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REVIEW 4 major objections 5 minor 32 references

MOODE: An R Package for Multi-Objective Optimal Design of Experiments

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper presents MOODE, an open-source R package that builds experimental designs by directly optimizing a user-weighted compound criterion covering precise estimation, lack-of-fit detection, and protection against model bias, with case…

desk verdict A useful software package with a real bug in the Plackett-Burman comparison and an unverified near-optimality claim; the package itself is worth refereeing. read the letter →

arxiv 2412.17158 v1 pith:5TP4FFJ5 submitted 2024-12-22 stat.CO stat.ME

Vasiliki Koutra , Olga Egorova , Steven G. Gilmour , Luzia A. Trinca This is my paper · ORCID
classification stat.COstat.ME MSC 62K0562K15
keywords optimaldesignofexperimentscompoundcriteriaresponsesurfacelackfitmodelmisspecificationpointexchangealgorithmcoordinateRpackage
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper presents MOODE, an R package for constructing experimental designs when several goals compete. It combines criteria for precise estimation of a primary model, detection of lack of fit from that model, and protection against bias from unmodeled terms into one weighted compound criterion. The package searches for exact designs that are near-optimal under this compound criterion, using point or coordinate exchange algorithms. Two case studies show the trade-offs: designs optimized for a single objective score poorly on the others, while compound designs keep high efficiencies on all three. The paper's claim is that this makes a multi-objective approach to experimental design accessible and usable in practice.

What carries the argument

The machinery is the compound optimality criterion built as a product of weighted efficiencies. Estimation and inference criteria ($DS$, $L$, $(DP)S$, $LP$) are based on the centred information matrix $X_1^\top(I_n - J_n/n)X_1$ and incorporate $F$-quantiles that depend on pure-error degrees of freedom $d = n - t$, which encourages replication. Lack-of-fit criteria use the posterior variance of potential terms, $\Sigma_2 = \sigma^2(R + \tau^{-2}I_q)^{-1}$, with $R$ the Schur complement for $X_2$ given $X_1$, and target designs with $X_1$ and $X_2$ near-orthogonal. MSE-robustness criteria minimize the determinant or trace of the MSE matrix of $\hat{\beta}_1$, with the expected log-determinant evaluated either by Monte Carlo or a point prior. The search is carried out by a modified Fedorov point-exchange algorithm (default for $k \le 4$ factors) or a coordinate-exchange algorithm (default for $k \ge 5$), with multiple random starts.

What would settle it

Solve a small instance exactly. For example, with $k=2$ two-level factors, $n=8$ runs, a main-effects primary model, and all two-factor interactions as potential terms, enumerate every multiset of eight runs (or use a global optimizer) and compute the compound criterion (12). If MOODE's best design over its default 10 starts has a criterion value more than, say, 1% worse than the global optimum, the near-optimality claim fails for that instance; repeating across several seeds and instances would show whether the failure is systematic.

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Extended reading notes

Core claim

The paper's claim is that MOODE provides the first accessible, open-source implementation of a multi-objective approach to optimal experimental design. Rather than optimizing a single alphabetic criterion, the package minimizes a weighted product of individual criteria: one component for precise estimation and inference under a primary polynomial model, one for detecting lack of fit in the direction of potential higher-order terms, and one for mean-squared-error robustness against bias from those potential terms. The compound criterion is $\phi_{\rm det}(D) = \phi_{(DP)S}^{\kappa_{DP}}(D)\,\phi_{\rm LoF-DP}^{\kappa_{\rm LoF-DP}}(D)\,\phi_{\rm MSE(DS)}^{\kappa_{\rm MSE(D)}}(D)$, with a trace-based analogue, and the weights are chosen by the user. The paper demonstrates that designs found under such compound criteria achieve high efficiencies on all three individual criteria simultaneously, whereas designs optimized for any single criterion have poor performance on at least one other, and that in 12-run two-level cases the compound designs maintain pure-error replication that Plackett-Burman designs lack.

Load-bearing premise

The load-bearing premise is that the point-exchange and coordinate-exchange heuristics, when started from the default number of random designs, actually find designs that are nearly optimal for the compound criterion; the paper states that convergence to the global optimum is not guaranteed, so if the algorithms routinely stop at poor local optima, the package's central promise fails.

Editorial extensions

If this is right

  • Experimenters can obtain, in one R function call, an exact design that explicitly trades off precise estimation, lack-of-fit detection, and protection against model bias; the weights $\kappa$ make the trade-off user-controlled.
  • Designs produced under compound criteria will routinely include replicated treatments, because the $F$-quantile components reward pure-error degrees of freedom, which supports model-robust inference.
  • The package should outperform existing single-criterion tools for response-surface problems with model uncertainty, since those tools focus only on precision-based criteria.
  • In screening settings with few runs, compound designs can beat the classic Plackett-Burman design on efficiency for the assumed model while gaining the ability to estimate pure error.
  • Because the code is modular, the same compound-criterion machinery can be extended to blocked designs, hybrid nonlinear models, or other optimization algorithms.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The real burden on users moves to choosing the weights $\kappa$ and the prior scale $\tau^2$; the paper gives defaults but no guidance for eliciting them from experimental priorities, and sensitivity to those choices could be explored systematically.
  • The Monte Carlo variant of $\rm MSE(DS)$ with the default $B=50$ samples may be noisy, so a user comparing two designs could mistake Monte Carlo error for a real efficiency difference; increasing $B$ or using the point-prior variant removes this.
  • Because the exchange algorithms are heuristic, the 'nearly optimal' claim is only as good as the number of random starts; a practical extension would be a built-in diagnostic comparing the best design's criterion value with the distribution over starts.
  • The compound approach could be adapted to nonlinear or Bayesian design settings, where multi-objective trade-offs are currently handled by separate criteria such as T-optimality or Bayesian D-optimality.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper introduces MOODE, an R package for constructing experimental designs under compound optimality criteria that combine inference-based criteria (DS, L, (DP)S, LP), lack-of-fit criteria (LoF-DP, LoF-LP), and mean-squared-error robustness criteria (MSE(DS), MSE(L)). The package implements point-exchange and coordinate-exchange algorithms and is demonstrated on two case studies: a 3-factor, 36-run response surface experiment and a comparison with Plackett-Burman designs for k=3 to 9 factors in 12 runs. The paper's central claim is that MOODE provides a practical, open-source tool for finding nearly optimal designs that balance estimation, lack-of-fit detection, and bias protection.

Significance. If the package performs as claimed, it would fill a practical gap by making multi-objective design methods from Egorova and Gilmour (2022) and Gilmour and Trinca (2012) accessible to experimenters. The open-source implementation and the two case studies are useful demonstrations of the intended functionality. However, the paper currently does not provide adequate evidence for the 'nearly optimal' claim: the only internal evidence consists of exchange-algorithm outputs, and one reported efficiency exceeds 100%, which indicates that the benchmark designs are themselves local optima. The code in the appendix also contains several errors that prevent the Plackett-Burman comparison from being reproducible. These issues are load-bearing because the paper's demonstrations and package credibility rest on the reliability of the reported efficiencies and the correctness of the implemented criteria.

major comments (4)
  1. [Section 4.1, Table 3] The near-optimality claim is not supported. In Table 3, the first compound design reports a LoF-DP efficiency of 100.54%, which is impossible if the row-5 design (kappa=(0,1,0)) is truly LoF-DP-optimal. The text attributes this to 'the empirical nature of the design optimisation,' but the correct implication is that the row-5 benchmark is a local optimum. The same issue is admitted in Section 4.2, where for k=7 the algorithm found a better LP design under a compound criterion than when directly optimizing LP. Because no exact verifier (enumeration, MILP, or branch-and-bound) is applied to any reported design, the paper provides no evidence that the package's outputs are near-optimal for the compound criteria, and the efficiency trade-offs in Table 3 and Figure 1 are not reliable benchmarks.
  2. [Section 4.2 and Appendix A.2] The code for the Plackett-Burman comparison is incorrect and non-reproducible. In the loop, the variable `pb` is first assigned a list via `pb <- list()`, and then the same name is called as a function in `undesign(pb(nruns = 12, nfactors = j))`, which would error in R because a list is not callable. Additionally, `mutate(trt = 1:12, intercept = rep(1, 12), .before = A)` inserts observation-number and intercept columns before the factor columns, so `X1` contains these extraneous predictors; `model.matrix(~ (.)^2, X1)` then builds interactions involving `trt` and `intercept`, and `[, -(1:(j + 1))]` removes the wrong columns. As a result, the Plackett-Burman baseline designs in Figure 1 and Table 5 are evaluated under a misspecified model, and the printed code cannot be run as shown.
  3. [Section 4.1 and Appendix A.1] The text and code disagree on the Monte Carlo sample size and the criterion used for the MSE(DS) benchmark. The text states that 'we use B = 1000 Monte Carlo samples' for the MSE(DS) component, but the code sets `control = list(Biter = 50)` and `criterion.choice = "MSE.P"`, which is the point-prior criterion (10), not the Monte Carlo criterion (9). The benchmark 'MSE(DS)-optimal design' in row 6 of Table 3 is therefore optimal for a different objective function than the one used to compute the efficiency column, so the 100% value in that column does not establish optimality under the reported criterion. Please reconcile the text, code, and evaluation.
  4. [Section 2.3, Eq. (8)] Equation (8) appears to assume that the columns of X1 sum to zero. From the definitions of M and A1, the term A1' M A1 equals X2' (I - J/n) X1 M^{-1} X1' (I - J/n) X2, not X2' X1 M^{-1} X1' X2 as written, unless (I - J/n)X1 = X1. The paper does not state a centered-coding assumption, and the example designs (e.g., Table 4) are not all balanced. This simplification underpins the MSE(DS) criterion, so the formula should be corrected or the centering assumption made explicit.
minor comments (5)
  1. [Section 4.1, code chunk] The loop is written as `for(i in 1:nrow(kappa1))`, but the matrix is defined as `kappa` in the preceding code block; as printed, the code will fail because `kappa1` is undefined.
  2. [Section 4.1, text] The sentence 'It is possible these design could be improved further do using more random starts of the algorithm' contains grammar errors; it should be 'It is possible these designs could be improved further by using more random starts of the algorithm.'
  3. [Table 1] In the LoF-DP and LoF-LP rows, the symbol L is used both as a matrix (in the trace expressions) and as part of the criterion names, which is confusing. In Section 2.2 the matrix is called R; please align the notation.
  4. [Section 2.1] The notation for the (DP)S criterion alternates between phi_D and phi_DS; please use one symbol consistently.
  5. [Section 5, Discussion] The claim that MOODE provides 'the first accessible, open source, implementation of a multi-objective approach to the problem' is strong. Please soften it to 'to our knowledge' or substantiate it with a systematic comparison of existing software capabilities.

Circularity Check

0 steps flagged · score 2.0 of 10

No definitional circularity found; the criteria and search algorithms are specified independently, with minor self-citations and a heuristic-benchmark caveat that are correctness risks rather than circular reductions.

full rationale

The core derivation chain is not circular. The compound objectives (11) and (12) are explicit products of individual criteria defined in Sections 2.1-2.3, with standard external sources (Box and Draper 1959a,b; DuMouchel and Jones 1994; Gilmour and Trinca 2012) and self-contained matrix formulas; the package is an implementation, not a fit. Efficiencies in Section 4 are computed against individual-criterion designs and, in the Plackett-Burman comparison, against an externally constructed 12-run PB design, so the central comparisons are not self-referential by construction. The manuscript's own caveat in Section 3 that 'convergence to the optimal solution is not guaranteed' and the 100.54% LoF-DP efficiency in Table 3, with the admission that 'the empirical nature of the design optimisation' produced it, show that the heuristic benchmarks are not certified optima; this is a genuine limitation of the near-optimality demonstration. The appendix PB code also inserts a 'trt' column before the intercept in X1, which would mis-specify that baseline. These are correctness and verification risks, not circularity in the definitional sense: no equation is defined in terms of the algorithm's output, and no fitted parameter is relabeled as a prediction. The self-citations to Egorova and Gilmour (2022) and Gilmour and Trinca (2012) are real intellectual provenance, but the criteria are restated and used on their own terms, so they are not load-bearing in a way that forces the paper's conclusions.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

MOODE introduces no new unobservable entities such as particles or forces. Its central claim rests on user-chosen weights, a Bayesian prior for potential model terms, standard linear regression assumptions, and the availability of good starting designs for heuristic exchange algorithms. The free parameters are explicit user inputs rather than hidden fitted values.

free parameters (2)
  • Compound-criterion weights kappa = User-specified; examples use 1/3,1/3,1/3; 0.4,0.2,0.4; 0.25,0.25,0.5; 1,0,0; 0,1,0; 0,0,1
    The kappa weights in equations (11) and (12) are chosen by hand to reflect experimental priorities. They are inputs, not fitted to data, but they control the design and the paper gives no data-driven way to select them.
  • Prior variance tau^2 for potential terms = 1 (default in the examples)
    The prior beta2 ~ N(0, sigma^2 tau^2 I) in Section 2.2 and the point prior beta2 = plus or minus sigma tau 1_q in Section 2.3 require the user to set tau^2. The paper sets tau^2 = 1 in the demonstrations and notes that the alias robustness is controlled by tau^2, but does not fit it.
assumptions (4)
  • domain assumption Standard linear model assumptions: independent errors with zero mean and constant variance sigma^2 (model (1), Section 1.1).
    The criteria and inference are derived under these assumptions, and the paper does not assess robustness to heteroscedasticity or correlated errors.
  • domain assumption Bayesian prior for potential terms: non-informative priors for beta0 and beta1, and beta2 ~ N(0, sigma^2 tau^2 I) (Section 2.2).
    The LoF and MSE criteria rest on this prior, motivated by DuMouchel and Jones (1994). If the prior is misspecified, the sensitivity and robustness criteria may not reflect the experimenter's actual uncertainty.
  • standard math The matrix determinant lemma, cited from Harville (2006), is used to derive the log-determinant of the MSE matrix in equation (8).
    This is a standard algebraic identity; no issue, but the simplification to X2' X1 M^{-1} X1' X2 implicitly assumes the design matrix X1 is centered so that (I - 1/n J)X1 = X1.
  • domain assumption Design points are restricted to a discrete set of levels for each factor (Section 1.1).
    The package searches over candidate lists built from discrete levels. The authors argue this causes no loss of efficiency for linear models with well-chosen levels, but the claim is not proven in the paper.

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Cite this review

Pith. "Pith review of MOODE: An R Package for Multi-Objective Optimal Design of Experiments." pith.science (2026). https://pith.science/paper/5TP4FFJ5

@misc{pith2026241217158,
  author       = {Pith},
  title        = {Pith review of: MOODE: An R Package for Multi-Objective Optimal Design of Experiments},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5TP4FFJ5}},
  note         = {Machine review of arXiv:2412.17158}
}
read the original abstract

We describe the R package MOODE and demonstrate its use to find multi-objective optimal experimental designs. Multi-Objective Optimal Design of Experiments (MOODE) targets the experimental objectives directly, ensuring that the full set of research questions is answered as economically as possible. In particular, individual criteria aimed at optimizing inference are combined with lack-of-fit and MSE-based components in compound optimality criteria to target multiple and competing objectives reflecting the priorities and aims of the experimentation. The package implements either a point exchange or coordinate exchange algorithm as appropriate to find nearly optimal designs. We demonstrate the functionality of MOODE through the application of the methodology to two case studies of varying complexity.

Figures

Figures reproduced from arXiv: 2412.17158 by the authors.

Figure 1
Figure 1. Design efficiencies under the LP, MSE(L) and L criteria, along with the pure error degrees of freedom (PE) for two-level designs with k = 3, . . . , 9 factors and n = 12 runs [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

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