Equivalence of weighted and partial optimality of experimental designs

Abstract

The recently introduced weighted optimality criteria for experimental designs allow one to place various emphasis on different parameters or functions of parameters of interest. However, various emphasis on parameter functions can also be expressed by considering the well-developed optimality criteria for estimating a parameter system of interest (the partial optimality criteria). We prove that the approaches of weighted optimality and of partial optimality are in fact equivalent for any eigenvalue-based optimality criterion. This opens up the possibility to use the large body of existing theoretical and computational results for the partial optimality to derive theorems and numerical algorithms for the weighted optimality of experimental designs. We demonstrate the applicability of the proven equivalence on a few examples. We also propose a slight generalization of the weighted optimality so that it can represent the experimental objective consisting of any system of linear estimable functions.

Appendix: Weighted optimality for any system of interest

The extension of the weighted optimality can be briefly described as follows. The weight matrix ${\mathbf {W}}$ is any nonnegative definite $v \times v$ matrix that satisfies ${\mathscr {L}}({\mathbf {W}}) \subseteq {\mathscr {E}}$, and the weight of ${\mathbf {q}}^T\tau$ is $({\mathbf {q}}^T {\mathbf {W}}^- {\mathbf {q}})^{-1}$ for ${\mathbf {q}}\in {\mathscr {L}}({\mathbf {W}})$. If ${\mathbf {q}}\not \in {\mathscr {L}}({\mathbf {W}})$, the weight of ${\mathbf {q}}^T\tau$ is zero. For any competing design, we require only ${\mathscr {L}}({\mathbf {W}}) \subseteq {\mathscr {L}}({\mathbf {C}}(\xi ))$ (instead of ${\mathscr {L}}({\mathbf {C}}(\xi )) = {\mathscr {E}}$).

Given ${\mathbf {W}}$ of rank d (say), we may write ${\mathbf {W}}={\mathbf {F}}{\mathbf {D}}{\mathbf {F}}^T$, where ${\mathbf {D}}$ is a diagonal $d \times d$ matrix of the positive eigenvalues of ${\mathbf {W}}$, and the columns of the $v \times d$ matrix ${\mathbf {F}}$ are the corresponding d eigenvectors of ${\mathbf {W}}$. Then, denote ${\mathbf {K}}_{\mathbf {W}}={\mathbf {F}}{\mathbf {D}}^{1/2}$ and the weighted information matrix of a competing $\xi$ is ${\mathbf {C}}_{\mathbf {W}}(\xi ) = ({\mathbf {K}}_{\mathbf {W}}^T {\mathbf {C}}^-(\xi ) {\mathbf {K}}_{\mathbf {W}})^{-1}$. The matrix ${\mathbf {K}}_{\mathbf {W}}$ is constructed so that ${\mathbf {K}}_{\mathbf {W}}{\mathbf {K}}_{\mathbf {W}}^T = {\mathbf {W}}$.

The proposed weight matrix is relevant with respect to weighted variances, analogously to that given by Stallings and Morgan (2015):

Proposition 1

Let ${\mathbf {W}}$ be a weight matrix of rank d, let $\xi$ be a competing design and let $\lambda _1, \ldots , \lambda _d$ be the positive eigenvalues of ${\mathbf {C}}_W(\xi )$. Then, the weighted variance of $\widehat{{\mathbf {q}}^T\tau }$ under $\xi$ for any ${\mathbf {q}}\in {\mathscr {L}}({\mathbf {W}})$ is a convex combination of $\lambda _1^{-1}, \ldots , \lambda _d^{-1}$.

Proof

Since ${\mathbf {W}}={\mathbf {K}}_{\mathbf {W}}{\mathbf {K}}_{\mathbf {W}}^T$, we have ${\mathscr {L}}({\mathbf {W}}) = {\mathscr {L}}({\mathbf {K}}_{\mathbf {W}})$, and thus ${\mathbf {q}}\in {\mathscr {L}}({\mathbf {W}}) = {\mathscr {L}}({\mathbf {K}}_{\mathbf {W}})$ yields ${\mathbf {q}}={\mathbf {K}}_{\mathbf {W}}{\mathbf {h}}$ for some ${\mathbf {h}}\in {\mathbb {R}}^d$. Therefore,

$$\begin{aligned} \mathrm {Var}(\widehat{{\mathbf {q}}^T\tau }) = {\mathbf {q}}^T {\mathbf {C}}^-(\xi ) {\mathbf {q}}= {\mathbf {h}}^T {\mathbf {C}}_{\mathbf {W}}^{-1}(\xi ) {\mathbf {h}}= {\mathbf {h}}^T {\mathbf {U}}\varLambda ^{-1} {\mathbf {U}}^T {\mathbf {h}}= \sum _{i=1}^d g_i^2\lambda _i^{-1}, \end{aligned}$$

where ${\mathbf {U}}\varLambda {\mathbf {U}}^T$ is the spectral decomposition of ${\mathbf {C}}_{\mathbf {W}}(\xi )$ and ${\mathbf {g}}:= {\mathbf {U}}^T {\mathbf {h}}$. Then, ${\mathbf {g}}^T{\mathbf {g}}= {\mathbf {h}}^T {\mathbf {U}}{\mathbf {U}}^T {\mathbf {h}}= {\mathbf {h}}^T {\mathbf {h}}$. The weight of ${\mathbf {q}}^T\tau$ is the reciprocal of ${\mathbf {q}}^T {\mathbf {W}}^- {\mathbf {q}}= {\mathbf {h}}^T {\mathbf {K}}_{\mathbf {W}}^T ({\mathbf {K}}_{\mathbf {W}}{\mathbf {K}}_{\mathbf {W}}^T)^- {\mathbf {K}}_{\mathbf {W}}{\mathbf {h}}= {\mathbf {h}}^T {\mathbf {h}}= {\mathbf {g}}^T{\mathbf {g}}$ because ${\mathbf {K}}_{\mathbf {W}}^T ({\mathbf {K}}_{\mathbf {W}}{\mathbf {K}}_{\mathbf {W}}^T)^- {\mathbf {K}}_{\mathbf {W}}$ is a symmetric idempotent matrix of full rank, i.e., ${\mathbf {I}}_d$. It follows that $\mathrm {Var}_{\mathbf {W}}(\widehat{{\mathbf {q}}^T\tau }) = \sum _{i=1}^d (g_i^2/{\mathbf {g}}^T{\mathbf {g}}) \lambda _i^{-1}$. $\square$

Such a definition of weighted optimality enables the construction of a matrix ${\mathbf {W}}_{\mathbf {Q}}$ for any system of estimable functions ${\mathbf {Q}}^T\tau$. Suppose that the experimental objective is expressed by a system of s estimable functions ${\mathbf {Q}}^T\tau$. Because we allow singular ${\mathbf {W}}$, there is no need for the normalizing term ${\mathbf {I}}-{\mathbf {P}}_\tau$ in the construction of ${\mathbf {W}}_{\mathbf {Q}}$; the corresponding weight matrix is of a simple form ${\mathbf {W}}_{\mathbf {Q}}= {\mathbf {Q}}{\mathbf {Q}}^T$. Then, ${\mathbf {W}}_{\mathbf {Q}}$ assigns weight 1 to each ${\mathbf {q}}_i^T\tau$ under the usual conditions:

Proposition 2

Let ${\mathbf {Q}}^T\tau$ be a system of s normalized estimable functions with $\mathrm {rank}({\mathbf {Q}})=s$. Then, ${\mathbf {W}}_{\mathbf {Q}}={\mathbf {Q}}{\mathbf {Q}}^T$ places weight 1 on each of the functions ${\mathbf {q}}_1^T\tau , \ldots , {\mathbf {q}}_s^T\tau$.

Proof

We have ${\mathbf {Q}}^T{\mathbf {W}}_{\mathbf {Q}}^-{\mathbf {Q}}= {\mathbf {Q}}^T ({\mathbf {Q}}{\mathbf {Q}}^T)^- {\mathbf {Q}}= {\mathbf {I}}_s$ because ${\mathbf {Q}}^T ({\mathbf {Q}}{\mathbf {Q}}^T)^- {\mathbf {Q}}$ is a symmetric idempotent matrix of full rank, i.e., ${\mathbf {I}}_s$. $\square$

As before, if primary weights $b_1, \ldots , b_s$ are specified, then one should work with $\tilde{{\mathbf {Q}}}^T\tau$, $\tilde{{\mathbf {q}}}_i = \sqrt{b_i} {\mathbf {q}}_i$ ($1 \le i \le s$).

The following theorem shows that the ‘new’ weighted optimality theory is equivalent to the standard one when the standard weight matrix by Stallings and Morgan (2015) exists.

Theorem 4

Let ${\mathbf {Q}}^T\tau$ be a system of functions of interest, such that $\mathrm {rank}({\mathbf {Q}})=\dim ({\mathscr {E}})$. Then, the weight matrices ${\mathbf {W}}_1 = {\mathbf {Q}}{\mathbf {Q}}^T$ and ${\mathbf {W}}_2 = ({\mathbf {I}}-{\mathbf {P}}_\tau ) + {\mathbf {Q}}{\mathbf {Q}}^T$ are equivalent with respect to the implied weights of the estimable functions. That is, ${\mathbf {q}}^T {\mathbf {W}}_1^- {\mathbf {q}}= {\mathbf {q}}^T {\mathbf {W}}_2^{-1} {\mathbf {q}}$ for any ${\mathbf {q}}\in {\mathscr {E}}$.

Proof

Let ${\mathbf {q}}\in {\mathscr {E}}$. Because $r=\dim ({\mathscr {E}})$, we have ${\mathscr {L}}({\mathbf {Q}})={\mathscr {E}}$, and thus ${\mathbf {q}}^T {\mathbf {W}}_1^- {\mathbf {q}}$ does not depend on the choice of ${\mathbf {W}}_1^-$. Then, ${\mathbf {q}}^T {\mathbf {W}}_2^{-1} {\mathbf {q}}= {\mathbf {q}}^T ({\mathbf {I}}-{\mathbf {P}}_\tau + ({\mathbf {Q}}{\mathbf {Q}}^T)^+){\mathbf {q}}= {\mathbf {q}}^T ({\mathbf {Q}}{\mathbf {Q}}^T)^+{\mathbf {q}}= {\mathbf {q}}^T {\mathbf {W}}_1^- {\mathbf {q}}$ since ${\mathbf {P}}_\tau {\mathbf {q}}={\mathbf {q}}$. $\square$

Equivalence analogous to Theorem 1 also holds for the optimality for ${\mathbf {Q}}^T\tau$ and the corresponding weighted optimality with respect to ${\mathbf {W}}_{\mathbf {Q}}={\mathbf {Q}}{\mathbf {Q}}^T$. Conversely, the system ${\mathbf {Q}}_{\mathbf {W}}\tau ={\mathbf {W}}^{1/2}\tau$ corresponds to an arbitrary weighted optimality with respect to ${\mathbf {W}}$, in the spirit of Theorem 2.