Expected Weighted D-optimal Designs for Experiments with Mixed Factors

Jie Yang; Siting Lin; Yifei Huang

arxiv: 2505.00629 · v4 · submitted 2025-05-01 · 📊 stat.ME · math.ST· stat.TH

Expected Weighted D-optimal Designs for Experiments with Mixed Factors

Siting Lin , Yifei Huang , Jie Yang This is my paper

Pith reviewed 2026-05-22 17:14 UTC · model grok-4.3

classification 📊 stat.ME math.STstat.TH

keywords D-optimal designexpected optimalitymixed factorsrobust designgeneralized linear modelsmultinomial logistic modelForLion algorithmexperimental design

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The pith

Expected weighted D-optimal designs give robustness to unknown parameters in experiments mixing discrete and continuous factors.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper characterizes expected weighted D-optimal designs for general parametric models that include both discrete and continuous factors. These designs maximize an average of the D-criterion over either empirical samples from a pilot study or a prior distribution on the unknown parameters, yielding designs less sensitive to parameter misspecification than classical locally optimal ones. The authors introduce the EW ForLion algorithm to locate such designs and supply a rounding procedure that converts an approximate design into an exact design with a fixed number of runs at prespecified grid points. Illustrations with multinomial logistic regression and generalized linear models show that the resulting designs retain high efficiency across plausible parameter ranges.

Core claim

Under a general parametric model with mixed factors, the expected weighted D-optimal design is the one that maximizes the expected value of the log-determinant of the Fisher information matrix, where the expectation is taken with respect to a distribution or sample of the parameter vector; the EW ForLion algorithm is shown to produce designs that attain this maximum.

What carries the argument

The expected weighted D-optimality criterion, which replaces the ordinary determinant with its expectation over parameter uncertainty to produce a single design that performs well on average.

If this is right

Experimenters obtain a single design whose performance does not collapse when the guessed parameter values are inaccurate.
When pilot data exist, the sample-based version can be recomputed after the pilot to guide the main experiment.
The rounding algorithm converts any approximate mixed-factor design into a feasible exact design with integer allocations and fixed grid points.
The same construction applies to both multinomial logistic models and generalized linear models with mixed factors.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same averaging idea could be applied to A- or E-optimality to protect against uncertainty in different estimation objectives.
Sequential updating of the parameter sample or prior after each batch of runs would turn the method into an adaptive design procedure.
For cost-sensitive studies the robustness gain may justify the modest extra computation required to evaluate the expected criterion.

Load-bearing premise

Either a pilot sample or an explicit prior distribution on the model parameters must be available so that the expected information matrix can be computed.

What would settle it

Run a Monte Carlo study in which true parameters are drawn from a distribution different from the one used to build the design and compare the realized estimation variances or efficiencies of the EW design against those of the locally D-optimal design at the guessed point.

Figures

Figures reproduced from arXiv: 2505.00629 by Jie Yang, Siting Lin, Yifei Huang.

read the original abstract

Optimal designs can help experimenters obtain more accurate parameter estimates with reduced experimental time and cost. In this paper, we characterize the Expected Weighted (EW) D-optimal designs as robust designs against unknown parameter values for experiments under a general parametric model with discrete and continuous factors. When a pilot study is available, we recommend sample-based EW D-optimal designs for subsequent experiments. Otherwise, we recommend EW D-optimal designs under a prior distribution for model parameters. We propose an EW ForLion algorithm for finding EW D-optimal designs with mixed factors, and justify that the designs found by our algorithm are EW D-optimal. To facilitate potential users in practice, we also develop a rounding algorithm that converts an approximate design with mixed factors to exact designs with prespecified grid points and the total number of experimental units. By applying our algorithms for real experiments under multinomial logistic models or generalized linear models, we show that our designs are highly efficient with respect to locally D-optimal designs and more robust against parameter value misspecifications.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper extends D-optimal design to an expected-weighted version for mixed discrete-continuous factors and supplies the EW ForLion algorithm plus rounding, but the optimality justification needs close checking on the averaged information matrix.

read the letter

The main point is that Lin, Huang, and Yang define an expected weighted D-criterion that averages the information matrix over a prior or pilot estimate. This produces designs that stay efficient even when the true parameters differ from what you assumed, and they target the common case of experiments with both discrete levels and continuous factors under general parametric models like GLMs or multinomial logistic regression.

Referee Report

1 major / 2 minor

Summary. The paper characterizes Expected Weighted (EW) D-optimal designs as robust alternatives to locally D-optimal designs for general parametric models that include both discrete and continuous factors. When a pilot study is available it recommends sample-based EW designs; otherwise it recommends designs computed under a prior on the parameters. It proposes the EW ForLion algorithm, claims to justify that the designs it returns are exactly EW D-optimal, supplies a rounding procedure that converts approximate mixed-factor designs into exact designs on a prespecified grid, and illustrates the approach on multinomial logistic and generalized linear models, reporting high efficiency relative to locally optimal designs and improved robustness to parameter misspecification.

Significance. If the central optimality justification holds, the work supplies a practical, implementable route to robust optimal designs for the common setting of mixed discrete-continuous factors. The provision of both a pilot-based and a prior-based version, together with an explicit rounding step for exact designs, addresses a genuine gap between theory and experiment; the reported efficiency gains in the GLM and multinomial examples suggest the method can deliver measurable improvements in real applications.

major comments (1)

[EW ForLion algorithm description] The abstract states that the EW ForLion algorithm is justified and that the returned designs are EW D-optimal. However, the optimality argument appears to invoke the standard Kiefer-Wolfowitz equivalence theorem directly on the expected information matrix without an explicit re-derivation of the sensitivity function that accounts for the product structure of the discrete levels and the continuous intervals. Because this step is load-bearing for the exact-optimality claim, the manuscript should supply the adapted directional derivative and the corresponding equivalence condition for the mixed-factor case.

minor comments (2)

Notation for the expected information matrix (integral or sum over the prior/pilot) should be introduced once and used consistently; currently the transition between the local and the EW versions is not always sign-posted.
The rounding algorithm section would benefit from a small numerical example that shows the grid, the approximate weights, and the final exact allocation side-by-side.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. We address the major comment below.

read point-by-point responses

Referee: The abstract states that the EW ForLion algorithm is justified and that the returned designs are EW D-optimal. However, the optimality argument appears to invoke the standard Kiefer-Wolfowitz equivalence theorem directly on the expected information matrix without an explicit re-derivation of the sensitivity function that accounts for the product structure of the discrete levels and the continuous intervals. Because this step is load-bearing for the exact-optimality claim, the manuscript should supply the adapted directional derivative and the corresponding equivalence condition for the mixed-factor case.

Authors: We appreciate the referee pointing out this aspect of our presentation. While the general Kiefer-Wolfowitz equivalence theorem applies to the expected information matrix in our setting, we acknowledge that an explicit adaptation to the mixed discrete-continuous factor structure would make the justification more transparent. In the revised manuscript, we will add a detailed derivation of the sensitivity function that incorporates the product structure of the discrete levels and continuous intervals, along with the corresponding equivalence condition. This addition will reinforce that the designs returned by the EW ForLion algorithm are indeed exactly EW D-optimal. revision: yes

Circularity Check

0 steps flagged

Minor self-citation for ForLion algorithm but EW D-optimality criterion and robustness claim remain independent of fitted values

full rationale

The paper defines the Expected Weighted D-criterion explicitly as an integral or sum of the information matrix over a prior distribution or pilot estimates. This definition does not reduce by any equation in the manuscript to a post-hoc fit of the target design performance itself. The EW ForLion algorithm is presented as an extension of prior work, but the central characterization of EW designs as robust against parameter misspecification follows directly from the expectation operator applied to the standard D-criterion and does not rely on a self-citation chain for its justification. No fitted-input-called-prediction or self-definitional reduction is exhibited. The derivation chain is therefore self-contained against external benchmarks once the prior or pilot is supplied.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard parametric model assumptions and the availability of either pilot data or a prior; no new physical entities are postulated and the only free choice is the form of the prior or weight function.

free parameters (1)

prior distribution or pilot estimate
Used to form the expected information matrix when parameters are unknown; its specific form is chosen by the user.

axioms (1)

domain assumption The underlying model is a general parametric model with discrete and continuous factors
Invoked to characterize the EW D-optimal designs and to apply the ForLion algorithm.

pith-pipeline@v0.9.0 · 5702 in / 1334 out tokens · 31017 ms · 2026-05-22T17:14:04.627887+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We propose an EW ForLion algorithm for finding EW D-optimal designs with mixed factors, and justify that the designs found by our algorithm are EW D-optimal... max x∈X d(x,ξ)≤p where d(x,ξ)=tr([E{F(ξ,Θ)}]^{-1}E{F(x,Θ)})

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages

[1]

Particle swarm optimization, in: Proceedings of ICNN’95-International Conference on Neural Networks, IEEE. pp. 1942–1948. Kiefer, J.,

work page 1942
[2]

of the 1976 Conf

Algorithms for computing D-optimal design on finite design spaces, in: Proc. of the 1976 Conf. on Information Science and Systems, John Hopkins University. pp. 213–216. Titterington, D.,

work page 1976
[3]

Statistica Sinica 27, 1879–1902

D-optimal designs with ordered categorical data. Statistica Sinica 27, 1879–1902. Yang, M., Biedermann, S., Tang, E.,

work page 1902
[4]

Biometrics 55, 437–444

Optimum experimental designs for multinomial logistic models. Biometrics 55, 437–444. 22 Expected Weighted D-optimal Designs for Experiments with Mixed Factors Siting Lin1, Yifei Huang 2, and Jie Yang 1 1University of Illinois at Chicago and 2Astellas Pharma Global Development, Inc. Supplementary Material S1Analytic Solutions for Multinomial Logistic Mode...

work page 2020
[5]

p jJ−1−p fi(1/(j+ 1))−j J−1 fi(0),j= 1, . . . , J−1. For typically applications,p≥J≥3. Sincef i(z) is an order-ppolynomial inz withf i(1) = 0, its maximum on [0,1] is attained either atz= 0 or at an interior point z∈(0,1) satisfyingf ′ i(z) = 0, that is, J−1X j=1 jbjzj−1(1−z) J−j−1 =p J−1X j=0 bjzj(1−z) J−j−1 ,0< z <1.(S1.1) In Bu et al. (2020), it was me...

work page 2020
[6]

ForJ= 5, we follow the arguments for equation (12) in Tong et al

withi= √−1, known as the imaginary unit, which in this case gets involved in (q3 +r 2)1/2 as well. ForJ= 5, we follow the arguments for equation (12) in Tong et al. (2014) and obtain the following results: Lemma 4.WhenJ= 5, equation(S1.1)is equivalent toA 4z4+A3z3+A2z2+A1z+A 0 = 0withA 0 =b 0p−b 1 ,A 1 =−4b 0p+b 1(3+p)−2b 2 ,A 2 = 6b0p−3b 1(1+p)+b 2(4+p)−...

work page 2014
[7]

IfA 4 ̸= 0,(S1.1)is equivalent toz 4 +a 3z3 +a 2z2 +a 1z+a 0 = 0witha i =A i/A4,i= 0,1,2,3. Then there are four solutions to(S1.1)calculated as complex numbers: z1, z2 =− a3 4 − √A∗ 2 ± √B∗ 2 , z 3, z4 =− a3 4 + √A∗ 2 ± √C∗ 2 with A∗ =− 2a2 3 + a2 3 4 + G∗ 3×2 1/3 , B∗ =− 4a2 3 + a2 3 2 − G∗ 3×2 1/3 − −8a1 + 4a2a3 −a 3 3 4√A∗ , C∗ =− 4a2 3 + a2 3 2 − G∗ 3...

work page 2014
[8]

That is, bothF SEW(X) andF EW(X) are convex

=λF EW (ξ1) + (1−λ)F EW (ξ2). That is, bothF SEW(X) andF EW(X) are convex. SinceXis compact under Assumption (A1), it can be verified thatΞ(X) is also compact under the topology of weak convergence. That is,ξ n converges weakly toξ 0 , asngoes to∞, if and only if lim n→∞ R X f(x)ξ n(dx) = R X f(x)ξ 0(dx) for all bounded continuous functionfonX(Billingsley...

work page 1999
[9]

Recall that we denoteF(ξ) = ˆE{F(ξ,Θ)}for sample-based EW D-optimality, and E{F(ξ,Θ)}for integral-based EW D-optimality;F x = ˆE{F(x,Θ)}for sample-based EW D-optimality, andE{F(x,Θ)}for integral-based EW D-optimality. Then the same set of assumptions listed in Section 2.4.2 of Fedorov and Leonov (2014) are provided in our notations as below: (B1) Ψ(F) is ...

work page 2014
[10]

, h p)T are continuous with respect to all continuous factors ofx∈ X, andΘis bounded

Proof of Theorem 4.Suppose all the predictor functionsh= (h 1, . . . , h p)T are continuous with respect to all continuous factors ofx∈ X, andΘis bounded. Since the number of level combinations of discrete factors is finite, there existM x >0 andM η >0, such that,∥h(x)h(x) T ∥ ≤M x for allx∈ X, and|h(x) T θ| ≤M η for allx∈ Xandθ∈Θ. Sinceνis continuous for...

work page 2025
[11]

(2024), we obtain tr (Estux sthx s (hx t )T ) =u x st tr (hx t )T Esthx s =u x st (hx t )T Esthx s

of Huang et al. (2024), we obtain tr (Estux sthx s (hx t )T ) =u x st tr (hx t )T Esthx s =u x st (hx t )T Esthx s . Note thatu x st here is eitherE{u x st(Θ)}or ˆE{ux st(Θ)}, which is different from Huang et al. (2024). Proof of Theorem 6.According to Theorem 2, a necessary and sufficient condi- tion for a designξto be EW D-optimal is max x∈X d(x,ξ)≤p, w...

work page 2024
[12]

Table S.1: Model comparison for paper feeder experiment Cumulative Continuation Adjacent Baseline po npo po npo po npo po npo AIC 1011.06937.871037.60 970.59 1027.72 962.99 1661.38 962.99 BIC 1104.341113.461130.88 1146.18 1121.00 1138.58 1754.66 1138.58 We first use AIC and BIC criteria to choose the most appropriate multinomial logistic model from eight ...

work page arXiv 2020
[13]

, mwithm= 183 for the original experimental design

+β 23xi1 +β 24xi2l +β 25xi2q +β 26xi3l +β 27xi3q +β 28xi4 +β 29xi5l +β 210xi5q +β 211xi6l +β 212xi6q +β 213xi7l +β 214xi7q +β 215xi8l +β 216xi8q , wherei= 1, . . . , mwithm= 183 for the original experimental design. S4.2 Locally D-optimal designs for paper feeder experiment Using the data listed in Table 3 of Joseph and Wu (2004), we fit the chosen model,...

work page 2004
[14]

ForLion exact grid2.5

to it with merging thresholdδ 2 = 3.2, grid levelL= 2.5, and the number of experimental unitsn= 1,785 (the same sample size as in the original experiment). The obtained exact design is listed as “ForLion exact grid2.5” in Tables S.2 and S.4. For comparison purpose, we also apply the lift-one and exchange algorithms of Bu et al. (2020) to two different set...

work page 2020
[15]

Original

The ForLion exact designs (with grid-0.1, grid-0.5, grid-1, and grid-2.5) in Table S.2 also show the convenience by adopting our rounding algorithm (Algorithm 2). Once an optimal approximate design is obtained, the exact design with a user-specified grid level can be obtained instantly with high relative efficiency. To show how we choose the merging thres...

work page 2020
[16]

with merging thresholdL= 1 and the number of experimental unitsn= 1,000, we obtain the corresponding exact designs, whose numbers of design points are still 15, 13, 21, 19, and 17, respectively. Their relative efficiencies are   l=1 l=2 l=3 l=4 l=5 r=1 1.00000 0.99144 0.99593 0.99299 0.98448 r=2 0.98287 1.00000 0.98535 0.98174 0.98128 r=3 0.99080 0....

work page 1995
[17]

(2020), which is slightly more robust than the corresponding Bayesian D-optimal design

By bootstrapping the original observations forB= 1,000 times, an EW D-optimal design ξBu ={(80,0.3120),(120,0.2911),(140,0.1087),(160,0.2882)}restricted to the original seven grid points was obtained by Bu et al. (2020), which is slightly more robust than the corresponding Bayesian D-optimal design. In this paper, we use this experiment with a continuous ...

work page 2020
[18]

ForLion”) obtained by Huang et al. (2024), and the d-QPSO design (“PSO

for electrostatic discharge experiment Support Support n= 100, L= 0.1, δ2 = 0.5 point Lot A Lot B ESD Pulse Voltagepi (%) point Lot A Lot B ESD Pulse Voltageni 1 -1 -1 -1 1 25.0000 8.75 1 -1 -1 -1 1 25.0 92 -1 1 1 1 25.0000 8.45 2 -1 1 1 1 25.0 83 -1 -1 -1 -1 25.0000 8.48 3 -1 -1 -1 -1 25.0 84 1 1 -1 1 25.0000 6.21 4 1 1 -1 1 25.0 65 -1 1 1 -1 38.9047 2.1...

work page 1975
[19]

We further apply our rounding algorithm to the approximate design withL= 0.0001 andn= 100, and obtain an exact design in just 0.45 second

By using our EW ForLion algorithm anyway, we obtain an EW S18 D-optimal approximate design, which costs 783 seconds. We further apply our rounding algorithm to the approximate design withL= 0.0001 andn= 100, and obtain an exact design in just 0.45 second. Remarkably, both designs consist of 7 design points, as detailed in Table S.8, which are different fr...

work page 2012
[20]

EW ForLion

for the three-continuous-factor example Support Support point x1 x2 x3 pi (%) point x1 x2 x3 ni 1 -2 -1 -3 7.231 1 -2 -1 -3 72 2 -1 -3 20.785 2 2 -1 -3 213 -2 1 -1.8 19.491 3 -2 1 -1.8 194 2 1 3 2.718 4 2 1 3 35 2 1 -0.3321 18.870 5 2 1 -0.3321 196 -2 -1 -0.0867 10.951 6 -2 -1 -0.0867 117 0.9467 -0.9969 2.9932 19.954 7 0.9467 -0.9969 2.9932 20 We first co...

work page 2020

[1] [1]

Particle swarm optimization, in: Proceedings of ICNN’95-International Conference on Neural Networks, IEEE. pp. 1942–1948. Kiefer, J.,

work page 1942

[2] [2]

of the 1976 Conf

Algorithms for computing D-optimal design on finite design spaces, in: Proc. of the 1976 Conf. on Information Science and Systems, John Hopkins University. pp. 213–216. Titterington, D.,

work page 1976

[3] [3]

Statistica Sinica 27, 1879–1902

D-optimal designs with ordered categorical data. Statistica Sinica 27, 1879–1902. Yang, M., Biedermann, S., Tang, E.,

work page 1902

[4] [4]

Biometrics 55, 437–444

Optimum experimental designs for multinomial logistic models. Biometrics 55, 437–444. 22 Expected Weighted D-optimal Designs for Experiments with Mixed Factors Siting Lin1, Yifei Huang 2, and Jie Yang 1 1University of Illinois at Chicago and 2Astellas Pharma Global Development, Inc. Supplementary Material S1Analytic Solutions for Multinomial Logistic Mode...

work page 2020

[5] [5]

p jJ−1−p fi(1/(j+ 1))−j J−1 fi(0),j= 1, . . . , J−1. For typically applications,p≥J≥3. Sincef i(z) is an order-ppolynomial inz withf i(1) = 0, its maximum on [0,1] is attained either atz= 0 or at an interior point z∈(0,1) satisfyingf ′ i(z) = 0, that is, J−1X j=1 jbjzj−1(1−z) J−j−1 =p J−1X j=0 bjzj(1−z) J−j−1 ,0< z <1.(S1.1) In Bu et al. (2020), it was me...

work page 2020

[6] [6]

ForJ= 5, we follow the arguments for equation (12) in Tong et al

withi= √−1, known as the imaginary unit, which in this case gets involved in (q3 +r 2)1/2 as well. ForJ= 5, we follow the arguments for equation (12) in Tong et al. (2014) and obtain the following results: Lemma 4.WhenJ= 5, equation(S1.1)is equivalent toA 4z4+A3z3+A2z2+A1z+A 0 = 0withA 0 =b 0p−b 1 ,A 1 =−4b 0p+b 1(3+p)−2b 2 ,A 2 = 6b0p−3b 1(1+p)+b 2(4+p)−...

work page 2014

[7] [7]

IfA 4 ̸= 0,(S1.1)is equivalent toz 4 +a 3z3 +a 2z2 +a 1z+a 0 = 0witha i =A i/A4,i= 0,1,2,3. Then there are four solutions to(S1.1)calculated as complex numbers: z1, z2 =− a3 4 − √A∗ 2 ± √B∗ 2 , z 3, z4 =− a3 4 + √A∗ 2 ± √C∗ 2 with A∗ =− 2a2 3 + a2 3 4 + G∗ 3×2 1/3 , B∗ =− 4a2 3 + a2 3 2 − G∗ 3×2 1/3 − −8a1 + 4a2a3 −a 3 3 4√A∗ , C∗ =− 4a2 3 + a2 3 2 − G∗ 3...

work page 2014

[8] [8]

That is, bothF SEW(X) andF EW(X) are convex

=λF EW (ξ1) + (1−λ)F EW (ξ2). That is, bothF SEW(X) andF EW(X) are convex. SinceXis compact under Assumption (A1), it can be verified thatΞ(X) is also compact under the topology of weak convergence. That is,ξ n converges weakly toξ 0 , asngoes to∞, if and only if lim n→∞ R X f(x)ξ n(dx) = R X f(x)ξ 0(dx) for all bounded continuous functionfonX(Billingsley...

work page 1999

[9] [9]

Recall that we denoteF(ξ) = ˆE{F(ξ,Θ)}for sample-based EW D-optimality, and E{F(ξ,Θ)}for integral-based EW D-optimality;F x = ˆE{F(x,Θ)}for sample-based EW D-optimality, andE{F(x,Θ)}for integral-based EW D-optimality. Then the same set of assumptions listed in Section 2.4.2 of Fedorov and Leonov (2014) are provided in our notations as below: (B1) Ψ(F) is ...

work page 2014

[10] [10]

, h p)T are continuous with respect to all continuous factors ofx∈ X, andΘis bounded

Proof of Theorem 4.Suppose all the predictor functionsh= (h 1, . . . , h p)T are continuous with respect to all continuous factors ofx∈ X, andΘis bounded. Since the number of level combinations of discrete factors is finite, there existM x >0 andM η >0, such that,∥h(x)h(x) T ∥ ≤M x for allx∈ X, and|h(x) T θ| ≤M η for allx∈ Xandθ∈Θ. Sinceνis continuous for...

work page 2025

[11] [11]

(2024), we obtain tr (Estux sthx s (hx t )T ) =u x st tr (hx t )T Esthx s =u x st (hx t )T Esthx s

of Huang et al. (2024), we obtain tr (Estux sthx s (hx t )T ) =u x st tr (hx t )T Esthx s =u x st (hx t )T Esthx s . Note thatu x st here is eitherE{u x st(Θ)}or ˆE{ux st(Θ)}, which is different from Huang et al. (2024). Proof of Theorem 6.According to Theorem 2, a necessary and sufficient condi- tion for a designξto be EW D-optimal is max x∈X d(x,ξ)≤p, w...

work page 2024

[12] [12]

Table S.1: Model comparison for paper feeder experiment Cumulative Continuation Adjacent Baseline po npo po npo po npo po npo AIC 1011.06937.871037.60 970.59 1027.72 962.99 1661.38 962.99 BIC 1104.341113.461130.88 1146.18 1121.00 1138.58 1754.66 1138.58 We first use AIC and BIC criteria to choose the most appropriate multinomial logistic model from eight ...

work page arXiv 2020

[13] [13]

, mwithm= 183 for the original experimental design

+β 23xi1 +β 24xi2l +β 25xi2q +β 26xi3l +β 27xi3q +β 28xi4 +β 29xi5l +β 210xi5q +β 211xi6l +β 212xi6q +β 213xi7l +β 214xi7q +β 215xi8l +β 216xi8q , wherei= 1, . . . , mwithm= 183 for the original experimental design. S4.2 Locally D-optimal designs for paper feeder experiment Using the data listed in Table 3 of Joseph and Wu (2004), we fit the chosen model,...

work page 2004

[14] [14]

ForLion exact grid2.5

to it with merging thresholdδ 2 = 3.2, grid levelL= 2.5, and the number of experimental unitsn= 1,785 (the same sample size as in the original experiment). The obtained exact design is listed as “ForLion exact grid2.5” in Tables S.2 and S.4. For comparison purpose, we also apply the lift-one and exchange algorithms of Bu et al. (2020) to two different set...

work page 2020

[15] [15]

Original

The ForLion exact designs (with grid-0.1, grid-0.5, grid-1, and grid-2.5) in Table S.2 also show the convenience by adopting our rounding algorithm (Algorithm 2). Once an optimal approximate design is obtained, the exact design with a user-specified grid level can be obtained instantly with high relative efficiency. To show how we choose the merging thres...

work page 2020

[16] [16]

with merging thresholdL= 1 and the number of experimental unitsn= 1,000, we obtain the corresponding exact designs, whose numbers of design points are still 15, 13, 21, 19, and 17, respectively. Their relative efficiencies are   l=1 l=2 l=3 l=4 l=5 r=1 1.00000 0.99144 0.99593 0.99299 0.98448 r=2 0.98287 1.00000 0.98535 0.98174 0.98128 r=3 0.99080 0....

work page 1995

[17] [17]

(2020), which is slightly more robust than the corresponding Bayesian D-optimal design

By bootstrapping the original observations forB= 1,000 times, an EW D-optimal design ξBu ={(80,0.3120),(120,0.2911),(140,0.1087),(160,0.2882)}restricted to the original seven grid points was obtained by Bu et al. (2020), which is slightly more robust than the corresponding Bayesian D-optimal design. In this paper, we use this experiment with a continuous ...

work page 2020

[18] [18]

ForLion”) obtained by Huang et al. (2024), and the d-QPSO design (“PSO

for electrostatic discharge experiment Support Support n= 100, L= 0.1, δ2 = 0.5 point Lot A Lot B ESD Pulse Voltagepi (%) point Lot A Lot B ESD Pulse Voltageni 1 -1 -1 -1 1 25.0000 8.75 1 -1 -1 -1 1 25.0 92 -1 1 1 1 25.0000 8.45 2 -1 1 1 1 25.0 83 -1 -1 -1 -1 25.0000 8.48 3 -1 -1 -1 -1 25.0 84 1 1 -1 1 25.0000 6.21 4 1 1 -1 1 25.0 65 -1 1 1 -1 38.9047 2.1...

work page 1975

[19] [19]

We further apply our rounding algorithm to the approximate design withL= 0.0001 andn= 100, and obtain an exact design in just 0.45 second

By using our EW ForLion algorithm anyway, we obtain an EW S18 D-optimal approximate design, which costs 783 seconds. We further apply our rounding algorithm to the approximate design withL= 0.0001 andn= 100, and obtain an exact design in just 0.45 second. Remarkably, both designs consist of 7 design points, as detailed in Table S.8, which are different fr...

work page 2012

[20] [20]

EW ForLion

for the three-continuous-factor example Support Support point x1 x2 x3 pi (%) point x1 x2 x3 ni 1 -2 -1 -3 7.231 1 -2 -1 -3 72 2 -1 -3 20.785 2 2 -1 -3 213 -2 1 -1.8 19.491 3 -2 1 -1.8 194 2 1 3 2.718 4 2 1 3 35 2 1 -0.3321 18.870 5 2 1 -0.3321 196 -2 -1 -0.0867 10.951 6 -2 -1 -0.0867 117 0.9467 -0.9969 2.9932 19.954 7 0.9467 -0.9969 2.9932 20 We first co...

work page 2020