary for any optimal design to attain the optimal treatment proportions, which may be obtained from the marginal model that excludes the nuisance effects. Moreover, we prove that for a design to be optimal, it is sufficient that it attains the optimal treatment proportions and satisfies the conditions for resistance to nuisance effects. For selected natural choices of treatment contrasts and optimality criteria, we calculate the optimal treatment proportions and provide an explicit form of optimal designs. In particular, we obtain optimal treatment proportions for the comparison of a set of test treatments with a set of controls. Once the optimal treatment proportions are determined, the results allow us to construct a method of calculating optimal approximate designs with small support sizes through linear programming. Consequently, we can construct efficient exact designs using a simple heuristic.

Appendix

Throughout Appendix, we assume that $K^T = (Q^T, 0_{s \times d})$.

Proof of Proposition 1

Let us partition the matrix L in

$$\begin{aligned} N_K(\xi )= \mathop \mathrm {min}\limits _{L \in \mathbb {R}^{s \times m}: LK=I_s} LM(\xi )L^T \end{aligned}$$

as $L=(L_1, L_2)$, where $L_1$ is an $s \times v$ matrix and $L_2$ is an $s \times d$ matrix. Then,

$$\begin{aligned} N_K(\xi )&=\mathop \mathrm {min}\limits _{LK=I_s} LM(\xi )L^T = \mathop \mathrm {min}\limits _{(L_1, L_2) (Q^T, 0)^T = I_s} (L_1, L_2)M(\xi )(L_1, L_2)^T \\&\preceq \mathop \mathrm {min}\limits _{L_1Q=I_s} L_1 M_{11}(\xi ) L_1^T = N_Q(w). \end{aligned}$$

$\square$

To prove Proposition 2, we will need two preliminary results formulated in the following lemmas.

Lemma 1

Let $\tilde{M}$ be a non-negative definite matrix. If a design $\xi$ satisfies $M(\xi )\tilde{M}^-K=K$ for some generalized inverse $\tilde{M}^-$ of $\tilde{M}$, then (i) $\xi$ is feasible for $K^T\beta$ and (ii) $K^T M^-(\xi )K = K^T \tilde{M}^-K$.

Proof

The steps of the proof follow the proof of Theorem 8.13 from Pukelsheim (2006). Let $G{:}=\tilde{M}^-$. Because $M(\xi )GK=K$, we obtain $M(\xi )X=K$, where $X=GK$. Therefore, $\mathcal {C}(K) \subseteq \mathcal {C}(M(\xi ))$, and hence, $\xi$ is feasible. Let us premultiply the equation $M(\xi )GK=K$ by $K^TM^-(\xi )$ such that we obtain $K^T M^-(\xi )K$ on the right-hand side. The left-hand side is then equal to $K^T M^-(\xi )M(\xi )GK$. Note that $K^T=X^TM^T(\xi )=X^TM(\xi )$, and hence, the following holds:

$$\begin{aligned} K^T M^-(\xi )M(\xi )GK = X^TM(\xi ) M^-(\xi )M(\xi )GK = X^T M(\xi ) G K = K^T G K. \end{aligned}$$

It follows that $K^T M^-(\xi )K = K^T \tilde{M}^- K$. $\square$

Lemma 2

Let $w>0$ be a treatment proportions design, and let $G:=\mathrm {diag}(w^{-1},0_d)$. Let $\xi$ be a design in model (1). Then, $\xi$ satisfies $M(\xi )GK=K$ if and only if (i) w is a treatment proportions design of $\xi$ and (ii) $\xi$ is resistant to nuisance effects.

Proof

We may express $M(\xi )GK {=} K$ as $M_{11}(\xi )\mathrm {diag}(w^{-1}) Q {=} Q$ and $M_{12}^T(\xi ) \mathrm {diag}(w^{-1}) Q = 0$. Because both $M_{11}(\xi )$ and $\mathrm {diag}(w^{-1})$ are diagonal matrices and all rows of Q are assumed to be non-zero vectors, the first equation is equivalent to $\frac{1}{w_u}\sum _t\xi (u,t) = 1$ for all u, which is (i). From the second equation, we find that every row of $M_{12}^T(\xi )\mathrm {diag}(w^{-1})= \big [ \frac{1}{w_1}\sum _t \xi (1,t)h(t), \ldots , \frac{1}{w_v}\sum _t \xi (v,t)h(t) \big ]$ needs to be in $\mathcal {N}(Q^T)$, which is (ii). $\square$

Proof of Proposition 2

Let $\tilde{M} := \mathrm {diag}(w,0_d)$. Then, $G := \mathrm {diag}(w^{-1},0_d)$ is a generalized inverse of $\tilde{M}$. From Lemma 2 it follows that $M(\xi )GK=K$ and from Lemma 1, it follows that (i) and (ii) hold. Statement (iii) is a direct consequence of (ii). $\square$

Proof of Theorem 1

Let $\xi$ be a feasible design in (1). Using Proposition 1, we find that $N_K(\xi ) \preceq N_Q(w)$, where w is the treatment proportions design of $\xi$. Moreover, because product designs are nuisance resistant, part (iii) of Proposition 2 implies that $N_Q(w) = N_K(w \otimes \alpha )$ for any nuisance conditions design $\alpha$. Therefore, $N_K(\xi ) \preceq N_K(w \otimes \alpha )$.

Suppose that $w^*$ is not a $\Phi$-optimal design. Then, there exists a design $w_{\mathrm {b}}$ in model (3) such that $\Phi (N_Q(w^*)) < \Phi (N_Q(w_{\mathrm {b}}))$. Then, $\Phi (N_K(\xi ^*)) \le \Phi (N_Q(w^*)) < \Phi (N_Q(w_{\mathrm {b}})) = \Phi (N_K(w_{\mathrm {b}} \otimes \alpha ))$ for any nuisance conditions design $\alpha$. This contradicts the $\Phi$-optimality of $\xi ^*$. $\square$

The following lemma (Theorem 8.13 from Pukelsheim 2006) enables a complete characterization of $\Phi$-optimal designs for a strictly concave $\Phi$.

Lemma 3

Let $\Phi$ be an information function that is strictly concave on $\mathfrak {S}^s_{++}$, and let $\xi ^*$ be $\Phi$-optimal for $K^T\beta$. Let G be a generalized inverse of $M(\xi ^*)$ that satisfies the normality inequality of the General Equivalence Theorem (Theorem 7.14 from Pukelsheim 2006), i.e., there exists a non-negative definite matrix D that solves the polarity equation

$$\begin{aligned} \Phi (N_K(\xi ^*))\Phi ^\infty (D) = \mathrm {tr}(N_K(\xi ^*)D)=1, \end{aligned}$$

where $\Phi ^\infty$ is the polar information function of $\Phi$ (see Pukelsheim 2006), and G satisfies the normality inequality

$$\begin{aligned} \mathrm {tr}(M(\xi )B) \le 1 \quad \text {for all feasible designs } \xi , \end{aligned}$$

where $B=GKN_K(\xi ^*)DN_K(\xi ^*)K^TG^T$. Then, a design $\xi$ is $\Phi$-optimal if and only if $M(\xi )GK=K$.

To use Lemma 3, we must obtain a matrix G that satisfies the normality inequality of the General Equivalence Theorem.

Lemma 4

Let $\Phi$ be an information function that is strictly concave on $\mathfrak {S}^s_{++}$. Then, there exists a unique $\Phi$-optimal treatment proportions design for $Q^T\tau$, say $w^*$. Let $G:=\mathrm {diag}((w^*)^{-1},0_d)$. Then, G satisfies the normality inequality of the General Equivalence Theorem for estimating $K^T\beta$ in model (1).

Proof

Let $w^*$ be a $\Phi$-optimal treatment proportions design, and let $G_{11}{:}{=}\,\mathrm {diag}((w^*)^{-1})$. Then, the matrix $G_{11}$, which is the unique generalized inverse of $M(w^*)$, satisfies the normality inequality of the General Equivalence Theorem for model (3). By Lemma 3, a treatment proportions design w is $\Phi$-optimal if and only if $M(w)M^{-1}(w^*)Q=Q$. By the same argument as that used in the proof of Lemma 2, $M(w)M^{-1}(w^*)Q=Q$ holds if and only if $w=w^*$, which proves the uniqueness of $w^*$. Let $N^*:=N_Q(w^*)$. Because $G_{11}$ satisfies the normality inequality in model (3), there exists a matrix D that satisfies the polarity equation $\Phi (N^*)\Phi ^\infty (D)=\mathrm {tr}(N^*D) = 1$ and the matrix $B_w=G_{11}QN^*DN^*Q^TG_{11}$ satisfies $\mathrm {tr}(M(\tilde{w})B_w) \le 1$ for all $\tilde{w}>0$.

Since $N_K(\xi ^*)$ is positive definite and $\Phi$ is strictly concave on $\mathfrak {S}^s_{++}$, there exists a unique $\Phi$-optimal information matrix $N_K(\xi ^*)$. The $\Phi$-optimality of $N_K(w^* \otimes \alpha ) = N_Q(w^*)=N^*$ for any nuisance conditions design $\alpha$ yields $N_K(\xi ^*) = N^*$. Thus, the polarity equality holds in model (1) for the same matrix D. Let $\tilde{\xi }$ be a feasible design. Then, the left-hand side of the normality inequality in model (1) is $\mathrm {tr}(M(\tilde{\xi })B)$, where

$$\begin{aligned} B=\begin{bmatrix} G_{11}&\quad 0 \\ 0&\quad 0 \end{bmatrix} \begin{bmatrix} Q \\ 0 \end{bmatrix} N^* D N^* \begin{bmatrix} Q^T&\quad 0 \end{bmatrix} \begin{bmatrix} G_{11}&\quad 0 \\ 0&\quad 0 \end{bmatrix} = \begin{bmatrix} B_w&\quad 0 \\ 0&\quad 0 \end{bmatrix}. \end{aligned}$$

Then, because $B_w$ satisfies the normality inequality in model (3), we obtain $\mathrm {tr}(M(\tilde{\xi })B) = \mathrm {tr}(M_{11}(\tilde{\xi })B_w) = \mathrm {tr}(M(\tilde{w})B_w) \le 1$, where $\tilde{w}$ is the treatment proportions design of $\tilde{\xi }$. $\square$

Proof of Theorem 2

Let $\xi$ be a feasible design in model (1), and let w be its treatment proportions design. Because $\Phi$ is isotonic, it follows from Proposition 1 that $\Phi (N_Q(w)) \ge \Phi (N_K(\xi ))$. Since $w^*$ is $\Phi$-optimal, $\Phi (N_Q(w^*)) \ge \Phi (N_Q(w))$ and it is feasible; thus, $w^* {>}0$. Using Proposition 2, we find that $\Phi (N_K(\xi ^*)) = \Phi (N_Q(w^*)) \ge \Phi (N_Q(w)) \ge \Phi (N_K(\xi )),$ i.e., $\xi ^*$ is $\Phi$-optimal. $\square$

Proof of Theorem 3

Lemma 4 ensures the uniqueness of $w^*$. Let $G=\mathrm {diag}((w^*)^{-1},0_d)$. Consider an extension of model (1), where $t \in \mathfrak T'$, $\mathfrak T'=\mathfrak T\cup \{ z\}$, and $h(z) = 0_d$, and let $\xi '$ be a design in this extended model, with its treatment proportions design $w^*$, that satisfies $\xi '(u,t) = 0$ for $t \ne z$. Then, $M(\xi ')=\mathrm {diag}(w^*,0_d)$ and G is a generalized inverse of $M(\xi ')$. From Lemma 4, it follows that G satisfies the normality inequality of the General Equivalence Theorem for any nuisance regressors and, consequently, also for the extended model. By Lemma 3, a design $\xi$ is $\Phi$-optimal in the extended model if and only if $M(\xi )GK=K$. The equality $M(\xi )GK=K$ holds if and only if $\xi$ satisfies (i) and (ii) from Lemma 2 in the extended model, i.e.,

$$\begin{aligned} \sum _{t \in \mathfrak T'} \xi (u,t) = w_u^*,\quad u=1,\ldots ,v, \end{aligned}$$

(16)

$$\begin{aligned} \begin{bmatrix} \frac{1}{w_1^*}\sum \limits _{t\in \mathfrak T'} \xi (1,t)h(t),&\ldots ,&\frac{1}{w_v^*}\sum \limits _{t\in \mathfrak T'} \xi (v,t)h(t) \end{bmatrix} Q = 0. \end{aligned}$$

(17)

Let $\Xi$ be the set of feasible designs in the extended model that satisfy $\xi (u,z)=0$ for all u, which represents the set of all feasible designs in model (1). The $\Phi$-optimal value in the extended model can be attained in $\Xi$, e.g., $\xi ^*=w^* \otimes \alpha$, where $\alpha$ is any nuisance conditions design that satisfies $\alpha (z)=0$. Thus, a design $\xi$ is $\Phi$-optimal in $\Xi$ if and only if $\xi \in \Xi$ and it satisfies (16) and (17), i.e., $\sum _{t \in \mathfrak T} \xi (u,t) = w_u^*$ for $u=1,\ldots ,v$, and (5). It follows that a design $\xi$ is $\Phi$-optimal in (1) if and only if its treatment proportions design is $\Phi$-optimal and $\xi$ is nuisance resistant.

Proof of Theorem 5

First, let us assume that Q is of full column rank. Let w be a feasible treatment proportions design, and let P be a $v \times v$ permutation matrix. We define Pw as the design obtained via the P-permutation of the treatments in w, i.e., $Pw(u)=w(\pi _P(u))$ for $u \in \{1,\ldots ,v\}$, where $\pi _P$ is the permutation of the elements $\{1,\ldots ,v\}$ that corresponds to the matrix P. Because $Pw>0$, it is feasible, its moment matrix is $M(Pw) = PM(w)P^T$, and it has the information matrix $N_Q(Pw) = (Q^TPM^{-1}(w)P^TQ)^{-1}$.

We will use the well-known fact that for any matrix X, the non-zero eigenvalues of the matrices $X^TX$ and $XX^T$ are the same (e.g., 6.54(c) in Seber 2008), including their multiplicities. Let us define $Y=Q^TM^{-1/2}(w)$ and $Z=Q^T P M^{-1/2}(w)$. Since $QQ^T$ is completely symmetric, $Y^TY = Z^TZ$. Furthermore, $YY^T=Q^T M^{-1}(w) Q = N_Q^{-1}(w)$ and $ZZ^T=Q^T P M^{-1}(w) P^T Q = N_Q^{-1}(Pw)$; thus, $N_Q(w)$ and $N_Q(Pw)$ have the same set of non-zero eigenvalues. Because they are of the same (full) rank, it follows that $N_Q(w)$ and $N_Q(Pw)$ are orthogonally similar and that $\Phi (Pw) = \Phi (w)$. Note that analogous results hold in the rank-deficient case for the matrices $C_Q(w)$ and $C_Q(Pw)$.

The uniform treatment proportions design satisfies

$$\begin{aligned} \Phi (\bar{w})&= \Phi \left( \frac{1}{v!}\sum _{P-\text {perm.}} Pw\right) \ge \frac{1}{v!}\sum _{P-\text {perm.}} \Phi (Pw) \\&= \frac{1}{v!}\sum _{P-\text {perm.}} \Phi (w) = \frac{1}{v!}v! \Phi (w) = \Phi (w), \end{aligned}$$

where the inequality follows from the concavity of $\Phi$. Thus, $\bar{w}$ is $\Phi$-optimal. $\square$

In the proof of the uniqueness of $\Phi _p$-optimal treatment proportions in Theorem 6, we will employ the following lemma.

Lemma 5

(Corollary 8.14 from Pukelsheim 2006) Let $p \in [-\infty ,0]$ and let $\xi ^*$ be a $\Phi _p$-optimal design. Then, if $p>-\infty$, any other design $\xi$ is also $\Phi _p$-optimal if and only if it satisfies $M(\xi )GK=K$, where G is a generalized inverse of $M(\xi ^*)$ that satisfies the normality inequality of the General Equivalence Theorem (see Lemma 3).

Proof of Theorem 6

Note that for $Q=(-I_g \otimes 1_{v-g}, 1_{g} \otimes I_{v-g})^T$, we have

$$\begin{aligned} QQ^T = \begin{bmatrix} (v-g)I_g&\quad -J_{g \times (v-g)} \\ -J_{(v-g) \times g}&\quad gI_{v-g} \end{bmatrix}. \end{aligned}$$

Let w be a treatment proportions design, let $P_1$ and $P_2$ be $g \times g$ and $(v-g) \times (v-g)$ permutation matrices, respectively, and let

$$\begin{aligned} \tilde{P}=\begin{bmatrix} P_1&\quad 0_{g \times (v-g)} \\ 0_{(v-g) \times g}&\quad P_2 \end{bmatrix}. \end{aligned}$$

(18)

Define $\tilde{P}w$ to be the design given by the $\tilde{P}$-permutations of the treatments. Then, $M(\tilde{P}w)= \tilde{P}M(w)\tilde{P}^T$ and $\tilde{P}^T QQ^T \tilde{P} = QQ^T$. From an argument analogous to that used in the proof of Theorem 5, $C_Q(\tilde{P}w)$ and $C_Q(w)$ are orthogonally similar; therefore $\Phi _p(\tilde{P}w) = \Phi _p(w)$.

Let $\tilde{w} := \frac{1}{(v-g)!g!} \sum _{\tilde{P}} \tilde{P} w$, where the sum is over all $v \times v$ permutation matrices $\tilde{P}$ of the form (18). Then, $\Phi _p(\tilde{w}) \ge \Phi _p(w)$. It follows that an optimal design exists in the class of designs that allocate one weight to each of the first g treatments, say $\gamma _1$ ($0<\gamma _1<1/g$), and another weight $\gamma _2:=(1-g\gamma _1)/(v-g)$ to each of the other treatments. We denote the total weight of the first g treatments as $\gamma :=g\gamma _1$ and for a given $\gamma$, we denote such designs as $w_\gamma$.

The non-zero eigenvalues of $C_Q(w_\gamma )$ are inverse to the non-zero eigenvalues of $V(w_\gamma ):=Q^TM^{-1}(w_\gamma )Q$, where $M(w_\gamma )=\mathrm {diag}(\gamma _1 1_g, \gamma _2 1_{v-g})$. Let $X=Q^TM^{-1/2}(w_\gamma )$. Then, the set of non-zero eigenvalues of $V(w_\gamma ) = XX^T$ coincides with the set of non-zero eigenvalues of

$$\begin{aligned} X^TX&= M^{-1/2}(w_\gamma )QQ^T M^{-1/2}(w_\gamma ) \\&= \begin{bmatrix} (v-g)\gamma _1^{-1}I_g&\quad - (\gamma _1\gamma _2)^{-1/2}J_{g \times (v-g)} \\ - (\gamma _1\gamma _2)^{-1/2}J_{(v-g) \times g}&\quad g\gamma _2^{-1}I_{v-g} \end{bmatrix}. \end{aligned}$$

It can be verified that $X^TX$ has the following eigenvalues, listed with the corresponding eigenvectors of the form $x=(x_1^T,x_2^T)^T$, where $x_1 \in \mathbb {R}^{g}$ and $x_2 \in \mathbb {R}^{v-g}$: $\mu _1 = g\gamma _2^{-1}$ with multiplicity (w.m.) $v-g-1$, $x_1=0_g$ and $1_{v-g}^Tx_2=0$; $\mu _2 = (v-g)\gamma _1^{-1}$ w.m. $g-1$, $1_g^Tx_1 = 0$ and $x_2=0_{v-g}$; $\mu _3 = (v-g)\gamma _1^{-1} + g\gamma _2^{-1}$ w.m. 1, $x_1=-(v-g)\gamma _2^{1/2}1_g$ and $x_2=g\gamma _1^{1/2}1_{v-g}$; and $\mu _4 = 0$ w.m. 1, $x_1=\gamma _1^{1/2}1_g$ and $x_2=\gamma _2^{1/2}1_{v-g}$.

Therefore, the non-zero eigenvalues of $C_Q(w_\gamma )$ are $\lambda _1 = \frac{1-g\gamma _1}{g(v-g)}$ w.m. $v-g-1$, $\lambda _2 = \frac{\gamma _1}{v-g}$ w.m. $g-1$ and $\lambda _3=\frac{\gamma _1(1-g\gamma _1)}{v-g}$ w.m. 1. Thus, for $p \in (-\infty ,0)$, the $\Phi _p$-optimal value of $w_\gamma$ is obtained by minimizing the convex function

$$\begin{aligned} f_p(\gamma _1) = (v-g-1)\left( \frac{1-g\gamma _1}{g(v-g)}\right) ^p + (g-1)\left( \frac{\gamma _1}{v-g}\right) ^p + \left( \frac{\gamma _1(1-g\gamma _1)}{v-g}\right) ^p \end{aligned}$$

for $\gamma _1 \in (0,1/g)$. Then, $f_p'(\gamma _1) = 0$ if and only if

$$\begin{aligned} -(v-g-1)(1-g\gamma _1)^{p-1} + (g-1)(g\gamma _1)^{p-1} + (1-2g\gamma _1)(g\gamma _1)^{p-1}(1-g\gamma _1)^{p-1}=0, \end{aligned}$$

which is equivalent to

$$\begin{aligned} -(v-g-1)(g\gamma _1)^{1-p} + (g-1)(1-g\gamma _1)^{1-p} + 1-2g\gamma _1 = 0. \end{aligned}$$

Using $\gamma =g\gamma _1$, we obtain (9).

For $p=0$ the unique maximum of the function $f_0(\gamma _1)=\lambda _1^{v-g-1}\lambda _2^{g-1}\lambda _3=(1-g\gamma _1)^{v-g}\gamma _1^g/[g^{v-g-1}(v-g)^{v-1}]$ for $\gamma _1 \in (0,1/g)$ is in $\gamma _1=1/v$, which means that $w_\gamma$ is a uniform design. This corresponds to $\gamma =g/v$, which is obtained if we set $p=0$ in (9).

The smallest non-zero eigenvalue of $C_Q(w_\gamma )$ is $\lambda _3$ and hence the $\Phi _{-\infty }$-optimal design can be obtained by maximizing

$$\begin{aligned} f_{-\infty }(\gamma ) = \frac{\gamma (1-\gamma )}{g(v-g)}, \end{aligned}$$

which has a unique maximum in $\gamma =\frac{1}{2}$.

For $p \in (-\infty ,0]$, the uniqueness follows from Lemma 5 (as stated in Sect. 8.18 of Pukelsheim 2006, the General Equivalence Theorem also holds for the rank deficient $\Phi _p$-criteria; however, instead of negative powers of $N_K(\xi ^*)$, we have positive powers of $K^TM^-(\xi ^*)K$ and instead of $\lambda _{\min }(N_K(\xi ^*))$, we have $1/\lambda _{\max }(K^TM^-(\xi ^*)K)$). Denote by $w^*$ the $\Phi _p$-optimal treatment proportions obtained from (9). The design $w^*$ is $\Phi _p$-optimal and the moment matrix $M(w^*)$ is non-singular, therefore there exists a unique generalized inverse $G=M^{-1}(w^*)$ satisfying the normality inequality. Then, $M(w)GQ=Q$ can be satisfied only by $w=w^*$, because M(w)G is a diagonal matrix and no row of Q is equal to zero.

For $p=-\infty$ let $w>0$. Then, the matrix $V(w)=Q^TM^{-1}(w)Q$ can be expressed as $V(w) = A_1(w) + A_2(w)$, where $A_1(w)$ is a block diagonal matrix with the blocks on the diagonal given by $w_1^{-1}J_{v-g}, \ldots , w_g^{-1}J_{v-g}$ and $A_2(w) = J_g \otimes \mathrm {diag}(w_{g+1}^{-1}, \ldots , w_v^{-1})$. Let us partition $u \in \mathbb R^{g(v-g)}$ as $u^T=(u_1^T,\ldots ,u_g^T)$, where $u_i \in \mathbb R^{v-g}$ for all i. Then, the largest eigenvalue $\mu _{\max }$ of V(w) satisfies

$$\begin{aligned} \mu _{\max }&= \max _{||u ||=1} u^T V(w) u \\&= \max _{||u ||=1} \left( \sum _{i=1}^g w_i^{-1}\Big (1_{v-g}^Tu_i\Big )^2 + \sum _{i=1}^g \sum _{j=1}^g u_i^T \mathrm {diag}\Big (w_{g+1}^{-1}, \ldots , w_v^{-1}\Big ) u_j \right) . \end{aligned}$$

For the particular choice of $u=1_{g(v-g)}/\sqrt{g(v-g)}$, we obtain

$$\begin{aligned} \mu _{\max }&\ge \sum \limits _{i=1}^g w_i^{-1}\left( \frac{v-g}{\sqrt{g(v-g)}}\right) ^2 + \sum \limits _{i=1}^g \sum \limits _{j=1}^g \sum \limits _{k=g+1}^v \left( \frac{1}{\sqrt{g(v-g)}}\right) ^2 w_k^{-1} \\&= \frac{v-g}{g} \sum \limits _{i=1}^g w_i^{-1} + \frac{g}{v-g} \sum \limits _{i=g+1}^v w_i^{-1}. \end{aligned}$$

If w does not satisfy $w_1 = \cdots = w_g$ and $w_{g+1} = \cdots = w_v$, then define $\tilde{w}$ such that $\tilde{w}_1 = \cdots = \tilde{w}_g = \sum _{i=1}^g w_i/g$ and $\tilde{w}_{g+1} = \cdots = \tilde{w}_v = \sum _{i=g+1}^v w_i/(v-g)$. Then, from the first part of the proof, we have $\mu _{\max }(\tilde{w}) = \mu _3(\tilde{w}) = (v-g)\tilde{w}_1^{-1} + g\tilde{w}_{g+1}^{-1}$. From the inequality of arithmetic and harmonic means, it follows that $\frac{v-g}{g}\sum _{i=1}^g w_i^{-1} \ge (v-g)\tilde{w}_{1}^{-1}$ and $\frac{g}{v-g}\sum _{i=g+1}^v w_i^{-1} \ge g\tilde{w}_{g+1}^{-1}$, and at least one of these inequalities is strict. Hence, $\mu _{\max }(w) > \mu _{\max }(\tilde{w})$ and thus any $\Phi _{-\infty }$-optimal w must satisfy $w_1 = \cdots = w_g$ and $w_{g+1} = \cdots = w_v$. From the first part of the proof it follows that the only $\Phi _{-\infty }$-optimal design that satisfies these conditions is $w^* = (1_g^T/(2g),1_{v-g}^T /(2(v-g)))^T$.

If $g=1$, the matrix Q is of full rank $v-1$, and the eigenvalues of V(w) are inverses of the eigenvalues of $N_Q(w)$. Therefore, the results hold also for $g=1$. $\square$

Proof of Theorem 7

This proof will closely follow the proof of Theorem 6. The covariance matrix of the least-square estimator is proportional to $V(w) = Q^TM^{-1}(w)Q$. Note that the $\textit{MV}$-optimality criterion $\Phi _{MV}$ is permutationally invariant, because it depends only on the diagonal of the variance matrix.

Let w be a treatment proportions design and let $\tilde{P}$, $\tilde{w}$, $\gamma _1$, $\gamma _2$, $\gamma$ and $w_\gamma$ be defined as in the proof of Theorem 6. Then, $Q^T\tilde{P} = BQ^T$, where $B=P_1 \otimes P_2$, which is a permutation matrix. Thus, $V(\tilde{P}w) = BV(w)B^T$, $\Phi _{MV}(\tilde{P}w) = \Phi _{MV}(w)$ and $\Phi _{MV}(\tilde{w}) \le \Phi _{MV}(w)$. It follows that an optimal design exists in the class of designs $w_\gamma$.

We have $V(w_\gamma )= \gamma _1^{-1}I_g \otimes J_{v-g} + \gamma _2^{-1} J_g \otimes I_{v-g}$ and all of its diagonal elements are $\gamma _1^{-1} + \gamma _2^{-1}$. Thus, the optimal $\gamma _1$ may be obtained by minimizing

$$\begin{aligned} f_{MV}(\gamma _1) = \gamma _1^{-1} + \frac{v-g}{1-g\gamma _1}, \end{aligned}$$

which has minimum in $\gamma _1^* = \frac{\sqrt{g(v-g)}-g}{g(v-2g)}$; therefore, $\gamma ^* = \frac{\sqrt{g(v-g)}-g}{v-2g}$. $\square$

The following lemma provides a method of constructing balanced designs in the model with trigonometric time trend described in (11).

Lemma 6

Let $l,a \in \mathbb {N}$, let $\xi _p$ be an exact design of size l and let $\xi =\xi _p\xi _p \ldots \xi _p$ be the exact design of size $n=la$ formed through a-fold replication of $\xi _p$. Assume that $b \in \mathbb {N}$ is not an integer multiple of a. Then, $\xi$ is balanced for the nuisance regressors of the form $\cos (b\phi _nt)$ and $\sin (b\phi _nt)$, $t=1,\ldots ,n$.

Proof

Let $u \in \{1,\ldots ,v\}$. Using the fact that $\xi (u,k+lj)=\xi _p(u,k)$ for all $k \in \{1,\ldots ,l\}$ and $j \in \{0,\ldots ,a-1\}$, we obtain

$$\begin{aligned} \sum _{t=1}^n \xi (u,t)\cos (b\phi _nt)+\mathrm {i} \sum _{t=1}^n \xi (u,t)\sin (b\phi _nt) = \sum _{t=1}^n \xi (u,t)e^{b\phi _nt\mathrm {i}} \\ = \sum _{j=0}^{a-1}\sum _{k=1}^l\xi (u,k+lj)e^{b\phi _n(k+lj)\mathrm {i}} =\left( \sum _{k=1}^l\xi _p(u,k)e^{b\phi _nk\mathrm {i}}\right) \left( \sum _{j=0}^{a-1}e^{(b\phi _nl \mathrm {i})j}\right) . \end{aligned}$$

Note that if b is not an integer multiple of a, then $b\phi _nl=2\pi (b/a)$ is not an integer multiple of $2\pi$, which implies that $e^{b\phi _nl\mathrm {i}} \ne 1$. In that case

$$\begin{aligned} \sum _{j=0}^{a-1}e^{(b\phi _nl\mathrm {i})j}=\frac{1-e^{b2\pi \mathrm {i}}}{1-e^{b\phi _n l\mathrm {i}}}=0. \end{aligned}$$

$\square$

Proof of Proposition 4

The proposition follows from Theorem 2 and Lemma 6. $\square$

Proof of Proposition 5

It is well known that a point x is a vertex of the set $\{x|Ax=b,x\ge 0\}$ if and only if the system $\{A_j| x_j > 0\}$, where $A_j$ is the jth column of A, is of full rank.

The matrix A consists of $v+(v-1)d+n$ rows, but these rows are linearly dependent. Let k be the affine dimension of $\{h(t)\}_{t \in \mathfrak T}$ and, without loss of generality, let $\mathfrak T=\{1,\ldots ,n\}$. Then, the matrix $[h(2)-h(1), \ldots , h(n)-h(1)]$ is of rank k and thus its row space has dimension k. That is, without the loss of generality, we obtain that $h_i(t)-h_i(1) = \sum _{j=1}^k c_j^{(i)}(h_j(t)-h_j(1))$ for some $c_1^{(i)}, \ldots , c_k^{(i)} \in \mathbb {R}$, for $i>k$ and $t\in \{1,\ldots ,n\}$ (for $t=1$, we formally obtain $0=0$). Let $u \in \{1,\ldots ,v\}$. Then, if (ii) is satisfied for $h_1, \ldots , h_k$, for all $i>k$ and $u \in \{1,\ldots ,v\}$ we have:

$$\begin{aligned} w_1^{-1} \sum _t \xi (1,t) h_i(t)= & {} w_1^{-1} \left( h_i(1) - \sum _{j=1}^k c_j^{(i)}h_j(1)\right) \sum _t \xi (1,t)\\&+ \sum _{j=1}^k c_j^{(i)}w_1^{-1} \sum _t \xi (1,t)h_j(t) \\= & {} h_i(1) - \sum _{j=1}^k c_j^{(i)}h_j(1) + \sum _{j=1}^k c_j^{(i)}w_u^{-1} \sum _t \xi (u,t)h_j(t) \\= & {} w_u^{-1} \sum _t \xi (u,t)\left( h_i(1) - \sum _{j=1}^k c_j^{(i)}h_j(1)\right) \\&+ \sum _{j=1}^k c_j^{(i)}w_u^{-1} \sum _t \xi (u,t)h_j(t) \\= & {} w_u^{-1}\sum _t \xi (u,t) \left[ \left( h_i(1) - \sum _{j=1}^k c_j^{(i)}h_j(1)\right) + \sum _{j=1}^k c_j^{(i)}h_j(t)\right] \\= & {} w_u^{-1} \sum _t \xi (u,t) h_i(t), \end{aligned}$$

where the second and the third equalities also use (i). It follows that (ii) provides at most $k(v-1)$ additional linearly independent equalities.

If $\xi$ satisfies (i), then $\sum _{u,t} \xi (u,t)=1$ holds. Thus, if $\xi$ satisfies (iii) for $t=1, \ldots , n-1$, we have $1= \sum _{t=1}^{n-1} \sum _u \xi (u,t) + \sum _u \xi (u,n) = \frac{n-1}{n} + \sum _u \xi (u,n)$ and therefore (iii) also holds for $t=n$. That is, (iii) provides only $n-1$ additional linearly independent equalities. Hence, A is of rank at most $v + (v-1)k + n-1$ and a vertex x contains at most $v + (v-1)k + n-1$ support points. $\square$