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.
Similar content being viewed by others
References
Atkinson AC, Donev A, Tobias R (2007) Optimum experimental designs, with SAS. Oxford University Press, New York
Bailey RA (2009) Designs for dose-escalation trials with quantitative responses. Stat Med 28(30):3721–3738. https://doi.org/10.1002/sim.3646
Greville TNE (1966) Note on the generalized inverse of a matrix product. SIAM Rev 8(4):518–521. https://doi.org/10.1137/1008107
Harman R, Sagnol G (2015) Computing D-optimal experimental designs for estimating treatment contrasts under the presence of a nuisance time trend. In: Steland A, Rafajlowicz E, Szajowski K (eds) Stochastic models, statistics and their applications. Springer, Wroclaw, pp 83–91
Hedayat AS, Jacroux M, Majumdar D (1988) Optimal designs for comparing test treatments with controls. Stat Sci 3(4):462–476. https://doi.org/10.1214/ss/1177012767
Jacroux M (1983) Some minimum variance block designs for estimating treatment differences. J R Stat Soc B Methodol 45:70–76. https://doi.org/10.1111/j.2517-6161.1983.tb01232.x
Majumdar D (1986) Optimal designs for comparisons between two sets of treatments. J Stat Plan Inference 14:359–372. https://doi.org/10.1016/0378-3758(86)90173-4
Majumdar D, Notz WI (1983) Optimal incomplete block designs for comparing treatments with a control. Ann Stat 11(1):258–266. https://doi.org/10.1214/aos/1176346076
Morgan JP, Stallings J (2017) Optimal experimental design that targets meaningful information. WIREs Comput Stat. https://doi.org/10.1002/wics.1393
Morgan JP, Wang X (2010) Weighted optimality in designed experimentation. J Am Stat Assoc 105(492):1566–1580. https://doi.org/10.1198/jasa.2010.tm10068
Morgan JP, Wang X (2011) E-optimality in treatment versus control experiments. J Stat Theory Pract 5(1):99–107. https://doi.org/10.1080/15598608.2011.10412053
Pronzato L, Pázman A (2013) Design of experiments in nonlinear models. Springer, New York
Pukelsheim F (1993) Optimal design of experiments. Wiley, New York
Rosa S (2018) Optimal designs for treatment comparisons represented by graphs. AStA Adv Stat Anal 102:479–503. https://doi.org/10.1007/s10182-017-0312-5
Rosa S, Harman R (2016) Optimal approximate designs for estimating treatment contrasts resistant to nuisance effects. Stat Pap 57(4):1077–1106. https://doi.org/10.1007/s00362-016-0809-0
Rosa S, Harman R (2017) Optimal approximate designs for comparison with control in dose-escalation studies. TEST 26:638–660. https://doi.org/10.1007/s11749-017-0529-3
Sagnol G (2011) Computing optimal designs of multiresponse experiments reduces to second-order cone programming. J Stat Plan Inference 141:1684–1708. https://doi.org/10.1016/j.jspi.2010.11.031
Sagnol G, Harman R (2015) Computing exact D-optimal designs by mixed integer second-order cone programming. Ann Stat 43(5):2198–2224. https://doi.org/10.1214/15-aos1339
Seber GA (2008) A matrix handbook for statisticians. Wiley, Hoboken
Senn S (2009) Commentary on ‘designs for dose-escalation trials with quantitative responses’. Stat Med 28:3754–3758. https://doi.org/10.1002/sim.3735
Senn S, Amin D, Bailey RA, Bird SM, Bogacka B, Colman P, Garrett A, Grieve A, Lachmann P (2007) Statistical issues in first-in-man studies. J R Stat Soc A Stat 170(3):517–579. https://doi.org/10.1111/j.1467-985X.2007.00481.x
Stallings JW, Morgan JP (2015) General weighted optimality of designed experiments. Biometrika 102(4):925–935. https://doi.org/10.1093/biomet/asv037
Yu Y (2010) Monotonic convergence of a general algorithm for computing optimal designs. Ann Stat 38(3):1593–1606. https://doi.org/10.1214/09-aos761
Acknowledgements
The author is grateful to Radoslav Harman for his comments and advice.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The author states that there is no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work was supported by the Slovak Scientific Grant Agency [Grant VEGA 1/0521/16].
Appendix: Weighted optimality for any system of interest
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,
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.
Rights and permissions
About this article
Cite this article
Rosa, S. Equivalence of weighted and partial optimality of experimental designs. Metrika 82, 719–732 (2019). https://doi.org/10.1007/s00184-019-00706-9
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00184-019-00706-9